Datasets:
Zyphra
/
Zyda

Dataset card Data Studio Files Community
6
source
stringclasses
1 value
text
stringlengths
152
659k
filtering_features
stringlengths
402
437
source_other
stringlengths
440
819k
arxiv
An Audit Framework for Adopting AI-Nudging on Children -2 An Audit Framework for Adopting AI-Nudging on Children -2 6. Acknowledgments 31 7. Bibliography 32 The term "nudges" is used to describe strategic tools designed to influence people's choices and behaviors, usually without subjects consciously recognizing that influence. The idea of "nudge theory" was introduced by Thaler and Sunstein in their 2008 book, Nudge, and is widely studied in both behavioral economics and psychology.To date, work on nudge theory in technology has focused mostly on choice architectures, designed by user experience (UX) experts, which have mostly predictable outcomes. However, artificial intelligence (AI) has revolutionized the world of nudging: today, an individual's decisions can be shaped by AI interfaces that constantly adapt and change according to the user's choices and detectable behavior. The outcomes in this process are myriad and unpredictable, and nudging may even be an unintended consequence, distinct from the original intended design for the AI interfaces.A detailed discussion of this form of AI-enhanced nudges is currently lacking. At present, commercial legislation (EU Unfair Commercial Practices Directive in the European Union, 2005) and design guidelines (La Forme Des Choix, 2019) exist, which are helpful in mitigating the use of the soft manipulation of behaviors. In addition, two separate groups of experts are working on the topic: IEEE P7008 (RAS/SC/Ethical Nudging) and CEN-CENELEC JTC21 (project on AIenhanced nudging in WG 4 on Foundational and societal aspects). However, few standards are presently able to address the risk of using personalized sequences of nudging mechanisms.Here, we address this gap, first by highlighting the ethical problems that emerge from adopting nudges in AI, focusing on protected categories such as children (Smith & de Villiers-Botha, 2021). This assessment is based on an analysis of the structural risk factors and the potential harms associated with the use of AI-nudging. More specifically, we argue that AI-nudging potentially increases the likelihood of harm when other risk-factors are in place. Second, we build an infrastructure of trust for AI-nudging in the contexts of games and social media. This framework indicates how risks can be assessed and mitigated by laying the groundwork for a third-party independent audit system (focusing on children and teenagers). Introduction At times, technological nudges are called "behavioral design" or "algorithmic nudging". Companies are increasingly using algorithms to manage and control individuals not by force, but rather by nudging them into desirable behavior -in other words, learning from their personalized data and altering their choices in some subtle way. (Möhlmann, 2021) Initial Clarifications Unlike the static form of nudging usually discussed in the literature, we focus here on a type of nudging that uses large amounts of data to provide personalized, dynamic feedback and interfaces. We call this AI-nudging (Lanzing, 2019, p. 549;Yeung, 2017). The ultimate goal of the audit outlined out here is to ensure that an AI system that uses nudges will maintain a level of moral inertia and neutrality by complying with the recommendations, requirements, or suggestions of the audit (in other words, the criteria of the audit). In the case of unintended negative consequences, the audit suggests risk mitigation mechanisms that can be put in place. In the case of unintended positive consequences, it suggests some reinforcement mechanisms. We do not rely on a specific ethical theory (with specific values and normative terms) and the goal we set (to reach an ethically neutral state) will be sufficiently broad to allow for the application of different ethical sensibilities and regulatory frameworks. The audit also refers to specific laws and regulations about protecting children: individual companies will need to ensure that they understand these laws and regulations in their own regions and cultures. Topic Goal Ethical framework This work is intended as an audit framework. This means that we offer guidelines and lay out the initial infrastructure for developing a full-fledged audit, but at this stage we do not establish definitive answers to our audit criteria or offer ways to evaluate companies and their risk levels. We recognize that there are already legislative frameworks for AI for children, in particular the UK Children's Code (2021) and the Children's Online Privacy Protection Act "COPPA" (2013). Additionally, the draft EU AI Act proposal (2021, article 5) acknowledges something similar to nudging when it prohibits the use of "subliminal techniques beyond a person's consciousness in order to materially distort a person's behaviour in a manner that causes or is likely to cause that person or another person physical or psychological harm." The proposal notes a specific concern about the use of these subliminal techniques on children (and other vulnerable groups of people). However, there is a lack of clear guidance on the impact of these techniques on children. Rather than appealing to people's conscious deliberation, nudges steer people's choices by influencing the non-deliberative, less rational parts of their cognitive architecture (Kahneman, 2011). Children and teenagers, because of the developmental state of their cognitive structures, are particularly vulnerable to the manipulative powers of nudges. Indeed, childhood and adolescence are key stages of life in which people grow and learn how to think, feel, and behave. These two stages are highly dynamic and, compared with adulthood, have specific cognitive importance and particular characteristics of their own, to which special attention must be paid. This significant cognitive difference from adulthood places children and adolescents in the category of users who must be carefully protected. Scope Motivation An ethical risk is any situation that is likely to lead us away from an intended ethical goal. The intended ethical goal here is to maintain a level of moral inertia (in the use of AInudging) to preserve a state of moral neutrality. Risk mitigation is a matter of eliminating (or reducing) the presence of those factors that lead us away from a state of moral neutrality or goodness. Thaler and Sunstein (2008, p. 6) define a nudge as "any aspect of the choice architecture that alters people's behavior in a predictable way without forbidding any options or significantly changing their economic incentives." Rather than simply letting people choose from different options, nudging is a way of incentivizing them to adopt (or refrain from adopting) one of those options. To obtain these effects, nudging mostly adopts insights from behavioral science that tells us that people often make choices in a predictable way based on bias As Figure 3 shows, nudges are those mechanisms that intervene in an interaction (or, "information process") between an agent and a receiver by interfering in the receiver's response process. Digital nudging uses digital means and is based on, and intentionally incorporated in, an explicit design (Selinger & Seager, 2012). In contrast, AI and machine learning-enhanced nudges are often not explicitly ex ante designed as part of the system: the design of these systems is partly autonomous and evolving in accordance with the goal the system was set When confronting difficult choices, humans tend to revert to the default: they prefer to go with the flow and adopt what has been set out for them rather than making a consequential and difficult choice that they might later regret. The choice of the "default" in these cases is a form of nudging on account of how it can influence people's behavior. People are usually not aware of this default-heuristic and, and heuristics. These are fast-and-frugal rules of thumb that people use to navigate an environment and reach quick decisions. However, because this type of choice-making is not a form of rational deliberation, it tends to be rigid and applied subconsciously and automatically. Nudges can exploit these features of our reasoning to influence our choices. Systems of nudges are often discussed and employed in healthcare settings (Vlaev et al., 2016), as in the following example of a nudge related to health: to achieve. Thus, nudging tools in AI are the result of reinforcement learning systems that constantly adapt to the behavior of the users in ways that are often unpredictable (Saetra, 2019;Yeung, 2017). So it is possible for a system to use nudges to achieve its goal, even if this goal cannot be traced back to the original design of the system in any clear way. In practice, this means that an individual's decisions can be shaped by AI interfaces that are personalized for each user without this being the result of intentional design. as a result, they are not aware that they are being nudged. Nudging triggers the adoption of our unconscious biases and shallow reasoning mechanisms to the extent that we do not really consciously deliberate about what we are doing. Classical Nudges Organ donation. In some countries, there is a surprisingly high number of people who consent to donate their organs after death. It has been noticed that these are the same countries where there is a default option in favor of organ donation: unless one explicitly refuses, the default is that their driver license or health insurance card will report their consent to donate their organs. In contrast, countries with a voluntary opt-in system (where the default is no organ donation), the number of donors is substantially lower (Blumenthal-Barby & Burroughs, 2012). Figure 1: The potential ethical risks associated with and made worse by the adopting of AI-nudges AI-nudging is associated with potential ethical risks (see Figure 1). However, that does not mean that nudges, even unintentional nudges, have a specific ethical value in themselves. Situations and actions can be morally charged as a result of the interaction of a multitude of (human and artificial) agents with the environment (Floridi, 2013). Similarly, nudging can have a moral valence depending on the nature of the interaction between different agents and the environment and the context in which this interaction takes place. Each one of these agents may produce morally negligible actions, but their interactions can cause morally laden results. More specifically, AInudging becomes morally problematic in relation to specific risk factors or macro-structures because of their tendency to heighten the level of risk and worsen the impact of these factors ( Figure 2). AI-enhanced nudges AI -NUDGE COMPLEXITY OF THE AGENT SYSTEM USER'S AUTONOMY AND DECISION MAKING SKILLS USER'S COMPETENCE AND COGNITIVE SKILLS ANOMALY DETECTIONS TIME AND FREQUENCY OPEN VS CLOSED ENVIRONMENT RISK FACTORS Figure 2: Risk factors associated with AI-nudges A) Social Media: risk factors and harmful consequences In this audit framework, we focus on AI-nudging that is specifically related to children and teenagers in social media and video games settings (Schneider et al., 2018). To develop our framework, in the following sections we show that AI-nudging becomes morally problematic when other risk factors are also present. In those cases, AI-nudging may present a serious risk of harm to the users by worsening the structural risks that already exist. No parent will be surprised to learn that teenagers spend on average at least three hours a day on mobile devices such as smartphones or tablets, and a good chunk of teenagers report that they use social media (Madden et al., 2013). Social media companies often implement AI-powered systems of recommendation to enhance engagement on their platforms. These systems are shaped by the data coming from users: AI is tracking users' preferences and delivers content that promotes further engagement as a result. The goal is to deliver content that is more and more relevant, gratifying, and therefore addictive (Deibert, 2019;DeVito, 2017;Quan-Haase & Young, 2010). Examples of AI-nudges in social media include the recommendation system and the ordering and nature of what we see in our feeds. Use Cases: social media and video games A data-driven, model based system is intrinsically more risky than a rule-based system because of the added unknowns associated with this type of technology. AI-enhanced nudges are thus intrinsically more risky as they are able to produce personalized outcomes and recommender systems that constantly evolve in ways that are difficult to foresee. They are also hard to escape: AI-nudges are constantly refining the user-experience, making it increasingly more interesting and engaging. Social media presents an open environment with a large quantity of different variables that may be impossible to control. Multiple agents (both human and artificial) are causally responsible for creating the content and the experience on the platforms. Multiple types of receiver agents are in play as well, who may use and interpret the content on the platform in radically different ways. All this represents a risk factor, namely a situation in which a deviation from some established ethical goals is likely to occur. Social media are at risk because they cannot-by their very nature-fully control and monitor their environment. Furthermore, an AI-powered nudging system makes this environment even more open and unstable because it introduces further variables: nudging tools that are not designed for the purpose will create content that cannot be easily controlled, allowing for more unknowns and anomalies. In this unstable environment, security may be very hard to ensure and harmful content can proliferate. For children and teenagers, without clear safeguards in place, this could mean being exposed to cyberbullying, for instance, or confronting highly inappropriate sexual material. To be sure, these risks cannot be fully eliminated. However, in the audit we ask social media companies to assess residual risks to show that they understand their impact and have safety mechanisms to contain unknown harmful effects as much as possible. The detection of anomalous online behavior and harmful content and experience is a particularly lengthy and difficult process on social media platforms because it requires detecting a standard deviation and thus collecting a sufficient amount of data. Data-sparsity is at times particularly problematic in this context. In certain situations, platforms can decide that a certain type of content is intrinsically risky and thus eliminate it as soon as it appears, but this cannot be done at scale and is often an imprecise measure. AI-enhanced nudging produces personalized content and experience on the platforms. The interactions produced are often highly individualized and difficult to analyze in terms of deviations and anomalies. As a result, it seems difficult to imagine that platforms would be able to detect risk constantly, in real time. And the harder it is to detect anomalies and intervene, the riskier the system becomes. Complexity of the Agent system Open environment Anomalydetection timing and frequency Children and teenagers are protected categories also because of their limited ability to fully understand the content of the messages and the experiences that they encounter (Smith & de Villiers-Botha, 2021). Social media platforms that adopt nudges to personalize content and online experience are riskier as they potentially amplify controversial content and information that may be difficult to process and contextualize for some users. By way of an example, consider a social media AI-system that nudges a girl, based on her interests, to engage with online influencers whose content promotes a stereotyped version of femininity. However, she may be unable to understand the nature of that content (e.g. its sexualized stereotypes) and she will be prone to believe that what she sees has normative value (to which she might feel compelled to conform). Thus, AI-nudges in a context in which receivers may have limited knowledge and skills are likely to produce more harm. Children and teenagers may be considered protected categories also on account of their limited ability to make autonomous decisions and to exercise self-control. Although nudges in general exploit our non-conscious decision mechanisms, adults are often able to overcome this influence, should they so wish. However, research indicates that this is hardly ever the case for children and adolescents (Albert & Steinberg, 2011;Dansereau et al., 2013). Children and teenagers might not yet have an established and clear set of values and personal choices to refer back to and they are thus more easily influenced. For instance, teenagers are easily nudged into sharing personal, private information that increases their risk of encountering sexual predators online. Thus, AI-nudges in a context in which receivers may have limited decisionmaking skills are intrinsically more risky. Risks on social media (heightened by AI-nudging) "Mental wellbeing" is a broad concept that is usually understood to refer to a subject's psychological health, mood states (e.g., depression), and abilities, including social abilities such as self-control and establishing close personal relationships. According to some researchers, there is evidence to suggest that social media use has direct negative effects on mental wellbeing, affecting teenagers and young adults in particular. Addictions, the inability to connect emotionally, depression, and falling into internet rabbit holes are some of the potential psychological consequences of social media use (Caplan, 2002;Ihm, 2018). Using AI-nudging to captivate the interest and attention of the user is potentially unethical as it may be even more likely to produce this kind of psychological harm. Online safety is a further key concern when it comes to children and teenagers using social media. Studies show that some young people lie about their age when they sign up for social media platforms (OFCOM, 2022). This means that their "age as a user online" is not their "real age." As a result, children and adolescents might face safety issues online that can affect their wellbeing. Online dangers include cyberbullying, the non-consensual sharing of personal content (e.g., pictures and videos), and encountering unwanted sexual material. Using AI-nudging makes it difficult to spot and intervene when these kinds of material are shared, thus potentially producing more harm for the child. Psychological harm Privacy violations and lack of transparency Discrimination and lack of inclusion The time that young people spend on social media platforms can decrease the amount of time they spend sleeping or exercising. They may be moved by the fear of missing out on what their friends are up to and thus might defer bedtime or sports activities in favor of keeping abreast of developments on social media. In addition, spending time in front of a screen before bedtime tends to wake people up and so teenagers may end up falling asleep later as a result (Alonzo et al., 2021;Scott et al., 2019). Using AI-nudging to captivate the attention of the user is potentially unethical as it is more likely to keep users on their phones and thus generate harm. It is hard to deny that privacy concerns should be key when assessing children's online experiences. Owing to a lack of transparency and knowledge concerning privacy settings, children often unintentionally divulge private information. It is also possible that, without their knowledge or consent, children may be diagnosed with a disability and the user interface and experience may be adapted to that diagnosis (How Do We Address Children and Disability Rights Online?, 2022). AI-nudging potentially amplifies this effect as it customizes experiences based on the user's interests and abilities while also making it more difficult to understand the choices made by the system. AI models may contain inaccuracies and inaccurate generalizations. This may lead to unjust biases, discrimination, and stigma against minorities and marginalized groups. Children and teenagers with certain types of disabilities may be at a disadvantage when processing the information they find on platforms. They might have difficulties processing violent content and inappropriate material, or they might find it more challenging to recognize and report abuses and inappropriate behavior. Audience's limited cognitive skills Parental supervision (Mosseri, 2022) Regional support for parents Requiring mandatory, age-appropriate course in media literacy when opening an account (for underaged users only) UX tools costumed for underaged users (Lehnert et al., 2022) Sponsoring independent research on the impact of social media for neurodivergent children The video game industry is employing AI and machine learning at an increasing rate (Skinner & Walmsley, 2019). And, yet, the risks of using this technology are particularly striking when it comes to video games. Through the use of AI, game design is explicitly geared towards engagement: games are made to be fun and entertaining and users are pushed to spend their time and money on (or in) the games. User engagement and the investment of time and money are partly achieved through manipulative techniques called "dark patterns" (van Rooij et al., 2021). Dark patterns also incentivize users (e.g., by activating their sense of competition or fear of missing out) to make-often poor-decisions for psychological and social reward (Gray et al., 2018;Hamari et al., 2017;Mathur et al., 2019;Peters, 2014;Zagal et al., 2013). B) Video games & dark patterns Below, we have compiled a list of possible risk factors and the harms associated with them: The more complex the digital system, the higher its risk of harm becomes. Video game design has become more and more sophisticated, especially because it is increasingly generated and developed by AI. In particular, the adoption of AI-nudges and behavioral design enhances the level of possible harms: Audience's limited decision-making skills Misinterpretation of the content and experience in the game Depression & anxiety Incapacity to use/understand privacy settings in place Discrimination and lack of inclusion for neuro-divergent children Sleep disruption AI-induced choices (limited autonomy) Self-harm "Take a break" strategies (Mosseri, 2021) Disclosing to the user (and their parents) the use of AI-nudges in real-time Slow & difficult anomaly-detection and mitigation procedures Feeling of loneliness, inability to find help Lack of inclusion and support Support system with developmental psychologists Promoting social-media literacy in schools Sponsoring independent research on the impact and risks of social media for underaged users Open/ uncontrollable environment Cyberbullying Misuse of scope (e.g. sharing of inappropriate material) Lack of privacy Stricter rules for following/be-friending underaged users Stricter rules for preventing underaged users from making their profile (photos/videos) visible to all Employing useful, positive nudges(Alemany et al., 2019) Transparency regarding data-gathering and manipulation as in Article 12 (EU GDPR, 2018) Complexity of the agent-system Encounter with inappropriate, harmful content Internet "rabbit holes" Addiction Lack of transparency for the user "Take a break" strategies (Mosseri, 2021) Employing positive nudges(Alemany et al., 2019) Signal to parents when AI-enhance nudges are in use on children's phones In this table we show the relationship between the risk factors mentioned above and the recognized harmful effects of social media: Risk Factors on social media Actual Risks (made worse by AI-nudging) Mitigation strategies for companies Complexity of the autonomous agent system Dark pattern-harms: Dark pattern design aims to enhance user-engagement and the companies' profits by using nudges and trade-offs that affect users' choices: temporal dark patterns push the user to spend more time engaging with the game than they would have otherwise; monetary dark patterns push the user to spend more money during the game by, for example, making purchases through micro-transactions; social capital-based dark patterns use social rewards and connections to push users to make choices that may not actually benefit them (e.g., subscribing to paymonthly gaming platforms offering bonuses for inviting friends). Lack of transparency: The complexity of the system and the many options available make it difficult to prevent harmful consequences. In the case of AI-nudging, it may become increasingly difficult to ensure transparency or to understand the structure of the AI-enhanced gaming design before it creates harm. Video games represent an open environment. This structure has intrinsic risks: Sexual abuse, psychological harm: Multiplayer video games are often accessed by sexual predators and are riskier because of the lack of clear boundaries around, and barriers to, entry. Given the difficulty in predicting the results of this technology and given the vast numbers of players and users involved in gaming, there is an intrinsic risk of harmful effects for users, particularly if they are children. Privacy violations: Automated profiling is a key tool for generating highly individualized usercontent that is important to the use of AI-enhanced nudges. At the level of data gathering, during play, users' private data are collected through the use of nudges and behavioral tools that incentivize users to act in a certain way (e.g., watching a particular advert to gain points to spend during the game). These private data are then analyzed and sold, without the user's knowledge or consent. Children and teenagers may not be able to understand the complexity of the game-experience they are going through and they may make rushed decisions. Some of the potential harms they might encounter include: Problematic content (psychological harms): An example of problematic content generation emerged in 2019 when the startup Latitude launched an adventure game inspired by the story of Dungeons & Dragons. They used OpenAI to design the interactive experience that allows the user to craft and invent part of the story. Unfortunately, it turned out that "some players were typing words that caused the game to generate stories depicting sexual encounters involving children" (Simonite, 2021). Addiction and gambling (psychological and financial harm-see "Gaming: Screening and Assessment Tools," 2021): In a recent case, a 19-year-old spent $17,000 on in-game purchases through micro-transactions (Gach, 2017). "Gaming disorder" is a compulsive disorder, "characterized by a severely reduced control over gaming, resulting in an increasing gaming time and leading to negative consequences in many aspects of the individual life: personal, family, social, occupational and other relevant areas of functioning (World Health Organization)" (Gros et al., 2020, p. 1 In this audit, we adopt the so-called "informational approach," which is a metaethical theory on how to analyze moral concepts and moral situations. According to this view, a moral situation is a "specific region of the 'infosphere' in space and time within which the moral action takes place" (Floridi, 2013, p. 108). The design of the audit framework proposed in this document is based on the features and behavior of the components of a moral situation that involves, among other elements, AI systems that adopt nudges and underaged users (more on this below). Here, we explain the reason for this choice, which represents a substantial change in the usual approach to building an AI audit. Previous practices have based their audit criteria primarily on the organization and production of the pipeline or the life-cycle of an AI system. The main limitation of these approaches is that they zoom in on specific structural and design elements of AI systems while losing sight of the morally relevant macro-structures in which these systems are immersed. As a result, they are technologyspecific and thus hardly generalizable. In contrast, the informational approach looks at AI systems on a more abstract, but ethically relevant, level where moral choices and actions take place and ethical consequences emerge. Hence, the audit potentially applies wherever and whenever these macrostructures are present, thus allowing our approach to be generalizable. Methodology and Background for the Audit Modelling the moral situation A moral situation is a "specific region of the infosphere in space and time within which the moral action takes place" (Floridi, 2013, p. 108). According to the informational approach, when the moral situation is modelled, its information entities become the objects that compose the system. These objects have properties and comply with behavioural rules. As shown in Figure 2, the moral situation is composed of an agent (1), a receiver (2), and the action or information process (3). In the original version, the receiver is called the patient; in this version, we have chosen to call him the receiver because non-ethicists generally misinterpret the term patient. The agent and the receiver may be human, artificial or hybrid, yet all have a personal representation of the world that constitutes their information shell (4). Therefore, for the agent, the shell is only a reduction of reality that may be influenced by the factual information about the moral situation (5) that the agent has. This is the envelope (6) within which the moral situation develops, which in turn is located in the information environment surrounding us, the infosphere (7). Now since all these elements are informational elements, an action (or information process) impacts the whole informational ecosystem. In principle, all the components may be impacted by an information process. In an informational world in which digital interfaces are used to interact, these are the two big families of nudges that intervene to induce the achievement of a goal: AI-unrelated nudges and AI-related nudges. To be consistent with the reference literature, we shall call the former "digital nudges" and the latter "AI-enhanced nudges." A digital nudge may be described as the individual mechanisms created by decisionmaker architects ( Figure 4) and it can be part of a repository of nudging mechanisms that can be used to power a funnel of nudges. In the last of these cases, the sequence of distribution is also designed by decisionmaker architects ( Figure 5). Practically, digital nudges may be embedded in visual interfaces (buttons, colors, robot facial expressions), sound interfaces (voices, music, noises), tactile interfaces (vibrations), sequences, and filtering (social network walls) in order to trigger emotions or shape available information that then enables another decision or set of decisions to be made. Schematically, there are two types of nudges that intervene to induce the achievement of a goal or a set of goals: 1. Digital nudges: 2. AI-enhanced nudges: a. The individual mechanisms created by decision-maker architects. b. The sequences of mechanisms (funnels of nudges) designed by decision-maker architects. a. The sequences of mechanisms (funnels of nudges) generated by AI systems from a repository of nudges designed by decision-maker architects. b. Mechanisms self-generated using correlative inferences. AI-enhanced nudges may be organized or generated by AI systems. In the first case, AI-enhanced nudges can utilize behavioral feedback from the receiver to generate mechanisms or sequences of mechanisms using data-driven statistical/correlational logic and not causal logic, as is the case with mechanisms or sequences designed by decision-maker architects ( Figure 6).They are the result of a sorting process that uses a repository of designed digital nudges. Practically, the AI system shapes the funnel of nudges to tune the frequency and the sequence in order to influence a decision or set of decisions. In the second case, AIenhanced nudges are mechanisms that are self-generated using correlative inferences ( Figure 7). Thus, an AI system manipulates all the variables at its disposal to enhance the achievement of a goal, such as increasing engagement time in a social network or encouraging the purchase of digital artefacts in video games. Mapping the audit categories to the moral situation This audit framework has been developed to verify that the moral situation generated by the use of AI-nudging in video games and social networks that interact with children and teenagers remains in a state of moral neutrality (in other words, mitigating risks while reinforcing positive practices). The criteria of the present audit framework focus on each element of the moral situation. However, given the novelty of the moral situation approach, we opted to divide the criteria into more commonly understood categories. 1. Oversight criteria. These criteria are related to the global approach to the moral situation, the factual information available to the agent about the moral situation, the agent's informational universe or shell, and the systemic impact on the entire informational ecosystem or infosphere. The collection view on the moral situation (6), the factual information held about it by the agent (5), and the choice in delimiting the shell of the artificial agent (4 for the agent) comprise the organization's oversight of the processes. This is generally called the "digital governance" of a company. Moreover, the agent generates a set of actions that have an impact on the entire system and not only on the receiver. In fact, the distributed nature of aggregate actions generated by a sociotechnical multi-agent system may generate systemic ethical disruptions in both the medium and the long term. This disruption can have a local impact on the company and a global impact on the world of information, data, knowledge, and communication, populated by informational entities: namely, the infosphere (7) (Floridi, 2001). Oversight criteria cover the agent's shell (4 for the agent), the agent's factual information about the moral situation (5), the moral situation (6), and the infosphere (7), as outlined in Figure 3. 2. Agent-related criteria. These criteria are related to the features and behaviors of the agent. The agent of a moral situation is the digital artefact that autonomously provides a series of suggestions during the interaction with the receiver. Given the interactive and multidimensional nature of video games and social networks, an agent should be considered a simplified reference to a conglomerate of artificial agents composing a multi-agent system. Owing to the sociotechnical complexity of these multiagent systems, sometimes the action can resolve unintended adversarial conflicts (Tsvetkova et al., 2016). The agent generates a set of actions that interact with the receiver, sometimes even in an adversarial manner. Agent-related criteria cover the agent A (1) in Figure 3. From an informational perspective, nudges can be defined as those mechanisms that intervene in the informational process to influence the decision architecture of the receiver of an action. This definition allows for an agnostic view of nudges, which at the same time frees them from the moral charge of paternalistic libertarianism (Thaler & Sunstein, 2008), from the behavioral taxonomy that categorizes them with respect to the decision-making autonomy of the recipient (Jesse & Jannach, 2020), and from the deceptive characteristics, such as those in dark patterns (Smith & de Villiers-Botha, 2021). Nudges can thus simply be regarded as mechanisms of decision-making deformation 3. Receiver-related criteria. These criteria are related to the features, behaviors and relations of the receivers. The receiver (2) is the one who receives the action and is understood as an individual or group of individuals or, in general, an information entity. In our case, the receivers are children and teenagers, who may have different characteristics related to age, minority affiliations, cognitive skills, and so on. Receivers interpret action with respect to their personal universe of information or shell (4 for the receiver). The protection of children and teenagers will be concerned with ensuring the relational, physical, and psychological integrity of the receivers and the informational integrity of their shells. Receiver-related criteria cover the receiver (2) and their personal universe of information or shell (4 for the receiver), in Figure 3. 4. Information process-related criteria. These criteria are related to the nudging mechanism embedded in the set of actions that affect a receiver's decision-making architecture. The message or action (2) is the informational process that affects the recipient. This process involves the use of nudge mechanisms whose behavior ( Figure 4) and distribution ( Figure 5) are directly designed by decision-maker architects, or whose behavior ( Figure 6) and distribution (Figure 7) are fully self-generated using quantitative inferential correlations made by AI systems. Information process-related criteria cover the information process (3), in Figure 3. Limitations With regard to the research methods, some limitations need to be acknowledged. One of the main difficulties with the informational line of reasoning when applied to nudging is that it may seem to be competing with other legislation that already currently addresses privacy protection (e.g., the EU GDPR), consumer protection (the EU UCPD Unfair Commercial Practices Directive 2005), or the future regulation on AI systems (the EU AI Act proposal). In practice, some regulations already prohibit the subliminal or unfair use of mechanisms that negatively influence users' decision-making. However, despite efforts to limit the risks, there is evidence that nudging mechanisms, or their functions as enhanced by AI systems, are heavily used in social networks and certain categories of video games (Zagal et al., 2013). Given the elusive nature of such mechanisms, we wish, with this framework, to contribute to the improvement of the ecosystem for children and teenagers. This framework aims to become an operational support tool that goes over and above the compliance with current legislation. Yet, the current framework has not proven its value on the ground, so a process of testing and validation will be necessary before reaching an operational capacity. Another arguable weakness may be the arbitrariness in our definitions and the choice of the informational approach. We are aware that this may not become the final ethicsbased audit for AI-enhanced nudges; however, we defend the idea that through a change of perspective, new critical ethical issues can be brought to light. There are still many unanswered questions about nudges and there is abundant room for further progress in determining terminologies and methodologies. Framework for an Audit for the Use of Nudging on Children and Teenagers 1. Children and teenagers' best interests are fundamental to the decision-making process. 1.1. To which public document (i.e., Code of Ethics) did the company refer to endorse children and teenagers' best interests? According to the provisions of the UNCRC (The United Nations Convention on the Rights of the Child), the best interests of the child include but are not limited to: safety; health; wellbeing; family relationships; physical, psychological, and emotional development; identity; freedom of expression; privacy; and agency to form their own views. Public Disclosure Document. 1.2. Has the company endorsed the UN Convention on the Rights of the Child? The company must be aware of, and have integrated into its processes, the fundamental rights expressed in the Declaration of the Rights of the Child (1959). https://www.ohchr.org/en/resources/educators/human-rights-education-training/1-declaration-rightschild-1959. Public Disclosure Document. 1.3. Has the ethical committee for children and teenagers established a mechanism to balance the child's best interests with the interests of the shareholders? The mechanism should involve an explicit procedure, a list of the pros and cons, the identification of tensions, and explanation of the choice. Public Disclosure Document. 1.4. Are the ethical decisions recorded, justified, and accessible? Verify the presence of the registers, the quality of the justification (not ethically but procedurally), and the ease of internal access by decision-makers. Ethical Risk Assessment, Chain of Custody, Internal Documents. 1.5. If the system uses nudging mechanisms based on cognitive abilities (i.e., language), has the set of mechanisms been assessed by experts (in developmental psychology, orthophony)? Ensure that the experts' level of competence is aligned to the task by analyzing their CVs. Contract. 1.6. If the system uses nudging mechanisms based on sensorial responses (i.e., colors, sounds, touch, etc.), has the set of mechanisms been assessed by experts (in neuropsychology for children)? Ensure that the experts' level of competence is sufficient for coping with high-sensitivity cognitive mechanisms. The level of expertise may have been assessed by third parties or sector associates. Contract. 2. The committee's specific expertise on children and teenagers. 2.1. When was the ethics committee created? The date of creation makes it possible to determine the level of experience of the committee with respect to the company's industrial strategies. Internal Procedure ManuaI. 2.2. Are the profiles of the committee and its members suited to the role? Not every type of committee is suitable for dealing with children and teenagers' interests. Committee profiles must be designed to create a balance between experts and non-experts. Internal Procedure Manual. How frequently does the committee have meetings? The frequency of committee meetings makes it possible to highlight their engagement in the decision-making process. Internal Procedure Manual. 2.4. Has the company integrated stakeholders (parents, teachers, children, teenagers) into the committee with specific competence for children and teenagers? The presence of non-experts representing the second-level receivers of the action is necessary to avoid paternalistic decision-making and to investigate latent ethical risks. Internal Procedure Manual. 2.5. If children or teenagers are part of the committee, has the company disclosed a document guaranteeing their anonymity, safety, or integrity? Integrating children or teenagers (namely, those directly affected by nudges) makes it possible to interpret nudging phenomena from another point of view. However, if minors are involved, it must be ensured that the process is appropriate to their age and that they can be accompanied psychologically. Internal Procedure Manual. 3. Ethical code based on an ethical framework for children and teenagers. 3.1. Has the company adopted a framework in the defense of children and teenagers' digital rights (e.g., UK Children's Code, UNICEF Children's rights-by-design)? Check whether an international framework has been adopted and indicate why. Public Disclosure Document. 3.2. What is the framework used? Check whether a theoretical framework has been adopted or has been developed internally to support ethics committee decisions. The absence of a theoretical framework may be an indicator of poor competence in the use of ethical tools and application of ethical principles. Public Disclosure Document. 3.3. When was the framework adopted? Indicate when it was adopted and what the internal dissemination procedures were. Abrupt adoption in reaction to a reputational problem could be considered ethical washing. Internal Procedure Manual. 3.4. By whom was the framework chosen? Determine whether the ethics committee was sufficiently experienced to choose an ethical framework suitable for the protection of children and teenagers' rights. Contract. 3.5. If an international ethical framework is used, has it been culturally aligned to the region of use? In the case of adopting an international framework, check whether cultural alignment work has been done. Cultural non-alignment can lead to the failure of a framework's implementation. Internal Procedure Manual. 3.6. If an ethical framework is used, has it been assessed by an independent third party? To ensure a high level of quality, it is advisable to certify the quality of work with an external independent group. Some companies use an external ethics committee to review the output of the internal ethics committee. Public Disclosure Document. 3.7. If the company uses children or teenagers as testers for their digital artefacts, has the company established risk control mechanisms and adequate psychological support for them? Verify that the use of minors in production processes always ensures the welfare of minors. Verify that ethical codes of child protection are used, as is the case in scientific experiments. Internal Procedure Manual. 3.8. If children or teenagers are part of the testing team, has the company disclosed a document guaranteeing their anonymity, safety, or integrity? The company must ensure that any data of minors used during the production phases will never be disseminated. The company also guarantees the maintenance of the chain of custody of the information used. Public Disclosure Document. 4. Diversity policy. 4.1. Has the company ensured gender diversity in ethics committees, development teams, and testers? Gender diversity at any level of the production phases and in decision-making committees (ethics committees, managers, designers, developers, etc.) is crucial for proper gender projection to occur in the development of digital artifacts. Public Disclosure Document. 4.2. Has the company ensured racial diversity in the ethics committees, development teams, and testers? Cultural diversity and the representation of social subgroups at any level of the production phases facilitates social representation. Public Disclosure Document. 4.3. Has the company ensured direct receiver, parent, or second-level stakeholder diversity in ethics committees, development teams, and testers? The presence and/or consultation of the receivers of the action at any level of the production phases reinforces the voice of recipients' arguments from the earliest stages of development. Public Disclosure Document. 5. Ethical awareness program. 5.1. Has the company trained the employees in ethical awareness? Verify the robustness of the training program for digital ethical awareness. Internal Procedure Manual. 5.2. How often are employees trained in ethical awareness? Verify the frequency of updates. Una Tantum training may be insufficient to detect new instances of ethical choice. Correspondence (Internal or External). 5.3. Is the ethical awareness of employees assessed by an independent third party? Confirm whether a third-party entity has verified the actual level of awareness. Taking a course on ethics, alone, may be insufficient for a satisfactory real-life application of the ethics. Contract. 6. Awareness of the cognitive relevance of the receiver's info-frame encapsulation. 6.1. If using personal data/proxy data/biometric data, has the system been assessed by the ethics committee? The use of personal data (which make a person identifiable), proxy data (which have been inferred from other data), and/or biometric data (which involve biometric information with other informational value, as in facial recognition) must be explicitly evaluated and justified by an ethics committee. Ethical Risk Assessment, Chain of Custody, Internal Documents. 6.2. If using personal data/proxy data/biometric data, is there a monitoring mechanism to assess their ethical use in time? The ethics committee must guarantee that it has evaluated and approved the monitoring and risk mitigation mechanisms involving personal data, proxy data, and biometric data. Ethical Risk Assessment, Chain of Custody, Internal Documents. 7. Compliance with the legal ecosystem. 7.1. Is the company compliant with the relevant legal frameworks of AI systems? Verify the result of an independent audit on the compliance of AI systems with local legislation (such as the EU AI Act, EU Accessibility Act, etc.). Contract. 7.2. If in the EU or working with children or teenagers from EU Member States, has the company been certified compliant with the GDPR by an independent third party? Verify the result of an independent audit on compliance with the General Data Protection Regulation (GDPR). The GDPR is currently one of the highest standards for private data protection. Compliance can also be achieved with other regulations (e.g., CCPA). Public Disclosure Document. 7.3. If in the UK, has the company been certified compliant with the Children's Code by an independent third party? Verify the result of an independent audit on compliance with the UK Children's Code. The UK Children's Code is currently one of the highest standards for protection of the rights of children and teenagers. Public Disclosure Document. 8. Density of agents. 8.1. How many possible agents can generate actions simultaneously? Examine the convergence of multiple agents on the same target. A high level of interaction between agents and/or multi-agent systems can generate adversarial conflicts. Internal Logs, Register, or Database. 9. Evaluation of the agent. 9.1. What is the nature of the agent? Identify the nature of the agent: artificial, human-only, or hybrid. Code, Design, Functionality. 9.2. Is the agent a multi-agent system? Check whether the agent entity is a multi-agent system, because a multi-agent system exhibits system-level behavior with a much broader scope than the behavior of its component agents. Code, Design, Functionality. 9.3. Is the agent a socio-technical system? I.e., is agency capability derived from the interaction of the social aspects of the recipients and the technical aspects of the system? Networked multi-player videogames, social networks, email, chat, bulletin boards, and blogs are all socio-technical systems. Code, Design, Functionality. 10. Semantic level (mediator). 10.1. What are the semantic capacities and competences of the mediators in the process? Are the mediators capable of identifying or resolving anomalies? A mediator is an entity (human, artificial, or hybrid) delegated to mitigate a risk. Mediators can be call centers, experts, chat bots, expert systems, etc. The higher the semantic capital of the mediator, the more relevant the mediation action will be. Semantic capital refers to the content that empowers the ability to make meaning and sense (semanticize) something (Floridi, 2018). Code, Design, Functionality. 10.2. How are the competences of the mediators assessed? Verify who determined and evaluated the competence of mediators. Contract. 10.3. If AI is used in the mediation process, is there an ethical explanation by the committee to justify its use? If mediation systems are enhanced by AI it is necessary to make a specific assessment of the ethical risks associated with this choice. Ethical Risk Assessment, Chain of Custody, Internal Documents. 10.4. Are there risk mitigations that are age-appropriate and child-friendly, including pause buttons and save features, as well as nudge techniques, conditioning, or persuasive tactics that support wellbeing? In the case of interface-mediated mitigation processes, verify their effectiveness in supporting child wellbeing. Physical Testing. 11. The company is aware of the age appropriateness of the receiver. 11.3. Has the age appropriateness of the interfaces and notification been assessed by experts in developmental psychology? The levels of age appropriateness of a social network or video game can change with respect to content and its evolution over time. Ethical Risk Assessment, Chain of Custody, Internal Documents. 12. The company is equipped to handle anomalies. 13. Anomaly detection frequency. 13.1. How frequently does the anomaly detection process take place? The frequency of anomaly detection should be proportionate to the intervention time and the degree of risk. Internal Procedure Manual. 13.2. If there is a need for a minimum critical mass to detect an anomaly, is there an ethical explanation by the committee to justify the choice of frequency? In some cases, as in social networks, it is necessary to collect a minimum amount of information before anomalies can be analyzed. In this case, the definition of the frequency threshold must be determined by taking into account possible negative ethical effects. Ethical Risk Assessment, Chain of Custody, Internal Documents. 14. Anomaly remediation. 14.1. Is there a remediation mechanism for predictable anomalies? Verify procedures to mitigate familiar anomalies (known unknowns) and their applicability. Internal Procedure Manual. 14.2. Is there a remediation mechanism for unpredictable anomalies? Verify what procedures are used in case of anomalies that are not covered by procedures (unknown unknowns) and the employees' readiness. Internal Procedure Manual. 14.3. Has the company identified local associations that could help children, teenagers, or parents? Verify that the organization has made a list of parent and child support associations in case of behavioral problems (such as suicidal tendencies, drug addiction, eating disorders, etc. 15.3. Has an assessment been carried out to verify the accuracy in identifying the age of users? Age-verification systems can be easily circumvented, so it is necessary to verify their quality with specific tests. Physical Testing. 15.4. Has there been an assessment of the recipients to see if they belong to protected categories? During registration processes, one must be able to identify users who belong to protected categories. This verification is particularly sensitive because it necessitates a trade-off between necessary personal data and the purpose of their use. The intervention of the ethics committee and strict data custody mechanisms are necessary to avoid possible drifts. Physical Testing. 15.5. Has there been an assessment of the recipients to identify their semantic capabilities? During registration processes, one must be able to identify the real understanding of the receiver. This verification is particularly sensitive because it necessitates a trade-off between necessary personal data and the purpose of their use. The intervention of the ethics committee and strict data custody mechanisms are necessary to avoid possible drifts. Physical testing. 15.6. Has there been an assessment of the recipients to identify their linguistic abilities? During registration processes, one must be able to identify the linguistic abilities of the user. This verification is particularly sensitive because it necessitates a trade-off between necessary personal data and the purpose of their use. The intervention of the ethics committee and strict data custody mechanisms are necessary to avoid possible drifts. Physical Testing. 16. Semantic decay (receiver). 16.1. What impact does the model have on the receiver's semantic capital? In the long term, the use of nudge systems can reduce a child's or teenager's decision-making, comprehension skills or ability to read the messages on the screen (semantic capital). Therefore, it is necessary to understand whether the mechanisms in use have detrimental effects. Ethical Risk Assessment, Chain of Custody, Internal Documents. Distribution (density of receivers). 17.1. How many possible receivers can be affected by an action? The higher the number of re-recipients, the higher the possibility of creating user subgroups with common characteristics. In these cases, it is necessary to increase monitoring systems for social subgroups. Ethical Risk Assessment, Chain of Custody, Internal Documents. 20.18. Is the influence of specific words or wording monitored? The use of certain words can harm children or teenagers. Physical Testing. 20.19. Is the content detrimental to the wellbeing of the child with a disability according to published guidelines, codes of practice, and equality legislation applicable to the personal data processing, information, content, services, applications, interfaces, and design. (e.g., CAP, Ofcom, Ipso, OFT)? Special attention should be given to children and teenagers with disabilities. Ethical Risk Assessment, Chain of Custody, Internal Documents. 20.20. Is profiling turned off by default, unless examined and justified for the system? Children and teenagers should not be profiled unless this is necessary for the purpose of the primary video game or social network. Physical Testing. 20.21. Does the system permit nudge techniques that lead a child to lower data privacy settings? Children and teenagers should not be prompted to share their personal information by the system. Physical Testing. 20.22. Have nudge techniques that are detrimental to children and teenagers' best interests, including unconscious psychological processes, been searched and assessed? Any organization using systems of nudges for children and teenagers should be linked with an independent center to monitor the effect on cognitive and behavioral processes over the medium to long term. Physical Testing. 20.23. Are nudge techniques in favor of the child's best interests? Many nudge mechanisms are developed to promote the learning and wellbeing of children and teenagers. Constant review is essential for nudges to be developed in favor of the best interests of children and teenagers, which include, but are not limited to, safety, health, wellbeing, family relationships, identity, freedom of expression, privacy, agency to form their own views, and physical, psychological, and emotional development. Physical Testing. Authors Acknowledgments We would like to express our sincere gratitude to IBM, Ethics Techs Lab, and to the Notre Dame University for selecting our project and giving us the opportunity to work on it. We are grateful to the World Economic Forum, Save the Children, and Telefono Azzurro for providing the platform to disseminate our work. Some people were particularly important in the process of writing this document directly or indirectly. In particular, we would like to thank Luciano Floridi for providing the theoretical foundations on which our work is based. While we have drawn heavily on his ideas, any errors or misinterpretations of his thought are entirely our own and in no way reflect his endorsement of our work. We are deeply grateful to Maud Stiernet who provided us with valuable feedback on children and disabilities. Special thanks go to Gianpiero Nughedu for bringing our ideas to life with his appealing graphic designs. We are thankful to Sundaraparipurnan Narayanan for the fruitful exchange on nudging. We would also like to express our appreciation to Laurence Devillers for sharing part of our work with WG4 of CEN-CENELEC JTC21 for standardization purposes. Finally, we are grateful to ForHumanity, and especially Ryan Carrier, for introducing us to the practice of independent auditing of autonomous systems. Figure 3 : 3The moral situation (elaborated according toFloridi, L. (2013). The Ethics ofInformation. Oxford University Press,. Figure 4 : 4Single designed nudge. Figure 5 : 5Sequence of designed nudges organized by the architect. Figure 6 : 6Sequence of designed nudges organized by the AI system. Figure 7 : 7Sequence of AI-generated nudges organized by the AI system. ). Gaming disorders have become such a serious problem that they are now included in the WHO's list of mental health disorders (Addictive Behaviours: Gaming Disorder, 2020).Age-appropriate content and explanationsRegional support for parents to mitigate the dangers of dark patterns Direct support for children with disability status Video games should have a pause-function and the possibility to take breaks Employing positive "design patterns": healthy choices need to be the default Signaling to parents when AI-enhance nudges may be in use on children's phones Don't focus exclusively on goals and incentives (e.g. profitability) that lead to dark patternsOpen/ uncontrollable environment Audience's limited competence and behavior skills Open/ uncontrollable environment (number of receivers, message- variability) Audience's limited decision-making skills Audience's limited cognitive skills and competence Privacy violation Sexual abuse Addiction Sleep disruption Sedentary habits AI-induced choices (no autonomy) Gambling, loss of money Sexual abuse Misinterpretation of the content and experience Self-harm Incapacity to use/understand privacy settings in place Problematic content generation Transparency regarding data-gathering and manipulation as in Article 12 (EU GDPR, 2018) Avoiding childrens' automated profiling For underage-user profiling, providing data protection impact assessment as in Article 35 (EU GDPR, 2018) Support system with developmental psychologists Disclosing to the user (and the parents) the use of AI- nudges in real-time Make sure the game has a pause-function and the possibility to take breaks Employing positive "design patterns" and constraints (e.g. spending limits) Risk Factors in video-games Complexity of the agent-system Risks (made worse by AI-nudges) Harms derives from 'dark patterns' Lack of transparency on the use and logic of dark patterns Suggested mitigation strategies for designers and companies Identify how the organization determines ethically relevant instances. Internal Procedure Manual.12.1. Does the company have a mechanism for tracing anomalies? Identify how unusual behaviors are monitored. Internal Procedure Manual. 12.2. Does the company have a procedure for assessing the moral relevance for an anomaly? 12.3. Does the company have a procedure for detecting non-predictable anomalies? Verify the qualitative monitoring procedures used to identify anomalies not included in the procedures (unknown unknowns). Internal Procedure Manual. 12.4. Does the company have a monitoring system for the global population? Verify what procedures are in place to monitor the overall user population. Internal Procedure Manual. 12.5. Does the company have a monitoring system for the child populations? Verify what procedures are dedicated to monitoring children and teenagers. Internal Procedure Manual. 12.6. Does the monitoring system include a detailed analysis about subgroups? Identify the sociological tools used to monitor subgroups (related to race, gender, etc.) of users. Internal Procedure Manual. 12.7. Does the company have a monitoring system for highlighting outlier behaviors? Identify how extremely anomalous behaviors (outliers) are monitored. Internal Procedure Manual. ). Public Disclosure Document. Verify whether the associations on the list are equipped to welcome and help language minorities. Public Disclosure Document. 15. Semantic level (receiver). 15.1. Does the recipient have an adequate level of understanding of the message? Age is a discriminator; however, other abilities (linguistic, cognitive, cultural) may come into play in the reception of a message. The lesser the ability to receive, the greater the negative ethical consequences may be. Ethical Risk Assessment, Chain of Custody, Internal Documents.15.2.Does the recipient of the action have the tools to counteract a decision (intellectually and/or financially)?Verify whether the organization has deployed legal resources disproportionate to those of the recipient of the action. A legal investment not balanced to a risk mitigation investment can be an indicator of an unethical strategic approach. Correspondence (Internal or External).14.4. Are parent support associations culturally and linguistically inclusive? Marianna B. Ganapini is a philosopher, an Assistant Professor at Union College and a Visiting Scholar at the Center for Bioethics at the New York University. She is Faculty Director for the Montreal AI Ethics Institute and she is a Fellow at ForHumanity a non-profit association dedicated to the development of Independent Audits for Artificial Intelligence. She is also technical coordinator and researcher in a joint agreement between Union College and the IBM T. J. Watson Research Center. For this join agreement, Marianna is doing research as part of the research project "Thinking Fast and Slow in AI" led by Francesca Rossi. Marianna is the author of several publications in philosophy and she is the receiver of numerous prestigious grants and awards. She was recently awarded the Alan Turing's funding call award for 'Online courses in Responsible AI' and the Notre Dame-IBM Technology Ethics Lab Call for Proposals Award. She is the cofounder of the instructional design start-up called Logicanow. Marianna has a PhD in philosophy from the Johns Hopkins University.Enrico Panai is an AI Ethicist and a Human Information Interaction Specialist. Following his studies in philosophy and a multi-year experience as a consultant in Italy, he taught for seven years as an adjunct professor of Digital Humanities in the Department of Philosophy at the University of Sassari. Since his move to France in 2007, he has been working as a consultant for large corporations. In 2017, he studied Strategies for Cyber Security Awareness at the Institut National de Hautes Etudes de la Sécurité et de la Justice [Institute for Advanced Studies in Security and Justice] at the Ecole Militaire in Paris. He holds a PhD in Cybergeography with a thesis on "Latent Cyber Battlefields in Tourism". He is the president of the Association of AI Ethicists, founder of the consultancy BeEthical.be andEthiqueNum. fr, professor of Responsible AI at EMlyon Business School in Paris, member of the French Standardisation Committee for IA -AFNOR, co-convenor on AI-enhanced nudging at CEN-CENELEC, and board member of ForHumanity, a non-profit association dedicated to the development of Independent Audits for Artificial Intelligence. His main research interests concern cybergeography, cyber wars, information ethics, data ethics, cybersecurity, humaninformation interaction, philosophy of information and semantic capital.Marianna B. Ganapini Enrico Panai Assistant Professor of Philosophy (Union College) [email protected] LinkedIn President of the Association of AI Ethicists [email protected] LinkedIn © 2023 University of Notre Dame 18. Semantic plausibility.18.1.How can the behavior be perceived as human?The greater the number of interfaces that simulate human behavior, the greater the likelihood of the receiver placing trust in the system and thus becoming increasingly susceptible to nudge mechanisms. Physical Testing.19. The company has a defined policy for handling nudging mechanisms.19.1.Is there a shared definition of nudges?When it comes to nudge or nudging functions, not everyone has the same view. For this reason, it is crucial that the organization uses a shared definition of nudges. Public Disclosure Document.19.2.Have developers been trained to recognize nudges?Given the sometimes unintended nature of AI-enhanced nudging mechanisms, it can be difficult for developers to spot them swiftly and know that they are in operation. For this reason, developers must be trained to recognize them. Employee Handbook.19.3.Have developers been trained to understand the ethical impacts of nudges?The ability to recognize the mechanism must be coupled with an awareness of the possible ethical consequences that a set of nudges may have for children and teenagers. Employee Handbook.19.4.Is there a catalog of designed nudges?If AI systems use a nudge repository to set the frequency and intensity of distribution, the nudges used must be cataloged and approved by an ethics committee. This document has been funded by the Tech Ethics Lab. The findings, interpretations and conclusions expressed herein represent the views of the authors.No part of this publication may be reproduced in any form or by any means without the written permission of the authors or the Tech Ethics Lab.ImageAdobe StockTech Ethics Lab Ethical Risk Assessment, Chain of Custody, Internal Documents. 20.6. If the system uses nudging mechanisms based on sensorial responses (e.g., colors, sounds, touch), has the set of multimodal interactions been assessed by experts (in neuropsychology)? Given the persuasive capacity of sensorial interfaces, it is necessary to ensure device control for the wellbeing of children and teenagers. Ethical Risk Assessment, Chain of Custody, Internal Documents. 20.7. Is there an element in the interface indicating that the nudging process is in progress? It would be useful to have graphical items in the interface (such as red lights or alert vignettes) that show that the system is activating nudge mechanisms. Physical Testing. 20.8. Is there an element in the interface that indicates the deviation value from standard behavior? It would be useful to have graphical items in the interface (such as red lights or alert vignettes) that show children and/or teenagers that they are displaying an unexpected behavior. Physical Testing. 20.9. Are there any user notifications on the interface on the effects of nudging? In the case of unexpected behavior, it would be appropriate to have a notification mechanism to alert the child or teenager. Physical Testing. 20.10. Are there any notifications on the interface to alert parents or guardians to the effects of nudging? In case of unexpected behavior, it would be appropriate to have a notification mechanism to alert parents or guardians, adapted according to the age of the child or teenager. Physical Testing. 20.11. Is each item of user notification accompanied by age-appropriate advice written in a manner appropriate to the user's age? Notifications may not be effective if. Contract. 203Ethical Risk Assessment, Chain of Custody, Internal Documents. 20.5. If the system uses nudging mechanisms based on cognitive abilities (e.g. language), has the set of mechanisms been assessed by experts (in developmental psychology, orthophony)? The presence of developmental specialists should always be a requirement during the content production phase. not written in age-appropriate language. Physical Testing. 20.12. Is every element of parental notification accompanied by appropriate advice? Notifications to parents or guardians can be effective if supported by age-appropriate counseling for the child or adolescent. Physical Testingthe content writer sufficiently prepared to write age-appropriate texts? Messages written for children and teenagers must be appropriate to the age of the recipient. Physical Testing. 20.2. Is there an assessment of the appropriateness of the messages (texts, images, sounds, etc.)? Messages sent must be appropriate to the intended purpose. Contract. 20.3. If messages are self-generated, is there an approval process before distribution (ex ante)? In some interactive processes, self-generated messages may be verified by experts before being injected into the informational flow. Ethical Risk Assessment, Chain of Custody, Internal Documents. 20.4. If messages are self-generated, is there a verification process after distribution (ex post)? Even after the distribution of messages, a quality-monitoring system is required. Ethical Risk Assessment, Chain of Custody, Internal Documents. 20.5. If the system uses nudging mechanisms based on cognitive abilities (e.g. language), has the set of mechanisms been assessed by experts (in developmental psychology, orthophony)? The presence of developmental specialists should always be a requirement during the content production phase. Ethical Risk Assessment, Chain of Custody, Internal Documents. 20.6. If the system uses nudging mechanisms based on sensorial responses (e.g., colors, sounds, touch), has the set of multimodal interactions been assessed by experts (in neuropsychology)? Given the persuasive capacity of sensorial interfaces, it is necessary to ensure device control for the wellbeing of children and teenagers. Ethical Risk Assessment, Chain of Custody, Internal Documents. 20.7. Is there an element in the interface indicating that the nudging process is in progress? It would be useful to have graphical items in the interface (such as red lights or alert vignettes) that show that the system is activating nudge mechanisms. Physical Testing. 20.8. Is there an element in the interface that indicates the deviation value from standard behavior? It would be useful to have graphical items in the interface (such as red lights or alert vignettes) that show children and/or teenagers that they are displaying an unexpected behavior. Physical Testing. 20.9. Are there any user notifications on the interface on the effects of nudging? In the case of unexpected behavior, it would be appropriate to have a notification mechanism to alert the child or teenager. Physical Testing. 20.10. Are there any notifications on the interface to alert parents or guardians to the effects of nudging? In case of unexpected behavior, it would be appropriate to have a notification mechanism to alert parents or guardians, adapted according to the age of the child or teenager. Physical Testing. 20.11. Is each item of user notification accompanied by age-appropriate advice written in a manner appropriate to the user's age? Notifications may not be effective if not written in age-appropriate language. Physical Testing. 20.12. Is every element of parental notification accompanied by appropriate advice? Notifications to parents or guardians can be effective if supported by age-appropriate counseling for the child or adolescent. Physical Testing. Does the system have a just-in-time interface for reporting an anomaly? Forms for reporting anomalies should be available in the interfaces for children and teenagers. Physical Testing13. Does the system have a just-in-time interface for reporting an anomaly? Forms for reporting anomalies should be available in the interfaces for children and teenagers. Physical Testing. According to the age of the child, does the system have an interface for reporting an anomaly? Forms for reporting anomalies should be available to parents or guardians. Physical Testing. 20.15. Is the monitoring process continuous? A group of experts should continuously monitor, if only by sampling, the informational flow sent to children and teenagers. 20.14Physical Testing20.14. According to the age of the child, does the system have an interface for reporting an anomaly? Forms for reporting anomalies should be available to parents or guardians. Physical Testing. 20.15. Is the monitoring process continuous? A group of experts should continuously monitor, if only by sampling, the informational flow sent to children and teenagers. Physical Testing. Do nudge techniques, conditioning, and persuasive tactics extend user engagement beyond key risk indicators? If thresholds of performance indicators have been set, then extra performances must be monitored. Code, Design, Functionality16. Do nudge techniques, conditioning, and persuasive tactics extend user engagement beyond key risk indicators? If thresholds of performance indicators have been set, then extra performances must be monitored. Code, Design, Functionality. Are keywords, hate speech, and bullying filtered? Based on the age of the child, a blacklist should be compiled to vote on hate speech and bullying actions. Physical Testing17. Are keywords, hate speech, and bullying filtered? Based on the age of the child, a blacklist should be compiled to vote on hate speech and bullying actions. Physical Testing. Bibliography • Addictive behaviours: Gaming disorder. Bibliography • Addictive behaviours: Gaming disorder. (2020, October 22). WHO. https://www.who.int/news-room/ questions-and-answers/item/addictive-behaviours-gaming-disorder. Judgment and Decision Making in Adolescence. D Albert, L Steinberg, Journal of Research on Adolescence. 21• Albert, D., & Steinberg, L. (2011). Judgment and Decision Making in Adolescence. In Journal of Research on Adolescence (Vol. 21). http://repository.brynmawr.edu/psych_pubs. Enhancing the privacy risk awareness of teenagers in online social networks through soft-paternalism mechanisms. J • Alemany, E Del Val, J Alberola, A García-Fornes, 10.1016/j.ijhcs.2019.03.008International Journal of Human Computer Studies. 129• Alemany, J., del Val, E., Alberola, J., & García-Fornes, A. (2019). Enhancing the privacy risk awareness of teenagers in online social networks through soft-paternalism mechanisms. International Journal of Human Computer Studies, 129, 27-40. https://doi.org/10.1016/j.ijhcs.2019.03.008. Interplay between social media use, sleep quality, and mental health in youth: A systematic review. R Alonzo, J Hussain, S Stranges, K K Anderson, 10.1016/j.smrv.2020.101414Sleep Medicine Reviews. W.B. Saunders Ltd56• Alonzo, R., Hussain, J., Stranges, S., & Anderson, K. K. (2021). Interplay between social media use, sleep quality, and mental health in youth: A systematic review. In Sleep Medicine Reviews (Vol. 56). W.B. Saunders Ltd. https://doi.org/10.1016/j.smrv.2020.101414. Seeking better health care outcomes: The ethics of using the "nudge. • Blumenthal-Barby, J S Burroughs, H , 10.1080/15265161.2011.634481American Journal of Bioethics. • Brighouse, H., & Swift, A.122Princeton University PressFamily Values: The Ethics of Parent-Child Relationships• Blumenthal-Barby, J. S., & Burroughs, H. (2012). Seeking better health care outcomes: The ethics of using the "nudge." American Journal of Bioethics, 12(2), 1-10. https://doi.org/10.1080/15265161.2011.634481. • Brighouse, H., & Swift, A. (2014). Family Values: The Ethics of Parent-Child Relationships. Princeton University Press. Problematic Internet use and psychosocial well-being: development of a theory-based cognitive-behavioral measurement instrument 2 ( 0 2 ) 0 0 0 0 4-3. S E • Caplan, Computers in Human Behavior. 18• Caplan, S. E. (2002). Problematic Internet use and psychosocial well-being: development of a theory-based cognitive-behavioral measurement instrument 2 ( 0 2 ) 0 0 0 0 4-3. In Computers in Human Behavior (Vol. 18). www.elsevier.com/locate/comphumbeh. Children's Online Privacy Protection Rule. COPPA"). • Children's Online Privacy Protection Rule ("COPPA"). (2013). https://www.ftc.gov/system/files/2012-31341. pdf. Improving adolescent judgment and decision making. D F Dansereau, D K Knight, P M Flynn, R J Deibert, 10.1037/a0032495The Road to Digital Unfreedom: Three Painful Truths About Social Media. 44• Dansereau, D. F., Knight, D. K., & Flynn, P. M. (2013). Improving adolescent judgment and decision making. Professional Psychology: Research and Practice, 44(4), 274-282. https://doi.org/10.1037/a0032495. • Deibert, R. J. (2019). The Road to Digital Unfreedom: Three Painful Truths About Social Media. https://www. journalofdemocracy.org/authors/ronald-j-deibert/]. From Editors to Algorithms: A values-based approach to understanding story selection in the Facebook news feed. M A Devito, 10.1080/21670811.2016.1178592Digital Journalism. 56• DeVito, M. A. (2017). From Editors to Algorithms: A values-based approach to understanding story selection in the Facebook news feed. Digital Journalism, 5(6), 753-773. https://doi.org/10.1080/21670811.2016.1178 592. . • Eu, A I , Act Proposal. • EU AI Act Proposal . (2021). https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=celex%3A52021PC0206. . • Eu Gdpr, • EU GDPR. (2018). https://eur-lex.europa.eu/eli/reg/2016/679/oj. Commercial Practices Directive in the European Union. • Eu Unfair, • EU Unfair Commercial Practices Directive in the European Union. (2005). https://eur-lex.europa.eu/legal- content/EN/TXT/?qid=1585324585932&uri=CELEX%3A02005L0029-20220528. TikTok-addicted students delete app during exams. J Evitts, • Evitts, J. (2022, September 4). TikTok-addicted students delete app during exams. BBC NEWS. https://www. bbc.com/news/uk-wales-62720657. Ethics in the Infosphere. The Philosophers Magazine. L Floridi, 6• Floridi, L. (2001). Ethics in the Infosphere. The Philosophers Magazine, 6, 18-19. The Ethics of Information. L Floridi, Oxford University Press• Floridi, L. (2013). The Ethics of Information. Oxford University Press. Semantic Capital: Its Nature, Value, and Curation. L Floridi, 10.1007/s13347-018-0335-1Philosophy and Technology. 314• Floridi, L. (2018). Semantic Capital: Its Nature, Value, and Curation. Philosophy and Technology, 31(4), 481-497. https://doi.org/10.1007/s13347-018-0335-1. Meet The 19-Year-Old Who Spent Over $17,000 On Microtransactions. E Gach, 17• Gach, E. (2017, November 30). Meet The 19-Year-Old Who Spent Over $17,000 On Microtransactions. 17. https://www.kotaku.com.au/2017/11/meet-the-19-year-old-who-spent-over-17000-on-microtransactions/. 10.1371/journal.pone.0110137• Gaming: Screening and Assessment Tools. 9Public Library of Science• Gaming: Screening and Assessment Tools. (2021). In PLoS ONE (Vol. 9, Issue 10). Public Library of Science. https://doi.org/10.1371/journal.pone.0110137. The dark (patterns) side of UX design. C M Gray, Y Kou, B Battles, J Hoggatt, A L Toombs, 10.1145/3173574.3174108Conference on Human Factors in Computing Systems -Proceedings. • Gray, C. M., Kou, Y., Battles, B., Hoggatt, J., & Toombs, A. L. (2018). The dark (patterns) side of UX design. Conference on Human Factors in Computing Systems -Proceedings, 2018-April. https://doi. org/10.1145/3173574.3174108. Video Game Addiction and Emotional States: Possible Confusion Between Pleasure and Happiness? Frontiers in Psychology. L Gros, N Debue, J Lete, C Van De Leemput, 10.3389/fpsyg.2019.0289410Gros, L., Debue, N., Lete, J., & van de Leemput, C. (2020). Video Game Addiction and Emotional States: Possible Confusion Between Pleasure and Happiness? Frontiers in Psychology, 10. https://doi.org/10.3389/ fpsyg.2019.02894. An Audit Framework for Adopting AI-Nudging on Children -33. An Audit Framework for Adopting AI-Nudging on Children -33 Why do players buy in-game content? An empirical study on concrete purchase motivations. J Hamari, K Alha, S Järvelä, J M Kivikangas, J Koivisto, J Paavilainen, 10.1016/j.chb.2016.11.045Computers in Human Behavior. 68• Hamari, J., Alha, K., Järvelä, S., Kivikangas, J. M., Koivisto, J., & Paavilainen, J. (2017). Why do players buy in-game content? An empirical study on concrete purchase motivations. Computers in Human Behavior, 68, 538-546. https://doi.org/10.1016/j.chb.2016.11.045. • How do we address children and disability rights online?. • How do we address children and disability rights online? (2022, October 26). Eurochild. https://eurochild.org/ news/how-do-we-address-children-and-disability-rights-online/. Social implications of children's smartphone addiction: The role of support networks and social engagement. J Ihm, 10.1556/2006.7.2018.48Journal of Behavioral Addictions. 72• Ihm, J. (2018). Social implications of children's smartphone addiction: The role of support networks and social engagement. Journal of Behavioral Addictions, 7(2), 473-481. https://doi. org/10.1556/2006.7.2018.48. M Jesse, D Jannach, 10.1016/j.chbr.2020.100052Digital Nudging with Recommender Systems: Survey and Future Directions. • Jesse, M., & Jannach, D. (2020). Digital Nudging with Recommender Systems: Survey and Future Directions. https://doi.org/10.1016/j.chbr.2020.100052. Thinking, Fast and Slow. D • Kahneman, Penguin Press• Kahneman, D. (2011). Thinking, Fast and Slow. Penguin Press. Strongly Recommended" Revisiting Decisional Privacy to Judge Hypernudging in Self-Tracking Technologies. • La, 10.1007/s13347-018-0316-4Philosophy and Technology. 323• La forme des choix. (2019). https://www.cnil.fr/sites/default/files/atoms/files/cnil_cahiers_ip6.pdf. • Lanzing, M. (2019). "Strongly Recommended" Revisiting Decisional Privacy to Judge Hypernudging in Self- Tracking Technologies. Philosophy and Technology, 32(3), 549-568. https://doi.org/10.1007/s13347-018- 0316-4. Child-Computer Interaction: From a systematic review towards an integrated understanding of interaction design methods for children. F K Lehnert, J Niess, C Lallemand, P Markopoulos, A Fischbach, V Koenig, 10.1016/j.ijcci.2021.100398International Journal of Child-Computer Interaction. 32Elsevier B• Lehnert, F. K., Niess, J., Lallemand, C., Markopoulos, P., Fischbach, A., & Koenig, V. (2022). Child- Computer Interaction: From a systematic review towards an integrated understanding of interaction design methods for children. In International Journal of Child-Computer Interaction (Vol. 32). Elsevier B.V. https:// doi.org/10.1016/j.ijcci.2021.100398. Teens, Social Media, and Privacy. M Madden, A Lenhart, S Cortesi, U Gasser, M Duggan, A Smith, M Beaton, Pew Reseach Center• Madden, M., Lenhart, A., Cortesi, S., Gasser, U., Duggan, M., Smith, A., & Beaton, M. (2013). Teens, Social Media, and Privacy. In Pew Reseach Center. Dark patterns at scale: Findings from a crawl of 11K shopping websites. A Mathur, G Acar, M J Friedman, E Lucherini, J Mayer, M Chetty, A Narayanan, 10.1145/3359183Proceedings of the ACM on Human-Computer Interaction. the ACM on Human-Computer InteractionAssociation for Computing Machinery3• Mathur, A., Acar, G., Friedman, M. J., Lucherini, E., Mayer, J., Chetty, M., & Narayanan, A. (2019). Dark patterns at scale: Findings from a crawl of 11K shopping websites. In Proceedings of the ACM on Human-Computer Interaction (Vol. 3, Issue CSCW). Association for Computing Machinery. https://doi. org/10.1145/3359183. Finding the 'nudge' in hypernudge. S Mills, 10.1016/j.techsoc.2022.102117Technology in Society. 71• Mills, S. (2022). Finding the 'nudge' in hypernudge. Technology in Society, 71. https://doi.org/10.1016/j. techsoc.2022.102117. Algorithmic Nudges Don't Have to Be Unethical. M • Möhlmann, Harvard Business Review. • Möhlmann, M. (April 2021) Algorithmic Nudges Don't Have to Be Unethical, Harvard Business Review, https://hbr.org/2021/04/algorithmic-nudges-dont-have-to-be-unethical. Hypernudging in the changing European regulatory landscape for digital markets. V • Morozovaite, 10.1002/poi3.329Policy and Internet. • Morozovaite, V. (2022). Hypernudging in the changing European regulatory landscape for digital markets. Policy and Internet. https://doi.org/10.1002/poi3.329. Back to Newsroom Instagram Pausing "Instagram Kids" and Building Parental Supervision Tools. A Mosseri, • Mosseri, A. (2021). Back to Newsroom Instagram Pausing "Instagram Kids" and Building Parental Supervision Tools. https://about.fb.com/news/2021/09/pausing-instagram-kids-building-parental-supervision- tools/. Introducing Family Center and Parental Supervision Tools on Instagram and in VR. A Mosseri, • Mosseri, A. (2022). Introducing Family Center and Parental Supervision Tools on Instagram and in VR. https://about.fb.com/news/2022/03/parental-supervision-tools-instagram-vr/. A practical guide to effective behavior change: How to identify what to change in the first place. G.-J Y Peters, European Health Psychologist. 165• Peters, G.-J. Y. (2014). A practical guide to effective behavior change: How to identify what to change in the first place. European Health Psychologist, 16(5), 142-155. Uses and Gratifications of Social Media: A Comparison of Facebook and Instant Messaging. • Quan-Haase, A Young, A L , 10.1177/0270467610380009Technology & Society. 305Bulletin of Science• Quan-Haase, A., & Young, A. L. (2010). Uses and Gratifications of Social Media: A Comparison of Facebook and Instant Messaging. Bulletin of Science, Technology & Society, 30(5), 350-361. https://doi. org/10.1177/0270467610380009. When nudge comes to shove: Liberty and nudging in the era of big data. H S Saetra, 10.1016/j.techsoc.2019.04.006Technology in Society. 59• Saetra, H. S. (2019). When nudge comes to shove: Liberty and nudging in the era of big data. Technology in Society, 59. https://doi.org/10.1016/j.techsoc.2019.04.006. Digital nudging: Guiding online user choices through interface design Designers can create designs that nudge users toward the most desirable option. C Schneider, M Weinmann, Brocke, 10.1145/3213765Communications of the ACM. 617J. vom.• Schneider, C., Weinmann, M., & Brocke, J. vom. (2018). Digital nudging: Guiding online user choices through interface design Designers can create designs that nudge users toward the most desirable option. Communications of the ACM, 61(7), 67-73. https://doi.org/10.1145/3213765. Social media use and adolescent sleep patterns: crosssectional findings from the UK millennium cohort study. H Scott, S M Biello, H C Woods, 10.1136/bmjopen-2019-031161BMJ Open. 99• Scott, H., Biello, S. M., & Woods, H. C. (2019). Social media use and adolescent sleep patterns: cross- sectional findings from the UK millennium cohort study. BMJ Open, 9(9), e031161. https://doi.org/10.1136/ bmjopen-2019-031161. Digital Jiminy Crickets: Do apps that promote ethical behavior diminish our ability to make just decisions? Slate. E Selinger, T Seager, • Selinger, E., & Seager, T. (2012, July 13). Digital Jiminy Crickets: Do apps that promote ethical behavior diminish our ability to make just decisions? Slate. https://slate.com/technology/2012/07/ethical-decision- making-apps-damage-our-ability-to-make-moral-choices.html. An Audit Framework for Adopting AI-Nudging on Children -34. An Audit Framework for Adopting AI-Nudging on Children -34 It Began as an AI-Fueled Dungeon Game. It Got Much Darker. T Simonite, • Simonite, T. (2021, May 5). It Began as an AI-Fueled Dungeon Game. It Got Much Darker. Wired. https:// www.wired.com/story/ai-fueled-dungeon-game-got-much-darker/1/12. Hey, Google, leave those kids alone-Against hypernudging children in the age of big data. G Skinner, T Walmsley, 4th International Conference on Computer and Communication Systems : ICCCS 2019. Smith, J., & de Villiers-BothaAI & SocietyArtificial Intelligence and Deep Learning in Video Games A Brief Review• Skinner, G., & Walmsley, T. (2019, February). Artificial Intelligence and Deep Learning in Video Games A Brief Review. 4th International Conference on Computer and Communication Systems : ICCCS 2019. • Smith, J., & de Villiers-Botha, T. (2021). Hey, Google, leave those kids alone-Against hypernudging children in the age of big data. AI & Society. Nudge: Improving Decisions About Health, Wealth, and Happiness. R H Thaler, C R Sunstein, Yale University Press• Thaler, R. H., & Sunstein, C. R. (2008). Nudge: Improving Decisions About Health, Wealth, and Happiness. Yale University Press. Even good bots fight: The case of Wikipedia. M Tsvetkova, R García-Gavilanes, L Floridi, T Yasseri, 10.1371/journal.pone.0171774PLoS ONE. 212• Tsvetkova, M., García-Gavilanes, R., Floridi, L., & Yasseri, T. (2016). Even good bots fight: The case of Wikipedia. PLoS ONE, 12(2). https://doi.org/https://doi.org/10.1371/journal.pone.0171774. . • Uk Children's Code, • UK Children's Code. (2021, September). ICO. https://ico.org.uk/for-organisations/guide-to-data-protection/ ico-codes-of-practice/age-appropriate-design-a-code-of-practice-for-online-services/. Behavioral design in video games. A J Van Rooij, M V Birk, S Van Der Hof, S Ouburg, S Van Hilten, • van Rooij, A. J., Birk, M. v., van der Hof, S., Ouburg, S., & van Hilten, S. (2021). Behavioral design in video games. https://www.trimbos.nl/wp-content/uploads/2021/12/AF1965-Behavioral-design-in-video-games.pdf. The Theory and Practice of "Nudging": Changing Health Behaviors. I Vlaev, D King, P Dolan, A Darzi, 10.1111/puar.12564Public Administration Review. 764• Vlaev, I., King, D., Dolan, P., & Darzi, A. (2016). The Theory and Practice of "Nudging": Changing Health Behaviors. Public Administration Review, 76(4), 550-561. https://doi.org/10.1111/puar.12564. Hypernudge': Big Data as a mode of regulation by design. K Yeung, 10.1080/1369118X.2016.1186713Information Communication and Society. 201• Yeung, K. (2017). 'Hypernudge': Big Data as a mode of regulation by design. Information Communication and Society, 20(1), 118-136. https://doi.org/10.1080/1369118X.2016.1186713. Dark Patterns in the Design of Games. J P Zagal, S Björk, C Lewis, • Zagal, J. P., Björk, S., & Lewis, C. (2013). Dark Patterns in the Design of Games. https://core.ac.uk/ display/301007767?utm_source=pdf&utm_medium=banner&utm_campaign=pdf-decoration-v1.
{'fraction_non_alphanumeric': 0.0443243011306344, 'fraction_numerical': 0.022290875544399392, 'mean_word_length': 4.767322444939372, 'pattern_counts': {'":': 3, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 44, 'lorem ipsum': 0, 'www.': 10, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 12, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'The term "nudges" is used to describe strategic tools designed to influence people\'s choices and behaviors, usually without subjects consciously recognizing that influence. The idea of "nudge theory" was introduced by Thaler and Sunstein in their 2008 book, Nudge, and is widely studied in both behavioral economics and psychology.To date, work on nudge theory in technology has focused mostly on choice architectures, designed by user experience (UX) experts, which have mostly predictable outcomes. However, artificial intelligence (AI) has revolutionized the world of nudging: today, an individual\'s decisions can be shaped by AI interfaces that constantly adapt and change according to the user\'s choices and detectable behavior. The outcomes in this process are myriad and unpredictable, and nudging may even be an unintended consequence, distinct from the original intended design for the AI interfaces.A detailed discussion of this form of AI-enhanced nudges is currently lacking. At present, commercial legislation (EU Unfair Commercial Practices Directive in the European Union, 2005) and design guidelines (La Forme Des Choix, 2019) exist, which are helpful in mitigating the use of the soft manipulation of behaviors. In addition, two separate groups of experts are working on the topic: IEEE P7008 (RAS/SC/Ethical Nudging) and CEN-CENELEC JTC21 (project on AIenhanced nudging in WG 4 on Foundational and societal aspects). However, few standards are presently able to address the risk of using personalized sequences of nudging mechanisms.Here, we address this gap, first by highlighting the ethical problems that emerge from adopting nudges in AI, focusing on protected categories such as children (Smith & de Villiers-Botha, 2021). This assessment is based on an analysis of the structural risk factors and the potential harms associated with the use of AI-nudging. More specifically, we argue that AI-nudging potentially increases the likelihood of harm when other risk-factors are in place. Second, we build an infrastructure of trust for AI-nudging in the contexts of games and social media. This framework indicates how risks can be assessed and mitigated by laying the groundwork for a third-party independent audit system (focusing on children and teenagers).', 'arxivid': '2304.14338', 'author': [], 'authoraffiliation': [], 'corpusid': 258352685, 'doi': '10.48550/arxiv.2304.14338', 'github_urls': [], 'n_tokens_mistral': 24155, 'n_tokens_neox': 20986, 'n_words': 13465, 'pdfsha': 'ad966ac829416541f5277d3f85aead89aa7bd72d', 'pdfurls': ['https://export.arxiv.org/pdf/2304.14338v1.pdf'], 'title': ['An Audit Framework for Adopting AI-Nudging on Children -2', 'An Audit Framework for Adopting AI-Nudging on Children -2'], 'venue': []}
arxiv
Optimal Control of Storage Regeneration with Repair Codes 8 Nov 2017 Francesco De Pellegrini Rachid El Azouzi ⋆ Alonso Silva Olfa Hassani Optimal Control of Storage Regeneration with Repair Codes 8 Nov 2017Index Terms-high availabilitycontainersregenerationre- pair codesoptimal control High availability of containerized applications requires to perform robust storage of applications' state. Since basic replication techniques are extremely costly at scale, storage space requirements can be reduced by means of erasure and/or repairing codes.In this paper we address storage regeneration using repair codes, a robust distributed storage technique with no need to fully restore the whole state in case of failure. In fact, only the lost servers' content is replaced. To do so, new clean-slate storage units are made operational at a cost for activating new storage servers and a cost for the transfer of repair data.Our goal is to guarantee maximal availability of containers' state files by a given deadline. Upon a fault occurring at a subset of the storage servers, we aim at ensuring that they are repaired by a given deadline. We introduce a controlled fluid model and derive the optimal activation policy to replace servers under such correlated faults. The solution concept is the optimal control of regeneration via the Pontryagin minimum principle. We characterize feasibility conditions and we prove that the optimal policy is of threshold type. Numerical results describe how to apply the model for system dimensioning and show the tradeoff between activation of servers and communication cost. I. INTRODUCTION Container technology has quickly become the most promising cloud virtualization technique for it is lightweight and portable to different hardware. The uptake of containerization is fast up to the point that containers have become the unique runnable entities supported by Google's infrastructure [1]. The main difference of containers with respect to traditional virtual machines is the fact they are executed in the application space of a server. In fact, container's deployment does not require the instantiation of a full operating system on top of the one ruling the host server, thus representing a lighter solution with faster setup time. However, performing high availability of containerized applications is still a developing concept, e.g., building blocks such as failure detection and failover management are missing [2]. Virtual machines and containers, in turn, may be supported by availability guarantees [3] corresponding to specific service level agreements (SLA) to remain continuously functional (staying operational 99.999% of the time is called the five nines rule [4]). High availability requires a large degree of fault tolerance, both at the software and the hardware level. In the case of containerized applications, whenever a container fails, such failure can be masked, while the related traffic and tasks are redirected to healthy replicas. Incidentally, this is also the standard technique for seamingless migration of containerized applications across cloud servers for load-balancing purposes. Cloud native applications to be containerized are ideally instantiated in a stateless fashion. This makes it simple to render container execution highly available. However, containerized applications not always can be made fully stateless. Instead, they can store the running state in a replicated distributed storage. One existing deployment in the literature is found in [5]. By using dedicated plug-ins, persistent volume from inside containers is made accessible. The state is hence saved onto the distributed file system before replacement or migration, and the new container can finally access the recorded state [2]. In order to maintain an up-to-date version for restoring or to migrate running containers, snapshot images of the containers' state have to be created. Commit commands available on container platforms [6] can be used and several optimizations are possible to this respect, e.g., by continuously synchronizing changes only. Furthermore, in this context many core aspects are relevant, including load balancing, replica synchronization, system monitoring, alarm generation, and configuration management. Such aspects are beyond the scope of this work. Instead, we focus on the mechanisms for failure recovery of storage serves. In fact, robustness of data storage becomes the bottleneck to ensure high availability for containers' state maintenance. Data loss events in data centers are reported as a common event by several operators, e.g, FaceBook [7] and Yahoo [8]. The traditional solution is to perform server content replication using three-way random replication, considered the standard good practice in distributed filesystem management [9], [10], [11], [12]. In the literature on distributed storage, nevertheless, there exist techniques to reduce redundancy, e.g., by means of erasure codes or by repairing codes. Erasure codes can achieve great savings in storage space, and are actually used by major cloud provides such as Facebook [13] and Google [9]. The basic idea with erasure codes is that a file is split into k chunks, and then encoded into n = k + h chunks. In case of r ≤ h server failures, the system state can be recovered by transferring the chunks from k of the n − r remaining servers and decoding those to retrieve the whole original file. Then, the file can be encoded all over again into n chunks and finally the lost encoded chunks are restored on a set of r replacement number of operational repair servers at time 0 X0(t) number of newly activated repair servers at time t X k (t) number of repair nodes having k servers at time t servers. We observe that in our context the servers may be either physical servers or virtual storage units, and faults may be due to simultaneous node failures due, e.g., to cluster-wide power outages [11]. When there exists a large number of containers, the data transfer phase can become bottleneck for fast recovery in private clouds and a costly service to offer at scale in a public cloud. A recent solution is the usage of repairing codes [14], [15], [16]. Several trade-offs for such technique are addressed in [17], showing a 10-fold improvement is possible over standard erasure coding. In this work, we investigate feasibility and cost of regeneration operations using repair codes under correlated faults, i.e., when several servers fail at once. State availability requirements are represented by a deadline T to regenerate all servers. The cost that it takes to maintain seamless operation of containers' involve both state storage, i.e., activating enough replacement servers, and communication costs, i.e., the cost of transferring coded data chunks to regenerate lost servers. In the rest of the paper, the limit performance of the system are derived using an optimal control framework. The paper is organized as follows. In Sec. II we review the related literature, whereas in Sec. III we introduce the system model. In Sec. IV we formulate the problem of state storage regeneration in the framework of optimal control. Sec. V details the solution. Sec. VI provides numerical results and Sec. VII concludes the paper. The complete proofs of the statements derived in this paper can be found in Appendix. II. RELATED WORKS Designing robust storage in the cloud is a classical problem. Random replication schemes appeared in the early Google filesystem [9] and in Facebook data centers [10]. Basic erasure codes achieve higher reliability compared to replication with same storage [18]. The cost reduction in datacenter footprints operations is dramatic, exceeding 50%, thus recommending their usage in next generation systems [19]. Hence, new specialized erasure codes appeared, such as the local reconstruction codes in Windows Azure Storage. [20], or piggybacked Reed-Solomon codes to reduce cross-racks restoration bandwidth in Facebook's datacenters [13]. The breakthrough in the field are the erasure codes introduced by Papailiopoulos and Dimakis in [15], a class of locally repairable codes of maximum distance type separable (MDS). Several follow up works, e.g., [14], [16] have explored the fundamental tradeoff of such codes. They can be either of the minimum storage (MSR) or the minimum bandwidth (MBR) regenerating type. When the code can be maintained in systematic form, simple repair by transfer with no decoding operations is possible. However, in general, regeneration involves also decoding and so computing-time [17], a facet of the problem that we leave as part of future works. In the rest of the paper, we consider an assigned deadline for failsafe operations, as proposed in [3]: in that work, recovery time limits are imposed on the parallel failover of virtual machines based on customers' SLA plans. Also, in this work we adopt a system perspective close to [17]. To the best of the authors' knowledge, this is the first paper describing optimal control of failsafe operations for storage regeneration. III. SYSTEM MODEL In order to perform repair coding, the containers' state is divided into k chunks and encoded into n = k + h ones, by using a repairing code C = (n, k, d), where n > d > k. Parameter d represents the number of chunks that can be used to repair a lost or corrupted one. Each chunk is hence stored by distributing the encoded chunks to n servers. At time t = 0, r-servers fail, with 0 < r ≤ n − d, whereas n − r servers are still operational. In case of a r-servers fault, there are two main restoration options: either full restoration or regeneration of failed servers. If r < h, full state restoration is possible from any set of k servers chunks: full restoration requires to transfer k data chunks, which have α bytes each, to reconstruct the whole state file, to perform the encoding process all over again and, finally, to transfer the re-encoded chunks to the destination servers (see Fig. 1). Instead, selective regeneration of failed servers is possible when r < n − d: each lost server is replaced by using the chunks of d repair servers, by transferring β bits of information from each encoded chunk. Clearly, repairing is possible as long as there exists at least d repair servers. Optimal repairing MSR codes set α = B/k and β = α/(d − k + 1), whereas optimal repairing MBR codes set α = 2dB/[k(2d − k + 1)] and β = α/[k(2d − k + 1)] [14]. In order to obey to availability constraints, we assume that repairing operations need to complete by time horizon T , i.e., it must hold X d (T ) = n. Once the regeneration procedure through repair codes is completed, the full set of n operational repairing nodes is restored. We model such procedure as follows. First, new repairing servers are activated, e.g., by adding a new physical node to the datacenter, or by installing dedicated storage virtual machines on servers already part of the fabric. They can be switched on at a maximum rate ζ; the activation process is a Poisson process with rate ζ, i.e., new servers can be activated at rate ζ > 0 new replacement servers per second. Once activated, a repairing server downloads parity information from d operational repairing servers. We assume that each chunk transfer requires an exponential random time with mean 1/λ > 0. The regeneration procedure has two cost components: i. activation cost: activating a new repairing server has a cost c 1 per repairing server, due to the usage of legacy hardware in the datacenter and the related setup costs; ii. transfer cost: data transfer has a cost c 2 per bit, hence a chunk transfer has a cost c 2 β. During the regeneration process, due to hardware and/or software issues, failure of repairing servers may occur as well; failure instants are modeled as exponential random variables of parameter µ. The number of newly activated servers is denoted by X 0 (t), whereas X k (t) denote the number of replacement servers that have k repair chunks, for k = 1, . . . , d. Only nodes retrieving d chunks are operational replacement nodes: for notation's sake, we shall consider X d (t) the whole set of repairing nodes, i.e., those include the n − r which have not crashed. Restoration of the system using repair codes is possible if and only if X d (t) ≥ d at each point in time (if k ≤ X d (t) < n − d only full restoration is possible, if X d (t) < k, containers' state is lost.). A. Markov model and fluid approximation We shall study how to optimally activate new repairing servers in order to successfully restore all n servers within finite time horizon T at minimum cost. We start by assuming a stochastic control, namely, the probability u that a replacement server is activated. The activation rate of new repairing servers is ζ · u(t). The control acts by thinning the maximum activation rate ζ, which can be easily implemented by randomly sampling servers to be activated. Thus, ζ · u(t) is the rate at which replacement servers become active subject to stochastic control u(t). Let us define the state of the system as X = (X 0 , X 1 , . . . , X d ), where X k denotes the number of servers which have retrieved the content from k repairing servers. The state X(t) has a dynamics described by a continuous time Markov decision process (MDP), where we observe that all states X such that X d < d are absorbing, since no repairing is possible. Let assume that once k chunks are acquired, the repair process proceeds by downloading from the remaining d − k repairing servers. Hence, for any initial state x, we can write the entries of the transition probability matrix P x ′ ,x (dt) = È {X t+dt = x ′ |X t = x} = =                ζ u(t) dt if x ′ = x + e 0 µ x 0 dt if x ′ = x − e 0 (d − k + 1)λx k−1 dt if x ′ = x + e k − e k−1 µx k dt if x ′ = x − e k−1 o(dt) otherwise(1) where with e k is the k-th element of the standard basis. The first row describes the event of activation and the second row the failure of a newly activated repairing server, respectively. The third row describes the acquisition of a repair chunk by a repairing node having k − 1 chunks, and the fourth row describes the failure of a node having retrieved k chunks. The last row states that multiple transitions are negligible in the corresponding infinitesimal generator. The process of regeneration of the servers can be studied using a fluid model. Due to the structure of system (1), the meanfield approximation can be proved tight for n in the order of a few tenths [21]. By using the resulting fluid approximation, in the next section we shall obtain an optimal control problem in continuous time. The control space U is the set of the piecewise continuous functions taking values in [0, 1]. The dynamics of the number of repairing servers thus writeṡ X 0 (t) = −µ 0 X 0 (t) + ζ u(t) = f 0 (X, u, t) X 1 (t) = −µ 1 X 1 (t) + dλX 0 (t) = f 1 (X, u, t) . . . X k (t) = −µ k X k (t) + (d − k + 1)λX k−1 (t) = f k (X, u, t) . . . X d (t) = −µ d X d (t) + λX d−1 (t) = f d (X, u, t)(2) The ODE system (2) represents the dynamics of the regeneration process. Here, µ k = µ + λ(d − k) is the rate at which servers with k chunks fail to repair plus the rate at which they receive a new chunk, thus joining those having k + 1 chunks. Also, the first equation of the ODE system (2), namely f 0 (·), incorporates the activation of new peers at controlled rate ζ u(t). IV. OPTIMAL CONTROL PROBLEM The objective is to minimize the cost to restore the system by deadline T : the storage regeneration dynamics (2) is controlled by activation control u. Hence, the objective function writes (3) where the first term appearing in the integral is the servers' activation cost whereas the second one is the cost for trans-ferring chunks to repair servers. We shall solve the following optimization problem: J(u) = T 0 c 1 ζu(v) + c 2 β d−1 i=0 λ(d − i) X i (v) dv Problem 1 (Optimal Storage Regeneration). Find a control policy u which solves: min u∈U J(u) s.t. X d (t) ≥ d ∀ 0 ≤ t ≤ T (4) X d (T ) = n where d ≤ X d (0) ≤ n. In order for the repairing procedure to succeed, at least d repair nodes must be present at all points in time. We observe that, because (2) describes the deterministic dynamics of the mean value of the underlying MDP, it is possible that some sample paths do not satisfy the constraints, an event that should occur with small probability. To this aim, is possible to tighten constraints appearing in (4), in the form d ′ = (1 + ǫ 1 )d n ′ = (1 + ǫ 2 )n, where ǫ 1 , ǫ 2 > 0 represent relative margins. In the rest of the paper, we shall refer to the case ǫ 1 = ǫ 2 = 0 without loss of generality. Hereafter, we shall determine the conditions when the problem is feasible, i.e., the set of solutions of the problem is not empty. Actually, we recall that, as long as k chunks exist in the system, full restoration is still possible. However, we focus solely on the cases when regeneration is feasible, which can be determined easily by analysis of the uncontrolled dynamics, as discussed next. A. Feasibility and System Dimensioning Let us denote X d (t) the dynamics corresponding to u(t) ≡ 1 in the interval [0, T ]. Because the activation control is basically slowing down the maximum activation rate ζ, it holds X d (t) ≤ X d (t) for all t ∈ [0, T ]. Hence, it is immediate to observe that the problem is feasible if and only the dynamics of X d is compatible with the constraints. Such condition can be derived in closed form. By writing the Laplace transform of (2), i.e., X k (s) = L{X k (t)} we obtain X0(s) = ζ s + µ0 , X1(s) = X1(s) s + µ1 , . . . , X d (s) = X d−1 (s) + X d (0) s + µ d which in turn provides X d (s) = λ d d!ζ d k=0 (s+µ k ) + X d (0) s+µ . As showed in the Appendix, the following closed form expression for the dynamics of the repairing servers holds: X d (t) = e −µt ζ 1 − e −λt d + X d (0) Feasibility conditions can be described in terms of the system parameters as follows: In the rest of the paper we assume µ > 0 and feasibility in the sense meant by the previous statement. System dimensioning. Lemma 1 provides indications for dimensioning the system in order to guarantee feasible regeneration. In particular, in the worst case we would need to transfer n − d chunks to newly activated repair nodes. In turn, one would choose the time horizon by which to repair, namely T , and λ, i.e., the rate at which chunks can be transferred, and the code's triple C = (n, k, d), such in a way to satisfy the assumptions of the above statement. B. Relaxed problem Constraint Relaxation. The terminal state constraint can be accounted by relaxing the problem in the form J γ (u) = J(u) + γ (n − X d (T ))(5) by means of the terminal cost function q(X) := γ (n−X d (T )). We note that γ ≥ 0 has the role of a multiplier, and when the constraint is active γ > 0. State Augmentation. In order to account for the first constraint, we operate the augmentation of the state space by introducing an auxiliary variablė X d+1 (t) = (X d (t) − d) 2 ½ {d − X d (t)} where the indicating function ½ {x} = 1 if x > 0 and ½ {x} = 0 if x < 0. Since X d+1 (t) = T 0 X d+1 (v)dv + X d+1 (0). We impose the auxiliary constraint X d+1 (T ) = X d+1 (0) = 0: because X d+1 (t) ≥ 0 for t ∈ [0, T ], when such two constraints are satisfied, then X d (t) ≥ d all over the interval [0, T ]. We denote the problem of minimizing J γ (u) the relaxed problem and it will be solved next. C. Hamiltonian formulation and Pontryagin Principle. Let denote g(X, u, t) the instantaneous cost appearing inside the integral cost (3). In order to solve the optimal control problem, it is possible to write the Hamiltonian for the optimal control problem in standard form H(X, u, p) = p(t) f (X, u) + g(X) where p is the vector of co-state variables Hence, according to the Pontryagin Minimum Principle [22], [23], the optimal control u needs to satisfy u(t) = arg min u∈U H(X, u, p) where the associated Hamiltonian system iṡ X k = H p k (X, u, p) (6) p k = −H X k (X, u, p)(7) We have d + 1 terminal conditions in the form p k (T ) = q X k (T ) = 0 for k = 0, 1, . . . , d − 1, d + 1. Also, terminal condition p d (T ) = q X d (T ) = −γ holds. V. SOLUTION In order to solve the storage regeneration problem, we can write the Hamiltonian as H(X, u, p) = ζ c 1 + p 0 (t) u(t) − µX 0 (t)p 0 (t) + +c 2 β d−1 i=0 λ(d − i) X i (t) + d k=1 − µX k (t) + λ(d − k + 1) · X k−1 (t) p k (t) +p d+1 (t) (X d (t) − d) 2 · ½ {d − X d (t)}(8) We can hence derive from (6) the adjoint ODE system in the costate variableṡ p 0 = −H X0 = µ 0 · p 0 − λd · p 1 − c 2 βλd (9) p 1 = −H X1 = µ 1 · p 1 − (d − 1)λ · p 2 − c 2 βλ(d − 1) . . . p k = −H X k = µ k · p k − (d − k)λ · p k+1 − c 2 βλ(d − k) . . . p d−1 = −H X d−1 = µ d−1 · p d−1 − λ · p d − c 2 βλ p d = −H X d = µ d p d − 2(X d (t) − d) · ½ {d − X d (t)} p d+1 p d+1 = 0 In what follows, we will derive the structure of the solutions of the optimal control problem. A bang-bang policy [22], [23] is one where u(t) takes only extreme values, that is u(t) = 1 or u(t) = 0 a.e. in [0, T ]. Notice that bang-bang policies are very convenient for implementation purposes since they rely only on a set of switching epochs, where the control switches from 1 to 0 or vice versa. A threshold policy is one in the form u(t) =      0 t on < t ≤ T 1 0 < t ≤ t off 0 t off < t < T(10) Threshold policies are convenient since they depend on a pair of parameters only, namely thresholds t on and t off . Bang-bang structure. We observe that (8) is linear in the control u. Hence, because the optimal activation control minimizes the Hamiltonian, the optimal policy has to satisfy u(t) = 1 if p 0 (t) < −c 1 0 if p 0 (t) > −c 1(11) which depends on the dynamics of p 0 , i.e., of the ODE system (9). Actually, in order to prove that the policy is bang-bang and non-degenerate, we need also to prove that the policy has a finite number of switches and that there are no singular arcs, i.e., no arcs where the Hamiltonian is null over an interval of positive measure. Lemma 2. If the problem is feasible, the optimal policy is bang-bang with no singular arcs. The dynamics of p 0 can be derived in closed form: Lemma 3. It holds p 0 (t) = −F (t) + G(t) where F (t) = γ 1 − e −λ(T −t) d e −µ(T −t) G(t) = c 2 βdλ d−1 k=0 d − 1 k T −t 0 (e λv − 1) k e −(µ+λd)v dv Next, we characterize solutions of the relaxed problem which correspond to feasible solutions. A. Pure Activation Cost We start our analysis from the simpler case when the transfer cost is negligible compared to the activation cost, i.e., c 2 = 0. It is hence possible to derive explicit relations on the structure of the optimal control. µ 0 := max{0, d T log( d γ/c 1 (1 − e −λT ))} while the switching epochs write t on = max{0, T + 1 λ log z on }, t off = T + 1 λ log z off , where z on ≤ z off are the two solutions for 0 ≤ z ≤ 1 of the equation (1 − z) = d c 1 γ z − µ λd B. General case In the general case, it is sufficient to characterize the dynamics of the multiplier p 0 (t) in terms of the extremal points attained in the interior of [0, T ]. Finally, as proved in the Appendix. Theorem 2. The optimal solution of the relaxed problem is a threshold control. The optimal control is hence a threshold policy for which the presence of an initial delay, i.e., t on > 0, depends on the parameters of the system. However, as a straightforward application of the optimality principle, given an optimal threshold policy with t on and t off , for a given pair T and r, the new threshold policy where t ′ on = 0, t ′ off = t off − t on is optimal for the problem where r ′ = n − X d (t m ) ≥ r and horizon T ′ = T − t on < T . Thus we obtain the optimal solution in threshold form with no initial delay for more conservative conditions, i.e., for smaller time horizon and larger number of failed servers, and yet having same cost. (11) is such that X d (T ) ≥ n 3: initialize: γR ← γ0, γL ← 0, i ← 0 4: while |X d (T ) − n| > ε do 5: Step i ← i + 1 6: γi ← (γL + γR)/2 7: Obtain p0(t), t ∈ [0, T ] solving backwards (9) 8: Calculate the optimal control ui according to (11) 9: Obtain X d (t), t ∈ [0, T ] solving forward (2) 10: if X d (T ) > n then 11: γR ← γi 12: else 13: γL ← γi 14: end if 15: end while 16: return (ui, γi) Note that, in the relaxed problem, we cannot exclude the null control u ≡ 0, i.e., when p(0) > −c 1 and m > −c 1 . But, it cannot solve the constrained problem: to do so we need to determine the optimal multiplier γ, as seen next. C. Optimal multiplier The discussion so far has addressed the relaxed problem, and the multiplier γ has been treated as a constant for the sake of discussion. However, determining the optimal solution requires to identify a pair (u * , γ * ) where u * solves the original constrained problem. The main result in this section is that we can calculate the value γ * using a simple bisection search as described in Algorithm 1, under the feasibility assumptions of Lemma 1. The algorithm starts by exploring the interval for γ ∈ [0, γ 0 ], where γ 0 > 0 is a suitably large value such that it holds X d (T ) ≥ n. At line 5, 6 and 7 it solves the optimal control problem determining finally the terminal value X d (T ) within a certain tolerance ε > 0. The search algorithm leverages the fact that the terminal number of repair servers is monotone in γ. In fact, when the target value number exceeds n, it explores on the left of the current interval, i.e., it searches in [γ L , γ ]. Viceversa, when the target value is below n, it explores the right interval [ γ, γ R ]. The formal justification of the correctness of the above search strategy, and the optimality of the output of the algorithm is resumed by the following result, proved in the Appendix. Theorem 3. Under the assumptions of Lemma 1, the optimal pair (u * , γ * ) which solves the relaxed problem is unique, u * solves Prob.1, and γ * can be approximated using a bisection search as in Alg. 1. VI. NUMERICAL RESULTS This section presents some numerical results on optimal storage regeneration under a realistic parameter setting. It also serves the purpose of explaining how to make use of the proposed model to characterize limit performance of the regeneration technique under prescribed deadline constraints. We have assumed a reference C = (n, k, d) MBR repairing code. The parameters of the code are n = 50, k = 10 and d = 20 [17] 1 . Also, the reference container state size is assumed B = 10 Gbytes. We recall that, based on the fundamental relation on MBR codes, we can derive the chunk size as β = 2B/(k(2d − k + 1)) [14], which in this case amounts to β = 64.5161 Mbytes. The numerical setting is completed by assuming that repairing servers may fail according to rate µ = 0.001s −1 (we remind that in our model server failures during restoration are exponential random variables of parameter µ). Furthermore, the maximum rate at which repairing servers can be activated is set as ζ = 10 servers/s. Also, we need to make assumptions on the available network throughput: in our scenario, the throughput available for repairing operations is 1 Gbit/s. This value matches link speeds of production datacenters: peak bitrates for repair chunks transfer can be attained when performing restoration in priority, i.e., giving highest priority to the traffic operating the transmission of repairing chunks. The resulting target horizon for repairing has been set to T = 3.5 s, which is feasible given the setting considered. Fig. 2a and Fig. 2b depict the results of the optimal activation control in case of simultaneous failure of r = 11 servers at time t = 0. We have reported on the dynamics of the costate variable p 0 (t), superimposed to the switching threshold value, namely −c 1 (upper graph), the graph of the corresponding optimal control dynamics (middle graph) and the one corresponding to the dynamics of the number of repairing servers X d (t) (bottom graph). In both cases, the optimal multiplier γ * has been determined using Algorithm 1 with tolerance ǫ = 0.05. In particular, in Fig. 2a we have considered the case of a null communication cost c 2 = 0, which corresponds to γ * = 12.7719 whereas in Fig. 2b we have considered c 2 = 100 dollars/Gbyte, for which the optimal cost is attained for γ * = 175.855. In both cases the threshold policy is such that the pair t on = 0 s and t off = 1.22 s identifies the unique control driving he dynamics to satisfy terminal state constraint X d (T ) = n. Fig. 2c contains two tables calculated for different values of the cost c 1 and c 2 . They report on the value of the optimal cost J * (u * ). We note that, as expected, it increases with both cost c 1 and c 2 . Also, we observe same behavior for γ * : the optimal multiplier value increases and we ascribe this behavior to the fact that the value of γ has to enforce the terminal state constraint against augmented running costs c 1 and c 2 . VII. CONCLUSIONS In this paper we have presented an analytical framework for the optimal control of state regeneration, a promising technology in order to offer high availability of containerized applications at scale and ease stateful containers' migration. The idea is that leveraging the network filesystem, it is possible to decouple the storage of containers' state and the execution of application images running in pods. We have studied optimal time-constrained regeneration, a crucial aspect to ensure high availability in the containers' state access. Under failure of a number of servers, regeneration is performed by transferring repairing chunks to newly deployed, clean slate repair servers. This occurs at a communication cost and at a server activation cost. The optimal activation strategy is of threshold-type and can be evaluated in closed form. This work has been motivated by the limited number of studies on storage regeneration at system level [17] and it is by no means conclusive. Indeed, several research directions are due in order to understand the potential of these novel restoration techniques in cloud systems. The first one relates to the frequency of updates of the containers' state, a design choice required in order to decide how often to dump the containers' state onto the network filesystem. Such rate determines how much of the computation already elapsed can be recovered using regeneration. Another relevant issue is the case of repeated failures. Actually, the information on where faults are more likely becomes available to the administrator over time, e.g., based on direct observation or online learning techniques. The optimal policy may in turn span several cycles of faults/restorations and would account for techniques to learn the aposteriori distribution of faults over which to operate the optimal control. Also, correlated faults described in this work are simultaneous. In reality, they may be scattered in time, e.g., due to cascading failures. Under such fault dynamics, the optimal control studied in this work may be suboptimal. New models should identify how to counter the effect of later additional faults occurring during regeneration. APPENDIX PROOF OF LEMMA. 1 Proof: Feasibility is indeed equivalent to X d to respect the constraints. Condition ζ 1 − e −λT d ≥ d − X d (0) ensures that sup µ X d (T ) ≥ n, which is attained for µ = 0. We observe that µ d is also well defined: g(µ) = inf t∈[0,T ] X d (t) is a continuous function of µ. Because inf µ g(µ) = 0 and g(0) = X d (0) ≥ d, there exists a value of µ that satisfies the definition. The statement follows immediately from the definition of µ d and µ n and from a continuity argument. PROOF OF LEMMA. 2 Proof: Preliminarily, let observe that a feasible solution must be such that X d (t) ≥ d, for t ∈ [0, T ]. Thus, the dual ODE system has to be solved as in the non-augmented case, where it holdsṗ d = µ d p d . Hence, since the Hamiltonian is linear in the control, a feasible policy is a bang-bang one. In order to exclude the presence of singular arcs, we need to exclude the possibility that c 1 +p 0 (t) = 0 over an interval I of positive measure. We shall prove that multiplier p 0 's cannot be a constant over any interval I of positive measure µ(I) > 0, and this guarantees that the control is actually bang-bang [22]. Let assume that p 0 is a constant p 0 = −c 1 over interval I: hence all its k-th order derivatives vanish in I. But, it follows from (9) that p 1 = µ0 dλ p 0 : thus p 1 is also a constant over I, and since p 2 = µ1 (d−1)λ p 1 , p 2 as well. We hence iteratively obtain that p i is a constant for i = 1, , 2, . . . , d. However,ṗ d = µ d ·p d , so that 0 = p d = p d−1 = . . . = p 1 . Finally, p 0 = 0, which is a contradiction. PROOF OF LEMMA. 3 Proof: The adjoint ODE system can be solved via Laplace transform. We make the replacement q k (v) = p k (T − t), thus considering the backward time variable v = T − t. It holdṡ q k (v) = −ṗ k (t), so that system (9) writeṡ q 0 = −µ 0 · q 0 + λd · q 1 + c 2 βλd . . . q k = −µ k · q k−1 + (d − k)λ · q k+1 + c 2 βλ(d − k), k = 1, . . . , d − 1 . . . q d = −µ d · q d + 2(X d (t) − d) · ½ {d − X d (t)} q d+1 q d+1 = 0 Let Q k (s) = L{q k (v)} for t = 0, 1, . . . , d be the Laplace transform of the k-th variable q k . The corresponding system writes sQ k (s) = −µ k Q k (s) + (d − k)λQ k+1 (s) + c 2 βλ(d − k) 1 s , for k = 0, 1, . . . , d − 1 sQ d (s) = −µ d Q d (s) − γ (12) from which Q k (s) = λ(d−k) (s+µ k ) Q k+1 (s) + c2βλ(d−k) s(s+µ k ) is obtained. By iterative replacement, and by accounting for the fact that i − j = n i=0 (−1) n−i λ n i!(n − i)! f i (t)(13) The statement follows after some algebraic manipulations of the above expression. from which it is immediate to observe that the absolute minimum over the real line is attained at t min = T − 1 λ log 1+ dλ µ ; the minimum writes m := −γ( µ λd ) d /(1 + µ λd ) d+ µ λ . Switching epochs t s are determined by the instants solving p 0 (t s ) = −c 1 . First, let observe that p 0 (T ) = 0 > −c 1 anḋ p 0 (T ) = λd, so that the control is indeed null in a left interval of T . In particular, it is possible to identify three cases: for a given value of γ > 0, there might exist either two, one or zero switching epochs in the interior of [0, T ]. We consider the three cases separately. Case i: single switch. The condition for a unique switching epoch is p 0 (0) < −c 1 , which writes γ(1 − e −λT ) d e −µT > c 1 , so that µ > d T log d γ/c 1 (1 − e −λT ) := µ 0 By inspection of (14), due to the continuity of p 0 , there exists switching epoch 0 < t off < T such that p 0 (t off ) = −c 1 . Because p 0 (t) has unimodal structure, such switch is unique so that the corresponding optimal control is in threshold form. Namely, u(t) = 1 for 0 ≤ t < t off and zero otherwise. Case iii: two switches. Condition p 0 (0) > −c 1 leads to a non-null control if and only if m < −c 1 . From the unimodal structure of p 0 , and from classic continuity arguments, there exist two real values, namely t on < t min < t off where p 0 (t on ) = −c 1 = p 0 (t off ), so that u(t) = 1 for t on < t < t off and zero otherwise. Case ii: no switch. This is the case t on = t off = 0, i.e., the optimal control is the null one. It occurs when p 0 (0) > −c 1 and m ≥ −c 1 . Finally, the explicit expression of the switching epochs is obtained by solving equation p 0 (t) = −c 1 , which concludes the proof. Figure 1 . 1Storage regeneration: repairing failures via erasure codes (left) and repairing codes (right); r = 1, d = n − 2 Lemma 1 . 1Problem 1 is feasible if and only if ζ 1−e −λT d ≥ n e µT − X d (0) and it is so for any µ ≤ µ, where µ := min{µ n , µ d } and µ n = max{µ ≥ 0|X d (T ) ≥ n} and µ d = max{µ ≥ 0| min t∈[0,T ] X d (t) ≥ d}. Theorem 1 . 1If c 2 = 0, then a solution of the relaxed problem is a threshold policy, in particular: i. Single switch: t on = 0 and 0 < t off < T iff µ ≤ µ 0 ; ii. Null control: 0 = t on = t off iff µ > µ 0 , and m ≥ −c 1 , where m = min v∈[0,T ] {p 0 (v)}; iii. Double switch: 0 < t on < t off ≤ T iff µ > µ 0 , and m < −c 1 The critical value Lemma 4 . 4Let S(γ) be the set of the interior extremal points of p 0 (t) for a given choice of the constraint multiplier γ. Then, S(γ) is one of the following forms: ∅, {M }, or {m, M }, where m := p 0 (t m ) denotes a minimum and M := p 0 (t M ) a maximum, and it holds 0 ≤ t m < t M < T . Figure 2 . 2Optimal regeneration control a) zero communication cost b) c 2 = 100 dollar/Gbyte c) The optimal cost and the optimal multiplier as function of costs c 1 and c 2 . s + µ) h have to be inverted. Let us denote f h (t) := e −µ h t ½ {t ≥ 0}, for the sake of notation. By recalling L{e −µt ½ {t ≥ 0}} = 1/(s + µ), From Lemma 3, if c 2 = 0, it followṡ p 0 (t) = γe −µ(T −t) (1−e −λ(T −t) ) d−1 −µ+(µ+λd)e −λ(T −t) Table I MAIN INOTATION USED THROUGHOUT THE PAPERSymbol Meaning B state size β repairing chunk size C = (n, d, k) repairing code c1 cost per repair node activation c2 cost per transferred repair chunk bit u(t) activation control ζ maximum activation rate λ number of repair chunks transferred per second µ repair server failure rate X d (0) Algorithm 1: Optimal Regeneration Control 1: input: T , β, λ, c1, c2, ε γ0 s.t. u from2: In[17] the code redundancy targets storage availability of 0.99 PROOF OF LEMMA. 4Proof: It is possible to write the derivative of the multiplier p 0 (t) in a convenient form. For notations' sake, we denote p 0 (t) the expression of p 0 (t) when c 2 = 0, and t m the point (on the real line) where the minimum of p 0 (t) is attained. We hence obtaiṅwhere we know that p 0 (t min ) = 0,ṗ 0 (t) < 0 for t < t min anḋ p 0 (t) > 0 for t > t min . However, p 0 (T ) = 0 andṗ 0 (t) = −c 2 βλd < 0, so that there exists a whole left neighborhood of T where p 0 (t) > 0 and decreasing. And,ṗ 0 (t) < 0 for t < t min .By taking into account the sign ofṗ 0 and the additional negative term appearing in(14), it is immediate to conclude that only the following three cases are possible:i S = ∅: in this case p 0 (t) is strictly decreasing in [0, T ]; ii S = {M } and the maximum is attained at 0 < t M < T : in this case p 0 (t) strictly increasing in [0, vi. S = {m, M } with p 0 (0) > −c 1 with m > −c 1 implies a two-switch control with t on > 0 and 0 < t on < t off < T .This concludes the proof, since in all cases the optimal bangbang control is a threshold policy.PROOF OF THM. 3Proof: In this proof we need to make the dependence on γ explicit in the notation: e.g., u * γ is the optimal control when multiplier γ is adopted in the relaxed objective function J γ (u). i. The fact that pair (u * , γ * ) minimizing J γ (u) is unique follows from the expression J γ (u) = J(u)+γ(n−X d (t)). Let assume by contradiction another pair (u, γ) is optimal, then it must hold J(u * ) = J(u). However, this implies that the two threshold policies must be identical, i.e., u * = u, and so also γ * = γ, because of the linear dependence with multiplier γ in (3).ii. The fact that the relaxed problem solves for the optimal solution of the original constrained minimization follows from the following argument. Let define U n f = { u| X d (T ) = n, X d (t) ≥ d, t ∈ [0, T ]} ⊂ U, let γ * be the optimal multiplier and u * the optimal solution of the constrained problem.where the equality follows from the fact that γ(n−X d (u)) = 0 over set U n f . iii. The correctness of the bisection search is due to the fact that J γ (u * ) is indeed monotone in γ. In fact, costate variable p γ 0 (t) = −γ F (t) + G(t) where we have made explicit the dependence on γ appearing in(3). Now, with respect to switching epoch t γ off , let us consider multiplier γ + δ, for some δ > 0. Then we can writeOpposite holds for t γ on : t γ on > t γ+δ on . From direct inspection of the cost function, it follows J γ (u * γ ) < J γ+δ (u * γ+δ ), which proves the claimed monotony argument. . B Burns, B Grant, D Oppenheimer, E Brewer, J Wilkes, Omega Borg, Kubernetes , Comm. of the ACM. 595B. Burns, B. Grant, D. Oppenheimer, E. Brewer, and J. Wilkes, "Borg, Omega, and Kubernetes," Comm. of the ACM, vol. 59, no. 5, pp. 1837- 1852, May 2016. Comparing containers versus virtual machines for achieving high availability. W Li, A Kanso, Proc. of IEEE IC2E. of IEEE IC2ETempe, USW. Li and A. Kanso, "Comparing containers versus virtual machines for achieving high availability," in Proc. of IEEE IC2E, Tempe, US, March 9-12 2015. ResilientVM: High performance virtual machine recovery in the cloud. V Salapura, R Harper, M Viswanathan, Proc. of ACM AIMC. of ACM AIMCBordeaux, FranceV. Salapura, R. Harper, and M. Viswanathan, "ResilientVM: High performance virtual machine recovery in the cloud," in Proc. of ACM AIMC, Bordeaux, France, Apr 21-24 2015, pp. 7-12. High-availability computer systems. J Gray, D P Siewiorek, Computer. 249J. Gray and D. P. Siewiorek, "High-availability computer systems," Computer, vol. 24, no. 9, p. 39-48, 1991. . Infinit International Inc. Infinit International Inc, https://infinit.sh/documentation/reference. Docker: The linux container engine. Docker, Docker, "Docker: The linux container engine," http://www.docker.io. Apache Hadoop goes realtime at Facebook. D Borthakur, Proc. of ACM SIGMOD PODS. of ACM SIGMOD PODSAthens, GreeceD. Borthakur et al., "Apache Hadoop goes realtime at Facebook," in Proc. of ACM SIGMOD PODS, Athens, Greece, June 12-16 2011. Data availability and durability with the hadoop distributed file system. R J Chansler, The USENIX Magazine. 371R. J. Chansler, "Data availability and durability with the hadoop dis- tributed file system," The USENIX Magazine, vol. 37, no. 1, February 2012. The Google file system. S Ghemawat, H Gobioff, S.-T Leung, SIGOPS Oper. Syst. Rev. 375S. Ghemawat, H. Gobioff, and S.-T. Leung, "The Google file system," SIGOPS Oper. Syst. Rev., vol. 37, no. 5, pp. 29-43, Oct. 2003. Cassandra: A decentralized structured storage system. A Lakshman, P Malik, SIGOPS Oper. Syst. Rev. 442A. Lakshman and P. Malik, "Cassandra: A decentralized structured storage system," SIGOPS Oper. Syst. Rev., vol. 44, no. 2, pp. 35-40, Apr. 2010. Copysets: Reducing the frequency of data loss in cloud storage. A Cidon, S Rumble, R Stutsman, S Katti, J Ousterhout, M Rosenblum, Proc. of USENIX ATC. of USENIX ATCSan Jose, USA. Cidon, S. Rumble, R. Stutsman, S. Katti, J. Ousterhout, and M. Rosenblum, "Copysets: Reducing the frequency of data loss in cloud storage," in Proc. of USENIX ATC, San Jose, US, June 26-28 2013. Tiered replication: A cost-effective alternative to full cluster geo-replication. A Cidon, R Escriva, S Katti, M Rosenblum, E G Sirer, Proc. of USENIX ATC. of USENIX ATCSanta Clara, CAA. Cidon, R. Escriva, S. Katti, M. Rosenblum, and E. G. Sirer, "Tiered replication: A cost-effective alternative to full cluster geo-replication," in Proc. of USENIX ATC, Santa Clara, CA, July 8-10 2015. A solution to the network challenges of data recovery in erasure-coded distributed storage systems: A study on the Facebook warehouse cluster. K V Rashmi, N B Shah, D Gu, Proc. of USENIX HotStorage. of USENIX HotStorageSan Jose, CAK. V. Rashmi, N. B. Shah, D. Gu et al., "A solution to the network challenges of data recovery in erasure-coded distributed storage systems: A study on the Facebook warehouse cluster," in Proc. of USENIX HotStorage, San Jose, CA, June 27-28 2013. Distributed storage codes with repair-by-transfer and nonachievability of interior points on the storage-bandwidth tradeoff. N B Shah, K V Rashmi, P V Kumar, K Ramchandran, IEEE Trans. Information Theory. 583N. B. Shah, K. V. Rashmi, P. V. Kumar, and K. Ramchandran, "Distributed storage codes with repair-by-transfer and nonachievability of interior points on the storage-bandwidth tradeoff," IEEE Trans. Information Theory, vol. 58, no. 3, pp. 1837-1852, 2012. Locally repairable codes. D S Papailiopoulos, A G Dimakis, IEEE Trans. Information Theory. 6010D. S. Papailiopoulos and A. G. Dimakis, "Locally repairable codes," IEEE Trans. Information Theory, vol. 60, no. 10, pp. 5843-5855, Oct 2014. Xoring elephants: novel erasure codes for big data. M Sathiamoorthy, M Asteris, D Papailiopoulos, A G Dimakis, Proc. of PVLDB. of PVLDBRiva del Garda, ItalyM. Sathiamoorthy, M. Asteris, D. Papailiopoulos, A. G. Dimakis et al., "Xoring elephants: novel erasure codes for big data," in Proc. of PVLDB, Riva del Garda, Italy, August 26-30 2013. Regenerating codes: A system perspective. S Jiekak, A.-M Kermarrec, N Le Scouarnec, G Straub, A Van Kempen, SIGOPS Oper. Syst. Rev. 472S. Jiekak, A.-M. Kermarrec, N. Le Scouarnec, G. Straub, and A. Van Kempen, "Regenerating codes: A system perspective," SIGOPS Oper. Syst. Rev., vol. 47, no. 2, pp. 23-32, Jul. 2013. Erasure coding vs. replication: A quantitative comparison. H Weatherspoon, J Kubiatowicz, Proc. of IPTPS. of IPTPSCambridge, MA, USAH. Weatherspoon and J. Kubiatowicz, "Erasure coding vs. replication: A quantitative comparison," in Proc. of IPTPS, Cambridge, MA, USA, March 7-8 2002. Rethinking erasure codes for cloud file systems: minimizing I/O for recovery and degraded reads. O Khan, R C Burns, J S Plank, W Pierce, C Huang, Proc. of USENIX FAST. of USENIX FASTSan Jose, USO. Khan, R. C. Burns, J. S. Plank, W. Pierce, and C. Huang, "Rethinking erasure codes for cloud file systems: minimizing I/O for recovery and degraded reads," in Proc. of USENIX FAST, San Jose, US, February 14-17 2012. Erasure coding in Windows Azure storage. C Huang, H Simitci, Y Xu, Proc. of USENIX ATC. of USENIX ATCBoston, MAC. Huang, H. Simitci, Y. Xu et al., "Erasure coding in Windows Azure storage," in Proc. of USENIX ATC, Boston, MA, June 26-28 2012. Dynamic control of coding for progressive packet arrivals in DTNs. E Altman, L Sassatelli, F. De Pellegrini, IEEE Trans. on Wireless Comm. 122E. Altman, L. Sassatelli, and F. De Pellegrini, "Dynamic control of coding for progressive packet arrivals in DTNs," IEEE Trans. on Wireless Comm., vol. 12, no. 2, pp. 725-735, 2013. An introduction to optimal control. G Leitmann, McGraw-HillG. Leitmann, An introduction to optimal control. McGraw-Hill, 1966. D E Kirk, Optimal Control Theory. An Introduction. Prentice Hall13th edD. E. Kirk, Optimal Control Theory. An Introduction., 13th ed. Prentice Hall, 2004.
{'fraction_non_alphanumeric': 0.06701297581583826, 'fraction_numerical': 0.01886019743932405, 'mean_word_length': 3.7332177106712914, 'pattern_counts': {'":': 0, '<': 40, '<?xml version=': 0, '>': 33, 'https://': 1, 'lorem ipsum': 0, 'www.': 1, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 32, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': "High availability of containerized applications requires to perform robust storage of applications' state. Since basic replication techniques are extremely costly at scale, storage space requirements can be reduced by means of erasure and/or repairing codes.In this paper we address storage regeneration using repair codes, a robust distributed storage technique with no need to fully restore the whole state in case of failure. In fact, only the lost servers' content is replaced. To do so, new clean-slate storage units are made operational at a cost for activating new storage servers and a cost for the transfer of repair data.Our goal is to guarantee maximal availability of containers' state files by a given deadline. Upon a fault occurring at a subset of the storage servers, we aim at ensuring that they are repaired by a given deadline. We introduce a controlled fluid model and derive the optimal activation policy to replace servers under such correlated faults. The solution concept is the optimal control of regeneration via the Pontryagin minimum principle. We characterize feasibility conditions and we prove that the optimal policy is of threshold type. Numerical results describe how to apply the model for system dimensioning and show the tradeoff between activation of servers and communication cost.", 'arxivid': '1711.03034', 'author': ['Francesco De Pellegrini ', 'Rachid El ', 'Azouzi ⋆ ', 'Alonso Silva ', 'Olfa Hassani ', 'Francesco De Pellegrini ', 'Rachid El ', 'Azouzi ⋆ ', 'Alonso Silva ', 'Olfa Hassani '], 'authoraffiliation': [], 'corpusid': 5669149, 'doi': '10.1109/cloudcom.2017.65', 'github_urls': [], 'n_tokens_mistral': 14083, 'n_tokens_neox': 12654, 'n_words': 8342, 'pdfsha': '85963d3221eebefe803725b77cdb91149e9bc0ff', 'pdfurls': ['https://arxiv.org/pdf/1711.03034v1.pdf'], 'title': ['Optimal Control of Storage Regeneration with Repair Codes', 'Optimal Control of Storage Regeneration with Repair Codes', 'Optimal Control of Storage Regeneration with Repair Codes', 'Optimal Control of Storage Regeneration with Repair Codes'], 'venue': []}
arxiv
On Hölder solutions to the spiral winding problem Jonathan M Fraser [email protected] School of Mathematics and Statistics The University of St Andrews KY16 9SSSt AndrewsUnited Kingdom On Hölder solutions to the spiral winding problem 10.1088/1361-6544/abe75eReceived 10 March 2020, revised 26 January 2021 Accepted for publication 17 February 2021 Published 7 May 2021London Mathematical Society Nonlinearity Nonlinearity 34 (2021) 3251-3270spiralwinding problemHolder exponentsAssouad dimensionbox dimensionAssouad spectrum Mathematics Subject Classification numbers: 28A8026A1637C4537C1028A7834C05 The winding problem concerns understanding the regularity of functions which map a line segment onto a spiral. This problem has relevance in fluid dynamics and conformal welding theory, where spirals arise naturally. Here we interpret 'regularity' in terms of Hölder exponents and establish sharp results for spirals with polynomial winding rates, observing that the sharp Hölder exponent of the forward map and its inverse satisfy a formula reminiscent of Sobolev conjugates. We also investigate the dimension theory of these spirals, in particular, the Assouad dimension, Assouad spectrum and box dimensions. The aim here is to compare the bounds on the Hölder exponents in the winding problem coming directly from knowledge of dimension (and how dimension distorts under Hölder image) with the sharp results. We find that the Assouad spectrum provides the best information, but that even this is not sharp. We also find that the Assouad spectrum is the only 'dimension' which distinguishes between spirals with different polynomial winding rates in the superlinear regime. In a different direction, spirals arise as solutions to various geometric problems. For example, the Lituus is the locus of points z ∈ C preserving the area of the circular sector {w : arg(w) ∈ (0, arg(z)), |w| |z|} (including multiplicity when arg(z) > 2π). Consider also the hyperbolic spiral which is the 'inverse' of the Archimedean spiral. A more sophisticated setting where spirals have proved important is in the theory of conformal welding. This considers the regularity of the induced self-homeomorphism of an oriented Jordan curve arising by composing a Jordan mapping on the interior with the inverse of a Jordan mapping on the exterior. Jordan curves defined using logarithmic spirals (see below) were shown in [KNS] to exhibit an interesting intermediate phenomenon (non-differentiable, but Lipschitz) not previously observed. Wherever spirals arise, be it via a dynamical system or as the solution to a geometric problem, the form and regularity of the spiral holds relevance for the underlying model or problem. Of course there are many ways to quantify regularity, for example, via various notions of fractal dimension since infinitely wound spirals can be viewed as fractals. Here we consider the winding problem, which characterizes the regularity of a spiral by the regularity of homeomorphisms mapping a line segment to the spiral-such a function is a solution to the winding problem, since it performs the task of winding the line segment to the spiral. A common formulation of this problem is to ask whether or not bi-Lipschitz solutions exist, see [FP, KNS]. Here we search for bi-Hölder solutions in situations where bi-Lipschitz solutions do not exist. This approach appears to be novel and has the advantage of applying to a larger class of spirals. In particular, spirals arising in nature via a dynamical process tend to have polynomial winding rates, which do not admit bi-Lipschitz solutions. Moreover, the Hölder version of the problem is more flexible since, once Hölder solutions are known to exist, one can consider the more refined problem of optimising the Hölder exponents of the solution. Given a winding function φ : [1, ∞) → (0, ∞), which we assume is continuous, strictly decreasing, and satisfies φ(x) → 0 as x → ∞, the associated spiral is the set (see figure 1) S(φ) = {φ(x) exp(ix) : 1 < x < ∞} ⊂ C. The winding problem concerns the regularity of S(φ) by asking how little distortion is required to map (0, 1) onto S(φ). A well-known and important example of this is that when φ(x) = e −cx for some c > 0, it is possible to map (0, 1) onto the so-called logarithmic spiral S(φ) via a bi-Lipschitz map. This was first established by Katznelson et al [KNS]. Moreover, if φ is sub-exponential, that is, if log φ(x) x → 0 (x → ∞), then this cannot be done, thus illustrating that the logarithmic family is sharp for the bi-Lipschitz problem. See Fish and Paunescu [FP] for an elegant proof of this latter fact. In this paper, we wish to understand the sub-exponential regime by considering the Hölder analogue of this problem. Namely, given a sub-exponential winding function, is it possible to map (0, 1) onto S(φ) via a Hölder map and if so what is the optimal Hölder exponent? It turns out that it is natural to consider bi-Hölder maps (Hölder maps with Hölder inverses), and the interplay between the two Hölder exponents is crucial. Indeed, if one wants to achieve the sharp Hölder exponent for a homeomorphism mapping (0, 1) to S(φ), then one must sacrifice the Hölder exponent of its inverse and vice versa. This is perhaps surprising since spirals are 'more complex' than (0, 1) and therefore one might naively expect that only the Hölder exponent of the forward map is relevant, with the inverse map Lipschitz for most reasonable homeomorphisms. Recall that, given α ∈ (0, 1] and X ⊂ C, a function f : X → C is α-Hölder if there exists a constant C > 0 such that, for all x, y ∈ X, | f(x) − f(y)| C|x − y| α . In the special case when α = 1, the map is called Lipschitz. Moreover, given 0 < α 1 β < ∞ we say that a homeomorphism f : X → Y is (α, β)-Hölder if there exists a constant C > 0 such that, for all x, y ∈ X, C −1 |x − y| β | f(x) − f(y)| C|x − y| α , that is, f is α-Hölder and f −1 : Y → X is β −1 -Hölder. Similar to above, (1, 1)-Hölder maps are called bi-Lipschitz. In order to simply our exposition we consider a particular family of sub-exponential spirals, namely the polynomial family φ p defined by φ p (x) = x −p for p > 0. Our results apply more generally, but we delay discussion of this until section 6. In fact the spirals associated with φ p are bi-Lipschitz equivalent to any spiral whose winding function is 'comparable' to φ p , see section 6, and thus behave in exactly the same way in the context of the winding problem. We write S p for the spiral associated to φ p , that is, we write S p = S(φ p ) for brevity (see figure 1). The spirals S p are sometimes referred to as generalized hyperbolic spirals (the hyperbolic spiral corresponds to p = 1) and are typically found in nature whenever there is an underlying dynamical process. In contrast, when spirals form in nature in a static setting, they tend to be logarithmic, that is, have exponential winding functions. A well-studied and important problem in the dimension theory of fractals is to consider how Hölder maps affect a given notion of fractal dimension, see [F]. More precisely, given a notion of dimension, such as the Hausdorff or box dimension, one can often relate the dimensions of f(K) and K for all sets K and Hölder maps f in terms of the Hölder exponents of f. As such, knowledge of the dimensions of S p give rise to bounds on the possible Hölder exponents in the winding problem. In section 4 we consider this problem thoroughly by considering a number of available notions of dimensions. In particular, we consider the Hausdorff, box, and Assouad dimensions of the spirals S p for p > 0, as well as the Assouad spectrum, which interpolates between the box and Assouad dimensions. We prove that the Assouad spectrum 'separates this class', that is, the Assouad spectra depends on p for all p > 0, whereas, the Hausdorff, box, and Assouad dimensions fail to do this. We establish precisely how much information can be extracted from dimension theory in the context of the Hölder version of the winding problem, proving that the best information comes from the Assouad spectrum, but even this is not sharp. Motivated by the above, we propose a general programme of research. Given two bounded homeomorphic sets X, Y ⊂ R d , first consider the Hölder mapping problem which asks for sharp estimates on α and β such that there exists an (α, β)-Hölder map f with f(X) = Y. Secondly, consider the estimates on α, β which come directly from knowledge of the dimensions of X and Y. The problem is then to determine in which situations the information provided by the dimensions is sharp and when it is not, as well as determining which notion of dimension 'performs best' in a given setting. In particular, the polynomial spirals we consider here are examples where sharp information is not provided by dimension theory, but where the Assouad spectrum performs the best. Main results: Hölder solutions to the winding problem We write X Y to mean that X cY for some universal constant c > 0. We also write X Y to mean Y X and X ≈ Y to mean that both X Y and X Y hold. We write |E| for the diameter of a set E. For real numbers x, y we write x ∧ y = min{x, y} and x ∨ y = max{x, y}. A useful trick which we will use throughout is to decompose S p into the disjoint union of 'full turns' S p = k 1 S k p , where S k p = {x −p exp(ix) : 1 + 2π(k − 1) < x 1 + 2πk} (2.1) for integer k 1. Also, given a homeomorphism f : (0, 1) → S p , we decompose (0, 1) into the corresponding half-open intervals I k = f −1 (S k p ). (2.2) Our first result brings in the interplay between the two Hölder exponents when one considers bi-Hölder functions. It also provides a simple upper bound for the forward Hölder exponent, α, considered in isolation. The inverse Hölder exponent, β, is trivially bounded below by 1 and this cannot be improved without considering α. Theorem 2.1. If f : (0, 1) → S p is an α-Hölder homeomorphism, then α < p. Moreover, if f : (0, 1) → S p is an (α, β)-Hölder homeomorphism, then β pα p − α . Proof. In the first instance suppose only that f : (0, 1) → S p is α-Hölder. We have k −p ≈ |S k p | = | f(I k )| |I k | α (2.3) Figure 2. Plots of the sharp Hölder exponents α, β for the map g t as functions of t (solid lines). Plots of 1 and min(1, p) are shown as dashed lines for reference. On the left p = 0.7 and on the right p = 1.3. The optimal β is achieved for t 1/p. Whereas, if p 1, then the optimal α is only obtained asymptotically as t → ∞ and, if p > 1, then the optimal α is obtained for t 1/(p − 1). and therefore 1 = ∞ k=1 |I k | ∞ k=1 k −p/α which forces α < p. Now suppose f : (0, 1) → S p is an (α, β)- Hölder homeomorphism, which we may assume satisfies f(x) → 0 as x → 0 and for convenience we extend f continuously to [0,1]. This extension preserves the Hölder property. By the above, we know p/α > 1. For integer l 1, let x l = ∞ k=l |I k |, where I k is as in (2.2). Note that x l is the right endpoint of I l . Combining this with (2.3) yields 1 | f(x l ) − f(0)| |x l | β l −p ∞ k=l |I k | β l −p ∞ k=l k −p/α β l −p l (1−p/α)β → 0 as l → ∞, if −p − (1 − p/α)β < 0. This forces β pα p − α as required. Despite how simple the proof of theorem 2.1 is, it turns out to be sharp. Moreover, there is a particularly natural family of examples demonstrating this sharpness, which we introduce now (see figure 2). Given t > 0, define g t : (0, 1) → S p by g t (x) = x tp exp(i/x t ) noting that each g t is clearly a homeomorphism between (0, 1) and S p . Theorem 2.2. For all p, t > 0, the map g t is a tp t + 1 ∧ 1, tp ∨ 1 − Hölder. homeomorphism between (0, 1) and S p , and these Hölder exponents are sharp. We delay the proof of theorem 2.2 until section 3. It follows immediately from theorem 2.2 that theorem 2.1 is sharp. We provide an alternative direct proof of this in section 5. Note we only consider α p/(p + 1) since β can be chosen to equal 1 for α = p/(p + 1) and so there is no need to consider weaker conditions on α. Corollary 2.3. For p > 0 and α ∈ [ p p+1 , p) ∩ (0, 1] , there exists an (α, pα p−α )-Hölder homeomorphism between (0, 1) and S p . Proof. Fix p > 0 and α ∈ [ p p+1 , p) ∩ (0, 1], consider the map g t with t = α p − α , and apply theorem 2.2. We remark that the sharp relationship between α and β given in theorem 2.1 and corollary 2.3 resembles that of Sobolev conjugates. In particular, for 1 p < d, the Sobolev embedding theorem states that W 1,p (R d ) ⊂ L q (R d ) for q = dp d − p , that is, q is the Sobolev conjugate of p. Here W 1,p (R d ) is the Sobolev space consisting of real-valued functions f on R d such that both f and all weak derivatives of f are in L p (R d ). A natural family of examples: proof of theorem 2.2 We first show that g t is α-Hölder, for α = tp t + 1 ∧ 1. Let 0 < x < y < 1 and let y * ∈ (0, y) be the largest value which satisfies arg(g t (y * )) = arg (g t (y)) + π. In order to prove that g t is α-Hölder, it suffices to show |g t (x) − g t (y)| |x − y| α 1. If x < y * , then both |x − y| > |y * − y| and |g t (x) − g t (y)| < |g t (y * ) − g t (y)| and hence it suffices to bound sup y * x<y |g t (x) − g t (y)| |x − y| α = sup y * x<y x 2tp + y 2tp − 2(xy) tp cos(x −t − y −t ) (y − x) α (3.1) from above by a constant independent of y. By Taylor's theorem cos z 1 − z 2 /2 and applying this estimate to the function inside the square root in (3.1), we get x 2tp + y 2tp − 2(xy) tp cos(x −t − y −t ) (y tp − x tp ) 2 + (y t − x t ) 2 (xy) t(p−2) and therefore applying the inequality √ a + b √ a + √ b for a, b 0 in (3.1) this gives sup y * x<y |g t (x) − g t (y) | |x − y| α sup y * x<y (y tp − x tp ) + (y t − x t )(xy) t(p/2−1) (y − x) α . Considering only the first term, we have sup y * x<y y tp − x tp (y − x) α sup y * x<y (y − x) tp∧1−α 1. For the remaining term, fix x ∈ (y * , y) and let ω = ω(x, y) ∈ (0, ∞) be such that arg(g t (x)) = arg(g t (y)) + πy ω . Directly from the definition of ω we have y t − x t (xy) t = πy ω . Moreover, this yields y − x = y − y (1 + πy ω+t ) 1/t ≈ y(1 + πy t+ω ) 1/t − y ≈ y 1+ω+t by Taylor's Theorem. This gives x y/2 for sufficiently small y and so may use the estimate x ≈ y for all x, y (with the implicit constants independent of x and y). Therefore we have (y t − x t )(xy) t(p/2−1) (y − x) α ≈ y ω y tp y (1+t+ω)α 1 since α = tp t + 1 ∧ 1 tp + ω t + 1 + ω for all ω > 0. Specifically, if α < 1, then the right-hand side is increasing in ω and so minimized at ω = 0, and if α = 1, then the right-hand side is uniformly bounded below by 1. This proves that g t is α-Hölder. It remains to show α = tp/(t + 1) ∧ 1 is the sharp Hölder exponent, that is, g t is not α -Hölder for α ∈ (α, 1]. Here we may assume that α = tp/(t + 1) < 1, since otherwise there is nothing to prove. To this end, let y ∈ (0, 1) and choose x = y * , and note that |g t (x) − g t (y)| |x − y| α = x tp + y tp (y − x) α . (3.2) Observe that, as above, x ≈ y and y − x ≈ y 1+t , noting that ω = ω(y * , y) = 0. Therefore |g t (x) − g t (y)| |x − y| α y tp−(1+t)α → ∞ as y → 0 if α > tp t + 1 proving the result. Next we show that g −1 t is β −1 -Hölder, for β = tp ∨ 1. It suffices to show that |g t (x) − g t (y)| |x − y| β 1 with implicit constants independent of x and y. Fix 0 < x < y < 1 and, for m 1, let y m ∈ (0, y) be the mth largest number satisfying arg(g t (y m )) = arg(g t (y)). If x ∈ [y m+1 , y m ) for m 1, then |x − y| |y m+1 − y| and |g t (x) − g t (y)| |g t (y m ) − g t (y)| |g t (y m+1 ) − g t (y)| with implicit constant independent of m. In particular, |g t (x) − g t (y)| |x − y| β |g t (y m+1 ) − g t (y)| |y m+1 − y| β . Recall the winding intervals I k = g −1 t (S k p ) ⊂ (0, 1), now defined for g t , see (2.1) and (2.2). In particular, I k = [a k+1 , a k ) where arg(g t (a k )) = a −t k = 1 + 2π(k − 1) and therefore |I k | ≈ (k − 1) −1/t − k −1/t ≈ k −1/t−1 . If y ∈ I l for some l 1, then y m+1 ∈ I l+m+1 and so |y m+1 − y| β l+m+1 k=l |I k | β ≈ l+m+1 k=l k −1/t−1 β ≈ l −1/t − (l + m + 1) −1/t β l −p − (l + m + 1) −p and |g t (y m+1 ) − g t (y)| ≈ l+m+1 k=l |g t (y k+1 ) − g t (y k )| ≈ l+m+1 k=l k −p−1 ≈ l −p − (l + m + 1) −p . Therefore |g t (x) − g t (y)| |x − y| β 1. Finally, suppose x ∈ (y 1 , y). If x ∈ [y * , y), where, as above, y * ∈ (0, y) is the largest value which satisfies arg(g t (y * )) = arg(g t (y)) + π, then |g t (x) − g t (y)| |x − y| β . If x ∈ [y 1 , y * ), and y ∈ I l for some l 1, then |g t (x) − g t (y)| |x − y| β |g t (y 1 ) − g t (y)| |y 1 − y| β l −p−1 l −(1/t+1)β 1, since β t(p + 1)/(t + 1). This completes the proof that g −1 t is β −1 -Hölder. The fact that β −1 is the sharp Hölder exponent for g −1 t follows from theorem 2.1 since α and β are 'winding conjugates', that is, they satisfy β = pα p − α . The proof of theorem 2.2 is complete. Hölder estimates from dimension theory If g : X → Y is an onto α-Hölder map, then dim Y dim X α , (4.1) where dim is the Hausdorff, packing, upper or lower box dimension, see [F]. Moreover, if g is Lipschitz, then H 1 (Y) H 1 (X), where H 1 is the one-dimensional Hausdorff measure. These estimates, and the analogous formulations for g −1 , give rise to bounds on α and β in the Hölder winding problem. We consider these estimates in this section, ultimately proving that they are not sharp. We refer the reader to [F, R, FY] for more background on dimension theory, including definitions and basic properties of the various dimensions. We recall the definitions of the box dimension and the Assouad spectrum here, which are the definitions we use directly. We omit the definition of H 1 since the only properties we need are that it is a measure and that the H 1 measure of the boundary of a circle is comparable to its radius. Let F ⊆ R d be a non-empty bounded set. The lower and upper box dimensions of F are defined by dim B F = lim inf r→0 log N r (F) − log r and dim B F = lim sup r→0 log N r (F) − log r , respectively, where N r (F) is the smallest number of sets required for an r-cover of F. If dim B F = dim B F, then we call the common value the box dimension of F and denote it by dim B F. The Assouad spectrum of F is defined as the function θ → dim θ A F where dim θ A F = inf α : there exists C > 0 such that, for all 0 < r < 1 and x ∈ F, N r B(x, r θ ) ∩ F C r θ r α . and θ ∈ (0, 1). This notion was introduced in [FY] and is similar in spirit to the Assouad dimension. The key difference is that the Assouad dimension considers all pairs of scales r < R, whereas here the parameter θ serves to fix the relationship between the big scale R = r θ and the small scale r. The result is that the Assouad spectrum captures more precise information about the set, and has the benefit of being easier to work with and better behaved (see applications below). It also continuously interpolates between the upper box and (quasi-) Assouad dimension in a meaningful way. The quasi-Assouad dimension is another related notion which can be defined by dim qA F = lim θ→1 dim θ A F. Note that this is not the original definition of the quasi-Assouad dimension, see [LX], but this formula (and the fact the limit exists) was established in [FHHTY]. Also, the Assouad dimension, dim A , which we will not use directly, satisfies dim qA F dim A F d. Moreover, it was proved in [FY] that dim B F dim θ A F dim B F 1 − θ ∧ dim qA F (4.2) and that dim θ A F is continuous in θ. We turn our attention now to the dimensions of spirals and the resulting applications to the winding problem. First, we note that the Hausdorff and packing dimensions of S p are 1 for all p > 0 and so no information can be gleaned from these dimensions since the dimensions of (0, 1) are also 1. One can get some weak information by considering the length (one-dimensional Hausdorff measure) of S p via the following simple result. Proof. Clearly H 1 (S k p ) ≈ k −p and so H 1 (S p ) = k 1 H 1 (S k p ) ≈ k 1 k −p from which the result follows. Next we consider the box dimensions of S p . These are strictly greater than 1 for p ∈ (0, 1), which therefore improves on the information contained in the previous theorem concerning the winding problem. The following result can be found in [T, VH], but we include our own proof since it informs the strategy in the more complicated setting of the Assouad spectrum which follows. See also [ZZ] for a treatment of the box dimensions of spirals in R 3 . Theorem 4.2. For all p > 0 dim B S p = 2 1 + p ∨ 1. Proof. Let r ∈ (0, 1) and k(r) be the unique positive integer satisfying k(r) −(p+1) r < (k(r) − 1) −(p+1) , noting that k(r) ≈ r −1/(p+1) . The importance of this parameter is that, decomposing S p as the disjoint union of two sets k>k(r) S k p ∪ ⎛ ⎝ k k(r) S k p ⎞ ⎠ , we see that B(0, k(r) −p ) is contained in the δ-neighbourhood of the first set for some δ ≈ r since this portion of the spiral is wound 'tighter' than ≈ r. However, a given r-ball may only cover part of second set with length r, since the turns in the spiral are still 'r-separated' at this point. It follows that N r S p ≈ N r S p ∩ B(0, k(r) −p ) + k(r) k=1 N r S k p ≈ k(r) −p r 2 + k(r) k=1 k −p r ≈ r − 2 1+p + r −1 k(r) k=1 k −p . Therefore, if p > 1, we get N r S p ≈ r −1 , if p = 1, we get N r S p ≈ r −1 + log k(r) ≈ r −1 1 + | log r| and, if p < 1, we get N r S p ≈ r − 2 1+p + r −1 k(r) 1−p ≈ r − 2 1+p . The result follows. Applying (4.1) for box dimension, we get the following corollary in the context of the winding problem. It was proved in [FY, theorem 7.2] that for a large class of spirals S(φ) (including the spirals S p which we study) that, if dim B S(φ) > 1, then the Assouad spectrum of S(φ) is given by dim θ A S(φ) = dim B S(φ) 1 − θ ∧ 2. Note that this is then the general upper bound from (4.2). In particular, this result combined with theorem 4.2 yields the Assouad spectrum of S p for p < 1. However, for p 1, dim B S p = 1 and so the Assouad spectrum is not derivable from [FY]. We compute it here and, surprisingly, it is not given by the general upper bound from (4.2) for p > 1 (see figure 3). Theorem 4.4. For p ∈ (0, 1) and θ ∈ (0, 1), we have dim θ A S p = 2 (1 + p)(1 − θ) ∧ 2 and, for p 1 and θ ∈ (0, 1), we have dim θ A S p = 1 + θ p(1 − θ) ∧ 2. In both cases, the Assouad spectrum has a single phase transition at θ = p 1+p and, if p > 1, then the Assouad spectrum is strictly smaller than the upper bound from (4.2) for 0 < θ < p 1+p . Proof. The p ∈ (0, 1) case follows from theorem 4.2 and [FY, theorem 7.2], and therefore we assume p 1. It suffices to prove the result for 0 < θ < p 1+p , since for θ = p 1+p it follows by continuity of the Assouad spectrum that dim θ A S p = 2 and therefore by [FY, corollary 3.6] dim θ A S p = 2 for all θ > p 1+p as required. We prove the upper and lower bound separately, starting with the lower bound. Let r ∈ (0, 1) and l(r), L(r) be the unique positive integers satisfying L(r) −(p+1) r < (L(r) − 1) −(p+1) and l(r) −p r θ < (l(r) − 1) −p , respectively. Note that L(r) ≈ r − 1 p+1 and l(r) ≈ r − θ p and so L(r) > l(r) for all sufficiently small r since we assume θ < p p+1 . Arguing as in the proof of theorem 4.2, we have N r B(0, r θ ) ∩ S p L(r) k=l(r) N r S k p ≈ L(r) k=l(r) k −p r . Therefore, if p > 1 we get N r B(0, r θ ) ∩ S p r −1 l(r) 1−p − L(r) 1−p ≈ r −1− θ(1−p) p = r θ r 1+ θ p(1−θ) and if p = 1 we get N r B(0, r θ ) ∩ S p r −1 | log r| = r θ r 1 1−θ | log r| and in both cases we get the desired lower bound. To prove the upper bound, we may assume that p > 1 since the upper bound follows from (4.2) in the p = 1 case. Let r ∈ (0, 1) and l(r), L(r) be as before. We first deal with the case z = 0. Once again arguing as in the proof of theorem 4.2, we have N r B(0, r θ ) ∩ S p N r B(0, L(r) −p ) ∩ S p + L(r) k=l(r) N r S k p ≈ L(r) −p r 2 + L(r) k=l(r) k −p r r p p+1 r 2 + r −1 l(r) 1−p ≈ r θ r 2 (1−θ)(p+1) + r θ r 1+ θ p(1−θ) . Finally we consider the case z = 0. If |z| 2r θ , then the above estimates hold up to uniform constants and so we assume |z| > 2r θ . We can make crude estimates in this case. If k 1 is such that B(z, r θ ) ∩ S k p = ∅, then k r −θ/p and N r B(z, r θ ) ∩ S k p r θ /r. This gives N r B(z, r θ ) ∩ S p r −θ/p r θ /r = r θ r 1+ θ p(1−θ) . We have proved dim θ A S p 2 (1 − θ)(p + 1) ∨ 1 + θ p(1 − θ) = 1 + θ p(1 − θ) as required. The following corollary follows from theorem 4.4. It was proved in the range p ∈ (0, 1) in [FY]. Corollary 4.5. For all p > 0, dim A S p = dim qA S p = 2. The Assouad dimension does not behave well under Hölder image, see [LX], and so we cannot glean any information from knowledge of the Assouad dimension, despite it being large. However, the Assouad spectrum is more regular, and can be controlled in this context, however the control is more complicated than (4.1). Lemma 4.6 (Theorem 4.11 [FY]). Let X, Y ⊆ R d and 0 < α 1 β < ∞. If g : X → Y is an (α, β)-Hölder homeomorphism, then dim A Y dim A X(1 − θ 0 ) β − αθ 0 , where θ 0 = inf{θ ∈ (0, 1) : dim θ A X = dim A X}. For convenience we write inf ∅ = 1. Applying lemma 4.6 with g = f −1 we obtain the following bounds. Corollary 4.7. If f : (0, 1) → S p is an (α, β)-Hölder homeomorphism, then β pα 1 + p − 2α ∨ 1. Proof. Noting that g = f −1 is a (β −1 , α −1 )-Hölder homeomorphism between S p and (0, 1), and θ 0 = inf{θ ∈ (0, 1) : dim θ A S p = dim A S p } = p p + 1 we obtain 1 2 1 − p p+1 α −1 − β −1 p p+1 directly from lemma 4.6. Rearranging this formula for β yields the desired bound, recalling that β 1 is trivial. For comparison, and in order to give an alternative expression of the bounds in terms of α, we summarize the various estimates obtained so far in the following corollary (see figure 4). α p + 1 2 ∧ 1, which is also not sharp. Notice that as we relax the restrictions on the inverse map, that is, we let β → ∞, the bounds obtained from the Assouad spectrum approach those obtained from the box dimension, and the sharp bounds approach those from the first part of theorem 2.1 which considers the forward Hölder exponent in isolation. An alternative proof of corollary 2.3 In this section we provide an alternative proof of corollary 2.3 where, instead of considering a natural family of explicit examples, we directly construct a function with the desired properties. We decided to include both proofs for the interested reader. Moreover, we find the proof via theorem 2.2 more natural and appropriate in this setting, but the proof presented here is less reliant on the precise setting of the problem and may be more straightforward to generalize. Certain details in the proof will be similar to those from the proof of theorem 2.2 and will be suppressed. Fix p > 0 and α ∈ [ p p+1 , p) ∩ (0, 1]. We construct a homeomorphism f : (0, 1) → S p in four steps. Step 1 Partition the interval (0, 1) into countably many half open intervals J k = [a, b) (k 1), labelled from right to left, which satisfy |J k | ≈ k −p/α where the implicit constants are independent of k. This can be done since k 1 k −p/α ≈ 1. These intervals will play the role of I k for the function f, see (2.2). Step 2 For k 1, let g k 0 : J k → [0, |J k | α ) be defined by g k 0 (x) = |J k | α − (sup J k − x) α . In particular, g k 0 is (α, 1)-Hölder, with implicit constants independent of k. Step 3 For k 1, let g k 1 : [0, |J k | α ) → S k p be a smooth homeomorphism satisfying H 1 (g k 1 (J)) ≈ |J| for all open intervals J ⊂ [0, |J k | α ), where the implicit constants are independent of k and J. Such a map exists because H 1 (S k p ) ≈ k −p ≈ |J k | α . Recall that S k p is the kth full turn, see (2.1), and that H 1 is the one-dimensional Hausdorff measure. Step 4 Let f : (0, 1) → S p be defined by f | J k (x) = g k 1 • g k 0 (x). By construction, f is a homeomorphism between (0, 1) and S p . Note that the maps (g k 1 • g k 0 ) k are compatible at the endpoints since the intervals J k were chosen to be half open. It remains to establish the Hölder exponents, which we separate into two claims. Claim 1. f is α-Hölder. Proof of Claim 1. Let 0 < x < y < 1 and, as before, y * ∈ (0, y) be the largest value satisfying arg( f(y * )) = arg( f(y)) + π. As in the proof of theorem 2.2, it suffices to prove sup y * <x<y | f(x) − f(y)| |x − y| α 1 where the implicit constant is independent of y. However, this follows immediately since (y * , y) can intersect at most 2 of the intervals J k . This relies on the fact that the maps g k 1 are Lipschitz and the maps g k 0 are α-Hölder, both with implicit constants independent of k. Claim 2. f −1 is p−α pα -Hölder. Proof of Claim 2. Let β = pα p−α and 0 < x < y < 1. Similar to above, for m 1, let y m ∈ (0, y) be the mth largest number satisfying arg( f(y m )) = arg( f(y)). If x ∈ [y m+1 , y m ) for m 1, then |x − y| |y m+1 − y| and | f(x) − f(y)| | f(y m ) − f(y)| | f(y m+1 ) − f(y)| with implicit constant independent of m. In particular, | f(x) − f(y)| |x − y| β | f(y m+1 ) − f(y)| |y m+1 − y| β . If y ∈ J l for some l 1, then y m+1 ∈ J l+m+1 and so |y m+1 − y| β l+m+1 k=l |J k | β ≈ l+m+1 k=l k −p/α β ≈ l 1−p/α − (l + m + 1) 1−p/α β l β(1−p/α) − (l + m + 1) β(1−p/α) = l −p − (l + m + 1) −p and | f(y m+1 ) − f(y)| ≈ l+m+1 k=l k −p−1 ≈ l −p − (l + m + 1) −p . Therefore | f(x) − f(y)| |x − y| β 1 as required. Finally, suppose x ∈ (y 1 , y). If x ∈ [y * , y), where y * is as above, then | f(x) − f(y)| |x − y| since (y * , y) can intersect at most 2 of the intervals J k . This relies on the fact that the maps g k 1 are bi-Lipschitz on the interval (y * , y) since it only corresponds to a half turn in S p , and the maps g k 0 are (α, 1)-Hölder. If x ∈ [y 1 , y * ), and y ∈ J l for some l 1, then | f(x) − f(y)| |x − y| β | f(y 1 ) − f(y)| |y 1 − y| β l −p−1 l −β p/α 1 completing the proof. Note that the final bound relies on the assumption that α p/(p + 1). Reduction to bi-Lipschitz classes In this section we prove a simple equivalence which extends our results to a much broader class of functions, as well as providing an example showing that our results do not generally hold in a slightly broader class still. Theorem 6.1. Let φ be a winding function such that the function ε : [1, ∞) → (0, ∞) defined by ε(x) = φ(x)x p is Lipschitz and uniformly bounded away from 0 and ∞. Then S p and S(φ) are bi-Lipschitz equivalent, that is, there is a bi-Lipschitz homeomorphism between S p and S(φ). Proof. Since ε is Lipschitz, there exists a constant L > 0 such that |ε(x) − ε(x)| L|x − y| for all x, y 1. Let F : S p → S(φ) be defined by F x −p exp(ix) = φ(x) exp(ix) and consider points x > y 1. If x − y(mod 2π) ∈ (π/2, 3π/2), then we immediately get |φ(x) exp(ix) − φ(y) exp(iy)| |x −p exp(ix) − y −p exp(iy)| ≈ φ(y) y −p = ε(y) ≈ 1. If x − y(mod 2π) / ∈ (π/2, 3π/2), then a slightly more complicated argument is needed. Applying the bounds 1 − z 2 /2 cos z 1 − z 2 /3 (|z| π/2) we get |φ(x) exp(ix) − φ(y) exp(iy)| |x −p exp(ix) − y −p exp(iy)| = φ(x) 2 + φ(y) 2 − 2φ(x)φ(y) cos(x − y) x −2p + y −2p − 2x −p y −p cos(x − y) (φ(y) − φ(x)) 2 + φ(x)φ(y)(x − y) 2 (y −p − x −p ) 2 + 2 3 x −p y −p (x − y) 2 (y −p ε(y) − x −p ε(x)) 2 + x −p y −p (x − y) 2 (y −p − x −p ) 2 + x −p y −p (x − y) 2 . (6.1) Using the fact that ε(x) ε(y) − L(x − y) we get (y −p ε(y) − x −p ε(x)) 2 (y −p ε(y) − x −p (ε(y) − L(x − y))) 2 = ε(y) 2 (y −p − x −p ) 2 + 2Lx −p ε(y)(y −p − x −p )(x − y) + L 2 x −2p (x − y) 2 (y −p − x −p ) 2 + x −p (y −p − x −p )(x − y) + x −p y −p (x − y) 2 (y −p − x −p ) 2 + x −p y −p (x − y) 2 . Here the final line follows since the middle of the three terms in the previous line is bounded above by the maximum of the other two. This proves that (6.1) is 1, proving that F is Lipschitz. The proof that (6.1) is also 1 is similar and omitted and establishes that F is bi-Lipschitz as required. An immediate consequence of theorem 6.1 is that we can replace S p with S(φ) in all of our main results in this paper where φ is any winding function such that φ(x)x p is Lipschitz and uniformly bounded away from 0 and ∞. In particular, in theorem 2.1, and corollary 2.3, as well as the dimension results in theorems 4.2 and 4.4. For example, φ(x) can be the reciprocal of any polynomial of degree p which is strictly increasing on [1, ∞). More complicated functions also work, including many non-differentiable functions or non-polynomial functions, such as φ(x) = 3 5x p x −1/x + x p/2 log(x) which is comparable to x −p in the above sense. It would be interesting to push theorem 6.1 further, with the most natural class to consider being φ such that φ(x)x p is not uniformly bounded away from 0 and ∞, but can be controlled by a lower order function, such as log(x). For example, are S p and S(φ) bi-Lipschitz equivalent for φ(x) = x −p log x? In fact this turns out to be false, which we show by adapting the arguments from theorem 2.1. In the following result, compare the strict lower bound for β with the corresponding bound from theorem 2.1. Theorem 6.2. Let p, γ > 0 and φ(x) = x −p (log x) γ . If f : (0, 1) → S(φ) is an (α, β)-Hölder homeomorphism, then α < p and β > pα p − α . Figure 1 . 1Three spirals: on the left, the logarithmic spiral with φ(x) = exp(−x/5), in the centre a hyperbolic spiral with φ(x) = x −1 , and on the right a Lituus with φ(x) = x −1/2 . which rotates with constant speed. In fluid dynamics spiral trajectories arise in various models of fluid turbulence and vortex formation. For example, in the well-studied α-models for fluid turbulence polynomial spirals appear as the evolution of the half-line [0, ∞) ⊂ R 2 under the resulting two-dimensional flow and the polynomial winding rate depends on the parameter α, see [FHT]. See [M, Mo, V, VH] for more specific examples of spirals appearing in fluid turbulence and [T, ZZ] for other dynamical examples giving rise to spiral trajectories where particular attention is paid to dimension. Theorem 4. 1 . 1If p > 1 then 0 < H 1 (S p ) < ∞, and if p 1 then H 1 (S p ) = ∞. Therefore, if p 1, then there cannot exist an onto Lipschitz map f : (0, 1) → S p . Corollary 4. 3 . 3If f : (0, 1) → S p is an onto α-Hölder map, then α p + 1 2 ∧ 1. Figure 3 . 3Plots of dim θ A S p for three different values of p. Moving from left to right, p = 1/2, p = 1, and p = 2. The general upper and lower bounds from (4.2) are shown as dashed lines. Figure 4 . 4Upper bounds on α, given the existence of an (α, β)-Hölder homeomorphism between (0, 1) and S p . On the left, β = 1, in the centre β = 2, and on the right β = 10. The sharp bounds are a solid line, the bounds obtained from the Assouad spectrum are a dashed line, and the bounds obtained from the box dimension are a dotted line. Corollary AcknowledgmentsThe author was supported by an EPSRC Standard Grant No. (EP/R015104/1) and a Leverhulme Trust Research Project Grant No. (RPG-2019-034). He thanks Han Yu for several stimulating conversations on spiral winding and the Assouad spectrum, and David Dritschel for a helpful explanation of spiral formation in the context of α-models for fluid turbulence. Finally, the author thanks an anonymous referee for carefully reading the paper and making several suggestions.ORCID iDsJonathan M Fraser https://orcid.org/0000-0002-8066-9120Proof. Analogous to (2.1) and (2.2), let S k = {φ(x) exp(ix) : 1 + 2π(k − 1) < x 1 + 2πk} for integer k 1 and I k = f −1 (S k ). We haveand thereforeand, for integer l 1, letExtending f continuously to [0, 1] and applying (6.2) yieldsa contradiction. To see the final convergence, note that the polynomial part will tend to 0, dominating the logarithmic part, unless β = pα/(p − α) in which case the polynomial part disappears. However, in this case β/α > 1 and the logarithmic part tends to 0. Proof. This follows immediately from theorem 6.2 since it is not possible to choose α = β = 1. . Y Dupain, Mendès France, M Tricot, C , 10.24033/bsmf.1985Dimensions des spirales Bul. Soc. Math. France. 79Dupain Y, Mendès France M and Tricot C 1983 Dimensions des spirales Bul. Soc. Math. France 79 193-201 K Falconer, Fractal Geometry: Mathematical Foundations and Applications. New YorkWiley3rd ednFalconer K J 2014 Fractal Geometry: Mathematical Foundations and Applications 3rd edn (New York: Wiley) The Navier-Stokes-alpha model of fluid turbulence Physica D 152-153. C Foias, D Holm, E S Titi, 10.1016/s0167-2789(01)00191-9Foias C, Holm D D and Titi E S 2001 The Navier-Stokes-alpha model of fluid turbulence Physica D 152-153 505-19 Unwinding spirals I. A Fish, Paunescu, 10.4310/maa.2018.v25.n3.a3Methods Appl. Anal. 25Fish A and Paunescu L 2018 Unwinding spirals I Methods Appl. Anal. 25 225-32 The Assouad spectrum and the quasi-Assouad dimension: a tale of two spectra Ann. J M Fraser, K E Hare, K G Hare, Troscheit S Yu, H , 10.5186/aasfm.2019.4419Acad. Sci. Fenn. Math. 44Fraser J M, Hare K E, Hare K G, Troscheit S and Yu H 2019 The Assouad spectrum and the quasi-Assouad dimension: a tale of two spectra Ann. Acad. Sci. Fenn. Math. 44 379-87 New dimension spectra: finer information on scaling and homogeneity. J Fraser, Yu , 10.1016/j.aim.2017.12.019Adv. Math. 329Fraser J M and Yu H 2018 New dimension spectra: finer information on scaling and homogeneity Adv. Math. 329 273-328 On conformal welding homeomorphisms associated to. Y Katznelson, S Nag, D P Sullivan, 10.5186/aasfm.1990.1517Jordan curves Ann. Acad. Sci. Fenn. Math. 15Katznelson Y, Nag S and Sullivan D P 1990 On conformal welding homeomorphisms associated to Jordan curves Ann. Acad. Sci. Fenn. Math. 15 293-306 . F Lü, L-F Xi, 10.4171/jfg/34Quasi-Assouad dimension of fractals J. Fractal Geometry. 3Lü F and Xi L-F 2016 Quasi-Assouad dimension of fractals J. Fractal Geometry 3 187-215 The Fractal Geometry of Nature. B Mandelbrot, FreemanSan Francisco, CAMandelbrot B B 1982 The Fractal Geometry of Nature (San Francisco, CA: Freeman) H Moffatt, Spiral structures in turbulent flow Wavelets, Fractals, and Fourier Transforms (The Institute of Mathematics and its Applications Conference Series. New YorkOxford University Press43Moffatt H K 1993 Spiral structures in turbulent flow Wavelets, Fractals, and Fourier Trans- forms (The Institute of Mathematics and its Applications Conference Series vol 43) (New York: Oxford University Press) pp 317-24 J Robinson, Embeddings, and Attractors. CambridgeCambridge University PressRobinson J C 2011 Dimensions, Embeddings, and Attractors (Cambridge: Cambridge University Press) Translated from the 1993 French original Vassilicos J C 1993 Fractals in turbulence Wavelets. C Tricot, Fractals, and Fourier Transforms (The Institute of Mathematics and its Applications Conference Series. New York; New YorkOxford University Press43Curves and Fractal DimensionTricot C 1995 Curves and Fractal Dimension (New York: Springer) Translated from the 1993 French original Vassilicos J C 1993 Fractals in turbulence Wavelets, Fractals, and Fourier Transforms (The Institute of Mathematics and its Applications Conference Series vol 43) (New York: Oxford University Press) pp 325-40 Fractal dimensions and spectra of interfaces with application to turbulence. J Vassilicos, J C R Hunt, 10.1098/rspa.1991.0158Proc. R. Soc. A. 435Vassilicos J C and Hunt J C R 1991 Fractal dimensions and spectra of interfaces with application to turbulence Proc. R. Soc. A 435 505-34 Box dimension of spiral trajectories of some vector fields in R 3 Qual. D Zubrinić, 10.1007/bf02972676Theory Dyn. Syst. 6Zubrinić D andŽupanović V 2005 Box dimension of spiral trajectories of some vector fields in R 3 Qual. Theory Dyn. Syst. 6 251-72
{'fraction_non_alphanumeric': 0.08774461406040353, 'fraction_numerical': 0.030480241006556795, 'mean_word_length': 3.328986301369863, 'pattern_counts': {'":': 0, '<': 52, '<?xml version=': 0, '>': 38, 'https://': 1, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 14, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': "The winding problem concerns understanding the regularity of functions which map a line segment onto a spiral. This problem has relevance in fluid dynamics and conformal welding theory, where spirals arise naturally. Here we interpret 'regularity' in terms of Hölder exponents and establish sharp results for spirals with polynomial winding rates, observing that the sharp Hölder exponent of the forward map and its inverse satisfy a formula reminiscent of Sobolev conjugates. We also investigate the dimension theory of these spirals, in particular, the Assouad dimension, Assouad spectrum and box dimensions. The aim here is to compare the bounds on the Hölder exponents in the winding problem coming directly from knowledge of dimension (and how dimension distorts under Hölder image) with the sharp results. We find that the Assouad spectrum provides the best information, but that even this is not sharp. We also find that the Assouad spectrum is the only 'dimension' which distinguishes between spirals with different polynomial winding rates in the superlinear regime.", 'arxivid': '1905.07563', 'author': ['Jonathan M Fraser [email protected] \nSchool of Mathematics and Statistics\nThe University of St Andrews\nKY16 9SSSt AndrewsUnited Kingdom\n'], 'authoraffiliation': ['School of Mathematics and Statistics\nThe University of St Andrews\nKY16 9SSSt AndrewsUnited Kingdom'], 'corpusid': 159042092, 'doi': '10.1088/1361-6544/abe75e', 'github_urls': [], 'n_tokens_mistral': 14339, 'n_tokens_neox': 12655, 'n_words': 7848, 'pdfsha': 'bf39ae3c9521ed65c2bd4828b75320b7932c13bb', 'pdfurls': None, 'title': ['On Hölder solutions to the spiral winding problem', 'On Hölder solutions to the spiral winding problem'], 'venue': []}
arxiv
Protecting and enhancing spin squeezing under decoherence using weak measurement 26 Jul 2016 Xiang-Ping Liao College of Science Hunan University of Technology 412008ZhuzhouHunanChina Man-Sheng Rong College of Science Hunan University of Technology 412008ZhuzhouHunanChina Mao-Fa Fang College of Physics and Information Science Hunan Normal University 410081ChangshaHunanChina Protecting and enhancing spin squeezing under decoherence using weak measurement 26 Jul 2016arXiv:1607.06530v2 [quant-ph] 1 Corresponding author. 1spin squeezingdecoherencesudden deathweak measurement PACS number(s): 0365Ud, 0367Mn, 0365Yz We propose an efficient method to protect spin squeezing under the action of amplitude-damping, depolarizing and phase-damping channels based on measurement reversal from weak measurement, and consider an ensemble of N independent spin-1/2 particles with exchange symmetry. We find that spin squeezing can be enhanced greatly under three different decoherence channels and spin-squeezing sudden death (SSSD) can be avoided undergoing amplitude damping and phase-damping channels. Introduction Spin squeezing has attracted a lot of attention in both the theoretical and experimental fields for decades [1,2,3,4,5,6,7,8] . An important application of spin squeezing is to detect quantum entanglement [9,10,11] . Due to the fact that spin squeezing is relatively easy to be generated and measured [2,12,13,14] , spin-squeezing parameters are multipartite entanglement witness in a general sense. Lots of efforts have been devoted to find relations between spin squeezing and entanglement [4,5,6,7,15,16,17] . Another application of spin squeezing is to improve the precision of measurements such as leading-noise reduction [18] and improving atomic sensor precision [19] . Thus, spinsqueezed states are useful resources for quantum information processing. However, the interactions between the system and its environment usually cause decoherence. In practice, decoherence is inevitable and harmful to spin squeezing and entanglement [20,21,22,23,24,25,26] . We find that, analogous to the definition of entanglement sudden death (ESD) [27] and distillability sudden death(DSD) [28] , spin squeezing can also suddenly vanish with different lifetimes for some decoherence channels, showing in general different vanishing times in multipartite correlations in quantum many-body systems. Wang et al. [25] have found that, under local decoherence, spin squeezing also appears as sudden death similar to the discovery of pairwise entanglement sudden death. An method to protecting and enhancing spin squeezing via continuous dynamical decoupling is proposed by Adam Zaman Chaudhry et. al [29] . In 1988, weak measurement was introduced by Aharonov, Albert, and Vaidman (AAV) [30] . Weak measurement is very useful and can help understand many counterintuitive quantum phenomena, for example, Hardy's paradoxes [31] . Recently, the weak measurement has been applied as a practically implementable method for protecting entanglement, quantum fidelity of quantum states undergoing decoherence [32,33,34,35,36,37] and improving payoffs in the quantum games in the presence of decoherence [38] . However, the study on protecting spin squeezing under the action of decoherence and avoiding spin-squeezing sudden death via using weak measurements is not involved so far. Motivated by recent studies of decoherence effects on spin squeezing and the application of weak measurement, we propose an efficient method to avoid spin-squeezing sudden death via measurement reversal from weak measurement, and consider an ensemble of N independent spin-1/2 particles with exchange symmetry. The definitions of spin squeezing and concurrence We consider an ensemble of N spin-1/2 particles with ground state |1 and excited state |0 . This system has exchange symmetry, and its dynamical properties can be described by the collective operators J α = 1 2 N k=1 σ kα(1) for α = x, y, z. Here, σ kα are the Pauli matrices for the kth qubit. We choose the initial state as a standard one-axis twisted state [1] |Ψ(0) = e −iθJ 2 x /2 | ↓ ... ↓(2) This state is prepared by the one-axis twisted Hamiltonian H = χJ 2 x , with the coupling constant χ , and θ = 2χt the twist angle. There are several spin-squeezing parameters, but we list only three typical and related ones as follows [1,2,3,4,5] : ξ 2 1 = 4(△J n ⊥ ) 2 min N (3) ξ 2 2 = N 2 4 J 2 ξ 2 1 (4) ξ 2 3 = λ min J 2 − N 2(5) Here, the minimization in the first equation is over all directions denoted by n ⊥ , perpendicular to the mean spin direction J / J 2 ; λ min is the minimum eigenvalue of the matrix Γ = (N − 1)γ + C where γ kl = C kl − J k J l f or k, l ∈ {x, y, z} = {1, 2, 3}(7) is the covariance matrix and C = [C kl ] with C kl = 1 2 J l J k + J k J l(8) is the global correlation matrix. The parameters ξ 2 1 , ξ 2 2 and ξ 2 3 were defined by Kitagawa and Ueda [1] , Wineland et al. [2] , and Tóth et al. [4] , respectively. If ξ 2 2 < 1 (ξ 2 3 < 1) spin squeezing occurs, and we can safely say that the multipartite state is entangled. For states with a well-defined parity (even or odd), we now express the squeezing parameters in terms of local expectations and correlations [7,25] ξ 2 1 = 1 + 2(N − 1)( σ 1+ σ 2− − | σ 1− σ 2− |)(9)ξ 2 2 = ξ 2 1 σ 1z 2(10)ξ 2 3 = min{ξ 2 1 , ς 2 } (1 − N −1 ) σ 1 · σ 2 + N −1(11) where ς 2 = 1 + (N − 1)( σ 1z σ 2z − σ 1z σ 2z )(12) For convenience, hereafter we use ζ 2 k = max(0, 1 − ξ 2 k ), k ∈ {1, 2, 3}(13) to characterize spin squeezing. With the above definition, spin squeezing occurs when ζ 2 k > 0. The concurrence is defined as [39] C = max(0, λ 1 − λ 2 − λ 3 − λ 4 )(14) where λ i are the square roots of eigenvalues ofρρ. Here ρ is the reduced density matrix of the system, andρ = (σ y ⊗ σ y )ρ * (σ y ⊗ σ y )(15) whereρ is the conjugate of ρ. The two-spin reduced density matrix for a parity state with the exchange symmetry can be written in a block-diagonal form [7] ρ 12 =     υ + u * u υ −     ⊕     w y y w    (16) where υ ± = 1 4 (1 ± 2 σ 1z + σ 1z σ 2z ) (17) w = 1 4 (1 − σ 1z σ 2z ) u = σ 1+ σ 2+ y = σ 1+ σ 2− the concurrence is given by C = max{0, 2(|u| − w), 2(y − √ υ + υ − )}(18) 3 Protecting spin squeezing under decoherence by using weak measurements We propose a scheme to protect spin squeezing under the action of decoherence channels by using weak measurement. The scheme is weak measurement M + decoherence channel + weak measurement N . The effect of quantum channels on the state of a system is a completely positive and trace-preserving map that is described in terms of Kraus operators. ρ in = |ψ ψ| → ε channel (ρ in ) = l E l |ψ ψ|E † l(19) The operator E l satisfies the CPTP relation l E † l E l = I. In order to protect and improve the spin squeezing, we should perform weak measurement M and measurement reversal N, before and after the decoherence channels, respectively. The two weak measurements can be written, respectively, as a non-unitary quantum operation [40] M =     1 0 0 m     N =     n 0 0 1    (20) where m and n are the measurement strengths. After these weak measurements being implemented, the state becomes Θ(ρ in ) = Nε channel (Mρ in M † )N † T r(Nε channel (Mρ in M † )N † )(21) where ε channel is defined by Eq. (19). By discussing the symmetry of the open system under consideration and the local decoherence and weak measurement are independent and identical. Thus, the exchange symmetry is not affected by the decoherence and weak measurement. We know that the spin squeezing can be expressed by the local expectations and correlations. The spin squeezing can then calculated by the dynamics of the local expectations and correlations. It is easy to check that an expectation value of the operator A can be calculated as A = T r[AΘ(ρ in )] = T r[Θ + (A)ρ in ](22) Thus, we can calculate the expectation value via the above equation, which is very similar to the standard Heisenberg picture. Amplitude-damping channel A single qubit Kraus operators for amplitude-damping channel(ADC) is E 0 = √ s|0 0| + |1 1|, E 1 = √ p|1 0|(23) where p = 1 − s, s = exp(−γt/2) and γ is the damping rate. Based on the above approach and the Kraus operators for the ADC given by Eq. (23), when sn 2 + p = m 2 , we find the evolutions of the following expectations under decoherence using weak measurement (see Appendix for details): σ 1z = [sn 2 σ 1z 0 − p]/M 1 (24) σ 1− σ 2− = sm 2 n 2 σ 1− σ 2− 0 /M 2 1 σ 1+ σ 2− = sm 2 n 2 σ 1+ σ 2− 0 /M 2 1 σ 1z σ 2z = [s 2 n 4 σ 1z σ 2z 0 − 2sn 2 p σ 1z 0 + p 2 ]/M 2 1 Q 1 = σ 1 . σ 2 = [sm 2 n 2 + sn 2 (sn 2 − m 2 ) σ 1z σ 2z 0 − 2sn 2 p σ 1z 0 + p 2 ]/M 2 1 where σ 1z 0 = −cos N −1 (θ/2), σ 1z σ 2z 0 = 2 −1 (1 + cos N −2 (θ)), M 1 = sn 2 + p = m 2 . Substituting the relevant expectation values and the correlation function into Eqs. (9), (10), and (11) leads to the explicit expression of the spin-squeezing parameters ξ 2 1 = 1 − sm 2 n 2 C r (0)/M 2 1 ;(25)ξ 2 2 = ξ 2 1 (sn 2 σ 1z 0 − p)/M 1 ) 2(26)ξ 2 3 = ξ 2 1 (1 − N −1 )Q 1 + N −1 (27) where C r (0) = (N − 1)C 0 , C 0 = 1 4 {[(1 − cos N −2 θ) 2 + 16sin 2 (θ/2)cos 2N −4 (θ/2)] 1 2 − 1 + cos N −2 θ}. The expression of concurrence can be simplified to [25] C r = 2(N − 1)max{0, |u|/M 2 1 − w} (28) where u = − 1 2 sm 2 n 2 Q 12y − sm 2 n 2 u 0 , w = 1 4 (1 − σ 1z σ 2z ), with Q 12y = 1 2 (1 − cos N −2 θ), u 0 = − 1 8 (1 − cos N −2 θ) − 1 2 i sin( θ 2 ) cos N −2 ( θ 2 ). In Fig. 1, we plot the spin-squeezing parameters and concurrence against the decoherence strength p under amplitude damping channel for different weak measurement strength m with θ = 0.1π, N = 12. It clearly shows that as the decoherence strength p increases, the spin squeezing decreases without weak measurement. For the smaller value of θ, there is no ESD and SSSD. They vanish only in the asymptotic limit (see Fig. 1(a)). However, we are able to enhance spin-squeezing parameters and the concurrence greatly by using weak measurement. Especially, they don't disappear in the asymptotic limit ( i.e. p = 1). Moreover, with the increase of m, spin-squeezing parameters and the concurrence becomes a fixed value respectively. The spin-squeezing parameters and the concurrence can be completely recovered to its initial value respectively regardless of the decoherence when weak measurement strength is large (see Fig. 1(d)). It seems that decoherence has no effect on the spin-squeezing parameters and the concurrence. This result can be explained as follows. According to sn 2 + p = m 2 , we have n 2 ≫ 1 when the weak measurement strength m 2 ≫ 1. And, we obtain sn 2 = m 2 . From Eq.(24), we can obtain the expectations as follows σ 1z σ 2z = σ 1z σ 2z 0(29)Q 1 = σ 1 . σ 2 = 1 Thus, the spin-squeezing parameters and concurrence can be calculated as ξ 2 1 = 1 − C r (0)(30)ξ 2 2 = ξ 2 1 σ 1z 2 0 ξ 2 3 = ξ 2 1 C r = ζ 2 3 = C r (0) So, the spin-squeezing parameters and the concurrence can be completely recovered to its initial value when weak measurement strength is very large. The overlap of the solid line and the dashed line in Fig. 1(d) due to the fact that the spin squeezing ζ 2 3 and the concurrence C r (0) are equivalent for the initial state Eq. (2). We plot the spin-squeezing parameters and concurrence against the decoherence strength p under amplitude damping channels for different weak measurement strength m with θ = 1.8π, N = 12 in Fig. 2. For larger values of θ, as the decoherence strength p increases, the spin squeezing decreases until it suddenly vanishes, so the phenomenon of ESD and SSSD occurs when there is no weak measurement (see Fig. 2(a)). However, the spin-squeezing parameters and concurrence can be improved greatly by using weak measurement. Moreover, with the increase of m, the phenomenon of ESD and SSSD can be avoided. When the measurement strength m is very large, the spinsqueezing parameters and the concurrence can be completely recovered to its initial value respectively no matter what the decoherence parameter is (see Fig. 2(d)). Depolarizing channel A single qubit Kraus operators for depolarizing channel(DPC) is E 0 = 1 − p ′ I, E 1 = p ′ 3 σ x(31)E 2 = p ′ 3 σ y , E 3 = p ′ 3 σ z where p ′ = 3p/4 and I is the identity operator. ξ 2 1 = 1 − s 2 n 2 C r (0)/M 2 2 ;(33)ξ 2 2 = ξ 2 1 { 1 2 [(n 2 s + s) σ 1z 0 + (n 2 − 1)]/M 2 } 2(34)ξ 2 3 = ξ 2 1 (1 − N −1 )Q 2 + N −1(35) The expression of concurrence can be simplified to [25] C r = 2(N − 1)max{0, |u|/M 2 2 − w}(36) where, u = − 1 2 s 2 n 2 Q 12y − s 2 n 2 u 0 . In Fig.3, we plot the spin-squeezing parameters and concurrence against the decoherence strength p under depolarizing channel with θ = 1.8π, N = 12. We can see that similar to amplitude damping channel, the spin squeezing decreases as the decoherence strength p increases without weak measurement. And, the phenomenon of ESD and SSSD occurs (see Fig.3(a)). However, we are able to improve the spin-squeezing parameters ζ 2 3 and the concurrence greatly by using weak measurement. The larger is the weak measurement strength n, the later is the vanishing time. And when weak measurement strength is very large, the spin-squeezing parameter ζ 2 3 and the concurrence vanish approximately in the asymptotic limit (see Fig.3(d)). We note that with the increase of weak measurement strength n, the spin-squeezing parameter ζ 2 2 becomes more and more weak until it is zero. This means that in our model, the parameter ξ 2 3 < 1 implies the existence of pairwise entanglement, while ξ 2 2 < 1 does not. Phase-damping channel A single qubit Kraus operators for phase-damping channel (PDC) is E 0 = √ sI, E 1 = √ p|0 0|, E 2 = √ p|1 1|(37) From Eq. (22) and the Kraus operators for the PDC given by Eq. (37), when n 2 − 1 = m 2 + 1, we find the evolutions of the following expectations under decoherence using weak measurement (see Appendix for details): σ 1z = [(m 2 + 1) σ 1z 0 + 1]/M 3 (38) σ 1− σ 2− = s 2 m 2 n 2 σ 1− σ 2− 0 /M 2 3 σ 1+ σ 2− = s 2 m 2 n 2 σ 1+ σ 2− 0 /M 2 3 σ 1z σ 2z = [(m 2 + 1) 2 σ 1z σ 2z 0 + 2(m 2 + 1) σ 1z 0 + 1]/M 2 3 Q 3 = σ 1 . σ 2 = [s 2 m 2 n 2 (1 − σ 1z σ 2z 0 ) + (m 2 + 1) 2 σ 1z σ 2z 0 + 2(m 2 + 1) σ 1z 0 + 1]/M 2 3 where M 3 = m 2 + 1 + σ 1zξ 2 2 = ξ 2 1 ((m 2 + 1) σ 1z 0 + 1)/M 3 ) 2 (40) ξ 2 3 = ξ 2 1 (1 − N −1 )Q 3 + N −1(41) The expression of concurrence can be simplified to [25] C r = 2(N − 1)max{0, |u|/M 2 3 − w}(42) where, u = − 1 2 s 2 m 2 n 2 Q 12y − s 2 m 2 n 2 u 0 . In Fig.4, we plot the spin-squeezing parameters and concurrence against the decoherence strength p under phase-damping channel with θ = 1.8π, N = 12. We can see that similar to amplitude-damping and depolarizing channel, the spin squeezing decreases as the decoherence strength p increases without weak measurement. And the phenomenon of ESD and SSSD occurs (see Fig.4(a)). However, we are able to enhance the spin-squeezing parameters ζ 2 3 and the concurrence greatly, and to avoid the phenomenon of ESD and SSSD by using weak measurement. Morover, when weak measurement strength m is small, the spin-squeezing parameter ζ 2 3 and the concurrence becomes a fixed value respectively regardless of the decoherence although the spin-squeezing parameter ζ 2 2 becomes zero(see Fig.4(d)). This result can be explained as follows. When the weak measurement strength m 2 ≪ 1, according to n 2 −1 = m 2 +1, we have n 2 = 2. So, we obtain M 3 = 1 + σ 1z 0 and s 2 m 2 n 2 ≪ M 2 3 . From Eq.(38), we can obtain the expectations as follows σ 1z σ 2z = [ σ 1z σ 2z 0 + 2 σ 1z 0 + 1]/M 2 3 (43) Q 3 = σ 1 . σ 2 = [ σ 1z σ 2z 0 + 2 σ 1z 0 + 1]/M 2 3 Thus, the spin-squeezing parameters and concurrence can be calculated as ξ 2 1 = 1 (44) ξ 2 2 = ξ 2 1 = 1 ξ 2 3 = 1 (1 − N −1 )Q 3 + N −1 C r = 1 2 (N − 1){[ σ 1z σ 2z 0 + 2 σ 1z 0 + 1]/M 2 3 − 1} So, the spin-squeezing parameter ζ 2 3 and the concurrence can be recovered to certain stationary value respectively and the spin-squeezing parameter ζ 2 2 = 0 when weak measurement strength m is very small. We also note that with the decrease of weak measurement strength m, the spinsqueezing parameter ζ 2 2 becomes more and more weak until it is zero. This means that in our model, the parameter ξ 2 3 < 1 implies the existence of pairwise entanglement, while ξ 2 2 < 1 does not. This result is the same as that discussed in the case of depolarizing channel. Conclusion In this paper, we have proposed an efficient method to protect spin squeezing under the action of amplitude-damping, depolarizing and phase-damping channels based on measurement reversal from weak measurement, and have considered an ensemble of N independent spin-1/2 particles with exchange symmetry. We have found that spin squeezing can be enhanced greatly under three different decoherence channels and spin-squeezing sudden death can be avoided undergoing amplitude-damping and phasedamping channels. Appendix: Derivation of the evolution of the correlations and expectations under decoherence by using weak measurements For an arbitrary matrix A =     a b c d     ,(45) from Eq. (22) and the Kraus operators (23) for the ADC, when sn 2 + p = m 2 , it is straight forward to find Θ + (A) =     asn 2 + dp bmn √ s cmn √ s dm 2     /(sn 2 + p),(46) The above equation imply that Θ + (σ µ ) = mn √ sσ µ /(sn 2 + p) f or µ = x, y(47)Θ + (σ z ) = (sn 2 σ z − p)/(sn 2 + p)(48) As we considered independent and identical decoherence channels and weak measurements acting separately on each spin, the evolution correlations and expectations in Eq. (24), are obtained directly from the above equations. From Eqs. (31) and (22), when m = 1, the evolution of the matrix A under the DPC is obtained as Θ + (A) =     d 2 p + an 2 − a 2 n 2 p bns cns ap 2 n 2 + d − d 2 p     /[ 1 2 (n 2 + 1) + 1 2 (n 2 s − s) σ z 0 ],(49) from which one finds Θ + (σ µ ) = nsσ µ /[ 1 2 (n 2 + 1) + 1 2 (n 2 s − s) σ z 0 ] f or µ = x, y(50)Θ + (σ z ) = [ 1 2 (n 2 s + s)σ z + 1 2 (n 2 − 1)]/[ 1 2 (n 2 + 1) + 1 2 (n 2 s − s) σ z 0 ](51) From Eq.(22) and the Kraus operators for the DPC given by Eq. (31), when m = 1, we find the evolutions of the following expectations under decoherence using weak measurement (see Appendix for details): n 2 s + s) 2 σ 1z σ 2z 0 + 2(n 2 − 1)(n 2 s + s) σ 1z 0 + (n 2 − 1) 2 ]/M 2 2 Q 2 = σ 1 . σ 2 = {s 2 n 2 (1 − σ 1z σ 2z 0 ) + 1 4 [(n 2 s + s) 2 σ 1z σ 2z 0 + 2(n 2 − 1)(n 2 s + s) σ 1z 0 + (n 2 − 1) 2 ]}/ 0 . 0Substituting the relevant expectation values and the correlation function into Eqs. (9), (10), and (11) leads to the explicit expression of the spin-squeezing parameters ξ 2 1 = 1 − s 2 m 2 n 2 C r ( FromFigure 1 :Figure 2 :Figure 3 :Figure 4 : 1234Eqs.(37) and(22), when n 2 − 1 = m 2 + 1, the evolution of the matrix A under the PDC is obtained asΘ + (A) /[(m 2 + 1) + σ z 0 ],Spin-squeezing parameters ς 2 2 (dash-dotted line), ς 2 3 (dashed line) and the concurrence C r (solid line) versus the decoherence strength p for the amplitudedamping channel with θ = 0.1π, N = 12. (a) Without weak measurement; (b) weak measurement strength m = 2; (c) m = 4; (d) Spin-squeezing parameters ς 2 2 (dash-dotted line), ς 2 3 (dashed line) and the concurrence C r (solid line) versus the decoherence strength p for the amplitudedamping channel with θ = 1.8π, N = 12. (a) Without weak measurement; (b) weak measurement strength m = 4; (c) m = 8; (d) Spin-squeezing parameters ς 2 2 (dash-dotted line), ς 2 3 (dashed line) and the concurrence C r (solid line) versus the decoherence strength p for the depolarizing channel with θ = 1.8π, N = 12. (a) Without weak measurement; (b) weak measurement strength n = 2; (c) n = 10; (d) Spin-squeezing parameters ς 2 2 (dash-dotted line), ς 2 3 (dashed line) and the concurrence C r (solid line) versus the decoherence strength p for the phase-damping channel with θ = 1.8π, N = 12. (a) Without weak measurement; (b) weak measurement strength m = 1; (c) m = 0.5; (d) m = 0.01. . M Kitagawa, M Ueda, Phys. Rev. A. 475138M. Kitagawa and M. Ueda, Phys. Rev. A 47, 5138 (1993) . D J Wineland, J J Bollinger, W M Itano, D J Heinzen, Phys. Rev. A. 5067D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen, Phys. Rev. A 50, 67 (1994). . A Sørensen, L.-M Duan, J I Cirac, P Zoller, Nature. 40963A. Sørensen, L.-M. Duan, J. I. Cirac, and P. Zoller, Nature (London) 409, 63 (2001). . G Toth, C Knapp, O Guhne, H J Briegel, Phys. Rev. Lett. 99250405G. Toth, C. Knapp, O. Guhne, and H. J. Briegel, Phys. Rev. Lett. 99, 250405 (2007). . G Toth, C Knapp, O Guhne, H J Briegel, Phys. Rev. A. 7942334G. Toth, C. Knapp, O. Guhne, and H. J. Briegel, Phys. Rev. A 79, 042334 (2009). . A Sørensen, K Mølmer, Phys. Rev. Lett. 832274A. Sørensen, K. Mølmer, Phys. Rev. Lett. 83, 2274 (1999). . X Wang, B C Sanders, Phys. Rev. A. 6812101X. Wang and B. C. Sanders, Phys. Rev. A 68, 012101 (2003). . J Ma, X Wang, C P Sun, F Nori, Phys. Rep. 50989J. Ma, X. Wang, C. P. Sun, and F. Nori, Phys. Rep. 509, 89 (2011). . O Guhne, G Toth, Phys. Rep. 4741O. Guhne, and G. Toth, Phys. Rep. 474, 1 (2009). . L Amico, R Fazio, A Osterloh, V Vedral, Rev. Mod. Phys. 80517L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod. Phys. 80, 517 (2008). . R Horodechi, P Horodechi, M Horodechi, K Horodechi, Rev. Mod. Phys. 81865R. Horodechi, P. Horodechi, M. Horodechi, and K. Horodechi, Rev. Mod. Phys. 81, 865 (2009). . C Genes, P R Berman, A G Rojo, Phys. Rev. A. 6843809C. Genes and P. R. Berman, and A. G. Rojo, Phys. Rev. A 68, 043809 (2003). . T Fernholz, H Krauter, K Jensen, J F Sherson, A S Sorensen, E S Polzik, Phys. Rev. Lett. 10173601T. Fernholz, H. Krauter, K. Jensen, J. F. Sherson, A. S. Sorensen, and E. S. Polzik, Phys. Rev. Lett. 101, 073601 (2008). . T Takano, M Fuyama, R Namiki, Y Takahashi, Phys. Rev. Lett. 10233601T. Takano, M. Fuyama, R. Namiki, and Y. Takahashi, Phys. Rev. Lett. 102, 033601 (2009). . D Ulam-Orgikh, M Kitagawa, Phys. Rev. A. 6452106D. Ulam-Orgikh and M. Kitagawa, Phys. Rev. A 64, 052106 (2001). . X Wang, Phys. Lett. A. 331164X. Wang, Phys. Lett. A 331, 164 (2004). . X Yin, X Wang, J Ma, X Wang, J. Phys. B: At. Mol. Opt. Phys. 4415501X. Yin, X. Wang, J. Ma, and X. Wang, J. Phys. B: At. Mol. Opt. Phys. 44, 015501 (2011). . Y C Liu, Z F Xu, G R Jin, L You, Phys. Rev. Lett. 10713601Y. C. Liu, Z. F. Xu, G. R. Jin, and L. You, Phys. Rev. Lett. 107, 013601 (2011). . Zilong Chen, Justin G Bohnet, Shannon R Sankar, Jiayan Dai, James K Thompson, Phys. Rev. Lett. 106133601Zilong Chen, Justin G. Bohnet, Shannon R. Sankar, Jiayan Dai, and James K. Thompson, Phys. Rev. Lett. 106, 133601 (2011). H.-P Breuer, F Petruccione, The Theory of Open QuantumSystems. New YorkOxford Univ. PressH.-P. Breuer and F. Petruccione, The Theory of Open QuantumSystems (Oxford Univ. Press, New York, 2002). C W Gardiner, P Zoller, Quantum Noise. BerlinSpringer-Verlag2nd ed.C. W. Gardiner and P. Zoller, Quantum Noise, 2nd ed. (Springer-Verlag, Berlin, 2000). . C Simon, J Kempe, Phys. Rev. A. 6552327C. Simon and J. Kempe, Phys. Rev. A 65, 052327 (2002). . W Dur, H. -J Briegel, Phys. Rev. Lett. 92180403W. Dur and H. -J. Briegel, Phys. Rev. Lett. 92, 180403 (2004). . A R R Carvalho, F Mintert, A Buchleitner, Phys. Rev. Lett. 93230501A. R. R. Carvalho, F. Mintert, and A. Buchleitner, Phys. Rev. Lett. 93, 230501 (2004). . X Wang, A Miranowicz, Y X Liu, C P Sun, F Nori, Phys. Rev. A. 8122106X. Wang, A. Miranowicz, Y. X. Liu, C. P. Sun, and F. Nori, Phys. Rev. A 81, 022106 (2010). . Z Sun, Phys. Rev. A. 8452307Z. Sun, Phys. Rev. A 84, 052307 (2011). . T Yu, J H Eberly, Phys. Rev. Lett. 93140404T. Yu and J. H. Eberly, Phys. Rev. Lett. 93, 140404 (2004); . Science. 323Science 323, 598 (2009). . W Song, L Chen, S L Zhu, Phys. Rev. A. 8012331W. Song, L. Chen,and S. L. Zhu, Phys. Rev. A, 80, 012331 (2009) . A Z Chaudhry, J B Gong, Phys. Rev. A. 8612311A. Z. Chaudhry and J. B. Gong, Phys. Rev. A 86, 012311 (2012) . Y Aharonov, D Z Albert, L Vaidman, Phys. Rev. Lett. 601351Y. Aharonov, D. Z. Albert, L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988) . Y Aharonov, A Botero, S Pospescu, B Reznik, J Tollaksen, Phys. Lett. A. 301130Y. Aharonov, A. Botero, S. Pospescu, B. Reznik, J. Tollaksen, Phys. Lett. A 301, 130 (2002); . J S Lundeen, A M Steinberg, Phys. Rev. Lett. 10220404J. S. Lundeen, A. M. Steinberg, Phys. Rev. Lett. 102 020404 (2009); . K Yokota, T Yamamoto, M Koashi, N Imoto, New J. Phys. 1133011K. Yokota, T. Yamamoto, M. Koashi, N. Imoto, New J. Phys. 11, 033011 (2009) . A N Korotkov, A N Jordan, Phys. Rev. Lett. 97166805A. N. Korotkov, A. N. Jordan, Phys. Rev. Lett. 97 166805 (2006) . Y S Kim, Y W Cho, Y S Ra, Y H Kim, Opt. Express. 1711978Y. S. Kim, Y. W. Cho, Y. S. Ra, Y. H. Kim, Opt. Express 17, 11978 (2009) . J C Lee, Y C Jeong, Y S Kim, Y H Kim, Opt. Express. 1916309J. C. Lee, Y. C. Jeong, Y. S. Kim, Y. H. Kim, Opt. Express 19, 16309 (2011) . Y S Kim, J C Lee, O Kwon, Y H Kim, Nature Phys. 8117Y. S. Kim, J. C. Lee, O. Kwon, Y. H. Kim, Nature Phys. 8, 117 (2012) . Z X Man, Y J Xia, N B An, Phys. Rev. A. 8652322Z. X. Man, Y. J. Xia, N. B. An, Phys. Rev. A 86, 052322 (2012) . X P Liao, M F Fang, J S Fang, Journal of Modern Optics. 611018X. P. Liao, M. F. Fang, J. S. Fang, Journal of Modern Optics, 61, 1018 (2014); . X P Liao, M F Fang, J S Fang, Q Q Zhu, Chin. Phys. B. 23X. P. Liao, M. F. Fang, J. S. Fang and Q. Q. Zhu, Chin. Phys. B. 23, 020304-1-7 (2014) . X P Liao, X Z Ding, M F Fang, Quantum Inf. Pro. 144395X. P. Liao, X. Z. Ding and M. F. Fang, Quantum Inf. Pro., 14, 4395 (2015) . W K Wootters, Phys. Rev. Lett. 802245W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998). . S C Wang, Z W Yu, W J Zou, X B Wang, Phys. Rev. A. 8922318S. C. Wang, Z. W. Yu, W. J. Zou, X. B. Wang, Phys. Rev. A 89, 022318 (2014)
{'fraction_non_alphanumeric': 0.09396077215288007, 'fraction_numerical': 0.07027676564074735, 'mean_word_length': 3.1922326941826453, 'pattern_counts': {'":': 0, '<': 6, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 44, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'We propose an efficient method to protect spin squeezing under the action of amplitude-damping, depolarizing and phase-damping channels based on measurement reversal from weak measurement, and consider an ensemble of N independent spin-1/2 particles with exchange symmetry. We find that spin squeezing can be enhanced greatly under three different decoherence channels and spin-squeezing sudden death (SSSD) can be avoided undergoing amplitude damping and phase-damping channels.', 'arxivid': '1607.06530', 'author': ['Xiang-Ping Liao \nCollege of Science\nHunan University of Technology\n412008ZhuzhouHunanChina\n', 'Man-Sheng Rong \nCollege of Science\nHunan University of Technology\n412008ZhuzhouHunanChina\n', 'Mao-Fa Fang \nCollege of Physics and Information Science\nHunan Normal University\n410081ChangshaHunanChina\n'], 'authoraffiliation': ['College of Science\nHunan University of Technology\n412008ZhuzhouHunanChina', 'College of Science\nHunan University of Technology\n412008ZhuzhouHunanChina', 'College of Physics and Information Science\nHunan Normal University\n410081ChangshaHunanChina'], 'corpusid': 119275762, 'doi': '10.1088/1612-202x/aa6dc7', 'github_urls': [], 'n_tokens_mistral': 10768, 'n_tokens_neox': 8697, 'n_words': 4929, 'pdfsha': '957ac108d011a4d092e580ccd25b23c9ed923197', 'pdfurls': ['https://arxiv.org/pdf/1607.06530v2.pdf'], 'title': ['Protecting and enhancing spin squeezing under decoherence using weak measurement', 'Protecting and enhancing spin squeezing under decoherence using weak measurement'], 'venue': []}
arxiv
Closed-loop stability analysis of a gantry crane with heavy chain and payload Dominik Stürzer Institute for Analysis and Scientific Computing Technische Universität Wien ViennaAustria Anton Arnold Institute for Analysis and Scientific Computing Technische Universität Wien ViennaAustria Andreas Kugi Automation and Control Institute Technische Universität Wien ViennaAustria ARTICLE HISTORY Closed-loop stability analysis of a gantry crane with heavy chain and payload Received  July  Accepted  May https://doi.org/./..Infinite-dimensional systemsstabilitygantry crane with heavy chains In this paper, we analyse a systematically designed and easily tunable backstepping-based boundary control concept for a gantry crane with heavy chain and payload. The corresponding closed-loop system is formulated as an abstract evolution equation in an appropriate Hilbert space. Non-restrictive conditions for the controller coefficients are derived, under which the solutions are described by a C 0semi-group of contractions, and are asymptotically stable. Moreover, by applying Huang's theorem we can finally even show that under these conditions the controller renders the closed-loop system exponentially stable. Introduction This paper deals with the rigorous stability analysis of a control concept presented by Thull, Wild, and Kugi (2006) applied to the infinite-dimensional model of a gantry crane with heavy chain and payload. The model consists of a cart of mass m c , which moves horizontally along a rail, a heavy chain of length L with mass per length ρ, attached to the cart 1 , and a payload of point mass m p at its end. The chain is assumed to be inextensible and perfectly flexible. For the derivation of the equations of motion, it is further assumed that no friction occurs in the system. The force F acting on the cart serves as the control input. The situation is sketched in Figure 1 on the left-hand side. Let w(t, x) denote the horizontal chain position. Then, under the assumption that the chain slopes x w(t, x) remain sufficiently small for all t > 0, the dynamics of the system are described by the following wave equation with dissipative higher order boundary conditions (see, e.g. Mifdal, 1997b;Petit & Rouchon, 2001;Thull et al., 2006) ρ∂ 2 t w(t, x) − ∂ x (P(x)∂ x w(t, x)) = 0, (1.1a) m p ∂ 2 t w(t, L) + P(L)∂ x w(t, L) = 0, (1.1b) m c ∂ 2 t w(t, 0) − P(0)∂ x w(t, 0) = F (t ). (1.1c) The function P(x) represents the tension in the chain at height x, given by P(x) = g[ρ(L − x) + m p ], where g denotes the gravitational acceleration. Note that CONTACT Andreas Kugi [email protected] P ࣙ g m p > 0 holds uniformly on [0, L]. In the following, it is only required that P ࢠ H 2 (0, L) and that P(x) ࣙ P 0 > 0 holds uniformly on [0, L] for some constant P 0 . Thus, the density of the chain does not need to be constant, as it was the case for instance in Thull et al. (2006). Moreover, the following notation v t w will be used in the sequel. For the system (1.1), many different control laws can be found in the literature (see, e.g. Conrad & Mifdal, 1998;Coron & d' Andrea Novel, 1998;Mifdal, 1997b;Thull et al., 2006), with v = w t . Basically, the common structure of these controllers looks like F (t ) = ϑ 1 v (t, 0) + ϑ 2 ∂ x v (t, 0) + ϑ 3 w (t, 0) + ϑ 4 ∂ x w (t, 0) ,(1.2) but they differ in the fact which parameters ϑ j , j = 1, … , 4 are equal to zero, which conditions have to be fulfilled to render the closed-loop system asymptotically (or even exponentially) stable, and how the controller parameters can be systematically tuned. Thus, for instance Conrad and Mifdal (1998) show that a (simple) passive controller with ϑ 2 = ϑ 4 = 0 in Equation (1.2) already ensures asymptotic stability of the closedloop system. While this is a nice result from a theoretical point of view, this controller is not able to damp vibrations of the chain in case of stick-slip effects in the cart, which are always present in the real experiment. To make this clear, let us assume that the cart is in a sticking position, which also entails that v(t, 0) = 0 and w(t, 0) = c, with a constant c, then the cart will not move as long as the absolute value of the sum of the internal force in the x L w (t, x) w(t, 0) pivot bearing carrying the chains F i = P(0) x w(t, 0) and the input force F(t), see Equation (1.1c), is smaller than the sticking friction. As a consequence the chain keeps on vibrating, but the cart stands still and the control law (1.2) with ϑ 2 = ϑ 4 = 0 only produces a constant input force F(t) = ϑ 3 c. F F i F i In Mifdal (1997b), the exponential stability of the closed-loop system is proven for the control law (1.2) with ϑ 4 = 0 under certain further conditions by using an energy multiplier approach. The proof of stability is performed in a rigorous and elegant way, however, there is no systematic design of the control law and it is not clear how to specifically tune the controller parameters. In contrast, the control law (1.2) used in this paper overcomes these deficiencies, but then the controller parameters for tuning appear in all parameters ϑ j , j = 1, … , 4, which will be shown subsequently, and ϑ 4 has to be unequal to zero. For the authors, it is not obvious and does not seem to be straightforward to extend the stability proof of Mifdal (1997b) to the case ϑ 4 ࣔ 0 under the same non-restrictive conditions on the controller parameters as presented in this paper. For a controller with the same structure as in Equation (1.2) but, compared to the controller considered in this paper, with totally different conditions on the parameters ϑ j , j = 1, … , 4, the asymptotic and exponential stability of the closed-loop system is shown by Coron and d' Andrea Novel (1998). In this paper, the dynamics of the payload (1.1b) was neglected. Moreover, the controller was designed based on a backstepping approach; however, the authors mention in their paper that this boundary feedback law ensures asymptotic stability, but it is not clear if it produces exponential stability. Therefore, they modify the backstepping controller and based on this they are able to show exponential stability of the closed-loop system. Inspired by Coron and d' Andrea Novel (1998), a controller was systematically designed and experimentally validated for the system (1.1) of Thull et al. (2006). In order to reveal the connection between the controller parameters for tuning and the parameters ϑ j , j = 1, … , 4 in Equation (1.2), the control design will be shortly revisited, see Thull et al. (2006) for more details. In contrast to the (simple) passive controller presented by Conrad and Mifdal (1998), the (damping) controller design by Thull et al. (2006) is based on the idea to specifically influence the energy flow between the cart and the chain, which is represented by the collocated variables cart velocity t w(t, 0) = v(t, 0) and internal force in the pivot bearing carrying the chains F i = P(0) x w(t, 0), thus forming an energy port, see right-hand side of Figure 1. The sum of the potential energy of the chain and the kinetic energy of the chain and the payload according to Equation (1.1) reads as H = 1 2 L 0 P(x) (∂ x w (t, x)) 2 + ρv 2 (t, x) dx + 1 2 m p v 2 (t, L) . (1.3) The change ofH along a solution of Equation (1.1) yields Thus,if v(t,0) were the (virtual) control input, the control law v(t, 0) χ 1 F i , with the controller parameter χ 1 > 0, would render the closed-loop system passive. However, this would ensure a good damping of the chain vibrations, but the cart position w(t, 0) remains unconsidered within this approach. Therefore, the energy functionalH from Equation (1.3) is extended by the potential energy of a virtual linear 2 spring attached to the cart in the form 3 d dtH = −v (t, 0) F i .V = χ 1H + χ 2 2 w 2 (t, 0) , (1.4) with the controller parameters χ 1 , χ 2 > 0. The change of V along a solution of Equation (1.1) d dtV = (χ 2 w (t, 0) − χ 1 F i ) v (t, 0) (1.5) immediately shows that the control law v(t, 0) −(χ 2 w(t, 0) − χ 1 F i ) for the virtual control input v(t, 0) makes the closed-loop system passive again. The controller parameters χ 1 and χ 2 facilitate a simple tuning of the closed-loop behaviour, because a larger χ 1 (wrt χ 2 ) brings along a higher damping of the chain vibrations at the cost of larger deviations of the cart position w(t, 0) from the equilibrium and for a larger χ 2 (wrt χ 1 ) the control of the cart position is becoming more important. Since v(t, 0) is not the real control input, a simple backstepping approach, see, e.g. Krstić, Kanellakopoulos, and Kokotović (1995), is applied to Equation (1.1c). In a nutshell, the functionalV from Equation (1.4) is extended in the form: V =V + 1 2 (v (t, 0) + χ 2 w (t, 0) − χ 1 F i ) 2 (1.6) and the control law is designed in such a way that d dt V = −v 2 (t, 0) − χ 3 (v (t, 0) + χ 2 w (t, 0) − χ 1 F i ) 2 ,(1.7) with the controller parameter χ 3 > 0. The control law finally takes the form (see also (1.2)): F (t ) = − (χ 3 + χ 2 + 1) m c ϑ 1 v (t, 0) + χ 1 P(0)m c ϑ 2 ∂ x v (t, 0) + (−χ 3 χ 2 m c ) ϑ 3 w (t, 0) + (χ 3 χ 1 m c − 1) P(0) ϑ 4 ∂ x w (t, 0) . (1.8) The third controller parameter χ 3 weights the deviation of the virtual control input v(t, 0) from its desired time evolution χ 2 w(t, 0) − χ 1 F i . Note that in addition to the feedback control law (1.8) the control concept of Thull et al. (2006) consists of a flatness-based feedforward controller as presented by Petit and Rouchon (2001). Due to the linearity of the system (1.1), the trajectory error dynamics are identical and thus also the stability proof remains the same. In Thull et al. (2006), energy dissipation of the closedloop system was shown for Equation (1.8). Thull, Wild, and Kugi (2005) attempted to show asymptotic stability by using LaSalle's invariance principle (see, Luo, Guo, & Morgül, 1999). This is common practice in the context of (hyperbolic) control systems, see, e.g. Miletić, Stürzer, and Arnold (2015), Miletić, Stürzer, Arnold, and Kugi (2016), Chentouf and Couchouron (1999), Conrad and Morgül (1998), Coron and d' Andrea Novel (1998), Kugi and Thull (2005) and Morgül (2001) for the control of an Euler-Bernoulli beam, and d' Andréa Novel, Boustany, and Conrad (1992), Morgül, Rao, and Conrad (1994) and d' Andréa Novel and Coron (2000) for the control of hanging cables. With the (energy) inner product chosen by Thull et al. (2005), Thull et al. (2006) the proof of the closed-loop stability did not work out, which was also correctly pointed out by Grabowski (2009). The backstepping approach presented by Thull et al. (2006) is quite intuitive from a control point of view, but it brings along the drawback that the control law depends on x v(t, 0), see Equation (1.8), which makes it impossible to analyse the closed-loop system in the space H 1 × L 2 . Therefore, an appropriate Hilbert space H and a convenient inner product, see Equation (2.6), are introduced in this paper which allow the application of the Lumer-Phillips theorem and the rigorous proof of the closed-loop stability of Equation (1.1) with Equation (1.8). Due to the change of the abstract model setting, the strategy to prove asymptotic stability does not make use of LaSalle's invariance principle. Instead, techniques from spectral analysis are used, see Luo et al. (1999). In order to show exponential stability, Huang's theorem, see, e.g. Huang (1985), is employed. In order to prove stability of the closed-loop system, in a first step (1.1) with Equation (1.8) is rewritten as an abstract evolution equationẏ = Ay in an appropriate Hilbert space H, where the generator A is a linear operator, see, e.g. d' Andréa Novel and Coron (2000), Grabowski (2008), Grabowski (2009), Morgül et al. (1994) and Luo et al. (1999) for the formulation of similar systems describing hanging cables. We start by showing that A generates a C 0 -semi-group of contractions, by using the Lumer-Phillips theorem. To this end an inner product, equivalent to the natural inner product in H, is used. Then, we show that the inverse A −1 exists and is compact. This implies that the spectrum σ (A) consists entirely of eigenvalues. Since A generates a C 0 -semi-group of uniformly bounded operators, the Hille-Yosida theorem implies that σ (A) ⊆ {ζ ∈ C : Re ζ ≤ 0}. We then show that iR lies in ρ(A), by demonstrating that for all λ ∈ R the eigenvalue equation Ay = iλy only has the trivial solution. According to Luo et al. (1999, Theorem 3.26), this proves the asymptotic stability of the system. Finally, we show uniform boundedness of the resolvent (iλ − A) −1 for λ ∈ R. Huang's Theorem (cf. Corollary 3.36 of Luo et al., 1999, see also Huang, 1985) then implies exponential stability of the closed-loop system. The paper is organised as follows. In Section 2, we prove that A generates a C 0 -semi-group of uniformly bounded operators. In Section 3, we show the asymptotic stability of this semi-group, and Section 4 is devoted to the proof of the exponential stability. Finally, Section 5 contains some conclusions. Formulation as a Dissipative Evolution Equation For the mathematical analysis of the system (1.1) with Equation (1.8) it is convenient to eliminate most numerical coefficients. To this end, we rescale length and time, i.e. we introduce new variablesx = P(L)ρ m p x andt = P(L) m p √ ρt. Withw(t,x) := w(t, x) andP(x) = P(x) the system (1.1) is equivalent to ∂ 2 tw (t,x) = ∂x(P(x)∂xw(t,x),x ∈ (0,L),t > 0, (1.1a) ∂ 2 tw (t,L) = −∂xw(t,L), (1.1b) ∂ 2 tw (t, 0) =θ 1ṽ (t, 0) +θ 2 ∂xṽ (t, 0) +θ 3w (t, 0) +θ 4 ∂xw(t, 0), (1.1c) in new coordinates. In Equation (1.1c'), m c and all additional factors arising from the change of coordinates as well as the termP(0)∂xw(t, 0) have been merged in the new coefficientsθ i , i = 1, … , 4. In the following, we only consider the system (1.1'). However, for the sake of readability, we will omit the superscript tilde in the sequel and simply write x, t, ϑ i , w, and P. For the analysis of Equation (1.1'), we define the (complex) Hilbert space: H = {z = (w, v, ξ, ψ ) : w ∈ H 2 (0, L), v ∈ H 1 (0, L), ξ = v (L), ψ = v (0)}, (2.1) which is a closed subspace of H 2 × H 1 × C × C. Here, H n (0, L) denotes the Sobolev space of functions whose derivatives up to order n are square-integrable (see Adams, 1975, for details). The auxiliary scalar variables ξ , ψ are introduced here in order to include the dynamical boundary conditions (1.1b') and (1.1c') into the initial value problem. H is equipped with the natural inner product: z 1 , z 2 = w 1 , w 2 H 2 + v 1 , v 2 H 1 + ξ 1ξ2 + ψ 1ψ2 , (2.2) whereξ denotes the complex conjugate of ξ . Let the linear operator A : D(A) ⊂ H → H be defined as A : ⎡ ⎢ ⎢ ⎣ w v ξ ψ ⎤ ⎥ ⎥ ⎦ → ⎡ ⎢ ⎢ ⎣ v (Pw ) −w (L) ϑ 1 v (0) + ϑ 2 v (0) + ϑ 3 w(0) + ϑ 4 w (0) ⎤ ⎥ ⎥ ⎦ , (2.3) where w denotes the spatial derivative of w, i.e. w = x w. The (dense) domain of A is defined as D(A) := z = (w, v, ξ, ψ ) : w ∈ H 3 (0, L), v ∈ H 2 (0, L), ξ = v (L), ψ = v (0),(Pw )(L) = −w (L), (Pw ) (0) = F[w, v] , (2.4) with F[w, v] := ϑ 1 v (0) + ϑ 2 v (0) + ϑ 3 w(0) + ϑ 4 w (0) due to Equation (1.1'). The boundary conditions stated in D(A) arise naturally from the requirement that ran A ⊂ H. With these definitions, we can rewrite the system (1.1') as the following initial value problem in H: ż(t ) = Az(t ), z(0) = z 0 ∈ H. (2.5) For some of the following proofs, the natural inner product ·, · on H is unpractical. Therefore, we define an equivalent inner product, which is more suitable for the considered problem: z 1 , z 2 H := α 1 L 0 γ (Pw 1 ) (Pw 2 ) + Pw 1w 2 dx + α 1 γ P(L)w 1 (L)w 2 (L) + α 2 w 1 (0)w 2 (0) + α 1 L 0 γ Pv 1v 2 + v 1v2 dx + α 1 P(L)ξ 1ξ2 + α 2 γ ψ 1ψ2 + 1 2 ψ 1 − 2α 1 P(0)w 1 (0) + 2α 2 w 1 (0) × ψ 2 − 2α 1 P(0)w 2 (0) + 2α 2w2 (0) , (2.6) where α 1 , α 2 , and γ are positive constants to be specified later (in Lemma 2.5 and the corresponding proof). We have the following lemma: Lemma 2.1: The norm · H is equivalent to the natural norm · on H. Proof: We have to prove the existence of constants c 1 , c 2 > 0 such that c 1 z ≤ z H ≤ c 2 z holds for all z ∈ H. To verify the first inequality, it remains to show the existence ofc 1 such that L 0 γ |(Pw ) | 2 + P|w | 2 dx ≥c 1 L 0 |w | 2 + |w | 2 dx (2.7) holds for all real-valued w ࢠ H 2 (0, L). Using the properties of P mentioned above, Lemma A.1 (see Appendix 1) can be applied pointwise in x with a = √ γ P (x), b = √ γ P(x), ε = P(x), x 1 = |w (x)| , and x 2 = |w (x)|, which directly yields the desired inequality (2.7). To verify the second inequality, it suffices to apply Cauchy's inequality ab ≤ a 2 2 + b 2 2 , a, b ∈ R, to the terms obtained by expansion of the last term in z 2 H . The main statement of this section is the following theorem, which will be proved in several steps: Theorem 2.1: Let there be constants a, b > 0 satisfying (a + b − 1) 2 < 4ab, such that ϑ 1 = ϑ 3 b − a, ϑ 2 = ϑ 4 b , (2.8) and ϑ 1 , ϑ 3 < 0 and ϑ 2 , ϑ 4 > 0. Then, the operator A is the infinitesimal generator of a C 0 -semi-group of uniformly bounded operators {T(t)} t ࣙ 0 on H. The conditions of Theorem 2.1 on the parameters ϑ j , j = 1, … , 4 from Equation (1.1c') are fulfilled if the controller parameters χ 1 , χ 2 , and χ 3 according to Equation (1.8) meet the following non-restrictive inequality constraints: χ 1 > 0, χ 2 > 0, and χ 3 > m p − P (L) √ ρ 2 4m p P (L) √ ρ , (2.9) where b =θ 4 ϑ 2 = m p α 3 P (L) √ ρ > 0 and a =θ 3 b −θ 1 = (α 3 + 1) m p P (L) √ ρ > 0. (2.10) In Theorem 2.1, the admissible parameters (a, b) lie inside a parabola in the first quadrant, and this parabola is tangent to the positive a − and b −axes. Remark 2.1: Let us briefly compare the model (1.1') subject to Equation (2.8) with the closed-loop system from Mifdal (1997b) and even its extension in §0(a) of Mifdal (1997a). We recall that the latter two models correspond to a control law with ϑ 4 = 0. Equations (1.1') and (2.8) can only be matched with the models analysed by Mifdal when using the special parameters a = 0 and b = P(0) P(L) M ϑ 2 m in Equation (2.8). Here, M and m denote parameters related to the masses of the payload and the cart in Mifdal (1997b). Since Theorem 2.1 requires a > 0, these two types of models and results are complementary to each other. Remark 2.2: With respect to the inner product specified in Lemma 2.5, this semi-group is even a semi-group of contractions, see the proof of Theorem 2.1 . We shall prove Theorem 2.1 by applying the Lumer-Phillips theorem (cf. Pazy, 1983). But before, we verify some basic properties of A. Lemma 2.2: The domain D(A) defined in Equation (2.3) is dense in H. Proof: Let z 0 = (w 0 , v 0 , ξ 0 , ψ 0 ) ∈ H. Since the inclusions H 3 (0, L)࣪H 2 (0, L)࣪H 1 (0, L) are dense, there exists a sequence z n = (w n , v n , ξ n , ψ n ) ∈ H 3 (0, L) × H 2 (0, L) × C 2 ∩ H such that z n → z 0 in H. Now, in general, the second derivatives 2 w n (0), 2 w n (L) will not satisfy the boundary conditions necessary for z n ࢠ D(A). The fact that H 1 0 (0, L) ⊂ L 2 (0, L) is dense ensures the existence of a sequence {u n }࣪H 1 (0, L) satisfying u n (0) = a for all n ∈ N and any fixed a ∈ C, with u n L 2 → 0. The sequence {y n } defined by y n := (0)) has a unique solution z ࢠ D(A) for every ( f , g, g(L), g (0)) ∈ H. This equation reads in detail: ⎡ ⎢ ⎢ ⎣ v (Pw ) −w (L) ϑ 1 v (0) + ϑ 2 v (0) + ϑ 3 w(0) + ϑ 4 w (0) ⎤ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎣ f g g(L) g(0) ⎤ ⎥ ⎥ ⎦ . (2.11) From the first line we immediately find v = f ࢠ H 2 (0, L), which also fixes the values v(0) and v (0). After integration of the second line we obtain w (x) = − P(L) P(x) g(L) + 1 P(x) x L g(y) dy, (2.12) where we used w (L) = −g(L) from the third line. Since 1/P ࢠ H 2 (0, L) and g ࢠ H 1 (0, L), we find w ࢠ H 2 (0, L). This equation also determines w (0). In combination with the already known values v(0) and v (0), we obtain w(0) from the fourth line in Equation (2.11), since ϑ 3 ࣔ 0. Hence, w(x) is uniquely determined as w(x) = w(0) − x 0 P(L) P(y) g(L) dy + x 0 1 P(y) y L g(ζ ) dζ dy. (2.13) All integrals exist, since P(x) > 0 holds uniformly. Finally, w ࢠ H 3 (0, L) holds. Thus, the inverse A −1 exists and is defined on H. Lemma 2.4: If ϑ 3 ࣔ 0, the operator A −1 is compact. Proof: We show that for ( f , g, g(L), g (0)) ∈ H the norm of z = A −1 (f, g, g(L), g(0)) in J := H 3 (0, L) × H 2 (0, L) × C 2 is uniformly bounded by ( f , g, g(L), g(0)) H . Due to the continuous embedding H 1 (0, L) →C[0, L] in one dimension (see, e.g. Adams, 1975), we have the estimates |g(L)|, |g(0)| ≤ C g H 1 . Here and in the sequel, C denotes positive, not necessarily equal constants. From the third line in Equation (2.11) we therefore get |w (L)| ≤ C g H 1 . With this and Equation (2.12) we find the estimate w L 2 ≤ C g H 1 . (2.14) Next we will apply this result to the identity Pw = g − P w , which is obtained from the second line in Equation (2.11), and use P ࢠ L Ý (0, L) and P(x) ࣙ P 0 > 0 (with P 0 introduced right after Equation (1.1)). This yields w L 2 ≤ C g H 1 . ( 2.15) Similarly, from (Pw ) = g we obtain the estimate: w L 2 ≤ C g H 1 . ( 2.16) For v we immediately get v H 2 = f H 2 using the first line in Equation (2.11). Due to the continuous embed- Adams, 1975), we find the following estimates: ding H k (0, L) →C k − 1 [0, L] in one dimension (cf.|v (0)| ≤ C f H 2 , (2.17) |v (0)| ≤ C f H 2 . (2.18) Using the above estimate for w (L) and Equation (2.12), we obtain |w (0)| ≤ C g H 1 . (2.19) Applying Equations (2.17)-(2.19) to the fourth line of Equation (2.11) and using |g(0)| ≤ C g H 1 yields |w(0)| 2 ≤ C( f 2 H 2 + g 2 H 1 ). (2.20) Altogether, we get w 2 H 3 ≤ C( f 2 H 2 + g 2 H 1 ). (2.21) Thus, we have w 2 H 3 + v 2 H 2 ≤ C( f 2 H 2 + g 2 H 1 ) , which shows that A −1 maps bounded sets in H into bounded sets in J . Since the embeddings H 3 (0, L)࣪࣪H 2 (0, L)࣪࣪H 1 (0, L) are compact, A −1 is a compact operator. From the previous lemma we know that A −1 is a closed operator, therefore we have: Corollary 2.1: For ϑ 3 ࣔ 0, the operator A is closed, and 0 ࢠ ρ(A), the resolvent set of A. Now we turn to the application of the Lumer-Phillips theorem in order to prove Theorem 2.1. To this end we shall prove the dissipativity of A with respect to the inner product ·, · H . Lemma 2.5: Let the assumptions of Theorem 2.1 hold, and let H be equipped with the inner product (2.6), where we choose the coefficients α 1 := ϑ 2 2P(0) , α 2 := − ϑ 2 ϑ 3 2ϑ 4 ,(2. 22) and γ > 0 is sufficiently small. Then, the operator A is dissipative in H. The proof is deferred to Appendix 3. Now, Theorem 2.1 follows directly from the above results: Proof of Theorem 2.1: First, we prove this result under the additional assumptions of Lemma 2.5 (on α 1 , α 2 , and γ ). Then, A is dissipative in H equipped with · H , and Corollary 2.1 implies 0 ࢠ ρ(A). Since ρ(A) is an open set, there exists some ζ ࢠ ρ(A) with positive real part. So the requirements of the Lumer-Phillips theorem are fulfilled, and we obtain that A generates a C 0 -semi-group of contractions on H with respect to · H . Next we drop the additional assumptions of Lemma 2.5, and consider an equivalent norm on H. Then, the semi-group {T(t)} t ࣙ 0 is not necessarily a contraction semi-group any more, but still a C 0 -semi-group of uniformly bounded operators. The following corollary follows as a consequence of Theorem 2.1, due to elementary properties of generators of C 0 -semi-groups of operators (for more details, see Pazy, 1983): Corollary 2.2: Under the assumptions of Theorem 2.1, the initial value problem ż(t ) = Az(t ) z(0) = z 0 (2.23) has a unique mild solution z(t) T(t)z 0 for all z 0 ∈ H, where {T(t)} t ࣙ 0 is the C 0 -semi-group generated by A. If z 0 ࢠ D(A), then z(t) is continuously differentiable on [0, Ý) and z(t) ࢠ D(A) for all t ࣙ 0, and therefore is a classical solution. Furthermore, the norm z(t ) H remains bounded as t → Ý. Asymptotic stability After having shown that the norm of every solution of the initial value problem (2.23) is uniformly bounded with respect to t ࣙ 0, we now prove that the norm even tends to zero as t → Ý, i.e. the C 0 -semi-group {T(t)} t ࣙ 0 generated by A is asymptotically stable, by applying the following theorem (see Luo et al., 1999, Theorem 3.26): Theorem 3.1: Let {S(t)} t ࣙ 0 be a uniformly bounded C 0semi-group in a Banach space X with generator A, and assume that the resolvent R(λ, A) is compact for some λ ∈ ρ(A). Then, {S(t)} t ࣙ 0 is asymptotically stable if and only if Re λ < 0 for all λ ∈ σ (A). Remark 3.1: The compactness of the resolvent R(λ, A) for one λ ∈ ρ(A) already implies its compactness for all λ ∈ ρ(A) (cf. Kato, 1966, Theorem III.6.29). Theorem 3.2: Let the assumptions of Theorem 2.1 hold. Then, the C 0 -semi-group {T(t)} t ࣙ 0 generated by A is asymptotically stable. Proof: According to Lemma 2.4 the operator A has compact resolvent, and the associated semi-group {T(t)} t ࣙ 0 is uniformly bounded due to Theorem 2.1. As a consequence of the Hille-Yosida theorem (see Pazy, 1983, Corollary 1.3.6), this implies σ (A) ⊆ {λ ∈ C : Re λ ≤ 0}. Hence, in order to apply Theorem 3.1, it remains to prove that iR ⊂ ρ(A). Since the resolvent is compact, σ (A) consists only of eigenvalues. Thus, it is sufficient to show that A − iτ is injective for all τ ∈ R, that is to show that the system ⎡ ⎢ ⎢ ⎣ v − iτ w (Pw ) − iτ v −w (L) − iτ v (L) ϑ 1 v (0) + ϑ 2 v (0) + ϑ 3 w(0) + ϑ 4 w (0) − iτ v (0) ⎤ ⎥ ⎥ ⎦ = 0 (3.1) only has the trivial solution in D(A). We can rewrite this system in terms of the following equivalent boundary value problem for w ࢠ H 3 (0, L) →C 2 [0, L]: (Pw ) + τ 2 w = 0, x ∈ (0, L), (3.2a) w (0) = c 0 w(0), (3.2b) w (L) = c L w(L), (3.2c) where c 0 := − ϑ 3 +τ 2 +iτ ϑ 1 ϑ 4 +iτ ϑ 2 and c L τ 2 . It is important to note that the conditions (2.8) on the ϑ i imply that c 0 / ∈ R for all τ ∈ R. We now multiply Equation (3.2a) by the complex conjugatew and integrate by parts, which yields − L 0 P|w | 2 dx + τ 2 w 2 L 2 + P(L)w (L)w(L) = P(0)w (0)w(0). Due to the boundary conditions (3.2b) and (3.2c) the lefthand side of above identity is real, but the right-hand side is either non-real or zero. Thus w (0)w(0) = 0, and Equation (3.2b) implies that w(0) = w (0) = 0. Therefore, every solution of the boundary value problem (3.2) also satisfies the initial value problem: (Pw ) + τ 2 w = 0, x ∈ (0, L), w(0) = 0, w (0) = 0. Hence, w ࣕ 0, and this shows that A − iτ is injective for all τ ∈ R. Exponential stability Here we show an even stronger result, namely the exponential stability of the semi-group {T(t)} t ࣙ 0 , i.e. we prove that every solution of the initial value problem (2.23) tends to zero exponentially. We follow a strategy similar to the one applied by Morgül (2001). Definition 4.1 (Exponential stability): A C 0 -semi- group {S(t)} t ࣙ 0 is said to be exponentially stable if there exist constants M ࣙ 1 and ω > 0 such that S(t) ࣘ Mexp ( − ωt) for all t ࣙ 0. To investigate exponential stability of a C 0 -semigroup, we use the following theorem (see Luo et al., 1999, Corollary 3.36): (4.1) Theorem 4.2: Assume that the conditions in Theorem 2.1 are satisfied. Then, the C 0 -semi-group {T(t)} t ࣙ 0 generated by A is exponentially stable. Proof: We know from Theorem 3.2 that {T(t)} t ࣙ 0 is asymptotically stable, and that iR ⊂ ρ(A). The map λ →R(λ, A) is analytic on ρ(A) (cf. Yosida, 1980), so, in particular, λ → R(λ, A) is continuous on iR. In order to apply Theorem 4.1, it therefore remains to prove that R(iτ , A) is uniformly bounded as |τ | → Ý. To this end, we need to find a τ -uniform estimate for the solution z = (w, v, v(L), v(0)) of the equation (A − iτ )z = ( f , g, g(L), g(0)) ∈ H (4.2) in terms of the right-hand side. The corresponding homogeneous problem (3.1) only has the trivial solution (cf. the proof of Theorem 3.2). Hence, we show that the unique solution (w, v) of the BVP v − iτ w = f , x ∈ (0, L), (4.3a) (Pw ) − iτ v = g, x ∈ (0, L), (4.3b) − w (L) − iτ v (L) = g(L), (4.3c) ϑ 1 v (0) + ϑ 2 v (0) + ϑ 3 w(0) + ϑ 4 w (0) − iτ v (0) = g(0) (4.3d) satisfies the estimate w H 2 + v H 1 ≤ C( f H 2 + g H 1 ) (4.4) uniformly for all f ࢠ H 2 (0, L), g ࢠ H 1 (0, L) and for all |τ | sufficiently large. Since v and w are directly related via Equation (4.3a), we replace v in Equations (4.3b)-(4.3d) by v = f + iτ w to obtain the following BVP for w: (Pw ) + τ 2 w = g + iτ f , x ∈ (0, L), (4.5a) − w (L) + τ 2 w(L) = (g + iτ f )(L), (4.5b) (ϑ 4 + iτ ϑ 2 ) =:γ 1 w (0) + (ϑ 3 + τ 2 + iτ ϑ 1 ) =:γ 2 w(0) = (g + iτ f )(0) − ϑ 1 f (0) − ϑ 2 f (0). (4.5c) With this, we first show the desired estimate for w. Step 1: Homogeneous boundary conditions To begin with, we shall transform Equation (4.5) into a BVP with homogeneous boundary conditions. To this end, we use Equation (4.5a) to eliminate the terms w(0) and w(L). This yields, after differentiating Equation (4.5a), the following BVP forỹ := Pw : y + τ 2 Pỹ = g + iτ f , x ∈ (0, L), (4.6a) y(L) + P(L)ỹ (L) = 0, (4.6b) γ 1 P(0)ỹ (0) − γ 2 τ 2ỹ (0) = − g(0) τ 2 (ϑ 3 + iτ ϑ 1 ) − iϑ 3 τ f (0) − ϑ 2 f (0) =:R 1 . (4.6c) In order to make the second boundary condition homogeneous, we determine a first order polynomial h(x) = a 1 x + a 0 , such that h(x) satisfies the boundary conditions (4.6b) and (4.6c). The coefficients can be determined uniquely: a 1 = − τ 2 P(0)R 1 γ 1 τ 2 (L + P(L)) + P(0)γ 2 , a 0 = −(L + P(L))a 1 . (4.7) We note that, as already mentioned in the proof of Theorem 3.2, γ 1 /γ 2 = 1/c 0 / ∈ R, and so a 1 is always well defined. For |τ | > 1 we find the estimate 1, (4.8) by using the continuous embedding H k (0, L) →C k − 1 [0, L] in one dimension (cf. Adams, 1975) to estimate the terms occurring in R 1 . Now, the function y :=ỹ − h satisfies the following problem with homogeneous boundary conditions: |a j | ≤ C τ 2 ( g H 1 + |τ | f H 2 ), for j = 0,y + τ 2 P y = H := g + iτ f − τ 2 P h, x ∈ (0, L), (4.9a) y(L) + P(L)y (L) = 0, (4.9b) γ 1 P(0) y(0) − γ 2 τ 2 y (0) = 0. (4.9c) Step 2: Solution estimate Now we determine the solution of Equation (4.9). Let {ϕ 1 , ϕ 2 } be a basis of solutions of the homogeneous equation y + τ 2 P y = 0. Then, the general solution of the inhomogeneous Equation (4.9a) can be obtained by variation of constants: (4.11) where J(x, t) is the Green's function introduced in Lemma B.1 in Appendix 2, and c j ∈ C are arbitrary constants. The derivative y (x) satisfies y(x) = c 1 ϕ 1 (x) + c 2 ϕ 2 (x) + x 0 H(t ) ϕ 1 (t )ϕ 2 (x) − ϕ 2 (t )ϕ 1 (x) ϕ 1 (t )ϕ 2 (t ) − ϕ 1 (t )ϕ 2 (t ) dt (4.10) = c 1 ϕ 1 (x) + c 2 ϕ 2 (x) + x 0 H(t )J(x, t ) dt,y (x) = c 1 ϕ 1 (x) + c 2 ϕ 2 (x) + x 0 H(t )∂ x J(x, t ) dt. (4.12) In order to determine the constants c j we now specify the initial conditions of the solutions ϕ 1 , ϕ 2 : ϕ 1 (0) = 0, ϕ 2 (0) = 1, ϕ 1 (0) = τ, ϕ 2 (0) = 0. These conditions imply that the functions ϕ j are realvalued. From the boundary conditions (4.9b) and (4.9c), we then find c 1 = − L 0 H(t )J(L, t ) dt − P(L) L 0 H(t )∂ x J(L, t ) dt ϕ 1 (L) + P(L)ϕ 1 (L) + γ 2 P(0) γ 1 τ [ϕ 2 (L) + P(L)ϕ 2 (L)] , c 2 = γ 2 P(0) γ 1 τ c 1 . (4.13) Again, since γ 2 /γ 1 / ∈ R and ϕ 1 , ϕ 2 are linearly independent, the coefficients c 1 , c 2 are well defined. Next, we estimate these coefficients. First, we find that lim |τ |→∞ γ 2 P(0) γ 1 τ = − iP(0) ϑ 2 . Therefore, we can find some constant C > 0, independent of |τ | > 1, such that the denominator N of c 1 can be estimated as follows: From the initial conditions of ϕ 1 , ϕ 2 and Lemma B.1 we find that the Wronskian satisfies ϕ 1 (L)ϕ 2 (L) − ϕ 1 (L)ϕ 2 (L) = τ . Since ϕ j L ∞ is uniformly bounded for all τ sufficiently large by Lemma B.2, this implies |ϕ 1 (L)| + |ϕ 2 (L)| ≥ Cτ , for some constant C > 0 independent of τ . With this result, we obtain the estimate |N| ≥ C|τ |, (4.14) for all |τ | > 1, and C independent of τ . Now it remains to estimate the integrals occurring in c 1 and those in Equations (4.11) and (4.12). To this end, we split these integrals according to H = (H − g ) + g . In order to estimate the integrals corresponding to the first term, we apply Theorem B.1 and use the estimates for h found in Equation (4.8). For the other integrals we apply Hölder's inequality, and obtain x 0 g (t )J(x, t ) dt L ∞ ≤ g L 1 J L ∞ ((0,L) 2 ) ≤ C |τ | g H 1 , where we used Lemma B.2 to estimate J L ∞ . The integrals with x J instead of J can be estimated analogously. Altogether we obtain x 0 H(t )J(x, t ) dt L ∞ ≤ C |τ | ( g H 1 + f H 2 ), (4.15) x 0 H(t )∂ x J(x, t ) dt L ∞ ≤ C( g H 1 + f H 2 ),(4.16) for all |τ | > 1, with C > 0 independent of τ . Therefore, we conclude that the estimate |c j | ≤ C |τ | ( g H 1 + f H 2 ) holds uniformly in τ . Applying these results and the estimates for the basis-functions ϕ 1 , ϕ 2 found in Lemma B.2 to Equations (4.11) and (4.12), we find that the following estimates hold uniformly for |τ | > 1: y L 2 ≤ C y L ∞ ≤ C |τ | ( g H 1 + f H 2 ), (4.17) y H 1 ≤ C( y L ∞ + y L ∞ ) ≤ C( g H 1 + f H 2 ). (4.18) Using Equation (4.8), we see that the same estimates hold forỹ. Furthermore, by usingỹ = Pw and the Equation (4.5a) to express w in terms of w and w , we find w H 1 ≤ C |τ | ( g H 1 + f H 2 ),(4.v H 1 ≤ C( g H 1 + f H 2 ), which completes the proof. Conclusions In Thull et al. (2006), a backstepping-based controller was proposed for the infinite-dimensional model of a gantry crane with heavy chain and payload. This controller shows excellent results, which was also verified experimentally by Thull et al. (2006). In particular the control law was designed in a systematic way, it features to be robust with respect to unmodelled (stick-slip) friction effects, which are always present in real applications, and it can be easily tuned. Though energy dissipation of the closed-loop system could be shown by Thull et al. (2006), the proof of the closed-loop stability did not work out, which was also correctly pointed out by Grabowski (2009). In this paper, a rigorous proof of the asymptotic and exponential stability of the closed-loop system is given. For this, it was necessary to formulate the dynamics of the closed-loop system as an abstract evolution equation in an appropriate Hilbert space which differs from the space H 1 × L 2 which is usually used in the context of heavy chain systems. Moreover, under very mild conditions on the controller parameters, which were explicitly derived, it was proven that the solutions of the closed-loop system are described by an asymptotically stable C 0 -semi-group of contractions. Finally, by employing Huang's theorem it was even possible to show that under the same conditions the backsteppingbased boundary controller renders the closed-loop system exponentially stable. Notes 1. Note that here only a single chain is considered, unlike the pair of parallel chains as used by Thull et al. (2006). This change corresponds to the substitution ρ → ρ/2. 2. Note that in this paper the virtual spring force f s ( · ) in Thull et al. (2006) is considered linear. 3. Here the equilibrium of the cart position is set to zero, but can, of course, take any other value in the operating range. (B.1). Then the Green's function of the equation is given by J(x, t ) := ϕ 1 (x)ϕ 2 (t ) − ϕ 2 (x)ϕ 1 (t ) ϕ 1 (t )ϕ 2 (t ) − ϕ 1 (t )ϕ 2 (t ) . (B.2) Furthermore, the Wronskian W (t ) := ϕ 1 (t )ϕ 2 (t ) − ϕ 1 (t )ϕ 2 (t ) is constant for t ࢠ [0, L]. Hence, Equation (B.2) simplifies to J(x, t) = C[ϕ 1 (x)ϕ 2 (t) − ϕ 2 (x)ϕ 1 (t)]. With the prescribed initial data ϕ(0) and ϕ (0), we shall denote the unique classical solution of Equation (B.1) by ϕ τ . The behaviour of solutions of Equation (B.1) is stated in the following lemma. For the proof, see Proposition 2.1 in Arnold et al. (2011). Lemma B.2: There exists a constant C > 0 such that for any family of solutions {ϕ τ } τ > 1 of Equation (B.1) the following estimates hold uniformly for τ > 1: ϕ τ L ∞ ≤ C τ τ |ϕ τ (0)| + |ϕ τ (0)| , ϕ τ L ∞ ≤ C τ |ϕ τ (0)| + |ϕ τ (0)| . Now we are able to prove the following theorem: Theorem B.1: Let {J τ } τ > 1 be the family of Green's functions defined in Lemma B.1. Then, there exists a constant C > 0 such that the following estimates hold uniformly for all f ࢠ H 1 (0, L) and τ > 1: with denoting here derivatives with respect to ξ . ψ x takes the initial values ψ x (ξ = 0) = 0 and ψ x (ξ = 0) = 1. Now, integrating by parts yields x 0 f (t )J τ (x, t ) dt L ∞ ≤ C τ 2 f H 1 , (B.3) x 0 f (t )∂ x J τ (x, t ) dt L ∞ ≤ C τ f Hx 0 f (x − ξ )ψ x (ξ ) dξ = − x 0 ∂ ξ [( f P)(x − ξ )] ξ 0 ψ x (ζ ) P(x − ζ ) dζ dξ + f (0)P(0) x 0 ψ x (ζ ) P(x − ζ ) dζ ≤ 2 ψ x L ∞ τ 2 x 0 |∂ ξ ( f P)(x − ξ )| dξ + | f (0)P(0)| ≤ C ψ x L ∞ f H 1 τ 2 , where we used Equation (B.5) in the second step. And in the last step we used the continuous embedding H 1 (0, L) →C[0, L]. From Lemma B.2 and the known initial conditions of ψ x we find that ψ x L ∞ is uniformly bounded for all τ > 1. Finally, we notice that ψ x (x − t) = J(x, t), so the above estimate also holds for J instead of ψ x , which proves Equation (B.3). Using the boundary conditions in D(A) to evaluate the term Pv (Pw ) | L 0 , we find that the real parts of all terms at x = L cancel against the real part of the third term of Figure  . Gantry crane with heavy chain and payload: schematics (left) and representation of the internal force F i (right). (ζ ) dζ dξ satisfies 2 y n (0) = a for all n ∈ N, and y n H 2 → 0.This shows that, for the sequence {w n }, the values 2 w n (0), 2 w n (L) can be modified such that the modified sequence {z n } ⊂ D(A), but stillz n → z 0 in H.Lemma 2.3: Under the condition ϑ 3 ࣔ 0, the operator A is injective and ran A = H, i.e. A −1 exists and D(A −1 ) = H. Proof: We prove this lemma by showing that the equation Az = (f, g, g(L), g Theorem 4. 1 ( 1Huang): Let {S(t)} t ࣙ 0 be a uniformly bounded C 0 -semi-group in a Hilbert space, and let A be its generator. Then, {S(t)} t ࣙ 0 is exponentially stable if and only if iR ⊂ ρ(A) and sup τ ∈R R(iτ, A) < ∞. |N| 2 : 2= ϕ 1 (L) + P(L)ϕ 1 (L) + γ2P(0) γ1τ [ϕ 2 (L) + P(L)ϕ 2 (L)] 2 ≥ C |ϕ 1 (L) + P(L)ϕ 1 (L)| 2 + |ϕ 2 (L) + P(L)ϕ 2 (L)| 2 . Proof: We are going to show Equation (B.3), the proof of Equation (B.4) can be done analogously. The index τ is omitted for the sake of simplicity. First, we make the substitution t = x − ξ in the left hand integral, and define the family of functions ψ x : ξ →J(x, x − ξ ) with parameter x. These functions are solutions of the equation ψ x + τ 2 P(x − ξ ) ψ x = 0, (B.5) Appendix 3 . 0 (= Re α 1 γ L 0 Pv (Pw ) dx + α 1 L 0 30010Deferred proofsProof of Lemma 2.5: For allz ࢠ D(A) we have Re z, Az H = Re α 1 γ L Pw ) (Pv ) dx + α 1 L 0 Pw v dx + α 1 γ P(L)w (L)v (L) + α 2 w(0)v (0) + α 1 γ L 0 Pv (Pw ) dx + α 1 L 0 v (Pw ) dx − α 1 P(L)v (L)w (L) + α 2 γ v (0)F + 1 2 v (0) − 2α 1 P(0)w (0) + 2α 2 w(0) × F − 2α 1 P(0)v (0) + 2α 2v (0) Pvw dx + α 1 γ P(L)w (L)v (L) + α 2 w(0)v (0) − α 1 P(L)v (L)w (L) + α 2 γ v (0)F + 1 2 v (0) − 2α 1 P(0)w (0) + 2α 2 w(0)× F − 2α 1 P(0)v (0) + 2α 2v (0) . (C.1) Finally, from Equation (4.3a) and by using Equation (4.19) we get the desired estimate19) w H 2 ≤ C( g H 1 + f H 2 ). (4.20) AcknowledgmentsThe authors acknowledge the TU Wien University Library for financial support through its Open Access Funding Program. The first author was supported by the FWF-project I395-N16 and the FWF doctoral school 'Dissipation and dispersion in nonlinear partial differential equations' with grant number W 1245. The second and the third authors were partially supported by the Doctoral School 'Partial differential equations in technical systems: modelling, simulation, and control' of Technische Universität Wien. The second author acknowledges a sponsorship by Clear Sky Ventures. We are grateful to the anonymous referee who drew our attention to the references(Conrad and Mifdal, 1998;Mifdal, 1997b).Disclosure statementNo potential conflict of interest was reported by the authors.Appendix 1. Useful inequalitiesLemma A.1: Let a 0 , b 0 , ε 0 > 0 be given. Then there exist positive constants c, d such thatholds uniformly for all x 1 , x 2 ∈ R and |a| ࣘ a 0 , b ࣙ b 0 and ε ࣙ ε 0 .Proof: Inequality (A.1) can be rewritten in the equivalent form:where the occurring matrix will be denoted as M. Since this inequality has to hold for all x 1 , x 2 ∈ R, it is equivalent to M being positive semi-definite. Applying the Sylvester criterion yields the following conditions:If a = 0, we can take c = ε 0 and d = b 2 . Otherwise, we see from the conditions (A.2) and (A.3) that d < b 2 , so that Equation (A.3) can be written asBecause of the monotonicity of the right-hand side we find the estimate:So, for Equation (A.4) to hold, it is sufficient that c, d satisfy the stricter inequality:For d sufficiently small, the right-hand side becomes positive, and therefore a c > 0 satisfying Equation (A.5) exists. Lemma A.2: Let α, β, δ ∈ R andand only if the coefficients satisfy the conditions:Proof:The polynomial can be written aswith M denoting the 3 × 3 matrix. Now the property P 3 (x 1 , x 2 , x 3 ) ≥ 0, ∀x 1 , x 2 , x 3 ∈ R is equivalent to M being positive semi-definite. Applying the Sylvester criterion to M yields the desired conditions.Appendix 2. ODEs with a parameter: Uniform estimatesIn this section, we discuss the behaviour of classical solutions y ࢠ C 2 [0, L] to the equation:where τ ∈ R and P ࢠ C 1 [0, L] is a real-valued function satisfying P 0 ࣘ P(x) ࣘ P 1 uniformly for x ࢠ [0, L] for some positive constants P 0 , P 1 . Since τ only occurs squared, we can assume that τ ࣙ 0 holds in the following.Lemma B.1(Birkhoff & Rota, 1962): Let (ϕ 1 , ϕ 2 ) be an arbitrary pair of linearly independent solutions of EquationBy introducing the functional J: w → − 2α 1 P(0)w (0) + 2α 2 w(0), we simplify the expression:Assuming the relations(Re z, Az H = 1 2 P 3 (Re y 1 , Re y 2 , Re y 3 )where P 3 is the polynomial defined byHence, A is dissipative ifAccording to Lemma A.2 this inequality is satisfied if there holdsSince γ > 0 has not yet been specified, we can choose γ arbitrarily small, so that the above conditions reduce to the single condition:(a + b − 1) 2 4ab < 1. (C.6) So, the relation (2.8) on the ϑ i together with the condition (C.6) on the a, b > 0 is sufficient for the dissipativity of A in H with respect to the inner product (2.6), with the choice (2.22) for α 1 and α 2 , and γ > 0 sufficiently small. R A Adams, Sobolev spaces. Pure and applied mathematics. New York, NYAcademic Press65Adams, R. A. (1975). Sobolev spaces. Pure and applied mathe- matics (Vol. 65). New York, NY: Academic Press. WKBbased schemes for the oscillatory 1D Schrödinger equation in the semiclassical limit. A Arnold, N Ben Abdallah, C Negulescu, SIAM Journal on Numerical Analysis. 494Arnold, A., Ben Abdallah, N., & Negulescu, C. (2011). WKB- based schemes for the oscillatory 1D Schrödinger equation in the semiclassical limit. SIAM Journal on Numerical Anal- ysis, 49(4), 1436-1460. Ordinary differential equations. Introductions to higher mathematics. G Birkhoff, G.-C Rota, Ginn and CompanyBostonBirkhoff, G., & Rota, G.-C. (1962). Ordinary differential equa- tions. Introductions to higher mathematics. Boston: Ginn and Company. Nonlinear feedback stabilization of a rotating body-beam without damping. ESAIM: Control, Optimisation and Calculus of Variations. B Chentouf, J.-F Couchouron, 4Chentouf, B., & Couchouron, J.-F. (1999). Nonlinear feedback stabilization of a rotating body-beam without damping. ESAIM: Control, Optimisation and Calculus of Variations, 4, 515-535. Strong stability of a model of an overhead crane. F Conrad, A Mifdal, Control and Cybernetics. 373Conrad, F., & Mifdal, A. (1998). Strong stability of a model of an overhead crane. Control and Cybernetics, 37(3), 363-374. On the stabilization of a flexible beam with a tip mass. F Conrad, Ö Morgül, SIAM Journal on Control and Optimization. 366Conrad, F., & Morgül, Ö. (1998). On the stabilization of a flexi- ble beam with a tip mass. SIAM Journal on Control and Opti- mization, 36(6), 1962-1986. Stabilization of a rotating body beam without damping. J.-M Coron, B Novel, IEEE Transactions on Automatic Control. 435Coron, J.-M., & d' Andrea Novel, B. (1998). Stabilization of a rotating body beam without damping. IEEE Transactions on Automatic Control, 43(5), 608-618. B Boustany, F Conrad, F , Control of an overhead crane: Stabilization of flexibilities. BerlinSpringer178d' Andréa Novel, B., Boustany, F., & Conrad, F. (1992). Control of an overhead crane: Stabilization of flexibilities (Vol. 178, pp. 1-26). Berlin: Springer. Exponential stabilization of an overhead crane with flexible cable via a backstepping approach. B Coron, J M , Automatica. 364d' Andréa Novel, B., & Coron, J. M. (2000). Exponential stabi- lization of an overhead crane with flexible cable via a back- stepping approach. Automatica, 36(4), 587-593. The motion planning problem and exponential stabilization of a heavy chain. P Grabowski, II. Opuscula Mathematica. 284Grabowski, P. (2008). The motion planning problem and expo- nential stabilization of a heavy chain. II. Opuscula Mathe- matica, 28(4), 481-505. The motion planning problem and exponential stabilisation of a heavy chain. I. International Journal of Control. P Grabowski, 82Grabowski, P. (2009). The motion planning problem and expo- nential stabilisation of a heavy chain. I. International Jour- nal of Control, 82(8), 1539-1563. Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. F L Huang, Annals of Differential Equations. 11Huang, F. L. (1985). Characteristic conditions for exponen- tial stability of linear dynamical systems in Hilbert spaces. Annals of Differential Equations, 1(1), 43-56. Perturbation theory for linear operators. T Kato, Die Grundlehren der mathematischen Wissenschaften. 132SpringerKato, T. (1966). Perturbation theory for linear operators. Die Grundlehren der mathematischen Wissenschaften (Vol. 132). New York, NY: Springer. Nonlinear and adaptive control design. M Krstić, I Kanellakopoulos, P Kokotović, John Wiley & SonsNew York; NYKrstić, M., Kanellakopoulos, I., & Kokotović, P. (1995). Nonlin- ear and adaptive control design. New York, NY: John Wiley & Sons. Infinite-dimensional decoupling control of the tip position and the tip angle of a composite piezoelectric beam with tip mass. A Kugi, D Thull, 322Kugi, A., & Thull, D. (2005). Infinite-dimensional decoupling control of the tip position and the tip angle of a composite piezoelectric beam with tip mass (Vol. 322, pp. 351-368). . Berlin Heidelberg, SpringerBerlin Heidelberg: Springer. Stability and stabilization of infinite dimensional systems with applications, Communications and control engineering series. Z.-H Luo, B.-Z Guo, Ö Morgül, SpringerLondonLuo, Z.-H., Guo, B.-Z., & Morgül, Ö. (1999). Stability and sta- bilization of infinite dimensional systems with applications, Communications and control engineering series. London: Springer. Etude de la stabilisation forte et uniforme de système hybride -application à un modèle de pont roulant (doctoral dissertation). A Mifdal, University Nancy IMifdal, A. (1997a). Etude de la stabilisation forte et uniforme de système hybride -application à un modèle de pont roulant (doctoral dissertation). University Nancy I. Stabilisation uniforme d'un système hybride. A Mifdal, Uniform stabilization of a hybrid systemMifdal, A. (1997b). Stabilisation uniforme d'un système hybride [Uniform stabilization of a hybrid system]. . Comptes Rendus de l'Académie des Sciences. 324Comptes Rendus de l'Académie des Sciences, 324, 37-42. An Euler-Bernoulli beam with nonlinear damping and a nonlinear spring at the tip. M Miletić, D Stürzer, A Arnold, Discrete and Continuous Dynamical Systems -Series B. 209Miletić, M., Stürzer, D., & Arnold, A. (2015). An Euler- Bernoulli beam with nonlinear damping and a nonlinear spring at the tip. Discrete and Continuous Dynamical Sys- tems -Series B, 20(9). Stability of an Euler-Bernoulli beam with a nonlinear dynamic feedback system. M Miletić, D Stürzer, A Arnold, A Kugi, IEEE Transactions on Automatic Control. 10Miletić, M., Stürzer, D., Arnold, A., & Kugi, A. (2016). Stabil- ity of an Euler-Bernoulli beam with a nonlinear dynamic feedback system. IEEE Transactions on Automatic Control, 61(10), 2782-2795. Stabilization and disturbance rejection for the beam equation. Ö Morgül, IEEE Transactions on Automatic Control. 4612Morgül, Ö. (2001). Stabilization and disturbance rejection for the beam equation. IEEE Transactions on Automatic Con- trol, 46(12), 1913-1918. On the stabilization of a cable with a tip mass. Ö Morgül, B P Rao, F Conrad, IEEE Transactions on Automatic Control. 3910Morgül, Ö., Rao, B. P., & Conrad, F. (1994). On the stabilization of a cable with a tip mass. IEEE Transactions on Automatic Control, 39(10), 2140-2145. Semigroups of linear operators and applications to partial differential equations. A Pazy, Applied mathematical sciences. 44SpringerPazy, A. (1983). Semigroups of linear operators and applications to partial differential equations. Applied mathematical sci- ences ( Vol. 44). New York, NY: Springer. Flatness of heavy chain systems. N Petit, P Rouchon, SIAM Journal on Control and Optimization. 402Petit, N., & Rouchon, P. (2001). Flatness of heavy chain systems. SIAM Journal on Control and Optimization, 40(2), 475-495. Infinit-dimensionale Regelung eines Brückenkranes mit schweren Ketten [Infinite-dimensional control of a gantry crane with heavy chains. D Thull, D Wild, A Kugi, 53at-AutomatisierungstechnikThull, D., Wild, D., & Kugi, A. (2005). Infinit-dimensionale Regelung eines Brückenkranes mit schweren Ketten [Infinite-dimensional control of a gantry crane with heavy chains]. at-Automatisierungstechnik, 53(8), 400-410. Application of a combined flatness-and passivity-based control concept to a crane with heavy chains and payload. D Thull, D Wild, A Kugi, Proceedings of the IEEE Conference on Control Application. the IEEE Conference on Control ApplicationThull, D., Wild, D., & Kugi, A. (2006). Application of a com- bined flatness-and passivity-based control concept to a crane with heavy chains and payload. In Proceedings of the IEEE Conference on Control Application, (pp. 656-661). . 10.1109/CACSD-CCA-ISIC.2006.4776723IEEEGermanyOctober 4-6, 2006, Germany: IEEE. doi: 10.1109/CACSD- CCA-ISIC.2006.4776723 K Yosida, Grundlehren der Mathematischen Wissenschaften. BerlinSpringer123Functional analysisYosida, K. (1980). Functional analysis (6th ed.). Grundlehren der Mathematischen Wissenschaften (Vol. 123). Berlin: Springer.
{'fraction_non_alphanumeric': 0.09452030501862033, 'fraction_numerical': 0.043348899529073316, 'mean_word_length': 3.4597539543058, 'pattern_counts': {'":': 0, '<': 6, '<?xml version=': 0, '>': 40, 'https://': 1, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 53, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': "In this paper, we analyse a systematically designed and easily tunable backstepping-based boundary control concept for a gantry crane with heavy chain and payload. The corresponding closed-loop system is formulated as an abstract evolution equation in an appropriate Hilbert space. Non-restrictive conditions for the controller coefficients are derived, under which the solutions are described by a C 0semi-group of contractions, and are asymptotically stable. Moreover, by applying Huang's theorem we can finally even show that under these conditions the controller renders the closed-loop system exponentially stable.", 'arxivid': '1607.07052', 'author': ['Dominik Stürzer \nInstitute for Analysis and Scientific Computing\nTechnische Universität Wien\nViennaAustria\n', 'Anton Arnold \nInstitute for Analysis and Scientific Computing\nTechnische Universität Wien\nViennaAustria\n', 'Andreas Kugi \nAutomation and Control Institute\nTechnische Universität Wien\nViennaAustria ARTICLE HISTORY\n'], 'authoraffiliation': ['Institute for Analysis and Scientific Computing\nTechnische Universität Wien\nViennaAustria', 'Institute for Analysis and Scientific Computing\nTechnische Universität Wien\nViennaAustria', 'Automation and Control Institute\nTechnische Universität Wien\nViennaAustria ARTICLE HISTORY'], 'corpusid': 205422410, 'doi': '10.1080/00207179.2017.1335439', 'github_urls': [], 'n_tokens_mistral': 19098, 'n_tokens_neox': 16654, 'n_words': 9591, 'pdfsha': 'bf89d6f0beb64c87222c634774dc99020236afeb', 'pdfurls': None, 'title': ['Closed-loop stability analysis of a gantry crane with heavy chain and payload', 'Closed-loop stability analysis of a gantry crane with heavy chain and payload'], 'venue': []}
arxiv
Comparisons of the core and mantle compositions of Earth analogs from different terrestrial planet formation scenarios Jesse T Gu Department of Earth and Planetary Sciences Harvard University CambridgeMAUSA Rebecca A Fischer Department of Earth and Planetary Sciences Harvard University CambridgeMAUSA Matthew C Brennan Department of Earth and Planetary Sciences Harvard University CambridgeMAUSA Matthew S Clement Earth and Planets Laboratory Carnegie Institution for Science Washington D.CUSA Seth A Jacobson Department of Earth and Environmental Sciences Michigan State University East LansingMIUSA Nathan A Kaib Department of Physics and Astronomy HL Dodge University of Oklahoma NormanOKUSA David P O&apos;brien Planetary Science Institute TucsonArizonaUSA Sean N Raymond Laboratoire d'Astrophysique de Bordeaux Université de Bordeaux CNRS BordeauxFrance Comparisons of the core and mantle compositions of Earth analogs from different terrestrial planet formation scenarios 1 Revision 2 The chemical compositions of Earth's core and mantle provide insight into the processes that led to their formation. N-body simulations, on the other hand, generally do not contain chemical information, and seek to only reproduce the masses and orbits of the terrestrial planets.These simulations can be grouped into four potentially viable scenarios of Solar System formation (Classical, Annulus, Grand Tack, and Early Instability) for which we compile a total of 433 N-body simulations. We relate the outputs of these simulations to the chemistry of Earth's core and mantle using a melt-scaling law combined with a multi-stage model of core formation.We find the compositions of Earth analogs to be largely governed by the fraction of equilibrating embryo cores (kcore_emb) and the initial embryo masses in N-body simulations, rather than the simulation type, where higher values of kcore_emb and larger initial embryo masses correspond to higher concentrations of Ni, Co, Mo, and W in Earth analog mantles and higher concentrations of Si and O in Earth analog cores. As a result, larger initial embryo masses require smaller values of kcore_emb to match Earth's mantle composition. On the other hand, compositions of Earth analog cores are sensitive to the temperatures of equilibration and fO2 of accreting material. Simulation type may be important when considering magma ocean lifetimes, where Grand Tack simulations have the largest amounts of material accreted after the last giant impact. However, we cannot rule out any accretion scenarios or initial embryo masses due to the sensitivity of Earth's mantle composition to different parameters and the stochastic nature of N-body simulations. We use our compiled simulations to explore the relationship between initial embryo masses and the melting history of Earth analogs, where the complex interplay between the timing between impacts, magma ocean lifetimes, and volatile delivery could affect the compositions of Earth analogs formed from different simulation types. Comparing the last embryo impacts experienced by Earth analogs to specific Moon-forming scenarios, we find the characteristics of the Moon-forming impact are dependent on the initial conditions in N-body simulations where larger initial embryo masses promote larger and slower Moon-forming impactors. Mars-sized initial embryos are most consistent with the canonical hit-and-run scenario onto a solid mantle. Our results suggest that constraining the fraction of equilibrating impactor core (kcore) and the initial embryo masses in N-body simulations could be significant for understanding both Earth's accretion history and characteristics of the Moon-forming impact. Introduction The terrestrial planets formed in several stages on timescales on the order of 10-100 Myr (Righter and O'Brien, 2011). First, nebular gas condensed into dust, which accreted into planetesimals likely due to gravitational collapse triggered by the streaming instability (as reviewed in Johansen et al., 2014). This was followed by runaway growth, where the largest protoplanetary bodies grew the fastest either due to planetesimal accretion by pairwise growth (Kokubo and Ida, 1996) or pebble accretion (Lambrechts and Johansen, 2012;Levison et al., 2015). When the local surface density of planetesimals was consumed and/or the pebble flux has decreased, runaway growth transformed into oligarchic growth, creating a bimodal distribution of massive embryos and smaller planetesimals (Kokubo and Ida, 2000;Lambrechts et al., 2019). Finally, massive collisions between embryos (i.e., giant impacts) resulted in the present-day terrestrial planets (Chambers and Wetherill, 1998). N-body simulations of late-stage terrestrial planet formation seek to reproduce the orbits and masses of the terrestrial planets by calculating the gravitational interactions between a prescribed set of bodies that are representative of the final giant impact stage of terrestrial planet formation (Chambers, 2001). Classical simulations can be split into two types-Eccentric Jupiter and Saturn (EJS) simulations begin with Jupiter and Saturn in their modern orbits in eccentric orbits while Circular Jupiter and Saturn (CJS) simulations begin with Jupiter and Saturn closer than they are now, but in circular orbits (Fischer and Ciesla, 2014;O'Brien et al., 2006;Raymond et al., 2009;Woo et al., 2022). CJS simulations generally produce Mars analogs that are too large, whereas EJS simulations do slightly better in this regard but are self-inconsistent (see Raymond et al., 2009 for details). A number of other models have been proposed to address the small-Mars problem. Here, we focus on three wellstudied scenarios which deplete mass in the Mars-forming region: 1) the "annulus" (or "lowmass asteroid belt") scenario, in which bodies are initially distributed in a narrow ring (Hansen, 2009;Kaib and Cowan, 2015;Raymond and Izidoro, 2017), 2) the Grand Tack scenario, in which Jupiter and Saturn migrate inward and then outward to their present locations (Walsh et al., 2011;Jacobson and Morbidelli, 2014;O'Brien et al., 2014), and 3) the Early Instability scenario, in which the gas giants undergo an orbital instability before Mars can grow (Clement et al., 2018(Clement et al., , 2021Liu et al., 2022). Earth analogs formed in each type of simulation are influenced by the dynamic evolution of the initial distribution of embryos and planetesimals. Different types of N-body simulations, which differ in terms of the growth rates and provenance of accreting bodies, will therefore form Earth analogs with unique accretion histories that accrete materials from distinct locations. In addition, different initial conditions prescribed between types of simulations, and even between simulations from the same suite, may result in different sizes and energies of accretionary impacts. Even though the probability of reproducing each terrestrial planet differs between simulation types, all simulation types have been shown to be plausibly capable of explaining the observed inner Solar System configuration (Raymond and Morbidelli, 2022). Current state-of-the-art models of Earth's accretion and core formation integrate growth histories from N-body simulations with self-consistent evolution of oxygen fugacity (fO2) to determine the partitioning of elements between metal and silicate following each impact (Fischer et al., 2017;Rubie et al., 2015). Experimental metal-silicate partitioning data are parameterized to capture the effects of pressure, temperature, composition, and fO2 on the concentrations of each element in the resulting core and mantle (e.g., Fischer et al., 2015;Siebert et al., 2012). The aim of these models has been to reproduce Earth's observed mantle composition (McDonough and Sun, 1995;Palme and O'Neill, 2013) in terms of a set of major and minor elements, such as Mg, Ca, Al, Fe, Ni, Co, Nb, and Ta (Fischer et al., 2017;Rubie et al., 2011). However, previous studies have only investigated core formation under a single suite of N-body simulations, using Classical or Grand Tack simulations from a single study, which prevents comparison of different simulation types and initial conditions (Fischer et al., 2017;Rubie et al., 2015). Moreover, these studies used simplified metal-silicate equilibration parameters, where equilibration pressures (Pequil) were assumed to increase linearly such that they represented equilibration at a constant fraction of the growing core-mantle boundary pressure. While embryo masses generally increase as Earth accretes due to ongoing oligarchic growth elsewhere in the Solar System, the stochastic nature of N-body simulations means that the size and timing of impacts vary greatly between simulations. Comparing simulations from multiple studies can therefore constrain the range of possible accretional events Earth could have experienced during its growth. Other equilibration parameters, such as the fractions of terrestrial mantle and impactor core (kcore) that participate in metal-silicate equilibration, were set as a constant multiple of the impactor's mass and at a constant fraction, respectively. Simplifying these parameters ignores the amount of energy delivered to the proto-Earth by individual accretionary impacts and the volume of melting produced by each event (Abramov et al., 2012;Nakajima et al., 2021). In addition, it has been suggested that kcore varies with impactor size, where larger impactor cores merge more efficiently with Earth's core and therefore experience lower degrees of equilibration (Marchi et al., 2018;Rubie et al., 2015). de Vries et al. (2016) used the melt-scaling law of Abramov et al. (2012) to relate the energies of individual impacts to Pequil, assuming Pequil occurred at the base of impact induced melt pools, over the course of accretion in Grand Tack simulations. They found Pequil to depend on initial conditions of N-body simulations and the lifetime of magma oceans. However, the relationship between Pequil and the resulting mantle compositions of Earth analogs remains unclear due to the absence of a core formation model. Furthermore, they only used Grand Tack simulations, yet it is possible that accretion histories of Earth analogs differ between different scenarios of Solar System formation. Here we build upon the study of de Vries et al. (2016) by compiling simulations from four different scenarios of Solar System formation and using a meltscaling law based on hydrodynamic simulations to determine Pequil and the fraction of the proto-Earth's mantle melted by each impact (fmelt). Our results are integrated with state-of-the-art models of core formation (Fischer et al., 2017;Rubie et al., 2015) to explore the effects of varying simulation type and initial conditions within these different simulation types on the chemistry of Earth analogs formed. The compiled simulations and the results from our model are then used to compare the relationship between different N-body simulations, the initial conditions used within them, and volatile delivery and melting histories of Earth analogs. Methods N-body simulations N-body simulations from different Solar System formation scenarios were compiled to use as inputs in our core formation model. A total of 48 Classical (CEJS) (O'Brien et al., 2006;Raymond et al., 2009), 110 Annulus (ANN) (Kaib and Cowan, 2015;Raymond and Izidoro, 2017), 142 Grand Tack (GT) (Jacobson and Morbidelli, 2014;O'Brien et al., 2014;Walsh et al., 2011), and 133 Early Instability (EI) (Clement et al., 2021) simulations were assembled to give a total of 433 simulations. Abbreviations of individual simulation names are given in Table 1 and will be used hereafter. Rows that contain simulation names in parentheses depict all simulations from a given study (e.g., CEJS-O06). Classical simulations from both studies were also split into CJS and EJS simulations to compare the two dynamical scenarios. GT simulations were split by both initial embryo to planetesimal mass ratio (e.g., GT 1:1) and initial embryo mass (e.g., GT-0.025) to compare different initial conditions. Earth analogs were defined as the body at the end of each simulation closest to 1 Earth mass and 1 AU within the ranges 0.8-1.25 Earth masses and 0.8-1.2 AU for a total of 109 Earth analogs. Such narrow ranges of masses and orbital radii were chosen to minimize the effects of planet mass and orbital radii of accreting material on Earth analog compositions (Fischer et al., 2017;Kaib and Cowan, 2015). We do not use the mass and orbit of Mars analogs as constraints because Mars' accretion history is unrelated to its final orbital parameters (Brennan et al., 2022). Table 1 also lists the initial conditions and number of Earth analogs produced from each suite of simulations. Initial embryo masses range from 0.005-0.48 Earth masses, with both the distributions and ranges of initial embryo masses varying between studies (Table 1). Most simulations used a bimodal distribution of larger embryos and smaller planetesimals, except for the ANN simulations from Kaib and Cowan (2015), in which simulations were run with equal-massed embryos only. For each simulation that formed an Earth analog, the impact histories of bodies that eventually accreted to form the Earth analog were tracked. All bodies were assigned an initial composition based on relative non-volatile elemental abundances in CI chondrite. The abundances of refractory elements were enriched, and each body was equilibrated at a given fO2 depending on its initial semi-major axis . The initial fO2 distribution followed a simple step-function with more reduced materials within the fO2 step, as in previous studies (Fischer et al., 2017;Rubie et al., 2015). The fO2 step was set to 2 AU with the outer fO2 set to 1.5 log units below the iron-wüstite buffer (∆IW-1.5), consistent with Mars' accretion (Brennan et al., 2022), while the inner fO2 was varied. Planetesimals were equilibrated at 0.1 GPa and 2000 K. To address differing initial embryo masses between simulation suites, we used a simple shell model to determine the initial pressure of equilibration in embryos. Here, we assumed a core mass fraction of 0.3, corresponding to equilibration of a CI chondrite composition without volatile elements at an fO2 of ~∆IW -3. Even though the core mass fraction is dependent on the bulk composition and fO2 of a differentiating body, we tested a case where embryos from beyond the fO2 cutoff were equilibrated at a higher Pequil, corresponding to a smaller core mass fraction, resulting in differences in mantle composition, on average, of <1%. The total mass, density, and pressure of mantle layers were calculated from outside in until mass in the silicate reached 70% of the embryo's mass, resulting in a core-mantle boundary pressure. Pequil was then set to be half of the core-mantle boundary pressure, which is consistent with the interpreted Pequil on Mars (Brennan et al., 2020;Rai and van Westrenen, 2013;Righter and Chabot, 2011), since Mars may be a stranded embryo itself (Dauphas and Pourmand, 2011). The resulting parameterization for Pequil of embryos was Pequil = 112.06*Memb + 0.37 GPa, where Memb was the mass of the embryo in Earth masses. Metal-silicate equilibration For impactors >0.01 Earth masses (embryos), the impactor and target masses, impact velocity, and impact angle of each collision were taken from the outputs of each simulation and were used as inputs in the melt-scaling law of Nakajima et al. (2021) (Fig. 1). The melt-scaling law parameterizes outputs from hydrodynamic simulations to determine the volume and geometry of melting in the target's mantle, along with the pressure at the base of the melt pool. Impact angles were rounded to the nearest angle for which results are available (0°, 30°, 60°, or 90°) (e.g., for a 44° impact angle, results for 30° were used). We note impact angles in all simulations show a uniform distribution centered around 45 degrees (Supplementary Fig. 1) and that rounding in this way could result in additional uncertainties in the determined melt fraction. The surface entropy was set to 1100 J/K/kg, corresponding to surface temperatures of ~300 K. Assuming metal-silicate equilibration occurred at the base of the melt pool, Pequil was set to the pressure at the base of the melt pool and fmelt was set to the fraction of the mantle that was melted. We introduce a parameter, kmantle_melt, to describe the fraction of the melted mantle that equilibrates with the impactor's core. A schematic representation of embryo equilibration is shown in Fig. 1. For the sake of simplicity, we assumed instantaneous crystallization of magma oceans, but discuss the possible effects of longer magma ocean lifetimes in Section 4.1. Following each impact, the mantle was homogenized such that portions of the mantle that did not melt were assumed to fully mix with the melted portions. Embryo cores were also assumed to sink and merge with the proto-Earth's existing core before the next embryo impact and remained isolated from further equilibration. Even though the actual physics of metal-silicate mixing and equilibration during Earth's accretion are more complex (Deguen et al., 2014(Deguen et al., , 2011Landeau et al., 2021), these simplifying assumptions allow us to relate impact energies to the conditions of core formation. Smaller bodies <0.01 Earth masses (planetesimals) were below the threshold of masses compatible with the melt-scaling law of Nakajima et al. (2021) and were small enough that they may have been stranded in the proto-Earth's mantle following an impact (de Vries et al., 2016). The time it takes for a planetesimal's core to sink through the solid mantle and merge with Earth's core exceeds the time between embryo impacts such that planetesimals were assumed to equilibrate with the subsequent embryo impact (Fleck et al., 2018). Planetesimals that accreted after the last embryo impact were equilibrated at an assigned low pressure (Pequil_ptsml) and targetto-impactor ratio of equilibrating silicate (Mmelt_ptsml) before the planetesimal's core merged with Earth's core and the equilibrated silicate mixed with Earth's mantle. Some embryos in CEJS simulations from Raymond et al. (2009) and ANN simulations from Kaib and Cowan (2015) were small enough that they were considered planetesimals during core formation (Table 1). Embryos in all other simulations were large enough that they were also considered embryos by the melt-scaling law. All embryos that eventually formed Earth analogs from CEJS simulations from Raymond et al. (2009) were >0.01 Earth masses, making this distinction only relevant for embryos from ANN simulations from Kaib and Cowan (2015). However, this discrepancy is only significant for small impactors that accrete after the last large impact, which our model is not very sensitive to (Table 2). We note that "planetesimals" in the context of core formation are all bodies <0.01 Earth masses, whereas it only apply to bodies <0.0025 Earth masses in the context of N-body simulations. In contrast to Pequil and fmelt, kcore cannot be easily constrained from melt-scaling laws, especially for large embryo impacts. Instead, we assume constant reference values for kcore, but used different values for embryos (kcore_emb = 0.3) and planetesimals (kcore_ptsml = 0.7) to reflect the dependence of kcore on the size of the impactor (Kendall and Melosh, 2016;Marchi et al., 2018). We note that kcore_ptsml is not set to 1 because it is possible that planetesimal cores remain in the proto-Earth's mantle and equilibrate during the next large impact. In this scenario, it is difficult to constrain the extent to which the original small impactor's core equilibrates. A compilation of equilibration parameters used, the ranges we tested, and the sensitivity of core and mantle compositions of Earth analogs are given in Table 2. We followed the methodology detailed in Supplementary S2 of Rubie et al. (2011), as revised by Fischer et al. (2017) and Brennan et al. (2020) to evolve fO2 self-consistently and calculate the composition of Earth's core and mantle following each equilibration event. The entire impactor mantle, along with a portion of the melted fraction of Earth's mantle (fmelt*kmantle_melt), participated in equilibration, where the fraction of the impactor's core that equilibrated was defined by kcore. In addition, material from planetesimals that impacted since the last embryo impact were added into the equilibrating mass. The core formation code of Brennan et al. (2020) was modified to incorporate impactor information from N-body simulations and was benchmarked against the results from Fischer et al. (2017). Parametrizations for Pequil and fmelt for each impactor were then incorporated according to the processes described above in place of the simple assumptions used in Fischer et al. (2017). The metal-silicate partitioning of elements were described by fits to experimental data as: !" ( # $ ) = $ + % ! & + ' ! ( &(1) for each element i, where T is the temperature in Kelvin, P is pressure in GPa, and ai, bi, and ci are fitting parameters. These parameters are detailed in Supplementary Table S1 and include the significant changes in partitioning at ~5 GPa for Si, O, Ni, and Co (Fischer et al., 2017 and references therein). We note that there may be more up-to-date partitioning parameterizations for Ni which have slightly different fitting coefficients (Huang and Badro, 2018). However, our results would not be affected significantly by incorporating these values. We also included fitting parameters for Nb, Ta, Mo, and W from Huang et al. (2020) and Huang et al. (2021) while Mg, Al, and Ca were assumed to be perfectly lithophile. KD is an exchange coefficient, defined in terms of partition coefficients (D), or in terms of the mole fractions (X) of elements and their oxides in the metal (met) and silicate (sil) as: # $ = # ! # "# $/& = ) ! '#( /) !) $/& *!+ () "# '#( /) "#) *!+ ) $/&(2) where n is the valence of element i. Pressures of equilibration were determined for each impactor as described above, and the temperature of equilibration was described by a polynomial fit to the liquidus of Andrault et al. (2011). First, Fe Si, Ni, and O were partitioned, where the concentrations of Fe, Si, Ni, and O in the core-forming liquid were described as a function of the moles of FeO in the mantle. The moles of FeO in the mantle were then iterated until the moles of FeO in the mantle and the moles of Fe and O in the core-forming liquid were self-consistent. This was followed by the partitioning of the trace elements (Co, Nb, Ta, Mo, and W), where partitioning of the trace elements was iterated to ensure self-consistent description of molar abundances. The compositions of the proto-Earth's core and mantle were updated after each equilibration event, and these steps were repeated until the last embryo impact, after which planetesimals that accreted were equilibrated at a pressure defined by Pequil_ptsml, with a portion of the mantle defined by Mmelt_ptsml, and a fraction of equilibrating core defined by kcore_ptsml. Results Accretion histories and equilibration parameters A comparison of the accretion histories from our different simulation suites shows the fast formation times of Earth analogs in GT and ANN simulations (Fig. 2, Supplementary Fig. S2). The time it takes for Earth analogs to reach 90% of their final mass (t90) are given in Table 1. Despite large variations, these timescales are all consistent with the 182 W anomaly of Earth's mantle due to the large effect of varying kcore on permissible timescales (Fischer and Nimmo, 2018). The distribution of impactor masses for each type of simulation are governed by initial embryo masses near 1 AU. As expected, simulations that begin with the largest embryos also have the largest median embryo masses regardless of simulation type ( Fig. 2c- which have more frequent large embryo collisions, are also able to reach the highest Pequil on average. However, regardless of initial embryo mass, simulations with longer accretion timescales (CEJS and EI) reach higher maximum average Pequil than those with shorter accretion timescales (ANN and GT). It is important to note that the curves shown in Fig. 3 are averaged over multiple Earth analogs and therefore do not fully capture the stochastic nature of the late stages of accretion, which results in large differences in Pequil between Earth analogs, and even between Earth analogs from the same simulations suite. Here, GT simulations are split by their initial masses because there is no dependence on the initial embryo to planetesimal mass ratio ( Supplementary Fig. S3). Compositions of Earth analog cores and mantles The core and mantle compositions of Earth analogs can be determined by combining the equilibration parameters and impact parameters determined from N-body simulations with our model of core formation. The metal-silicate partitioning of elements, as described by KD, are sensitive to the partitioning of Fe between silicate and metal, or the fO2 of equilibrating material (relative to the iron-wüstite (IW) buffer). For a constant set of equilibration parameters, average FeO concentrations vary greatly between Earth analogs from different simulations because of differences in the initial semi-major axis of accreting material (Fig. 4). Overall, Earth analogs produced by simulations from the same suite have similar mass-weighted average semi-major axes, regardless of initial embryo mass (Table 1, Supplementary Fig. S4). We adjust the Ni, Co, Nb, Ta, Mo, and W are siderophile elements that are either moderately refractory or refractory and have been used to trace core formation processes (Fischer et al., 2017;Huang et al., 2021Huang et al., , 2020Jennings et al., 2021;Rubie et al., 2015). The partitioning behavior of each element is given in Supplementary Table S1 combined make up ~2-7 wt% of the core, within the ranges allowed from geophysical and geochemical constraints (Fischer et al., , 2011. Other light elements, such as H, C, and S, could also contribute to the density deficit of Earth's core (Blanchard et al., 2022;Fischer et al., 2020;Suer et al., 2017;Tagawa et al., 2021). (2015) (smallest embryos) and those from EI simulations (largest embryos). These differences result from the larger average Pequil that correspond to large initial embryo masses (Fig. 3). The effects of kcore_emb on Ni and Co concentrations remedy the discrepancy in mantle composition between many Earth analogs and Earth. For example, larger values of kcore_emb would be required for Earth analogs with lower average Pequil. Mo and W, on the other hand, are systematically higher than Earth's mantle composition, even when considering the sensitivity of different parameters. It is possible that including the effects of C on Mo and W partitioning could make both elements more siderophile, reducing their mantle abundances (Jennings et al., 2021). In addition, the mantle compositions of Mo and W are highly uncertain (Liang et al., 2017). In contrast to mantle compositions, core compositions are more sensitive to certain core formation parameters (temperature for O and fO2 for Si). The large deviations in core compositions indicate that Earth's core composition could vary significantly depending on the chosen conditions of core formation and could be more difficult to constrain. We emphasize that our goal is not to find the best set of parameters to match Earth's composition but show the compositions of Earth analogs as evidence that our model can reproduce Earth's mantle composition reasonably well, in addition to producing plausible core Si and O concentrations. Another compositional effect not shown in Table 2 or Fig. 6 comes from planetesimals that accrete after the last embryo impact. A large fraction of this material must equilibrate, or else Earth's mantle siderophile element composition would greatly exceed mass estimates of the late veneer (Holzheid et al., 2000). Simulations with large percentages of material accreting after the last large impact have lower average Pequil due to the equilibration of these planetesimals at low pressures. When looking specifically at GT simulations, it is difficult to distinguish any trend in composition with initial embryo mass due to larger initial embryo masses correlating with large percentages of material accreted after the last large impact (Fig. 5a-b and Table 1). In contrast, smaller percentages of material accreting after the last large impact in simulations with larger initial embryo to planetesimal mass ratios causes mantle NiO and CoO concentrations to increase (Table 1 and Supplementary Fig. S6). Nevertheless, the effects of material accreting after the last large impact, which are dependent on the type of N-body simulation, are not as significant as varying initial embryo masses. Discussion We have compiled N-body simulations and combined them with models of core formation in which Pequil and fmelt are parameterized using a melt-scaling law. We find Earth's mantle composition to be most sensitive to the initial embryo masses in N-body simulations and the chosen value of kcore. The sensitivity of Earth's mantle composition to these parameters allows Earth's mantle composition to be reproduced for different scenarios of Solar System formation and different initial conditions within these scenarios. Below, we explore the effects of assuming the crystallization timescale of magma oceans and potential implications of different accretion histories for the Moon-forming impact. Magma ocean lifetimes and Earth's melting history The results presented above assume instant magma ocean crystallization and planetesimals equilibrating with the next embryo impact. To test the effects of long-lived magma oceans, we use the opposite endmember scenario of infinite magma ocean lifetimes. Here all planetesimals equilibrate with magma oceans generated by the previous embryo impact. Planetesimals that accrete before the first embryo impact are equilibrated with the initial embryo that grew into the Earth analog. Following an embryo impact, the melt pool will isostatically adjust to form a global magma ocean on the order of 10 2 -10 5 years (Reese and Solomatov, 2006). Therefore, subsequent planetesimals will equilibrate at the base of the global magma ocean rather than with the melt pool. In contrast to the instant crystallization scenario, surface entropy in the long-lived magma ocean case was set to 3160 J/K/kg, corresponding to surface temperatures of ~2000 K. Table 2 Compared to the two endmember scenarios we explored, realistic magma ocean lifetimes depend on the timing of embryo impacts and the efficiency of heat loss from the proto-Earth's interior to space. The presence or absence of an atmosphere, which hinges on the complex interplay between volatile delivery and atmosphere erosion, would therefore strongly influence the equilibration of planetesimals during accretion and could affect the compositions of Earth analogs formed in certain simulation types (Elkins-Tanton, 2008;Lebrun et al., 2013;Sakuraba et al., 2021). Assuming persistent atmospheres throughout accretion, the different timescales of Earth's accretion between different simulation types could be related to varying proportions of magma oceans that persist until the following embryo impact (Fig. 7). Specifically, fast accretion timescales in Grand Tack and Annulus result in 70.1% and 76.9% of embryos impacting within 2 Myrs of each other. These fast accretion timescales could promote persistent magma oceans and planetesimal impacts onto existing magma oceans (de Vries et al., 2016). In contrast, CEJS and EI simulations have less frequent embryo impacts, with only 30.0% and 5.3% of impacts occurring within <2Myrs apart. Despite these correlations, the stochastic nature of N-body simulations complicates the interpretation of these data. For example, small amounts of mass originating from large semi-major axes could contribute a significant quantity of Earth's volatiles at specific times during accretion. Furthermore, magma ocean lifetimes are dependent on the mass and composition of existing atmospheres, which could be related to the delivery of different volatile species and differences in their solubilities (Gaillard et al., 2022;Lichtenberg et al., 2021). Whether the proto-Earth could sustain an atmosphere during the giant impact stage of accretion and its relationship to the composition of Earth's core and mantle still needs to be explored. The masses assigned to embryos at the start of the giant impact stage of accretion define the initial conditions of N-body simulations. Jacobson and Morbidelli (2014) predicted that Mars-sized embryos would best match the Solar System's architecture. It is often assumed in Nbody simulations that all embryos begin with equal masses. Recent advances in simulating the formation of embryos suggests that the presence of a dissipating gas disk promotes formation of the largest embryos inside 1 AU, with the largest embryos reaching up to ~10-50% of Earth's mass (Clement et al., 2020;Walsh and Levison, 2019;Woo et al., 2021). Our results suggest that all initial conditions can match Earth's mantle composition due to the unconstrained value of kcore. However, the number of embryo impacts is independent of simulation type and decreases with the average initial embryo mass from which Earth analogs form (Fig. 8). Within GT simulations, increasing the initial embryo to planetesimal mass ratio increases the number of embryo impacts, because planetesimals supply less mass during Earth's formation. For CEJS and EI simulations, where magma oceans are more likely to crystallize before the next embryo impact, the number of embryo impacts is also more likely to be representative of the number of magma oceans experienced by Earth analogs. For GT and ANN simulations, which are more likely to have persistent magma oceans, the number of embryo impacts corresponds to the maximum number of magma oceans Earth analogs would have had. Earth analogs in EI simulations, which use initial conditions from the outputs of an embryo growth model with ~1:1 embryo to planetesimal mass ratios, likely experienced between 2-5 magma ocean events. Geochemical estimates suggest that Earth experienced at least two magma oceans during its accretion (Tucker and Mukhopadhyay, 2014). We show how the initial embryo masses and embryo to planetesimal mass ratios can be used to place constraints on the maximum number of magma oceans and outgassing events Earth experienced. Future constraints on the geochemical consequence of magma oceans may therefore place constraints on the masses of embryos in the early Solar System. Implications for Moon formation The likelihood of specific Moon-forming impact scenarios and the resulting melting of Earth's mantle can be evaluated by focusing on the last embryo impact, or sequence of embryo impacts, experienced by each Earth analog ( Fig. 9) (Jacobson and Morbidelli, 2014). Increasing initial embryo masses results in larger last embryo impacts that melt a large fraction (>90%) of Earth's mantle (fmelt). Equal-massed embryos of 0.005 Earth masses used in ANN-KC15 simulations result in Moon-forming impactors that are too small to match any Moon-forming scenario. The most probable Moon-forming scenarios based on impactor masses are the canonical hit-and-run and rapidly rotating Earth scenarios (Canup and Asphaug, 2001;Ćuk and Stewart, 2012;Reufer et al., 2012). Simulations that begin with Mars-sized embryos are most consistent with the canonical hit-and-run scenario, whereas those with smaller embryos have masses most consistent with the rapidly rotating Earth scenario. On the other hand, equal size impactors are rarely achieved, although the probability of such a scenario could be increased with larger initial embryos, or during pebble accretion (Canup, 2012;Johansen et al., 2021). By also considering impact velocities, it becomes difficult to simultaneously match the scaled impactor masses and high impact velocities required by the rapidly rotating Earth scenario (Kaib and Cowan, 2015). However, we do note that smaller initial embryo masses correspond to higher likelihoods of fast (vrel > 2) last embryo impacts ( Fig. 9e-g). Our results thus suggest that the probability of each Moon-forming scenario is dependent on the initial conditions in N-body simulations, where larger initial embryo masses promote larger and slower impactors. Mars-sized initial embryos are most consistent with the canonical hit-and-run scenario. Constraining the initial conditions in N-body simulations will therefore aid in understanding the likelihood of last embryo impacts that fall within the range allowed by each Moon-forming scenario. Recent theories of Moon formation have emerged that add to the range of possible scenarios presented above. A canonical impact onto an existing magma ocean aids in matching the compositional similarities between the Earth and the Moon (Hosono et al., 2019). Even though impactor masses and impact velocities allowed in such a scenario are similar to the canonical hit-and-run, the probability that the last embryo impact occurs onto an existing magma ocean depends on the time between the last two embryo impacts. We find this probability to be <5% for CEJS and EI simulations, and <15% for GT and ANN simulations, assuming a maximum magma ocean lifetime of 2 Myrs (Supplementary Fig. S8). When focusing only on GT simulations, this probability increases to 22.5%. It is also a possibility that the Moon formed from a series of impacts throughout Earth's accretion (Rufu et al., 2017). Interestingly, 99 out of 109 (90.8%) of Earth analogs experience complete mantle melting at some point during accretion. Therefore, Earth analogs that don't experience large Moon-forming impacts are still likely to have experienced complete mantle melting from a prior large embryo impact. Such impacts could aid in the formation of moonlets. Evaluating the likelihood of Moon formation from multiple impacts is beyond the scope of the current work but should be investigated by future studies that incorporate realistic impact histories from N-body simulations with hydrodynamic impact simulations. Conclusions We have compiled N-body simulations covering four models of Solar System formation. Building upon previous models of accretion and core formation (Fischer et al., 2017;Rubie et al., 2011), we incorporate the melt-scaling law of Nakajima et al. (2021) Kaib (1846388). We thank M. Nakajima for providing the publicly available melt-scaling law and assistance with its use in our model. We also thank J. Dong and H. Fu for helpful discussions and advice throughout the duration of this project. Finally, we thank D. Rubie and an anonymous reviewer for their suggestions that have substantially improved the manuscript. Pequil is the pressure at the base of the melt pool, fmelt is the fraction of the target's mantle that is melted, kmantle_melt is the fraction of the melted mantle that equilibrates with the impactor's core (such that fmelt*kmantle_melt = fraction of the whole mantle that equilibrates), and kcore is the fraction of the impactor's core that equilibrates. Table 2 (expressed as new value minus reference divided by reference). For each set of parameters, the compositions of all Earth analogs are averaged and compared to the average core and mantle compositions from the reference model. Signs ("+" and "-") indicate the direction the parameter is varied to result in the percent change shown. Parameters without signs are those with contrasting effects depending on the element of interest. See Table 2 for the sensitivities of all elements to all parameters. Disk surface density is defined as defined as Σ = Σ 0 r -⍺ where, alpha is the value shown in the column. 2 Embryo-to-planetesimal mass ratio is the total mass of embryos to the total mass of planetesimals. Parameters derived from the melt-scaling law of Nakajima et al. (2021). 2 Equilibration along the liquidus of Andrault et al. (2011). 3 Inner f O 2 was set for each simulation based on the average FeO of Earth analogs (Fig. S5). 4 Endmember simulations are those with the lowest and highest average P equil in Fig. 5 the averaged mantle and core compositions of Earth analogs when one parameter is varied. f). Furthermore, simulations containing large embryos also have more impacting embryos with mass >0.2 Earth masses. The impact velocities of embryo impacts are more consistent between different simulations and may be mainly controlled by the local gravitational environment at the time of the impact (Fig. 2g-j). Compared to other simulation types, impact velocities in GT and ANN-RI17 simulations are skewed towards smaller values due to dynamical friction from abundant unaccreted planetesimals, owing to their fast formation timescales. In contrast, ANN-KC15 simulations have fast formation times, don't have planetesimals, and as a result, are not affected by this dynamical friction. Using the impactor masses, impact velocities, and impact angles of individual embryo impacts shown in Fig. 2 and Fig. S1 (Nakajima et al., 2021), Pequil and fmelt were determined for each impact and parameterized over the course of accretion for each Earth analog. Fig. 3 shows the distributions of Pequil and fmelt for all embryo impacts from each simulation suite. The combination of impactor masses and velocities shapes these distributions, such that simulations with larger impactor masses also have correspondingly higher Pequil and fmelt. Larger initial embryo masses are therefore associated with more frequent high Pequil and large mantle melting events (fmelt > 0.8). Pequil averaged over all Earth analogs from each simulation type in 0.01 increments of mass fraction accreted are shown in Fig. 3i-j. These figures ignore the equilibration of planetesimals after the last embryo impact and are therefore representative of embryo collisions only. Incorporation of planetesimals would decrease the average Pequil towards the end of accretion in all simulations depending on the fraction of material delivered after the last embryo impact. Average Pequil are remarkably similar between different suites of simulations during the first 40% of accretion. EI simulations are an exception, owing to the large initial proto-Earth masses used in these simulations, which result in high values of average Pequil due to our parameterization of embryo differentiation. Simulations with large initial embryo masses, fO2 of accreting material by changing the inner fO2 of the disk, such that on average, the FeO concentrations of Earth analogs formed match Earth's actual mantle FeO concentration (Supplementary Fig. S5). Adjusting the inner fO2 to more reducing values has the same effect as moving the fO2 step out in our model. On average, matching Earth's mantle FeO requires an inner fO2 of ΔIW-3.16, -2.89, -2.59, and -2.60 for CEJS, GT, EI, and ANN simulations, respectively. The inner fO2 for CEJS and GT simulations are slightly more oxidized than those used inRubie et al. (2015) andFischer et al. (2017), which used inner fO2 values of IW-4 and -3.5, respectively. These differences can be attributed to the different simulations used in each study, whereFischer et al. (2017) used CEJS simulations from Fischer and Ciesla(2014),whereasRubie et al. (2015) only used a subset of GT simulations fromJacobson and Morbidelli (2014). and their concentrations for each Earth analog formed are shown in Fig. 5, along with the concentrations of Si and O in Earth analog cores (Fig. 5 g-h). In Fig. 5, average Pequil corresponds to Pequil averaged over the growth history of each Earth analog within each simulation such that each point represents one Earth analog. In general, larger initial embryo masses result in higher NiO, CoO, MoO2, and WO3 in Earth analog mantles and higher Si and O in Earth analog cores. The concentrations of light elements in Earth's core remain debated, but we show core compositions from Hirose et al. (2021) for reference. Si and O and Fig. 6 show the sensitivities of Earth's core and mantle compositions to variations in magma ocean lifetime. Supplementary Fig. S7 shows the differences (long-lived magma oceans -instant crystallization) in average Pequil and mantle NiO and CoO concentrations for each simulation that produced an Earth analog. In general, average Pequil is not shifted significantly, except in Grand Tack simulations, which have large percentages of material accreted after the last giant impact, due to the equilibration of planetesimals at the base of a deep magma ocean formed by the previous embryo impact. For other simulation types, these effects are overshadowed by kcore_emb and initial embryo mass for mantle composition but can be significant for the O concentration in the core, due to the large effect of temperature on O partitioning. to explore the relationships between the compiled N-body simulations, equilibration parameters, and Earth's mantle composition. We find Earth's mantle composition to be most sensitive to the initial embryo mass in N-body simulations and the chosen value of kcore_emb. These sensitivities allow Earth's mantle composition to be matched for different scenarios of Solar System formation and initial conditions within. Larger initial embryo masses require smaller values of kcore, on average, to match Earth's mantle composition. Considering interactions between accretion timescales, magma oceans, volatile delivery, Earth's mantle composition may be affected by magma ocean lifetimes depending on the time between embryo impacts. Characteristics of last embryo impacts suggest that Mars-sized embryos are most consistent with the canonical hit-and-run scenario onto a solid mantle. However, future constraints on the initial embryo masses in N-body simulations and the values of kcore_emb could yield further insights into Earth's accretion history and the Moon-forming impact. 6. Acknowledgements This work was supported in part by National Science Foundation Graduate Research Fellowships awarded to J.T. Gu and M.C. Brennan (DGE1745303), a NASA Emerging Worlds grant (80NSSC21K0388) and an NSF grant (EAR-2054912) both awarded to R.A. Fischer, and an NSF CAREER Award to N.A Fig. 1 . 1Schematic representation of the melt-scaling law from Nakajima et al. (2021) and equilibration of embryos at the base of the melt pool formed from an embryo impact. Masses (m1 and m2), velocities (v1 and v2), and impact angle (θ) are labeled on the left. Fig. 2 . 2Properties of Earth's accretion history from different N-body simulations and scenarios of Solar System formation. a) Accretion histories of Earth analogs where dashed vertical lines represent the average time at which Earth analogs from each scenario reach 90% of their final mass (t90). b) Distributions of initial embryo masses used as initial conditions in N-body simulations. The symbols on the figure represent initial conditions for only one simulation from each suite. c-f) Normalized histograms of impactor masses and g-j) impact velocities of embryo collisions onto Earth analogs. Impactor masses and impact velocities are binned in increments of 0.05 Earth masses and 0.1, respectively. Abbreviations are as follows: O06(O'Brien et al., 2006), R09(Raymond et al., 2009), KC15(Kaib and Cowan, 2015), RI17(Raymond and Izidoro, 2017a), and 0.025, 0.05, and 0.08 represent the initial embryo masses, in Earth masses, of Grand Tack simulations fromJacobson and Morbidelli (2014). Fig. 3 . 3Pequil and fmelt determined from the melt-scaling law ofNakajima et al. (2021) for embryo impacts. a-d) Normalized histograms of Pequil and e-h) fmelt for each scenario. Pequil and fmelt are binned in increments of 20 GPa and 0.2, respectively. i-j) Pequil averaged as a function of mass fraction accreted across all Earth analogs from each simulation type excluding the equilibration of planetesimals at 1 GPa to highlight the effects of embryo impacts. Simulation suites are shown separately in i) and grouped by dynamical type in j). The colors of lines in i) correspond to the bar colors in a-h). Abbreviations are as follows: O06(O'Brien et al., 2006), R09(Raymond et al., 2009), KC15(Kaib and Cowan, 2015), RI17(Raymond and Izidoro, 2017a), and 0.025, 0.05, and 0.08 represent the initial embryo masses, in Earth masses, of Grand Tack simulations fromJacobson and Morbidelli (2014). Fig. 4 . 4Feeding zones of Earth analogs. a-b) Mass-weighted cumulative distribution of where Earth-forming material originates. c-d) Moving average of the initial semi-major axis of Earthforming material over the course of Earth's accretion. The initial semi-major axes of Earth accreting material are averaged over windows of 0.05 in mass fraction accreted. Simulation suites are shown separately in a) and c) and are grouped by dynamical type in b) and d).Abbreviations are as follows: CJS (Circular Jupiter and Saturn), EJS (Eccentric Jupiter and Saturn), KC15(Kaib and Cowan, 2015), RI17(Raymond and Izidoro, 2017a), and 1:1, 2:1, 4:1, and 8:1 represent the initial embryo to planetesimal mass ratios of Grand Tack simulations fromJacobson and Morbidelli (2014). Fig. 5 . 5Core and mantle compositions of individual Earth analogs. Data points are core and mantle compositions of individual Earth analogs plotted as a function of Pequil averaged over the course of accretion. a-f) Mantle and g-h) core concentrations of investigated elements when model parameters are set to their reference values. Gray shaded regions represent estimated ranges of core and mantle compositions encompassed byHirose et al. (2021) for the core and from bothMcDonough and Sun (1995) andPalme and O'Neill (2013) for the mantle. Abbreviations are as follows: O06(O'Brien et al., 2006), R09(Raymond et al., 2009), KC15(Kaib and Cowan, 2015), RI17(Raymond and Izidoro, 2017a), and 0.025, 0.05, and 0.08 represent the initial embryo masses, in Earth masses, of Grand Tack simulations fromJacobson and Morbidelli (2014). Fig. 6 . 6Effects of varying select parameters on core and mantle compositions of Earth analogs. Percent change shows the effect of varying each parameter from the reference value within the ranges specified in Fig. 7 . 7Comparison of a typical magma ocean lifetime with the time between embryo impacts. Normalized histograms of the time between embryo impacts in each simulation suite are shown. Gray shaded regions represent a typical magma ocean lifetime of 2 Myrs(Lichtenberg et al., 2021). Abbreviations are as follows: O06(O'Brien et al., 2006), R09(Raymond et al., 2009), KC15(Kaib and Cowan, 2015), RI17(Raymond and Izidoro, 2017a), and 0.025, 0.05, and 0.08 represent the initial embryo masses of Grand Tack simulations in Earth masses fromJacobson and Morbidelli (2014). Fig. 8 . 8Relationship between the number of embryo impacts experienced by Earth analogs and the average initial embryo mass in each simulation. Average initial embryo masses were calculated for each Earth analog by averaging the initial masses of embryos that eventually collided with the proto-Earth. Abbreviations are as follows: O06(O'Brien et al., 2006), R09(Raymond et al., 2009), KC15(Kaib and Cowan, 2015), RI17(Raymond and Izidoro, 2017a), and 1:1, 2:1, 4:1, and 8:1 represent the initial embryo to planetesimal mass ratios of Grand Tack simulations fromJacobson and Morbidelli (2014). Fig. 9 . 9Characteristics of the last embryo impact in each simulation. a-d) Scaled impactor mass(Mimp/(Mimp + Mtar)), e-h) impact velocity, and i-l) fmelt. Shaded regions show the a-d) scaled impactor masses and e-h) impact velocities of different moon-forming impactors from the rapidly rotating Earth (gray)(Ćuk and Stewart, 2012), canonical hit-and-run onto a solid or liquid mantle (blue)(Canup and Asphaug, 2001;Hosono et al., 2019;Reufer et al., 2012), and equal sized impactor (green)(Canup, 2012) scenarios. The blue-green regions in e-h) are where the canonical hit-and-run and equal sized impactors overlap. Scaled impactor masses, impact velocities, and fmelt are binned in increments of 0.05 Earth masses and 0.25, and 0.1, respectively. Abbreviations are as follows: O06(O'Brien et al., 2006), R09(Raymond et al., 2009), KC15(Kaib and Cowan, 2015), RI17(Raymond and Izidoro, 2017a), and 0.025, 0.05, and 0.08 represent the initial embryo masses, in Earth masses, of Grand Tack simulations fromJacobson and Morbidelli (2014). Some elements, such as Nb and Ta, match Earth's mantle(McDonough and Sun, 1995;Palme and O'Neill, 2013) for a wide range of average Pequil, whereas other elements, such as Ni, Co, Mo, and W, are more sensitive to average Pequil.Fig. 6shows the sensitivity of Earth analog core and mantle compositions to select parameters from Table 2. In addition to kcore_emb, we find that the chosen initial conditions in N-body simulations play an important role in determining the composition of Earth's mantle. The changes resulting from N-body simulations are determined by taking difference between the average Earth analog compositions from ANN simulations from Kaib and Cowan Table 1 1Simulations and characteristics of Earth analogs used in this study. All information in the table is determined only Simulation type Earth analogs Simulations Disk limits (AU) Disk surface density ⍺ 1Classical (CEJS-O06) 4 8 0.3-4.0 -3/2 Classical (CEJS-R09) 8 40 0.5-4.5 -3/2 CJS 5 20 - - EJS 7 28 - - Grand Tack (GT) 40 142 0.7-3.0, 6-13 4 -3/2 GT 1:1 9 34 - GT 2:1 9 30 - GT 4:1 8 30 - GT 8:1 14 48 - GT-0.025 14 42 - GT-0.05 17 52 - GT-0.08 9 48 - Annulus (ANN-KC15) 24 50 0.7-1.0 0 Annulus (ANN-RI17) 12 60 0.7-1.5 -1 Early Instability (EI) 21 133 0.48-4.0 -3/2 1 Avg t 90 is the time it takes Earth analogs, on average, to reach 90% of their final mass.Grand Tack simulations contain mass distributed from 0.7-3.0 AU and planetesimals beyond 6 AU. No mass is ini le is determined only from simulations that formed an Earth analog. Classical simulations are split into CJS and EJS simulatio Embryo mass (M Earth ) AU. No mass is initially placed between 3-6 AU.3 4 # embryos Planetesimal mass (M Earth ) # planetesimals 0.0933 25 0.0025 1000 0.005-0.15 85-90 0.0025 1000-2000 0.005-0.15 - 0.0025 - 0.005-0.15 - 0.0025 - 0.025-0.08 29-214 0.000138-0.0025 0.025-0.08 28-74 0.000138-0.0025 727-7209 0.025-0.08 42-131 0.000138-0.0025 5850 0.025-0.08 55-170 0.000138-0.0025 2125-4346 0.025-0.08 68-213 0.000138-0.0025 2250 0.025 74-213 0.000138-0.0025 727-2250 0.05 37-107 0.000138-0.0025 727-2250 0.08 28-68 0.000138-0.0025 2250-7209 0.005 400 - - 0.05-0.15 20 0.000167 3000 0.02-0.48 23-25 0.0025 954-1000 6 Table 2 2Model parameters, reference values, and the effects of changing each parameter on mantle and core compositionParameter Reference Range NiO CoO P equil (GPa) Melt-scaling 1 - - - Temperature (K) A11 liquidus 2 -300 +1000 -4.8% +6.4% -8.0% +15.1% f melt Melt-scaling 1 - - - k mantle_melt 1 0.1 1 -10.7% - -14.5% - k core_emb 0.3 0.1 1 -35.4% +73.1% -27.1% +45.9% P equil_ptsml (GPa) 1 1 15 - +5.1% - +8.0% M melt_ptsml (impactor mass) 5 1 10 +18.7% -15.4% +14.5% -10.9% k core_ptsml 0.7 0.3 1 -6.5% +4.7% -3.3% +3.0% f O 2 inner (ΔIW) Match FeO 3 -3.5 -2.5 -11.8% +5.0% -19.0% +8.1% f O 2 outer (ΔIW) -1.5 - - - f O 2 step (AU) 2 - - - Initial embryo mass 4 - ANN-KC15 EI -41.8% +54.5% -28.4% +39.2% Magma ocean lifetime Instant crystallization Instant crystallization Infinite MO - +42.4% - +27.5% 1 . Compositions are averaged n mantle and core compositions of Earth analogs. Changes in composition are calculated from the averaged mantle and core Nb 2 O 5 7% g. 5. Compositions are averaged over all Earth analogs from each endmember scenario.Ta 2 O 5 MoO 2 WO 3 Core Si Core O - - - - - - -0.2% +4.2% -0.1% +5.4% +18.1% -34.2% +32.3% -34.2% -17.2% +117.5% -31.6% +397.7% - - - - - - +0.2% - +0.2% - +3.5% - -6.8% - -5.8% - +1.3% - -0.0% +1.1% -0.2% +0.5% -41.6% +128.4% -28.8% +68.0% -23.2% +60.4% -41.5% +155.2% - -0.0% - -0.0% - +2.3% - -0.5% - +0.1% - -0.4% -0.0% +0.0% -0.0% +0.0% +19.8% -16.2% +6.2% -2.4% +0.1% -0.0% -0.1% +0.1% +0.0% -0.0% +0.1% -0.0% -9.5% +7.0% -1.6% +2.4% -4.5% +3.2% -7.7% +6.0% -3.9% +1.2% +10.7% -2.2% -10.5% +6.6% -37.6% +24.4% +153.9% -25.1% -30.6% +17.1% - - - - - - - - - - - - -0.8% +2.8% -1.8% +2.9% -49.0% +61.1% -32.4% +20.3% -60.0% +79.8% -54.8% +79.0% - +3.4% - +3.4% - +48.3% - +19.6% - +50.2% - +226. Differential melt scaling for oblique impacts on terrestrial planets. O Abramov, S M Wong, D A Kring, 10.1016/j.icarus.2011.12.022Icarus. 218Abramov, O., Wong, S.M., Kring, D.A., 2012. Differential melt scaling for oblique impacts on terrestrial planets. Icarus 218, 906-916. https://doi.org/10.1016/j.icarus.2011.12.022 Solidus and liquidus profiles of chondritic mantle: Implication for melting of the Earth across its history. D Andrault, N Bolfan-Casanova, G Nigro, Lo, M A Bouhifd, G Garbarino, M Mezouar, 10.1016/j.epsl.2011.02.006Earth Planet Sci Lett. Andrault, D., Bolfan-Casanova, N., Nigro, G. lo, Bouhifd, M.A., Garbarino, G., Mezouar, M., 2011. Solidus and liquidus profiles of chondritic mantle: Implication for melting of the Earth across its history. Earth Planet Sci Lett. https://doi.org/10.1016/j.epsl.2011.02.006 The metal-silicate partitioning of carbon during Earth's accretion and its distribution in the early solar system. I Blanchard, D C Rubie, E S Jennings, I A Franchi, X Zhao, S Petitgirard, N Miyajima, S A Jacobson, A Morbidelli, 10.1016/j.epsl.2022.117374Earth Planet Sci Lett. 580Blanchard, I., Rubie, D.C., Jennings, E.S., Franchi, I.A., Zhao, X., Petitgirard, S., Miyajima, N., Jacobson, S.A., Morbidelli, A., 2022. The metal-silicate partitioning of carbon during Earth's accretion and its distribution in the early solar system. Earth Planet Sci Lett 580. https://doi.org/10.1016/j.epsl.2022.117374 Core formation and geophysical properties of Mars. M C Brennan, R A Fischer, J C E Irving, 10.1016/j.epsl.2019.115923Earth Planet Sci Lett. 530Brennan, M.C., Fischer, R.A., Irving, J.C.E., 2020. Core formation and geophysical properties of Mars. Earth Planet Sci Lett 530, 115923. https://doi.org/10.1016/j.epsl.2019.115923 Timing of Martian core formation from models of Hf-W evolution coupled with N-body simulations. M C Brennan, R A Fischer, F Nimmo, D P O&apos;brien, 10.1016/j.gca.2021.09.022Geochim Cosmochim Acta. 316Brennan, M.C., Fischer, R.A., Nimmo, F., O'Brien, D.P., 2022. Timing of Martian core formation from models of Hf-W evolution coupled with N-body simulations. Geochim Cosmochim Acta 316, 295-308. https://doi.org/10.1016/j.gca.2021.09.022 Forming a moon with an Earth-like composition via a giant impact. R M Canup, 10.1126/science.1226073Science. 338Canup, R.M., 2012. Forming a moon with an Earth-like composition via a giant impact. Science (1979) 338. https://doi.org/10.1126/science.1226073 Origin of the Moon in a giantimpact near the end of theEarth's formation. R M Canup, E Asphaug, Nature. 412Canup, R.M., Asphaug, E., 2001. Origin of the Moon in a giantimpact near the end of theEarth's formation. Nature 412, 708-712. Making More Terrestrial Planets. J E Chambers, 10.1006/icar.2001.6639Icarus. 152Chambers, J.E., 2001. Making More Terrestrial Planets. Icarus 152. https://doi.org/10.1006/icar.2001.6639 Making the Terrestrial Planets: N-Body Integrations of Planetary Embryos in Three Dimensions. J E Chambers, G W Wetherill, 10.1006/icar.1998.6007Icarus. 136Chambers, J.E., Wetherill, G.W., 1998. Making the Terrestrial Planets: N-Body Integrations of Planetary Embryos in Three Dimensions. Icarus 136. https://doi.org/10.1006/icar.1998.6007 Embryo Formation with GPU Acceleration: Reevaluating the Initial Conditions for Terrestrial Accretion. M S Clement, N A Kaib, J E Chambers, 10.3847/psj/ab91aaPlanet Sci J. 1Clement, M.S., Kaib, N.A., Chambers, J.E., 2020. Embryo Formation with GPU Acceleration: Reevaluating the Initial Conditions for Terrestrial Accretion. Planet Sci J 1, 18. https://doi.org/10.3847/psj/ab91aa The early instability scenario: Mars' mass explained by Jupiter's orbit. M S Clement, N A Kaib, S N Raymond, J E Chambers, 10.1016/j.icarus.2021.114585Icarus. 367114585Clement, M.S., Kaib, N.A., Raymond, S.N., Chambers, J.E., 2021. The early instability scenario: Mars' mass explained by Jupiter's orbit. Icarus 367, 114585. https://doi.org/10.1016/j.icarus.2021.114585 Mars' growth stunted by an early giant planet instability. M S Clement, N A Kaib, S N Raymond, K J Walsh, 10.1016/j.icarus.2018.04.008Icarus. 311Clement, M.S., Kaib, N.A., Raymond, S.N., Walsh, K.J., 2018. Mars' growth stunted by an early giant planet instability. Icarus 311, 340-356. https://doi.org/10.1016/j.icarus.2018.04.008 Making the moon from a fast-spinning earth: A giant impact followed by resonant despinning. M Ćuk, S T Stewart, 10.1126/science.1225542Science. 338Ćuk, M., Stewart, S.T., 2012. Making the moon from a fast-spinning earth: A giant impact followed by resonant despinning. Science (1979) 338, 1047-1052. https://doi.org/10.1126/science.1225542 Hf-W-Th evidence for rapid growth of Mars and its status as a planetary embryo. N Dauphas, A Pourmand, 10.1038/nature10077Dauphas, N., Pourmand, A., 2011. Hf-W-Th evidence for rapid growth of Mars and its status as a planetary embryo. Nature. https://doi.org/10.1038/nature10077 Impact-induced melting during accretion of the Earth. J De Vries, F Nimmo, H J Melosh, S A Jacobson, A Morbidelli, D C Rubie, 10.1186/s40645-016-0083-8Prog Earth Planet Sci. 3de Vries, J., Nimmo, F., Melosh, H.J., Jacobson, S.A., Morbidelli, A., Rubie, D.C., 2016. Impact-induced melting during accretion of the Earth. Prog Earth Planet Sci 3. https://doi.org/10.1186/s40645-016-0083-8 Turbulent metal-silicate mixing, fragmentation, and equilibration in magma oceans. R Deguen, M Landeau, P Olson, 10.1016/j.epsl.2014.02.007Earth Planet Sci Lett. Deguen, R., Landeau, M., Olson, P., 2014. Turbulent metal-silicate mixing, fragmentation, and equilibration in magma oceans. Earth Planet Sci Lett. https://doi.org/10.1016/j.epsl.2014.02.007 Experiments on turbulent metal-silicate mixing in a magma ocean. R Deguen, P Olson, P Cardin, 10.1016/j.epsl.2011.08.041Earth Planet Sci Lett. Deguen, R., Olson, P., Cardin, P., 2011. Experiments on turbulent metal-silicate mixing in a magma ocean. Earth Planet Sci Lett. https://doi.org/10.1016/j.epsl.2011.08.041 Linked magma ocean solidification and atmospheric growth for Earth and Mars. L T Elkins-Tanton, 10.1016/j.epsl.2008.03.062Earth Planet Sci Lett. 271Elkins-Tanton, L.T., 2008. Linked magma ocean solidification and atmospheric growth for Earth and Mars. Earth Planet Sci Lett 271, 181-191. https://doi.org/10.1016/j.epsl.2008.03.062 Equations of state in the Fe-FeSi system at high pressures and temperatures. R A Fischer, A J Campbell, R Caracas, D M Reaman, D L Heinz, P Dera, V B Prakapenka, 10.1002/2013JB010898J Geophys Res Solid Earth. Fischer, R.A., Campbell, A.J., Caracas, R., Reaman, D.M., Heinz, D.L., Dera, P., Prakapenka, V.B., 2014. Equations of state in the Fe-FeSi system at high pressures and temperatures. J Geophys Res Solid Earth. https://doi.org/10.1002/2013JB010898 Sensitivities of Earth's core and mantle compositions to accretion and differentiation processes. R A Fischer, A J Campbell, F J Ciesla, 10.1016/j.epsl.2016.10.025Earth Planet Sci Lett. 458Fischer, R.A., Campbell, A.J., Ciesla, F.J., 2017. Sensitivities of Earth's core and mantle compositions to accretion and differentiation processes. Earth Planet Sci Lett 458, 252-262. https://doi.org/10.1016/j.epsl.2016.10.025 Equation of state and phase diagram of FeO. R A Fischer, A J Campbell, G A Shofner, O T Lord, P Dera, V B Prakapenka, 10.1016/j.epsl.2011.02.025Earth Planet Sci Lett. Fischer, R.A., Campbell, A.J., Shofner, G.A., Lord, O.T., Dera, P., Prakapenka, V.B., 2011. Equation of state and phase diagram of FeO. Earth Planet Sci Lett. https://doi.org/10.1016/j.epsl.2011.02.025 Dynamics of the terrestrial planets from a large number of Nbody simulations. R A Fischer, F J Ciesla, 10.1016/j.epsl.2014.02.011Earth Planet Sci Lett. 392Fischer, R.A., Ciesla, F.J., 2014. Dynamics of the terrestrial planets from a large number of N- body simulations. Earth Planet Sci Lett 392, 28-38. https://doi.org/10.1016/j.epsl.2014.02.011 The carbon content of Earth and its core. R A Fischer, E Cottrell, E Hauri, K K M Lee, M Le Voyer, 10.1073/pnas.1919930117Proceedings of the National Academy of Sciences. 117Fischer, R.A., Cottrell, E., Hauri, E., Lee, K.K.M., le Voyer, M., 2020. The carbon content of Earth and its core. Proceedings of the National Academy of Sciences 117, 8743-8749. https://doi.org/10.1073/pnas.1919930117 High pressure metal-silicate partitioning of Ni. R A Fischer, Y Nakajima, A J Campbell, D J Frost, D Harries, F Langenhorst, N Miyajima, K Pollok, D C Rubie, V Co, Si Cr, O , 10.1016/j.gca.2015.06.026Geochim Cosmochim Acta. 167Fischer, R.A., Nakajima, Y., Campbell, A.J., Frost, D.J., Harries, D., Langenhorst, F., Miyajima, N., Pollok, K., Rubie, D.C., 2015. High pressure metal-silicate partitioning of Ni, Co, V, Cr, Si, and O. Geochim Cosmochim Acta 167, 177-194. https://doi.org/10.1016/j.gca.2015.06.026 Effects of core formation on the Hf-W isotopic composition of the Earth and dating of the Moon-forming impact. R A Fischer, F Nimmo, 10.1016/j.epsl.2018.07.030Earth Planet Sci Lett. Fischer, R.A., Nimmo, F., 2018. Effects of core formation on the Hf-W isotopic composition of the Earth and dating of the Moon-forming impact. Earth Planet Sci Lett. https://doi.org/10.1016/j.epsl.2018.07.030 Iron diapirs entrain silicates to the core and initiate thermochemical plumes. J R Fleck, C L Rains, D S Weeraratne, C T Nguyen, D M Brand, S M Klein, J M Mcgehee, J M Rincon, C Martinez, P L Olson, 10.1038/s41467-017-02503-2Nat Commun. 9Fleck, J.R., Rains, C.L., Weeraratne, D.S., Nguyen, C.T., Brand, D.M., Klein, S.M., McGehee, J.M., Rincon, J.M., Martinez, C., Olson, P.L., 2018. Iron diapirs entrain silicates to the core and initiate thermochemical plumes. Nat Commun 9, 1-12. https://doi.org/10.1038/s41467- 017-02503-2 Redox controls during magma ocean degassing. F Gaillard, F Bernadou, M Roskosz, M A Bouhifd, Y Marrocchi, G Iacono-Marziano, M Moreira, B Scaillet, G Rogerie, 10.1016/j.epsl.2021.117255Earth Planet Sci Lett. 577Gaillard, F., Bernadou, F., Roskosz, M., Bouhifd, M.A., Marrocchi, Y., Iacono-Marziano, G., Moreira, M., Scaillet, B., Rogerie, G., 2022. Redox controls during magma ocean degassing. Earth Planet Sci Lett 577. https://doi.org/10.1016/j.epsl.2021.117255 Formation of the terrestrial planets from a narrow annulus. B M S Hansen, 10.1088/0004-637X/703/1/1131Astrophysical Journal. 703Hansen, B.M.S., 2009. Formation of the terrestrial planets from a narrow annulus. Astrophysical Journal 703, 1131-1140. https://doi.org/10.1088/0004-637X/703/1/1131 Light elements in the Earth's core. K Hirose, B Wood, L Vočadlo, 10.1038/s43017-021-00203-6Nat Rev Earth Environ. Hirose, K., Wood, B., Vočadlo, L., 2021. Light elements in the Earth's core. Nat Rev Earth Environ. https://doi.org/10.1038/s43017-021-00203-6 Evidence for a late chondritic veneer in the Earth's mantle from high-pressure partitioning of palladium and platinum. A Holzheid, P Sylvester, H S C O&apos;neill, D C Ruble, H Palme, 10.1038/35019050Nature. 406Holzheid, A., Sylvester, P., O'Neill, H.S.C., Ruble, D.C., Palme, H., 2000. Evidence for a late chondritic veneer in the Earth's mantle from high-pressure partitioning of palladium and platinum. Nature 406. https://doi.org/10.1038/35019050 Terrestrial magma ocean origin of the Moon. N Hosono, S Karato, J Makino, T R Saitoh, 10.1038/s41561-019-0354-2Nat Geosci. 12Hosono, N., Karato, S. ichiro, Makino, J., Saitoh, T.R., 2019. Terrestrial magma ocean origin of the Moon. Nat Geosci 12, 418-423. https://doi.org/10.1038/s41561-019-0354-2 The niobium and tantalum concentration in the mantle constrains the composition of Earth's primordial magma ocean. D Huang, J Badro, J Siebert, 10.1073/pnas.2007982117Proc Natl Acad Sci U S A. 117Huang, D., Badro, J., Siebert, J., 2020. The niobium and tantalum concentration in the mantle constrains the composition of Earth's primordial magma ocean. Proc Natl Acad Sci U S A 117, 27893-27898. https://doi.org/10.1073/pnas.2007982117 High pressure partitioning behavior of Mo and W and late sulfur delivery during Earth's core formation. D Huang, J Siebert, J Badro, 10.1016/j.gca.2021.06.031Geochim Cosmochim Acta. 310Huang, D., Siebert, J., Badro, J., 2021. High pressure partitioning behavior of Mo and W and late sulfur delivery during Earth's core formation. Geochim Cosmochim Acta 310. https://doi.org/10.1016/j.gca.2021.06.031 Lunar and terrestrial planet formation in the Grand Tack scenario. S A Jacobson, A Morbidelli, 10.1098/rsta.2013.0174Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. Jacobson, S.A., Morbidelli, A., 2014. Lunar and terrestrial planet formation in the Grand Tack scenario. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. https://doi.org/10.1098/rsta.2013.0174 Metal-silicate partitioning of W and Mo and the role of carbon in controlling their abundances in the bulk silicate earth. E S Jennings, S A Jacobson, D C Rubie, Y Nakajima, A K Vogel, L A Rose-Weston, D J Frost, 10.1016/j.gca.2020.09.035Geochim Cosmochim Acta. 293Jennings, E.S., Jacobson, S.A., Rubie, D.C., Nakajima, Y., Vogel, A.K., Rose-Weston, L.A., Frost, D.J., 2021. Metal-silicate partitioning of W and Mo and the role of carbon in controlling their abundances in the bulk silicate earth. Geochim Cosmochim Acta 293. https://doi.org/10.1016/j.gca.2020.09.035 The Multifaceted Planetesimal Formation Process. A Johansen, J Blum, H Tanaka, C Ormel, M Bizzarro, H Rickman, 10.2458/azu_uapress_9780816531240-ch024Protostars and Planets VI. Johansen, A., Blum, J., Tanaka, H., Ormel, C., Bizzarro, M., Rickman, H., 2014. The Multifaceted Planetesimal Formation Process, in: Protostars and Planets VI. https://doi.org/10.2458/azu_uapress_9780816531240-ch024 A pebble accretion model for the formation of the terrestrial planets in the solar system. A Johansen, T Ronnet, M Bizzarro, M Schiller, M Lambrechts, A Nordlund, H Lammer, 10.1126/sciadv.abc0444Sci Adv. 7Johansen, A., Ronnet, T., Bizzarro, M., Schiller, M., Lambrechts, M., Nordlund, A., Lammer, H., 2021. A pebble accretion model for the formation of the terrestrial planets in the solar system. Sci Adv 7. https://doi.org/10.1126/sciadv.abc0444 The feeding zones of terrestrial planets and insights into Moon formation. N A Kaib, N B Cowan, 10.1016/j.icarus.2015.01.013Icarus. 252Kaib, N.A., Cowan, N.B., 2015. The feeding zones of terrestrial planets and insights into Moon formation. Icarus 252, 161-174. https://doi.org/10.1016/j.icarus.2015.01.013 Differentiated planetesimal impacts into a terrestrial magma ocean: Fate of the iron core. J D Kendall, H J Melosh, 10.1016/j.epsl.2016.05.012Earth Planet Sci Lett. 448Kendall, J.D., Melosh, H.J., 2016. Differentiated planetesimal impacts into a terrestrial magma ocean: Fate of the iron core. Earth Planet Sci Lett 448, 24-33. https://doi.org/10.1016/j.epsl.2016.05.012 Formation of Protoplanets from Planetesimals in the Solar Nebula. E Kokubo, S Ida, 10.1006/icar.1999.6237Icarus. 143Kokubo, E., Ida, S., 2000. Formation of Protoplanets from Planetesimals in the Solar Nebula. Icarus 143. https://doi.org/10.1006/icar.1999.6237 On runaway growth of planetesimals. E Kokubo, S Ida, 10.1006/icar.1996.0148Icarus. 123Kokubo, E., Ida, S., 1996. On runaway growth of planetesimals. Icarus 123. https://doi.org/10.1006/icar.1996.0148 Rapid growth of gas-giant cores by pebble accretion. M Lambrechts, A Johansen, 10.1051/0004-6361/201219127Astron Astrophys. 544Lambrechts, M., Johansen, A., 2012. Rapid growth of gas-giant cores by pebble accretion. Astron Astrophys 544. https://doi.org/10.1051/0004-6361/201219127 Formation of planetary systems by pebble accretion and migration: How the radial pebble flux determines a terrestrial-planet or super-Earth growth mode. M Lambrechts, A Morbidelli, S A Jacobson, A Johansen, B Bitsch, A Izidoro, S N Raymond, 10.1051/0004-6361/201834229Astron Astrophys. 627Lambrechts, M., Morbidelli, A., Jacobson, S.A., Johansen, A., Bitsch, B., Izidoro, A., Raymond, S.N., 2019. Formation of planetary systems by pebble accretion and migration: How the radial pebble flux determines a terrestrial-planet or super-Earth growth mode. Astron Astrophys 627. https://doi.org/10.1051/0004-6361/201834229 Metalsilicate mixing by large Earth-forming impacts. M Landeau, R Deguen, D Phillips, J A Neufeld, V Lherm, S B Dalziel, 10.1016/j.epsl.2021.116888Earth Planet Sci Lett. 564Landeau, M., Deguen, R., Phillips, D., Neufeld, J.A., Lherm, V., Dalziel, S.B., 2021. Metal- silicate mixing by large Earth-forming impacts. Earth Planet Sci Lett 564, 116888. https://doi.org/10.1016/j.epsl.2021.116888 Thermal evolution of an early magma ocean in interaction with the atmosphere. T Lebrun, H Massol, E Chassefière, A Davaille, E Marcq, P Sarda, F Leblanc, G Brandeis, 10.1002/jgre.20068Journal of Geophysical Research E: Planets. 118Lebrun, T., Massol, H., Chassefière, E., Davaille, A., Marcq, E., Sarda, P., Leblanc, F., Brandeis, G., 2013. Thermal evolution of an early magma ocean in interaction with the atmosphere. Journal of Geophysical Research E: Planets 118. https://doi.org/10.1002/jgre.20068 Growing the gas-giant planets by the gradual accumulation of pebbles. H F Levison, K A Kretke, M J Duncan, 10.1038/nature14675Nature. 524Levison, H.F., Kretke, K.A., Duncan, M.J., 2015. Growing the gas-giant planets by the gradual accumulation of pebbles. Nature 524. https://doi.org/10.1038/nature14675 Molybdenum isotope fractionation in the mantle. Y H Liang, A N Halliday, C Siebert, J G Fitton, K W Burton, K L Wang, J Harvey, 10.1016/j.gca.2016.11.023Geochim Cosmochim Acta. Liang, Y.H., Halliday, A.N., Siebert, C., Fitton, J.G., Burton, K.W., Wang, K.L., Harvey, J., 2017. Molybdenum isotope fractionation in the mantle. Geochim Cosmochim Acta 199. https://doi.org/10.1016/j.gca.2016.11.023 Vertically Resolved Magma Ocean-Protoatmosphere Evolution: H2, H2O, CO2, CH4, CO, O2, and N2 as Primary Absorbers. T Lichtenberg, D J Bower, M Hammond, R Boukrouche, P Sanan, S M Tsai, R T Pierrehumbert, 10.1029/2020JE006711J Geophys Res Planets. 126Lichtenberg, T., Bower, D.J., Hammond, M., Boukrouche, R., Sanan, P., Tsai, S.M., Pierrehumbert, R.T., 2021. Vertically Resolved Magma Ocean-Protoatmosphere Evolution: H2, H2O, CO2, CH4, CO, O2, and N2 as Primary Absorbers. J Geophys Res Planets 126. https://doi.org/10.1029/2020JE006711 Early Solar System instability triggered by dispersal of the gaseous disk. B Liu, S N Raymond, S A Jacobson, 10.1038/s41586-022-04535-1Nature. 604Liu, B., Raymond, S.N., Jacobson, S.A., 2022. Early Solar System instability triggered by dispersal of the gaseous disk. Nature 604, 643-646. https://doi.org/10.1038/s41586-022- 04535-1 Heterogeneous delivery of silicate and metal to the Earth by large planetesimals. S Marchi, R M Canup, R J Walker, 10.1038/s41561-017-0022-3Nat Geosci. 11Marchi, S., Canup, R.M., Walker, R.J., 2018. Heterogeneous delivery of silicate and metal to the Earth by large planetesimals. Nat Geosci 11, 77-81. https://doi.org/10.1038/s41561-017- 0022-3 The composition of the Earth. W F F Mcdonough, S. -S S Sun, 10.1016/0009-2541(94)00140-4Chem Geol. 120McDonough, W.F.F., Sun, S. -s. s., 1995. The composition of the Earth. Chem Geol 120, 223- 253. https://doi.org/10.1016/0009-2541(94)00140-4 Scaling laws for the geometry of an impact-induced magma ocean. M Nakajima, G J Golabek, K Wünnemann, D C Rubie, C Burger, H J Melosh, S A Jacobson, L Manske, S D Hull, 10.1016/j.epsl.2021.116983Earth Planet Sci Lett. 568Nakajima, M., Golabek, G.J., Wünnemann, K., Rubie, D.C., Burger, C., Melosh, H.J., Jacobson, S.A., Manske, L., Hull, S.D., 2021. Scaling laws for the geometry of an impact-induced magma ocean. Earth Planet Sci Lett 568, 116983. https://doi.org/10.1016/j.epsl.2021.116983 Terrestrial planet formation with strong dynamical friction. D P O&apos;brien, A Morbidelli, H F Levison, 10.1016/j.icarus.2006.04.005Icarus. 184O'Brien, D.P., Morbidelli, A., Levison, H.F., 2006. Terrestrial planet formation with strong dynamical friction. Icarus 184, 39-58. https://doi.org/10.1016/j.icarus.2006.04.005 Water delivery and giant impacts in the "Grand Tack" scenario. D P O&apos;brien, K J Walsh, A Morbidelli, S N Raymond, A M Mandell, 10.1016/j.icarus.2014.05.009Icarus. 239O'Brien, D.P., Walsh, K.J., Morbidelli, A., Raymond, S.N., Mandell, A.M., 2014. Water delivery and giant impacts in the "Grand Tack" scenario. Icarus 239, 74-84. https://doi.org/10.1016/j.icarus.2014.05.009 Cosmochemical Estimates of Mantle Composition. H Palme, H O&apos;neill, 10.1016/B978-0-08-095975-7.00201-1Elsevier Ltd2nd ed, Treatise on Geochemistry: Second EditionPalme, H., O'Neill, H., 2013. Cosmochemical Estimates of Mantle Composition, 2nd ed, Treatise on Geochemistry: Second Edition. Elsevier Ltd. https://doi.org/10.1016/B978-0- 08-095975-7.00201-1 Core-mantle differentiation in Mars. N Rai, W Van Westrenen, 10.1002/jgre.20093J Geophys Res Planets. 118Rai, N., van Westrenen, W., 2013. Core-mantle differentiation in Mars. J Geophys Res Planets 118. https://doi.org/10.1002/jgre.20093 The empty primordial asteroid belt. S N Raymond, A Izidoro, 10.1126/sciadv.1701138Sci Adv. 3Raymond, S.N., Izidoro, A., 2017. The empty primordial asteroid belt. Sci Adv 3. https://doi.org/10.1126/sciadv.1701138 Planet Formation: Key Mechanisms and Global Models. S N Raymond, A Morbidelli, 10.1007/978-3-030-88124-5_1Raymond, S.N., Morbidelli, A., 2022. Planet Formation: Key Mechanisms and Global Models. https://doi.org/10.1007/978-3-030-88124-5_1 Building the terrestrial planets: Constrained accretion in the inner Solar System. S N Raymond, D P O&apos;brien, A Morbidelli, N A Kaib, 10.1016/j.icarus.2009.05.016Icarus. 203Raymond, S.N., O'Brien, D.P., Morbidelli, A., Kaib, N.A., 2009. Building the terrestrial planets: Constrained accretion in the inner Solar System. Icarus 203, 644-662. https://doi.org/10.1016/j.icarus.2009.05.016 Fluid dynamics of local martian magma oceans. C C Reese, V S Solomatov, 10.1016/j.icarus.2006.04.008Icarus. 184Reese, C.C., Solomatov, V.S., 2006. Fluid dynamics of local martian magma oceans. Icarus 184, 102-120. https://doi.org/10.1016/j.icarus.2006.04.008 A hit-and-run giant impact scenario. A Reufer, M M M Meier, W Benz, R Wieler, 10.1016/j.icarus.2012.07.021Icarus. 221Reufer, A., Meier, M.M.M., Benz, W., Wieler, R., 2012. A hit-and-run giant impact scenario. Icarus 221. https://doi.org/10.1016/j.icarus.2012.07.021 Moderately and slightly siderophile element constraints on the depth and extent of melting in early Mars. K Righter, N L Chabot, 10.1111/j.1945-5100.2010.01140.xMeteorit Planet Sci. 46Righter, K., Chabot, N.L., 2011. Moderately and slightly siderophile element constraints on the depth and extent of melting in early Mars. Meteorit Planet Sci 46. https://doi.org/10.1111/j.1945-5100.2010.01140.x Terrestrial planet formation. K Righter, D P O&apos;brien, 10.1073/pnas.1013480108Proc Natl Acad Sci. Righter, K., O'Brien, D.P., 2011. Terrestrial planet formation. Proc Natl Acad Sci U S A. https://doi.org/10.1073/pnas.1013480108 Heterogeneous accretion, composition and core-mantle differentiation of the Earth. D C Rubie, D J Frost, U Mann, Y Asahara, F Nimmo, K Tsuno, P Kegler, A Holzheid, H Palme, 10.1016/j.epsl.2010.11.030Earth Planet Sci Lett. 301Rubie, D.C., Frost, D.J., Mann, U., Asahara, Y., Nimmo, F., Tsuno, K., Kegler, P., Holzheid, A., Palme, H., 2011. Heterogeneous accretion, composition and core-mantle differentiation of the Earth. Earth Planet Sci Lett 301, 31-42. https://doi.org/10.1016/j.epsl.2010.11.030 Accretion and differentiation of the terrestrial planets with implications for the compositions of early-formed Solar System bodies and accretion of water. D C Rubie, S A Jacobson, A Morbidelli, D P O&apos;brien, E D Young, J De Vries, F Nimmo, H Palme, D J Frost, 10.1016/j.icarus.2014.10.015Icarus. 248Rubie, D.C., Jacobson, S.A., Morbidelli, A., O'Brien, D.P., Young, E.D., de Vries, J., Nimmo, F., Palme, H., Frost, D.J., 2015. Accretion and differentiation of the terrestrial planets with implications for the compositions of early-formed Solar System bodies and accretion of water. Icarus 248, 89-108. https://doi.org/10.1016/j.icarus.2014.10.015 A multiple-impact origin for the Moon. R Rufu, O Aharonson, H B Perets, 10.1038/ngeo2866Nat Geosci. 10Rufu, R., Aharonson, O., Perets, H.B., 2017. A multiple-impact origin for the Moon. Nat Geosci 10, 89-94. https://doi.org/10.1038/ngeo2866 Numerous chondritic impactors and oxidized magma ocean set Earth's volatile depletion. H Sakuraba, H Kurokawa, H Genda, K Ohta, 10.1038/s41598-021-99240-wSci Rep. 11Sakuraba, H., Kurokawa, H., Genda, H., Ohta, K., 2021. Numerous chondritic impactors and oxidized magma ocean set Earth's volatile depletion. Sci Rep 11. https://doi.org/10.1038/s41598-021-99240-w Metal-silicate partitioning of Ni and Co in a deep magma ocean. J Siebert, J Badro, D Antonangeli, F J Ryerson, 10.1016/j.epsl.2012.01.013Earth Planet Sci Lett. 321Siebert, J., Badro, J., Antonangeli, D., Ryerson, F.J., 2012. Metal-silicate partitioning of Ni and Co in a deep magma ocean. Earth Planet Sci Lett 321-322, 189-197. https://doi.org/10.1016/j.epsl.2012.01.013 A sulfur-poor terrestrial core inferred from metal-silicate partitioning experiments. T.-A Suer, J Siebert, L Remusat, N Menguy, G Fiquet, 10.1016/j.epsl.2017.04.016Earth Planet Sci Lett. 469Suer, T.-A., Siebert, J., Remusat, L., Menguy, N., Fiquet, G., 2017. A sulfur-poor terrestrial core inferred from metal-silicate partitioning experiments. Earth Planet Sci Lett 469, 84-97. https://doi.org/10.1016/j.epsl.2017.04.016 Experimental evidence for hydrogen incorporation into Earth's core. S Tagawa, N Sakamoto, K Hirose, S Yokoo, J Hernlund, Y Ohishi, H Yurimoto, 10.1038/s41467-021-22035-0Nat Commun. 12Tagawa, S., Sakamoto, N., Hirose, K., Yokoo, S., Hernlund, J., Ohishi, Y., Yurimoto, H., 2021. Experimental evidence for hydrogen incorporation into Earth's core. Nat Commun 12. https://doi.org/10.1038/s41467-021-22035-0 Evidence for multiple magma ocean outgassing and atmospheric loss episodes from mantle noble gases. J M Tucker, S Mukhopadhyay, 10.1016/j.epsl.2014.02.050Earth Planet Sci Lett. Tucker, J.M., Mukhopadhyay, S., 2014. Evidence for multiple magma ocean outgassing and atmospheric loss episodes from mantle noble gases. Earth Planet Sci Lett. https://doi.org/10.1016/j.epsl.2014.02.050 Planetesimals to terrestrial planets: Collisional evolution amidst a dissipating gas disk. K J Walsh, H F Levison, 10.1016/j.icarus.2019.03.031Icarus. 329Walsh, K.J., Levison, H.F., 2019. Planetesimals to terrestrial planets: Collisional evolution amidst a dissipating gas disk. Icarus 329, 88-100. https://doi.org/10.1016/j.icarus.2019.03.031 A low mass for Mars from Jupiter's early gas-driven migration. K J Walsh, A Morbidelli, S N Raymond, D P O&apos;brien, A M Mandell, 10.1038/nature10201Walsh, K.J., Morbidelli, A., Raymond, S.N., O'Brien, D.P., Mandell, A.M., 2011. A low mass for Mars from Jupiter's early gas-driven migration. Nature. https://doi.org/10.1038/nature10201 The terrestrial planet formation paradox inferred from high-resolution N-body simulations. J M Y Woo, R Brasser, S L Grimm, M L Timpe, J Stadel, 10.1016/j.icarus.2021.114692Icarus. 371Woo, J.M.Y., Brasser, R., Grimm, S.L., Timpe, M.L., Stadel, J., 2022. The terrestrial planet formation paradox inferred from high-resolution N-body simulations. Icarus 371. https://doi.org/10.1016/j.icarus.2021.114692 Growing Mars fast: High-resolution GPU simulations of embryo formation. J M Y Woo, S Grimm, R Brasser, J Stadel, 10.1016/j.icarus.2021.114305Icarus. 359Woo, J.M.Y., Grimm, S., Brasser, R., Stadel, J., 2021. Growing Mars fast: High-resolution GPU simulations of embryo formation. Icarus 359. https://doi.org/10.1016/j.icarus.2021.114305 CJS and EJS simulations and GT simulations are split by initial conditions to show the effects of simulation type and initia Embryo:planetesimal mass ratio 2 Avg t. 903nto CJS and EJS simulations and GT simulations are split by initial conditions to show the effects of simulation type and initia Embryo:planetesimal mass ratio 2 Avg t 90 (Myr) 3 t by initial conditions to show the effects of simulation type and initial conditions on Earth analog characteristics. Numbers in Average mass-weighted semi-major axis (AU) Percent of material after last giant impact (%). t by initial conditions to show the effects of simulation type and initial conditions on Earth analog characteristics. Numbers in Average mass-weighted semi-major axis (AU) Percent of material after last giant impact (%)
{'fraction_non_alphanumeric': 0.06727418922540873, 'fraction_numerical': 0.061191194807313574, 'mean_word_length': 4.285740683708038, 'pattern_counts': {'":': 0, '<': 7, '<?xml version=': 0, '>': 6, 'https://': 74, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 4, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': "The chemical compositions of Earth's core and mantle provide insight into the processes that led to their formation. N-body simulations, on the other hand, generally do not contain chemical information, and seek to only reproduce the masses and orbits of the terrestrial planets.These simulations can be grouped into four potentially viable scenarios of Solar System formation (Classical, Annulus, Grand Tack, and Early Instability) for which we compile a total of 433 N-body simulations. We relate the outputs of these simulations to the chemistry of Earth's core and mantle using a melt-scaling law combined with a multi-stage model of core formation.We find the compositions of Earth analogs to be largely governed by the fraction of equilibrating", 'arxivid': '2302.11020', 'author': ['Jesse T Gu \nDepartment of Earth and Planetary Sciences\nHarvard University\nCambridgeMAUSA\n', 'Rebecca A Fischer \nDepartment of Earth and Planetary Sciences\nHarvard University\nCambridgeMAUSA\n', 'Matthew C Brennan \nDepartment of Earth and Planetary Sciences\nHarvard University\nCambridgeMAUSA\n', 'Matthew S Clement \nEarth and Planets Laboratory\nCarnegie Institution for Science\nWashington D.CUSA\n', 'Seth A Jacobson \nDepartment of Earth and Environmental Sciences\nMichigan State University\nEast LansingMIUSA\n', 'Nathan A Kaib \nDepartment of Physics and Astronomy\nHL Dodge\nUniversity of Oklahoma\nNormanOKUSA\n', 'David P O&apos;brien \nPlanetary Science Institute\nTucsonArizonaUSA\n', "Sean N Raymond \nLaboratoire d'Astrophysique de Bordeaux\nUniversité de Bordeaux\nCNRS\nBordeauxFrance\n"], 'authoraffiliation': ['Department of Earth and Planetary Sciences\nHarvard University\nCambridgeMAUSA', 'Department of Earth and Planetary Sciences\nHarvard University\nCambridgeMAUSA', 'Department of Earth and Planetary Sciences\nHarvard University\nCambridgeMAUSA', 'Earth and Planets Laboratory\nCarnegie Institution for Science\nWashington D.CUSA', 'Department of Earth and Environmental Sciences\nMichigan State University\nEast LansingMIUSA', 'Department of Physics and Astronomy\nHL Dodge\nUniversity of Oklahoma\nNormanOKUSA', 'Planetary Science Institute\nTucsonArizonaUSA', "Laboratoire d'Astrophysique de Bordeaux\nUniversité de Bordeaux\nCNRS\nBordeauxFrance"], 'corpusid': 255701796, 'doi': '10.1016/j.icarus.2023.115425', 'github_urls': [], 'n_tokens_mistral': 29611, 'n_tokens_neox': 23678, 'n_words': 12365, 'pdfsha': 'af6389e3d095295fe3039c2d956aa8f37381e72a', 'pdfurls': ['https://export.arxiv.org/pdf/2302.11020v1.pdf'], 'title': ['Comparisons of the core and mantle compositions of Earth analogs from different terrestrial planet formation scenarios', 'Comparisons of the core and mantle compositions of Earth analogs from different terrestrial planet formation scenarios'], 'venue': []}
arxiv
Generative Adversarial Networks for Malware Detection: a Survey Aeryn Dunmore Cybersecurity Lab Massey University AlbanyNew Zealand Aeryn Dunmore Julian Jang-Jaccard Cybersecurity Lab Massey University AlbanyNew Zealand Fariza Sabrina Cybersecurity Lab Massey University AlbanyNew Zealand School of Engineering and Technology Central Queensland University SydneyNSWAustralia Jin Kwak Department of Cyber Security Ajou University SuwonRepublic of Korea Generative Adversarial Networks for Malware Detection: a Survey RESEARCH * Correspondence: [email protected] Full list of author information is available at the end of the articleGenerative Adversarial Networks (GAN)machine learningcurrent research surveythreat detectionmalwaredata augmentationadversarial examples Since their proposal in the 2014 paper by Ian Goodfellow [1], there has been an explosion of research into the area of Generative Adversarial Networks. While they have been utilised in many fields, the realm of malware research is a problem space in which GANs have taken root. From balancing datasets to creating unseen examples in rare classes, GAN models offer extensive opportunities for application. This paper surveys the current research and literature for the use of Generative Adversarial Networks in the malware problem space. This is done with the hope that the reader may be able to gain an overall understanding as to what the Generative Adversarial model provides for this field, and for what areas within malware research it is best utilised. It covers the current related surveys, the different categories of GAN, and gives the outcomes of recent research into optimising GANs for different topics, as well as future directions for exploration. Introduction Generative Adversarial Networks, or GANs, are a type of deep learning neural network model based on the Game Theory premise of a zero-sum game [1]. These networks have become popular in many fields as a Machine Learning (ML) model which has great success at synthesising large samples of dataset classes based on learning classes and features from an existing dataset. They are particularly good arXiv:2302.08558v2 [cs.CR] 24 Feb 2023 at synthesising images [2], making them both popular in computer vision tasks, and excellent at generating malware 'images' for training systems to detect malicious files and applications. They also offer a chance to augment the rarest classes in a dataset [3]. While they have received attention from many disciplines and research topics, the research into GANs for synthesising images of different malware classes is very promising, and as such, is where we have chosen to focus our survey. The research in this study concerns of the state of the art in GANs and Malware, and we have found a space for an up-to-date examination of where this research discipline is and where it appears to be headed. As is explained in Section 3, on related works, while there are surveys or studies which are similar in aspects of their research, to our knowledge there is no updated survey on this topic. This is of distinct importance because of how much growth we have seen in this area of research in recent years. We have done our utmost to present a balanced examination, with both breadth and depth, which can be of use to both researchers new to the area, and those wanting an update on the current problem space. We have also attempted to approach this survey in a way that makes it accessible for the machine learning or cybersecurity researcher both. The rest of the paper is structured as follows: Section 2 describes the structure of GAN models, and how they are built and trained, as well as defining malware for the purposes of this paper. Section 3 gives a relatively brief breakdown on the recent works with the most similarities to our survey, and explains how we have developed something different in content and structure. Section 4 clarifies the different methods by which researchers measure the performance of their respective GAN models. Section 5 explains the datasets used in the experiments and research we have surveyed and what types of data they contain as well as their origins. Section 6 delves into the different types of Generative Adversarial Network -both the most commonly implemented models and the innovations that have come from recent researchers developing new ways in which to use the model. Section 7 will explain the types of uses GAN models are being used in malware research, along with the specific areas within the area to which GAN research is contributing. Section 8 contains our in-depth discussion of how GANs are functioning within malware research and what this means for researchers both in cybersecurity and machine learning. Finally, Section 9 discusses the opportunities for future research, and then Section 10 goes on to the conclusions we believe can be drawn from the survey of papers within this discipline. Terms, Definitions, and Explanations A Generative Adversarial Network, at its base, is a machine learning algorithm built out of two separate deep learning networks which work together, competing to win a zero-sum game. One network takes in noise and then attempts to create samples with the right characteristics to have them seem like real or 'genuine' samples. The second network takes as input both real and generated samples, and then classifies them as either real or fake samples. The back-propagation that occurs then is the backbone of the model. If the Discriminator network is right, information is sent back to the Generator network, so that it will adjust its weights and probability distributions to improve the quality of its forgeries. If the Discriminator instead gets it wrong, that information is sent to the Discriminator to make the necessary adjustments. The games end when the Discriminator has the accuracy of a coin flip -when the forgeries are all but impossible to separate from the genuine samples. As discussed above, the Generator creates data that is meant to look as real as possible. The Discriminator has only one job -determine if the data provided to it is generated data (created by the Generator) or if it is in fact genuine data. The Generator is considered to have "won" when the Discriminator has a success rate of 50%. Meaning that the Generator is so good at producing almost real data, that the Discriminator is left with the same accuracy as tossing a coin. The GAN is considered an unsupervised machine learning method, and it was developed in 2013 to help model the behaviour of wildlife [4]. GANs are an alternative generative model to Variational Autoencoders (VAE), which can also be used to create new samples from a given dataset. In many of the papers we surveyed, VAE were used as a point of comparison/control in the experiments for improved GAN models. How does a GAN model work? Having given a simplistic overview, we now explain the architecture of the GAN model for machine learning in detail. The architecture is innovative for the way it processes and creates both information and datasets. In a world where we need exceptionally large datasets to train the machine learning algorithms that are now slipping into so much of our technology -and therefore into our lives -the ability [5] to create new data for training purposes is invaluable. This is, of course, provided the data is, or is at least almost, authentic. Creating exceptional forgeries, in the way that GANs do, is therefore a reason in and of itself to employ them in most problem domains. Cybersecurity especially has need of as many samples as possible of the different types of malware, in order to be able to train defensive technologies, to detect when a file or action is malicious. A Standard GAN Model Structure The standard/vanilla Goodfellow GAN design is simple but ingenious. It relies on the use of two main deep learning neural nets, with back-propagation and feedback. The Generator and Discriminator play a "two-player minimax game" [1] using the value function found in Equation 9. This equation is taken from Goodfellow's original 2014 paper [1] introducing Generative Adversarial Networks [1]. Figure 1, from [5], shows the structure of a standard model GAN, and one of its derivatives, Least Squares GAN (LSGAN) discussed later in Section 6.8. Generator The Generator portion of the network is arguably the more complex. The Generator Network within the GAN model starts with a random seed or noise as input, and produces an output which starts understandably far from the goal. However, as the Discriminator Network feeds information through back-propagation to the Generator, it slowly achieves convergence on the target samples. Each iteration, each epoch, the Generator syntheses data that is more and more realistic. Convergence and training of the Generator Network is finished when the Discriminator cannot tell if the synthesised data is real or manufactured with any more than 50% accuracy -effectively becoming as useful in classification as a coin toss. The advantages to being able to do this are many, and this generation of data is what makes the GAN model so popular in the world of machine learning. The ability to extract and analyse important features of each dataset class are a highly sought after trait in machine learning research. Discriminator The Discriminator's job essentially comes down to a binary option. Is the data real, or is it synthesised? This part of the network, like most deep neural networks, is about feedback -in this case, the back-propagation of the result to the Generator Network. When the Discriminator guesses incorrectly, that alters the internal weights of the system as the information back-propagates to both Generator and Discriminator. The Discriminator is also playing to lose, as the target is to have the Discriminator as effective at distinguishing real data from generated data as a simple coin toss. At that point in the game, the Generator has graduated to building data for use in other scenarios. The data provided is considered so close in featurespace that it may as well be treated like the real deal. The Discriminator is the part of the dual network that recognises when the model has been sufficiently trained to produce high-quality synthesised data. It is, also, no longer required once the Generator is producing the samples at that level of 'perfection'. It does not need to be used outside of the training of the Generator, as that is its singular purpose. Types of feedback loops Essential to every GAN implementation is the type of feedback response network it utilises. Being able to feed the results of the tests run by the Discriminator back to the Generator is key in the development of an effective Generator. With every wrong choice, the Generator gets to adjust the weights that little bit closer to where they want to be. Likewise, the Discriminator choosing correctly means that the Generator needs to update its neural weights differently, to change the emphasis on particular neurons, and to change the outputs of the network for a more favourable outcome. The back-propagation of the network allows both systems to carefully adjust the weights and probabilities of their internal deep learning neural networks. Because of the feedback loop needed by GAN architecture, it is a back-propagation network. There are networks that change the types of feedback loops and what is presented to cause changes in the algorithm. The function to alter the weights is essential to the training period, and is as such customised in different models and applications such that the network can reach optimal function as efficiently as possible. It is important to remember that this adjustment over time occurs in a black-box. The results and inputs are what we as researchers can control, while the actual operations can only be modelled and estimated. Malware Malware, in the general definition for the purposes of this paper, is code written to cause malicious behaviour. Trojans and ransomware are both examples of malware. In its purest form, malware is programming intended to in some way break or disrupt the regular operation of an operating system [6]. One of the most wellknown malware attacks is WannaCry's ransomware in 2017, infecting computers across the globe with hackers locking users out of their PCs. As ransomware has evolved, it has become a billion dollar business [7]. This is simply another reason to use every means at our disposal to create a new generation of detection systems, such that individuals (who may be mostly unaware of how best to protect themselves) need not attempt to protect their devices on their own. This paper aims to show the ways in which GAN models can and are being used in order to help create these new systems. Related Work The areas in which the use of GAN is both possible and implemented is exhaustive. The classification, generalisation, and feature extraction abilities of the GAN models make them useful in too many fields to reasonably keep track of, but there are many surveys that have tried to enumerate all the ways in which GAN models helped in their fields. These include steganography [8], the cracking of cryptographic methods [9], e-commerce [10], and cross-lingual methods for detecting hackers [11]. There are however, some areas in which GAN models are more common than others. We have summarised the types of GAN models examined in the related works in Table 1. In Berman et al., 2019[12], the authors take the chance to explore the ways in which deep learning methods have been integrated into different realms of cybersecurity. Our paper is similar, though it focuses specifically on the work done using Generative Adversarial Networks for implementation in the research space of malware, both generation and detection. Berman et al. [12] is intended to familiarise the reader with research into machine learning for cybersecurity. The authors are careful to emphasize the difference between the deep versus shallow neural network machine learning models. This survey is far greater in depth than spread -the types of uses for GAN models in the survey are a small number primarily about identifying attacks, but they are all examined in great detail over the course of many papers in each area. Their survey is indeed intended to familiarise the reader first with machine learning, neural networks, and deep learning, before moving on to Generative Adversarial Networks and then to their applications in cybersecurity. In this, the paper does achieve its aims. However, the lack of variety in the tasks in which GAN models can be used in cybersecurity is a note on which there is certainly room for more breadth. This survey is very careful to ensure the reader understands The focus on Adversarial Machine Learning, as opposed to Generative Adversarial Networks, for network intrusion detection systems mean that, while this paper is very in-depth and covers a variety of machine learning models including GANs, our paper fits within the gap between AML for IDS and GANs for malware research. The survey, Navidan et al., 2021 [14], covers familiar ground in the research surveyed, but in exceptional depth. The authors are quick to note that, as is evident by the content of these surveys, while GAN for cybersecurity is a relatively new field, it is already being extensively researched. However, this paper has a very small related works section, with only two other surveys mentioned briefly. The survey covers some interesting areas in which GANs are currently being utilised. In the paper [15], the survey authors note the creation of an ingenious GAN model for morphing traffic flow data, called FlowGAN. The purpose of this model is to train it as to what benign or normal traffic flow patterns look like. Then it morphs the traffic data that needs to pass by undetected, into something with the right features and patterns to be labelled as benign/normal traffic. Such a model would be of interest in regards to malware research as a method to defend against malicious files and applications being disguised as normal and benign traffic. In Future Work, the authors of the survey note that improving ways to avoid image translations are of great significance. This would ease the ability of researchers when finding or developing a new method to use with a different type of data, making it not so cost-expensive with regards to overhead. We have found a space within these existing surveys to fill gaps with regards to how Generative Adversarial networks are used in areas of malware research for cybersecurity. We have done so with a focus on creating an overview that will be of use to those individuals in need of a primer on GANs and their potential applications in research into malware. We have attempted to balance depth with breadth of content, and to point readers to other papers that may give them further information on the use of GAN models in these research areas. While our paper might be considered of a parallel topic to [16], the latter paper is much more compact, and as well as dealing more broadly in cybersecurity as a whole, it involves a detailed case study which leaves it little room to discuss the topics in the depth which we Measuring Performance In most papers related to GAN schemes, there are expected metrics for evaluating a machine learning systems like this one [17]. The most popular are listed here, as are the ways these show the performance of the GAN. Table 2. True Positive The True Positive/TP is the number of correctly predicted positive results, or the total number of correctly classified benign samples. False Positive The False Positive/FP is the number of incorrectly predicted positive results, or the total number of incorrectly classified benign samples. True Negative The True Negative/TN is the number of correctly predicted negative results, or the total number of correctly classified malicious samples. False Negative The False Negative/FN is the number of incorrectly predicted negative results, or the total number of incorrectly classified malicious samples. Accuracy The accuracy is the average of correct predictions -of both positive and negative varieties -when classified. Thus, it is the correct predictions divided by the total predictions, or: = + + + +(1) This is the assumed metric in papers or articles which talk only about averages and score. Precision Also known as Positive Predictive Value or PPV, this is the samples that were classed correctly as benign over all samples that have been classified as benign. = + (2) Recall The recall, also known as true positive ratio, or sensitivity, is the ratio of samples classed as benign over the total samples classed as benign. = + (3) F1-Score This is the Harmonic Mean of the precision and the recall values. A harmonic mean is one of three types of Pythagorean averages. It is heavily influenced by the lowest of the values, when applied to real numbers, meaning it holds an important place to check the minority classes' accuracy. § 1 = 2 + (4) Inception Score When is the Generator, is the Discriminator, and there are two finite, label sets, Ω and Ω . As such, is a distribution over Ω . : Score over all probability distributions, Ω , and Ω [18]. Ω → (Ω ) is( , ) = exp(E ∼ [ ( (· | ) ∫ (· | ) ( ) )])(5) There is a pre-trained network which measures the Inception Score, and the higher the score of the model, the higher the quality of the images produced [19]. The Inception Score and Network were introduced in 2016 for Convolutional Neural Networks by Szegedy et al. [20]. It was originally developed to remove human subjectivity in computer vision research. Mode Score The Mode Score is meant to be an improved version of the Inception Score. It still measures the quality and diversity of images, but it counts the prior distribution of labels [21]. (KL) = exp(E [KL( ( | ) ( )] − KL( ( ) ( )))(6) Fréchet Inception Distance There is another equation derived from the Inception Score. The Fréchet Inception Distance (FID) and the Inception Score (IS) together can be used as an attempt to solve overfitting [i] . The FID is shown below in Equation 7. The purpose of the FID is to examine the distance between groups. It was also developed for the specific task of image processing in machine learning [20]. Frechet Inception Distance for any two probability distributions, and , over the set of real numbers, R , is calculated as follows: ( , ) := ∈Γ( , ) ∫ R ×R − 2 ( , ) 1/2(7) The set used in the FID here, Γ( , ) is actually the 2-Wasserstein distance over R [21]. There is a second calculation for the FID score, but it works only over two Gaussian, multi-dimensional distributions, N ( , ) -symbolised below as -and N ( , ) -shown below as . ( , ) = − 2 2 + ( ∑︁ + ∑︁ −2( ∑︁ ∑︁ ) 1/2 )(8) Dataset The different datasets on which the Machine Learning algorithms are trained have a significant effect on how they read the given data, what their biases or preconceived ideas may be, and how they are trained to recognise different integral features. In [i] Overfitting occurs in statistical analysis when too few samples are present and the model is fitted too closely to this small selection of samples, making its ability to generalise low. this section we have attempted to cover the main datasets used in the papers we have surveyed. DGArchive The DGArchive is a set of domains, of 43 families, classes, or variants, with more than 20 million domains as of 2015 [22]. These domains are from models in This data is extremely important in creating new machine learning methods for identifying botnet C&C centres (as in [23]). The compilation of this information into such a large and comprehensive database is an important research tool. The DGArchive dataset is also used to create adversarial machine learning models, such as MaldomDetector [24], which undertake the generation of malicious domain names itself, and allows researchers to test defensive machine learning algorithms on an adversary. VirusTotal This repository of both anti-virus software and a database of files, both benign and malicious, is known as VirusTotal. It can scan a given file using 70 antivirus systems as well as checking with URL and domain blacklisting programs [25]. Each uploaded file -as well as resulting in a report stating the findings and results labelling it either benign or malicious and how these results were arrived at -is also kept and added to the overall database of VirusTotal files. The service is free for research and non-commercial use, and licences can be purchased for commercial users or those needing a large sample set of data [26]. In addition to scanning, users can request a subset of the database for use in training and testing their own algorithms. This is a often used service in machine learning research, such as [27], because of the depth and breadth of malware covered by the VirusTotal dataset. It is also useful because of the constant updating the servers get as users upload their own programs and files to scan. This is an unusual dataset in that regard, where other datasets covered in this section are static and set, while VirusTotal is continually changing and updating. VirusTotal contains files, programs, Android applications [28], applications for Windows, Mac, Linux, iOS, and so on. This is another point in favour of the database -the type of data available for training and testing is extensive and covers a lot of ground, where other datasets discussed only cover one type of information. Contagio Contagio is a publicly available dataset of malware, specifically samples of Android malware and benign applications [29]. The dataset was updated periodically between 2011 and 2018, and can be found for open access online at a range of places [30]. The fact that this dataset is focused on Android malware makes it extremely useful, as overall, openly available databases of mobile malware are not as prevalent as those for desktop malware or traffic flow data. As of 2021, Contagio contained 11,960 malicious and 16,800 benign samples of Android software [29], with a total size of approximately 9GB [31]. This dataset can be accessed in .zip format for researchers and white-hat activities. Drebin Drebin is a repository for Android malware, similar to Contagio. Drebin contains 123,453 applications and 5,560 malicious APKs for Android, in a variety of malware families, totalling about 6GB in size [31]. It was collected from 2010 to 2012. It was originally proposed as part of a paper in which Drebin -a new algorithm -was proposed to catch malware on Android smartphones. As part of this, a database of 5,560 malicious Android APKs were collected, which now make up the Drebin dataset [32]. It is important, therefore, to differentiate between the Drebin staticanalysis detection software, and the Drebin dataset. Both were organised around the use of eight main feature sets for analysis [32]. These sets are as follows: • Types of GAN models The papers we have surveyed have used a range of variants of the traditional GAN. As a reference and refresher, we have included this section. in which we cover the different types of GANs we will be discussing, and the points of difference in each. We also wanted to clearly illustrate the issues inherent in the standard GAN model, so that the variations which are developed specifically for overcoming them are understood. and was based on adversarial nets as a framework, with back-propagation. This model uses a two-player minmax game to adjust the weights, as per Figure [1]. An important distinction is that while the discriminator has access to both real and generated data, the generator has no access to either, and so has to rely on the value functions and the back-propagation to change the weights and take the model closer to producing realistic generated output [47]. The generator and discriminator are both able to be non-linear mapping functions [49]. The GAN model came through the adversarial nets framework, a way of dealing with weights without Markov chains, and instead using back-propagation [49]. For an in depth overview of how the Goodfellow GAN operates, please refer to Section 2.1. Inherent Problems in the Goodfellow GAN Architecture Mode Collapse Problem The complexities of the MinMax game that are essential to the standard/Vanilla GAN result in an optimisation problem. This is solved in the standard version using the gradient descent-ascent (GDA) method [50]. However, this can lead to serious errors in convergence resulting in failure of the GAN, including a problem known as mode collapse [51]. Combating this problem is one of the reasons there are so many variations of GAN models -many are developed to help avoid the convergence problems in optimisation of the minmax function as much as possible. Catastrophic Forgetting Catastrophic Forgetting (CF) occurs when "knowledge of previously learned tasks is abruptly destroyed by the learning of the current task" ( [51]). CF can prevent proper convergence in the model, and limit it from finding the necessary local maxima optimum for the task it is set. Remembering the location and features of the real samples used to train the generator is essential -when the generator loses these samples, catastrophic forgetting occurs as the new generated samples are not created with the real samples as a guide [51]. It is important to note that mode collapse and catastrophic forgetting are interlinked -they make the other worse in situations where both problems arise. The equation for describing the optimal Discriminator in Goodfellow's GAN model is shown in Equation 9. The training criteria for a given discriminator, , and a generator, , are shown in Equation 10. * ( ) = ( ) ( ) + ( ) (9) ( , ) = ∫ ( ) ( ( ))(10)+ ∫ ( ) (1 − ( ( )))(11)( , ) = ∫ ( ) ( ( ))(12)+ ( ) (1 − ( ))(13) Conditional GAN The Conditional GAN (CGAN), proposed in [49], modify the original vanilla GAN. In the original model, the generative process could not be controlled or conditioned. It was unsupervised entirely. CGANs allow the generation process to be controlled and directed, meaning that the model can be steered towards a focus on a particular class, or feature. The original paper proposing a CGAN model utilised the MNIST dataset (see Section 5.8) to test its capabilities. The change in control occurs when the focus is put on some element , which can be a class, value, feature, so on, and this is fed into both the generator and discriminator as an additional layer of input. In the original proposal, the generator is fed not only the focus , but also a noise function, ( ) [49]. In a subsequent study, a CGAN for facial recognition was proposed [52], which used sampled random noise for ( ) and a random sampling for which is taken from the training dataset, utilising a Pazan window, ( ). Deep Convolutional GAN The Deep Convolutional Generative Adversarial Network, or DCGAN, was proposed in a 2015 paper titled "Unsupervised representation learning with deep convolutional generative adversarial networks" [53]. The model for DCGAN was based in research around convolutional neural networks, and how they might offer opportunities for growth in other machine learning models. The paper was focused on the generation of sudo-natural images, as GAN models are so highly effective in image generation tasks. While most deep learning algorithms are black-box methods, it is possible through careful tuning to examine the underlying functions of a Convolutional Neural Network (CNN) model. The authors made use of several changes to traditional CNN architecture, from the following papers: • Striving for simplicity: The all convolutional net [54] • Inceptionism: Going deeper into neural networks [55] • Bi-directional GAN The Bi-directional Generative Adversarial Network (BiGAN) was proposed in a 2017 paper called "Adversarial feature learning" [59]. Similarly to MGAN (see Section 6.15) this is a three party model, consisting of an encoder, a generator, and a discriminator. The role of the encoder is to map data to a latent space representation . Donahue et al specify that the encoder is taught to invert the generator, even though the modules do not interact with one another or directly process the other module's outputs. The BiGAN model is meant to excel at tasks that involve semantic data and representation. They are also an entirely unsupervised model in machine learning. Interestingly, BiGAN was brought into the spotlight in Bioinformatics in a 2021 paper titled "BiGAN: LncRNA-disease association prediction based on bidirectional generative adversarial network" [60], the BiGAN model proved highly effective. When compared against the three gold-standard algorithms for detecting the "associations of IncRNA-disease pairs" [60], BiGAN achieved the highest scores, including 93.1% for the AUC. That was several percentage points higher than the standard methods. BiGAN models are now found in many different fields, including research into malware. MalFox The creation of MalFox, a GAN model for creating attacks and new malware [61] gave an important and powerful tool to those testing or attacking existing systems. MalFox is an amalgamation of parser, generator, and discriminator layers, which takes as input Windows Portable Executable files, or PEs, and outputs the same executables. This makes it a more practical tool that the more common adversarial example generators which often take an image created by a feature extraction process and don't create functioning malware in pre-approved file types. Since its inception, MalFox has undergone more than one transformation, but objectives functions for the LSGAN model, from [65]. ( ) = 1 2 E ∼ [( ( ) − ) 2 ] + 1 2 E ∼ ( ) [ ( ( )) − ) 2 ](14)( ) = 1 2 E ∼ ( ) [( ( ( )) − ) 2 ](15) AC-GAN The auxiliary classifier GAN (ACGAN) was proposed in 2016 by Odena et al [66]. The ACGAN variant was proposed, at the time, for use in image generation, but has since moved into other subjects, as many GAN models do due to their easily transferable nature. The variant distinguishes itself by the use of an auxiliary decoder network within the discriminator. As a result, the algorithm can: • Give as output the label of the class for training samples. • Output a subset of the set of latent variables used to generate the samples. According to Navidan et al. [14], the strongest point of difference between the ACGAN model and the CGAN variant is that in order to determine class labels, the CGAN relies on the conditioning of the generator. In contrast, the ACGAN predicts class labels due to the auxiliary decoder network. The way the ACGAN predicts class labels can be found in Equation 16. IS GAN The Identity-Sensitive Generative Adversarial Network was proposed and focused on face photo-sketch synthesis [67]. This is the process by which a photo of a face is turned into a sketch through machine learning. The goal of creating the ISGAN model was to create a formal image translation task that addresses the problem of turning a photo into a pseudo-hand drawn sketch. This area does have a security and police angle -on occasion, when given a poor quality image of the face of a suspect, turning it into a sketch can help to idetify features that may not be as prominent in the original photo. Especially when machine learning is involved, allowing the ISGAN to understand and augment the original image. ISGAN is not the only GAN model that has been applied to this task, but, when the benchmark tests were run against the current state-of-the-art methods, ISGAN was either on par or above them in score [67]. InfoGAN The rather than a vanilla GAN. fvGAN In [68], the authors develop a method by which they can utilise GAN to build malicious code into PDF files in such a way as to evade detection by even schemes dedicated to the detection of PDF malware. The proposed method, feature vector GAN or fvGAN, took in the fact that features in PDF files are highly interconnected and interdependent, meaning one cannot simply change the features to the ones required. First they had to pull out the features that were most essential. Using mimicus, an invention of their own design, they were able to pull feature vectors with 135-dimensions from the files. Once the fvGAN had been trained on the Contagio and Surrogate datasets (from the original PDFRate study [69]) of both malicious and benign PDFs, they used it to create PDFs with content injection attacks to great effect. CycleGAN The CycleGAN model is foremost an image translation mechanism [70]. Cycling an 'unpaired' image from one domain to another is its main purpose. Proposed in 2017 by Zhu et al [71], CycleGANs have become a reliable tool in image processing, and have assisted researchers in many domains. With regards to security, it is important to note that CycleGAN models can be used for biometrics -particularly facial recognition. More recently, a CycleGAN variant was proposed for video-to-video translations -Mocycle-GAN [72]. This raises the possibility of using this type of GAN to build facial recognition into CCTV software. ProGAN The Proximity Generative Adversarial Network, or ProGAN [73], is meant to preserve the proximities of instances and samples that are reduced in dimensionality. The original proximity in the space prior to dimensionality reduction must be preserved, and thus ProGan was created. The proximity of nodes in this subject is classed as first-order, second-order, and so on. If a node is connected to another node with an edge, it is considered first-order. These relationships are preserved through network embedding. MGAN Mixture Generative Adversarial Networks (MGAN) was proposed focusing on overcoming the mode collapse problem in vanilla GAN models (see Section 6.1.1). This problem is a serious risk for standard GAN models. MGAN seeks to address that issue by using multiple generators to create generated output based on the real data given to the discriminator [74]. The generators are trained simultaneously, rather than sequentially, and the resulting distributions can be mixed to achieve a realistic distribution. The ultimate goal is to create a three-party minmax game, Areas of Use There are many areas in which Generative Adversarial Networks are of use, and these include a selection of cybersecurity related topics. A broad overview of the types of use each different model of GAN is used in is shown in the table in Section 6.15. Classification and Images One of the first, and enduring, tasks for which GANs are used is that of image classification [75]. The ease with which GANs can compare and create images with the necessary similarities and contextual elements, meaning the creation of many images that belong to clear classes, is a task that GAN models do so well even nonimage based tasks are often translated into images [76] for images. While the authors did not achieve a high increase in classification, the principle of their work has been examined by others as well. One study, [78] found they could increase classification accuracy of malware samples by 6% through the generation of synthetic examples of malware for training purposes. In [79], authors implemented GAN for the classification of greyscale images that were created by transforming malware files with feature extraction. This task is suited to GAN schemes because GAN, more so than any other problem space, excels at image classification. In this study, the classifier's performance improved by approximately 6%. In [80], the authors again translate malware files into greyscale images in order to use GAN models on them for classification. GAN systems are excellent at picking out images that have significant similarity, which in this case means that they belong to the same malware families. Using the Microsoft Malware Classification Challenge [81], the system is run against AE-SVM [82], tDCGAN [83], Strand [84], and MCSCasm [85], It performs better when classifying the malware family the images belong to than these state of the art classifiers, with the lowest error rate. Data Augmentation, Rare Classes, and Balancing a Dataset Machine learning models suffer from a desperate need for training data. The amount of data needed to train complex systems to the point at which they know how to deal with the data coming into their systems is enormous. And, unless they are an unsupervised learning model, that data requires logging and labelling. This task is an enormous undertaking, and one that requires human input, hours upon hours of sitting at computers and labelling each piece of data that will be sent to the model for training and verification processes. GAN is an effective tool to potentially solve some of the issues which arise out of a need for data augmentation [86]. Instead of an individual creating hashes of new malware as it arrives, GANs can be used to generate and generalise synthetic malware examples for machine learning models. Rather than learning through the hashed values of malware files, the GAN can produce images of malware files and rare families for training purposes. The importance of feature extraction for learning in malware detection is of particular weight in this scenario. This is shown clearly in [78], in which they augment their malware datasets with synthetic samples created by GAN models. There are other problems with the data necessary for machine learning models too. Augmenting an existing dataset such that the different types of data, the different classes, are able to be trained for the highest levels of accuracy is highly important in classifying data. It is a major use of GAN schemes (see [87][88][89][90][91][92][93][94]). Taking a dataset that is perhaps too small, or has classes that are too imbalanced to learn robustly, is an area GAN models shine. As an example, [95] focuses on using GAN models to create new samples of different classes of malware in order to balance a dataset on which to train machine learning models. In many existing datasets, especially those related to cybersecurity, there exists a significant imbalance in the classes of data within a dataset. There are many sets where the class ratios are significantly imbalanced. As seen in [96], a dataset with very unbalanced classes can be made more even across types using a GAN to solve the rarity of certain classes. The involved datasets were malware designed as Android APKs, which the authors translated into greyscale images. By supplementing these datasets of Android malware with GAN methods, the authors were able to achieve increases of 5-20% in the F1-score. This shows the power of GAN models in augmenting rare data. Zero Day Malware The accurate and speedy classification of malware files and malicious code in computer systems is a expansive task, and one that has remained an enduring problem in the cybersecurity domain [83]. This is a task that is getting bigger and more pressing, not losing significance. The ease with which malicious code can be edited once antivirus software has been updated to detect that particular type of malware means that the creators of this malware have the edge in the battle between attackers and defenders. The sophisticated attacks using polymorphic malware (see 8.2, which morphs and changes itself in order to escape detection, make the task of identifying the malicious code even more difficult [84]. One of the bigger stumbling blocks in malware detection is the unseen or 'zero-day' attacks [99] and Hollowmal as techniques to perturb the data for attacking, when used against the online malware repository Virus Share [36], the detection rate was minimised to 56.2%. In [104], the authors use a standard GAN model against a Deep Neural Network defence system in order to create Windows malware that is perturbed just enough to evade detection and retain its original purpose and function. The authors achieved this using raw byte sequences and training the GAN on existing byte sequences of malware. They achieved evasion rates of more than 50%. Applied Attack GAN Models The flipside of using GAN to generate unseen malware and use it to train antivirus and IDS models, is the use of GAN to create highly effective, theoretical attacks on existing systems. The possibilities offered by GANs for creating new 'adversarial examples' to use against detection systems, mean that there is significant opportunity to build new malware with the same functionality, but in a way that is fast and avoids detection with high rates of success. As we discussed in Section 6, the creation of models like MalGAN and MalFox (see Section 6.7 and 6.6), are clear demonstrations of the attack potential of GAN models in generating malware. In [105], the authors managed to build a model which evaded firewalls in order to attack Android systems with a success rate of 95%. This is a troubling note for security against new malware creation techniques. The use of these systems to train the new generation of malware detection schemes should be therefore a high priority for security researchers and developers. Discussion As popular as GAN research is, there are almost as many types of proposed models for GAN as there are papers discussing GANs. There are some which can be easily seen and categorised as types of the same genus, and then there are those rare new examples, which offer a fresh method for implementing GANs within a research or work setting. Surveying as many papers proposing solutions to difficulties in particular sectors results in surveying almost as many problems in others. We have gathered some particular points of interest and problems which may offer future lines of research or simply act to temper future models with an eye towards creation of new malware detection schemes. Malware and the Sandbox One issue with identifying malware based on features and contextual relationships is that it may require the malware be run before it can be accurately identified. This type of analysis is dynamic analysis, whereas detection systems which use the file without running it are performing static analysis. The majority of the papers we have assessed in this study propose static analysis [106]. However, this then causes concern for the implementation of these methods -can they run successfully in realtime, stopping the malware from running even before it has been classified? This puts a level of vulnerability into the scheme. What damage will be done while the machine identifies the currently executing program as malware? A mitigation for this type of problem is to run the programs in a sandbox before classification. This means they cannot affect the performance of the system or exploit any vulnerabilities prior to identification [107]. Another suggestion is that the program is not run until the binaries have been extracted and used to try to identify the malware family to which it belongs. This problem is not specific to Deep Learning or Machine Learning based detectors -it has been of concern for many years, and has often been addressed using the sandbox option. However, this simply lead to the creation of "environmental detection" in malware [108], allowing the code to detect when it was being run in a sandbox and when it was in the real system environment. Polymorphism, Evolution and the Dangers of Malware Malware developers have taken stock of the current state of research in the area and used it to their advantage. New types of polymorphic malware, which twists and turns itself into code-based pretzels in order to avoid detection, are able to fool signature-based detection systems [106]. GANs schemes have the ability to perturb the code of existing malware code and use these unseen examples to train new models. This ability has potential to defend against polymorphic malware. One method, discussed in [106], is to use malware behaviour to teach new detection systems. Of course, this has innate risk, because the malware has to be executed in order to complete the behaviour analysis. In such a case, the idea would be to run the malware sample in a 'sandbox', as discussed in Section 8.1. Another potential problem is fighting the method by which the malware is spread. Social engineering hacking, sending malicious files to email, or using watering hole attacks [iii] to download the malware onto the target computer [110] are all tasks that are reliant on the ways in which the user interacts with their devices. Ideally, a new and improved machine learning based detection scheme would protect users even when they clicked on email links or went to download a PDF from a new site. This makes training effective and accurate classifiers are priority in cybersecurity, and one of the preeminent areas of malware research. Optimisation and Nature Inpired Computing In all machine learning models based on Neural Networks, there is a need to optimise the model. The number of layers, the nodes in each layer, the activation function, the weights. All of these make the model significantly better or worse at its task, and for the most part, they are altered and improved through trial and error. One group of researchers, however, has taken this optimisation problem and made it the focus of their research into GAN models [111]. Using Genetic Algorithms, they run through the different options for optimisation automatically, with the non- [iii] A watering hole attack involves infecting a site the target is likely to visit, rather than directly attacking the target's device. See [109] for a detailed examination of waterholing attacks. denominated sorting genetic algorithm (NSGA-II) finding the best values for the GAN model. This true positive score of the optimised GAN was more than 98% on the MNIST dataset, introduced in [112], which is made up of images of handwritten letters. The same format for optimisation was employed when the model ran on the malware dataset taken from the Vision Institute [113], and achieved a true positive rate of 97.87%, a score several points higher than the GAN run on the same dataset without the optimisation of the NSGA-II algorithm. This suggests there is a road to take for optimising the layout and structure of each of the GAN models, and that it is possible another Nature-Inspired Computing area may be how that is achieved. As such, it is an interesting direction for malware researchers to explore. Data Format and Translation In many of the examples profiled in this paper, the data, be it traffic flow, machine language, API calls, or malware files, is often translated into different types of images. This makes the job of the GAN model easier because they work so well with image classification tasks, but it also means that there is additional complexity in the algorithms due to the necessity of translating the data into the correct format. There are papers, however, such as [95], [114], [62], [61], [115], or [68] which use the raw data, without translating it to a type more palatable to a GAN scheme. This is important because of the overhead of these different models. When investigating to find an efficient model for use in a given domain, ensuring that there is little to no additional or unnecessary overhead seems a significant consideration. One thing common amongst almost all papers discussed is the need to change the form in which the data is used. The pre-processing, referred to in [116] as data triage, is a cost-expensive and high overhead requirement in both time and processing power. There are those that have managed to avoid translating the data overmuch -such as the research presented in [100] -but most translate data into images or carefully sectioned bytes of data. This is an area of concern while real-time application of a GAN based IDS or firewall is a goal. To further the idea of a real-time GAN scheme, the methods of data pre-processsing need to be carefully examined. They offer unacceptably large overheads for real world application. Because malware files are generally simple and popular to translate into images, the task is not as arduous as it might be in other areas of interest. Ethics and Responsibility in Malware Research The ethical questions posed in [117] offer an interesting path for future researchthere is a general lack of discussion in GAN research papers about the ramifications of possible misappropriation or misclassification of the work contained within. This is an important step to consider in all academic research, and the ethical implications of things like Deepfake [118], or AdvandMal [64] are likely to cause extreme and often unintended effects. Whether by GAN or VAE, the type of research in papers which create new models for malware synthesis are an excellent example of why researchers need to be careful in balancing the quest for knowledge and the security/ethical risks of their research. Knowing this type of attack is possible, and with the results they achieved, is important for those who develop defensive mechanisms for this type of attack, but it is also useful to black hat individuals and malicious operators. The amount of detail over the model created, how it operates, and was trained is where that balance of ethics and information comes into play. These things must be looked at carefully and applied with consideration. The situation of neural network applications as they are demands that we take stock of what has been built, how the dataset has been labelled, how the features have been established, and how/where the neural network is going to be deployed. While this work is starting to take place in the field -see [119], [120] -there is much still to cover, and researchers like those in [121] are offering frameworks for incorporating ethics into the fabric of machine learning research. Therefore, like many research topics, it has a flipside, and both sides need serious ethical review and consideration to ensure that the research benefits those who are most vulnerable. Future Research Directions There are still many avenues for potential research. The methods employed by the authors in [103] to avoid the popular step of translating the dataset into sequences or images and instead working on the data directly using the n-gram feature extraction method is certainly an area worthy of future research for more applications. The incredibly low detection rate achieved by The use of Genetic Algorithms to optimise the performance metrics of GAN models [111] is an avenue which could prove very fruitful. The development of real-time, dynamic analysis and detection is a challenge researchers are still only beginning to scratch the surface of [106], and requires further research into the types of secure environments in which this analysis can safely take place. The realm of malware research contains so many possible avenues for research when it comes to GAN algorithms, and this has been illustrated in this paper to the best of our ability. Conclusion This paper has presented a wide range of research in the current malware research space, using different types of Generative Adversarial Models. Our aim is to have other surveys done in related areas, particularly to demonstrate that there has yet to be an in-depth survey paper on the uses of GAN in cubersecurity and malware; we have explained the different metrics used for evaluation; we gave an in-depth review of the datasets currently favoured in the problem space; we included a list on the different GAN models that are discussed throughout the survey; we have delved in depth to the areas of use that are currently most popular, we have endeavoured to provide a discussion and survey that goes into detail so as to give the reader the full picture; and we have presented potential future avenues for research in the area. We hope that this survey of malware research through the lens of Generative Adversarial Networks, and the way in which they can be employed, has given the reader an idea of where to start with their own research in this area, or given them an update state of the art. The field of GANs for malware research is only getting started, and there is much to do, and many questions to be answered. Declarations Figure 1 : 1A standard GAN, SGAN, or LSGAN model, reproduced based on diagrams from Wang et al., 2017 the metrics, and the results for each of the different papers included in the survey are clearly outlined. The metrics are important and effective at communicating why researchers should further investigate GAN methods for different tasks. In Liu, et al., 2022, [13], the authors undertake a thorough, extensive, and careful examination of the state of the art in Adversarial Machine Learning -of which GANs are one type. The stated purpose of this survey is to examine the weaknesses of Machine Learning Intrusion Detection Systems, and look at ways AML models are being implemented to assist in defending them. The over-arching theme, of course, is through the creation of adversarial examples such that Machine Learning approaches can train on unseen data, and learn to spot the necessary contextual and semantic relationships that can signal malicious code. Like our paper, it includes a table of related and similar works and their main features. The paper leans heavily on the uses for machine learning to protect mobile networks from attacks. Interestingly, this paper separates into two distinct parts with regards to the surveying of papers, coded as the Offensive Perspective, and the Defensive Perspective. the discriminator function and (Ω ) is the set of all possible probability distributions over the set Ω . Any image can be , while is any label. Thus, writing ( | ) is calculating the probability that the image , has the label -as calculated by the Discriminator Network. The below shows the equation for calculating the Inception 6. 1 1Vanilla GAN The traditional, or Vanilla, Generative Adversarial Network is the original proposed model from Goodfellow et al's 2014 paper [1]. The authors proposed the GAN model as an alternative to Variational Autoencoders for adversarial machine learning. This original version of the Generative Adversarial Network is a deep learning model, even the original version shows the power of GAN-based schemes for attack purposes. The initial experiments were used against pre-trained classifiers -Decision Tree; Random Forest; Logistic Regression; Support Vector Machine; Multi-Layer Perceptron; Vote; Long Short-Term Memory; Bi-directional LSTM; LSTM Average; Bi-LSTM Average; LSTM Attention; and BiLSTM Attention. The evasion rate -the percentage of times the program was classified as benign by the systems -MalFox achieved was 99% minimum across the board. This is a stunning display of the power of these schemes. Furthermore, when tested against the open-access giant VirusShare, the detection rate was only 29.7% on average. The evasion rate on the same was averaged as 56.2%. MalFox and the experiments done show exactly how powerful these schemes can be. 6.7 MalGAN In 2017, the authors [62] created a GAN which proposed black-box adversarial examples for attacking via Windows binaries. This scheme, called MalGAN, is now widespread, with multiple different variants and branches of development. The use of binaries, and portable executable files, is one of the ways this is so successful at showing the potential of GAN-based attacks. The authors of the original MalGAN were able to get the detection rate down to almost zero. This clearly demonstrates the danger that is posed by GAN attack systems. Since its inception, MalGAN has been modified and improved, under the auspices of creating the best database for the training of robust detection schemes. [63] proposed using an LSGAN model to address weaknesses like mode collapse. The model they propose still uses the MalGAN scheme, after the use of the LSGAN method, which involves implementing a Least Square function. Their purpose was making MalGAN more robust and avoiding the potential fallout from limiter problems or mode collapse. The authors are still focused on creating adversarial examples, though they focus more on poisoning attacks as well as the traditional 'perturbation' attacks, which change only a small portion of the code while still retaining the capabilities or functionality of the original sample. They are not the only researchers to build a version of MalGAN with Least Square functions to increase the robustness of the function. [64] also suggested the use of a Least Square function in order to minimize mode collapse. The authors of this paper also changed up the activation functions, and added LeakyReLU to the mix. They achieved an 85% success rate over seven different ML classifiers. 6.8 Least Square GAN Mao et al propose the creation of a GAN variant called the Least Square GAN, (LSGAN)[65]. Named for its innovation, the model uses the Least Squares equation for the discriminator. This helps to minimise the Pearson divergence of 2 . The Pearson 2 Divergence is a variant of the -divergence, The LS function can help distance correctly classified samples from the genuine data, improving the performance of the classifier, and thus increasing the training level of the generator. The objective functions of the LSGAN model are presented in Equation 14, The Information Maximising Generative Adversarial Network, or InfoGAN, model was first proposed in Chen et al, in 2016 [46]. The authors noted the ability of InfoGAN's model to untangle images of handwritten characters. The model was tested and trained using the MNIST dataset (see Section 5.8). It was also utilised on 3-dimensional images of faces, and on pictures of house street numbers. In performance, the InfoGAN model adds 'negligible' complexity to the vanilla GAN (6.1) model. The training itself was based on the training done for a DCGAN (6.3), rather than the traditional two party game with vanilla GAN. The parties involved in a MGAN minmax game are: the discriminator, the classifier, and the set of generators. The different generators are meant to work harmoniously, and the authors point out the importance of having the different generators specialise in different data modes. GAN Models Balancing datasets Attack & Security Malware Functional Malware Adversarial Examples Malicious Traffic Feature Extraction Phishing URLs the ease it provides when creating augmented datasets using GAN schemes. The implications for the us in malware are self-evident. The ability to generate new samples of malware families in order to train machine learning based detection schemes on what the features of a family of malware are, is a leap forwards in terms of defensive technology. for an example, in [77], the authors use deep learning GAN models to generate unseen malware examples and train schemes against the signatures of these new malware Traditional antivirus software relies on the use of hash codes. Once the authors of the antivirus software have identified a piece of malicious code, it is hashed into a value string that specifically identifies that piece of code. The antivirus definitions are updated, and the program knows how to recognize that code, should it come into contact with the computer on which the defense software is installed. This does, however, rely on the idea that this malware has been identified before it turns up on the target system. Because the slightest change in the malicious code causes cascading changes in the hash code, all a malware developer needs to do to escape the antivirus system is move a portion of code to a different place within the program. This retains functionality, while changing the hash code of the program enough that it will no longer be picked up by traditional antivirus schemes. Until the antivirus developers find and identify the new variant of the malware, the traditional scheme will not notice the altered malware and it will be able to operate undetected. GAN has become a tool for building new adversarial examples that the defender can train with to prevent this outcome[97]. Similarly, in[98], the authors address this problem using GAN-based adversarial examples to train their blockchain method of intrusion detection. Their LSTM-CGAN model for generating these examples allowed the resulting classifiers to jump several percentage points in accuracy when the classifier was trained on the AEs as opposed to when it was trained on the unenhanced dataset. provided an explanation of not only what Generative Adversarial Networks are and how they are trained and assessed, that we have also given an effective grounding in the applications within the malware research community, which GANs may work with both in the current literature and in any potential future research. To that end, we have iterated through the explanation of GAN's basic functions; the work on Table 1 : 1Types of GAN Research in Related Workshave attempted in this survey. Overall, we believe that we have contributed valuable commentary on the state of the art in GAN models for malware research. Table 2 : 2Metrics in Machine Learning Classifiers4.1 Evaluation Metrics The TP, FP, TN, FN scores are often tabulated as a Confusion Matrix to show the performance of the ML algorithm. An example Confusion Matrix can be found in The Comodo Database[33], maintained by Comodo Antivirus, contains sample files of malware. As part of a program to encourage research, Comodo partnered with universities and launched Comodemia[34]. This gave access to Comodo's tools to researchers internationally, for research purposes only. The Comodo malware database primarily contains files classified as unknown malware -totalling 147,103instances. The other categories are Trojan viruses (462 instances) and Unwanted Applications (13 instances). It is a clearly unbalanced dataset. However, like the other datasets, it is used to train and test different machine learning classifiers, asThe VirusShare database[36] is a large online, open-source repository for malware.A user account is required for access, but anyone with an account can access and download the live viruses in the database. The database is found at VirusShare.com, and is maintained by Corvus Forensics, though anyone can submit files to be added to the dataset. Some researchers, like[37], in which 14,616 unique examples were taken from the VirusShare database, have taken portions of the VirusShare database and melded them with other datasets in order to balance and augment datasets as necessary. In[38], the VirusShare dataset was augmented with malware obtained from Kaggle, in order to test the author's proposed malware detection scheme, and was used as a benchmark when the new model was run against VirusTotal's (seeSection 5.2) antivirus detection program. The Microsoft Malware Classification Challenge [39], made available an open source database of malware examples for Windows. The challenge was part of a general push to coders into creating their own deep learning methods for malware classification [40]. It can be found primarily on Kaggle [ii] . It was an open competition on the site, for teams to come together and develop their own solutions to the challenge. It was completed by 377 teams during the open competition time between April 13 -18, 2015. The dataset contains more than 20,000 malware examples, and [ii] Dataset and competition information can be found here: https://www.kaggle.com/c/malware-classification according to the authors of the challenge [41], as of 2018 it had been cited in over 50 research papers. It is now a widely used dataset for machine learning research, The MNIST dataset, or Modified NIST dataset, a collection of images of handwritten characters, was introduced in 1998 by LeCun et al[45], for the primary purpose of computer vision and recognition tasks in machine learning. It contains characters which are clear representations, as well as those which have been perturbed to examine the extent to which a computer vision algorithm can recognise deformed figures. Many models dealing with the challenges of computer vision in machine learning have utilised this dataset. For example, the InfoGAN model,discussed in Section 6.11, was trained and tested on the MNIST dataset,[46], before moving on to 3D renderings. Another study, surveying the effectiveness of different models of GANs, used the MNIST dataset to benchmark the performance of each model[47]. Since its inception, new, updated versions of MNIST have been proposed. One such dataset is EMNIST (Extended MNIST), which takes the dataset from digits only into handwritten alphanumeric characters[48], with a total of 814,255 samples in all classes combined.Hardware components • Requested permissions • App components • Filtered intents • Restricted API calls • Used permissions • Suspicious API calls • Network addresses 5.5 Comodo in [35]. 5.6 VirusShare 5.7 Microsoft Malware Classification Challenge (2015) as in [42], [43], [44]. 5.8 MNIST the cDCGAN model is a deep CNN. The Leaky activation function is used in this model. The generator matches its picture quality to 32x32 pixels, with a LeakyReLu function.Batch normalization: Accelerating deep network training by reducing internal covariate shift[56] 6.4 cDCGAN The conditional Deep Convolutional Generational Adversarial Network, or cDCGAN, takes the properties of both the CGAN (See section 6.2) and DCGAN (see section 6.3) models. It was proposed as part of a paper focusing on handwritten Arabic characters [57], a significantly more complicated task than identifying English alphanumeric characters. Arabic characters are distinctly different in that Arabic lettering can have similar characters to the extent that they "are only distinguishable by dots"[57]. The database utilised was the AHDBase/MADBase [58], containing 70,000 digits, chosen because it was the database that best matched the MNIST database of numeric handwritten digits. (5.8). The discriminator in Table 3 : 3The different types of GAN models surveyed in this paper with their most common uses. . Because of the types of data that a GAN model can create, generating about the actual structure of the system they are attacking. In a black box attack, the attack must be generalised enough that it can be employed successfully against any defender. In MalFox, which was trained specifically with Obfusmal, Stealmal,new adversarial examples and zero day malware is a possible way to train a security model on unseen data. In [100], the authors use a GAN model they name TrafficGAN to create new malicious traffic patterns for zero-day attacks, partially by including noise into the data as substitution for some unseen traffic, In [101], a standard variation GAN with a LeakyReLU activation function is implemented to train IDS models on unseen malware. While their contribution on the whole is an increase of only 1% accuracy overall, the idea of using the data generated by GAN to increase the robust nature of a model on data it hasn't seen in training sets is a useful and practical one. In fact, the authors of [62] use GAN modelling to create black- box attack methods. Their chosen model in particular, MalGAN (see Section 6.7), was created entirely towards the view that GAN schemes could create functional and unseen examples of malware to attack machine learning systems. The attacks they conducted on non-neural network systems managed to achieve a True Positive Rate of zero, while on the neural network based models (RF, LR, DT, SVM, MLP, VOTE), which had achieved a TPR of 92% minimum prior to the attack tests, managed to achieve a TPR of 0.19% maximum when the generated attack data was integrated into the testing set. This study shows how vulnerable IDS or malware detection systems are to the data and attacks generated by a malicious GAN. On the other had, when the authors of [102] generated their own zero-day attacks and added them into the training datasets for their IDS, they were able to achieve a success rate of 84% when classifying unseen malware. Utilising the Microsoft Malware Classification Challenge (2015) dataset [81], they outperformed 14 other state-of-the-art systems, and achieved a 98% success rate when classifying the zero day attacks. This shows that using GAN to create unseen examples and zero-day attacks can be a powerful force for creating truly robust classifiers and detectors. 7.4 Detection Evasion Given the success at generating unseen examples in order to train more robust systems, the use of GAN methods to create malicious code which evades detection is a logical step. Building GAN schemes which can generate malicious files that evade current detection schemes is necessary for the training of the next generation of malware detection schemes. In [103], the authors create a scheme using GAN to evade detection from a broad range of machine learning models -Multi-layer Perceptron, Decision Tree, Logistical Regression, Support Vector Machine, Random Forest -and found that feature extraction was the key in making their scheme achieve the minimal True Positive Rate (TPR). As they increased the selection of features in both attacker and defender, the TPR rose. While the authors take the steps of adding benign features into the malicious examples they create, they achieve a slightly less impressive TPR than the previously mentioned scheme -the lowest TPR is under 11%. Unlike the previous model, the authors utilise n-gram feature selection, a method borrowed primarily from Natural Language Schemes, but which removes the step of translating the malware into a different form -such as the popular method of making malware 'images' -and allows the model to work on the raw data itself. The creation of the MalFox GAN scheme in [61] shows the ability of GAN schemes to create black-box attacks, in which the attacker knows nothing Competing interestsThe authors declare that they have no competing interests.FundingThe authors would like to thank the Ministry of Business, Innovation, and Employment (MBIE) from the New Zealand Government to support our work with the grant (MAUX1912) which made it possible for us to conduct the research. Generative Adversarial Nets. I Goodfellow, J Pouget-Abadie, M Mirza, B Xu, D Warde-Farley, S Ozair, A Courville, Y Bengio, Advances in Neural Information Processing Systems. 27Goodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., Courville, A., Bengio, Y.: Generative Adversarial Nets. Advances in Neural Information Processing Systems 27 (2014) Generative adversarial network for global image-based local image to improve Malware classification using convolutional neural network. S Jang, S Li, Y Sung, 10.3390/app10217585Applied Sciences (Switzerland). 1021Jang, S., Li, S., Sung, Y.: Generative adversarial network for global image-based local image to improve Malware classification using convolutional neural network. Applied Sciences (Switzerland) 10(21), 1-14 (2020). doi:10.3390/app10217585 Exploring Efficiency of GAN-based Generated URLs for Phishing URL Detection. T D Pham, T T T Pham, S T Hoang, V C Ta, 10.1109/mapr53640.2021.95852872021 International Conference on Multimedia Analysis and Pattern Recognition (MAPR) 00. Pham, T.D., Pham, T.T.T., Hoang, S.T., Ta, V.C.: Exploring Efficiency of GAN-based Generated URLs for Phishing URL Detection. 2021 International Conference on Multimedia Analysis and Pattern Recognition (MAPR) 00, 1-6 (2021). doi:10.1109/mapr53640.2021.9585287 Generative adversarial network: An overview of theory and applications. A Aggarwal, M Mittal, G Battineni, 10.1016/j.jjimei.2020.100004International Journal of Information Management Data Insights. 11100004Aggarwal, A., Mittal, M., Battineni, G.: Generative adversarial network: An overview of theory and applications. International Journal of Information Management Data Insights 1(1), 100004 (2021). doi:10.1016/j.jjimei.2020.100004 Generative adversarial networks: introduction and outlook. K Wang, C Gou, Y Duan, Y Lin, X Zheng, F.-Y Wang, IEEE/CAA Journal of Automatica Sinica. 44Wang, K., Gou, C., Duan, Y., Lin, Y., Zheng, X., Wang, F.-Y.: Generative adversarial networks: introduction and outlook. IEEE/CAA Journal of Automatica Sinica 4(4), 588-598 (2017) Android malware detection through generative adversarial networks. M Amin, B Shah, A Sharif, T Ali, K.-I Kim, S Anwar, 10.1002/ett.3675Transactions on Emerging Telecommunications Technologies. 332Amin, M., Shah, B., Sharif, A., Ali, T., Kim, K.-I., Anwar, S.: Android malware detection through generative adversarial networks. Transactions on Emerging Telecommunications Technologies 33(2) (2022). doi:10.1002/ett.3675 Evolution of ransomware. P O&apos;kane, S Sezer, D Carlin, 10.1049/iet-net.2017.0207IET Networks. 75O'Kane, P., Sezer, S., Carlin, D.: Evolution of ransomware. IET Networks 7(5), 321-327 (2018). doi:10.1049/iet-net.2017.0207 Automatic Steganographic Distortion Learning Using a Generative Adversarial Network. W Tang, S Tan, B Li, J Huang, 10.1109/lsp.2017.2745572IEEE Signal Processing Letters. 2410Tang, W., Tan, S., Li, B., Huang, J.: Automatic Steganographic Distortion Learning Using a Generative Adversarial Network. IEEE Signal Processing Letters 24(10), 1547-1551 (2017). doi:10.1109/lsp.2017.2745572 I Corley, J Lwowski, J Hoffman, 1912.10070Destruction of Image Steganography using Generative Adversarial Networks. arXiv. Corley, I., Lwowski, J., Hoffman, J.: Destruction of Image Steganography using Generative Adversarial Networks. arXiv (2019). 1912.10070 A Kumar, A Biswas, S Sanyal, 1801.03244eCommerceGAN : A Generative Adversarial Network for E-commerce. arXiv. Kumar, A., Biswas, A., Sanyal, S.: eCommerceGAN : A Generative Adversarial Network for E-commerce. arXiv (2018). 1801.03244 Cross lingual cybersecurity analytics in the international dark web with adversarial deep representation learning. M Ebrahimi, Y Chai, S Satani, H Chen, MIS Quarterly Forthcoming. Ebrahimi, M., Chai, Y., Satani, S., Chen, H.: Cross lingual cybersecurity analytics in the international dark web with adversarial deep representation learning. MIS Quarterly Forthcoming (2021) A survey of deep learning methods for cyber security. D S Berman, A L Buczak, J S Chavis, C L Corbett, 10.3390/info10040122Information (Switzerland). 104122Berman, D.S., Buczak, A.L., Chavis, J.S., Corbett, C.L.: A survey of deep learning methods for cyber security. Information (Switzerland) 10(4), 122 (2019). doi:10.3390/info10040122 Adversarial Machine Learning: A Multilayer Review of the State-of-the-Art and Challenges for Wireless and Mobile Systems. J Liu, M Nogueira, J Fernandes, B Kantarci, 10.1109/comst.2021.3136132IEEE Communications Surveys and Tutorials. 241Liu, J., Nogueira, M., Fernandes, J., Kantarci, B.: Adversarial Machine Learning: A Multilayer Review of the State-of-the-Art and Challenges for Wireless and Mobile Systems. IEEE Communications Surveys and Tutorials 24(1), 123-159 (2022). doi:10.1109/comst.2021.3136132 Generative Adversarial Networks (GANs) in networking: A comprehensive survey & evaluation. H Navidan, P F Moshiri, M Nabati, R Shahbazian, S A Ghorashi, V Shah-Mansouri, D Windridge, 10.1016/j.comnet.2021.108149Computer Networks. 194108149Navidan, H., Moshiri, P.F., Nabati, M., Shahbazian, R., Ghorashi, S.A., Shah-Mansouri, V., Windridge, D.: Generative Adversarial Networks (GANs) in networking: A comprehensive survey & evaluation. Computer Networks 194, 108149 (2021). doi:10.1016/j.comnet.2021.108149. 2105.04184 Dynamic traffic feature camouflaging via generative adversarial networks. J Li, L Zhou, H Li, L Yan, H Zhu, 2019 IEEE Conference on Communications and Network Security (CNS). IEEELi, J., Zhou, L., Li, H., Yan, L., Zhu, H.: Dynamic traffic feature camouflaging via generative adversarial networks. In: 2019 IEEE Conference on Communications and Network Security (CNS), pp. 268-276 (2019). IEEE Shantanu: A Review on Application of GANs in Cybersecurity Domain. A Arora, 10.1080/02564602.2020.1854058IETE Technical Review (Institution of Electronics and Telecommunication Engineers. 392India)Arora, A., Shantanu: A Review on Application of GANs in Cybersecurity Domain. IETE Technical Review (Institution of Electronics and Telecommunication Engineers, India) 39(2), 433-441 (2022). doi:10.1080/02564602.2020.1854058 Performance metrics. R C Eberhart, R W Dobbins, L V Hutton, 10.1016/B978-0-12-228640-7.50013-1Neural Network PC Tools. Academic PressEberhart, R.C., Dobbins, R.W., Hutton, L.V.: Performance metrics. In: Neural Network PC Tools, pp. 161-176. Academic Press, ??? (1990). doi:10.1016/B978-0-12-228640-7.50013-1 S Barratt, R Sharma, 10.48550/arxiv.1801.01973A Note on the Inception Score. arXiv. Barratt, S., Sharma, R.: A Note on the Inception Score. arXiv (2018). doi:10.48550/arxiv.1801.01973. 1801.01973 Generative adversarial networks (gans): An overview of theoretical model, evaluation metrics, and recent developments. P Salehi, A Chalechale, M Taghizadeh, arXiv:2005.13178arXiv preprintSalehi, P., Chalechale, A., Taghizadeh, M.: Generative adversarial networks (gans): An overview of theoretical model, evaluation metrics, and recent developments. arXiv preprint arXiv:2005.13178 (2020) Rethinking the inception architecture for computer vision. C Szegedy, V Vanhoucke, S Ioffe, J Shlens, Z Wojna, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionSzegedy, C., Vanhoucke, V., Ioffe, S., Shlens, J., Wojna, Z.: Rethinking the inception architecture for computer vision. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2818-2826 (2016) Pros and Cons. A Borji, 10.48550/arxiv.1802.03446GAN Evaluation Measures. arXiv. Borji, A.: Pros and Cons of GAN Evaluation Measures. arXiv (2018). doi:10.48550/arxiv.1802.03446. 1802.03446 DGArchive A deep dive into domain generating malware. D Plohmann, Plohmann, D.: DGArchive A deep dive into domain generating malware (2015). Revised Selected Papers. C Choudhary, R Sivaguru, M Pereira, B Yu, A C Nascimento, M D Cock, 10.1007/978-981-13-5826-5_50Security in Computing and Communications, 6th International Symposium. Bangalore, IndiaChoudhary, C., Sivaguru, R., Pereira, M., Yu, B., Nascimento, A.C., Cock, M.D.: Security in Computing and Communications, 6th International Symposium, SSCC 2018, Bangalore, India, September 19-22, 2018, Revised Selected Papers. Communications in Computer and Information Science, 640-655 (2019). doi:10.1007/978-981-13-5826-5_50 MaldomDetector: A system for detecting algorithmically generated domain names with machine learning. A O Almashhadani, M Kaiiali, D Carlin, S Sezer, 10.1016/j.cose.2020.101787Computers & Security. 93101787Almashhadani, A.O., Kaiiali, M., Carlin, D., Sezer, S.: MaldomDetector: A system for detecting algorithmically generated domain names with machine learning. Computers & Security 93, 101787 (2020). doi:10.1016/j.cose.2020.101787 . Virustotal -Home, VirusTotal -Home. https://www.virustotal.com/gui/home/upload Accessed 2022-08-26 H S Anderson, P Roth, 1804.04637EMBER: An Open Dataset for Training Static PE Malware Machine Learning Models. arXiv. Anderson, H.S., Roth, P.: EMBER: An Open Dataset for Training Static PE Malware Machine Learning Models. arXiv (2018). 1804.04637 Evaluation of Supervised Machine Learning Techniques for Dynamic Malware Detection. H Zhao, M Li, T Wu, F Yang, 10.2991/ijcis.11.1.87International Journal of Computational Intelligence Systems. 111Zhao, H., Li, M., Wu, T., Yang, F.: Evaluation of Supervised Machine Learning Techniques for Dynamic Malware Detection. International Journal of Computational Intelligence Systems 11(1), 1153-1169 (2018). doi:10.2991/ijcis.11.1.87 New Techniques in Profiling Big Datasets for Machine Learning with a Concise Review of Android Mobile Malware Datasets. G Canbek, S Sagiroglu, T T Temizel, 10.1109/ibigdelft.2018.8625275International Congress on Big Data, Deep Learning and Fighting Cyber Terrorism (IBIGDELFT) 00. Canbek, G., Sagiroglu, S., Temizel, T.T.: New Techniques in Profiling Big Datasets for Machine Learning with a Concise Review of Android Mobile Malware Datasets. 2018 International Congress on Big Data, Deep Learning and Fighting Cyber Terrorism (IBIGDELFT) 00, 117-121 (2018). doi:10.1109/ibigdelft.2018.8625275 A survey of intrusion detection from the perspective of intrusion datasets and machine learning techniques. G Singh, N Khare, 10.1080/1206212x.2021.1885150International Journal of Computers and Applications. 447Singh, G., Khare, N.: A survey of intrusion detection from the perspective of intrusion datasets and machine learning techniques. International Journal of Computers and Applications 44(7), 1-11 (2021). doi:10.1080/1206212x.2021.1885150 M Parkour, Contagio Malware Dataset. Parkour, M.: Contagio Malware Dataset (2020). http://contagiodump.blogspot.com/ Accessed 2022-08-26 The Paradox of Choice: Investigating Selection Strategies for Android Malware Datasets Using a Machine-learning Approach. S Sharma, N Kumar, R Kumar, C R Krishna, 10.17705/1cais.04626Communications of the Association for Information Systems. 46Sharma, S., Kumar, N., Kumar, R., Krishna, C.R.: The Paradox of Choice: Investigating Selection Strategies for Android Malware Datasets Using a Machine-learning Approach. Communications of the Association for Information Systems 46, 619-637 (2020). doi:10.17705/1cais.04626 Drebin: Effective and Explainable Detection of Android Malware in Your Pocket. D Arp, M Spreitzenbarth, M Hübner, H Gascon, K Rieck, 10.14722/ndss.2014.23247Proceedings 2014 Network and Distributed System Security Symposium. 2014 Network and Distributed System Security SymposiumArp, D., Spreitzenbarth, M., Hübner, M., Gascon, H., Rieck, K.: Drebin: Effective and Explainable Detection of Android Malware in Your Pocket. Proceedings 2014 Network and Distributed System Security Symposium (2014). doi:10.14722/ndss.2014.23247 Comodo Anti Malware Database Latest Version & Additions. Comodo Anti Malware Database Latest Version & Additions 2022. Your Academics to Collaborate with Comodo. | Comodemia, Get, Comodemia | Get Your Academics to Collaborate with Comodo. https://comodemia.comodo.com/ Accessed 2022-08-26 DeepAM: a heterogeneous deep learning framework for intelligent malware detection. Y Ye, L Chen, S Hou, W Hardy, X Li, 10.1007/s10115-017-1058-9Knowledge and Information Systems. 542Ye, Y., Chen, L., Hou, S., Hardy, W., Li, X.: DeepAM: a heterogeneous deep learning framework for intelligent malware detection. Knowledge and Information Systems 54(2), 265-285 (2018). doi:10.1007/s10115-017-1058-9 VirusShare -Because Sharing is Caring. C Forensics, Forensics, C.: VirusShare -Because Sharing is Caring. www.virusshare.com B Düzgün, A Çayır, F Demirkıran, C N Kahya, B Gençaydın, H Dağ, 2111.15205Benchmark Static API Call Datasets for Malware Family Classification. arXiv. Düzgün, B., Çayır, A., Demirkıran, F., Kahya, C.N., Gençaydın, B., Dağ, H.: Benchmark Static API Call Datasets for Malware Family Classification. arXiv (2021). 2111.15205 The arms race: Adversarial search defeats entropy used to detect malware. H D Menéndez, S Bhattacharya, D Clark, E T Barr, 10.1016/j.eswa.2018.10.011Expert Systems with Applications. 118Menéndez, H.D., Bhattacharya, S., Clark, D., Barr, E.T.: The arms race: Adversarial search defeats entropy used to detect malware. Expert Systems with Applications 118, 246-260 (2019). doi:10.1016/j.eswa.2018.10.011 R Ronen, M Radu, C Feuerstein, E Yom-Tov, M Ahmadi, 10.48550/ARXIV.1802.10135Microsoft Malware Classification Challenge. arXiv. Ronen, R., Radu, M., Feuerstein, C., Yom-Tov, E., Ahmadi, M.: Microsoft Malware Classification Challenge. arXiv (2018). doi:10.48550/ARXIV.1802.10135. https://arxiv.org/abs/1802.10135 Empirical Study on Microsoft Malware Classification. R Chivukula, M Vamsi, T Jaya, M Harini, 10.14569/ijacsa.2021.0120361International Journal of Advanced Computer Science and Applications. 123Chivukula, R., Vamsi, M., Jaya, T., Harini, M.: Empirical Study on Microsoft Malware Classification. International Journal of Advanced Computer Science and Applications 12(3) (2021). doi:10.14569/ijacsa.2021.0120361 R Ronen, M Radu, C Feuerstein, E Yom-Tov, M Ahmadi, 1802.10135Microsoft Malware Classification Challenge. arXiv. Ronen, R., Radu, M., Feuerstein, C., Yom-Tov, E., Ahmadi, M.: Microsoft Malware Classification Challenge. arXiv (2018). 1802.10135 Performance analysis of machine learning and pattern recognition algorithms for malware classification. B N Narayanan, O Djaneye-Boundjou, T M Kebede, 10.1109/NAECON.2016.78568262016 IEEE National Aerospace and Electronics Conference (NAECON) and Ohio Innovation Summit (OIS). Narayanan, B.N., Djaneye-Boundjou, O., Kebede, T.M.: Performance analysis of machine learning and pattern recognition algorithms for malware classification. In: 2016 IEEE National Aerospace and Electronics Conference (NAECON) and Ohio Innovation Summit (OIS), pp. 338-342 (2016). doi:10.1109/NAECON.2016.7856826 Classification of Malware programs using autoencoders based deep learning architecture and its application to the microsoft malware Classification challenge (BIG 2015) dataset. T M Kebede, O Djaneye-Boundjou, B N Narayanan, A Ralescu, D Kapp, 10.1109/naecon.2017.8268747IEEE National Aerospace and Electronics Conference (NAECON). Kebede, T.M., Djaneye-Boundjou, O., Narayanan, B.N., Ralescu, A., Kapp, D.: Classification of Malware programs using autoencoders based deep learning architecture and its application to the microsoft malware Classification challenge (BIG 2015) dataset. 2017 IEEE National Aerospace and Electronics Conference (NAECON), 70-75 (2017). doi:10.1109/naecon.2017.8268747 Malware Classification with Deep Convolutional Neural Networks. M Kalash, M Rochan, N Mohammed, N D B Bruce, Y Wang, F Iqbal, 10.1109/ntms.2018.83287499th IFIP International Conference on New Technologies, Mobility and Security (NTMS). Kalash, M., Rochan, M., Mohammed, N., Bruce, N.D.B., Wang, Y., Iqbal, F.: Malware Classification with Deep Convolutional Neural Networks. 2018 9th IFIP International Conference on New Technologies, Mobility and Security (NTMS), 1-5 (2018). doi:10.1109/ntms.2018.8328749 Gradient-based learning applied to document recognition. Y Lecun, L Bottou, Y Bengio, P Haffner, 10.1109/5.726791Proceedings of the IEEE. 8611Lecun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278-2324 (1998). doi:10.1109/5.726791 X Chen, Y Duan, R Houthooft, J Schulman, I Sutskever, P Abbeel, InfoGAN: Interpretable Representation Learning by Information Maximizing Generative Adversarial Nets. arXiv. Chen, X., Duan, Y., Houthooft, R., Schulman, J., Sutskever, I., Abbeel, P.: InfoGAN: Interpretable Representation Learning by Information Maximizing Generative Adversarial Nets. arXiv (2016). An analysis of generative adversarial networks and variants for image synthesis on MNIST dataset. K Cheng, R Tahir, L K Eric, M Li, 10.1007/s11042-019-08600-2Multimedia Tools and Applications. 79Cheng, K., Tahir, R., Eric, L.K., Li, M.: An analysis of generative adversarial networks and variants for image synthesis on MNIST dataset. Multimedia Tools and Applications 79(19-20), 13725-13752 (2020). doi:10.1007/s11042-019-08600-2 EMNIST: Extending MNIST to Handwritten Letters. G Cohen, S Afshar, J Tapson, A V Schaik, 10.1109/ijcnn.2017.7966217International Joint Conference on Neural Networks (IJCNN). Cohen, G., Afshar, S., Tapson, J., Schaik, A.v.: EMNIST: Extending MNIST to Handwritten Letters. 2017 International Joint Conference on Neural Networks (IJCNN), 2921-2926 (2017). doi:10.1109/ijcnn.2017.7966217 M Mirza, S Osindero, 1411.1784Conditional Generative Adversarial Nets. arXiv. Mirza, M., Osindero, S.: Conditional Generative Adversarial Nets. arXiv (2014). 1411.1784 Combating Mode Collapse in GAN training: An Empirical Analysis using Hessian Eigenvalues. R Durall, A Chatzimichailidis, P Labus, J Keuper, arXiv (2020). 2012.09673Durall, R., Chatzimichailidis, A., Labus, P., Keuper, J.: Combating Mode Collapse in GAN training: An Empirical Analysis using Hessian Eigenvalues. arXiv (2020). 2012.09673 Catastrophic forgetting and mode collapse in GANs. H Thanh-Tung, T Tran, 10.1109/ijcnn48605.2020.92071812020 International Joint Conference on Neural Networks (IJCNN) 00. Thanh-Tung, H., Tran, T.: Catastrophic forgetting and mode collapse in GANs. 2020 International Joint Conference on Neural Networks (IJCNN) 00, 1-10 (2020). doi:10.1109/ijcnn48605.2020.9207181 Conditional generative adversarial nets for convolutional face generation. Class project for Stanford CS231N: convolutional neural networks for visual recognition. J Gauthier, 2Gauthier, J.: Conditional generative adversarial nets for convolutional face generation. Class project for Stanford CS231N: convolutional neural networks for visual recognition, Winter semester 2014(5), 2 (2014) Unsupervised representation learning with deep convolutional generative adversarial networks. A Radford, L Metz, S Chintala, arXiv:1511.06434arXiv preprintRadford, A., Metz, L., Chintala, S.: Unsupervised representation learning with deep convolutional generative adversarial networks. arXiv preprint arXiv:1511.06434 (2015) J T Springenberg, A Dosovitskiy, T Brox, M Riedmiller, arXiv:1412.6806Striving for simplicity: The all convolutional net. arXiv preprintSpringenberg, J.T., Dosovitskiy, A., Brox, T., Riedmiller, M.: Striving for simplicity: The all convolutional net. arXiv preprint arXiv:1412.6806 (2014) A Mordvintsev, C Olah, M Tyka, Inceptionism: Going deeper into neural networks. Mordvintsev, A., Olah, C., Tyka, M.: Inceptionism: Going deeper into neural networks (2015) Batch normalization: Accelerating deep network training by reducing internal covariate shift. S Ioffe, C Szegedy, PMLRInternational Conference on Machine Learning. Ioffe, S., Szegedy, C.: Batch normalization: Accelerating deep network training by reducing internal covariate shift. In: International Conference on Machine Learning, pp. 448-456 (2015). PMLR Conditional Deep Convolutional Generative Adversarial Networks for Isolated Handwritten Arabic Character Generation. I B Mustapha, S Hasan, H Nabus, S M Shamsuddin, 10.1007/s13369-021-05796-0Arabian Journal for Science and Engineering. 472Mustapha, I.B., Hasan, S., Nabus, H., Shamsuddin, S.M.: Conditional Deep Convolutional Generative Adversarial Networks for Isolated Handwritten Arabic Character Generation. Arabian Journal for Science and Engineering 47(2), 1309-1320 (2022). doi:10.1007/s13369-021-05796-0 Arabic handwritten digit recognition. S Abdleazeem, E El-Sherif, 10.1007/s10032-008-0073-5International Journal of Document Analysis and Recognition (IJDAR). 113Abdleazeem, S., El-Sherif, E.: Arabic handwritten digit recognition. International Journal of Document Analysis and Recognition (IJDAR) 11(3), 127-141 (2008). doi:10.1007/s10032-008-0073-5 Adversarial feature learning. J Donahue, P Krähenbühl, T Darrell, 5th International Conference on Learning Representations. International Conference on Learning Representations. Donahue, J., Krähenbühl, P., Darrell, T.: Adversarial feature learning. In: 5th International Conference on Learning Representations. International Conference on Learning Representations (2017) BiGAN: LncRNA-disease association prediction based on bidirectional generative adversarial network. Q Yang, X Li, 10.1186/s12859-021-04273-7BMC Bioinformatics. 221357Yang, Q., Li, X.: BiGAN: LncRNA-disease association prediction based on bidirectional generative adversarial network. BMC Bioinformatics 22(1), 357 (2021). doi:10.1186/s12859-021-04273-7 MalFox: Camouflaged Adversarial Malware Example Generation Based on Conv-GANs Against Black-Box Detectors. F Zhong, X Cheng, D Yu, B Gong, S Song, J Yu, arXiv (2020). 2011.01509Zhong, F., Cheng, X., Yu, D., Gong, B., Song, S., Yu, J.: MalFox: Camouflaged Adversarial Malware Example Generation Based on Conv-GANs Against Black-Box Detectors. arXiv (2020). 2011.01509 Generating Adversarial Malware Examples for Black-Box Attacks Based on GAN. W Hu, Y Tan, 1702.05983Hu, W., Tan, Y.: Generating Adversarial Malware Examples for Black-Box Attacks Based on GAN. arXiv (2017). 1702.05983 LSGAN-AT: enhancing malware detector robustness against adversarial examples. J Wang, X Chang, Y Wang, R J Rodríguez, J Zhang, 10.1186/s42400-021-00102-9Cybersecurity. 4138Wang, J., Chang, X., Wang, Y., Rodríguez, R.J., Zhang, J.: LSGAN-AT: enhancing malware detector robustness against adversarial examples. Cybersecurity 4(1), 38 (2021). doi:10.1186/s42400-021-00102-9 Mal-LSGAN: An Effective Adversarial Malware Example Generation Model. J Wang, X Chang, J Mišić, V B Mišić, Y Wang, J Zhang, 10.1109/globecom46510.2021.96854422021 IEEE Global Communications Conference. 00GLOBECOM 2021 -ProceedingsWang, J., Chang, X., Mišić, J., Mišić, V.B., Wang, Y., Zhang, J.: Mal-LSGAN: An Effective Adversarial Malware Example Generation Model. 2021 IEEE Global Communications Conference, GLOBECOM 2021 -Proceedings 00, 1-6 (2021). doi:10.1109/globecom46510.2021.9685442 Least Squares Generative Adversarial Networks. X Mao, Q Li, H Xie, R Y K Lau, Z Wang, S P Smolley, 10.1109/iccv.2017.304IEEE International Conference on Computer Vision (ICCV). Mao, X., Li, Q., Xie, H., Lau, R.Y.K., Wang, Z., Smolley, S.P.: Least Squares Generative Adversarial Networks. 2017 IEEE International Conference on Computer Vision (ICCV), 2813-2821 (2017). doi:10.1109/iccv.2017.304 A Odena, C Olah, J Shlens, 1610.09585Conditional Image Synthesis With Auxiliary Classifier GANs. arXiv. Odena, A., Olah, C., Shlens, J.: Conditional Image Synthesis With Auxiliary Classifier GANs. arXiv (2016). 1610.09585 IsGAN: Identity-sensitive generative adversarial network for face photo-sketch synthesis. L Yan, W Zheng, C Gou, F.-Y Wang, 10.1016/j.patcog.2021.108077Pattern Recognition. 119108077Yan, L., Zheng, W., Gou, C., Wang, F.-Y.: IsGAN: Identity-sensitive generative adversarial network for face photo-sketch synthesis. Pattern Recognition 119, 108077 (2021). doi:10.1016/j.patcog.2021.108077 A feature-vector generative adversarial network for evading PDF malware classifiers. Y Li, Y Wang, Y Wang, L Ke, Y.-A Tan, 10.1016/j.ins.2020.02.075Information Sciences. 523Li, Y., Wang, Y., Wang, Y., Ke, L., Tan, Y.-A.: A feature-vector generative adversarial network for evading PDF malware classifiers. Information Sciences 523, 38-48 (2020). doi:10.1016/j.ins.2020.02.075 Practical evasion of a learning-based classifier: A case study. P Laskov, 2014 IEEE Symposium on Security and Privacy. Laskov, P., et al.: Practical evasion of a learning-based classifier: A case study. In: 2014 IEEE Symposium on Security and Privacy, pp. 197-211 A Brief Review on Cycle Generative Adversarial Networks. M M Zhu, S Gong, Z Qian, L Zhang, 10.12792/icisip2019.046Proceedings of The 7th International Conference on Intelligent Systems and Image Processing. The 7th International Conference on Intelligent Systems and Image ProcessingZhu, M.M., Gong, S., Qian, Z., Zhang, L.: A Brief Review on Cycle Generative Adversarial Networks. Proceedings of The 7th International Conference on Intelligent Systems and Image Processing 2019, 235-242 (2019). doi:10.12792/icisip2019.046 Unpaired Image-to-Image Translation Using Cycle-Consistent Adversarial Networks. J.-Y Zhu, T Park, P Isola, A A Efros, 10.1109/iccv.2017.244IEEE International Conference on Computer Vision (ICCV). Zhu, J.-Y., Park, T., Isola, P., Efros, A.A.: Unpaired Image-to-Image Translation Using Cycle-Consistent Adversarial Networks. 2017 IEEE International Conference on Computer Vision (ICCV), 2242-2251 (2017). doi:10.1109/iccv.2017.244 L Amsaleg, B Huet, M Larson, G Gravier, H Hung, C.-W Ngo, W T Ooi, Y Chen, Y Pan, T Yao, X Tian, T Mei, 10.1145/3343031.3350937Proceedings of the 27th ACM International Conference on Multimedia. the 27th ACM International Conference on MultimediaMocycle-GANAmsaleg, L., Huet, B., Larson, M., Gravier, G., Hung, H., Ngo, C.-W., Ooi, W.T., Chen, Y., Pan, Y., Yao, T., Tian, X., Mei, T.: Mocycle-GAN. Proceedings of the 27th ACM International Conference on Multimedia, 647-655 (2019). doi:10.1145/3343031.3350937 Progan: Network embedding via proximity generative adversarial network. H Gao, J Pei, H Huang, Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining. the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data MiningGao, H., Pei, J., Huang, H.: Progan: Network embedding via proximity generative adversarial network. In: Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pp. 1308-1316 (2019) MGAN: Training generative adversarial nets with multiple generators. Q Hoang, T D Nguyen, T Le, D Phung, International Conference on Learning Representations. Hoang, Q., Nguyen, T.D., Le, T., Phung, D.: MGAN: Training generative adversarial nets with multiple generators. International Conference on Learning Representations (2018) Generative adversarial networks for hyperspectral image classification. L Zhu, Y Chen, P Ghamisi, J A Benediktsson, IEEE Transactions on Geoscience and Remote Sensing. 569Zhu, L., Chen, Y., Ghamisi, P., Benediktsson, J.A.: Generative adversarial networks for hyperspectral image classification. IEEE Transactions on Geoscience and Remote Sensing 56(9), 5046-5063 (2018) Defending IT systems against intelligent malware. J Kargaard, T Drange, A.-L Kor, H Twafik, E Butterfield, 10.1109/dessert.2018.8409169Proceedings of 2018 IEEE 9th International Conference on Dependable Systems, Services and Technologies. 2018 IEEE 9th International Conference on Dependable Systems, Services and TechnologiesKargaard, J., Drange, T., Kor, A.-L., Twafik, H., Butterfield, E.: Defending IT systems against intelligent malware. Proceedings of 2018 IEEE 9th International Conference on Dependable Systems, Services and Technologies, DESSERT 2018, 411-417 (2018). doi:10.1109/dessert.2018.8409169 Discovering Future Malware Variants By Generating New Malware Samples Using Generative Adversarial Network. Z Moti, S Hashemi, A Namavar, 10.1109/iccke48569.2019.89649139th International Conference on Computer and Knowledge Engineering (ICCKE) 00. Moti, Z., Hashemi, S., Namavar, A.: Discovering Future Malware Variants By Generating New Malware Samples Using Generative Adversarial Network. 2019 9th International Conference on Computer and Knowledge Engineering (ICCKE) 00, 319-324 (2019). doi:10.1109/iccke48569.2019.8964913 Generative Adversarial Network for Improving Deep Learning Based Malware Classification. Y Lu, J Li, Lu, Y., Li, J.: Generative Adversarial Network for Improving Deep Learning Based Malware Classification. 10.1109/wsc40007.2019.9004932Winter Simulation Conference (WSC) 00. Winter Simulation Conference (WSC) 00, 584-593 (2019). doi:10.1109/wsc40007.2019.9004932 Generative Adversarial Network for Improving Deep Learning Based Malware Classification. Y Lu, J Li, 10.1109/wsc40007.2019.9004932Proceedings -Winter Simulation Conference 2019-December. -Winter Simulation Conference 2019-DecemberLu, Y., Li, J.: Generative Adversarial Network for Improving Deep Learning Based Malware Classification. Proceedings -Winter Simulation Conference 2019-December, 584-593 (2019). doi:10.1109/wsc40007.2019.9004932 A weak coupling of semi-supervised learning with generative adversarial networks for malware classification. S Wang, Q Wang, Z Jiang, X Wang, R Jing, 10.1109/icpr48806.2021.9412832Proceedings -International Conference on Pattern Recognition. -International Conference on Pattern Recognition00Wang, S., Wang, Q., Jiang, Z., Wang, X., Jing, R.: A weak coupling of semi-supervised learning with generative adversarial networks for malware classification. Proceedings -International Conference on Pattern Recognition 00, 3775-3782 (2020). doi:10.1109/icpr48806.2021.9412832 R Ronen, M Radu, C Feuerstein, E Yom-Tov, M Ahmadi, Microsoft Malware Classification Challenge (BIG 2015). Ronen, R., Radu, M., Feuerstein, C., Yom-Tov, E., Ahmadi, M.: Microsoft Malware Classification Challenge (BIG 2015) (2018). https://www.kaggle.com/c/malware-classification Accessed 2022-7-23 Autoencoder-based feature learning for cyber security applications. M Yousefi-Azar, V Varadharajan, L Hamey, U Tupakula, 2017 International Joint Conference on Neural Networks (IJCNN). Yousefi-Azar, M., Varadharajan, V., Hamey, L., Tupakula, U.: Autoencoder-based feature learning for cyber security applications. In: 2017 International Joint Conference on Neural Networks (IJCNN), pp. 3854-3861 Zero-day malware detection using transferred generative adversarial networks based on deep autoencoders. J.-Y Kim, S.-J Bu, S.-B Cho, 10.1016/j.ins.2018.04.092Information Sciences. 460Kim, J.-Y., Bu, S.-J., Cho, S.-B.: Zero-day malware detection using transferred generative adversarial networks based on deep autoencoders. Information Sciences 460, 83-102 (2018). doi:10.1016/j.ins.2018.04.092 Polymorphic malware detection using sequence classification methods. J Drew, T Moore, M Hahsler, IEEE Security and Privacy Workshops. SPWDrew, J., Moore, T., Hahsler, M.: Polymorphic malware detection using sequence classification methods. In: 2016 IEEE Security and Privacy Workshops (SPW), pp. 81-87 Malware identification using visualization images and deep learning. S Ni, Q Qian, R Zhang, Computers & Security. 77Ni, S., Qian, Q., Zhang, R.: Malware identification using visualization images and deep learning. Computers & Security 77, 871-885 (2018) Using fake text vectors to improve the sensitivity of minority class for macro malware detection. M Mimura, 10.1016/j.jisa.2020.102600Journal of Information Security and Applications. 54102600Mimura, M.: Using fake text vectors to improve the sensitivity of minority class for macro malware detection. Journal of Information Security and Applications 54, 102600 (2020). doi:10.1016/j.jisa.2020.102600 G Mariani, F Scheidegger, R Istrate, C Bekas, C Malossi, arXiv:1803.09655Data augmentation with balancing gan. BaganarXiv preprintMariani, G., Scheidegger, F., Istrate, R., Bekas, C., Malossi, C.: Bagan: Data augmentation with balancing gan. arXiv preprint arXiv:1803.09655 (2018) Enhancing Robustness of Malware Detection using Synthetically-adversarial Samples. W L Tan, T Truong-Huu, 10.1109/globecom42002.2020.9322377GLOBECOM 2020 -2020 IEEE Global Communications Conference. 00Tan, W.L., Truong-Huu, T.: Enhancing Robustness of Malware Detection using Synthetically-adversarial Samples. GLOBECOM 2020 -2020 IEEE Global Communications Conference 00, 1-6 (2020). doi:10.1109/globecom42002.2020.9322377 Dual Generative Adversarial Networks Based Unknown Encryption Ransomware Attack Detection. X Zhang, J Wang, S Zhu, 10.1109/access.2021.3128024IEEE Access. 10Zhang, X., Wang, J., Zhu, S.: Dual Generative Adversarial Networks Based Unknown Encryption Ransomware Attack Detection. IEEE Access 10, 900-913 (2022). doi:10.1109/access.2021.3128024 Advandmal: Adversarial training for android malware detection and family classification. C Wang, L Zhang, K Zhao, X Ding, X Wang, 10.3390/sym13061081Symmetry. 1361081Wang, C., Zhang, L., Zhao, K., Ding, X., Wang, X.: Advandmal: Adversarial training for android malware detection and family classification. Symmetry 13(6), 1081 (2021). doi:10.3390/sym13061081 Adversarial Training Against Adversarial Attacks for Machine Learning-Based Intrusion Detection Systems. M S Haroon, H M Ali, 10.32604/cmc.2022.029858Materials & Continua. 732ComputersHaroon, M.S., Ali, H.M.: Adversarial Training Against Adversarial Attacks for Machine Learning-Based Intrusion Detection Systems. Computers, Materials & Continua 73(2), 3513-3527 (2022). doi:10.32604/cmc.2022.029858 Conservative Novelty Synthesizing Network for Malware Recognition in an Open-Set Scenario. J Guo, S Guo, S Ma, Y Sun, Y Xu, 10.1109/tnnls.2021.3099122IEEE Transactions on Neural Networks and Learning Systems PP(99). Guo, J., Guo, S., Ma, S., Sun, Y., Xu, Y.: Conservative Novelty Synthesizing Network for Malware Recognition in an Open-Set Scenario. IEEE Transactions on Neural Networks and Learning Systems PP(99), 1-15 (2021). doi:10.1109/tnnls.2021.3099122 R H Randhawa, N Aslam, M Alauthman, H Rafiq, 2109.08026EVAGAN: Evasion Generative Adversarial Network for Low Data Regimes. arXiv. Randhawa, R.H., Aslam, N., Alauthman, M., Rafiq, H.: EVAGAN: Evasion Generative Adversarial Network for Low Data Regimes. arXiv (2021). 2109.08026 Optimized Generative Adversarial Networks for Adversarial Sample Generation. D M Alghazzawi, S H Hasan, S Bhatia, 10.32604/cmc.2022.024613Computers, Materials and Continua. 722Alghazzawi, D.M., Hasan, S.H., Bhatia, S.: Optimized Generative Adversarial Networks for Adversarial Sample Generation. Computers, Materials and Continua 72(2), 3877-3897 (2022). doi:10.32604/cmc.2022.024613 A Deep Learning-based Malware Hunting Technique to Handle Imbalanced Data. Z Moti, S Hashemi, A N Jahromi, 10.1109/iscisc51277.2020.926191317th International ISC Conference on Information Security and Cryptology (ISCISC) 00. Moti, Z., Hashemi, S., Jahromi, A.N.: A Deep Learning-based Malware Hunting Technique to Handle Imbalanced Data. 2020 17th International ISC Conference on Information Security and Cryptology (ISCISC) 00, 48-53 (2020). doi:10.1109/iscisc51277.2020.9261913 Using Generative Adversarial Networks for Data Augmentation in Android Malware Detection. Y.-M Chen, C.-H Yang, G.-C Chen, 10.1109/dsc49826.2021.93462772021 IEEE Conference on Dependable and Secure Computing, DSC 2021 00. Chen, Y.-M., Yang, C.-H., Chen, G.-C.: Using Generative Adversarial Networks for Data Augmentation in Android Malware Detection. 2021 IEEE Conference on Dependable and Secure Computing, DSC 2021 00, 1-8 (2021). doi:10.1109/dsc49826.2021.9346277 Resilience against Adversarial Examples: Data-Augmentation Exploiting Generative Adversarial Networks. M Kang, H Kim, S Lee, S Han, 10.3837/tiis.2021.11.013KSII Transactions on Internet and Information Systems. 1511Kang, M., Kim, H., Lee, S., Han, S.: Resilience against Adversarial Examples: Data-Augmentation Exploiting Generative Adversarial Networks. KSII Transactions on Internet and Information Systems 15(11), 4105-4121 (2021). doi:10.3837/tiis.2021.11.013 LSTM-CGAN: Towards Generating Low-Rate DDoS Adversarial Samples for Blockchain-Based Wireless Network Detection Models. Z Liu, X Yin, 10.1109/access.2021.3056482IEEE Access. 9Liu, Z., Yin, X.: LSTM-CGAN: Towards Generating Low-Rate DDoS Adversarial Samples for Blockchain-Based Wireless Network Detection Models. IEEE Access 9, 22616-22625 (2021). doi:10.1109/access.2021.3056482 Before we knew it: an empirical study of zero-day attacks in the real world. L Bilge, T Dumitraş, Proceedings of the 2012 ACM Conference on Computer and Communications Security. the 2012 ACM Conference on Computer and Communications SecurityBilge, L., Dumitraş, T.: Before we knew it: an empirical study of zero-day attacks in the real world. In: Proceedings of the 2012 ACM Conference on Computer and Communications Security, pp. 833-844 (2012) Efficient Malware Originated Traffic Classification by Using Generative Adversarial Networks. Z Liu, S Li, Y Zhang, X Yun, Z Cheng, 10.1109/iscc50000.2020.9219561Proceedings -IEEE Symposium on Computers and Communications. -IEEE Symposium on Computers and CommunicationsLiu, Z., Li, S., Zhang, Y., Yun, X., Cheng, Z.: Efficient Malware Originated Traffic Classification by Using Generative Adversarial Networks. Proceedings -IEEE Symposium on Computers and Communications 2020-July, 1-7 (2020). doi:10.1109/iscc50000.2020.9219561 Discovering future malware variants by generating new malware samples using generative adversarial network. Z Moti, S Hashemi, A Namavar, 10.1109/iccke48569.2019.89649139th International Conference on Computer and Knowledge Engineering. Moti, Z., Hashemi, S., Namavar, A.: Discovering future malware variants by generating new malware samples using generative adversarial network. 2019 9th International Conference on Computer and Knowledge Engineering, ICCKE 2019 00, 319-324 (2019). doi:10.1109/iccke48569.2019.8964913 A GAN Based Malware Adversaries Detection Model. M Umer, Y Saleem, M Saleem, N Aman, 10.1109/icosst53930.2021.968386315th International Conference on Open Source Systems and Technologies (ICOSST) 00. Umer, M., Saleem, Y., Saleem, M., Aman, N.: A GAN Based Malware Adversaries Detection Model. 2021 15th International Conference on Open Source Systems and Technologies (ICOSST) 00, 1-9 (2021). doi:10.1109/icosst53930.2021.9683863 N-gram MalGAN: Evading machine learning detection via feature n-gram. E Zhu, J Zhang, J Yan, K Chen, C Gao, 10.1016/j.dcan.2021.11.007Digital Communications and NetworksZhu, E., Zhang, J., Yan, J., Chen, K., Gao, C.: N-gram MalGAN: Evading machine learning detection via feature n-gram. Digital Communications and Networks (2021). doi:10.1016/j.dcan.2021.11.007 Feature selection-based android malware adversarial sample generation and detection method. X Li, K Kong, S Xu, P Qin, D He, 10.1049/ise2.12030IET Information Security. 156Li, X., Kong, K., Xu, S., Qin, P., He, D.: Feature selection-based android malware adversarial sample generation and detection method. IET Information Security 15(6), 401-416 (2021). doi:10.1049/ise2.12030 Adversarial-Example Attacks Toward Android Malware Detection System. H Li, S Zhou, W Yuan, J Li, H Leung, 10.1109/jsyst.2019.2906120IEEE Systems Journal. 141Li, H., Zhou, S., Yuan, W., Li, J., Leung, H.: Adversarial-Example Attacks Toward Android Malware Detection System. IEEE Systems Journal 14(1), 653-656 (2018). doi:10.1109/jsyst.2019.2906120 An Approach for Malware Behavior Identification and Classification. M F Zolkipli, A Jantan, 10.1109/iccrd.2011.57640013rd International Conference on Computer Research and Development. 1Zolkipli, M.F., Jantan, A.: An Approach for Malware Behavior Identification and Classification. 2011 3rd International Conference on Computer Research and Development 1, 191-194 (2011). doi:10.1109/iccrd.2011.5764001 Detecting environment-sensitive malware. M Lindorfer, C Kolbitsch, P Milani Comparetti, International Workshop on Recent Advances in Intrusion Detection. SpringerLindorfer, M., Kolbitsch, C., Milani Comparetti, P.: Detecting environment-sensitive malware. In: International Workshop on Recent Advances in Intrusion Detection, pp. 338-357 (2011). Springer Sandprint: Fingerprinting malware sandboxes to provide intelligence for sandbox evasion. A Yokoyama, K Ishii, R Tanabe, Y Papa, K Yoshioka, T Matsumoto, T Kasama, D Inoue, M Brengel, M Backes, International Symposium on Research in Attacks, Intrusions, and Defenses. SpringerYokoyama, A., Ishii, K., Tanabe, R., Papa, Y., Yoshioka, K., Matsumoto, T., Kasama, T., Inoue, D., Brengel, M., Backes, M., et al.: Sandprint: Fingerprinting malware sandboxes to provide intelligence for sandbox evasion. In: International Symposium on Research in Attacks, Intrusions, and Defenses, pp. 165-187 (2016). Springer A STUDY ON WHA (WATERING HOLE ATTACK) -THE MOST DANGEROUS THREAT TO THE ORGANISATION. N Krithika, International Journal of Innovations in Scientific and Engineering Research. 48Krithika, N.: A STUDY ON WHA (WATERING HOLE ATTACK) -THE MOST DANGEROUS THREAT TO THE ORGANISATION. International Journal of Innovations in Scientific and Engineering Research 4(8), 196-198 (August 2017) Comparative analysis of various ransomware virii. A Gazet, 10.1007/s11416-008-0092-2Journal in Computer Virology. 61Gazet, A.: Comparative analysis of various ransomware virii. Journal in Computer Virology 6(1), 77-90 (2010). doi:10.1007/s11416-008-0092-2 Structure tuning method on deep convolutional generative adversarial network with nondominated sorting genetic algorithm II. L Du, Z Cui, L Wang, J Ma, 10.1002/cpe.5688Concurrency and Computation: Practice and Experience. 3214Du, L., Cui, Z., Wang, L., Ma, J.: Structure tuning method on deep convolutional generative adversarial network with nondominated sorting genetic algorithm II. Concurrency and Computation: Practice and Experience 32(14) (2020). doi:10.1002/cpe.5688 Gradient-based learning applied to document recognition. Y Lecun, L Bottou, Y Bengio, P Haffner, Proceedings of the IEEE. 8611LeCun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278-2324 (1998) Malware images: visualization and automatic classification. L Nataraj, S Karthikeyan, G Jacob, B S Manjunath, Proceedings of the 8th International Symposium on Visualization for Cyber Security. the 8th International Symposium on Visualization for Cyber SecurityNataraj, L., Karthikeyan, S., Jacob, G., Manjunath, B.S.: Malware images: visualization and automatic classification. In: Proceedings of the 8th International Symposium on Visualization for Cyber Security, pp. 1-7 GenAtSeq GAN with Heuristic Reforms for Knowledge Centric Network with Browsing Characteristics Learning, Individual Tracking and Malware Detection with Website2Vec. C Sur, 10.1007/s42979-020-00234-8SN Computer Science. 14228Sur, C.: GenAtSeq GAN with Heuristic Reforms for Knowledge Centric Network with Browsing Characteristics Learning, Individual Tracking and Malware Detection with Website2Vec. SN Computer Science 1(4), 228 (2020). doi:10.1007/s42979-020-00234-8 IoT-based Android Malware Detection Using Graph Neural Network With Adversarial Defense. R Yumlembam, B Issac, S M Jacob, L Yang, 10.1109/jiot.2022.3188583IEEE Internet of Things Journal PP. 99Yumlembam, R., Issac, B., Jacob, S.M., Yang, L.: IoT-based Android Malware Detection Using Graph Neural Network With Adversarial Defense. IEEE Internet of Things Journal PP(99), 1-1 (2022). doi:10.1109/jiot.2022.3188583 A Survey on Cyber Situation Awareness Systems: Framework, Techniques, and Insights. H Alavizadeh, J Jang-Jaccard, S Y Enoch, H Al-Sahaf, I Welch, S A Camtepe, D D Kim, 10.1145/3530809ACM Computing SurveysAlavizadeh, H., Jang-Jaccard, J., Enoch, S.Y., Al-Sahaf, H., Welch, I., Camtepe, S.A., Kim, D.D.: A Survey on Cyber Situation Awareness Systems: Framework, Techniques, and Insights. ACM Computing Surveys (2022). doi:10.1145/3530809 Semi-adversarial networks: Convolutional autoencoders for imparting privacy to face images. V Mirjalili, S Raschka, A Namboodiri, A Ross, 2018 International Conference on Biometrics (ICB). IEEEMirjalili, V., Raschka, S., Namboodiri, A., Ross, A.: Semi-adversarial networks: Convolutional autoencoders for imparting privacy to face images. In: 2018 International Conference on Biometrics (ICB), pp. 82-89 (2018). IEEE The emergence of deepfake technology: A review. M Westerlund, Technology Innovation Management Review. 911Westerlund, M.: The emergence of deepfake technology: A review. Technology Innovation Management Review 9(11) (2019) The responsibility gap: Ascribing responsibility for the actions of learning automata. A Matthias, Ethics and information technology. 63Matthias, A.: The responsibility gap: Ascribing responsibility for the actions of learning automata. Ethics and information technology 6(3), 175-183 (2004) Responsibility beyond design: Physicians' requirements for ethical medical ai. M Sand, J M Durán, K R Jongsma, Bioethics. 362Sand, M., Durán, J.M., Jongsma, K.R.: Responsibility beyond design: Physicians' requirements for ethical medical ai. Bioethics 36(2), 162-169 (2022) Designing ethical algorithms. K E Martin, MIS Quarterly Executive. Martin, K.E.: Designing ethical algorithms. MIS Quarterly Executive June (2019)
{'fraction_non_alphanumeric': 0.05080552553363956, 'fraction_numerical': 0.04340887597946501, 'mean_word_length': 4.741370952103936, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 5, 'lorem ipsum': 0, 'www.': 4, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 14, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'Since their proposal in the 2014 paper by Ian Goodfellow [1], there has been an explosion of research into the area of Generative Adversarial Networks. While they have been utilised in many fields, the realm of malware research is a problem space in which GANs have taken root. From balancing datasets to creating unseen examples in rare classes, GAN models offer extensive opportunities for application. This paper surveys the current research and literature for the use of Generative Adversarial Networks in the malware problem space. This is done with the hope that the reader may be able to gain an overall understanding as to what the Generative Adversarial model provides for this field, and for what areas within malware research it is best utilised. It covers the current related surveys, the different categories of GAN, and gives the outcomes of recent research into optimising GANs for different topics, as well as future directions for exploration.', 'arxivid': '2302.08558', 'author': ['Aeryn Dunmore \nCybersecurity Lab\nMassey University\nAlbanyNew Zealand\n', 'Aeryn Dunmore ', 'Julian Jang-Jaccard \nCybersecurity Lab\nMassey University\nAlbanyNew Zealand\n', 'Fariza Sabrina \nCybersecurity Lab\nMassey University\nAlbanyNew Zealand\n\nSchool of Engineering and Technology\nCentral Queensland University\nSydneyNSWAustralia\n', 'Jin Kwak \nDepartment of Cyber Security\nAjou University\nSuwonRepublic of Korea\n'], 'authoraffiliation': ['Cybersecurity Lab\nMassey University\nAlbanyNew Zealand', 'Cybersecurity Lab\nMassey University\nAlbanyNew Zealand', 'Cybersecurity Lab\nMassey University\nAlbanyNew Zealand', 'School of Engineering and Technology\nCentral Queensland University\nSydneyNSWAustralia', 'Department of Cyber Security\nAjou University\nSuwonRepublic of Korea'], 'corpusid': 257020089, 'doi': '10.48550/arxiv.2302.08558', 'github_urls': [], 'n_tokens_mistral': 34768, 'n_tokens_neox': 28938, 'n_words': 17350, 'pdfsha': '327b696502dd95bba030a466f532ac533130dce8', 'pdfurls': ['https://export.arxiv.org/pdf/2302.08558v2.pdf'], 'title': ['Generative Adversarial Networks for Malware Detection: a Survey', 'Generative Adversarial Networks for Malware Detection: a Survey'], 'venue': []}
arxiv
Coherent States and a Path Integral for the Relativistic Linear Singular Oscillator arXiv:math-ph/0702003v1 1 Feb 2007 S M Nagiyev Institute of Physics National Academy of Sciences Javid av. 33 AZ1143BakuAzerbaijan, Azerbaijan E I Jafarov Institute of Physics National Academy of Sciences Javid av. 33 AZ1143BakuAzerbaijan, Azerbaijan Department of Applied Mathematics and Computer Science Ghent University Krijgslaan 281-S9B-9000GentBelgium M Y Efendiyev Azerbaijan Cooperation University Narimanov av. 86 AZ1106BakuAzerbaijan Coherent States and a Path Integral for the Relativistic Linear Singular Oscillator arXiv:math-ph/0702003v1 1 Feb 2007(Dated: October 11, 2018)numbers: 0365Ge0270Bf4225Kb0365Pm0365-w Keywords: Relativistic linear singular oscillatorSU (1, 1) coherent statesPath integral * Corresponding AuthorElectronic address: azhep@physicsabaz The SU (1, 1) coherent states for a relativistic model of the linear singular oscillator are considered. The corresponding partition function is evaluated. The path integral for the transition amplitude between SU (1, 1) coherent states is given. Classical equations of the motion in the generalized curved phase space are obtained. It is shown that the use of quasiclassical Bohr-Sommerfeld quantization rule yields the exact expression for the energy spectrum. I. INTRODUCTION Coherent States (CS) are a useful tool for studying quantum systems [1,2,3]. The use of the CS makes it possible to apply more transparent classical language to describe the quantum phenomena [4,5]. The concept of CS was first introduced for the boson oscillator [6,7]. In this case they are closely related with the unitary representations of the Heisenberg-Weyl group. Later on, the generalized CS, associated with the unitary representations of an arbitrary Lie group, have been defined [8]. The notion of generalized CS arises, when we attempt to construct quasi-classical states for dynamical systems other than the harmonic oscillator [9,10]. In the present work the technique of constructing a path integral representation for the transition amplitude (propagator) between SU(1, 1) coherent states, developed in [8,11,12,13,14] is applied to the relativistic model of the linear singular oscillator [15]. The same problem for the relativistic model of the harmonic oscillator was considered in [16]. This paper has following structure: Section 2 presents a brief description of the relativistic linear singular oscillator and its SU(1, 1) dynamical symmetry group. The explicit form of SU(1, 1) CS for this problem is given and the corresponding partition function is evaluated in Section 3. In Section 4 we consider a path integral expression of the propagator in SU(1, 1) CS and examine the corresponding classical limit. It is shown that the use of the quasiclassical Bohr-Summerfield quantization rule yields the exact expression for the energy spectrum of the considered relativistic linear singular oscillator. II. THE RELATIVISTIC LINEAR SINGULAR OSCILLATOR AND SU (1, 1) DY- NAMICAL SYMMETRY GROUP Recently, we constructed a relativistic model of the quantum linear singular oscillator [15], which can be applied for studying relativistic physical systems as well as systems on a lattice. This model is formulated in the framework of the finite-difference relativistic quantum mechanics, which was developed in several papers and applied to the solution of a lot of problems in particle physics [17,18,19,20,21,22,23]. The Hamiltonian of the relativistic model of the linear singular oscillator under consideration is a finite-difference operator [15] H = mc 2 cosh i∂ ρ + 1 2 ω 2 0 ρ (2) e i∂ρ + g 0 ρ (2) e i∂ρ ,(1) where ρ = x/λ is a dimensionless variable,λ = mc is the Compton wavelength of the particle, ω 0 = ω mc 2 , g 0 = mg , and ρ (2) is the generalized degree [24] ρ (δ) = i δ Γ (δ − iρ) Γ (−iρ) . The eigenfunctions of the Hamiltonian (1) in the interval 0 < ρ < ∞ are expressed in terms of the continuous dual Hahn polynomials S n (x 2 ; a, b, c), i.e. ψ n (ρ) = c n ω iρ 0 (−ρ) (α) Γ (ν + iρ) S n ρ 2 ; α, ν, 1 2 ,(2) c n = 2 n!Γ (n + α + ν) Γ (n + α + 1/2) Γ (n + ν + 1/2) . Here we have introduced the notations α = 1 2 + 1 2 1 + 2 ω 2 0 1 − 1 − 8g 0 ω 2 0 ,(3)ν = 1 2 + 1 2 1 + 2 ω 2 0 1 + 1 − 8g 0 ω 2 0 . The functions (2) are orthonormal ∞ 0 ψ * n (ρ) ψ m (ρ) dρ = δ nm .(4) A dynamical symmetry group for the model of the relativistic linear singular oscillator under consideration is the SU(1, 1) group. The corresponding Lie algebra is formed by the generators K 0 = 1 2 ω H, K − = A − f −1 (H) , K + = f −1 (H) A + ,(5) where f (H) = H mc 2 + ω 0 (α − ν − 1) H mc 2 + ω 0 (ν − α − 1) 1/2 . Having defined a generalized momentum operator P = −mc sinh (i∂ ρ ) + 1 2 ω 2 0 ρ (2) e i∂ρ + g 0 ρ (2) e i∂ρ by means of the commutator [ρ, H] = icP, the operators A ± may be written as A ± = 1 2ω 0 ω 0 ρ ∓ i mc P 2 − 2g 0 ρ 2 + 1 .(6) The generators (5) satisfy the commutation relations K 0 , K ± = ±K ± , K − , K + = 2K 0 .(7) The operator K 0 has a discrete spectrum in a infinite-dimensional unitary irreducible representation D + (k) such that K 0 ψ n = (n + k) ψ n ,(8) where n = 0, 1, 2, . . ., and k > 0. The Casimir invariant is K 2 = K 2 0 − 1 2 K + K − + K − K + = k (k − 1)Î. For the operators (5) one has K 2 = α+ν 2 α+ν 2 − 1 , so that k = (α + ν) /2. Thus from (5) and (8) we determine the energy levels as E n = 2 ω (n + k) = ω (2n + α + ν) .(9) Let us emphasize that due to the commutation relations (7) the action of the generators K ± on the wavefunctions ψ n is given by K − ψ n = k n ψ n−1 , K + ψ n = k n+1 ψ n+1 ,(10)k n = n(n + 2k − 1) = n(n + α + ν − 1). From (10) follows that ψ n = N n K + n ψ 0 , N −1 n = k 1 k 2 · · · k n = n! (α + ν) n ,(11) (a) n = Γ (n + a) /Γ(a). In the non-relativistic limit, when c → ∞ the wave-functions ψ n (ρ) coincide with the wavefunctions of the non-relativistic linear singular oscillator. In this limit we also have lim c→∞ H − mc 2 = H nonrel = ω − 1 2 ∂ 2 ξ + 1 2 ξ 2 + g 0 ξ 2 , lim c→∞ E n − mc 2 = E nonrel n = ω (2n + d + 1) , lim c→∞ 1 2 A − = K − nonrel = 1 2 a − 2 − g 0 ξ 2 , lim c→∞ 1 2 A + = K + nonrel = 1 2 a + 2 − g 0 ξ 2 ,(12)lim c→∞ Π = −i √ m ω∂ ξ = −i ∂ x = p x , lim c→∞ α = d + 1/2, lim c→∞ (ν − µ) = 1/2, where d = 1 2 √ 1 + 8g 0 , ξ = mω x and a + = 1 √ 2 (ξ − ∂ ξ ) , a + = 1 √ 2 (ξ − ∂ ξ ) are the usual creation and annihilation operators. |ζ, k = D (β) ψ 0 (ρ) = 1 − |ζ| 2 k e ζK + ψ 0 (ρ) ,(13) where β = − τ 2 e −iϕ and ζ = − tanh τ 2 e −iϕ , τ and ϕ are group parameters. From (10) and (13) it follows that the decomposition of |ζ, k over the wavefunctions ψ n (ρ) (2) has the form |ζ, k = 1 − |ζ| 2 k ∞ n=0 (2k) n n! ζ n ψ n (ρ).(14) Using (2) one can rewrite (14) as follows Mentioned above condition allows us to rewrite (15) as |ζ, k = 1 − |ζ| 2 k 2 Γ (α + ν) Γ (α + 1/2) Γ (ν + 1/2) ω iρ 0 (−ρ) (α) Γ (ν + iρ) ×(15)|ζ, k = 1 − |ζ| 2 k 2 Γ (α + ν) Γ (α + 1/2) Γ (ν + 1/2) ω iρ 0 (−ρ) (α) Γ (ν + iρ) ×(16) ∞ n=0 ζ n n! (|α| + 1/2) n S n ρ 2 ; |α| , |α| , 1 2 . By the use of the following generation function for the continuous dual Hahn polynomials [25] ∞ n=0 S n (x 2 ; a, b, c) (a + c) n n! t n = (1 − t) −b+ix 2 F 1   a + ix, c + ix a + c t   one can simplify (16) as |ζ, k = 1 − |ζ| 2 k 2 Γ (α + ν) Γ (α + 1/2) Γ (ν + 1/2) ω iρ 0 (−ρ) (α) Γ (ν + iρ) ×(17)(1 − ζ) −|α|+iρ 2 F 1   |α| + iρ, 1 2 + iρ |α| + 1 2 ζ   . The SU(1, 1) CS (13) are orthogonal and the overlap of two states |ζ, k and |ζ ′ , k is given as ζ ′ , k | ζ, k = 1 − |ζ ′ | 2 k 1 − |ζ| 2 k (1 − ζ ′ * ζ) −2k .(18) The important property of these states is the completeness relation dµ k (ζ) |ζ, k ζ ′ , k| = 1,(19) where dµ k (ζ) = 2k − 1 π d 2 ζ 1 − |ζ| 2 2 .(20) The matrix elements of the generators K − , K + , K 0 in the SU(1, 1) CS have the form ζ ′ , k| K − |ζ, k = 2kζ 1 − ζ ′ * ζ ζ ′ , k | ζ, k , ζ ′ , k| K + |ζ, k = 2kζ ′ * 1 − ζ ′ * ζ ζ ′ , k | ζ, k ,(21)ζ ′ , k| K 0 |ζ, k = k (1 + ζ ′ * ζ) 1 − ζ ′ * ζ ζ ′ , k | ζ, k . The transition amplitude (propagator) between SU(1, 1) CS is defined as K (ζ ′ ; ζ; T ) = ζ ′ ; k| exp − i T H |ζ; k , = ζ ′ ; k| exp [−2iω 0 T K 0 ] |ζ; k . Using (14) and (18) it is easy to show that K (ζ ′ , ζ, T ) = e −2iωkT 1 − |ζ| 2 k 1 − |ζ ′ | 2 k (1 − ζ ′ * ζe −2iωkT ) 2k .(22) The partition function for the relativistic model of the linear singular oscillator under consideration is given as Z rel = Tr K (ζ, ζ ′ ; −i β) = e −2k ωβ 1 − e −2k ωβ = e −2 ω(α+ν−d−1) Z nonrel , where Z nonrel is the partition function for the nonrelativistic linear singular oscillator. IV. PATH INTEGRAL AND CLASSICAL EQUATIONS OF MOTION IN THE GENERALIZED PATH SPACE Following the paper [11,12] we now derive the path integral expression for the amplitude (22). Defining ε = T /N and using the completeness relation (19) it is possible to represent (22) as K (ζ ′ , ζ; T ) = · · · N −1 j=1 dµ k (ζ j ) ζ ′ , k| e − i εH |ζ N −1 , k ζ N −1 , k| e − i εH |ζ N −2 , k · · · ζ 1 , k| e − i εH |ζ, k .(23) With the help of (21) it is easy to show that for small ε each factor in the integrand (23) can be written as ζ j , k| e − i εH |ζ j−1 , k ∼ = exp [ln ζ j , k | ζ j−1 , k ] − iε H k ζ * j , ζ j−1 , where H k ζ * j , ζ j−1 = ζ j , k| H |ζ j−1 , k ζ j , k | ζ j−1 , k = 2k ω 1 + ζ * j ζ j−1 1 − ζ * j ζ j−1 .(24) If we take into account (18) and fact that ζ j−1 = ζ j − ∆ζ j , we can write ln ζ j , k | ζ j−1 , k ∼ = k 1 − |ζ| 2 ζ j ∆ζ * j − ζ * j ∆ζ j . Thus, when N → ∞ (or ε → 0) we arrive at the following path integral for the amplitude (22) K (ζ ′ , ζ; T ) = Dµ k (ζ) exp    i T 0 L k ζ,ζ, ζ * ,ζ * dt    ,(25) with the classical Lagrangian L k ζ,ζ, ζ * ,ζ * = i k 1 − |ζ| 2 ζ * ζ − ζζ * − H k (ζ * , ζ)(26) in a generalized curved phase space in the form of a Lobachevsky plane. The corresponding classical Euler-Lagrange equations have the form d dt ∂L k ∂ζ = ∂L k ∂ζ , d dt ∂L k ∂ζ * = ∂L k ∂ζ * .(27) Using (26) we can represent (27) in the form of Hamiltonian's equations: ζ = 1 − |ζ| 2 2 2i k ∂H k ∂ζ * ,ζ * = 1 − |ζ| 2 2 2i k ∂H k ∂ζ .(28) If we define a Poisson bracket by {A, B} = 1 − |ζ| 2 2 2i k ∂A ∂ζ ∂B ∂ζ * − ∂A ∂ζ * ∂B ∂ζ ,(29) then we can write the equations (28) in a more compact form aṡ ζ = {ζ, H k } ,ζ * = {ζ * , H k } .(30) Since in our case H k (ζ * , ζ) ≡ H k (τ ) = 2 ωk cosh (τ ), the equations (30) written in terms of the group parameters τ and ϕ will be reduced tȯ τ = {τ, H k (τ )} = 0,φ = {ϕ, H k (τ )} = 2ω.(31) The solutions of (31) are τ = const and ϕ = 2ωtϕ 0 . Therefore, the classical motion in the curved phase space is oscillator like. In terms of τ and ϕ the Lagrangian (26) becomes L k = k [(cosh (τ ) − 1)φ − 2ω cosh (τ )] ≡ kL (τ, ϕ) .(32) Using the momentum p = ∂L/∂φ = cosh (τ ) − 1, canonically conjugate to the coordinate ϕ we may writeL (p, ϕ) ≡L (τ, ϕ) = pφ − 2ω (p + 1) . Substituting (33) into (25) we arrive at the path-integral expression K (ζ ′ , ζ; T ) = Dµ k (p, ϕ) exp    ik T 0L (p, ϕ) dt    .(34) Since in the → 0 limit the parameter k = (α + ν) /2 characterizing the irreducible representation D + (k) of the dynamical symmetry group SU(1, 1) behaves as k ∼ = mc 2 ω , from (34) it follows that for k sufficiently large, the motion of the relativistic linear singular oscillator in the curved phase space becomes quasiclassical. Thus, when k → ∞ we can use Bohr-Sommerfeld quantization rule to find the energy spectrum E k = H k (τ ), i.e. pdϕ = 2π k n, n = 0, 1, 2, . . . .(35) From (35) follows that p = n/k and therefore E k = H k (τ ) = 2 ωk cosh(τ ) = 2 ωk(p + 1) = 2 ω(n + k) = ω(2n + α + ν).(36) Therefore, as in the non-relativistic case, the Bohr-Sommerfeld quantization rule yields for the energy spectrum of the relativistic linear singular oscillator the exact expression (36). V. CONCLUSION In spite of many attractive papers devoted to construction of CS for non-relativistic quantum systems, the number of works studying relativistic approaches to CS and path integral formulation of the quantum systems is still rather few [16,26,27,28]. In this paper we have considered the CS for a relativistic model of the linear singular oscillator and obtained their explicit form. Thereafter, a path integral expression of the transition amplitude between CS has been studied and corresponding classical limits are shown. By the use of path integral approach the classical equations of the motion in the generalized curved phase space are obtained. It was shown that the use of quasiclassical Bohr-Sommerfeld quantization rule yields the exact expression for the energy spectrum. |ζ, k are defined by acting with the displacement operator D (β) = exp (βK + − β * K − ) on the ground state wavefunctions ψ 0 (ρ), i.e. . One can look for the explicit expression of CS (15) taking into account Hermiticity conditions of the Hamiltonian. Hermiticity condition of the Hamiltonian imposes a restriction on the values of the quantity g 0 . Therefore, eigenvalues (9) are real only in case when α and ν are real or complex-conjugate. We will calculate series (15) for the case, when α and ν are equal or complex-conjugate, which imposes the condition g 0 The behavior of α and ν are presented inFigs. 1 and 2. FIG. 1 : 1The behavior of α and ν: real parts . Y Hassouni, E M F Curado, M A Rego-Monteiro, Phys. Rev. 7122104Y. Hassouni, E.M.F. Curado and M.A. Rego-Monteiro, Phys. Rev. A71 (2005) 022104. . T Shreecharan, P K Panigrahi, J Banerji, Phys. Rev. 6912102T. Shreecharan, P.K. Panigrahi and J. Banerji, Phys. Rev. A69 (2004) 012102. . Y Wu, X X Yang, Commun. Theor. Phys. 37539Y. Wu and X.X. Yang, Commun. Theor. Phys. 37 (2002) 539. . R F Fox, M H Choi, Phys. Rev. 6132107R.F. Fox and M.H. Choi, Phys. Rev. A61 (2000) 032107. . J C Su, F H Zheng, Commun. Theor. Phys. 43641J.C. Su and F.H. Zheng, Commun. Theor. Phys. 43 (2005) 641. . R J Glauber, Phys. Rev. 1302529R.J. Glauber, Phys. Rev. 130 (1963) 2529. J R Klauder, E C G Sudarshan, Fundamentals of Quantum Optics. Benjamin, New YorkJ.R. Klauder and E.C.G. Sudarshan, Fundamentals of Quantum Optics, Benjamin, New York (1968). . A M Perelomov, Commun. Math. Phys. 26222A.M. Perelomov, Commun. Math. Phys. 26 (1972) 222. . V Chernyak, S Choi, S Mukamel, Phys. Rev. 6753604V. Chernyak, S. Choi and S. Mukamel, Phys. Rev. A67 (2003) 053604. . B W Xu, F Ye, G H Ding, Commun. Theor. Phys. 39151B.W. Xu, F. Ye and G.H. Ding, Commun. Theor. Phys. 39 (2003) 151. . H Kuratsuji, T Suzuki, J. Math. Phys. 21472H. Kuratsuji and T. Suzuki, J. Math. Phys. 21 (1980) 472. . C C Gerry, S Silverman, J. Math. Phys. 23C.C. Gerry and S. Silverman, J. Math. Phys. 23 (1982) 1995. . C C Gerry, J B Todeas, S Silverman, Phys. Rev. 281939C.C. Gerry, J.B. Todeas and S. Silverman, Phys. Rev. D28 (1983) 1939. . C C Gerry, A Benmoussa, Phys. Rev. 6233812C.C. Gerry and A. Benmoussa, Phys. Rev. A62 (2000) 033812. . S M Nagiyev, E I Jafarov, R M Imanov, J. Phys. A: Math. Gen. 367813S.M. Nagiyev, E.I. Jafarov and R.M. Imanov, J. Phys. A: Math. Gen. 36 (2003) 7813. . N M Atakishiyev, R M Mir-Kasimov, J. Math. Phys. 301631N.M. Atakishiyev and R.M. Mir-Kasimov, J. Math. Phys. 30 (1989) 1631. . V G Kadyshevsky, R M Mir-Kasimov, N B Skachkov, Nuovo Chim. 55233V.G. Kadyshevsky, R.M. Mir-Kasimov and N.B. Skachkov, Nuovo Chim. A55 (1968) 233. . V G Kadyshevsky, R M Mir-Kasimov, M Freeman, Yad. Fiz. 9646in RussianV.G. Kadyshevsky, R.M. Mir-Kasimov and M. Freeman, Yad. Fiz. 9 (1969) 646 [in Russian]. . M Freeman, M D Mateev, R M Mir-Kasimov, Nucl. Phys. 12197M. Freeman, M.D. Mateev and R.M. Mir-Kasimov, Nucl. Phys. B12 (1969) 197. . A Klein, T.-S H Lee, Phys. Rev. 104308A. Klein and T.-S.H. Lee, Phys. Rev. D10 (1974) 4308. . N M Atakishiyev, R M Mir-Kasimov, S M Nagiyev, Theor. Math. Phys. 44592N.M. Atakishiyev, R.M. Mir-Kasimov and S.M. Nagiyev, Theor. Math. Phys. 44 (1980) 592. . N M Atakishiyev, R M Mir-Kasimov, S M Nagiyev, Ann. Phys. Lpz. 4225N.M. Atakishiyev, R.M. Mir-Kasimov and S.M. Nagiyev, Ann. Phys. Lpz 42 (1985) 25. . N M Atakishiyev, R M Mir-Kasimov, Theor. Math. Phys. 67362N.M. Atakishiyev and R.M. Mir-Kasimov, Theor. Math. Phys. 67 (1986) 362. . S M Nagiyev, J. Phys. A: Math. Gen. 212559S.M. Nagiyev, J. Phys. A: Math. Gen. 21 (1988) 2559. The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. R Koekoek, R F Swarttouw, no. 98-17Delft University of TechnologyReportR. Koekoek and R.F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Delft University of Technology, Report no. 98-17 (1998). . V G Bagrov, I L Buchbinder, D M Gitman, J. Phys. A: Math. Gen. 91955V.G. Bagrov, I.L. Buchbinder and D.M. Gitman, J. Phys. A: Math. Gen. 9 (1976) 1955. . B I Lev, A A Semenov, C V Usenko, J R Klauder, Phys. Rev. 6622115B.I. Lev, A.A. Semenov, C.V. Usenko and J.R. Klauder, Phys. Rev. A66 (2005) 022115. . M Haghlighat, A Dadkhah, Phys. Lett. 316271M. Haghlighat and A. Dadkhah, Phys. Lett. A316 (2003) 271 The behavior of α and ν: imaginary parts. FIG. 2: The behavior of α and ν: imaginary parts
{'fraction_non_alphanumeric': 0.11223291852547547, 'fraction_numerical': 0.06004930734914299, 'mean_word_length': 3.016265912305516, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 32, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'The SU (1, 1) coherent states for a relativistic model of the linear singular oscillator are considered. The corresponding partition function is evaluated. The path integral for the transition amplitude between SU (1, 1) coherent states is given. Classical equations of the motion in the generalized curved phase space are obtained. It is shown that the use of quasiclassical Bohr-Sommerfeld quantization rule yields the exact expression for the energy spectrum.', 'arxivid': 'math-ph/0702003', 'author': ['S M Nagiyev \nInstitute of Physics\nNational Academy of Sciences Javid av. 33\nAZ1143BakuAzerbaijan, Azerbaijan\n', 'E I Jafarov \nInstitute of Physics\nNational Academy of Sciences Javid av. 33\nAZ1143BakuAzerbaijan, Azerbaijan\n\nDepartment of Applied Mathematics and Computer Science\nGhent University\nKrijgslaan 281-S9B-9000GentBelgium\n', 'M Y Efendiyev \nAzerbaijan Cooperation University Narimanov av. 86\nAZ1106BakuAzerbaijan\n'], 'authoraffiliation': ['Institute of Physics\nNational Academy of Sciences Javid av. 33\nAZ1143BakuAzerbaijan, Azerbaijan', 'Institute of Physics\nNational Academy of Sciences Javid av. 33\nAZ1143BakuAzerbaijan, Azerbaijan', 'Department of Applied Mathematics and Computer Science\nGhent University\nKrijgslaan 281-S9B-9000GentBelgium', 'Azerbaijan Cooperation University Narimanov av. 86\nAZ1106BakuAzerbaijan'], 'corpusid': 119142869, 'doi': '10.1088/0253-6102/49/2/13', 'github_urls': [], 'n_tokens_mistral': 7498, 'n_tokens_neox': 6404, 'n_words': 3318, 'pdfsha': '27bd9c2b6227368b5f0e02d2b96fb5fe920ff330', 'pdfurls': ['https://arxiv.org/pdf/math-ph/0702003v1.pdf'], 'title': ['Coherent States and a Path Integral for the Relativistic Linear Singular Oscillator', 'Coherent States and a Path Integral for the Relativistic Linear Singular Oscillator'], 'venue': []}
arxiv
Exploring interacting topological insulator of extended Su-Schrieffer-Heeger model Xiaofan Zhou Institute of Laser spectroscopy State Key Laboratory of Quantum Optics and Quantum Optics Devices Shanxi University 030006TaiyuanChina Collaborative Innovation Center of Extreme Optics Shanxi University 030006TaiyuanShanxiChina Jian-Song Pan College of Physics Sichuan University 610065ChengduChina Key Laboratory of High Energy Density Physics and Technology of Ministry of Education Sichuan University 610065ChengduChina Suotang Jia Institute of Laser spectroscopy State Key Laboratory of Quantum Optics and Quantum Optics Devices Shanxi University 030006TaiyuanChina Collaborative Innovation Center of Extreme Optics Shanxi University 030006TaiyuanShanxiChina Exploring interacting topological insulator of extended Su-Schrieffer-Heeger model Exploring topological phases in interacting systems is a challenging task. We investigate manybody topological physics of interacting fermions in an extended Su-Schrieffer-Heeger (SSH) model, which extends the two sublattices of SSH model into four sublattices and thus is dubbed SSH4 model, based on the density-matrix renormalization-group numerical method. The interactiondriven phase transition from topological insulator to charge density wave (CDW) phase can be identified by analyzing the variations of entanglement spectrum, entanglement entropies, energy gaps, CDW order parameter, and fidelity. We map the global phase diagram of the many-body ground state, which contains nontrivial topological insulator, trivial insulator and CDW phases, respectively. In contrast to interacting SSH model, in which the phase transitions to the CDW phase are argued to be first-order phase transitions, the phase transitions between the CDW phase and topologically trivial/nontrivial phases are shown to be continuous phase transitions. Finally, we also show the phase diagram of interacting spinful SSH4 model, where the attractive (repulsive) on-site spin interaction amplifies (suppresses) the CDW phase. The models analyzed here can be implemented with ultracold atoms on optical superlattices. arXiv:2208.00390v3 [cond-mat.quant-gas] I. INTRODUCTION Understanding the topological properties of band insulator and interacting topological insulator is one of the most fundamental and challenging tasks in the studies of condensed matter materials and ultracold atomic gases [1][2][3][4][5][6][7][8]. As a highly controllable and disorder-free system, ultracold atoms in optical lattices provide a powerful platform for quantum simulation of topological states of matter [6][7][8]. One of the most basic and easiest models in describing band topology is the celebrated Su-Schrieffer-Heeger (SSH) model [9], which has been experimentally implemented with ultracold atoms in onedimensional (1D) dimerized optical superlattices [10][11][12][13][14]. The SSH model describes noninteracting quantum particles hopping in a 1D lattice with alternating hopping coefficients. Varying the hopping ratio, the topological trivial phase or nontrivial phase appears, depending on the hopping term on the end of SSH model is strong or weak [15]. For a noninteracting topological insulator, edge degeneracy comes directly from the zero-energy edge mode, which is protected by its topological invariants of the bulk crystal through the bulk-edge correspondence. After considering the interaction, the SSH model exhibits a rich phase diagram [16][17][18][19][20][21][22][23] and interesting topological bound states [24], where the single-particle picture is not applicable. On the other hand, stimulated by experimental progresses, many variations and extensions of the SSH model * Electronic address: [email protected] have been proposed and explored, such as driven SSH model [25,26], SSH model with long-range hopping [27][28][29][30][31], two-leg SSH model [32], Creutz ladder model [33][34][35][36] and extended SSH model [37]. One typical extended example is to change the site period of the unit cell from two to four, thus one can transform the standard SSH model into the considerably richer SSH4 model with four hopping coefficients [37]. The wider parameter space of the SSH4 model is useful for studying topological properties of system with higher dimensions including synthetic dimension [38][39][40][41][42]. The SSH4 model has the chiral symmetry and belongs to the same topological class of the SSH model, and the winding number can characterize its band topology [37]. With open boundary condition, there exist topological edge states at the boundary of the system [43]. For a SSH4 model with infinite sites, the topological trivial and nontrivial phases are determined by the tunneling ratio. So far, the single-particle topological characterizations of SSH4 have been investigated clearly [37,44,45]. However, to the best of our knowledge, a detailed study of interacting SSH4 model is still lacked. In this paper, we investigate interacting topological properties of spinless and spin-1/2 SSH4 models in 1D optical superlattices, based on density-matrix renormalization-group (DMRG) numerical method [46,47]. For interacting SSH4 model, the topological invariant and classification of interacting TI become Z 4 , which are different from the single-particle TI classified with Z group. The nearest-neighbor interaction can drive the topological insulator (TI) and the topologically trivial insulator phases to the charge density wave (CDW) phase, which is characterized with the entanglement spectrum, entanglement entropies, energy gaps, and CDW order parameter. We numerically work out the many-body phase diagrams, and show the typical features of the appeared quantum phases. Although the phase diagram is similar to that of interacting SSH model, we find the phase transitions to the CDW phase are continuous phase transitions, unlike those in the SSH model, which are argued to be first-order phase transitions based on variational study [49]. The central charges at the phase boundaries between the CDW and TI/trivial insulator (TI and trivial insulator) phase are shown to be 2 (1). Further, we analyze the ground states of interacting spinful SSH4 model. It shows that the repulsive on-site interaction of spin-1/2 SSH4 model can enhance the TI phase and suppress the CDW phase, but the attractive on-site interaction plays the opposite role. Our results may stimulate a new avenue for simulating interacting fermionic topological phases using cold atom in optical lattices. t 1 t 2 t 3 t 4 t 1 t 2 t 3 t 4 … … j-2 j-1 j j+1 j+2 … … A 1 B 1 A 2 B 2 FIG II. MODEL AND HAMILTONIAN In cold atom experiment, three superposed optical lattices with lattice constant a/2, a and 2a effectively realize an SSH4 model, as shown in Fig. 1. The three optical lattices may be obtained from a single laser working at λ laser = 1064nm. The a/2 optical lattice can be obtained by retroreflecting the frequency-doubled laser, with a = λ laser /2. The a lattice may be obtained by retroreflecting the laser λ laser . The lattice at 2a may be obtained by crossing two λ laser beams with a small angle [37]. This superlattice exhibits four sites per unit cell (see Fig. 1), and hence is called as SSH4 model. The tight-binding interacting SSH4 Hamiltonian can be writ- ten as H = L/4 j=1 [t 1ĉ † 4j−3ĉ 4j−2 + t 2ĉ † 4j−2ĉ 4j−1 + t 3ĉ † 4j−1ĉ 4j +t 4ĉ † 4jĉ 4j+1 + H.c.] + V L jn jnj+1 ,(1) whereĉ j (ĉ † j ) are fermionic annihilation (creation) operators of the jth site,n j =ĉ † jĉ j , t n are the tunneling rates, and V measures the nearest-neighbor density-density interaction. In this configuration, t 1 = t 3 , but t 2 and t 4 can be tuned independently by varying the three lattice strengths and relative phase between the three lattices. For single-particle items of Hamiltonian (1), the timereversal, particle-hole, and chiral symmetries exist, then the topological insulator belongs to the symmetry class BDI of the Altland-Zirnbauer classification and is characterized by a Z invariant [4,48,50,51]. When t 1 = t 3 and t 2 = t 4 , the SSH4 model reduces to the common SSH model. The SSH4 model has four bands with more mid-gap states located inside the three gaps. However, only the zero-energy state is protected by the chiral symmetry and associated with the band-topology. Thus, the winding number is defined to identify the property of the two negative (or positive) energy bands, which all contribute to its value. The winding number w = 1 when |t 1 t 3 | < |t 2 t 4 |, and w = 0 when |t 1 t 3 | > |t 2 t 4 |. In the presence of weak interaction V , the topological invariant become Z 4 [52]. In the following, we focus on the two negative topology bands with zero-energy states, which corresponds to the half-filling occupation, i.e., N/L = 0.5, with the atom number N and lattice length L. The Hamiltonian (1) shows the spinless SSH4 model. In cold atoms experiment, the hyperfine states of the atoms usually can be treated as components of spin. After considering the two hyperfine states, we can get the spin-1/2 SSH4 Hamiltonian, which can be written as H = L/4 j=1,σ [t 1ĉ † 4j−3,σĉ 4j−2,σ + t 2ĉ † 4j−2,σĉ 4j−1,σ +t 3ĉ † 4j−1,σĉ 4j,σ + t 4ĉ † 4j,σĉ 4j+1,σ + H.c.] +V jn jnj+1 + U jn j,↑nj,↓ ,(2) where σ presents the spin-up and spin-down, ↑, ↓ for spin-1/2 fermion, U is a on-site interaction strength between opposite spin due to s-wave scattering withn j = σĉ † j,σĉ j,σ the number operator. With the spin degree of freedom, the model exhibits eight energy bands. The topological insulator is presented at the filling N/L = 1. In order to quantitatively reveal the SSH4 models, we will perform DMRG numerical method with lattice length up to L = 320 for spinless SSH4 model and L = 128 for spin-1/2 SSH4 model, for which we retain 400 truncated states per DMRG block and perform 30 sweeps with acceptable truncation errors [53]. -1 0 -5 ( h ) V d 2 E / d V 2 -3 5 -2 0 -5 1 0 d 3 E / d V 3 0 . 0 0 0 0 . 0 0 5 0 . 0 1 0 2 1 0 x 1 0 -2 ( i ) D M R G F i t 1 / D 1 / L 1 III. ORDER PARAMETERS The strong-correlated topological properties can be well described by the degeneracy in entanglement spectrum of ground-state, entanglement entropy, and excited energy gap. The system is topological nontrivial if the entanglement spectrum is degenerate since the entanglement spectrum is associated with the energy spectrum of edge excitations [16,[54][55][56][57][58][59]. The entanglement spectrum is defined as a logarithmic rescaling of the Schmidt values [54] ξ i = − ln(ρ i ),(3) with ρ i being the eigenvalue of the reduced density matrixρ l = Tr L−l |ψ ψ|, where |ψ is the ground-state wave-function of Hamiltonian (1) and l is the length of the left block for a specific bipartition. The quantum criticality of the interaction-driven topological phase transition can be characterized with the von Neumann entropy [59][60][61][62][63][64] S vN = −Tr l [ρ l logρ l ],(4) with l = L/2 the half part of the lattice. It is believed that the property underlying the long-range correlations is entanglement [65], and, on the other hand, the correlation length becomes divergent at the critical point of continuous phase transition [66]. The divergence of the von Neumann entropy at the critical point thus indicates a continuous transition [57]. Besides, the von Neumann entropy also reveals the central charge of the conformal field theory underlying the critical behavior, which generally determines the effective field theory and reflects the universality class of phase transition [67]. For a critical system under periodic boundary conditions, the von Neumann entropy of a subchain of length l scales as S vN (l) = c 3 ln sin πl L + const,(5) in which, the slope at large distance gives the central charge c of the conformal field theory [68][69][70][71]. The formula for the case under open boundary condition is obtained by replacing 3 by 6. One also can use the fidelity of the wavefunction of the ground states to identify the phase transition, which can be defined as the modulus of the overlap between two states(ψ , ψ) [74] F (ψ , ψ) = | ψ |ψ |, where |ψ and |ψ are the input and output states respectively, and both of them are normalized. The topological ground state of extended SSH model under periodic boundary condition is nondegenerate and separated from the first excited state by a finite gap, which closes and reopens across a topological phase transition. The excited energy gap is defined as ∆ e = E p e (N ) − E p g (N ),(7) where E p e (N ) [E p g (N )] is the first-excited (ground) state energy of N atoms under periodic boundary condition. As we all known, the nearest-neighbor interaction V can induce the CDW phase, in which the CDW order parameters can be defined as C = 1 L L i=1 (−1) i n i .(8) For TI under open boundary condition, the presence of localized density of edge mode is its typical feature. The density distribution of the edge modes can be calculated as ∆n j = n j (N + 1) − n j (N ) ,(9) with n j (N ) the density distribution for N atoms under the open boundary condition. IV. MANY-BODY QUANTUM PHASES A. Spinless SSH4 We first characterize the many-body properties of spinless SSH4 Hamiltonian (1). Based on the experimental setup, we here fixed t 1 = t 3 = t 4 = 1 and vary t 2 . When increasing t 2 from 0 to 2 in the absence of interaction, the phase is trivial band insulator (BI) when t 2 < 1, and TI t 2 > 1 with critical point t c 2 = 1. Here we consider the topological phase transition driven by the nearest-neighbor interaction V for a fixed tunneling, i.e., t 2 = 1.6. For zero and weak nearest-neighbor interaction strength V , the lowest entanglement spectrum ξ i is twofold degenerate for finite lattice size L = 320, as shown in Fig. 2(a). But the ξ i show the different features for noninteracting and weak interaction topological insulators. For noninteracting topological insulator, some of the high levels of ξ i show the four-fold degeneracy. However, all levels of ξ i for weak interacting topological insulator are two-fold degenerate [see Fig. 2(b)]. Further increasing the interaction strength V , ξ i is no longer degenerate beyond a critical interaction strength V c ∼ 4.34, as shown in Fig. 2(a). The von Neumann entropy S vN also exhibits a sharp peak at around the critical point, as shown in Fig. 2(c). Moreover, the excited energy gap ∆ e under the periodic boundary condition closes at the critical point and then reopens, as shown in Figs. 2(d) and 2(e). In this processing of the phase transition, the CDW order parameter C become finite values from zero when V beyond the critical strength V c , as shown in Fig. 2(f). Above all, one can conclude that the nearest-neighbor interaction V drives the TI into the CDW phase through a phase transition. In the CDW phase, the chiral symmetry protecting the nontrivial topological phase has been spontaneously broken by the CDW order [49]. The ground states of interacting SSH model are approximately equivalent to that of a non-interacting SSH model plus an additional onsite staggered term in the CDW phase [49]. Hence, the phase transitions from the topologically trivial/nontrivial phases to the CDW phase can be classified with Landau's paradigm. Specifically, the local order parameter characterizing the phase transition is the CDW order defined in Eq. (8). Although, the CDW phase is a topologically trivial phase, evidenced by the lacking of entanglemententropy degeneracy, as shown in Fig. 2(a). The phase transition to the CDW phase is not a standard topological phase transition in the common sense, because it can be described with local order parameters and ac- companies the spontaneous breaking of symmetries. It is consistent with the transition from the topological band insulator to the antiferromagnetic Mott insulator in twodimensional Kane-Mele-Hubbard model [72], which is in the universality class of three-dimensional XY model that also accompanies spontaneous symmetry breaking [73]. As expected from the previous literature [74], a sharp dip on the curve of fidelity | ψ(V )|ψ(V + δV ) | (δV = 0.02) accompanies with the quantum phase transition to emerge, and the dip becomes sharper and sharper as the system size increases, as shown in Fig. 2(g). The behavior is associated with the dramatic change of ground state in the critical regime. It is argued that the phase transition to CDW phase in interacting SSH model is a first-order phase transition when the difference between the alternating hopping coefficients is not too large, based on the variational study [49]. In contrast, we find this transition in the interacting SSH4 model considered here is a continuous (third-order) phase transition, as directly evidenced by the discontinuousness of the third-order derivative of ground-state energy [see Fig. 2(h)]. Actually, the sharp peak of entanglement entropy shown in Fig. 2(c) also indicates the phase transition is continuous. Note that the discontinuousness of the third-order derivative of groundstate energy looks not so obvious in the figure is due to that the system size is finite in the numerical calculation, but it obviously shows a sharp jump at around the critical point. As shown in Fig. 2(i), the finite-size scaling indicates that the fourth-order derivative of ground-state energy moves toward the divergent regime in the thermodynamic limit (the relatively large deviation is due to the derivatives are calculated with numerical difference order by order). This observation further confirms the thirdorder derivative of energy is not continuous and the CDW phase transition is a third-order phase transition. At the critical point between the TI and CDW, the energy spectrum is gapless in the thermodynamic limit [i.e. 46. By using finite-size scaling, we get the critical points of interaction-driven Landau's phase transitions between TI and CDW V c = 4.53 when t 2 = 1.6 in the thermodynamic limit for interacting SSH4 model, as shown in Fig. 2(l). We use similar methods to identify the critical points V c for several t 2 . According to the calculated degeneracy of entanglement spectrum, entanglement entropy, energy gaps, CDW parameter order, fidelity, and derivatives of ground-state energy, we can draw the phase diagram in the t 2 − V plane, as shown in Fig. 3(a). This phase diagram contains three phases such as TI, BI and CDW. By scaling the von Neumann entropy S vN (l) of the critical lines, we find that the critical line between TI and CDW (BI) is the Luttinger liquid with central charge c = 0.46. For large nearest-neighbor interaction strength V , the density profile n j of the ground-state always modulates along real lattice space with periodic 2, the corresponding phase is CDW, as shown in Fig. 3(b). For weak V , the ground-state is TI (BI) when t 2 > 1 (t 2 < 1). The TI not only exhibits two-fold degenerate entanglement spectrum but also has two-fold degenerate ground-state under the open boundary condition, in which only one edge model occupied on one edge side for each degenerate ground-state, as shown in Figs. 3(c) and 3(d). For BI, the density profile is uniform (i.e. n j = 0.5), with the entanglement spectrum almost completely non-degenerate, which are not shown. B. Spin-1/2 SSH4 Here, we consider the spin-1/2 SSH4 Hamiltonian (2), which contains the on-site interaction. Similar as the Fig. 3, we calculate the entanglement spectrum, entanglement entropy, energy gaps, CDW orders, fidelity, derivatives of ground-state energy, and central charge, as shown in Fig. 4. Combing with the finite-size scaling, we map out the phase diagram of the spin-1/2 SSH4 model, as shown in Figs. 5(a) and 5(b). For the filling N/L = 1, the atoms fully occupy the lower half of the eight energy bands in the BI and TI regime. The repulsive on-site interaction favors less density on the same site, while the CDW phase has twice density than the uniform BI and TI cases. Therefore, repulsive on-site interaction enhances the TI (BI) phases and suppresses the CDW phase. In contrast, the attractive on-site interaction suppresses the TI (BI) phases but enhances the CDW phase. We would like to note that, in contrast to the spinless case, the inter-spin nearest-neighbor interaction terms in the spinful case also enhance the CDW phase, even when the on-site interaction is absent. The spinful case thus has a lower critical value of nearest-neighbor interaction for the CDW phase transition. For example, as shown in Fig. 5 (Fig. 3), the spinful (spinless) system already enters (have not entered) the CDW phase when U = 0 and V = 2 (V = 2). For TI in repulsive on-site interaction regime, only the one component atoms can be localized at one side of the edge. The TI features four-fold degeneracies groundstate with the four-fold localized density distributions at the edge, such as | ↑ located at left edge, | ↑ located at right edge, | ↓ located at left edge, and | ↓ located at right edge, as shown in Figs. 5(c) and 5(d). For TI in attractive on-site interaction regime, both the two component atoms with equal number localized at one side of the edge. The ground-state are two-fold degeneracies, one is | ↑ + ↓ located at left edge, another is | ↑ + ↓ located at right edge. V. CONCLUSIONS In conclusion, we have studied theoretically the interacting SSH4 models at half filling using DMRG method. We find that the nearest-neighbor interaction can drive the TI (BI) to CDW phase. Varying the tunnelings, there exist the topological phase transition between TI and BI. The critical lines of the topological phase transitions correspond to the Luttinger liquids with integer central charges. We have calculated entanglement spectrum, entanglement entropy, energy gaps, and CDW order parameter, to identify the interaction-driven CDW phase transitions and phase diagrams. The phase transition to the CDW phase driven by interaction is shown to be a continuous phase transition. The central charges at the phase boundaries are fixed. We also have studied the topological properties of the spin-1/2 interacting SSH4 model, and find that the repulsive on-site interaction can enhance the TI and suppress the CDW phase. But the attractive on-site interaction plays the oppositive role. We also investigate the edge-model of the TIs. In experiment, the entanglement entropy can be measured using quantum interference of many-body twins of ultracold atoms in optical lattices [64].The CDW phase can be detected by time-of-flight in cold atom experiment. Our work provides new insights into the many-body physics in systems with topological properties, and may stimulate the quantum simulation of strong-correlated topological insulators with cold atoms in optical superlattices. FIG. 2 : 2(a) The lowest four levels in the entanglement spectrum ξi (i = 0, 1, 2, 3) as a function of the interaction strength V . (b) The 16 lower levels in the entanglement spectrum for two values of V = 0, 1. (c) The von Neumann entropy SvN, and (d) the excited energy gap ∆e versus V . (e) The finite-size scaling of the ∆e at the critical point, and the red solid line is a linear fit with ∆e ∼ 0 in the large-L limit. (f) The CDW order parameter C, (g) fidelity ψ(V )|ψ(V + δV ) (δV = 0.02) of finite lattice lengths, and (h) the derivatives of ground-state energy d n E/dV n (n = 2, 3) versus V . (i) The finite-size scaling of the inverse of peak value of the fourth-order derivative of ground-state energy (not show here)], and (j) The fitting central charge c (note that only at the critical point c can be defined, although we always can fit the formula and obtain a value) as functions of the interaction strength V . (k) The scaling of the von Neumann entropy SvN(l) as a function of subchain l at the critical point. The green line is SvN(l) = c 3 ln[sin(πl/L)] + const with c = 0.46 (extracted from the fitting of the mean values of SvN). (l) The finite-size scaling of the critical point for phase transition between TI and CDW with t2 = 1.6. The critical point Vc = 4.53 in the thermodynamic limits. In all subfigure, we have t2 = 1.6 and L = 320 except the finite-size scalings. All subfigure are under periodic boundary condition. FIG. 3 : 3(a) The phase diagram of spinless SSH4 Hamiltonian (1) in t2 − V plane in the thermodynamic limit under periodic condition, which contains TI (topological insulator), BI (band insulator), and CDW (charge density wave). The critical line between TI and CDW (black line with circle symbol) is the Luttinger liquid with central charge c = 0.46. The critical line between TI and BI (red line with diamond symbol) is the Luttinger liquid with central charge c = 0.46. (b) The density profile nj of CDW with t2 = 1.2 and V = 5.0 under periodic boundary condition. (c) and (d) The edge-model density distributions ∆nj of two-fold degenerate TI with t2 = 1.6 and V = 2.0 under open boundary condition. In (b)-(d), we have L = 320 and N = 160. FIG. 4 : 4Similar as the Fig. 3. (a) The entanglement spectrum ξi versus U . (b) The several lower levels in the entanglement spectrum for two values of U = 0, 5. (c) The von Neumann entropy SvN, and (d) the excited energy gap ∆e versus U . (e) The finite-size scaling of the ∆e at the critical point. (f) The CDW order parameter C, (g) the fidelity Fidelity ψ(U )|ψ(U + δU ) (δU = 0.02), and (h) the derivatives of ground-state energy d n E/dU n (n = 2, 3) versus U . (i) The finite-size scaling of the inverse of peak value of the fourth-order derivative of ground-state energy, and (j) the fitting central charge c as functions of the interaction strength U . (k) The scaling of the von Neumann entropy SvN(l) as a function of subchain l at the critical point. (l) The finite-size scaling of the critical point for phase transition between TI and CDW. In all subfigure, we have t2 = 1.6, V = 2.0 and L = N = 128 except the finite-size scalings. All subfigure are under periodic boundary condition. ∆ e = 0; see Fig. 2(e)] and the scaling of von Neumann entropy S vN (l) = c 3 ln[sin(πl/L)] + const with a central charge c = 0.46, as shown in Figs. 2(j) and 2(k). The critical line is the Luttinger liquid with central charge c = 0. FIG. 5 : 5The phase diagram of spin-1/2 SSH4 Hamiltonian (2) (a) in t2 − U plane with V = 2, and (b) in V − U plane with t2 = 1.6 in the thermodynamic limit under periodic condition. The edge-model density distributions of different spin ∆n σ j (c) σ =↑ and (d) σ =↓, with t2 = 1.6, U = 5.0, V = 2.0, and L = N = 128 under open boundary condition. In (c) and (d), the number in the figure legends label different degenerate ground-state. . 0 0 0 0 . 0 0 5 0 . 0 1 0 3 . 6 4 . 0 4 . 4 4 . 8 ( l ) D M R G F i t 2020SCUNL210. Our simulations make use of the ALPSCore library[75], based on the original ALPS project[76]. Colloquium: Topological insulators. M Z Hasan, C L Kane, Rev. Mod. Phys. 823045M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82, 3045 (2010). Topological insulators and superconductors. X.-L Qi, S.-C Zhang, Rev. Mod. Phys. 831057X.-L. Qi and S.-C. Zhang, Topological insulators and su- perconductors, Rev. Mod. Phys. 83, 1057 (2011). Colloquium: Topological band theory. A Bansil, H Lin, T Das, Rev. Mod. Phys. 8821004A. Bansil, H. Lin, and T. Das, Colloquium: Topological band theory, Rev. Mod. Phys. 88, 021004 (2016). Classification of topological quantum matter with symmetries. C.-K Chiu, J C Y Teo, A P Schnyder, S Ryu, Rev. Mod. Phys. 8835005C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Classification of topological quantum matter with sym- metries, Rev. Mod. Phys. 88, 035005 (2016). Weyl and Dirac semimetals in three-dimensional solids. N P Armitage, E J Mele, A Vishwanath, Rev. Mod. Phys. 9015001N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyl and Dirac semimetals in three-dimensional solids, Rev. Mod. Phys. 90, 015001 (2018). Topological quantum matter with ultracold gases in optical lattices. N Goldman, J C Budich, P Zoller, Nat. Phys. 12639N. Goldman, J. C. Budich, and P. Zoller, Topological quantum matter with ultracold gases in optical lattices, Nat. Phys. 12, 639 (2016). Topological quantum matter with cold atoms. D.-W Zhang, Y.-Q Zhu, Y X Zhao, H Yan, S.-L Zhu, Adv. Phys. 67253D.-W. Zhang, Y.-Q. Zhu, Y. X. Zhao, H. Yan, and S.-L. Zhu, Topological quantum matter with cold atoms, Adv. Phys. 67, 253 (2018). Topological bands for ultracold atoms. N R Cooper, J Dalibard, I B Spielman, Rev. Mod. Phys. 9115005N. R. Cooper, J. Dalibard, and I. B. Spielman, Topo- logical bands for ultracold atoms, Rev. Mod. Phys. 91, 015005 (2019). Solitons in polyacetylene. W P Su, J R Schrieffer, A J Heeger, Phys. Rev. Lett. 421698W. P. Su, J. R. Schrieffer, and A. J. Heeger, Solitons in polyacetylene, Phys. Rev. Lett. 42, 1698 (1979). Direct measurement of the Zak phase in topological Bloch bands. M Atala, M Aidelsburger, J T Barreiro, D Abanin, T Kitagawa, E Demler, I Bloch, Nat. Phys. 9795M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E.e Demler, and I. Bloch, Direct measurement of the Zak phase in topological Bloch bands, Nat. Phys. 9, 795 (2013). Topological charge pumping in a one-dimensional optical lattice. L Wang, M Troyer, X Dai, Phys. Rev. Lett. 11126802L. Wang, M. Troyer, and X. Dai, Topological charge pumping in a one-dimensional optical lattice. Phys. Rev. Lett. 111, 026802 (2013). A Thouless quantum pump with ultracold bosonic atoms in an optical superlattice. M Lohse, C Schweizer, O Zilberberg, M Aidelsburger, I Bloch, Nat. Phys. 12350M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger, and I. Bloch, A Thouless quantum pump with ultracold bosonic atoms in an optical superlattice, Nat. Phys. 12, 350 (2016). Topological Thouless pumping of ultracold fermions. S Nakajima, T Tomita, S Taie, T Ichinose, H Ozawa, L Wang, M Troyer, Y Takahashi, Nat. Phys. 12296S. Nakajima, T.i Tomita, S. Taie, T. Ichinose, H. Ozawa, L. Wang, M. Troyer, and Y. Takahashi, Topological Thouless pumping of ultracold fermions, Nat. Phys. 12, 296 (2016). Real-space imaging of a topologically protected edge state with ultracold atoms in an amplitudechirped optical lattice. M Leder, C Grossert, L Sitta, M Genske, A Rosch, M Weitz, Nat. Comm. 713112M. Leder, C. Grossert, L. Sitta, M. Genske, A. Rosch, and M. Weitz, Real-space imaging of a topologically pro- tected edge state with ultracold atoms in an amplitude- chirped optical lattice, Nat. Comm. 7, 13112 (2016). Topological Insulators-Dirac Equation in Condensed Matters. S Q Shen, SpringerNew YorkS. Q. Shen, Topological Insulators-Dirac Equation in Condensed Matters (Springer, New York, 2012). Characterization of a Topological Mott Insulator in One Dimension. T Yoshida, R Peters, S Fujimoto, N Kawakami, Phys. Rev. Lett. 112196404T. Yoshida, R. Peters, S.i Fujimoto, and N. Kawakami, Characterization of a Topological Mott Insulator in One Dimension, Phys. Rev. Lett. 112, 196404 (2014). Sedlmayr, Boundary fidelity and entanglement in the symmetry protected topological phase of the SSH model. J Sirker, M Maiti, N P Konstantinidis, N , J. Stat. Mech. 10032J. Sirker, M. Maiti, N. P. Konstantinidis, and N. Sedl- mayr, Boundary fidelity and entanglement in the sym- metry protected topological phase of the SSH model, J. Stat. Mech. P10032 (2014). Detecting edge degeneracy in interacting topological insulators through entanglement entropy. D Wang, S Xu, Y Wang, C Wu, Phys. Rev. B. 91115118D. Wang, S. Xu, Y. Wang, and C. Wu, Detecting edge degeneracy in interacting topological insulators through entanglement entropy, Phys. Rev. B 91, 115118 (2015). Two-body physics in the Su-Schrieffer-Heeger model. M Di Liberto, A Recati, I Carusotto, C Menotti, Phys. Rev. A. 9462704M. Di Liberto, A. Recati, I. Carusotto, and C. Menotti, Two-body physics in the Su-Schrieffer-Heeger model, Phys. Rev. A 94, 062704 (2016). Entanglement spectrum of Su-Schrieffer-Heeger-Hubbard model. B.-T Ye, L.-Z Mu, H Fan, Phys. Rev. B. 94165167B.-T. Ye, L.-Z. Mu, and H. Fan, Entanglement spec- trum of Su-Schrieffer-Heeger-Hubbard model, Phys. Rev. B 94, 165167 (2016). Diagnosing topological edge states via Entanglement monogamy. K Meichanetzidis, J Eisert, M Cirio, V Lahtinen, J K Pachos, Phys. Rev. Lett. 116130501K. Meichanetzidis, J. Eisert, M. Cirio, V. Lahtinen, and J. K. Pachos, Diagnosing topological edge states via En- tanglement monogamy, Phys. Rev. Lett. 116, 130501 (2016). Multihole edge states in Su-Schrieffer-Heeger chains with interactions. A M Marques, R G Dias, Phys. Rev. B. 95115443A. M. Marques and R. G. Dias, Multihole edge states in Su-Schrieffer-Heeger chains with interactions, Phys. Rev. B 95, 115443 (2017) Phase structure of the interacting Su-Schrieffer-Heeger model and the relationship with the Gross-Neveu model on lattice. Y Kuno, Phys. Rev. B. 9964105Y. Kuno, Phase structure of the interacting Su-Schrieffer- Heeger model and the relationship with the Gross-Neveu model on lattice, Phys. Rev. B 99, 064105 (2019). A M Marques, R G Dias, Topological bound states in interacting Su-Schrieffer-Heeger rings. 30305601A. M. Marques and R. G. Dias, Topological bound states in interacting Su-Schrieffer-Heeger rings, Journ. of Phys.: Cond. Matt. 30, 305601 (2018). Floquet-bloch theory and topology in periodically driven lattices. A Gómez-León, G Platero, Phys. Rev. Lett. 110200403A. Gómez-León and G. Platero, Floquet-bloch theory and topology in periodically driven lattices, Phys. Rev. Lett. 110, 200403 (2013). Floquet topological transitions in a driven one-dimensional topological insulator. V Lago, M Atala, L E F F Torres, Phys. Rev. A. 9223624V. Dal Lago, M. Atala, and L. E. F. F. Torres, Floquet topological transitions in a driven one-dimensional topo- logical insulator, Phys. Rev. A 92, 023624 (2015). Topological phases of generalized Su-Schrieffer-Heeger models. L Li, Z Xu, S Chen, Phys. Rev. B. 8985111L. Li, Z. Xu, and S. Chen, Topological phases of gen- eralized Su-Schrieffer-Heeger models, Phys. Rev. B 89, 085111 (2014). Engineering a flux-dependent mobility edge in disordered zigzag chains. F A An, E J Meier, B Gadway, Phys. Rev. X. 831045F. A. An, E. J. Meier, and B. Gadway, Engineering a flux-dependent mobility edge in disordered zigzag chains, Phys. Rev. X 8, 031045 (2018). Interplay between long-range hopping and disorder in topological systems. B Perez-Gonzalez, M Bello, A Gomez-Leon, G Platero, Phys. Rev. B. 9935146B. Perez-Gonzalez, M. Bello, A. Gomez-Leon, and G. Platero, Interplay between long-range hopping and dis- order in topological systems, Phys. Rev. B 99, 035146 (2019). Topological and nontopological features of generalized Su-Schrieffer-Heeger models. N Ahmadi, J Abouie, D Baeriswyl, Phys. Rev. B. 101195117N. Ahmadi, J. Abouie, and D. Baeriswyl, Topological and nontopological features of generalized Su-Schrieffer- Heeger models, Phys. Rev. B 101, 195117 (2020). R R Kumar, N Roy, Y R Kartik, S Rahul, S Sarkar, arXiv:2112.02485Topological phase transition at quantum criticality. arXiv preprintR. R. Kumar, N. Roy, Y. R. Kartik, S. Rahul, and S. Sarkar, Topological phase transition at quantum critical- ity, arXiv preprint arXiv:2112.02485 (2021). Two-leg Su-Schrieffer-Heeger chain with glide reflection symmetry. S.-L Zhang, Q Zhou, Phys. Rev. A. 95R61601S.-L. Zhang and Q. Zhou, Two-leg Su-Schrieffer-Heeger chain with glide reflection symmetry, Phys. Rev. A 95, 061601(R) (2017). Quantum charge pumps with topological phases in a Creutz ladder. N Sun, L.-K Lim, Phys. Rev. B. 9635139N. Sun and L.-K. Lim, Quantum charge pumps with topological phases in a Creutz ladder, Phys. Rev. B 96, 035139 (2017). Exploring interacting topological insulators with ultracold atoms: the synthetic creutzhubbard model. J Jünemann, A Piga, S.-J Ran, M Lewenstein, M Rizzi, A Bermudez, Phys. Rev. X. 731057J. Jünemann, A. Piga, S.-J. Ran, M. Lewenstein, M. Rizzi, and A. Bermudez, Exploring interacting topologi- cal insulators with ultracold atoms: the synthetic creutz- hubbard model, Phys. Rev. X 7, 031057 (2017). Realization of a Cross-Linked Chiral Ladder with Neutral Fermions in a 1D Optical Lattice by Orbital-Momentum Coupling. J H Kang, J H Han, Y Shin, Phys. Rev. Lett. 121150403J. H. Kang, J. H. Han, and Y. Shin, Realization of a Cross-Linked Chiral Ladder with Neutral Fermions in a 1D Optical Lattice by Orbital-Momentum Coupling, Phys. Rev. Lett. 121, 150403 (2018). Flat-Band Localization in Creutz Superradiance Lattices, Realization of a Cross-Linked Chiral Ladder with Neutral Fermions in a 1D Optical Lattice by Orbital-Momentum Coupling. J Y He, R Mao, H Cai, J.-X Zhang, Y Li, L Yuan, S.-Y Zhu, D.-W Wang, Phys. Rev. Lett. 126103601J Y. He, R. Mao, H. Cai, J.-X. Zhang, Y. Li, L. Yuan, S.-Y. Zhu, and D.-W. Wang, Flat-Band Localization in Creutz Superradiance Lattices, Realization of a Cross- Linked Chiral Ladder with Neutral Fermions in a 1D Optical Lattice by Orbital-Momentum Coupling, Phys. Rev. Lett. 126,103601 (2021). Topological characterization of chiral models through their long time dynamics. M Maffei, A Dauphin, F Cardano, M Lewenstein, P Massignan, New J. Phys. 2013023M. Maffei, A. Dauphin, F. Cardano, M. Lewenstein, and P. Massignan, Topological characterization of chiral mod- els through their long time dynamics. New J. Phys. 20, 013023 (2018). Synthetic gauge fields in synthetic dimensions. A Celi, P Massignan, J Ruseckas, N Goldman, I B Spielman, G Juzeliūnas, M Lewenstein, Phys. Rev. Lett. 11243001A. Celi, P. Massignan, J. Ruseckas, N. Goldman, I. B. Spielman, G. Juzeliūnas, and M. Lewenstein, Synthetic gauge fields in synthetic dimensions. Phys. Rev. Lett. 112, 043001 (2014). Four-dimensional quantum Hall effect with ultracold atoms. H M Price, O Zilberberg, T Ozawa, I Carusotto, N Goldman, Phys. Rev. Lett. 115195303H. M. Price, O. Zilberberg, T. Ozawa, I. Carusotto, and N. Goldman, Four-dimensional quantum Hall effect with ultracold atoms, Phys. Rev. Lett. 115, 195303 (2015). Observation of chiral edge states with neutral fermions in synthetic Hall ribbons. M Mancini, G Pagano, G Cappellini, L Livi, M Rider, J Catani, C Sias, P Zoller, M Inguscio, M Dalmonte, L Fallani, Science. 3491510M. Mancini, G. Pagano, G. Cappellini, L. Livi, M. Rider, J. Catani, C. Sias, P. Zoller, M. Inguscio, M. Dalmonte, and L. Fallani, Observation of chiral edge states with neutral fermions in synthetic Hall ribbons, Science 349, 1510 (2015). Visualizing edge states with an atomic Bose gas in the quantum Hall regime. B K Stuhl, H.-I Lu, L M Aycock, D Genkina, I B Spielman, Science. 3491514B. K. Stuhl, H.-I. Lu, L. M. Aycock, D. Genkina, and I. B. Spielman, Visualizing edge states with an atomic Bose gas in the quantum Hall regime, Science 349, 1514 (2015). Photonic topological insulator in synthetic dimensions. E Lustig, Nature. 567356E. Lustig, et al. Photonic topological insulator in syn- thetic dimensions. Nature 567, 356 (2019). Analytical solution of open crystalline linear 1D tight-binding models. A M Marques, R G Dias, Journ. of Phys. A: Math. and Theor. 5375303A. M. Marques and R. G. Dias, Analytical solution of open crystalline linear 1D tight-binding models, Journ. of Phys. A: Math. and Theor. 53, 075303 (2020). Topological characterizations of an extended Su-Schrieffer-Heeger model. D Xie, W Gou, T Xiao, B Gadway, B Yan, npj Quantum Inf. 555D. Xie, W. Gou, T. Xiao, B. Gadway, and B. Yan, Topo- logical characterizations of an extended Su-Schrieffer- Heeger model, npj Quantum Inf 5, 55 (2019). Non-Hermitian generalizations of extended Su-Schrieffer-Heeger models. Y He, C.-C Chien, J. Phys.: Condens. Matter. 3385501Y. He and C.-C. Chien, Non-Hermitian generalizations of extended Su-Schrieffer-Heeger models, J. Phys.: Con- dens. Matter 33, 085501 (2021). Density matrix formulation for quantum renormalization groups. S R White, Phys. Rev. Lett. 692863S. R. White, Density matrix formulation for quan- tum renormalization groups, Phys. Rev. Lett. 69, 2863 (1992). The density-matrix renormalization group. U Schollwök, Rev. Mod. Phys. 77259U. Schollwök, The density-matrix renormalization group, Rev. Mod. Phys. 77, 259 (2005). Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures. A Altland, M R Zirnbauer, Phys. Rev. B. 551142A. Altland and M. R. Zirnbauer, Nonstandard symme- try classes in mesoscopic normal-superconducting hybrid structures, Phys. Rev. B 55, 1142 (1997). Variational study of the interacting, spinless Su-Schrieffer-Heeger model. M Yahyavi, L Saleem, B Hetényi, J. Phys.: Condens. Matter. 30445602M. Yahyavi, L. Saleem and B. Hetényi, Variational study of the interacting, spinless Su-Schrieffer-Heeger model, J. Phys.: Condens. Matter 30, 445602 (2018). Ludwig, Classification of topological insulators and superconductors in three spatial dimensions. A P Schnyder, S Ryu, A Furusaki, A W W , Phys. Rev. B. 78195125A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Lud- wig, Classification of topological insulators and super- conductors in three spatial dimensions, Phys. Rev. B 78, 195125 (2008). Topological phases: classification of topological insulators and superconductors of noninteracting fermions, and beyond. A W W Ludwig, Phys. Scr., T. 16814001A. W. W. Ludwig, Topological phases: classification of topological insulators and superconductors of non- interacting fermions, and beyond, Phys. Scr., T 168, 014001 (2016). Breakdown of the topological classification Z for gapped phases of noninteracting fermions by quartic interactions. T Morimoto, A Furusaki, C Mudry, Phys. Rev. B. 92125104T. Morimoto, A. Furusaki, and C. Mudry, Breakdown of the topological classification Z for gapped phases of non- interacting fermions by quartic interactions, Phys. Rev. B 92, 125104 (2015). In spinless model with lattice length L = 320, the ground states have truncation errors smaller than 10 −31 . The excited states even still have truncation errors smaller than 10 −11 . For the spinful model with lattice length L = 128, the truncation errors for the ground state are smaller than 10 −9 . Even for the excited states in the spinful model. Although the truncation errors (the sum of all discarded eigenvalues in the reduced density matrix) for the excited states are far larger than those for the ground states, the maximam truncation errors are basically smaller than 10 −6 and are acceptible. we still have truncation errors smaller than 10 −6Although the truncation errors (the sum of all discarded eigenvalues in the reduced density matrix) for the excited states are far larger than those for the ground states, the maximam truncation errors are basically smaller than 10 −6 and are acceptible. In spinless model with lattice length L = 320, the ground states have truncation er- rors smaller than 10 −31 . The excited states even still have truncation errors smaller than 10 −11 . For the spin- ful model with lattice length L = 128, the truncation errors for the ground state are smaller than 10 −9 . Even for the excited states in the spinful model, we still have truncation errors smaller than 10 −6 . Entanglement spectrum as a generalization of entanglement entropy: Identification of topological order in non-Abelian fractional quantum Hall effect states. H Li, F D M Haldane, Phys. Rev. Lett. 10110504H. Li and F. D. M. Haldane, Entanglement spectrum as a generalization of entanglement entropy: Identification of topological order in non-Abelian fractional quantum Hall effect states, Phys. Rev. Lett. 101, 010504 (2008). Symmetry-protected topological phase in a one-dimensional correlated bosonic model with a synthetic spin-orbit coupling. J.-Z Zhao, S.-J Hu, P Zhang, Phys. Rev. Lett. 115195302J.-Z. Zhao, S.-J. Hu, and P. Zhang, Symmetry-protected topological phase in a one-dimensional correlated bosonic model with a synthetic spin-orbit coupling, Phys. Rev. Lett. 115, 195302 (2015). Topological phases of one-dimensional fermions: An entanglement point of view. A M Turner, F Pollmann, E Berg, Phys. Rev. B. 8375102A. M. Turner, F. Pollmann, and E. Berg, Topological phases of one-dimensional fermions: An entanglement point of view, Phys. Rev. B 83, 075102 (2011). Entanglement spectrum of a topological phase in one dimension. F Pollmann, A M Turner, E Berg, M Oshikawa, Phys. Rev. B. 8164439F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa, Entanglement spectrum of a topological phase in one di- mension, Phys. Rev. B 81, 064439 (2010). Entanglement spectrum of topological insulators and superconductors. L Fidkowski, Phys. Rev. Lett. 104130502L. Fidkowski, Entanglement spectrum of topological in- sulators and superconductors, Phys. Rev. Lett. 104, 130502 (2010). Topological entanglement Renyi entropy and reduced density matrix structure. S T Flammia, A Hamma, T L Hughes, X.-G Wen, Phys. Rev. Lett. 103261601S. T. Flammia, A. Hamma, T. L. Hughes, and X.-G. Wen, Topological entanglement Renyi entropy and re- duced density matrix structure, Phys. Rev. Lett. 103, 261601 (2009). Measuring Rényi entanglement entropy in quantum monte carlo simulations. M B Hastings, I González, A B Kallin, R G Melko, Phys. Rev. Lett. 104157201M. B. Hastings, I. González, A. B. Kallin, and R. G. Melko, Measuring Rényi entanglement entropy in quan- tum monte carlo simulations, Phys. Rev. Lett. 104, 157201 (2010). Measuring entanglement growth in quench dynamics of bosons in an optical lattice. A J Daley, H Pichler, J Schachenmayer, P Zoller, Phys. Rev. Lett. 10920505A. J. Daley, H. Pichler, J. Schachenmayer, and P. Zoller, Measuring entanglement growth in quench dynamics of bosons in an optical lattice, Phys. Rev. Lett. 109, 020505 (2012). Measuring entanglement entropy of a generic many-body system with a quantum switch. D A Abanin, E Demler, Phys. Rev. Lett. 10920504D. A. Abanin and E. Demler, Measuring entanglement entropy of a generic many-body system with a quantum switch, Phys. Rev. Lett. 109, 020504 (2012). Identifying topological order by entanglement entropy. H.-C Jiang, Z.-H Wang, L Balents, Nat. Phys. 8902H.-C. Jiang, Z.-H. Wang, and L. Balents, Identifying topological order by entanglement entropy, Nat. Phys. 8, 902 (2012). Measuring entanglement entropy in a quantum many-body system. R Islam, R Ma, P M Preiss, M E Tai, A Lukin, M Rispoli, M Greiner, Nature. 52877R. Islam, R. Ma, P. M. Preiss, M. E. Tai, A. Lukin, M. Rispoli, and M, Greiner, Measuring entanglement en- tropy in a quantum many-body system, Nature 528, 77 (2015). Entanglement, quantum phase transitions, and density matrix renormalization. T J Osborne, M A Nielsen, Quant. Inform. Proc. 1T. J. Osborne, and M. A. Nielsen, Entanglement, quan- tum phase transitions, and density matrix renormaliza- tion, Quant. Inform. Proc. 1, 45-53 (2002). S Sachdev, Quantum Phase Transitions. CambridgeCambridge University PressS. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, 1999). P Calabrese, J Cardy, Entanglement Entropy and Quantum Field Theory. 6002P. Calabrese, and J. Cardy, Entanglement Entropy and Quantum Field Theory, J. Stat. Mech. 2004, P06002(2014) Entanglement in Quantum Critical Phenomena. G Vidal, J Latorre, E Rico, A Kitaev, Phys. Rev. Lett. 90227902G. Vidal, J. Latorre, E. Rico, and A. Kitaev, Entangle- ment in Quantum Critical Phenomena, Phys. Rev. Lett. 90, 227902 (2003). Entanglement entropy and quantum field theory. P Calabrese, J Cardy, J. Stat. Mech. 066002P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 06 (2004) P06002. Entanglement entropy and conformal field theory. P Calabrese, J Cardy, J. Phys. A. 42504005P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42, 504005 (2009). Violation of the area law and long-range correlations in infinitedimensional-matrix product states. A E B Nielsen, G Sierra, J I Cirac, Phys. Rev. A. 8353807A. E. B. Nielsen, G. Sierra, and J. I. Cirac, Violation of the area law and long-range correlations in infinite- dimensional-matrix product states, Phys. Rev. A 83, 053807 (2011). Quantum phase transitions in the Kane-Mele-Hubbard model. M Hohenadler, Z Y Meng, T C Lang, S Wessel, A Muramatsu, F F Assaad, Phys. Rev. B. 85115132M. Hohenadler, Z. Y. Meng, T. C. Lang, S. Wessel, A. Muramatsu, and F. F. Assaad, Quantum phase transi- tions in the Kane-Mele-Hubbard model, Phys. Rev. B 85, 115132(2012). Critical behaviour of the 3D XY-model: a Monte Carlo study. Aloysius P Gottlob, Martinhasenbusch , Physica A: Statistical Mechanics and its Applications. 201Aloysius P.Gottlob, and MartinHasenbusch, Critical be- haviour of the 3D XY-model: a Monte Carlo study, Phys- ica A: Statistical Mechanics and its Applications 201, 593-613(1993). Fidelity approach to quantum phase transitions. Shi-Jian Gu, Int. J. Mod. Phys. B. 244371Shi-Jian Gu, Fidelity approach to quantum phase tran- sitions, Int. J. Mod. Phys. B 24, 4371(2010). . A Gaenko, A Antipov, G Carcassi, T Chen, X Chen, Q Dong, L Gamper, J Gukelberger, R Igarashi, S Iskakov, M Konz, J Leblanc, R Levy, P Ma, J Paki, H Shinaoka, S Tödo, M Troyer, E Gull, Computer Physics Communications. 213235Updated core libraries of the ALPS projectA. Gaenko, A. Antipov, G. Carcassi, T. Chen, X. Chen, Q. Dong, L. Gamper, J. Gukelberger, R. Igarashi, S. Iskakov, M. Konz, J. LeBlanc, R. Levy, P. Ma, J. Paki, H. Shinaoka, S. Tödo, M. Troyer, and E. Gull, Updated core libraries of the ALPS project, Computer Physics Communications 213, 235 (2017). The ALPS project release 2.0: open source software for strongly correlated systems. B Bauer, L D Carr, H G Evertz, A Feiguin, J Freire, S Fuchs, L Gamper, J Gukelberger, E Gull, S Guertler, A Hehn, R Igarashi, Journal of Statistical Mechanics: Theory and Experiment. 055001B. Bauer, L. D. Carr, H. G. Evertz, A. Feiguin, J. Freire, S. Fuchs, L. Gamper, J. Gukelberger, E. Gull, S. Guertler, A. Hehn, R. Igarashi et al., The ALPS project release 2.0: open source software for strongly correlated systems, Journal of Statistical Mechanics: Theory and Experiment 2011(05), P05001 (2011).
{'fraction_non_alphanumeric': 0.06837487537387836, 'fraction_numerical': 0.04081754735792622, 'mean_word_length': 4.114839367669556, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 16, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'Exploring topological phases in interacting systems is a challenging task. We investigate manybody topological physics of interacting fermions in an extended Su-Schrieffer-Heeger (SSH) model, which extends the two sublattices of SSH model into four sublattices and thus is dubbed SSH4 model, based on the density-matrix renormalization-group numerical method. The interactiondriven phase transition from topological insulator to charge density wave (CDW) phase can be identified by analyzing the variations of entanglement spectrum, entanglement entropies, energy gaps, CDW order parameter, and fidelity. We map the global phase diagram of the many-body ground state, which contains nontrivial topological insulator, trivial insulator and CDW phases, respectively. In contrast to interacting SSH model, in which the phase transitions to the CDW phase are argued to be first-order phase transitions, the phase transitions between the CDW phase and topologically trivial/nontrivial phases are shown to be continuous phase transitions. Finally, we also show the phase diagram of interacting spinful SSH4 model, where the attractive (repulsive) on-site spin interaction amplifies (suppresses) the CDW phase. The models analyzed here can be implemented with ultracold atoms on optical superlattices. arXiv:2208.00390v3 [cond-mat.quant-gas]', 'arxivid': '2208.00390', 'author': ['Xiaofan Zhou \nInstitute of Laser spectroscopy\nState Key Laboratory of Quantum Optics and Quantum Optics Devices\nShanxi University\n030006TaiyuanChina\n\nCollaborative Innovation Center of Extreme Optics\nShanxi University\n030006TaiyuanShanxiChina\n', 'Jian-Song Pan \nCollege of Physics\nSichuan University\n610065ChengduChina\n\nKey Laboratory of High Energy Density Physics and Technology of Ministry of Education\nSichuan University\n610065ChengduChina\n', 'Suotang Jia \nInstitute of Laser spectroscopy\nState Key Laboratory of Quantum Optics and Quantum Optics Devices\nShanxi University\n030006TaiyuanChina\n\nCollaborative Innovation Center of Extreme Optics\nShanxi University\n030006TaiyuanShanxiChina\n'], 'authoraffiliation': ['Institute of Laser spectroscopy\nState Key Laboratory of Quantum Optics and Quantum Optics Devices\nShanxi University\n030006TaiyuanChina', 'Collaborative Innovation Center of Extreme Optics\nShanxi University\n030006TaiyuanShanxiChina', 'College of Physics\nSichuan University\n610065ChengduChina', 'Key Laboratory of High Energy Density Physics and Technology of Ministry of Education\nSichuan University\n610065ChengduChina', 'Institute of Laser spectroscopy\nState Key Laboratory of Quantum Optics and Quantum Optics Devices\nShanxi University\n030006TaiyuanChina', 'Collaborative Innovation Center of Extreme Optics\nShanxi University\n030006TaiyuanShanxiChina'], 'corpusid': 251224152, 'doi': '10.1103/physrevb.107.054105', 'github_urls': [], 'n_tokens_mistral': 16526, 'n_tokens_neox': 13895, 'n_words': 7933, 'pdfsha': '15bdfcd016c6061c9590e92af44e4f66e80117d9', 'pdfurls': ['https://export.arxiv.org/pdf/2208.00390v3.pdf'], 'title': ['Exploring interacting topological insulator of extended Su-Schrieffer-Heeger model', 'Exploring interacting topological insulator of extended Su-Schrieffer-Heeger model'], 'venue': []}
arxiv
Possible dibaryon production atPanda with a Lagrangian approach * May 2022. 2022 Yubing Dong Institute of High Energy Physics Chinese Academy of Sciences 100049BeijingChina School of Physical Sciences University of Chinese Academy of Sciences 101408BeijingChina Pengnian Shen Institute of High Energy Physics Chinese Academy of Sciences 100049BeijingChina Possible dibaryon production atPanda with a Lagrangian approach * Chinese Physics C No. X, Xxx; Bc2022XMay 2022. 2022Panda experimentsd * (2380)Phenomenological effective Lagrangian approachpp anni- hilationProduction In order to confirm the existence of the dibaryon state d * (2380) observed at WASA@COSY, we estimate the production of the possible dibaryon and anti-dibaryon pair d * d * at the energy region of the upcoming experiments atPanda. Based on some qualitative properties of d * extracted from the analysez in the non-relativistic quark model, the production cross section for this spin-3 particle pair are calculated with the help of an phenomenological effective relativistic and covariant Lagrangian approach.No. X 2 quantum numbers being 2370 ∼ 2380 MeV, 70 ∼ 80 MeV, and I(J P ) = 0(3 + ), respectively. Since the baryon number of this resonance is 2, one believes it may just be the light-quark-only dibaryon d * (2380) that has been hunted for several decades.According to the mass of d * and the relevant thresholds of the two-baryon (∆∆), two-baryon plus one-meson (N N π), and two-baryon plus two-meson (N N ππ) channels near by, one could believe the threshold (or cusp) effect should be much smaller in the case of this resonance than that in the cases of the exotic XYZ resonances[7][8][9][10]. And due to its narrow width, one may think of this dibaryon as a state with, at least, a hexaquark dominated structure. Up to now, several theoretical proposals for its internal structure have been investigated. Among them, two proposed structures have attracted much attention. The first one comes from the study in QDF. The calculation showed that d * has a compact structure and is a candidate of an exotic hexaquark-dominated resonant state[11][12][13][14][15][16][17]. (A similar assumption regards the d * state being a deeply bound state of two ∆s[18,19], however its width is larger than the measured value.) The other one considers it, in HDF, as a molecular-like hadronic state, which originates from an assumption of a three-body resonance ∆N π or a molecular-like state D 12 π[20][21][22]25]. Although the mass and the partial widths of the double pionic decays of such a hypothetic dibaryon resonance can be reasonably reproduced by both proposed structures, the interpretations of it in two proposals are entirely different. Of course, there are many other studies for understanding the structure of d * and pp → π + d reaction, for instance, the triple di-quark model [23] and the triangle singularity mechanism [24]. Whether they can systematically explain all existing experimental data still needs to be further tested. Therefore, it is necessary to look for other physical observables in some sophisticated kinematics regions or some processes other than p-p (or p-d) collision, which might explicitly provide remarkably different results for different structure models, especially the two d * structure models highlighted earlier. Actually, such theoretical analyses have been carried out on the electromagnetic form factors of d * [26, 27] and on the possible evidence in the γ + d processes[28].Up to now, the dibaryon d * has been observed by WASA@COSY Collaborations in the process of pn → dππ and the fusion process of pd → 3 He+ππ. It seems that d * was also observed in another process, say γ+d → dππ at ELPH[29][30][31]. It should be stressed that the forthcoming experiments atPanda (Pbar ANnihilation at DArmstadt) are expected to provide a confirmation of this dibaryon state if it does exist. This is because that atPanda, the antiproton beam collides with the proton target and the momentum of p could be in the range from 1 GeV/c to 15 GeV/c. It corresponds to a range of the total center-of-mass (CM) energy √ s of the proton-antiproton system being from ∼ 2.25 GeV to ∼ 5.5 GeV [32-34], which covers 2M d * ∼ 4.76 GeV . Therefore, the future experiments based on the pp annihilation reaction can provide another way to produce the dibaryon and anti-dibaryon pairs, say d * d * , and can further give the information of this d * resonance.In this work, a phenomenological effective Lagrangian approach (PELA) is employed to study the production of a spin-3 particle d * . It should be mentioned that this approach has been successfully applied to many weakly bound state problems[9]in the exotic meson sectors of X(3872), Z b (10610), and Introduction It is known that the history of the study of dibaryons, such as H and d * particles, can be traced back to about 60 years ago (refer to the review articles by Clement [1,2]). Theoretically, the possible dibaryon states have been carefully investigated with many approaches from the hadronic degrees of freedom (HDF) to the quark degrees of freedom (QDF). In 2009, the evidence of d * was firstly reported by CELSIUS/WASA and WASA@COSY Collaborations [3][4][5][6]. Observations of the existence of such a dibaryon were claimed in their series of experiments. This is because that their observed peak cannot simply be understood by the role from either the intermediate Roper excitation or from the t-channel intermediate ∆∆ state, except by introducing a new intermediate resonance with its mass, width, and Z b (10650) [35][36][37][38] and the exotic baryon sector of Λ c (2940) [39,40], and also the deuteron (S=1) [41]. For the pion meson, which is different from the above-mentioned loosely bound states, its properties can also be reasonably obtained by this approach [42]. Moreover, this approach has been applied to the study of the dibaryon candidate of N Ω (S=2) [43], predicted by Ref. [44] and the HAL QCD collaboration [45]. Therefore, as an extrapolation, PELA could be adopted as a reasonable tool to estimate the cross section of pp → d * d * in the energy region of √ s ∈ [4.8, 5.50] GeV atPanda. This paper is organized as follows. In section 2, we show the description of the spin-3 dibaryon states by PELA. Then, a brief discussion for the cross sections of the pp → ∆∆ and pp → ∆∆ → d * d * processes is given in section 3. The numerical results are displayed in section 4. Finally, section 5 is devoted for short summary and discussions. 2 Description of the spin-3 particle d * (2380) in PELA By considering the interpretation of d * in the nonrelativistic quark model in Refs. [11,[13][14][15][16][17], here, we write the effective Lagrangian of d * (3 + ) and its two constituents (for example two ∆s) as L d * ∆∆ (x) = g d * ∆∆ d 4 yΦ(y 2 )∆ α (x + y/2)Γ α,(µ 1 µ 2 µ 3 ),β ∆ C β (x − y/2)d * µ 1 µ 2 µ 3 (x; λ) + H.C.(1) where ∆ α is the spin-3/2 ∆ field, and ∆ C α stands for its charge-conjugated with ∆ C α = C∆ T α and C = iγ 2 γ 0 . In the above equation, d * µ 1 µ 2 µ 3 (x; λ) represents the spin-3 d * field with the polarization λ. It is a rank-3 field. The coupling of the two ∆s to d * relates to the two spin-3/2 particles and a spin-3 particle. The three-particle vertex reads [46] Γ α,(µ 1 µ 2 µ 3 ),β = 1 6 γ µ 1 g µ 2 α g µ 3 β + g µ 2 β g µ 3 α + γ µ 2 g µ 3 α g µ 1 β + g µ 1 β g µ 3 α (2) + γ µ 3 g µ 1 α g µ 2 β + g µ 1 β g µ 2 α . The introduced correlation function Φ(y 2 ) in eq. (1) describes the distribution of the two constituents in the system and makes the integral of Feynman diagrams finite in ultraviolet. This function is related to its Fourier transform in momentum space,Φ(−p 2 ), by Φ(y 2 ) = d 4 p (2π) 4 e −ipyΦ (−p 2 ) where p stands for the relative Jacobi momentum between the two constituents of d * . For simplicity,Φ is phenomenologically chosen in a Gaussian-like form asΦ (−p 2 ) = exp(p 2 /Λ 2 ),(3) where Λ is a model parameter, relating to the scale of the distribution of the constituents inside d * , and has dimension of mass. All calculations for the loop integral, hereafter, are performed in Euclidean space after the Wick transformation, and all the external momenta go like p µ = (p 0 , p ) → p µ E = (p 4 , p ) (where the subscript "E" stands for the momentum in Euclidean space) with p 4 = −ip 0 . In Euclidean space the Gaussian correlation function ensures that all loop integrals are ultraviolet finite (details can be found in Ref. [9]). d * (P) ∆(k) ∆(k 1 ) d * (P) β ′ α ′ α β Then, one can determine the coupling of d * to its constituents by using the Weinberg-Salam compositeness condition [47][48][49][50]. This condition means that the probability of finding the dressed bound state as a bare (structureless) state is equal to zero. In the case of d * , our previous calculation in QDF [11,[13][14][15][16][17] shows that d * contains a |∆∆ > and also a |CC > components, which are orthogonal to each other. As a rough estimation, a simplest chain approximation is used. Then this condition can be written as Z d * = 1 − ∂Σ (1) (∆∆) (P 2 ) ∂P 2 P 2 =M 2 d * − ∂Σ (1) (CC) (P 2 ) ∂P 2 P 2 =M 2 d * = Z d * ,(∆∆) + Z d * ,(CC) = 0 ,(4) where P is the momentum of d * (2380), Σ (1) (∆∆) or (CC) (M 2 d * ) is the non-vanishing part of structural integral of the mass operator of d * with spin-parity 3 + (the detailed derivation can be found in Refs. [51,52]). Here we assume these Z d * ,(∆∆) and Z d * ,(CC) are independent. Since the probabilities of the ∆∆ and CC components are about P ∆∆ ∼ 1/3 and P CC ∼ 2/3, respectively in quark model calculation, therefore, Z d * ,(∆∆) = 1 3 − ∂Σ (1) (∆∆) (P 2 ) ∂P 2 P 2 =M 2 d * = 0, (4a) Z d * ,(CC) = 2 3 − ∂Σ (1) (CC) (P 2 ) ∂P 2 P 2 =M 2 d * = 0,(4b) and the coupling g d * ∆∆ can be extracted from the compositeness condition of (4a). The mass operator of the d * dressed by the ∆∆ channel is given in Fig. 1. It should be stressed that the coupling constant determined from the compositeness condition of eq. (4a) contains the renormalization effect since the chain approximation is considered (also refer to Refs. [51,52]). The explicit expression of the full mass operator can be written as Σ (µ ′ i ),(µ j ) ∆ (P) = g d * ∆∆ (Λ) 2 d 4 k (2π) 4 i exp − 2(k − P 2 ) 2 E Λ 2 × T r Γ (µ ′ i ) α ′ β ′ (5) × / k + M ∆ k 2 − M 2 ∆ − g ββ ′ + γ β γ β ′ 3 + 2k β k β ′ 3M 2 ∆ + γ β k β ′ − γ β ′ k β 3M ∆ × Γ (µ j ) β α × / k 1 − M ∆ k 2 1 − M 2 ∆ − g α ′ α + γ α ′ γ α 3 + 2k α ′ 1 k α 1 3M 2 ∆ − γ α ′ k α 1 − γ α k α 1 3M ∆ k 1 =P−k , with µ ′ i and µ j being the abbreviations of (µ ′ 1 , µ ′ 2 , µ ′ 3 ) and (µ 1 , µ 2 , µ 3 ), respectively. In general, according to its Lorentz structure, the mass operator Σ (µ ′ i ),(µ j ) (c) (P) takes the form of Σ (µ ′ i ),(µ j ) (c) (P) = 8 l=1 L (µ ′ i ),(µ j ) (l),(c) Σ (l) (c) (P 2 ),(6) with Σ (l) (c) (P 2 ) being the structural integrals appeared in the expression of the full mass operator, and the Lorentz structures being L (µ ′ i ),(µ j ) (1) = 1 6 g µ ′ 1 µ 1 g µ ′ 2 µ 2 g µ ′ 3 µ 3 + g µ ′ 2 µ 3 g µ ′ 3 µ 2 + g µ ′ 1 µ 2 g µ ′ 2 µ 1 g µ ′ 3 µ 3 + g µ ′ 2 µ 3 g µ ′ 3 µ 1 (7) + g µ ′ 1 µ 3 g µ ′ 2 µ 2 g µ ′ 3 µ 1 + g µ ′ 2 µ 1 g µ ′ 3 µ 2 , = 1 6 g α 1 β 1 (g α 2 β 2 g α 3 β 3 + g α 2 β 3 g α 3 β 2 ) + ...... with α i ∈ (µ ′ 1 , µ ′ 2 , µ ′ 3 ) and β j ∈ (µ 1 , µ 2 , µ 3 ), L (µ ′ i ),(µ j ) (2) = 1 9 g α 1 α 2 g α 3 β 1 g β 2 β 3 + g α 3 β 2 g β 3 β 1 + g α 3 β 3 g β 1 β 2 + ...... ,(8)L (µ ′ i ),(µ j ) (3) = 1 18 P α 1 P β 1 g α 2 β 2 g α 3 β 3 + g α 2 β 3 g α 3 β 2 + ...... ,(9)L (µ ′ i ),(µ j ) (4) = 1 9 P α 1 P β 1 g α 2 α 3 g β 2 β 3 + ...... ,(10)L (µ ′ i ),(µ j ) (5) = 1 18 P α 1 P α 2 g α 3 β 1 g β 2 β 3 + g α 3 β 2 g β 1 β 3 + g α 3 β 3 g β 1 β 2 + ......(11)+ P β 1 P β 2 g α 1 β 3 g α 2 α 3 + g α 2 β 3 g α 2 α 3 + g α 3 β 3 g α 1 α 2 + ...... , L (µ ′ i ),(µ j ) (6) = 1 9 P α 1 P α 2 P β 1 P β 2 g α 3 β 3 + ...... ,(12)L (µ ′ i ),(µ j ) (7) = 1 6 P α 1 P α 2 P α 3 P β 1 g β 2 β 3 + P β 1 P β 2 P β 3 P α 1 g α 2 α 3 + ...... ,(13)L (µ ′ i ),(µ j ) (8) = P µ ′ 1 P µ ′ 2 P µ ′ 3 P µ 1 P µ 2 P µ 3 .(14) Clearly, due to the property of the polarization vector of the spin-3 particle, like ǫ µ 1 µ 2 µ 3 (P, λ) shown in Ref. [26], only the first term on the right-hand side of eq. (6) gives the contribution while the other terms do not. We introduce Lorentz projector T (µ ′ i ),(µ j ) ⊥ = 1 42 g µ ′ 1 µ 1 g µ ′ 2 µ 2g µ ′ 3 µ 3 +g µ ′ 2 µ 3g µ ′ 3 µ 2 (15) +g µ ′ 1 µ 2 g µ ′ 2 µ 1g µ ′ 3 µ 3 +g µ ′ 2 µ 3g µ ′ 3 µ 1 +g µ ′ 1 µ 3 g µ ′ 2 µ 2g µ ′ 3 µ 1 +g µ ′ 2 µ 1g µ ′ 3 µ 2 − 1 105 g µ ′ 1 µ ′ 2 g µ ′ 3 µ 1g µ 2 µ 3 +g µ ′ 3 µ 2g µ 1 µ 3 +g µ ′ 3 µ 3g µ 1 µ 2 +g µ ′ 1 µ ′ 3 g µ ′ 2 µ 1g µ 2 µ 3 +g µ ′ 2 µ 2g µ 1 µ 3 +g µ ′ 2 µ 3g µ 1 µ 2 +g µ ′ 2 µ ′ 3 g µ ′ 1 µ 1g µ 2 µ 3 +g µ ′ 1 µ 2g µ 1 µ 3 +g µ ′ 1 µ 3g µ 1 µ 2 , withg µν = g µν ⊥ = −g µν + P µ P ν M 2 d * .(16) It satisfies following relations P i T (µ ′ i ),(µ j ) ⊥ = 0, µ ′ i ∈ (µ ′ 1 , µ ′ 2 , µ ′ 3 ) or µ j ∈ (µ 1 , µ 2 , µ 3 );(17)L (1) (µ ′ i ),(µ i ) T (µ ′ i ),(µ j ) ⊥ = 1(18) and L (i) (µ ′ i ),(µ j ) T (µ ′ i ),(µ j ) ⊥ = 0, (i = 2, 3, ..., 8).(19) Thus, when the full mass operator Σ (µ ′ i ),(µ j ) (P) acts with the Lorentz projector T (µ ′ i ),(µ j ) ⊥ , the product gives the scalar function Σ (1) (P 2 ) in eq.(4), and it will contribute to the compositeness condition. Finally the coupling constant |g| 2 d * ∆∆ can be determined from eq. (4a). It should be stressed that here we have adopted the Gaussian-type correlation function of eq.(3), Φ(−p 2 ) = exp(p 2 /Λ 2 ), the model-dependent parameter Λ relates to the size of the system in the nonrelativistic approximation, at least in physical meaning. Thus, one may roughly connect b, representing the size of the d * in the non-relativistic wave function, to the parameter Λ by b 2 2 ∼ 1 Λ 2 . According to the quark model calculation in Refs. [14], b ∼ 0.8f m, and we roughly choose the parameter Λ ∼ 0.34 GeV . 3 Cross sections for pp → ∆∆ and pp → ∆∆ → d * d * Cross section for pp → ∆∆ There are only a few experiments of the pp → ∆(1232)∆(1232) process in the literature [53][54][55][56][57]. Refs. [56,57] studied pp → ∆∆ at 7.23 GeV and 12 GeV . The samples were obtained from the large exposures of the 2m hydrogen bubble-chamber (HBC) experiment to the U5 antiproton beam at CERN. The account of ppπ + π − was thought to come dominantly from the ∆ ++ ∆ ++ channel. It was believed that the process can be described by the t-channel pion or reggeized pion exchange. A good description of the mass and t-distributions for the reaction at 3.6 GeV and 5.7 GeV was given by the one-pion exchange model [58]. Moreover the cross section of the process, in terms of the Mandelstam variable of s, is parameterized as σ(s) = As −n with A = (67 ± 20) mb and n = 1.5 ± 0.1, respectively [56]. This pp → ∆∆ process can also be estimated theoretically by using an effective Lagrangian [59] L (t ∆ z t N z ) πN∆ = g πN ∆ F (p t )∆ (t ∆ z ) µ I t ∆ z t N z · ∂ µ π (t π z ) N (t N z ) + h.c.,(20) where g πN ∆ and F (p t ) are the effective coupling constant and phenomenological form factor, respectively, the latter function is chosen to be F (p t ) = Λ * 2 M − m 2 π Λ * 2 M − p 2 t n exp(αp 2 t ),(21) with the parameters Λ * M ∼ 1 GeV and n = 1. In eq. (20), I t ∆ z t N z = C 3/2t ∆ z 1tπ ,1/2t N zê * tπ is the isospin transition operator. Then, the cross section is σ = (2π) 4 δ 4 (p 1 + p 2 − p 3 − p 4 ) 4 (p 1 · p 2 ) − m 2 1 m 2 2 × P ol. M if 2 d 3 p 3 (2π) 3 2E p 3 d 3 p 4 (2π) 3 2E p 4 ,(22) where p 1,2 (or p 3,4 ) are the momenta of the incoming (or outgoing) particles, M if 2 stands for averaging over the polarizations of the initial states and summing over the polarizations of final states. We can write the matrix element M if , representing the contribution of the tree-diagram to pp → ∆ ++ ∆ ++ , via π exchange with the Lagrangian of eq. (20), as M pp→∆ ++ ∆ ++ if = g 2 πN ∆ F 2 (p t ) Ū ∆ α p α t u(p 1 ) 1 p 2 t − m 2 π v(p 2 )p β t V ∆ β (p 4 ) .(23) The resultant cross section is shown and compared with the parameterized empirical cross section in Fig. 2. It should be mentioned that in Ref. [60] F πN ∆ = f πN ∆ /m π , and in order to fit the decay width of ∆ → πN , where the initial momentum of ∆ is set to be zero, the value of f πN ∆ is taken as 2.2 ± 0.04. Thus, their F πN ∆ ∼ (15.7 ± 0.285) GeV −1 . In our present numerical calculation, to fit the parameterized cross section, we introduce an addition trajectory function exp(0.2t) (p 2 t = t < 0) and take F πN ∆ ∼ 10.75 GeV −1 . Here, we find that the 10% variation in F πN ∆ may cause about 50% change in the total cross section since the cross section is proportional to F 4 πN ∆ . In addition, the change of the estimated √ s-dependent cross section with respect to the variations of the parameters Λ * M and α are shown in this figure as well. Those curves tell that the cross section with smaller √ s becomes larger when Λ * M deceases or α increases. The combined effect of Λ * M and α, namely the effect of the phenomenological form factor, on the cross section is more pronounced in the small √ s region. Therefore, the current lagrangian is flexible enough to fit the experimental data. It should be mentioned that in this calculation, we only consider the one-pion exchange, insert a phenomenological form factor, and take the coupling of F πN ∆ as a free parameter. It seems that our tree diagram result is reasonable to reproduce the total cross section of pp → ∆ ++∆++ , although we do not consider the contributions from other meson exchange, for instance the ρ meson. In conclusion, the effective Lagrangian L (t ∆ z t N z ) πN ∆ mentioned above is appropriate for describing the cross section of the pp → ∆∆ process, so it should also be acceptable and reasonable to be further used in the investigation of the d * d * generation in the pp → ∆∆ → d * d * process. Cross section for pp → ∆∆ → d * d * The Feynman diagram of the pp → d * d * process via ∆∆ intermediate is shown in Fig. 3. In this diagram, d * d * pair is generated from the pp → ∆∆ annihilation reaction. It should be noted that in the loop, in the higher order approximation, when pp annihilation generates a ∆∆ pair, it can also create a corresponding CC pair, therefore, when ∆ interacts with ∆ (or∆ interacts with∆), a corresponding hidden-color component CC (orCC) would exist. According to the conclusion in our previous quark model calculations, about 1/3 of ∆∆ (∆∆) and 2/3 of CC (CC) can form a d * (d * ),as |d * >∼ 1 3 |∆∆ > + 2 3 |CC >, with the spin and isospin quantum numbers of the colored cluster C being 3/2 and 1/2. Thus, to estimate events of d * (d * ) creation, we can only use 1/3 of ∆∆ (∆∆) component, because it corresponds to one d * (d * ). It should be further stressed that the process in this diagram can occur only when the Mandelstam variable satisfies √ s > 2M d * ∼ 4.8 GeV . It is clear that the threshold of this production channel is lower than the upper limit of the CM energy of thePanda device. To calculate the matrix element of Fig. 3, we have to use the vertices of L πN∆ in eq. (20) and L d * ∆∆ in eq. (1). The matrix element of M if for the process of pp → d * d * reads d * (p 3 ) p(p 2 ) k 1 k 2 p(p 1 ) k 3 π(p t ) α 2 β 2 α 1 β 1 α ′ 1 β ′ 1d * (p 4 )M (pp→d * d * ) if =v N (p 2 )Π (ν i ),(µ j ) u N (p 1 ) d * (p 3 ) (µ j ) (λ) d * (p 4 ) (ν i ) (λ),(24) with Π (ν i ),(µ j ) = d 4 p t (2π) 4 i p α 2 t S C 3/2,(α 2 β 2 ) (k 2 )Γ β 2 ,(ν i ),β 1 S C 3/2,(β 1 β ′ 1 ) (k 3 ) (25) ×Γ β ′ 1 ,(µ j ),α ′ 1 S 3/2,(α ′ 1 α 1 ) (k 1 )p t,α 1 F 2 (p t ) p 2 t − m 2 π × exp − (k 1 − k 3 ) 2 E 4Λ 2 + (k 2 − k 3 ) 2 E 4Λ 2 × C Iso , where the exponential factors in the last arrow on the right side of eq. (25) come from the consideration of the phenomenological bound state problem of d * discussed explicitly in section 2, and the subscripts "E" and "M " denote "Euclidean" and "Minkowski", respectively. The propagators of a spin-3/2 particle ∆ and its charge conjugate are S 3/2,µν (p, M ∆ ) = (p / − m) −1 × − g µν + γ µ γ ν 3 + 2p µ p ν 3M 2 ∆ + γ µ p ν − γ ν p µ 3M ∆ ,(26)S C 3/2,νµ (p, M ∆ ) = CS T 3/2,µν (p, M ∆ )C , with the charge conjugate operator being C = iγ 2 γ 0 . Moreover, the constant C Iso. = 7 18 represents the isospin factor since the intermediate state can be either ∆ ++ ∆ ++ , or ∆ + ∆ + , or ∆ 0 ∆ 0 (here we only consider the pion-exchange in the pp → ∆∆ process). Then, the cross section of such a process is formally expressed by eq. (22), where the matrix element is replaced by M (pp→d * d * ) if given in eq. (24). Noticed that the square of the matrix element is proportional to g 4 d * ∆∆ and g 4 πN ∆ , respectively. Here, since the d * is a spin-3 particle, its field can be described by a traceless rank-3 polarization vector like ǫ µ 1 µ 2 µ 3 (P, λ). This polarization vector has the properties of ǫ ααβ = 0, ǫ αβγ = ǫ βαγ , and P α ǫ αβγ = 0. Therefore, in the summation calculation, we have pol. ǫ µνσ ǫ * αβγ = 1 6 g µα g νβgσγ +g νγgσβ +g µβ g ναgσγ +g νγgσα +g µγ g ναgσβ +g νβgσα (27) − 1 15 g µν g σαgβγ +g σβgαγ +g σγgαβ +g µσ g ναgβγ +g νβgαγ +g νγgαβ +g νσ g µαgβγ +g µβgαγ +g µγgαβ , withg µν showed in eq. (16). Numerical results and discussions Discussion of g d * ∆∆ In this work, we employ our phenomenological effective Lagrangian approach to describe the spin-3 resonance d * . Consequently, the Feynman diagram of Fig. 3 can be calculated covariantly and relativistically. In the calculation, there is only one unique model parameter Λ. We fix this parameter according to the qualitative conclusions obtained from the dynamical calculation in the non-relativistic constituent quark model [14,15]: (1) d * contains two components |∆∆ > and |CC > with probabilities 1/3 and 3/2, respectively; (2) d * is a compact system with a size about b ∼ 0.8 f m; and (3) In the quark model approach, the strong decay widths of d * , in the leading order approximation, are dominantly contributed by the ∆∆ component. Thus, Λ 2 ∼ 2 b 2 , which gives Λ ∼ 0.34 GeV when b ∼ 0.8 f m. Further taking P ∆∆ ∼ 1/3, we can calculate the coupling constant of d * to ∆∆ by using the formulas shown in section 2. The result shows g d * ∆∆ ∼ 3.35. We present the change of the dimensionless coupling constant g d * = g d * ∆∆ √ P ∆∆ with respect to the variation of the model-parameter Λ in the region of [0.25, 0.45] GeV in Fig. 4. The curve in Fig. 4 shows that the dimensionless coupling g d * relates to the model parameter Λ and to the integral of the mass operator structure. When Λ increases, the integral of the loop structure increases, and consequently, the obtained g d * decreases. In addition, although g d * does not depend on P ∆∆ , the g d * ∆∆ is proportional to the square root of the channel probability √ P ∆∆ . Finally, we would mention that we cannot dynamically determined the size parameter as well as the probability in this approach. Instead, to proceed the calculation without contradicting the results given by the quark model, we simply borrow the corresponding qualitative conclusions given in those dynamical quark model calculations. Cross section for pp → ∆∆ → d * d * In the CM energy region of √ s ∈ [4.8 − 5.5] GeV , the evaluated total cross section of the process pp → ∆∆ → d * d * , shown by the Feynman diagram in Fig. 3, is given in Fig. 5. Here, we reiterate that the cross section is evaluated based on the qualitative interpretations of d * in the non-relativistic quark model approach, with which all observed properties of d * can be well described. The cross section curve in Fig. 5 tells us that the total cross section in the d * d * pair production process is about 4-6 orders of magnitude smaller than that in the pp → ∆∆ reaction. We know that the cross section in Fig.5 is dependent on the phase space as well as the matrix element M if . The phase space increases with the increasing √ s. The matrix element of M if relates to model parameter Λ as well as to √ s. The estimated total cross section of pp → ∆∆ → d * d * is also subject to the impact of the interpretation of the d * state, namely its size and the probability of its ∆∆ component. The resultant cross section (with a fixed value of P ∆∆ ∼ 1/3) in Fig. 5 shows its dependence on Λ. Actually the coupling g d * ∆∆ is proportional to √ P ∆∆ and the matrix element M if is proportional to P ∆∆ . Thus, the obtained cross section changes with respect to P 2 ∆∆ . Moreover, the coupling g d * ∆∆ and the matrix element M if are closely related to the structure of the mass operator in the structural integral and to the loop calculations of Fig. 3, respectively. Here, we only display the Λ dependence explicitly in Fig. 5. It shows that in the small √ s region, say less than 5.2 GeV , the cross section is distinctly suppressed, because the coupling constant g d * decreases due to the increase of Λ. However, when √ s is greater than 5.5 GeV , the production cross section with a smaller Λ value, say less than 0.34 GeV , may increase dramatically with the increase of √ s due to the larger structural integral, caused by a larger Λ-dependent g d * ∆∆ value, and a larger phase space. It should be reiterated that as a rough estimate, we only consider the production cross section for the d * -d * pair in this paper, and not take the complicated background contribution into account. When the CM energy is about 5.2 GeV , the Λ dependence of the cross section becomes small, and the estimated cross section becomes meaningful. It should be noted that the obtained cross section of pp → ∆∆ → d * d * is in the order of nb. According to the designed luminosity and integrated luminosity ofPDanda, which are about ∼ 2 × 10 32 cm −2 /s and ∼ 10 4 nb −1 /day, respectively, we expect that about (0.51, 0.71, 1.19) × 10 4 d * d * events can be observed per-day at √ s = (5.0, 5.1, 5.2) GeV , if the overall efficiency is 100%. On the other hand, from a technical point of view, d * cannot be directly observed. Observation of d * is usually achieved through the measurements of its strong decay processes, namely measuring various mesons and baryons, such as π, proton, neutron and etc., and measuring some invariant mass spectra and Dalitz plots and etc.. It is noticed that the dominated decay channels of d * are d * → dππ and d * → pnππ with their partial decay widths of about 27 M eV and 31 M eV , respectively, which correspond to the branching ratios of about 36% and 41%, respectively. As a consequence, the possible events of pp → d * d * →d * dππ (or pp → d * d * → d * d ππ) and pp → d * d * →d * pnππ (or pp → d * d * → d * pn ππ) can roughly be estimated. They are respectively about ((0.18, 0.21), (0.26, 0.29), (0.43, 0.49)) × 10 4 per-day at √ s = (5.0, 5.1, 5.2) GeV (if the overall efficiency is being assumed 100%). Finally, it should be further mentioned that in order to avoid the interference caused by the background of a large number of produced pions and nucleons, according to our previous discussion [61,62], it may be more practical to confirm the existence of d * by looking ford * via the decay channels in above brackets. Summary and discussions In this work, we estimate the cross section of the pp → ∆∆ → d * d * reaction, which might possibly be measured at forthcoming experiments atP anda in the CM energy of √ s ∈ [4.8, 5.5] GeV . A relativistic and covariant phenomenological effective Lagrangian approach is employed in the practical calculation. To describe the structure of the outgoing d * d * pair, qualitative conclusions from the sophisticated and dynamical calculations in the non-relativistic constituent quark model, with which all existing data can be well explained, are directly adopted to fix the model-parameter Λ approximately. The estimated production cross section for d * d * should be a lower bound, since in our assumption, only 1/3 of ∆∆ is considered to be an ingredient of d * . The result shows that the estimated production cross section of this reaction is in the order of nb which is much smaller than the known cross section of pp → ∆∆ whose value is in the order of mb. Nevertheless, among a huge amount of events of produced hadron pair atP anda, there may still exist a certain amount of events of produced d * d * pair. These events are expected to be observed through measuring the final baryons and mesons in some strong decay processes of d * , such as d * → dππ (ord * →dππ) and d * → pnππ (ord * →pnππ). We also rough estimate the possible events of these processes from the branching ratios of the d * strong decays as a reference. Fig. 1 . 1The Mass operator of the d * (2380) → ∆∆. (Fig. 2 . 2Λ* M , α) (1.0, 0.2) (1.15, 0.25) (1.15, 0.2) The estimated cross sections for pp → ∆ ++ ∆ ++ compared to that with a parameterized form of σ(mb) = 67s −1.5 (red curve). The black solid, dashed, and dotted curves represent the calculated results with the parameters of (λ * M (GeV ), α(GeV 2 )) being (1.15, 0.2), (1.0, 0.2), and (1.15, 0.25), respectively. Fig. 3 . 3The Feynman diagram for the pp → d * (2380) +d * (2380) process, where the red bold line and double line stand for the internal ∆ (or ∆ C ) field and the outgoing d * , respectively. Fig. 4 . 4g d * = g d * ∆∆ √ P ∆∆ in PELA versus Λ ∈ [0.25, 0.45] GeV . Fig. 5 . 5The estimated cross section for the reaction of pp → d * d * in units of (nb). On the History of Dibaryons and their Final Observation. H Clement, arXiv:1610.05591Prog. Part. Nucl. Phys. 93195nucl-exH. Clement, "On the History of Dibaryons and their Final Observation,", arXiv:1610.05591 [nucl-ex], Prog. Part. Nucl. Phys. 93, 195 (2017). H Clement, T Skorodko, arXiv:2008.07200Dibaryons: Molecular versus Compact Hexaquarks. 4522001nucl-thH. Clement and T. Skorodko, "Dibaryons: Molecular versus Compact Hexaquarks", arXiv:2008.07200 [nucl-th], Chin. Phys. C 45, 022001 (2021). . M Bashkanov, Phys. Rev. Lett. 10252301M. Bashkanov et al., Phys. Rev. Lett. 102 052301 (2009). ABC Effect in Basic Double-Pionic Fusion -Observation of a new resonance. P Adlarson, WASA-at-COSY CollaborationPhys. Rev. Lett. 106242302P. Adlarson et al. [WASA-at-COSY Collaboration], "ABC Effect in Basic Double-Pionic Fusion -Observation of a new resonance,", Phys. Rev. Lett. 106, 242302 (2011). Isospin Decomposition of the Basic Double-Pionic Fusion in the Region of the ABC Effect. P Adlarson, WASA-at-COSY CollaborationPhys. Lett. 721229P. Adlarson et al. [WASA-at-COSY Collaboration], "Isospin Decomposition of the Basic Double-Pionic Fusion in the Region of the ABC Effect," Phys. Lett. B721, 229 (2013). . P Adlarson, WASA-at-COSY CollaborationPhys. Rev. Lett. 11220202301P. Adlarson et al. [WASA-at-COSY Collaboration], Phys. Rev. Lett. 112, no. 20, 202301 (2014). The hidden-charm pentaquark and tetraquark states. H X Chen, W Chen, X Liu, S L Zhu, arXiv:1601.02092Phys. Rept. 6391hep-phH. X. Chen, W. Chen, X. Liu and S. L. Zhu, "The hidden-charm pentaquark and tetraquark states," arXiv:1601.02092 [hep-ph], Phys. Rept. 639, 1 (2016). Hadronic molecules. F K Guo, C Hanhart, U G Meissner, Q Wang, Q Zhao, B S Zou, arXiv:1705.00141Rev. Mod. Phys. 9015004hep-phF. K. Guo, C. Hanhart, U. G. Meissner, Q. Wang, Q. Zhao and B. S. Zou, "Hadronic molecules," arXiv:1705.00141 [hep-ph], Rev. Mod. Phys. 90, 015004 (2018). . Y Dong, A Faessler, V E Lyubovitskij, Prog. Part. Nucl. Phys. 94282Y. Dong, A. Faessler and V. E. Lyubovitskij, Prog. Part. Nucl. Phys. 94 (2017) 282. R F Lebed, R E Mitchell, E S Swanson, arXiv:1610.04528Heavy-Quark QCD Exotica. 93143hep-phR. F. Lebed, R. E. Mitchell and E. S. Swanson, "Heavy-Quark QCD Exotica," arXiv: 1610.04528 [hep-ph], Prog. Part. Nucl. Phys. 93, 143 (2017). . X Q Yuan, Phys. Rev. C. 6045203X. Q. Yuan et al., Phys. Rev. C 60, 045203 (1999). . M Bashkanov, Stanley J Brodsky, H Clement, Phys. Lett. B. 727438M. Bashkanov, Stanley J. Brodsky, and H. Clement, Phys. Lett. B 727, 438 (2013). . F Huang, Z Y Zhang, P N Shen, W L Wang, Chin. Phys. C. 39771001F. Huang, Z. Y. Zhang, P. N. Shen and W. L. Wang, Chin. Phys. C 39, no. 7, 071001 (2015). . F Huang, P N Shen, Y B Dong, Z Y Zhang, Sci. China Phys. Mech. Astron. 592622002F. Huang, P. N. Shen, Y. B. Dong and Z. Y. Zhang, Sci. China Phys. Mech. Astron. 59, no. 2, 622002 (2016). . Yubing Dong, Pengnian Shen, Fei Huang, Zongye Zhang, Phys. Rev. 9164002Yubing Dong, Pengnian Shen, Fei Huang, and Zongye Zhang, Phys. Rev. C91, 064002 (2015). . Yubing Dong, Fei Huang, Pengnian Shen, Zongye Zhang, Phys. Rev. 9414003Yubing Dong, Fei Huang, Pengnian Shen, and Zongye Zhang, Phys. Rev. C94, 014003 (2016). . Yubing Dong, Fei Huang, Pengian Shen, Zongye Zhang, Chinese Physics. 41101001Yubing Dong, Fei Huang, Pengian Shen, and Zongye Zhang, Chinese Physics C41 (2017) 101001. Dynamical calculation of the ∆∆ dibaryon candidates. H Huang, J Ping, F Wang, arXiv:1312.7756Phys. Rev. C. 89334001hep-phH. Huang, J. Ping and F. Wang, "Dynamical calculation of the ∆∆ dibaryon candidates," arXiv:1312.7756 [hep-ph], Phys. Rev. C 89, no. 3, 034001 (2014). Nucleon-nucleon interaction in a chiral SU(3) quark model revisited. F Huang, W L Wang, Phys. Rev. D. 98774018F. Huang, and W. L. Wang, "Nucleon-nucleon interaction in a chiral SU(3) quark model revisited", Phys. Rev. D, 98, No. 7, 074018, 2018. . A Gal, H Garcilazo, Phys. Rev. Lett. 111172301A. Gal and H. Garcilazo, Phys. Rev. Lett. 111, 172301 (2013). . A Gal, H Garcilazo, Nucl. Phys. A. 92873A. Gal and H. Garcilazo, Nucl. Phys. A 928, 73 (2014). Hidden dibaryons in one-and two-pion production in NN collisions. M N Platonova, V I Kukulin, arXiv:1412.4574Nucl. Phys. A. 946117nucl-thM. N. Platonova and V. I. Kukulin, "Hidden dibaryons in one-and two-pion production in NN collisions," arXiv:1412.4574 [nucl-th] Nucl. Phys. A 946, 117 (2016). d * (2380) and its partners in a diquark model. Pan-Pan, Fei Shi, Wen-Ling Huang, Wang, Eur. Phys. J. C. 794314Pan-Pan Shi, Fei Huang, and Wen-Ling Wang, "d * (2380) and its partners in a diquark model", Eur. Phys. J. C, 79, No. 4, 314 (2019). Triangle singularity mechanism for the pp → π + d fusion reaction. Natsumi Ikeno, Raquel Molina, Eulogio Oset, Phys. Rev. C. 1041Natsumi Ikeno, Raquel Molina, and Eulogio Oset, "Triangle singularity mechanism for the pp → π + d fusion reaction", Phys. Rev. C, 104, No. 1, 014614, (2021). ABC-effect as a signal of chiral symmetry restoration in hadronic collisions. M N Platonova, V I Kukulin, arXiv:1211.0444Phys. Rev. C. 87225202nucl-thM. N. Platonova and V. I. Kukulin, "ABC-effect as a signal of chiral symmetry restoration in hadronic collisions," arXiv: 1211.0444 [nucl-th], Phys. Rev. C 87, no. 2, 025202 (2013). . Yubing Dong, Fei Huang, Pennian Shen, Zongye Zhang, Phys. Rev. D. 9694001Yubing Dong, Fei Huang, Pennian Shen and Zongye Zhang, Phys. Rev. D 96, 094001 (2017). . Yubing Dong, Pennian Shen, Zongye Zhang, Phys. Rev. D. 9711114002Yubing Dong, Pennian Shen and Zongye Zhang, Phys. Rev. D 97, no. 11, 114002 (2018). . Yubing Dong, Fei Huang, Pennian Shen, Zongye Zhang, Int. J. Mod. Phys. A. 34181950100Yubing Dong, Fei Huang, Pennian Shen and Zongye Zhang, Int. J. Mod. Phys. A 34, no. 18, 1950100 (2019). First measurement of coherent double neutral-pion photoproduction on the deuteron at incident energies below 0.9 GeV. T Ishikawa, arXiv:1610.05532Phys. Lett. B. 772nucl-exT. Ishikawa et al., "First measurement of coherent double neutral-pion photoproduction on the deuteron at incident energies below 0.9 GeV", arXiv: 1610.05532 [nucl-ex], Phys. Lett. B 772, 398-402 (2017). Non-strange dibaryons studied in the γd → π 0 π 0 d reaction. T Ishikawa, Phys. Lett. B. 789T. Ishikawa et al., "Non-strange dibaryons studied in the γd → π 0 π 0 d reaction", Phys. Lett. B 789, (2019), 413-318. Study of Non-strange Dibaryon Resonances Via Coherent Double Neutral-Meson Photoproduction from the Deuteron. T Ishikawa, Springer Proc. Phys. 238T. Ishikawa et al., "Study of Non-strange Dibaryon Resonances Via Coherent Double Neutral-Meson Photoproduction from the Deuteron," Springer Proc. Phys., 238 (2020), 609-613. Experimental overview of the P anda experiment. Elisa Fioravanti, J. Phys. Conf. Ser. 50312030Elisa Fioravanti, "Experimental overview of the P anda experiment", J. Phys. Conf. Ser. 503, 012030 (2014). Hadron physics experiments in antiproton proton reactions with the planned PANDA detector. P Hawranek, Int. J. Mod. Phys. A. 22P. Hawranek, "Hadron physics experiments in antiproton proton reactions with the planned PANDA detector", Int. J. Mod. Phys. A 22, 574-577 (2007). Hadron physics with the PANDA experiment at the FAIR. Diego Bettoni, Prof. Part. Nucl. Phys. 67Diego Bettoni, "Hadron physics with the PANDA experiment at the FAIR", Prof. Part. Nucl. Phys. 67, 502-510 (2012). Estimate for the X(3872) to gamma J/ψ decay width. Y B Dong, A Faessler, T Gutsche, V E Lyubovitskij, Phys. Rev. D. 7794013Y. B. Dong, A. Faessler, T. Gutsche and V. E. Lyubovitskij, "Estimate for the X(3872) to gamma J/ψ decay width," Phys. Rev. D 77, 094013 (2008); X(3872) as a hadronic molecule and its decays to charmonium states and pions. Y B Dong, A Faessler, T Gutsche, S Kovalenko, V E Lyubovitskij, Phys. Rev. D. 7994013Y. B. Dong, A. Faessler, T. Gutsche, S. Kovalenko and V. E. Lyubovitskij, "X(3872) as a hadronic molecule and its decays to charmonium states and pions," Phys. Rev. D 79, 094013 (2009). J/ψ gamma and ψ(2S) gamma decay modes of the X(3872). Yubing Dong, Amand Faessler, Thomas Gutsche, Valery E Lyubovitskij, arXiv:0909.0380J. Phys. G. 3815001hep-phYubing Dong, Amand Faessler, Thomas Gutsche, and Valery E. Lyubovitskij, "J/ψ gamma and ψ(2S) gamma decay modes of the X(3872)", arXiv:0909.0380 [hep-ph], J. Phys. G. 38, (2011), 015001. Strong decays of molecular states Z + c and Z ′ + c. Yubing Dong, Amand Faessler, Thomas Gutsche, Valery E Lyubovitskij, arXiv:1306.0824Phys. Rev. D. 8814030hep-phYubing Dong, Amand Faessler, Thomas Gutsche, and Valery E. Lyubovitskij, "Strong decays of molecular states Z + c and Z ′ + c ", arXiv:1306.0824 [hep-ph], Phys. Rev. D 88, (2013), 014030. Strong two-body decays of the Λc(2940)+ in a hadronic molecule picture. Y Dong, A Faessler, T Gutsche, V E Lyubovitskij, arXiv:0910.1204Phys. Rev. D. 8114006hep-phY. Dong, A. Faessler, T. Gutsche and V. E. Lyubovitskij, "Strong two-body decays of the Λc(2940)+ in a hadronic molecule picture," arXiv:0910.1204 [hep-ph] Phys. Rev. D 81, 014006 (2010). Radiative decay of Λc(2940) + in a hadronic molecule picture. Y Dong, A Faessler, T Gutsche, S Kumano, V E Lyubovitskij, Phys. Rev. D. 8234035Y. Dong, A. Faessler, T. Gutsche, S. Kumano and V. E. Lyubovitskij, "Radiative decay of Λc(2940) + in a hadronic molecule picture" Phys. Rev. D 82, 034035 (2010). Phenomenological Lagrangian approach to the electromagnetic deuteron form factors. Y Dong, A Faessler, T Gutsche, S Kumano, V E Lyubovitskij, arXiv:0806.3679Phys. Rev. C. 7835205hep-phY. Dong, A. Faessler, T. Gutsche, S. Kumano and V. E. Lyubovitskij, "Phenomenological Lagrangian approach to the electromagnetic deuteron form factors", arXiv:0806.3679 [hep-ph], Phys. Rev. C 78, 035205 (2008). Pion and sigma meson properties in a relativistic quark model. Amand Faessler, M A Thomas Gutsche, Valery E Ivanov, P Lyubovitskij, Wang, arXiv:0304031Phys. Rev. D. 6814011hep-phAmand Faessler, Thomas Gutsche, M. A. Ivanov, Valery E. Lyubovitskij and P. Wang, "Pion and sigma meson properties in a relativistic quark model", arXiv:0304031[hep-ph], Phys. Rev. D 68, 014011(2003). Towards the decay properties of deuteron-like state d NΩ. Yubing Cheng-Jian Xiao, Thomas Dong, Valery E Gutsche, Dian-Yong Lyubovitskij, Chen, arXiv:2004.12415Phys. Rev. D. 101114032hep-phCheng-Jian Xiao, Yubing Dong, Thomas Gutsche, Valery E. Lyubovitskij, and Dian-Yong Chen, "Towards the decay properties of deuteron-like state d NΩ ", arXiv: 2004.12415 [hep-ph], Phys. Rev. D 101, (2020), 114032. N Ω and ∆Ω dibaryons in a SU(3)chiral quark model. Q B Li, P N Shen, Euro. Phys. J. A8. 417Q. B. Li, P.N.Shen, "N Ω and ∆Ω dibaryons in a SU(3)chiral quark model", Euro. Phys. J. A8,(2000), 417. N Ω dibaryon from lattice QCD near the physical point. T Iritani, HAL QCD CollaborationPhys. Lett. B. 792284T. Iritani et al. (HAL QCD Collaboration), "N Ω dibaryon from lattice QCD near the physical point", Phys. Lett. B 792, 284 (2019). Covariant Propagators and Vertex Functions for Any Spin. D , Michael Scadron, Phys. Rev. 165D. Michael Scadron, "Covariant Propagators and Vertex Functions for Any Spin", Phys. Rev. 165, (1968), 1640-1647. Lagrangian Theory Of Composite Particles. A Salam, Nuovo Cim. 25224A. Salam, "Lagrangian Theory Of Composite Particles," Nuovo Cim. 25, 224 (1962). Elementary Particle Theory Of Composite Particles. S Weinberg, Phys. Rev. 130776S. Weinberg, "Elementary Particle Theory Of Composite Particles," Phys. Rev. 130, 776 (1963). . K Hayashi, M Hirayama, T Muta, N Seto, T Shirafuji, Fortsch. Phys. 15625K. Hayashi, M. Hirayama, T. Muta, N. Seto and T. Shirafuji, Fortsch. Phys. 15, 625 (1967). The Quark Confinement Model of Hadrons. G V Efimov, M A Ivanov, IOP PublishingBristol & PhiladelphiaG. V. Efimov and M. A. Ivanov, The Quark Confinement Model of Hadrons, (IOP Publishing, Bristol & Philadelphia, 1993). Compositeness criteria of particles in quantum field theory and S-matrix theory. Kenji Hayashi, Minoru Hirayama, Taizo Muta, Noriaki Seto, Noriaki , Takeshi Shirafuji, Fortsch. Phys. 1510Kenji Hayashi, Minoru Hirayama, Taizo Muta, Noriaki Seto, Noriaki, and Takeshi Shirafuji, "Compositeness criteria of particles in quantum field theory and S-matrix theory", Fortsch. Phys., 15, No. 10, 625-660, 1967. Strong nucleon and ∆-isobar form factors in quark confinement model. G V Efimov, M A Ivanov, V E Lyubovitskij, Few Body Syst. 6G. V. Efimov, M. A. Ivanov, and V. E. Lyubovitskij, "Strong nucleon and ∆-isobar form factors in quark confinement model", Few Body Syst., 6, 17-43, 1989. Final States with a Neutral Meson and a π + π − orpp Pair in anti-p p Interactions at 7.2-GeV/c. G W Van Apeldoorn, R L F Gruendeman, D Harting, Nucl. Phys. B. 133G. W. Van Apeldoorn, R. L. F. Gruendeman, D. Harting, et al., "Final States with a Neutral Meson and a π + π − orpp Pair in anti-p p Interactions at 7.2-GeV/c", Nucl. Phys. B. 133 245-265 (1978). ∆ ++ and ∆ ++ production in 100 GeV/cpp interactions. D R E Ward?? R, P Ansorge?? C, Bust, Nucl. Phys. B. 141D. R. Ward?? R. E. Ansorge?? C. P. Bust, et al., "∆ ++ and ∆ ++ production in 100 GeV/cpp interactions", Nucl. Phys. B. 141, 203-219 (2978). General Features of the Anti-p p Interaction at 12-GeV/c. P Johnson, Nucl. Phys. B. 173P. Johnson, et al., "General Features of the Anti-p p Interaction at 12-GeV/c", Nucl. Phys. B 173, 77-92 (1980). . G W Van Apeldoorn, Z. Phys. C. 12G. W. Van Apeldoorn et al., Z. Phys. C 12, (1985), 95-98. . Mohammad Saleem, Fazal-E-Aleem, Prog. Theor. Phys. 701156Mohammad Saleem and Fazal-e-Aleem, Prog. Theor. Phys. 70, 1156, (1983). . E , Guenter Wolf, Phys. Rev. 182E. Guenter Wolf, Phys. Rev. 182, (1969), 1538-1560. Xu Cao, Bing-Song Zou, Bing - Song, Hu-Shan Xu, arXiv:1009.1060Phenomenological study on thepN →N N ππ reactions. 861nucl-thXu Cao, Bing-Song Zou, Bing-Song, Hu-Shan Xu, "Phenomenological study on thepN →N N ππ reactions," arXiv: 1009.1060 [nucl-th], Nucl. Phys. A 861, (2011), 23-36. Nucleon resonance transition couplings to vector mesons. D O Riska, G E Brown, arXiv:0005049Nucl. Phys. A. 679nucl-thD. O. Riska and G. E. Brown, "Nucleon resonance transition couplings to vector mesons," arXiv:0005049 [nucl-th], Nucl. Phys. A 679, (2011), 577-596. Ping Chao-Yi Lü, Yubing Wang, Dong, Zong-Ye Peng-Nian Shen, Zhang, arXiv:1810.02138The opportunity to findd * (2380) in the Υ(nS) decay. 9936015hep-phChao-Yi Lü, Ping Wang, Yubing Dong, Peng-nian Shen, and Zong-ye Zhang, "The opportunity to findd * (2380) in the Υ(nS) decay", arXiv: 1810.02138 [hep-ph], Phys. Rev. D 99, 036015 (2019). Phenomenological study on the decay widths of Υ(nS) →d * (2380) + X. Ping Chao-Yi Lü, Yubing Wang, Dong, Zong-Ye Peng-Nian Shen, D M Zhang, Li, arXiv:1803.07795Chinese Phys. C. 4264102hep-phChao-Yi Lü, Ping Wang, Yubing Dong, Peng-nian Shen, Zong-ye Zhang, and D. M. Li, "Phenomenological study on the decay widths of Υ(nS) →d * (2380) + X", arXiv: 1803.07795 [hep-ph], Chinese Phys. C 42, 064102 (2018).
{'fraction_non_alphanumeric': 0.09319230336446495, 'fraction_numerical': 0.05988522561044222, 'mean_word_length': 3.385708645874457, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 8, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 109, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'In order to confirm the existence of the dibaryon state d * (2380) observed at WASA@COSY, we estimate the production of the possible dibaryon and anti-dibaryon pair d * d * at the energy region of the upcoming experiments atPanda. Based on some qualitative properties of d * extracted from the analysez in the non-relativistic quark model, the production cross section for this spin-3 particle pair are calculated with the help of an phenomenological effective relativistic and covariant Lagrangian approach.No. X 2 quantum numbers being 2370 ∼ 2380 MeV, 70 ∼ 80 MeV, and I(J P ) = 0(3 + ), respectively. Since the baryon number of this resonance is 2, one believes it may just be the light-quark-only dibaryon d * (2380) that has been hunted for several decades.According to the mass of d * and the relevant thresholds of the two-baryon (∆∆), two-baryon plus one-meson (N N π), and two-baryon plus two-meson (N N ππ) channels near by, one could believe the threshold (or cusp) effect should be much smaller in the case of this resonance than that in the cases of the exotic XYZ resonances[7][8][9][10]. And due to its narrow width, one may think of this dibaryon as a state with, at least, a hexaquark dominated structure. Up to now, several theoretical proposals for its internal structure have been investigated. Among them, two proposed structures have attracted much attention. The first one comes from the study in QDF. The calculation showed that d * has a compact structure and is a candidate of an exotic hexaquark-dominated resonant state[11][12][13][14][15][16][17]. (A similar assumption regards the d * state being a deeply bound state of two ∆s[18,19], however its width is larger than the measured value.) The other one considers it, in HDF, as a molecular-like hadronic state, which originates from an assumption of a three-body resonance ∆N π or a molecular-like state D 12 π[20][21][22]25]. Although the mass and the partial widths of the double pionic decays of such a hypothetic dibaryon resonance can be reasonably reproduced by both proposed structures, the interpretations of it in two proposals are entirely different. Of course, there are many other studies for understanding the structure of d * and pp → π + d reaction, for instance, the triple di-quark model [23] and the triangle singularity mechanism [24]. Whether they can systematically explain all existing experimental data still needs to be further tested. Therefore, it is necessary to look for other physical observables in some sophisticated kinematics regions or some processes other than p-p (or p-d) collision, which might explicitly provide remarkably different results for different structure models, especially the two d * structure models highlighted earlier. Actually, such theoretical analyses have been carried out on the electromagnetic form factors of d * [26, 27] and on the possible evidence in the γ + d processes[28].Up to now, the dibaryon d * has been observed by WASA@COSY Collaborations in the process of pn → dππ and the fusion process of pd → 3 He+ππ. It seems that d * was also observed in another process, say γ+d → dππ at ELPH[29][30][31]. It should be stressed that the forthcoming experiments atPanda (Pbar ANnihilation at DArmstadt) are expected to provide a confirmation of this dibaryon state if it does exist. This is because that atPanda, the antiproton beam collides with the proton target and the momentum of p could be in the range from 1 GeV/c to 15 GeV/c. It corresponds to a range of the total center-of-mass (CM) energy √ s of the proton-antiproton system being from ∼ 2.25 GeV to ∼ 5.5 GeV [32-34], which covers 2M d * ∼ 4.76 GeV . Therefore, the future experiments based on the pp annihilation reaction can provide another way to produce the dibaryon and anti-dibaryon pairs, say d * d * , and can further give the information of this d * resonance.In this work, a phenomenological effective Lagrangian approach (PELA) is employed to study the production of a spin-3 particle d * . It should be mentioned that this approach has been successfully applied to many weakly bound state problems[9]in the exotic meson sectors of X(3872), Z b (10610), and', 'arxivid': '2205.11712', 'author': ['Yubing Dong \nInstitute of High Energy Physics\nChinese Academy of Sciences\n100049BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n101408BeijingChina\n', 'Pengnian Shen \nInstitute of High Energy Physics\nChinese Academy of Sciences\n100049BeijingChina\n'], 'authoraffiliation': ['Institute of High Energy Physics\nChinese Academy of Sciences\n100049BeijingChina', 'School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n101408BeijingChina', 'Institute of High Energy Physics\nChinese Academy of Sciences\n100049BeijingChina'], 'corpusid': 242045340, 'doi': '10.1088/1674-1137/ac3567', 'github_urls': [], 'n_tokens_mistral': 17777, 'n_tokens_neox': 14606, 'n_words': 8260, 'pdfsha': 'e70c487e3dc1bcce7bb8c463efde88ddfcc5df5d', 'pdfurls': ['https://arxiv.org/pdf/2205.11712v1.pdf'], 'title': ['Possible dibaryon production atPanda with a Lagrangian approach *', 'Possible dibaryon production atPanda with a Lagrangian approach *'], 'venue': ['Chinese Physics C']}
arxiv
The nanoscale structure of the Pt-water double layer under bias revealed 28 May 2019 Rémi Khatib Ashwinee Kumar Stefano Sanvito Marialore Sulpizi Clotilde S Cucinotta [email protected] of Physics ‡School of Physics Advanced Materials and Bioengineering Research Centre (AMBER) and Centre for Research on Adaptive Nanostructures and Nanodevices (CRANN) Institute Johannes Gutenberg University Staudingerweg 755128MainzGermany ¶Department of Chemistry Trinity College Dublin 2Ireland Imperial College London W12 0BZUK and TYC The nanoscale structure of the Pt-water double layer under bias revealed 28 May 2019 The nanoscopic mass and charge distribution within the double layer at electrified interfaces plays a key rôle in electrochemical phenomena of huge technological relevance for energy production and conversion. However, in spite of its importance, the nanoscopic structure of the double layer and its response to an applied potential is still almost entirely unknown, even for Pt-water, the most fundamental electrochemical interface. Using a general ab initio methodology which advances previous models towards a dynamic and more realistic description of an electrode/electrolyte interface, we simulate for the first time the nanoscopic structure of the Pt-water double layer and its response to an applied potential, in realistic solution conditions. We reveal that the 1 arXiv:1905.11850v1 [physics.chem-ph] 28 May 2019 nanoscopic metal/surface structure and charging are not captured by traditional capacitor models, as the electrode polarization is associated with a charge oscillation within the double layer and a densification of the water layer in contact with the electrode, both of which strongly depend on the applied potential. Furthermore, we demonstrate that the interface dipole is not determined by the reorientation of the first water layer in contact with the electrode, but by its charging state in combination with its number density, while water reorientation becomes relevant only in the second water layer. Our findings will be essential to develop highly realistic models for the catalytic processes at the Pt-water interface.The last decades have witnessed an extensive research effort aimed at expanding energy production, transformation and storage, through the design and optimisation of a variety of electrochemical (EC) devices. These include fuel cells, solar cells, super capacitors and batteries. These devices could potentially enable a cleaner, cheaper and safer energy technology, but their performance is still not adequate for commercial applications and their development is slow. The fundamental reason behind this is that the optimization of these devices still proceeds by trial and error, due to a lack of atomistic understanding of the basic processes across the electrified interfaces (EI)s which underlie the functioning of such devices. The formation of a double layer (DL) is a fundamental process occurring at EIs. Its equilibrium nano-structure and its response to an applied potential strongly affect the activation barriers for the mass and charge transfer at EIs which converts chemical into electrical energy and vice versa.Modern experimental techniques, such as scanning tunnel microscopy (STM), atomic force microscopy (AFM) and near field spectroscopy (NFS), can achieve atomic resolution.However, in most experimental situations it is only possible to observe the macroscopic manifestation of EC conversion processes. Therefore, in spite of the vast amount of empirical data on EC systems accumulated since Faraday's ground-breaking work, 1 there is still a wide gap between experimental observation of EC processes and their microscopic interpretation. On the theoretical side, computer simulations can achieve atomistic and electronic resolution and are therefore essential to developing better devices. However, despite a long history of empirical [2][3][4] and ab initio 5-8 theoretical surface science [9][10][11][12] investigations, 13,14 resulting in deep insight into the structure and reactivity of interfaces, 15 a comprehensive mechanistic description of atomic and electronic scale processes at electrode surfaces under applied potential is still lacking, even for the most fundamental interfaces, such as the Pt-water interface. In order to describe the EC processes at EIs it is necessary to model the effect of the application of a potential to the cell, since the electrode potential, U, is an essential parameter in electrochemistry. A milestone in the modelling of the effect of the EC potential is represented by the work of Nørskov, who in 2004, using the fact that for a standard (pH=0, p=1 bar, T=298 K) hydrogen electrode (SHE), U=0 is defined by the equilibrium condition H 2 (g) −− −− H + (aq) + e − ,(1) demonstrated 16 that an implicit incorporation of the applied potential to a cell can be modeled. This incorporation is realised through the concept of the computational hydrogen electrode, where a computational reference to the SHE is established by imposing condition (1), without explicitly calculating any solvation energies. Nørskov's method has been successful in predicting trends in various EC reactions. 16,17 However, in his method the electrode potential only affects the Fermi level of the electrons, and the EC environment is only considered as the adsorbates' reservoir. The explicit representation of the atomistic structure of the EI, and its effect on adsorption energetics, as well as the effect of the applied potential on the interfacial geometry, charge redistribution and electronic structure, are neglected. These are all crucial components of the EC phenomenon, whose modelling is essential for a realistic description of electrode activity. 18 A more realistic method is therefore needed. More recent simulation methods explicitly addressing the description of the EI, typically use supercell geometries and periodic boundary conditions (PBC) in three dimensions to speed up the calculations and minimise the effect of finite sample sizes. Since PBC make it difficult to apply an electrostatic potential to the cell, this effect is modelled by imposing charges on the electrodes; cell neutrality is achieved by adding a distribution of fixed counter-charges to the cell, which can take the form of a homogeneous background filling the simulation cell, 19 of charged planes, 20 a doped semiconductor electrode 8 or hydronium ions placed at a fixed distance from the electrode surface. 16 Most of these methods reproduce the localised electric field and the potential energy drop within a nanoscopic distance from the metal surface. 19 However, none of them has provided a comprehensive atomistic description of interface structure and charge polarisation within the DL under applied potential. This is not only due to the small size of the samples used; it is also due to the lack of an extended dynamic description of the DL nano-structure based on first principles that can describe the liquid nature of the electrolyte and the emergence of huge interfacial electric fields. A more realistic modelling of the EC environment, in combination with well controlled experiments, is crucial to develop more reliable models for catalytic processes at EIs. 18,21 An alternative computational SHE method based on ab initio molecular dynamics (AIMD), but relying on an approach that might be seen as intermediate between simulation and theory, was developed by Sprik and co-workers. 7,[22][23][24] This method makes it possible to refer the computed potential to the SHE by using the solvation free energy of the proton in water. The method is sophisticated and powerful, but one downside is its high computational cost. Furthermore, the method does not provide a description of the over-potential associated to the transfer of species to, from, or across the interface under bias, and does not include an explicit description of the electrolytic solution or address the problem of controlling the charging of the electrode. For these reasons we abandon part of the theoretical sophistications of the previous methods and adopt a simpler, intuitive description of the electrode/electrolyte interface. We here use AIMD simulations to carry out a virtual experiment and directly reproduce the nanostructure of a Pt-water EI under applied potential, in realistic solution conditions ( Fig. 1). We evaluate the capacitance and the potential of zero charge of this interface as well as the electrode potentials (wrt. the SHE) associated to different charging states of the Pt electrode. We reproduce the Sum Frequency Generation (SFG) spectrum for this interface. To achieve this, we develop a highly realistic, general ab initio methodology, the ion unbalance methodology, which enables the direct simulation of a metal/water EI under bias and accounts explicitly for charge polarisation effects at both sides of the EI, the full dynamics of solvent rearrangement and the electronic structures details. Results and Discussion The ion unbalance methodology In ab initio simulations, the density functional theory (DFT) approach determines selfconsistently the charge of each species by filling the Kohn-Sham electron states according to their relative energy with respect to the system's Fermi energy, which is in turn determined by the metal. If the metal electrode is large enough to act as a suitable charge reservoir, the charge of each ion will be determined by the transfer of electrons to/from the electrode, such that overall charge neutrality will be maintained. On the left, the initial condition for the occupation of the energy levels centred around Na and Cl atomic species immediately after they are immersed in solution and before the charge transfer from the electrode to the electrolyte occurs. The highest occupied state for Na in solution, here HOMO(Na) (by analogy with the nomenclature for highest occupied molecular orbitals) is above E F , thus it will be empty and Na species is expected to be fully ionised. Correspondingly, the lowest unoccupied state for Cl in solution, here SOMO(Cl) (the degenerate semi occupied molecular orbital) is below E F , thus it will be filled and Cl species is expected to be negatively charged.Filled and empty water molecular states lie well below and above E F , respectively. Note that a frozen picture of the single electron energy levels in our system is adopted here. In the ion unbalance methodology, the electrostatic potential drop for each electrode charging state (i.e. each unbalance in the population of ions) is obtained a posteriori, from the corresponding equilibrium charge density distribution. Different potential drops can be associated to electrodes at different potentials, measurable e.g. with respect to the SHE. The bulk electrolyte energy level, which is located in the middle of each electrolyte region, represents a common reference for the systems in different electrode states, and will be used to align the respective electronic structures, eliminating the need for artificially introducing any vacuum slabs in the middle of the cell. 19 Note that to exploit the ion unbalance methodology proposed, we need the aqueous electrolyte to approach bulk conditions in its central region, therefore system's size and the simulation times are highly relevant in our methodology. Our methodology uses PBC approach. An advantage of this approach is that it will produce a single type of interface (the half-cell ), even in the case of the charged electrodes, since the screening between the two electrode surfaces in the cell will decouple them and the excess of ions will be equally partitioned between the two electrode surfaces. This makes it possible both to disentangle the rôle of anode and cathode, and to improve the statistical sampling from the AIMD simulations, by averaging over the two interfaces present in the simulation cell. Overall, with our methodology we provide a realistic framework which will enable the ab initio estimation of the internal energy, the entropy and the free energy along the reaction paths for chemical transformation at EIs, simulated in a realistic charged environment. This estimate can also be used to develop and tune semi-empirical models for the simulation of the steady state of large systems over time scales longer than those currently achievable with fully ab initio simulations. The model system In particular, in our model system (see Fig. 1) a Pt slab in contact with a NaCl aqueous solution is realistically represented by about 1500 atoms and 5250 electrons. The entire model system is neutral and there is no charge background. We simulate the ab initio dynamics of three interfaces of nearly equivalent solvent composition (≈ 2 M), differing only slightly in the relative number of Na and Cl ions included in the solution. We name these interfaces after their solvent compositions, thus, 10Na:10Cl, 12Na:10Cl and 10Na:12Cl. The system size is adequate for our purposes for a number of reasons. Firstly, at the electrolyte concentration considered (≈2 M of NaCl in water) the separation between the two electrode surfaces (d=28 Å) corresponds to ≈8 Debye screening lengths. This reduced Debye length ensures that the electrolyte bulk condition is reached in the centre of the cell, i.e. at a relatively small distance from the surface of the electrode. Secondly, the large cell cross section (286 Å 2 ) reduces the effects of the 2D periodicity along the electrode surface, justifying the minimal k-point sampling (Γ-point only) of the system Brillouin zone. Thirdly, the Pt slab alone contains nearly 3000 electrons. Thus it represents a sizeable charge reservoir able to correctly position the system Fermi energy and to stabilise its value even in the case of transfer of charge from the electrolyte to the electrodes. Finally, the overall size of the model system allows us to accumulate sufficient statistics even during the relatively short timescales accessible to ab initio simulations. We have simulated each of the charged systems for the unprecedented time of ≈50 ps after equilibration. Overall, more than 200 ps of Born Oppenheimer AIMD have been simulated on our systems. Note that the AIMD simulation of this huge, yet relatively simple model system constitutes a substantial computational challenge, well beyond current standard, which was only managed due to the efficient implementation of DFT based AIMD in CP2K code. 27 In the following, we demonstrate how our methodology and model system make possible to overcome the traditional picture for the charge and mass distribution at the electrodeelectrolyte interface and obtain a direct observation of the strong dependence of DL charge, water structure, density and orientation within the DL on an applied potential. The new picture for the Pt-water DL under applied potential In the conventional Gouy-Chapman-Stern picture of the DL, the electrode charge is strictly localised on the metal electrode, and is counterbalanced by the charge of the counter ions adsorbed on the metal surface and those present in the diffuse layer, the region with an ion concentration gradient. The first feature of the Pt-water DL revealed by our methodology is that the surface charging at the metal-water solution interface cannot be simply described Within this picture the electrode charge is not exclusively localised on the electrode moiety of the interface, but includes the electronic charge on the electrode and in the water molecules in its proximity, whose number changes as a function of the applied potential. This picture is consistent with and provides the computational baseline for the general definition of the electrode total charge concept, which was developed by Frumkin 28 Secondly, our Bader analysis shows that the charge on the electrode moiety of the DL is mostly localised on the external electrode's surface layers and is zero in the centre of every metal slab. We find that the charge on the surface layer of the Pt slab strongly depends on the number of ions in solution, whilst the charge in the subsurface layer of the slab is the same for all interfaces under consideration (see "Surface" and "Subsurface" in Tab. 1). Note Table 1: Bader excess (nominal-calculated) valence charges in units of |e|. In column 'System', the name of the interface under consideration, defined in terms of the number of Na + and Clions in solution and the type of metal electrode. In columns 2-6 the total, subsurface and external surface charge, and the charge on first and second layer of water in contact with the electrode. These values are averaged along the trajectory and between the two interfaces present in our cell. Columns "Cl + H" and "Na" report the charge localised around Cl and Na ions (see SI). More specifically, "Cl + H" reports the sum of the electronic charge around Cl plus some charge artificially associated to the H atoms surrounding Cl (due to the crude Bader definition of atomic volume boundaries, part of the charge actually around Cl is artificially associated by the Bader scheme to the H atoms pointing towards the Cl anion). The last two columns "Solvation Shells" and "DL", report the total charge in the first solvation shells of both Na and Cl ions, and the total charge on the double layer, which includes the charge on electrode, first and second water layer. A full picture of Bader charge distribution is reported in the SI. Therefore, the overall charge on the water layer in contact with the differently charged electrodes is determined by the different number of equally charged molecules composing this layer (see Tab. 1 and next paragraph). This means that the electrode charge is screened by changing the water density at the interface. Overall, we find that the negative charge on the Pt electrode and within the DL is larger than that expected from traditional models 29 (see SI). In particular, the Bader charge on the electrode side of the DL amounts to -1.1e, -0.6e and -0.03e and the overall charge on the DL amounts to 0.65e, -0.28e and 1.5e, for the 12Na:10Cl, 10Na:10Cl and 10Na:12Cl systems, respectively (see Tab. 1). In addition, the absolute charge around every ion is on average 0.8 |e|, rather than exactly 1 |e|, as expected for ions in bulk solution. Notably, a negatively charged electrode and a charged water layer is also observed in the presence of nominally neutral electrolyte, i.e. when a balanced population of ions A discussion about the suitability of the PBE functional to describe the metallic moiety of the Pt-water interface and the charge transfer to and from the electrolyte 30,31 can be found in SI. In the following paragraphs we will show that our methodology and model system also make possible to go beyond current models for the atomic structure of the Pt-water DL and reveal that mass density, structure and orientation of the water layers in contact with the electrode strongly depend on the applied potential. Interfacial water structure and orientation Current models describing the metal/water interface 32-34 atomic structure have every second water molecule oriented parallel to the surface and the other pointing one H atom either toward or away from the surface. In these models it is assumed that the electrode polarization is not associated to any mass redistribution in the water layer in contact with the metal electrode. 16,29,35 Here we reveal that exactly the opposite is the case. Our most remarkable finding is the strong dependence on the applied potential of the mass density of first and second water layers in contact with the electrode. In our calculations, first and second water layers in contact with the electrode are unequivocally individuated by two well-defined peaks in the electrolyte mass density profile along the direction perpendicular to the electrode surface (see Fig. 7 in SI). Fig.3 (c) shows that the intensities of the first O and H density peaks increase as the metal electrode becomes more positively charged, namely as we move from system 12Na:10Cl, to 10Na:10Cl, to 10Na:12Cl. The analysis of these density peaks also show that all the water molecules in the first layer have, on average, almost the same orientation in every system studied, and lie on a plane nearly parallel to the metal, with the O atom nearer to the metal, and the H atoms slightly out of plane, pointing towards the bulk water. Thus, as the negative charge on the electrode decreases, a higher number of similarly oriented and positively charged water molecules is present in the first water layer in contact with it (see above and Tab. ??). The mass density of the second water layer in our systems (see Fig. 3 (d)) is less affected by the electrode charging state. However, here water orientation strongly depends on the electrode charging state. The mass density profile for this water layer is characterized by broader density peaks and shows that one H atom per molecule is always located almost in the same plane, defined by the O atoms, and the other always points towards the Pt surface. As the negative charge on the electrode increases, water molecules progressively reorient, pointing a higher number of H atoms towards the metal electrode. Note that our electrodes are all negative. To complement this picture, our analysis shows that the O atoms in the first layer form intralayer chemical bonds with the H atoms and interlayer H bonding, whose density depends on the electrode state, as highlighted by a constant coordination number of only n=2 under 1.5 Å, and n=2.2-2.8 at larger distances, up to 3 Å (See Fig.3(b)). Finally, we studied the time evolution of the position of the water molecules in contact with the electrode at different potentials. The positions of the O atoms belonging to first and second water layers are reported as a function of time in Fig. 4(a). This figure clearly shows that the water molecules belonging to the first layer have their O atoms on the atop sites of the Pt(111) surface (blue traces in Fig. 4) and do not diffuse within the time scale of our simulations (≈ 50 ps). In contrast, the water molecules belonging to the second water layer are more mobile, as shown by the red traces in Fig. 4. We also observed that interfacial water is arranged in hexagonal and pentagonal water rings, which are stable on Pt surface over the simulation time and have a different densities and distributions, depending on the electrode charging. Such water rings are formed by molecules belonging to the first and second water layer and are present even if the symmetry of the simulation supercell we used is not compatible in all directions with the formation of a closely packed hexagonal (or pentagonal) water arrangement on the surface. The observation of hexagonal and pentagonal water rings on Pt is in agreement with other works 6,9,36,37 and has been detected experimentally. 33,36,38 We show here that the structure of the water layers near the Pt electrode (see Fig. 4(a)) is potential dependent and results from the balance between the mass density of the first charged water layer and the orientation of the second one. The results presented above establish a new model for the nanostructure of the DL at metal/water interfaces, which goes beyond both the standard Gouy-Chapman-Stern capacitor picture and current computational models. We have revealed a complex dependence of interfacial charge screening and polarization as well as atomistic structure and dynamics on the applied potential. We expect that our model will represent the baseline for future more realistic evaluations of kinetic overpotentials associated to electrocatalytic transformation at the Pt-water interface of huge technological relevance. Our model for the Pt-water DL is new and is different from existing models; it is nevertheless consistent with the experimental observation of key EC quantities such as the interface capacitance, the electrode potential and the potential of zero charge. In the following, before illustrating how we computationally evaluated such important EC quantities, we will show how our methodology can be used to predict SFG spectra, which cannot be easily extracted experimentally. Computational Sum Frequency Generation Spectra The water orientation at the interface could in principle be insightfully probed by interface sensitive spectroscopy, such as SFG. Despite a couple of pioneering attempts to investigate the Pt-water interface with SFG, 39,40 the experimental investigation is made challenging by the the presence of a strong non-resonant signal from the metal, which makes it impossible to directly access the water structure and orientation. 39 However, calculating the resonant SFG spectra at the Pt-water interface is computationally possible, and is here obtained evaluating the surface sensitive Vibrational Density of States (VDOS) 40,41 (see Methods section for details). Using this approach, we observe in the first adsorbed layer (Fig. 3(e)) a single positive band located at around 2800-3000 cm −1 , whose intensity increases as the positive charge on the electrode increases. The more intense the signal, the higher the density of the water molecules pointing towards the water bulk. The surface sensitive VDOS for the second adsorbed water layer, shown in Fig. 3(f), has a completely different behaviour; for all the interfaces studied, we find that a positive and a negative band are present. We observe that as the negative charge on the electrode decreases, the intensity of the positive band at lower frequencies increases and the negative bend is reduced. These trends indicate a higher density of dipoles reoriented as in the first water layer. These VDOS data are consistent with the results presented above on mass density distribution and molecular reorientation under bias (illustrated in Fig. 3(c-d) ) and provide a useful tool to experimentalists to distinguish and interpret the geometric structures of water at Pt interfaces under bias. The positive band at 2800-3000 is also in agreement with some very recent calculations by Cheng of simple VDOS, which showed similar peaks for the water molecules directly adsorbed on the Pt(111) surface. 42 Cheng's calculations were performed at the PZC and the significant redshift observed, as compared to bulk water, was attributed to the combined effects of charge transfer and strong H-bonds. The Interface Capacitance We will now illustrate how with our ion unbalance methodology we directly evaluated the capacitance, the zero point charge and the absolute electrode potential for each interface. In our methodology the capacitance C = ∆Q/∆V of the Pt-water interface is obtained a posteriori, after all the equilibrium AIMD trajectories are produced, from the evaluation of the average interfacial potential drops, ∆V , associated to the average DL charges, ∆Q, for each charging state of the electrode. The DL charges are here evaluated from Bader analysis and include for each system the charge on the electrode and in the charged water layers in its proximity; the associated potential drops can be compared as in our systems the bulk electrolyte condition is reached in the middle of each electrolyte region, and therefore the associated potential energy level can be used as a common reference; this bulk electrolyte condition is clearly marked in all our systems by the locally flat average electrostatic potential energy and density profiles; and by a locally neutral electrolyte solution in the centre of each cell (see Fig. 1 in SI). In our calculations, the interfacial potential drops associated to different charging states of the electrode are clearly distinguishable during the simulation time and depend linearly on the interfacial charge ( Fig. 3 and Fig. 4 of the SI). The capacitance is here calculated as the slope of the linear fit of the potential drop/charge relation (see Fig. 5) and amounts to C = 8.29 µF·cm −2 . This is close to the experimental value 43 for the capacitance of a Pt-water electrolyte solution interface at high frequency and low NaCl concentration, which is expected to be lower than 10 µF·cm −2 . Note that this experimental value does not include the low frequency modes determined by the presence of the diffused layer. 44 From the calculated potential drop trend we also straightforwardly determined the potential of zero charge (PZC) for our Pt-water interface, by linearly fitting our data and finding the potential corresponding to the zero charge point. By aligning this potential value to the Note that to evaluate the capacitance with our methodology, there is no need to place the electrode potential on an absolute scale, as only differences between the potential drops corresponding to different electrode charges are needed. Finally, we evaluated the P ZC computationally, by relating the potential drop to the vacuum potential, and using the linear fit procedure described above (see Fig. 3 in the SI). To evaluate the potential drop with respect to the vacuum potential we inserted a vacuum slab in the middle of cell and measured the difference between the Fermi level and the vacuum potential. 19 The P ZC with respect to the SHE calculated in this way amounts to P ZC = 0.31 V, a value extremely close to the experimental one. 7,45 A discussion about possible different ways to evaluate the potential drop ∆V , that are in principle equivalent, can be found in SI. In Summary, we have presented the first realistic model for the nanoscopic structure of the Pt-water DL under applied potential. This was obtained by developing an ab initio methodology, the ion unbalance methodology, which advances previous models towards a more realistic and dynamic description of EI, and uses simulations to carry out virtual experiments. Our methodology has enabled the direct observation of the nanoscopic structure of the DL under applied potential and revealed that electrode potarization is associated with a mass redistribution and a charge oscillation within the DL, both of which strongly depend on the applied potential. We have shown that the metal/surface charging cannot be simply described in terms of the Gouy-Chapman-Stern capacitor model and have proposed a more complex picture, where the electronic charge distribution oscillates at the interface and spills over the aqueous electrolyte in contact with the metal. Furthermore we have demonstrated that the interfacial dipole under bias is not merely determined by reorientation of the first water layer in contact with the metal surface, but by its charging state, in combination with its number density, while water reorientation becomes relevant only in the second layer. Finally, our methodology has provided an intuitive approach to evaluate the electrode potential wrt the SHE and key EC quantities such as the potential of zero charge and the capacitance of the EI. bsolute electrode potential and key EC quantities such as the potential of zero charge and the capacitance of the EI. The development of our comprehensive microscopic model for the DL under applied potential is extremely significant for fundamental chemistry, physics and interface science. Perhaps most importantly, it makes possible for the first time to design highly realistic models for catalytic processes under bias at interfaces of huge technological relevance. These processes play a key role e.g. in solar, chemical fuel production and fuel cell technology. Thus, our results pave the way for developing new energy transformation models. Methods DFT-MD MD based on DFT is performed for all systems using the Quickstep module of the CP2K package. 27 The electronic structure is obtained at the PBE level. 25 Assessment of the Ion unbalance methodology The analysis of the Kohn-Sham density of states projected on the metal moiety and the Na and Cl ionic species in our systems (PDOS, represented in Fig.3 in SI), confirms that the position of the ions centred energy levels with respect to the Pt Fermi level, E F , is qualitatively correct, as it reproduces the one schematically represented in Fig. 2. In particular, the lowest unoccupied energy level on Cl is below the Pt Fermi level, while the highest occupied energy level centered on Na is above it. This indicates that electrons localise around the Cl atom, forming a Clanion and leave Na + in the cationic form. Note that we did not observe any water centred states in the gap between the ion centred energy levels; this confirms that the correct energy alignment is qualitatively reproduced by DFT in our systems. We also observe that the PDos on Cl and Na ions in the three systems studied (12Na:10Cl, 10Na:10Cl and 10Na:12Cl) are aligned. Thus, even if the three interfaces have slightly different ionic composition, their ions in solution have a common status of charge and can be considered as three expressions of the same Pt-water interface under different bias conditions. Finally, we find that the structure and the charge of the ion solvation shells in our systems only depend on the nature of the solvated ions, and are not affected by varying the number of ions in solution. This further confirms that creating an unbalance of ions in the solution (adding or subtracting few ions) can be used as a strategy to model different electrode states which does not affect the electrolyte properties. Note that in every system studied the overall charge associated to the first solvation shells around Na and Cl ions is nearly equal in magnitude, but opposite in sign (see below for a full picture of Bader charges in our system). Thus, when an equal population of anions and cations is present in solution their overall charge almost cancels out (see Tab.1). Data Availability Authors can confirm that all relevant data are included in the paper and/ or its supplementary information files. Acknowledgement Author Contributions CSC wrote the manuscript with RK contributing and all authors contributing to a minor extent. RK, AK and CSC analysed the data with contribution from MS. RK made the pictures with contribution from CSC, performed ss-VDOS analysis with contribution from MS, and calculated density profiles with contribution from AK. AK performed Bader analysis with contribution from RK and CSC. All authors discussed the data. CSC performed the AIMD simulations and guided and supervised the project. Supporting Information Available The following files are available free of charge. List of the contents of SI: • Bulk electrolyte condition Figure 1 : 1A snapshot taken from the AIMD trajectory of the Pt-water solution system characterised by an electrode charge of -0.03 |e|, and 10 Na and 12 Cl ions in solution (10Na:12Cl). The horizontal bar on the top of the figure marks the external boundary of the first and second water layer (Z 1 and Z 2 , respectively, as per Tab. 1 in SI). Z = 0 labels the average position of the surface Pt layers in contact with the aqueous electrolyte. Red, white, olive green, cyan and blue spheres represent O, H, Pt, Cl and Na atoms, respectively. ion unbalance methodology enables the direct observation of the effect of an applied potential on the nanostructure of an EI under bias, by simulating and comparing EIs with the electrode in different charging states. Different charging states of the electrode are obtained by introducing different unbalanced populations of neutral atoms of different types (Na and Cl) in the electrolyte solution. This introduction then leads to the formation of cations and anions in solution and a charged electrode. Figure 2 : 2The introduction of different imbalances in the number of cations and anions in the solution will thus result in the transfer of different amounts of charge between the electrolyte and the Pt electrode. For instance, adding an imbalance between atomic species to the solution leading to the formation of more cations than anions will result in a more negatively charged electrode and an excess of cations in solution. These ionic species in solution will, in turn, contribute to the DL screening of the electrode. The validity of the proposed ion unbalance methodology to charge the electrode, relies on the correct DFT description of the single electron energy level alignment, which may change for different approximations of the exchange and correlation functional. Correct energy level alignment should reflect the ordering shown inFig. 2. We find that in our Pt-water interface this is achieved already by the generalised gradient approximation at the level of the Perdew-Burke-Ernzerhof 25 (PBE) potential, 7 despite the fact that other properties, such as the water band gap and the band gap of several semiconductors may be severely underestimated. Single electron energy level alignment and schematic picture of a Pt-electrolyte half-cell. E F (Pt) separates filled and empty electronic states. On the right, the equilibrium condition for the occupation of the energy levels of Na+ and Cl-ions in solution. in terms of the Gouy-Chapman-Stern capacitor model. A more complex picture emerged from our calculations, where (1) electrode charge spills over the water molecules in contact with the metal, therefore the charge on the electrolyte moiety of the DL is not determined uniquely by the counter ions; (2) an oscillating charge distribution is present at the metalwater interface; (3) electrode polarization is associated with a change in the number of molecules in the first water layer in contact with the electrode, and a reorientation of the molecules of the second water layer. (4) the interfacial charge and molecular distribution strongly depends on the applied potential. based on thermodynamic considerations, and without relying on any traditional model for the DL. In this thermodynamic definition, rather then in terms of localisation on the electrode, charge is defined operationally, as the amount of electricity which needs to be supplied to the electrode when its surface is increased by 1 cm 2 , and the concentration of all the solution components remains constant. 28 The charge distribution A quantitative foundation to the new model for the DL nano-structure emerged from our calculations is first of all provided by the analysis of the excess (nominal -calculated) Bader charges distribution across the interface. Firstly, Bader analysis confirms the viability of the ion unbalance methodology, showing that the electrode state can be effectively controlled by the relative unbalance in the population of anions and cations in solution. Indeed, in our calculations a larger excess of positive (negative) ions in solution always corresponds to a more negatively (positively) charged electrode (see Tab. 1 and SI). Pt atoms of the subsurface layers are kept fixed during the siimulation.Thirdly, moving out from the negatively charged metal surface, we encounter a net positive electronic excess charge in the first water layer in contact with the electrode and a negative electronic excess charge in the second water layer, before reaching neutrality again in the bulk electrolyte, in the central part of the cell (see Tab. 1 andFig. 3(a)).The observed oscillating trend in the charge distribution at the DL is qualitatively the same for all interfaces studied; however, the amount of charge on each water layer is strongly dependent on the electrode state. Note that the electronic charge spillover to the first water layer in contact with the electrode occurs such that each molecule always carries the same amount of positive charge (≈ 0.1 |e|), irrespective of the electrode state. Our calculations show that the number of equally charged molecules composing the first water layer progressively increases when the electrode becomes more positive (see Tab. 2 in SI). ( 10Na:10Cl) or no ions (0Na:0Cl) are present in solution (see Tab. 1 and SI). This is in clear contrast with the assumption in standard computational approaches that the electrode remains neutral in the presence of a neutral electrolyte, and should be considered when developing future explicit models for neutral Pt-water interfaces. Figure 3 : 3(a) Overall excess (nominal-calculated) Bader valence charge distribution along the double layer. The black dashed line (joining the Pt subsurface layers and the region in their middle) signals that the in the middle of the metal slab the actual charge is 0 (see Fig. 1 in SI); (b) integrated O-H radial distribution profle, where O belongs to the molecules of the frst water layer in contact with the electrode; (c-d) average atomic density profiles for the O and H atoms belonging to the first (c) and second (d) water layers in contact with the electrode (in units of atoms per Å); The plain and dashed lines stand for the O and H atoms' contributions, respectively; (e-f) layer resolved aurface sensitive vibrational density of states (VDOS) obtained only from the molecules of first (e) and second (f) water layer. The charge on the metal moiety of each DL is reported in brackets, in units of |e|. 12Na:10Cl, 10Na:10Cl, and 10Na:12Cl systems are represented in black, red and blue, respectively. For each system, all the represented quantities are averaged over 50 ps long trajectories at 330K, obtained performing DFT based Born Oppenheimer AIMD. First and second water layers are defined as described in Tab. 1 in SI. Figure 4 : 4(a) Prototypical configuration of first and second water layer for 10Na:12Cl, 10Na:10Cl and 12Na:10Cl systems, from left to right. Red lines highlight hexagonal and pentagonal water motives on the Pt surface. (b) Trajectories of O atoms belonging to first and second water layers in contact with the electrode, marked with blue and red lines, respectively. The charge on the metal electrode is also reported (units of |e|). Wat. nb. indicates the average number of molecules in the first water layer. Figure 5 : 5Double layer charge versus potential drop ∆V , as averaged along the trajectory for 12Na:10Cl (electrode charge -1.1 |e|), 10Na:10Cl (electrode charge -0.6 |e|) and 10Na:12Cl (electrode charge -0.03 |e|) systems. Charge and potential drop are provided in units of |e| and Volts, respectively. Also reported, the system's capacitance (C = 8.29 µF·cm −2 ), evaluated from the slope of the charge/potential linear fit (dashed black line). The continuous vertical red line represents the experimental P ZC. Aligning this potential to the point where the linear fit crosses zero charge, allowed to define ad absolute potential scale for our electrodes, referred to the standard hydrogen electrode potential. The dashed vertical red line represents our evaluation of the P ZC, where ∆V is evaluated relating the interfacial Fermi energy in every system to the vacuum level, and this latter to the SHE. See SI for more detail on the evaluation of ∆V . experimental 7,45 P ZC for a Pt-water interface, it is then possible to define an absolute scale for the electrode potential in our systems.In this way, the absolute electrode potentials with respect to the SHE of the 12Na:10Cl, 10Na:10Cl and 10Na:12Cl systems are calculated at -0.48, 0.27 and 0.64 V, respectively. Tether, Goedecker and Hutter (GTH) pseudopotentials, 46 valence triple-zeta TZV basis set for Pt, TZVP (TZV polarization) basis sets for the other atoms and a cutoff energy of 300 Ry for the plane waves are used. For every atomic configuration and every system studied, the potential energy is computed by minimising the electronic DFT functional, while time evolution is simulated by Born-Oppenheimer MD, using the gradients of the DFT potential energy surface to provide the forces entering Newton's equations of motion. The time-step used to integrate the equations of motion is 0.7 fs within the NVT ensemble. The temperature is controlled by using a Langevin thermostat 26,47,48 set at 340 K. For all systems PBC are used. The in-plane cell parameters are a = 14.568 Å and b = 19.626 Å. In order to keep the density of the aqueous solution consistent with the studied system (≈ 1 kg·L −1 or slightly higher, depending on the salt concentration), the normal axis, c changes in the interval 35.942 Å ≤ c ≤ 39.603 Å. The Pt-water half-cell is realistically modelled with nearly 1000 atoms. In particular, we have 768 Pt atoms, 250 water molecules, a variable number of Na + and Clions, depending on composition, and about 5250 electrons. The Pt(111) electrode surface is modelled by using a 4-layer slab (42 Pt atoms per layer). The coordinates of the central layers of the Pt slab are constrained to the bulk electrode values. Using a four layer slab introduces an error in the evaluation of the Pt work function of less than 0.04 eV. This is acceptable in view of the reduced computational cost. Extensive tests have been performed over the slab size against sampling of the Brillouin zone. This work was enabled by EPSRC (EP/P033555/1), Science Foundation Ireland (SFI) funded centre AMBER (SFI/12/RC/2278) and Irish Research Council (IRC) postgraduate grant GOIPG/2014/1392, DFG Research Grant SU 752/2-1,TRR146. The calculations performed for this work were performed using ARCHER, UK National Supercomputing Service (http://www.archer.ac.uk, via our membership of the UK's HEC Materials Chemistry Consortium, which is funded by EPSRC (EP/R029431)), the Kelvin and Boyle clusters, maintained by the Trinity Centre for High Performance Computing (project id: HPC-16-00932, these clusters were funded through grants from Science Foundation Ireland) and HRLS, Stuttgart, DE. We gratefully acknowledge Prof. Pietro Ballone for helpful discussions about the ion unbalance methodology and Dr. Akinlolu Akande, for assistance with some figures at an early stage of the preparation of this paper. • Evaluation of the electrode/electrolyte potential drop • Ability of PBE functional to describe qualitative energy level alignment, the metallic moiety of the Pt-water interface and the density of states of the Ag-water solution interface • Water structure and orientation • ss-VDOS • Bader charges • Van der Waals interactions Experimental researches in electricity.âĂŤSixth series. Philosophical Transactions of the Royal Society of London 1834. M Faraday, Vi, 124Faraday, M. VI. Experimental researches in electricity.âĂŤSixth series. Philosophical Transactions of the Royal Society of London 1834, 124, 77-122. Structure and dynamics of water at the Pt(111) interface: Molecular dynamics study. K Raghavan, K Foster, K Motakabbir, M Berkowitz, The Journal of Chemical Physics. 94Raghavan, K.; Foster, K.; Motakabbir, K.; Berkowitz, M. Structure and dynamics of water at the Pt(111) interface: Molecular dynamics study. The Journal of Chemical Physics 1991, 94, 2110-2117. Accurate Simulation of Surfaces and Interfaces of Face-Centered Cubic Metals Using 12-6 and 9-6 Lennard-Jones Potentials. H Heinz, R A Vaia, B L Farmer, R R Naik, The Journal of Physical Chemistry C. 112Heinz, H.; Vaia, R. A.; Farmer, B. L.; Naik, R. R. Accurate Simulation of Surfaces and Interfaces of Face-Centered Cubic Metals Using 12-6 and 9-6 Lennard-Jones Potentials. The Journal of Physical Chemistry C 2008, 112, 17281-17290. An atomistic force-field to describe the interaction of proteins with Au(111) surfaces in water. F Iori, R Di Felice, E Molinari, S Corni, Golp, Journal of Computational Chemistry. 30Iori, F.; Di Felice, R.; Molinari, E.; Corni, S. GolP: An atomistic force-field to describe the interaction of proteins with Au(111) surfaces in water. Journal of Computational Chemistry 2009, 30, 1465-1476. Water Confined in Nanotubes and between Graphene Sheets: A First Principle Study. G Cicero, J C Grossman, E Schwegler, F Gygi, G Galli, Journal of the American Chemical Society. 130Cicero, G.; Grossman, J. C.; Schwegler, E.; Gygi, F.; Galli, G. Water Confined in Nanotubes and between Graphene Sheets: A First Principle Study. Journal of the American Chemical Society 2008, 130, 1871-1878. Vibrational Recognition of Hydrogen-Bonded Water Networks on a Metal Surface. S Meng, L F Xu, E G Wang, S Gao, Physical Review Letters. 89176104Meng, S.; Xu, L. F.; Wang, E. G.; Gao, S. Vibrational Recognition of Hydrogen-Bonded Water Networks on a Metal Surface. Physical Review Letters 2002, 89, 176104. Determining Potentials of Zero Charge of Metal Electrodes versus the Standard Hydrogen Electrode from Density-Functional-Theory-Based Molecular Dynamics. J Le, M Iannuzzi, A Cuesta, J Cheng, Physical Review Letters. 11916801Le, J.; Iannuzzi, M.; Cuesta, A.; Cheng, J. Determining Potentials of Zero Charge of Metal Electrodes versus the Standard Hydrogen Electrode from Density-Functional- Theory-Based Molecular Dynamics. Physical Review Letters 2017, 119, 016801. First-Principles Approach to Model Electrochemical Reactions: Understanding the Fundamental Mechanisms behind Mg Corrosion. S Surendralal, M Todorova, M W Finnis, J Neugebauer, Physical Review Letters. 120246801Surendralal, S.; Todorova, M.; Finnis, M. W.; Neugebauer, J. First-Principles Approach to Model Electrochemical Reactions: Understanding the Fundamental Mechanisms be- hind Mg Corrosion. Physical Review Letters 2018, 120, 246801. Properties of metal-water interfaces studied from first principles. S Schnur, A Groß, New Journal of Physics. 11125003Schnur, S.; Groß, A. Properties of metal-water interfaces studied from first principles. New Journal of Physics 2009, 11, 125003. Structure of water layers on hydrogen-covered Pt electrodes. T Roman, A Groß, Catalysis Today. 202Roman, T.; Groß, A. Structure of water layers on hydrogen-covered Pt electrodes. Catalysis Today 2013, 202, 183 -190. Electronic Structure of Liquid Water and a Platinum Surface. I Tamblyn, Tamblyn, I. Electronic Structure of Liquid Water and a Platinum Surface. ArXiv e- prints 2014, Extending the Concept of Defect Chemistry from Semiconductor Physics to Electrochemistry. M Todorova, J Neugebauer, Physical Review Applied. Todorova, M.; ; Neugebauer, J. Extending the Concept of Defect Chemistry from Semi- conductor Physics to Electrochemistry. Physical Review Applied 2018, 1, 014001. Electrolysis of water on (oxidized) metal surfaces. J Rossmeisl, A Logadottir, J Nørskov, Molecular Charge Transfer in Condensed Media -from Physics and Chemistry to Biology and Nanoengineering in honour of Alexander M. Kuznetsov on his 65th birthday. 319Rossmeisl, J.; Logadottir, A.; Nørskov, J. Electrolysis of water on (oxidized) metal surfaces. Chemical Physics 2005, 319, 178 -184, Molecular Charge Transfer in Con- densed Media -from Physics and Chemistry to Biology and Nanoengineering in honour of Alexander M. Kuznetsov on his 65th birthday. Modeling the electrified solidâĂŞliquid interface. J Rossmeisl, E Skãžlason, M E Bjãűrketun, V Tripkovic, J K Nãÿrskov, Chemical Physics Letters. 466Rossmeisl, J.; SkÞlason, E.; BjÃűrketun, M. E.; Tripkovic, V.; NÃÿrskov, J. K. Mod- eling the electrified solidâĂŞliquid interface. Chemical Physics Letters 2008, 466, 68 - 71. . W Schmiclker, E Santos, Interfacial Electrochemistry. SpringerSchmiclker, W.; Santos, E. Interfacial Electrochemistry; Springer: New York City, 2010. Origin of the Overpotential for Oxygen Reduction at a Fuel-Cell Cathode. J K Nørskov, J Rossmeisl, A Logadottir, L Lindqvist, J R Kitchin, T Bligaard, H Jónsson, The Journal of Physical Chemistry B. 108Nørskov, J. K.; Rossmeisl, J.; Logadottir, A.; Lindqvist, L.; Kitchin, J. R.; Bligaard, T.; Jónsson, H. Origin of the Overpotential for Oxygen Reduction at a Fuel-Cell Cathode. The Journal of Physical Chemistry B 2004, 108, 17886-17892. Trends in the Exchange Current for Hydrogen Evolution. J K Nørskov, T Bligaard, A Logadottir, J R Kitchin, J G Chen, S Pandelov, U Stimming, Journal of The Electrochemical Society. 152Nørskov, J. K.; Bligaard, T.; Logadottir, A.; Kitchin, J. R.; Chen, J. G.; Pandelov, S.; Stimming, U. Trends in the Exchange Current for Hydrogen Evolution. Journal of The Electrochemical Society 2005, 152, J23-J26. Water as a Promoter and Catalyst for Dioxygen Electrochemistry in Aqueous and Organic Media. J Staszak-Jirkovskã¡, ACS Catalysis. 5Staszak-Jirkovská, J. et al. Water as a Promoter and Catalyst for Dioxygen Electro- chemistry in Aqueous and Organic Media. ACS Catalysis 2015, 5, 6600-6607. Elucidation of the Electrochemical Activation of Water over Pd by First Principles. J.-S Filhol, M Neurock, Angewandte Chemie International Edition. 45Filhol, J.-S.; Neurock, M. Elucidation of the Electrochemical Activation of Water over Pd by First Principles. Angewandte Chemie International Edition 2006, 45, 402-406. Ab initio simulation of charged slabs at constant chemical potential. A Y Lozovoi, A Alavi, J Kohanoff, R M Lynden-Bell, The Journal of Chemical Physics. 115Lozovoi, A. Y.; Alavi, A.; Kohanoff, J.; Lynden-Bell, R. M. Ab initio simulation of charged slabs at constant chemical potential. The Journal of Chemical Physics 2001, 115, 1661-1669. Enhancing Hydrogen Evolution Activity in Water Splitting by Tailoring Li + -Ni(OH) 2 -Pt Interfaces. R Subbaraman, D Tripkovic, D Strmcnik, K.-C Chang, M Uchimura, A P Paulikas, V Stamenkovic, N M Markovic, Science. 334Subbaraman, R.; Tripkovic, D.; Strmcnik, D.; Chang, K.-C.; Uchimura, M.; Paulikas, A. P.; Stamenkovic, V.; Markovic, N. M. Enhancing Hydrogen Evolution Activity in Water Splitting by Tailoring Li + -Ni(OH) 2 -Pt Interfaces. Science 2011, 334, 1256-1260. Redox potentials and pKa for benzoquinone from density functional theory based molecular dynamics. J Cheng, M Sulpizi, M Sprik, The Journal of Chemical Physics. 131154504Cheng, J.; Sulpizi, M.; Sprik, M. Redox potentials and pKa for benzoquinone from density functional theory based molecular dynamics. The Journal of Chemical Physics 2009, 131, 154504. The oxidation of tyrosine and tryptophan studied by a molecular dynamics normal hydrogen electrode. F Costanzo, M Sulpizi, R G D Valle, M Sprik, The Journal of Chemical Physics. 134Costanzo, F.; Sulpizi, M.; Valle, R. G. D.; Sprik, M. The oxidation of tyrosine and tryptophan studied by a molecular dynamics normal hydrogen electrode. The Journal of Chemical Physics 2011, 134, 244508. Redox Potentials and Acidity Constants from Density Functional Theory Based Molecular Dynamics. J Cheng, X Liu, J Vandevondele, M Sulpizi, M Sprik, Accounts of Chemical Research. 47Cheng, J.; Liu, X.; VandeVondele, J.; Sulpizi, M.; Sprik, M. Redox Potentials and Acid- ity Constants from Density Functional Theory Based Molecular Dynamics. Accounts of Chemical Research 2014, 47, 3522-3529. Generalized Gradient Approximation Made Simple. J P Perdew, K Burke, M Ernzerhof, Physical Review Letters. 77Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Physical Review Letters 1996, 77, 3865-3868. Towards an assessment of the accuracy of density functional theory for first principles simulations of water. II. E Schwegler, J C Grossman, F Gygi, G Galli, Journal of Chemical Physics. 121Schwegler, E.; Grossman, J. C.; Gygi, F.; Galli, G. Towards an assessment of the accuracy of density functional theory for first principles simulations of water. II. Journal of Chemical Physics 2004, 121, 5400-5409. Quickstep: Fast and accurate density functional calculations using a mixed Gaussian and plane waves approach. J Vandevondele, M Krack, F Mohamed, M Parrinello, T Chassaing, J Hutter, Computer Physics Communications. 167VandeVondele, J.; Krack, M.; Mohamed, F.; Parrinello, M.; Chassaing, T.; Hutter, J. Quickstep: Fast and accurate density functional calculations using a mixed Gaussian and plane waves approach. Computer Physics Communications 2005, 167, 103-128. In Comprehensive Treatise of Electrochemistry: The Double Layer. A N Frumkin, O A Petrii, B B Damaskin, Frumkin, A. N.; Petrii, O. A.; Damaskin, B. B. In Comprehensive Treatise of Electro- chemistry: The Double Layer ; . J O Bockris, B E Conway, E Yeager, SpringerUS; Boston, MABockris, J. O., Conway, B. E., Yeager, E., Eds.; Springer US: Boston, MA, 1980; pp 221-289. Electrolysis of water on oxide surfaces. J Rossmeisl, Z.-W Qu, H Zhu, G.-J Kroes, J Nørskov, Journal of Electroanalytical Chemistry. 607Theoretical and Computational ElectrochemistryRossmeisl, J.; Qu, Z.-W.; Zhu, H.; Kroes, G.-J.; Nørskov, J. Electrolysis of water on oxide surfaces. Journal of Electroanalytical Chemistry 2007, 607, 83 -89, Theoretical and Computational Electrochemistry. Modeling Adsorption and Reactions of Organic Molecules at Metal Surfaces. W Liu, A Tkatchenko, M Scheffler, Accounts of Chemical Research. 47Liu, W.; Tkatchenko, A.; Scheffler, M. Modeling Adsorption and Reactions of Organic Molecules at Metal Surfaces. Accounts of Chemical Research 2014, 47, 3369-3377. Localization and Delocalization Errors in Density Functional Theory and Implications for Band-Gap Prediction. P Mori-Sánchez, A J Cohen, W Yang, Physical Review Letters. 100146401Mori-Sánchez, P.; Cohen, A. J.; Yang, W. Localization and Delocalization Errors in Density Functional Theory and Implications for Band-Gap Prediction. Physical Review Letters 2008, 100, 146401. Water adsorption and the wetting of metal surfaces. A Hodgson, S Haq, Surface Science Reports. 64Hodgson, A.; Haq, S. Water adsorption and the wetting of metal surfaces. Surface Science Reports 2009, 64, 381 -451. A onedimensional ice structure built from pentagons. J Carrasco, A Michaelides, M Forster, S Haq, R Raval, A Hodgson, Nature Materials. 8Carrasco, J.; Michaelides, A.; Forster, M.; Haq, S.; Raval, R.; Hodgson, A. A one- dimensional ice structure built from pentagons. Nature Materials 2009, 8, 427-431. A molecular perspective of water at metal interfaces. J Carrasco, A Hodgson, A Michaelides, Nature Materials. 11Carrasco, J.; Hodgson, A.; Michaelides, A. A molecular perspective of water at metal interfaces. Nature Materials 2012, 11, 667-674. Theory of solid/electrolyte interfaces. Ψ k Scientific Highlight Of The Month. Gross2014, Gross2014, Theory of solid/electrolyte interfaces. Ψ k Scientific Highlight Of The Month 2014, 1-32. Thürmer, K. Pentagons and Heptagons in the First Water Layer on Pt(111). S Nie, P J Feibelman, N C Bartelt, Physical Review Letters. 26102Nie, S.; Feibelman, P. J.; Bartelt, N. C.; Thürmer, K. Pentagons and Heptagons in the First Water Layer on Pt(111). Physical Review Letters 2010, 105, 026102. Density functional theory study of the electrochemical interface between a Pt electrode and an aqueous electrolyte using an implicit solvent method. S Sakong, M Naderian, K Mathew, R G Hennig, A Groß, The Journal of Chemical Physics. 142234107Sakong, S.; Naderian, M.; Mathew, K.; Hennig, R. G.; Groß, A. Density functional theory study of the electrochemical interface between a Pt electrode and an aqueous electrolyte using an implicit solvent method. The Journal of Chemical Physics 2015, 142, 234107. Orientational ordering of twodimensional ice on Pt(111). A Glebov, A P Graham, A Menzel, J P Toennies, The Journal of Chemical Physics. 106Glebov, A.; Graham, A. P.; Menzel, A.; Toennies, J. P. Orientational ordering of two- dimensional ice on Pt(111). The Journal of Chemical Physics 1997, 106, 9382-9385. On the Role of Fresnel Factors in Sum-Frequency Generation Spectroscopy of MetalâĂŞWater and Metal-OxideâĂŞWater Interfaces. E H G Backus, N Garcia-Araez, M Bonn, H J Bakker, The Journal of Physical Chemistry C. 116Backus, E. H. G.; Garcia-Araez, N.; Bonn, M.; Bakker, H. J. On the Role of Fres- nel Factors in Sum-Frequency Generation Spectroscopy of MetalâĂŞWater and Metal- OxideâĂŞWater Interfaces. The Journal of Physical Chemistry C 2012, 116, 23351- 23361. Water orientation and hydrogen-bond structure at the fluorite/water interface. R Khatib, E H G Backus, M Bonn, M.-J Perez-Haro, M.-P Gaigeot, M Sulpizi, Scientific Reports. 624287Khatib, R.; Backus, E. H. G.; Bonn, M.; Perez-Haro, M.-J.; Gaigeot, M.-P.; Sulpizi, M. Water orientation and hydrogen-bond structure at the fluorite/water interface. Scien- tific Reports 2016, 6, 24287. Sum Frequency Generation Spectra from VelocityâĂŞVelocity Correlation Functions. R Khatib, M Sulpizi, The Journal of Physical Chemistry Letters. 8Khatib, R.; Sulpizi, M. Sum Frequency Generation Spectra from VelocityâĂŞVelocity Correlation Functions. The Journal of Physical Chemistry Letters 2017, 8, 1310-1314. Theoretical insight into the vibrational spectra of metalâĂŞwater interfaces from density functional theory based molecular dynamics. J Le, Q Fan, L Perez-Martinez, A Cuesta, J Cheng, Phys. Chem. Chem. Phys. 20Le, J.; Fan, Q.; Perez-Martinez, L.; Cuesta, A.; Cheng, J. Theoretical insight into the vibrational spectra of metalâĂŞwater interfaces from density functional theory based molecular dynamics. Phys. Chem. Chem. Phys. 2018, 20, 11554-11558. Electrode polarization impedance in weak NaCl aqueous solutions. P Mirtaheri, S Grimnes, G Martinsen, IEEE Transactions on Biomedical Engineering. 52Mirtaheri, P.; Grimnes, S.; Martinsen, G. Electrode polarization impedance in weak NaCl aqueous solutions. IEEE Transactions on Biomedical Engineering 2005, 52, 2093- 2099. Double layer capacitance of Pt(111) single crystal electrodes. T Pajkossy, D Kolb, Electrochimica Acta. 46Pajkossy, T.; Kolb, D. Double layer capacitance of Pt(111) single crystal electrodes. Electrochimica Acta 2001, 46, 3063 -3071. Measurement of the surface charge density of CO-saturated Pt(111) electrodes as a function of potential: the potential of zero charge of Pt(111). A Cuesta, Surface Science. 572Cuesta, A. Measurement of the surface charge density of CO-saturated Pt(111) elec- trodes as a function of potential: the potential of zero charge of Pt(111). Surface Science 2004, 572, 11-22. Separable dual-space Gaussian pseudopotentials. S Goedecker, M Teter, J Hutter, Physical Review B. 54Goedecker, S.; Teter, M.; Hutter, J. Separable dual-space Gaussian pseudopotentials. Physical Review B 1996, 54, 1703-1710. Free energy of liquid water on the basis of quasichemical theory and ab initio molecular dynamics. D Asthagiri, L R Pratt, J D Kress, Physical Review E. 6841505Asthagiri, D.; Pratt, L. R.; Kress, J. D. Free energy of liquid water on the basis of quasichemical theory and ab initio molecular dynamics. Physical Review E 2003, 68, 041505. I.-F W Kuo, C J Mundy, M J Mcgrath, J I Siepmann, J Vandevondele, M Sprik, J Hutter, B Chen, M L Klein, F Mohamed, M Krack, Parrinello, M. Liquid Water from First Principles: Investigation of Different Sampling Approaches. 108Kuo, I.-F. W.; Mundy, C. J.; McGrath, M. J.; Siepmann, J. I.; VandeVondele, J.; Sprik, M.; Hutter, J.; Chen, B.; Klein, M. L.; Mohamed, F.; Krack, M.; ; Parrinello, M. Liquid Water from First Principles: Investigation of Different Sampling Approaches. Journal of Physical Chemistry B 2004, 108, 12990-12998.
{'fraction_non_alphanumeric': 0.039133473095737246, 'fraction_numerical': 0.022171399529890095, 'mean_word_length': 4.758642765685019, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 1, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': "The nanoscopic mass and charge distribution within the double layer at electrified interfaces plays a key rôle in electrochemical phenomena of huge technological relevance for energy production and conversion. However, in spite of its importance, the nanoscopic structure of the double layer and its response to an applied potential is still almost entirely unknown, even for Pt-water, the most fundamental electrochemical interface. Using a general ab initio methodology which advances previous models towards a dynamic and more realistic description of an electrode/electrolyte interface, we simulate for the first time the nanoscopic structure of the Pt-water double layer and its response to an applied potential, in realistic solution conditions. We reveal that the 1 arXiv:1905.11850v1 [physics.chem-ph] 28 May 2019 nanoscopic metal/surface structure and charging are not captured by traditional capacitor models, as the electrode polarization is associated with a charge oscillation within the double layer and a densification of the water layer in contact with the electrode, both of which strongly depend on the applied potential. Furthermore, we demonstrate that the interface dipole is not determined by the reorientation of the first water layer in contact with the electrode, but by its charging state in combination with its number density, while water reorientation becomes relevant only in the second water layer. Our findings will be essential to develop highly realistic models for the catalytic processes at the Pt-water interface.The last decades have witnessed an extensive research effort aimed at expanding energy production, transformation and storage, through the design and optimisation of a variety of electrochemical (EC) devices. These include fuel cells, solar cells, super capacitors and batteries. These devices could potentially enable a cleaner, cheaper and safer energy technology, but their performance is still not adequate for commercial applications and their development is slow. The fundamental reason behind this is that the optimization of these devices still proceeds by trial and error, due to a lack of atomistic understanding of the basic processes across the electrified interfaces (EI)s which underlie the functioning of such devices. The formation of a double layer (DL) is a fundamental process occurring at EIs. Its equilibrium nano-structure and its response to an applied potential strongly affect the activation barriers for the mass and charge transfer at EIs which converts chemical into electrical energy and vice versa.Modern experimental techniques, such as scanning tunnel microscopy (STM), atomic force microscopy (AFM) and near field spectroscopy (NFS), can achieve atomic resolution.However, in most experimental situations it is only possible to observe the macroscopic manifestation of EC conversion processes. Therefore, in spite of the vast amount of empirical data on EC systems accumulated since Faraday's ground-breaking work, 1 there is still a wide gap between experimental observation of EC processes and their microscopic interpretation.", 'arxivid': '1905.11850', 'author': ['Rémi Khatib ', 'Ashwinee Kumar ', 'Stefano Sanvito ', 'Marialore Sulpizi ', 'Clotilde S Cucinotta [email protected] ', '\nof Physics\n‡School of Physics\nAdvanced Materials and Bioengineering Research Centre (AMBER) and Centre for Research on Adaptive Nanostructures and Nanodevices (CRANN) Institute\nJohannes Gutenberg University\nStaudingerweg 755128MainzGermany\n', '\n¶Department of Chemistry\nTrinity College\nDublin 2Ireland\n', '\nImperial College London\nW12 0BZUK and TYC\n'], 'authoraffiliation': ['of Physics\n‡School of Physics\nAdvanced Materials and Bioengineering Research Centre (AMBER) and Centre for Research on Adaptive Nanostructures and Nanodevices (CRANN) Institute\nJohannes Gutenberg University\nStaudingerweg 755128MainzGermany', '¶Department of Chemistry\nTrinity College\nDublin 2Ireland', 'Imperial College London\nW12 0BZUK and TYC'], 'corpusid': 167217328, 'doi': '10.1016/j.electacta.2021.138875', 'github_urls': [], 'n_tokens_mistral': 16823, 'n_tokens_neox': 14346, 'n_words': 9719, 'pdfsha': 'ae21f60c84f9d1ef84fe93e31408ca1d6e67ebbb', 'pdfurls': ['https://arxiv.org/pdf/1905.11850v1.pdf'], 'title': ['The nanoscale structure of the Pt-water double layer under bias revealed', 'The nanoscale structure of the Pt-water double layer under bias revealed'], 'venue': []}
arxiv
NONLINEAR NONLOCAL DOUGLAS IDENTITY 10 Jan 2023 Krzysztof Bogdan Tomasz Grzywny Katarzyna Pietruska-Pa ANDArtur Luba Rutkowski NONLINEAR NONLOCAL DOUGLAS IDENTITY 10 Jan 2023 We give Hardy-Stein and Douglas identities for nonlinear nonlocal Sobolev-Bregman integral forms with unimodal Lévy measures. We prove that the corresponding Poisson integral defines an extension operator for the Sobolev-Bregman spaces. As an application, we obtain the boundedness of the Dirichlet-to-Neumann operator on weighted L p spaces. We also show that the Poisson integrals are quasiminimizers of the Sobolev-Bregman forms. Introduction In 1931 J. Douglas [25] established a connection of the energy of the harmonic function u on the unit disc B(0, 1) with the "energy" of its boundary trace g, regarded as a function on [0, 2π): (1.1) B(0,1) |∇u(x)| 2 dx = 1 8π [0,2π)×[0,2π) (g(η) − g(ξ)) 2 sin 2 ((η − ξ)/2) dηdξ. The formula arose in the study of the so-called Plateau problem -the problem of existence of minimal surfaces posed by J.-L. Lagrange. The identity holds true provided that the left-hand side is finite -for details see, e.g., Chen and Fukushima [15, (2.2.60)]. Thus, under the integrability condition, (1.1) is valid for the solutions of the Dirichlet problem, ∆u = 0 in B(0, 1), u = g in ∂B(0, 1). In our paper we propose a variant of (1.1), which we call nonlinear nonlocal Douglas identity. The term "nonlocal" means that the Laplace operator ∆ above is replaced by a nonlocal operator L. Specifically, we adopt the following setting. Let d = 1, 2, . . .. Suppose that the function ν : [0, ∞) → (0, ∞] is nonincreasing and, with a slight abuse of notation, let ν(z) = ν(|z|) for z ∈ R d . In particular, ν is symmetric, i.e., ν(z) = ν(−z), z ∈ R d . Assume further that (1.2) R d ν(z) dz = ∞ and R d |z| 2 ∧ 1 ν(z) dz < ∞. Thus, ν is a strictly positive density function of an infinite isotropic unimodal Lévy measure on R d (in short, ν is unimodal). For u : R d → R and x ∈ R d we let Lu(x) = lim (u(x + z) + u(x − z) − 2u(x))ν(z) dz. Here, ν(x, y) := ν(y − x), and the limit exists, e.g., for u in C ∞ c (R d ), the smooth functions with compact support. Operators of the form (1.3) are called nonlocal, because the value of Lφ(x) also depends on the values of φ outside of a neighborhood of x. Furthermore, the operators satisfy the maximum principle, meaning that if φ(x 0 ) = sup{φ(x) : x ∈ R d }, then Lφ(x 0 ) ≤ 0. It is well known that such operators may be used to describe transportation of mass, charge, etc. in elliptic and parabolic equations; especially to pose boundary-value problems. To our nonlocal setting we bring a judicious way of measuring the smoothness of functions for a given set. Let D ⊂ R d be open. For the sake of gradual introduction we first consider the quadratic form (1.4) E D [u] = 1 2 R d ×R d \D c ×D c (u(x) − u(y)) 2 ν(x, y) dxdy. Such forms appeared in Servadei and Valdinoci [55,56], where the set R d ×R d \D c ×D c was denoted Q, then in Ros-Oton [52, (3.1)] and Dipierro, Ros-Oton and Valdinoci [24, p. 379]. Similar forms were also used in Felsinger, Kassmann and Voigt [29,Definition 2.1 (ii)]. E D is the energy functional of the nonlocal Dirichlet problem (1.5) Lu = 0 in D, u = g on D c , see [55,56] and Bogdan, Grzywny, Pietruska-Pa luba and Rutkowski [8]. It should be noted that E D is better than the vanilla form E R d for solving (1.5), because it allows for more general external conditions g due to the restriction of integration in (1.4) [8, p. 39]. Therefore E D constitutes an important step forward in nonlocal variational problems; we refer the reader to [8] for more details and to [55,56] for applications to nonlinear equations. We note that our results also have consequences for the Dirichlet problem for L on D when E R d is used, see Corollary 4.4 below. Numerous papers study the nonlocal Dirichlet problem by variational methods for nonlocal operators -in the present setting we should note [29], [52], and Rutkowski [53]. It is known for many Lévy and Lévy-type kernels ν and bounded D [29,53], [8,Section 5] that a unique weak solution of (1.5) exists provided that g : D c → R can be extended to a function u ∈ L 2 (D) from the Sobolev class to Q = R d × R d \ D c × D c , cf.(1.6) V D := {u : R d → R | E D [u] < ∞}. It is therefore important to determine conditions on g that allow for such an extension -in other words -to determine the trace space, say, X D , of V D . We note in passing that by [8,Lemma 3.4], the functions from V D are automatically square integrable on D. For the fractional Laplacian ∆ α/2 := −(−∆) α/2 (see Subsection 2.1 for a definition) a solution to this problem was proposed by Dyda and Kassmann [26] by using the Whitney decomposition and the method of reflection. In fact, [26,Theorem 3] concerns general p-increments, i.e., |u(x) − u(y)| p with p ≥ 1. In [8] we resolved the extension and trace problem for p = 2 for a wide class of unimodal Lévy operators by a different approach based on the (quadratic) nonlocal Douglas identity. Namely, [8,Theorem 2.3] asserts that the trace space X D consists of functions g : D c → D for which the following form on D c is finite, H D [g] := 1 2 D c ×D c (g(z) − g(w)) 2 γ D (z, w) dzdw. Here and afterwards we call γ D (w, z) := D D ν(w, x)G D (x, y)ν(y, z) dxdy, w, z ∈ D c , the kernel of interaction via D, or interaction kernel, and G D is the Green function of L for D; see Section 2.1 for details. We note that γ D is the nonlocal normal derivative of the Poisson kernel of L, see (6.1) and (2.8) below, similarly as the kernel in the classical Douglas identity, see Bogdan, Fafu la, and Rutkowski [7,Subsection 2.3]. The nonlocal Douglas identity of [8] can be stated as follows, (1.7) E D [u] = H D [g], where g : D c → R, H D [g] < ∞, and u = P D [g] is the Poisson integral of g, see Section 2.1. Notably, P D [g] is a harmonic function of L, so the identity (1.7) explains the energy of a harmonic function by the energy of its external values. In the language of Chen and Fukushima [15,Chapter 5], the right-hand side of (1.7) is the trace form and γ D (z, w) dzdw is the Feller measure for (E R d , V R d ) on D c , but the extension and trace problem for V D were not investigated in [15]. We also note that Jacob and Schilling [41] studied Douglas identities for nonlocal censored-type Dirichlet forms. Our present goal is to extend the nonlocal Douglas formula (1.7) to a more general nonlinear case. The possibility of such a setting occurred to us owing to the recent Hardy-Stein identities of Bogdan, Dyda and Luks [6,Theorem 2]. To this end we will use the following notion, the French power : x κ = |x| κ sgn(x), x ∈ R, κ ∈ R. More precisely, x κ = x κ if x > 0, x κ = −|x| κ if x < 0, and 0 κ = 0. For example, x 0 = sgn(x) and x 2 = x 2 as functions on R. In what follows we fix 1 < p < ∞, the exponent of the "nonlinearity" alluded to in the title of the paper. Our nonlinear nonlocal Douglas identity is as follows: (1.8) where u = P D [g] and g : D c → R. For a precise statement see Theorem 4.1 and Remark 4.2 below, since the result hinges on suitable additional assumptions on ν, D and g. No analogue of (1.8) seems to exist in the literature for p = 2, even for ∆ α/2 . However, related nonlinear forms u p−1 Lu appear often in the literature concerning Markovian semigroups of operators on L p spaces, see also (2.27) and (7.3) below. This is because for p ∈ (1, ∞) the dual space of L p is L p/(p−1) and for u ∈ L p we have u p−1 ∈ L p/(p−1) , and |u| p = |u p−1 | p/(p−1) = u p−1 u. Therefore in view of the Lumer-Phillips theorem, u p−1 yields a linear functional on L p appropriate for testing dissipativity of generators, see, e.g., Pazy [48,Section 1.4]. In this connection we note that Davies [18,Chapter 2 and 3] gives some fundamental calculations with forms and powers. For the semigroups generated by local operators we refer to Langer and Maz'ya [45] and Sobol and Vogt [57, Theorem 1.1]. Liskevich and Semenov [46] use the L p setting to analyze perturbations of Markovian semigroups. For nonlocal operators we refer to Farkas, Jacob and Schilling [28, (2.4)], and to the monograph of Jacob [40, (4.294)]. 1 2 R d ×R d \D c ×D c (u(x) p−1 − u(y) p−1 )(u(x) − u(y)) ν(x, y) dxdy = 1 2 D c ×D c (g(w) p−1 − g(z) p−1 )(g(w) − g(z))γ D (w, z) dwdz, The following variant of (1.8) is also true, see (2.21), (2.22), and (2.24) below, 1 2 R d ×R d \D c ×D c (|u(y)| p − |u(x)| p − pu(x) p−1 (u(y) − u(x))) ν(x, y) dxdy = 1 2 D c ×D c (|g(z)| p − |g(w)| p − pg(w) p−1 (g(z) − g(w)))γ D (w, z) dwdz. (1.9) The integrands in (1.9) come from the second order Taylor remainder of the convex function x → |x| p , see (2.13), which leads us to the notion of Bregman divergence; see Subsection 2.2, see also Bregman [12] for the original contribution or Sprung [58]. Bregman divergence is important for statistical learning, see Nielsen and Nock [47] or Frigyik, Gupta and Srivastava [31] and the references therein. The Bregman divergence based on the power function |x| p defines the free energy functionals in the studies of Sobolev and Gagliardo-Nirenberg-Sobolev inequalities by Carrillo et al. [14, p. 71] and Bonforte, Dolbeault, Nazaret, and Simonov [11]. It also commonly appears in entropy inequalities, see, e.g., Wang [60]. The present paper indicates further uses of Bregman divergence in PDEs. As we show in Section 6, γ D is the kernel of the Dirichlet-to-Neumann map (6.2) for L. Over the last few years, Dirichletto-Neumann map related to nonlocal operators was intensively studied in the context of inverse problems, see, e.g., [34,33,3,17]. The forms in (1.8) are suitable for studying the Dirichlet-to-Neumann map as an operator in L p . In particular, using our Douglas identity we show that the normalized Dirichlet-to-Neumann operator (6.5) is bounded on a certain weighted L p space. Results in this direction were obtained by Vondraček [59] and Foghem and Kassmann [30] for p = 2, but even in this case our approach gives new insights. As another motivation, we mention that the form on the left-hand side of (1.8) with D = R d is appropriate for studying L p properties of Markovian semigroups. For instance, it was used by Bogdan, Jakubowski, Lenczewska, and Pietruska-Pa luba [9] to characterize the contractivity on L p (R d ) of the semigroups generated by the fractional Laplacian with Hardy-type potentials. The interested reader may find insights into the technique in [9, Lemma 7 and Proof of Theorem 3], or even in (2.27) and (7.3) below. The paper is organized as follows. Section 2 contains definitions and basic facts. Subsection 2.1 introduces notions from the probabilistic potential theory and Subsection 2.2 introduces our nonlinear setting and novel Sobolev-Bregman spaces V p D and X p D defined by the condition of finiteness of the respective sides of (1.8). In (2.19) we collect in one place four (equivalent) approximations for our Bregman divergence, which appear in the literature. In Section 3 we generalize the Hardy-Stein identities of [6] and [8] to our present context. This is instrumental for the proof of the Douglas identity in Section 4. In Corollary 4.3 we conclude that the Poisson integral P D and the restriction to D c are the extension and trace operators between the Sobolev-Bregman spaces. In view toward applications in variational problems, in Section 5 we prove the Douglas formula with the remainder for the energy of sufficiently regular nonharmonic functions. We also show that harmonic functions are quasi-minimizers of the considered nonlinear nonlocal forms, but in general not minimizers. In Section 6 we apply our results for the analysis of the Dirichlet-to-Neumann operator in L p for p ≥ 2. Finally, in Section 7 we give, for p ≥ 2, the following result for Poisson integrals u = P D [g] and the more usual integral forms based on the p-increments of functions: (1.10) R d ×R d \D c ×D c |u(x) − u(y)| p ν(x, y) dxdy ≤ c D c ×D c |g(w) − g(z)| p γ D (w, z) dwdz . It follows that g → P D [g] is an extension operator for nonlocal Sobolev-type spaces W p D , defined by the finiteness of the left-hand side. In the remainder of Section 7 we compare V p D and W p D . Preliminaries All the considered functions, sets and measures are tacitly assumed to be Borel. When we write f ≈ g (resp. f g), we mean that there is a number c > 0, i.e. a constant, such that (1/c)f (x) ≤ g(x) ≤ cf (x) (resp. f (x) ≤ cg(x) ) for all arguments x. Important constants will be capitalized: C 1 , C 2 , . . . , and their values will not change throughout the paper. 2.1. Processes and potential-theoretic notions. Let L and ν be as in the Introduction. Following [8], we additionally assume that: (A1) ν is twice continuously differentiable on (0, ∞) and there is a constant C 1 such that |ν ′ (r)|, |ν ′′ (r)| ≤ C 1 ν(r), r > 1. (A2) There exist constants β ∈ (0, 2) and C 2 > 0 such that ν(λr) ≤ C 2 λ −d−β ν(r), 0 < λ, r ≤ 1, (2.1) ν(r) ≤ C 2 ν(r + 1), r ≥ 1. (2.2) A prominent representative of unimodal Lévy operators L is the fractional Laplacian ∆ α/2 := −(−∆) α/2 . In this case we have ν(x, y) = c d,α |y − x| −d−α , where α ∈ (0, 2), x, y ∈ R d , and c d,α = 2 α Γ((d + α)/2) π d/2 |Γ(−α/2)| . We refer the reader to Bogdan and Byczkowski [5], Di Nezza, Palatucci and Valdinoci [22], Garofalo [32], and Kwaśnicki [44] for more information on ∆ α/2 . Clearly, ν(r) = c d,α r −d−α satisfies both (A1) and (A2). Our results depend in part on martingale properties of harmonic functions, so we introduce the Lévy process (X t , t ≥ 0) on R d whose generator is given by (1.3). Let ψ(ξ) = R d (1 − cos ξ · x)ν(|x|) dx, ξ ∈ R d , the Lévy-Khinchine exponent of (X t ). Since ν(R d ) = ∞, by Sato [54,Theorem 27.7] and Kulczycki and Ryznar [43, Lemma 2.5], the densities p t (x) of (X t ) are continuous on R d \ {0} for t > 0, and satisfy R d e iξ·x p t (x) dx = e −tψ(ξ) , t > 0, ξ ∈ R d . For t > 0 and x, y ∈ R d denote p t (x, y) = p t (y − x), the transition density of (X t ) considered as Markov process on R d . Namely, for starting point x ∈ R d , times 0 ≤ t 1 < t 2 < . . . t n and sets A 1 , A 2 , . . . A n ⊂ R d we let, as usual, P x (X t 1 ∈ A 1 , . . . , X tn ∈ A n ) = A 1 A 2 . . . An p t 1 (x, x 1 )p t 2 −t 1 (x 1 , x 2 ) · · · p tn−t n−1 (x n−1 , x n ) dx 1 dx 2 · · · dx n . This determines P x , the distribution of the process (X t ) starting from x, and E x , the corresponding expectation. In the wording of [54,Section 11], (X t ) is the symmetric Lévy process in R d with (0, ν, 0) as the Lévy triplet. Without losing generality we actually assume that each X t is the canonical projection X t (ω) = ω(t) on the space of càdlàg functions ω : [0, ∞) → R d . We will also use the standard complete right-continuous filtration (F t , t ≥ 0) to analyze (X t ), see Protter [50,Theorem I.31]. In passing we recall that every Lévy process is a Feller process [50]. Let ∅ = D ⊂ R d be an open set. The time of the first exit of X from D is, as usual, τ D = inf{t > 0 : X t / ∈ D}. The Dirichlet heat kernel p D t (x, y) is defined by Hunt's formula, cf. Chung and Zhao [16, Chapter 2.2], p D t (x, y) = p t (x, y) − E x (p t−τ D (X τ D , y); τ D < t), t > 0, x, y ∈ R d . It is the transition density of the process (X t ) killed upon exiting D, i.e., E x [t < τ D ; f (X t )] = R d f (y)p D t (x, y)dy, x ∈ R d , t > 0 , for integrable functions f . The Green function of D is the potential of p D t : G D (x, y) = ∞ 0 p D t (x, y) dt, x, y ∈ R d , and by Fubini-Tonelli we have (2.3) E x τ D = R d G D (x, y) dy, x ∈ R d . The Poisson kernel of D for L is defined by (2.4) P D (x, z) = D G D (x, y)ν(y, z) dy, x ∈ D, z ∈ D c . With (A2) for bounded set D we easily see that for all x, y ∈ D and z ∈ D c with dist(z, D) ≥ ρ > 0, (2.5) ν(x, z) ≈ ν(y, z), where comparability constants depend on ν, D and ρ. Consequently, (2.4) implies (2.6) P D (x, z) ≈ ν(x, z)E x τ D , x ∈ D, dist(z, D) ≥ ρ > 0, with the same proviso on comparability constants. Note that if D is bounded and x ∈ D is fixed, then E x τ D is bounded by a positive constant, see Pruitt [51]. We further note that for w, z ∈ D c the interaction kernel satisfies γ D (w, z) = D D ν(w, x)G D (x, y)ν(y, z) dxdy (2.7) = D ν(w, x)P D (x, z) dx = D ν(z, x)P D (x, w) dx = γ D (z, w). (2.8) Finally, the L-harmonic measure of D for x ∈ R d is, as usual, (2.9) ω x D (dz) = P x [X τ D ∈ dz], the distribution of the random variable X τ D with respect to P x . From the Ikeda-Watanabe formula (see, e.g., Bogdan, Rosiński, Serafin and Wojciechowski [10, Section 4.2]) it follows that P D (x, z) dz is the part of ω x D (dz) which results from the discontinuous exit from D (by a jump). Below, by suitable assumptions on D and ν, we assure that P D is the density of the whole harmonic measure, that is (2.10) D c P D (x, z) dz = 1, x ∈ D. This is true, e.g., if D is bounded, ν satisfies (A2), |∂D| = 0 and D c has the property (VDC). The latter means that there is c > 0 such that for every r > 0 and x ∈ ∂D, (2.11) |D c ∩ B(x, r)| ≥ cr d . Here, as usual, B(x, r) = {y ∈ R d : |y − x| < r}. For the proof of (2.10) under the above conditions, see [8, Corollary A.2]. Observe that for U ⊂ D we have p U ≤ p D and G U ≤ G D . Therefore, P U (x, z) ≤ P D (x, z) for x ∈ U , z ∈ D c , and γ U (z, w) ≤ γ D (z, w) for z, w ∈ D c . These inequalities may be referred to as domain monotonicity. For g : D c → R we define the Poisson extension of g: P D [g](x) = g(x) for x ∈ D c , D c g(z)P D (x, z) dz for x ∈ D,(2.12) and we call D c g(z)P D (x, z) dz the Poisson integral, as long as it is convergent. 2.2. Function F p and related function spaces. We depend on the two humble real functions: x → |x| κ and x → x κ , x ∈ R, κ ∈ R. Clearly, |x| κ is symmetric, x κ is antisymmetric: (−x) κ = −x κ , and their derivatives obey (|x| κ ) ′ = κx κ−1 and (x κ ) ′ = κ|x| κ−1 , x = 0. Recall that p > 1. We let F p (a, b) = |b| p − |a| p − pa p−1 (b − a), a, b ∈ R. (2.13) For instance, if p = 2, then F 2 (a, b) = (b − a) 2 , and if p = 4, then F 4 (a, b) = (b − a) 2 (b 2 + 2ab + 3a 2 ). As the second-order Taylor remainder of the convex function |x| p , F p is nonnegative. In fact, Lemma 6]. In particular, for p ≥ 2 we have (2.14) F p (a, b) ≈ (b − a) 2 (|b| ∨ |a|) p−2 , a, b ∈ R, see [6,(2.15) F p (a, b) ≈ (b − a) 2 (|a| p−2 + |b| p−2 ), a, b ∈ R. Recall that if X is a random variable with the first moment finite and a ∈ R, then (2.16) E(X − a) 2 = E(X − EX) 2 + (EX − a) 2 = Var X + (EX − a) 2 . Here we do not exclude the case EX 2 = ∞, in which case both sides of (2.16) are infinite, hence equal. This variance formula has the following analogue for F p . Lemma 2.1. Let p > 1. Suppose that X is a random variable such that E|X| < ∞. Then, (i) EF p (EX, X) = E|X| p − |EX| p ≥ 0, (ii) EF p (a, X) = F p (a, EX) + EF p (EX, X) ≥ EF p (EX, X), a ∈ R, (iii) EF p (a, X) = EF p (b, X) + F p (a, b) + (pa p−1 − pb p−1 )(b − EX), a, b ∈ R. Proof. The verification is elementary, but we present it to emphasize that the finiteness of the first moment suffices. We have EF p (EX, X) = E |X| p − |EX| p − p(EX) p−1 (X − EX) = E|X| p − |EX| p , where E|X| p = ∞ is permitted, too. The expression in (i) is nonnegative by Jensen's inequality or because F p is nonnegative. For all a ∈ R we have, EF p (a, X) = E |X| p − |a| p − pa p−1 (X − a) = E |X| p − |EX| p − p(EX) p−1 (X − EX) + |EX| p − |a| p − pa p−1 (EX − a) = EF p (EX, X) + F p (a, EX) ≥ EF p (EX, X), as claimed in (ii). Finally, for all a, b ∈ R the right-hand side of (iii) is E|X| p − |b| p − pb p−1 (EX − b) + |b| p − |a| p − pa p−1 (b − a) + (pa p−1 − pb p−1 )(b − EX), which simplifies to the left-hand side of (iii). Needless to say, (ii) is a special case of (iii). We next propose a simple lemma concerning the p-th moments of random variables, which is another generalization of (2.16). Lemma 2.2. For every p ≥ 1 there exist constants 0 < c p ≤ C p such that for every random variable X with E|X| < ∞ and every number a ∈ R, (2.17) c p (E|X − EX| p + |EX − a| p ) ≤ E|X − a| p ≤ C p (E|X − EX| p + |EX − a| p ) . Proof. If E|X| p = ∞, then all the sides of (2.17) are infinite. Otherwise, by convexity, E|X − a| p = E|(X − EX) + (EX − a)| p ≤ 2 p−1 (E|X − EX| p + |EX − a| p ) . For the lower bound we make two observations: |EX − a| p ≤ E|X − a| p (Jensen's inequality), and E|X − EX| p = E|(X − a) − (EX − a)| p ≤ 2 p−1 (E|X − a| p + |EX − a| p ) ≤ 2 p E|X − a| p . Adding the two, we get that |EX − a| p + E|X − EX| p ≤ (1 + 2 p )E|X − a| p . The function F p (a, b) is not symmetric in a, b, but the right-hand side of (2.14) is, so it is natural to consider the symmetrized version of F p , given by the formula: H p (a, b) = 1 2 (F p (a, b) + F p (b, a)) = p 2 (b p−1 − a p−1 )(b − a), a, b ∈ R. (2.18) We can relate H p to a "quadratic" expression as follows. Lemma 2.3. For every p > 1 we have F p (a, b) ≈ H p (a, b) ≈ (b p/2 − a p/2 ) 2 . Proof. The first comparison follows from (2.14): we have F p (a, b) ≈ F p (b, a), hence F p ≈ H p . As for the second statement, if either a or b are equal to 0, then the expressions coincide up to constants depending on p. If a, b = 0, then a = tb with t = 0. Using this representation we see that the second comparison is equivalent to the following: (t p−1 − 1)(t − 1) ≈ (t p/2 − 1) 2 , t ∈ R. The latter holds because both sides are continuous and positive except at t = 1; at infinity both are power functions with the leading term |t| p , and at t = 1 their ratio converges to a positive constant. Summarizing, by (2.14) and Lemma 2.3 for each p ∈ (1, ∞) we have (2.19) F p (a, b) ≈ H p (a, b) ≈ (b − a) 2 (|b| ∨ |a|) p−2 ≈ (b p/2 − a p/2 ) 2 , a, b ∈ R. It is hard to trace down the first occurrence of such comparisons in the literature. The one-sided inequality |b p/2 − a p/2 | 2 ≤ p 2 4(p−1) (b − a)(b p−1 − a p−1 ) for a, b ≥ 0 can be found in connection with logarithmic Sobolev inequalities, e.g., in Davies [18, (2.2.9)] for 2 < p < ∞, and Bakry [2, p. 39] for p > 1. The opposite inequality (b − a)(b p−1 − a p−1 ) ≤ (b p/2 − a p/2 ) 2 with a, b > 0 and p > 1 appears, e.g., in [46,Lemma 2.1]. In fact the following inequalities hold for all p ∈ (1, ∞) and a, b ∈ R: (2.20) 4(p − 1) p 2 (b p/2 − a p/2 ) 2 ≤ (b − a)(b p−1 − a p−1 ) ≤ 2(b p/2 − a p/2 ) 2 . Indeed, if a and b have opposite signs then it is enough to consider b = t ≥ 1 and a = −1, and to compare (t + 1)( t p−1 + 1) = t p + t p−1 + t + 1 with (t p/2 + 1) 2 = t p + 2t p/2 + 1. We have t p/2 = √ t p−1 t ≤ (t p−1 + t)/2 , which verifies the left-hand side inequality in (2.20) with constant 1, which is better than 4(p − 1)/p 2 . We further get the right-hand side inequality in (2.20), and the constant 2 suffices, because t p−1 + t − (t p + 1) = (1 − t)(t p−1 − 1) ≤ 0. Note that the constant 2 is not optimal for individual values of p, e.g., for p = 2, but the constant 1 does not suffice for p ∈ (1, 2) ∪ (2, ∞) because then 1 ∨ (p − 1) > p/2, and so t p−1 + t > 2t p/2 for large t. If a and b have the same sign, then we may assume b = ta, a > 0, t ≥ 1, and consider the quotient H(t) = (t p−1 − 1)(t − 1) (t p/2 − 1) 2 = 1 − t(t (p−2)/2 − 1) 2 (t p/2 − 1) 2 = 1 − h(s) 2 , where s = √ t, h(s) = s(s p−2 − 1)/(s p − 1) . We see that h(s) is strictly positive for p > 2, s > 1 and negative for p ∈ (1, 2). We claim that it decreases in the former case and increases in the latter. The sign of the derivative of h is the same as the sign of the function l(s) = −s 2p−2 + (p − 1)s p − (p − 1)s p−2 + 1. Now, since l(1) = 0, the sign of l on (1, ∞) is in turn equal to the sign of l ′ (s) = (p − 1)s p−3 (−2s p + ps 2 − (p − 2)) , and further equal to the sign of −2p(s p−1 − s). Since the last function is negative on (1, ∞) if p > 2 and positive for p ∈ (1, 2), the claim is proved. Consequently, the function s → h(s) 2 is decreasing on (1, ∞), so we get lim t→1 + H(t) = 4(p − 1) p 2 < H(t) < 1, t > 1, and (2.20) follows. The above also shows that the constant 4(p−1)/p 2 in (2.20) cannot be improved. We would like to note that for p = 2, F p (a + t, b + t) is not comparable with F p (a, b). Indeed, for a, r > 0 one has F p (a, a + r) ≈ r 2 (a ∨ (a + r)) p−2 = r 2 (a + r) p−2 , which is not comparable with F p (0, r) = r 2 for large values of a. Here are one-sided comparisons of F p (a, b) with the more usual p-increments, see, e.g., Zeidler [61, p. 503]. Lemma 2.4. If p ≥ 2 then F p (a, b) |b − a| p , and if 1 < p ≤ 2, then |b − a| p F p (a, b). Proof. If a = b, then the inequalities are trivial, so assume that a = b and consider the quotient F p (a, b) |b − a| p ≈ (|a| ∨ |b|) p−2 |b − a| p−2 . Both parts of the statements now follow from the inequality |b − a| r ≤ 2 r (|a| ∨ |b|) r , r > 0. In analogy to (1.4) for u : R d → R we define (2.21) E (p) D [u] := 1 p R d ×R d \D c ×D c F p (u(x), u(y))ν(x, y) dxdy. By the symmetry of ν and (2.18), E (p) D [u] = 1 p R d ×R d \D c ×D c H p (u(x), u(y))ν(x, y) dxdy = 1 2 R d ×R d \D c ×D c (u(y) p−1 − u(x) p−1 )(u(y) − u(x))ν(x, y) dxdy. (2.22) Of course, E (2) D = E D . For D = R d we have (2.23) E (p) R d [u] = 1 2 R d ×R d (u(y) p−1 − u(x) p−1 )(u(y) − u(x))ν(x, y) dxdy. Clearly, for p = 2 we retrieve the classical Dirichlet form of the operator L. Let g : D c → R. To quantify the increments of g, we use the form: H (p) D [g] = 1 p D c ×D c F p (g(w), g(z))γ D (w, z) dwdz = 1 p D c ×D c H p (g(w), g(z))γ D (w, z) dwdz = 1 2 D c ×D c (g(z) p−1 − g(w) p−1 )(g(z) − g(w))γ D (w, z) dwdz. (2.24) The spaces V D and X D discussed in the Introduction lend themselves to the following generalizations: (2.25) V p D := {u : R d → R | E (p) D [u] < ∞}, and (2.26) X p D := {g : D c → R | H (p) D [g] < ∞}. We call them Sobolev-Bregman spaces, since they involve the Bregman divergence. Our development below indicates that V p D and X p D provide a viable framework for nonlocal nonlinear variational problems. In view of (2.22) for all u : R d → R we have (2.27) E (p) D [u] = E D (u p−1 , u), where E D (v, u) := 1 2 R d ×R d \D c ×D c (v(x) − v(y))(u(x) − u(y))ν(x, y) dxdy, if the integral is well defined, which is the case in (2.27) for v = u p−1 .D [g] ≈ H D [g p/2 ], for all u : R d → R and g : D c → R with the comparability constants depending only on p. Below, however, we focus on genuine equalities. Hardy-Stein identity We first collect properties of harmonic functions that are needed in the proof of the identity (1.8). We mostly follow [8], so our presentation will be brief. We write U ⊂⊂ D if the closure of U is a compact subset of D. u(x) = E x u(X τ U ), x ∈ U. If u(x) = E x u(X τ D ) for all x ∈ D, then we say that u is regular harmonic. In the above we assume that the expectations are absolutely convergent. The strong Markov property of (X t ) implies that if u is regular L-harmonic in D, then it is L-harmonic in D. By [8,Section 4], if u is L-harmonic in D, then u ∈ L 1 loc (R d ) ∩ C 2 (D), Lu(x) can be computed pointwise for x ∈ D as in (1.3), and Lu(x) = 0 for x ∈ D. We also note that the Harnack inequality holds for L-harmonic functions (see Grzywny and Kwaśnicki [39,Theorem 1.9]; the assumptions of that theorem follow from (A2)). We will use the following Dynkin-type lemma, proven in our setting in [8,Lemma 4.11]. Lemma 3.2. Let the set U ⊂⊂ D be open and Lipschitz. If R d |φ(y)|(1 ∧ ν(y)) dy < ∞ and φ ∈ C 2 (U ), then Lφ is bounded on U and for every x ∈ R d , (3.1) E x φ(X τ U ) − φ(x) = U G U (x, y)Lφ(y) dy, where the integrals converge absolutely. The following Hardy-Stein formula extends [6,Lemma 8] and [8,Lemma 4.12], where it was proved, for the fractional Laplacian and p > 1, and for unimodal operators L and p = 2, respectively. E x |u(X τ U )| p = |u(x)| p + U G U (x, y) R d F p (u(y), u(z))ν(y, z) dzdy, x ∈ U. (3.2) Proof. As a guideline, the result follows by taking φ = |u| p in the Dynkin formula (3.1). We combine the methods of [6] and [8]. By [8,Lemma 4.9] if u is harmonic in D, then u ∈ C 2 (D). Thus, in particular, |u| p is bounded in a neighborhood of U . Let x ∈ U . Consider the complementary cases: (i) U c |u(z)| p ν(x, z) dz = ∞, or (ii) U c |u(z)| p ν(x, z) dz < ∞. Since |u| p is bounded in a neighborhood of U , this dichotomy can be reformulated as In case (i), we show that the right-hand side of (3.2) is infinite as well. Assume first that |u| > 0 on a subset of U of positive measure. Pick y ∈ U satisfying |u(y)| > 0, and let A = {z ∈ U c : |u(z)| ≥ (2 + √ 2)|u(y)|}. Now, since x, y ∈ U are fixed and ν is positive, continuous, and satisfies (2.2), we have ν(x, z) ≈ ν(y, z) for z ∈ U c . Therefore, by (i), (i) E x |u(X τ U )| p = ∞, or (ii) E x |u(X τ U )| p < ∞,U c |u(z)| p ν(y, z) dz = ∞ as well. Furthermore, U c \A |u(z)| p ν(y, z) dz ≈ U c \A |u(z)| p ν(x, z) dz ≤ (2 + √ 2) p |u(y)| p ν(x, U c ) < ∞, and consequently we must have A |u(z)| p ν(y, z) dz = ∞. By the definition of A, for z ∈ A we have (3.3) (u(z) − u(y)) 2 ≥ 1 2 u(z) 2 and |u(z)| ≥ |u(y)|. By (2.14) and (3.3) we therefore obtain R d F p (u(y), u(z))ν(y, z) dz ≈ R d (u(z) − u(y)) 2 (|u(y)| ∨ |u(z)|) p−2 ν(y, z) dz ≥ A (u(z) − u(y)) 2 |u(z)| p−2 ν(y, z) dz ≥ 1 2 A |u(z)| p ν(y, z) dz = ∞. This is true for all points y in a set of positive Lebesgue measure, which proves that the right-hand side of (3.2) is infinite. If, on the other hand, u ≡ 0 in U , then F p (u(y), u(z)) = c|u(z)| p for all z ∈ R d , y ∈ U , and by (i) the right-hand side of (3.2) is infinite again. We now consider the case (ii). Thus E x |u(X τ U )| p < ∞ and the integrability condition of Lemma 3.2 is satisfied for φ = |u| p . We will first prove (3.2) for p ≥ 2. Then φ is of class C 2 on D, so we are in a position to use Lemma 3.2 and we get (3.4) E x |u(X τ U )| p = |u(x)| p + U G U (x, y)L|u| p (y) dy, x ∈ U. The integral on the right-hand side is absolutely convergent. Furthermore, since u is L-harmonic, L|u| p (y) = L|u| p (y) − pu(y) p−1 Lu(y) = lim ǫ→0+ |z−y|>ǫ (|u(z)| p − |u(y)| p − pu(y) p−1 (u(z) − u(y)))ν(y, z) dz = R d F p (u(y), u(z))ν(y, z) dz ≥ 0. Inserting this to (3.4) gives the statement. When p ∈ (1, 2), the function R ∋ r → |r| p is not twice differentiable, and the above argument needs to be modified. We work under the assumption (ii), and we follow the proof of [6, Lemma 3]. Consider ε ∈ R and the function R d ∋ x → (x 2 + ε 2 ) p/2 . Let (3.5) F (ε) p (a, b) = (b 2 + ε 2 ) p/2 − (a 2 + ε 2 ) p/2 − pa(a 2 + ε 2 ) (p−2)/2 (b − a) , a, b ∈ R. Since 1 < p < 2, by [6, Lemma 6], (3.6) 0 ≤ F (ε) p (a, b) ≤ 1 p − 1 F p (a, b) , ε, a, b ∈ R , Let ε > 0. We note that (u 2 + ε 2 ) p/2 ∈ C 2 (D). Also, the integrability condition in Lemma 3.2 is satisfied for φ = (u 2 + ε 2 ) p/2 since it is satisfied for φ = |u| p by (ii), and (3.7) (u 2 + ε 2 ) p/2 ≤ (|u| + ε) p ≤ 2 p−1 (|u| p + ε p ), see also (1.2). Furthermore, E x (u(X τ U ) 2 + ε 2 ) p/2 < ∞. As in the first part of the proof, L(u 2 + ε 2 ) p/2 (y) = L(u 2 + ε 2 ) p/2 (y) − pu(y)(u(y) 2 + ε 2 ) (p−2)/2 Lu(y) (3.8) = R d F (ε) p (u(y), u(z))ν(y, z) dz, therefore by Lemma 3.2, (3.9) E x (u(X τ U ) 2 + ε 2 ) p/2 = (u(x) 2 + ε 2 ) p/2 + U G U (x, y) R d F (ε) p (u(y), u(z))ν(y, z) dzdy. From the Dominated Convergence Theorem the left-hand side of (3.9) goes to E ) as ε → 0 + . Furthermore, by Fatou's lemma and (3.9), x |u(X τ U )| p < ∞ as ε → 0 + . Of course, F (ε) p (a, b) → F p (a, bU G U (x, y) R d F p (u(y), u(z))ν(y, z) dzdy ≤ lim inf ε→0 + U G U (x, y) R d F (ε) p (u(y), u(z))ν(y, z) dzdy = E x |u(X τ U )| p − |u(x)| p < ∞. By (3.6) and the Dominated Convergence Theorem, we obtain (3.2) for p ∈ (1, 2). As a consequence, we obtain the the Hardy-Stein identity for D, generalizing and strengthening [6, (16) If additionally u is regular harmonic, then E x |u(X τ U )| p = |u(x)| p + D G D (x, y) R d F p (u(y), u(z))ν(y, z) dzdy.E x |u(X τ D )| p = |u(x)| p + D G D (x, y) R d F p (u(y) , u(z))ν(y, z) dzdy. (3.11) This is delicate. Indeed, by [8,Remark 4.4], the martingale {u(X τ U ), U ⊂⊂ D} is closed by the integrable random variable u(X τ D ). Therefore Lévy's Martingale Convergence Theorem yields that u(X τ U ) converges almost surely, and in L 1 to a random variable Z, as U ↑ D, and we have Z = E x [u(X τ D )|σ( U ⊂⊂D F τ U )] , see, e.g., Dellacherie and Meyer [21, Theorem 31 a,b, p. 26]. We claim that the σ-algebra σ( U ⊂⊂D F τ U ) is equal to F τ D . Indeed, by Proposition 25.20 (i),(ii), and Proposition 25.19 (i),(ii) in Kallenberg [42, p. 501], the filtration of (X t ) is quasi-left continuous. Therefore τ U increases to τ D as U increases to D, and our claim follows from Dellacherie and Meyer [20, Theorem 83, p. 136]. Consequently, Z = u(X τ D ). Now, if sup x∈U ⊂⊂D E x |u(X τ U )| p < ∞, then [21, Theorem 31 c, p. 26] yields (3.11). Else, if the supremum is infinite, then E x |u(X τ D )| p = ∞ by Jensen's inequality, and (3.11) holds, too. We note in passing that the case p = 2 of (3.11) was stated for less general sets D in the first displayed formula following (5.2) in the proof of Theorem 2.3 in [8]. Accordingly, the proof in [8] was easier. The Douglas identity We now present our main theorem. It is a counterpart of (1.7) with square increments of the function replaced by "increments" measured in terms F p or H p . (ii) Furthermore, if u : R d → R satisfies E (p) D [u] < ∞, then H (p) D [u| D c ] < ∞. Here, as usual, u| D c is the restriction of u to D c , but in what follows we will abbreviate: To the best of our knowledge the present Douglas identities are completely new, and our approach is original. The proof of Theorem 4.1 is given below in this section. Recall the space V p D , defined in (2.25), which is a natural domain of E It is well justified to call Ext the extension operator and Tr the trace operator for V p D . We next give the Douglas identity for the Poisson extension on D and the form E (p) R d (with the integration over the whole of R d × R d ). Corollary 4.4. If P D [|g|] < ∞ on D, in particular if H (p) D [g] < ∞, then E (p) R d [P D [g]] = 1 p D c ×D c F p (g(z), g(w))(γ D (z, w) + ν(z, w)) dzdw. We note that the kernel on the right-hand side of the above identity also appears in [15, Theorem 5.6.3] for p = 2, but even the form E Proof. Denote I = D c |g(z)| p P D (x, z) dz. If H (p) D [g] < ∞, then D c ×D c F p (g(w), g(z))γ D (w, z) dwdz = D D c D c F p (g(w), g(z))ν(w, x)P D (x, z) dzdwdx < ∞. (4.2) Since ν > 0, for almost all (hence for some) pairs (w, x) ∈ D c × D we get (4.3) D c F p (g(w), g(z))P D (x, z) dz < ∞. For the remainder of the proof, we only consider pairs (w, x) satisfying the above condition. We will use different approaches for p ≥ 2 and p ∈ (1, 2). Let p ≥ 2. From (2.15) we obtain A := D c (g(z) − g(w)) 2 |g(z)| p−2 P D (x, z) dz < ∞. For z ∈ D c , let g n (z) = −n ∨ g(z) ∧ n. Clearly |g n (z)| ≤ |g(z)| and |g n (z)| ր |g(z)| when n → ∞. Since |g n (z)| ≤ n, the integral I n := D c |g n (z)| p P D (x, z) dz is finite. It is also true that the increments of g n do not exceed those of g, that is |g n (z) − g n (w)| ≤ |g(z) − g(w)|. Consequently, I n = D c g n (z) 2 |g n (z)| p−2 P D (x, z) dz ≤ 2 D c (g n (z) − g n (w)) 2 |g n (z)| p−2 P D (x, z) + 2g n (w) 2 D c |g n (z)| p−2 P D (x, z) dz ≤ A + 2g(w) 2 D c |g n (z)| p P D (x, z) dz p−2 p . The last inequality is obvious for p = 2, and follows from Jensen's inequality if p > 2. Thus, (4.4) I n ≤ A + 2g(w) 2 (I n ) 1− 2 p , hence the sequence (I n ) is bounded. By the Monotone Convergence Theorem, I < ∞. By Jensen's inequality we also get D c |g(z)|P D (x, z) dz < ∞. By the Harnack inequality, the finiteness of the Poisson integral of |g| or |g| p at any point x ∈ D guarantees its finiteness at every point of D, see, e.g., [8,Lemma 4.6], therefore the proof is finished for p ≥ 2. Now let p ∈ (1, 2). If g ≡ 0 a.e. on D c , then the statement is trivial. Otherwise, pick w ∈ D c such that 0 < |g(w)| < ∞. Let B = {z ∈ D c : |g(z)| > |g(w)|}. We have D c \B |g(z)| p P D (x, z) dz ≤ |g(w)| p < ∞. Using (2.14) and (4.3) we get B |g(z)| p P D (x, z) dz = B g(z) 2 |g(z)| p−2 P D (x, z) dz ≤ 2 B (g(z) − g(w)) 2 |g(z)| p−2 P D (x, z) dz + 2g(w) 2 B |g(z)| p−2 P D (x, z) dz ≈ B F p (g(w), g(z))P D (x, z) dz + 2|g(w)| p < ∞. Thus, P D [|g| p ](x) < ∞. The rest of the proof is the same as in the case p ≥ 2. Proof of Theorem 4.1. To prove (i) we let H (p) D [g] < ∞ and we have (4.2). Let u = P D [g]. By (2.12), u = g on D c . By Lemma 4.5, u is well-defined, and it is regular L-harmonic in D, that is E x [u(X τ D )] = u(x) for x ∈ D, cf. Definition 3.1 and (2.10). In particular, we have E x |u(X τ D )| < ∞. For x ∈ D consider the integral D c F p (u(w), u(z))P D (x, z) dz. By (2.10), P D (x, z) is the density of the distribution of X τ D under P x , hence D c F p (u(w), u(z))P D (x, z) dz = E x [F p (u(w), u(X τ D ))]. By Lemma 2.1 (ii) applied to a = u(w), X = u(X τ D ) and E = E x , the above expression is equal to (4.5) F p (u(w), E x u(X τ D )) + E x F p (u(x), u(X τ D )) = F p (u(w), u(x)) + E x F p (u(x), u(X τ D )). By integrating the first term on the right-hand side of (4.5) against ν(x, w) dxdw we obtain (4.6) D c ×D F p (u(w), u(x))ν(x, w) dxdw. For the second term in (4.5) we use Lemma 2.1 (i) and Proposition 3.4: E x F p (u(x), u(X τ D )) = E x |u(X τ D )| p − |u(x)| p = D G D (x, y) R d F p (u(y), u(z))ν(y, z) dzdy. We integrate the latter expression against ν(x, w) dxdw. By Fubini-Tonelli, (2.4) and (2.10), Since the sum of (4.6) and (4.7) equals pE D c D D R d G D (x, y)F p (u(y), u(z))ν(y, z)ν(x, w) dzdydxdw = D R d F p (u(y), u(z)) D c D G D (x, y)ν(x, w) dx dw ν(y, z) dzdy = D R d F p (u(y), u(z)) D c P D (y, w) dw ν(y, z) dzdy = D R d(p) D [u] , we obtain the Douglas identity. We now prove (ii). It is not obvious how to directly conclude that E Douglas and Hardy-Stein identities with remainders Throughout this section we assume that D is bounded. In the (quadratic) case p = 2, under a mild additional assumption on D, the Poisson integral P D [g] was shown to be the minimizer of the form E D among all Borel functions with a fixed exterior condition g ∈ X D (see [8,Proposition 5.4 and Theorem 5.5]). This needs not be the case when p = 2, and in this section we give an example of D and g ∈ X p D for which P D [g] does not minimize E (p) D among functions in V p D equal to g on D c . However, P D [g] is always a quasiminimizer, if we adopt the following definition: Definition 5.1. Let K ≥ 1. Function u ∈ V p D is a K-quasiminimizer of E (p) D , if E (p) U [u] ≤ KE (p) U [v] for every nonempty open set U ⊂D satisfying (VDC) and |∂U | = 0, and every v ∈ V p U equal to u on U c . We say that u is a quasiminimizer if it is a K-quasiminimizer for some K ∈ [1, ∞). The definition is inspired by the classical one given by Giaquinta and Giusti [35, (5.26)]. To avoid technical complications and to make this digression short we require regular test sets U above. However, to be prudent we note that the choice of admissible sets U may affect the definition of quasiminimizers and should be carefully considered, cf. Giusti [36,Example 6.5]. In the classical PDEs, quasiminimizers display many regularity properties similar to minimizers, see, e.g., Adamowicz and Toivanen [1], DiBenedetto and Trudinger [23], and Ziemer [62]. The main motivation for studying quasiminimizers is the fact that the solution of a complicated variational problem may be a quasiminimizer of a better understood functional see, e.g., [35,Theorem 2.1]. Proposition 5.2. Suppose that the assumptions of Theorem 4.1 are satisfied, D is bounded, and let g ∈ X p D . Then P D [g] is a K-quasiminimizer of E (p) D with K independent of g. Proof. Fix a subset U ⊂ D satisfying (VDC) and |∂U | = 0, and let v ∈ V p U be equal to u := P D [g] on U c . According to (2.28) we have v p/2 ∈ V U and E (p) U [v] ≈ E U [v p/2 ], with constants independent of U and v. Note that v p/2 agrees with u p/2 on U c . Since U c satisfies (VDC), by [8, Proposition 5.4 and Theorem 5.5], (5.1) E U [v p/2 ] ≥ E U [P U [u p/2 ]]. By applying the Douglas identity for the set U , first with exponent 2, and then with exponent p, and by (2.29) we get that the right-hand side of (5.1) is equal to H U [u p/2 ] ≈ H (p) U [u] = E (p) U [P U [u]] = E (p) U [u]. In the last equality we use the identity P U [u] = u, see (2.10). The proof is complete. To prove that Poisson integrals need not be minimizers, we first extend the Hardy-Stein and Douglas identities to functions that are not harmonic. The results are new even for p = 2 and ∆ α/2 . Recall that D is bounded, hence E x τ D is bounded. In what follows by lim U ↑D we denote the limit over an arbitrary ascending sequence of Lipschitz open sets U n ⊂⊂ D such that n U n = D. Here is an extended version of the Hardy-Stein formula. lim U ↑D E x |u(X τ U )| p = |u(x)| p + D G D (x, y) R d F p (u(y), u(z))ν(y, z) dzdy (5.2) + p D G D (x, y)u(y) p−1 Lu(y) dy. (5.3) If in addition D c satisfies (VDC) and |∂D| = 0, then lim U ↑D E x |u(X τ U )| p = E x |u(X τ D )| p . Proof. Let x ∈ D. Since u, Lu, and E x τ D are bounded on D, by (2.3) we get that the integral in (5.3) is finite. Therefore, using the arguments from the proof of Proposition 3.4, in what follows we may and do assume that R d |u(x)| p (1 ∧ ν(x)) dx < ∞, because otherwise both sides of (5.2) are infinite. With this in mind we first consider open Lipschitz U ⊂⊂ D so large that x ∈ U. Let p ≥ 2. Since u ∈ C 2 (D), we get that L|u| p (x) and E x |u(X τ U )| p are finite for x ∈ U , and (3.4) holds. Furthermore, since Lu is finite in D, the following manipulations are justified for y ∈ D: L|u| p (y) = L|u| p (y) − pu(y) p−1 Lu(y) + pu(y) p−1 Lu(y) (5.4) = lim ǫ→0 + |z−y|>ǫ |u(z)| p − |u(y)| p − pu(y) p−1 (u(z) − u(y)) ν(z, y) dz + pu(y) p−1 Lu(y) = R d F p (u(y), u(z))ν(y, z) dz + pu(y) p−1 Lu(y). Consequently, (3.4) takes on the form E x |u(X τ U )| p = |u(x)| p + U G U (x, y) R d F p (u(y), u(z))ν(y, z) dzdy (5.5) + U G U (x, y)u(y) p−1 Lu(y) dy. (5.6) For clarity we note that the left-hand side of (5.5) is finite and the integral in (5.6) is absolutely convergent, so the integral in (5.5) is finite as well. For p ∈ (1, 2) we proceed as in the proof of Proposition 3.3, that is, instead of |u(x)| p we consider ε > 0 and the function x → (u(x) 2 + ε 2 ) p/2 . We obtain (cf. (3.8) and (5.4)), E x (u(X τ U ) 2 + ε 2 ) p/2 = (u(x) 2 + ε 2 ) p/2 + U G U (x, y) R d F (ε) p (u(y), u(z))ν(y, z) dzdy (5.7) + p U G U (x, y)u(y)(u(y) 2 + ε 2 ) (p−2)/2 Lu(y) dy. (5.8) As in the proof of Proposition 3.4, the left-hand side tends to E x |u(X τ U )| p as ε → 0 + . Furthermore, since Lu and u are bounded in D, the integral in (5.8) converges to that in (5.6). Then we apply Fatou's lemma and the Dominated Convergence Theorem to the integral on the right-hand side of (5.7) and we obtain (5.5) for p ∈ (1, 2), too. We let U ↑ D in (5.5). By the boundedness of u and Lu in D, the integral in (5.6) tends to the one in (5.3), which is absolutely convergent. The integral on the right-hand side of (5.5) converges to the one on the right-hand side of (5.2) by the domain monotonicity and the Monotone Convergence Theorem. Since the limit on the right-hand side of (5.2) exists, the limit on the left-hand side must exist as well. This proves (5.2). If D c satisfies (VDC) and |∂D| = 0, then (2.10) holds true. Furthermore, we have E x |u(X τ U )| p = E x (|u(X τ U )| p ; τ U = τ D ) + E x (|u(X τ D )| p ; τ U = τ D ). The first term on the right converges to 0 by the boundedness of u on D and the fact that P x (τ U = τ D ) decreases to 0 as U ↑ D (see the remark preceding (2.10); see also the proof of Lemma 17 in Bogdan [4] and the proof of Lemma A.1 in [8]). The second term converges to E x |u(X τ D )| p by the Monotone Convergence Theorem. Thus the left-hand side of (5.5) tends to E x |u(X τ D )| p . We next provide a Douglas-type identity for a class of nonharmonic functions: D [u] + A D (u), where A D (u) = D u(x) p−1 Lu(x) dx + D D c u(w) p−1 (u(x) − P D [u](x)) ν(w, x) dwdx. Proof. Since u is bounded on R d , we have R d |u(x)|(1 ∧ ν(x)) dx < ∞. Assume first that H D [u] = D D c D c F p (u(w), u(z))P D (x, z)ν(x, w) dzdwdx. We apply Lemma 2.1 (iii) to a = u(w), b = u(x), with w ∈ D c and x ∈ D, X = u(X τ D ), and E = E x . Note that EX = P D [u](x). This yields: D c F p (u(w), u(z))P D (x, z) dz = D c F p (u(x), u(z))P D (x, z) dz + F p (u(w), u(x)) + (pu(w) p−1 − pu(x) p−1 )(u(x) − P D [u](x)). After integration, we obtain pH (p) D [u] = D D c D c F p (u(x), u(z))P D (x, z)ν(x, w) dzdwdx + D D c F p (u(w), u(x))ν(x, w) dwdx + D D c (pu(w) p−1 − pu(x) p−1 )(u(x) − P D [u](x)) ν(x, w) dwdx =: A 1 (u) + A 2 (u) + A 3 (u). Note that every term above is finite. Indeed, by the boundedness of u, |A 3 (u)| D D c |u(x) − P D [u](x)|ν(x, w) dwdx. To prove that this is finite, This allows us to further estimate A 3 : let v = u − P D [u]. We have Lv = Lu = f ∈ L ∞ (D) and v = 0 on D c . Note that v ∈ C 2 (D) and R d |v(x)|(1 ∧ ν(x)) dx < ∞, cf. [8, Lemma 3.6]. Let U ⊂⊂ D. By Lemma 3.2, E x v(X τ U ) − v(x) = U G U (x, y)f (y) dy, x ∈ U. Since u is bounded on R d , we have E x u(X τ U ) → E x u(X τ D ) = P D [u](x) as U ↑ D,|A 3 (u)| D D c D G D (x, y)ν(w, x) dydwdx = D D c P D (y, w) dwdy = |D| < ∞. Since A 1 (u) and A 2 (u) are nonnegative, they must be finite as well, because H (p) D [u] < ∞. We then have D c F p (u(x), u(z))P D (x, z) dz = E x F p (u(x), u(X τ D )) = E x |u(X τ D )| p − |u(x)| p − pu(x) p−1 (P D [u](x) − u(x)). Thus, by Proposition 5.3 we obtain A 1 (u) = A 4 (u) + p D D c D G D (x, y)u(y) p−1 Lu(y)ν(x, w) dydwdx (5.10) − p D D c u(x) p−1 (P D [u](x) − u(x))ν(x, w) dwdx, where A 4 (u) is the integral in (4.7). Note that A 2 (u) + A 4 (u) = pE (p) D [u] . Also, all the expressions in (5.10) are finite, see the discussion of A 3 (u). To finish the proof of (5.9) in the case H (p) D [u] < ∞, we simply note that pA D (u) = A 1 (u) − A 4 (u) + A 3 (u). The situation H (p) D [u] = ∞ remains to be considered. Since P D [u] is bounded in D, by arguments similar to those in the estimates of A 3 (u) above, we prove that A D (u) is finite. Therefore by Theorem 4.1 the identity (5.9) holds with both sides infinite. Knowing the form of the remainder A D (u) in the Douglas identity (5.9), we may provide an example which shows that Poisson integral need not be a minimizer of E (p) D for p = 2; it is only a quasiminimizer by Proposition 5.2. Example 5.5 (The Poisson extension need not be a minimizer for p = 2). Let p > 2 and consider 0 < R < R 1 such that D ⊂⊂ B R . Define g n (z) = ((|z| − R) −1/(p−1) ∧ n)1 B R 1 \B R (z). Since each g n is bounded with support separated from D, we have g n ∈ X p D ∩ X D ; see the discussion following Example 2.4 in [8]. By (2.6) there exists c > 0 such that (5.11) P D (x, z) ≤ c, x ∈ D, z ∈ B R 1 \ B R . Furthermore, for every U ⊂⊂ D there is ǫ > 0 such that (5.12) P D (x, z) ≥ ǫ, x ∈ U, z ∈ B R 1 \ B R . For x ∈ D we let u n (x) = G D [1](x) + P D [g n ](x). Obviously u n are bounded on R d . We will verify that G D [1] ∈ C 2 (D). For this purpose we let f be a smooth, compactly supported, nonnegative function equal to 1 on D. By the Hunt's formula and Fubini-Tonelli we get (5.13) G D [f ](x) = G D [1](x) = R d G(x − y)f (y) dy − E x R d G(X τ D , y)f (y) dy, x ∈ R d . Here G is either the potential kernel or the compensated potential kernel of (X t ); see Grzywny A D (u n ) = − D u n (x) p−1 dx + D D c u n (w) p−1 G D [1](x)ν(x, w) dwdx = D (E x u n (X τ D ) p−1 − (E x u n (X τ D ) + G D [1](x)) p−1 ) dx = U + D\U . (5.14) We claim that A D (u n ) > 0 for large n. Indeed, recall that G D [1](x) = E x τ D is bounded. Since the integrals D c g n (x) dx are bounded, by (5.11) there is M > 0 such that E x u n (X τ D ) < M for every x ∈ D and n ∈ N. Therefore the integral D\U in (5.14) is bounded from below, independently of n. Note that D c g n (x) p−1 dx → ∞ as n → ∞. Thus, by (5.12) we obtain that U → ∞ in (5.14) as n → ∞. Hence, for sufficiently large n we get that A D (u n ) > 0, which proves that E Applications to Dirichlet-to-Neumann map In this section we adopt the assumptions of Theorem 4.1. In addition, we assume that the set D is bounded and p ∈ [2, ∞). We define the nonlocal normal derivative as an analogue of the fractional version of Dipierro, Ros-Oton, and Valdinoci [24, (1.2)], see also Vondraček [59]: N f (z) = D (f (z) − f (x))ν(x, z) dx. (6.1) Note that the increments of f are integrated on D, but the integral is evaluated for z ∈ D c , if convergent. For instance, if f ∈ L 1 (R d , 1 ∧ ν), then N f ∈ L 1 loc (D c ). Assume that g ∈ X p D . Then u = P D [g] solves the Dirichlet problem (1.5). By definition, the Dirichlet-to-Neumann operator DN maps the exterior condition g to the nonlocal normal derivative h := N u. So, u solves the Neumann problem Lu = 0 in D, N u = h on D c , and DN := N • P D on X p D . In fact, for almost every z ∈ D c , DN g(z) = N u(z) = D (u(z) − u(x))ν(x, z) dx (6.2) = D D c (u(z) − u(w))P D (x, w)ν(x, z) dwdx = D c (u(z) − u(w))γ D (z, w) dw = D c (g(z) − g(w))γ D (z, w) dw,m(z) = c d,α D |z − x| −d−α dx ≈ δ D (z) −α , δ D (z) ≤ 1, δ D (z) −d−α , δ D (z) > 1. Back to general L, we note that sharp estimates of γ D are known for bounded C 1,1 domains and the half-space, see [8, Theorem 2.6 and 6.1]. We next define the normalized Dirichlet-to-Neumann operator, for g ∈ X p D and a.e. z ∈ D c , DN g(z) = DN g(z) m(z) = D c (g(z) − g(w)) γ D (z, w) m(z) dz. (6.5) In what follows we give several results for the Dirichlet-to-Neumann operator on L p . In particular, we show that DN is well-defined: X p D → L p (D c , m 1−p ) and DN is bounded on L p (D c , m). We also relate the form H (p) D to the operator DN in (6.9). Proposition 6.1. Assume that g ∈ X p D . Then DN g ∈ L p (D c , m 1−p ) and DN g ∈ L p (D c , m). Furthermore, there exists a constant C, independent of g, such that DN g p L p (D c ,m 1−p ) = DN g p L p (D c ,m) ≤ CH (p) D [g]. Proof. Using (6.3) and Jensen's inequality we get D c |DN g(z)| p m(z) 1−p dz = D c |DN g(z)| m(z) p m(z) dz ≤ D c D c |g(w) − g(z)| p γ D (z, w) dzdw. Since p ≥ 2, we have |a − b| p ≤ (a − b) 2 (|a| p−2 + |b| p−2 ). So, by (2.19), D c D c |g(w) − g(z)| p γ D (z, w) dzdw H (p) D [g] < ∞, (6.6) which ends the proof. Proposition 6.2. If g ∈ L p (D c , m), then g ∈ X p D and there is C > 0, independent of g, such that H (p) D [g] ≤ C g p L p (D c ,m) . Proof. Following [30, Remark 2.37], we let g be the function g extended to D by 0. Then, E (p) D [ g] = 1 p (D c ×D c ) c F p ( g(z), g(w))ν(z, w) dzdw = D D c |g(z)| p ν(z, w) dwdz = D c |g(z)| p m(z) dz < ∞. (6.7) In particular, g ∈ V p D . By Proposition 5.2 we get that there exists a constant C, independent of g, such that E D [g] = E (p) D [P D [g]] ≤ CE (p) D [ g] = C D c |g(z)| p m(z) dz, which proves the result. D ; see also [9,Lemma 7] for a detailed discussion of E (p) R d for L = ∆ α/2 . Further discussion As usual, D is a nonempty open set in R d . We define Proof. Assume that g ∈ Y p D , i.e., the right-hand side of (1.10) is finite. By a simple modification of the proof of [8,Lemma 4.6] we get that g ∈ L p (D c , P D (x, z) dz) for every x ∈ D, in particular the Poisson integral P D [g](x) converges absolutely. By (2.8), the right-hand side of (1.10) equals (7.1) W p D = u : R d → R R d ×R d \D c ×D c |u(x) − u(y)| p ν(x, y) dxdy < ∞ , and Y p D = g : D c → R D c ×D c |g(w) − g(z)| p γ D (w, z) dwdz < ∞ .D c D c D |g(w) − g(z)| p ν(w, x)P D (x, z) dxdwdz. We use Fubini-Tonelli and consider the integral D c |g(w) − g(z)| p P D (x, z) dz = E x |u(X τ D ) − g(w)| p . By Lemma 2.2 we get that for x ∈ D and w ∈ D c , E x |u(X τ D ) − g(w)| p ≈ E x |u(X τ D ) − u(x)| p + |u(x) − g(w)| p ≥ E x |u(X τ D ) − u(x)| p . We apply Proposition 3.4, to u(z) := u(z) − u(x). It is L-harmonic on D and u(x) = 0, therefore E x |u(X τ D ) − u(x)| p = D G D (x, y) R d F p ( u(y), u(z))ν(z, y) dzdy. For p = 2 it is not true that F p (a + t, b + t) is comparable with F p (a, b), but since p ≥ 2, by Lemma 2.4 we have F p (a + t, b + t) ≥ c|a + t − b − t| p = c|a − b| p . It follows that F p ( u(y), u(z)) |u(y) − u(z)| p , and thus E x |u(X τ D ) − g(w)| p D G D (x, y) R d |u(y) − u(z)| p ν(z, y) dzdy. We integrate the inequality on D c × D against ν(w, x) dwdx as in (4.7), and the right-hand side is D R d |u(x) − u(y)| p ν(x, y) dxdy ≥ 1 2 R d ×R d \(D c ×D c ) |u(x) − u(y)| p ν(x, y) dxdy. The result follows. We remark that in general (1.10) fails for p ∈ (1, 2); see Lemma 7.4 and Example 7.5. In the remainder of this section we compare W p D and V p D , see (2.25), by using C ∞ c (R d ). Lemma 7.2. For every p > 1 we have C ∞ c (R d ) ⊆ V p R d ⊆ V p D . Proof. The inclusion V p R d ⊆ V p D follows from the definition. To prove that C ∞ c (R d ) ⊆ V p R d , we let φ ∈ C ∞ c (R d ). We have |φ(x + z) + φ(x − z) − 2φ(x)| ≤ M (1 ∧ |z| 2 ), x, z ∈ R d . It follows that Lφ is bounded on R d , cf. (1.3) and (1.2). Thus, (7.2) R d |φ(x)| p−1 |Lφ(x)| dx < ∞. Furthermore, by the Dominated Convergence Theorem and the symmetry of ν, R d φ(x) p−1 Lφ(x) dx = 1 2 R d φ(x) p−1 lim ǫ→0 + |z|>ǫ (φ(x + z) + φ(x − z) − 2φ(x))ν(z) dzdx = lim ǫ→0 + R d |z|>ǫ φ(x) p−1 (φ(x + z) − φ(x))ν(z) dzdx. By Fubini's theorem, the substitutions z → −z and x → x + z, and the symmetry of ν, R d |z|>ǫ φ(x) p−1 (φ(x + z) − φ(x))ν(z) dzdx = R d |z|>ǫ φ(x + z) p−1 (φ(x) − φ(x + z))ν(z) dzdx = − 1 2 |z|>ǫ R d (φ(x + z) p−1 − φ(x) p−1 )(φ(x + z) − φ(x)) dx ν(z) dz for every ǫ > 0. By (2.23), the Monotone Convergence Theorem and the above, E (p) R d [φ] = 1 2 R d R d (φ(x + z) p−1 − φ(x) p−1 )(φ(x + z) − φ(x))ν(z) dxdz = − R d φ(x) p−1 Lφ(x) dx. (7.3) The result follows from (7.2) and (2.25). The inclusion C ∞ c (R d ) ⊆ V p D indicates that the Sobolev-Bregman spaces will be useful in variational problems posed in L p . The situation with the spaces W p D is more complicated. While for p ≥ 2 we have a result similar to that of Lemma 7.2, for p ∈ (1, 2) it is not so. More precisely, we have the following two lemmas: Lemma 7.3. For p ≥ 2 we have C ∞ c (R d ) ⊆ W p R d ⊆ W p D . Proof. For φ ∈ C ∞ c (R d ) let K = supp φ. Then we have |φ(x) − φ(y)| = 0 on K c × K c and |φ(x) − φ(y)| p 1 ∧ |x − y| p ≤ 1 ∧ |x − y| 2 , x, y ∈ R d × R d \ K c × K c . It follows that φ ∈ W p R d . The inclusion W p R d ⊆ W p D is clear from the definition of the spaces. Lemma 7.4. Let p ∈ (1, 2) and assume that for some r > 0 we have ν(y) |y| −d−p for |y| < r. If u ∈ W p D has compact support in R d and vanishes on D c , then u ≡ 0. Results of this type are well-known for the spaces with integration over D × D, where D is connected. Brezis [13,Proposition 2] shows that any measurable function must be constant in this case; a simpler proof of this fact was given by De Marco, Mariconda and Solimini [19,Theorem 4.1]. Lemma 7.4 follows by taking Ω = R d in the aforementioned results, but we present a different proof. Such facts also hold true in the context of metric spaces, see, e.g., Pietruska-Pa luba [49]. We will see in the proof of Lemma 7.4 that the result reduces to that with D = R d . Proof of Lemma 7.4. We may assume that u is bounded, because the p-increments of (0 ∨ u) ∧ 1 do not exceed those of u. Thus, since u is compactly supported, we get that u ∈ L p (R d ) ∩ L 2 (R d ). Let u(ξ) = R d u(x)e −2πiξx dx, ξ ∈ R d . The Hausdorff-Young inequality asserts that for u ∈ L p (R d ) we have (7.4) u p ≥ u p ′ , where p ′ = p p−1 , see, e.g., Grafakos [37,Proposition 2.2.16]. We estimate the left-hand side of (1.10) by using (7.4): R d ×R d \D c ×D c |u(x) − u(y)| p ν(x, y) dxdy = R d R d |u(x) − u(x + y)| p ν(y) dxdy ≥ R d R d |(u(·) − u(· + y)) ∧ (ξ)| p ′ dξ p p ′ ν(y) dy = R d R d |1 − e −2πiξy | p ′ | u(ξ)| p ′ dξ p p ′ ν(y) dy. By (7.4), | u(ξ)| p ′ dξ is a finite measure on R d . As we have p/p ′ < 1, by Jensen and Fubini-Tonelli, R d R d |1 − e −2πiξy | p ′ | u(ξ)| p ′ dξ p p ′ ν(y) dy R d ν(y) R d |1 − e −2πiξy | p | u(ξ)| p ′ dξdy = R d | u(ξ)| p ′ R d |1 − e 2πiξy | p ν(y) dydξ. Since |1 − e 2πiξy | ≥ | sin 2πξy| and ν(y) |y| −d−p for small |y|, the integral is infinite, unless u = 0 a.e. in R d . As a comment to Lemmas 7.2 and 7.4 we recall that V p D is defined in terms of F p . When a is close to b then, regardless of p > 1, the Bregman divergence F p (a, b) is of order (b − a) 2 rather than |b − a| p . Thus V p D agrees with the Lévy measure condition (1.2) better than W p D does. The following example indicates that the scale of linear spaces W p D may not be suitable for analysis of harmonic functions when p ≤ 2: By the boundedness of u, the boundedness of D and the separation of D and B, the last integral is finite. Furthermore, since u is not constant and vanishes on D c , the left-hand side is infinite by Lemma 7.4. Therefore the left-hand side of (7.6) is infinite, which yields (7.5). Definition 3. 1 . 1We say that the function u :R d → R is L-harmonic (or harmonic, if L is understood) in D ifit has the mean value property inside D, that is: for every open set U ⊂⊂ D, Proposition 3. 3 . 3If u : R d → R is L-harmonic in D, p > 1, and U ⊂⊂ D is open Lipschitz, then see the end of the proof of [8, Lemma 4.11] and (2.6). . Let p > 1 be given. If u is L-harmonic in D and x ∈ D, then sup x∈U ⊂⊂D If u is regular L-harmonic in D, then the left-hand side can be replaced with E x |u(X τ D )| p .Proof. As noted in[8, Remark 4.4], {u(X τ U ), U ⊂ D}is a martingale ordered by the inclusion of open subsets of D. By domain monotonicity of the Green function and the nonnegativity of F p , both sides of (3.2) increase if U increases. Since every open set U ⊂⊂ D is included in an open Lipschitz set U ⊂⊂ D, the supremum in (3.10) may be taken over open Lipschitz sets U ⊂⊂ D. The first part of the statement follows from the monotone convergence theorem. Theorem 4 . 1 (D 41Douglas identity). Let p > 1. Assume that the Lévy measure ν satisfies (A1) and (A2), D ⊂ R d is open, D c satisfies (VDC), and |∂D| = 0. (i) Let g : D c → R be such that H (p) D [g] < ∞. Then P D [g] is well-defined and satisfies [P D [g]]. P D [u| D c ] = P D [u]. Remark 4. 2 . 2The more explicit expression of the Douglas identity (1.8) stated in the Introduction follows from (4.1), (2.22) and (2.24). D , and the space X p D , defined in (2.26), which is a natural domain of H(p) D . From Theorem 4.1 we immediately obtain the following trace and extension result in the nonquadratic setting. Corollary 4 . 3 . 43Let Ext g = P D [g], the Poisson extension, and Tr u = u| D c , the restriction to D c . Then Ext : X p D → V p D , Tr : V p D → X p D , and Tr Ext is the identity operator on X p D . D and the Douglas identity of Theorem 4.1 with p = 2 on full domain V 2 D do not appear in[15]. The proof of Theorem 4.1 uses the following lemma, which asserts that the condition H D [g] < ∞ implies the finiteness of P D [|g| p ] and P D [|g|] on D. Lemma 4 . 5 . 45Suppose that g : D c → R satisfies H (p) D [g] < ∞.Then for every x ∈ D we have D c |g(z)| p P D (x, z) dz < ∞. In particular, the Poisson integral of g is well-defined. F p (u(y), u(z))ν(y, z) dzdy. (4.7) D [u] < ∞ implies P D [|u|] < ∞ on D, thus we cannot apply Lemma 2.1. Instead we use another approach: by Lemma 2.3, E (p) D [u] < ∞ is equivalent to E D [u p/2 ] < ∞. By the trace theorem for p = 2, see [8, Theorem 2.3], H D [u p/2 ] < ∞. By Lemma 2.3 we get (ii). Proposition 5. 3 . 3Let p > 1 and assume that ν satisfies (A1) and (A2). Let u : R d → R. If u ∈ C 2 (D) and u and Lu are bounded in D, then for every x ∈ D, Theorem 5. 4 . 4Suppose that the assumptions of Theorem 4.1 hold with the addition that D is bounded. Let u : R d → R be bounded, u ∈ C 2 (D), and Lu be bounded in D. Then(5.9) E (p) D [P D [u]] = E (p) cf. the last part of the proof of Proposition 5.3. Hence, the boundedness of f , the domain monotonicity, and the Dominated Convergence Theorem yield v(x) = − D G D (x, y)f (y) dy, x ∈ D. D [u n ] for some n, as needed. The case p ∈ (1, 2) may be handled similarly, by using g n (z) = ((|z| − R) −1 ∧ n)1 B R 1 \B R (z) and u n = P D [g n ] − G D[1]. have used the definition of γ D , the fact that u = P D [g], and the Fubini-Tonelli theorem (justified by the estimates in the proof of Proposition 6.1). For z ∈ Int(D c ) = R d \ D we let m(z) := D c γ D (w, z) dw = D ν(x, z) dx < ∞. (6.4) For example, for L = ∆ α/2 and (bounded) D of class C 1,1 , with δ D (z) := d(z, D) we have Corollary 6 . 3 . 63The normalized Dirichlet-to-Neumann map DN is bounded on L p (D c , m).The following is an analogue of the formula (7.3) below.this connection, the reader may compare (6.9), DN and H Example 7. 5 . 5Let ν and p be as inLemma 7.4. Let B = B(0, 1) and assume that D is bounded and dist(D, B) > 0. Then there is g ∈ Y p D such that u := P D [g] / ∈ W p D , i.e.,(7.5) R d ×R d \D c ×D c |u(x) − u(y)| p ν(x, y) dxdy = ∞. Let g(z) = 1 B (z) for z ∈ D c . Then g ∈ Y p D ,cf. the arguments following [8, Example 2.4]. Clearly, u is bounded in D. By the positivity of P D [39, Lemma 2.2], u(x) > 0 for every x ∈ D. Of course, B, D c \ B = B c \ D and D form a partition of R d . Therefore their Cartesian products partition R d × R d ; in fact also B c × B c and R d × R d \ D c × D c (see below). Since u vanishes on D c \ B, u(x) − u(y) vanishes on (D c \ B) × (D c \ B). It follows that (7.6) B c B c |u(x) − u(y)| p ν(x, y) dxdy ≤ R d ×R d \D c ×D c |u(x) − u(y)| p ν(x, y) dxdy. Define u = u on B c and u = 0 on B. Then, u = u on D and u = 0 on D c , and R d ×R d \D c ×D c Proposition 7.1. If p ≥ 2 then (1.10) holds true under the assumptions on D and ν from Theorem 4.1, and the Poisson extension acts from Y p D to W p D . Proposition 6.4. Let f ∈ L p (D c , m) and g ∈ X p D . Then D c |DN g(z)||f (z)| p−1 dz < ∞ andProof. By Hölder's inequality with exponents p and p ′ = p p−1 , and by Proposition 6.1,It suffices to prove (6.8). By the symmetry of γ D ,The above application of the Fubini-Tonelli theorem is justified by using Hölder's inequality with exponents p and p/(p−1), and (6.6); see also(6.4). The first and the last lines above yield (6.8).Let us discuss related results for p = 2. In [59, Proposition 3.2], Vondraček shows that the normalized Dirichlet-to-Neumann operator map is bounded on L 2 (D c , m); our Corollary 6.3 extends this result to L p . As observed by Foghem and Kassmann[30,Remark 2.37], the space L 2 (D c , m) can be smaller than the trace space X D . In[30,Section 4.4], the authors investigate the Dirichletto-Neumann operator for the equation Lu = λu + f , where λ ∈ R is not a Dirichlet eigenvalue of L in D. They prove the boundedness of the Dirichlet-to-Neumann operator from the trace space into its dual. If we let DN F K be the Dirichlet-to-Neumann operator defined in[30]for λ = 0 and f = 0, then using our Douglas identity and v = u g = P D [g] in[30,Definition 4.18], for g ∈ X D we getHere ·, · is the pairing between X D and its dual, see[30,Section 2.6]. Then, by polarization,w)dzdw, (6.10) for g 1 , g 2 ∈ X D . Both (6.3) and (6.10) give explicit integral representations for the Dirichlet-to-Neumann operator, which are more direct than (6.2). They were not stated in[30,59], although similar formulas appear in[59,Section 3 and (4.2)] and the author of[59]was probably aware of the explicit versions.On an informal level, (6.3) and (6.10) mean that the Dirichlet-to-Neumann map is the negative of the Lévy-type operator on D c with jump kernel γ D , and H D is the corresponding Dirichlet form. Despite being smaller than X 2 D , the space L 2 (D c , m), used by Vondraček, is suitable for studying the (negative of the) normalized Dirichlet-to-Neumann operator DN as a generator of a Markov process on D c . In fact, − DN is the generator of the so-called trace process, see[59, (4.2)]. In Hölder continuity of quasiminimizers with nonstandard growth. T Adamowicz, O Toivanen, Nonlinear Anal. 125T. Adamowicz and O. Toivanen. Hölder continuity of quasiminimizers with nonstandard growth. Nonlinear Anal., 125:433-456, 2015. L'hypercontractivité et son utilisation en théorie des semigroupes. D Bakry, Lectures on probability theory. Saint-Flour; BerlinSpringer1581D. Bakry. L'hypercontractivité et son utilisation en théorie des semigroupes. In Lectures on probability theory (Saint-Flour, 1992), volume 1581 of Lecture Notes in Math., pages 1-114. Springer, Berlin, 1994. Inverse problems for the fractional-Laplacian with lower order non-local perturbations. S Bhattacharyya, T Ghosh, G Uhlmann, Trans. Amer. Math. Soc. 3745S. Bhattacharyya, T. Ghosh, and G. Uhlmann. Inverse problems for the fractional-Laplacian with lower order non-local perturbations. Trans. Amer. Math. Soc., 374(5):3053-3075, 2021. The boundary Harnack principle for the fractional Laplacian. K Bogdan, Studia Math. 1231K. Bogdan. The boundary Harnack principle for the fractional Laplacian. Studia Math., 123(1):43-80, 1997. Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains. K Bogdan, T Byczkowski, Studia Math. 1331K. Bogdan and T. Byczkowski. Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains. Studia Math., 133(1):53-92, 1999. On Hardy spaces of local and nonlocal operators. K Bogdan, B Dyda, T Luks, Hiroshima Math. J. 442K. Bogdan, B. Dyda, and T. Luks. On Hardy spaces of local and nonlocal operators. Hiroshima Math. J., 44(2):193-215, 2014. K Bogdan, D Fafu La, A Rutkowski, arXiv:2207.07431The Douglas formula in L p . arXiv e-prints. K. Bogdan, D. Fafu la, and A. Rutkowski. The Douglas formula in L p . arXiv e-prints, page arXiv:2207.07431, 2022. Extension and trace for nonlocal operators. K Bogdan, T Grzywny, K Pietruska-Pa Luba, A Rutkowski, J. Math. Pures Appl. 1379K. Bogdan, T. Grzywny, K. Pietruska-Pa luba, and A. Rutkowski. Extension and trace for nonlocal operators. J. Math. Pures Appl. (9), 137:33-69, 2020. Optimal Hardy inequality for the fractional Laplacian on L p. K Bogdan, T Jakubowski, J Lenczewska, K Pietruska-Pa Luba, Paper No. 109395. 2822022K. Bogdan, T. Jakubowski, J. Lenczewska, and K. Pietruska-Pa luba. Optimal Hardy inequality for the fractional Laplacian on L p . J. Funct. Anal., 282(8):Paper No. 109395, 31, 2022. Lévy systems and moment formulas for mixed Poisson integrals. K Bogdan, J Rosiński, G Serafin, L Wojciechowski, Stochastic analysis and related topics. ChamBirkhäuser/Springer72K. Bogdan, J. Rosiński, G. Serafin, and L. Wojciechowski. Lévy systems and moment formulas for mixed Poisson integrals. In Stochastic analysis and related topics, volume 72 of Progr. Probab., pages 139-164. Birkhäuser/Springer, Cham, 2017. Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method. M Bonforte, J Dolbeault, B Nazaret, N Simonov, To appear in Memoirs AMSM. Bonforte, J. Dolbeault, B. Nazaret, and N. Simonov. Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method. To appear in Memoirs AMS. A relaxation method of finding a common point of convex sets and its application to the solution of problems in convex programming. L M Bregman, Ž. Vyčisl. Mat i Mat. Fiz. 7L. M. Bregman. A relaxation method of finding a common point of convex sets and its application to the solution of problems in convex programming.Ž. Vyčisl. Mat i Mat. Fiz., 7:620-631, 1967. How to recognize constant functions. A connection with Sobolev spaces. H Brezis, Uspekhi Mat. Nauk. 574H. Brezis. How to recognize constant functions. A connection with Sobolev spaces. Uspekhi Mat. Nauk, 57(4(346)):59-74, 2002. Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities. J A Carrillo, A Jüngel, P A Markowich, G Toscani, A Unterreiter, Monatsh. Math. 1331J. A. Carrillo, A. Jüngel, P. A. Markowich, G. Toscani, and A. Unterreiter. Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities. Monatsh. Math., 133(1):1-82, 2001. Symmetric Markov processes, time change, and boundary theory. Z.-Q Chen, M Fukushima, Princeton University Press35Princeton, NJZ.-Q. Chen and M. Fukushima. Symmetric Markov processes, time change, and boundary theory, volume 35 of London Mathematical Society Monographs Series. Princeton University Press, Princeton, NJ, 2012. From Brownian motion to Schrödinger's equation. K L Chung, Z X Zhao, Grundlehren der Mathematischen Wissenschaften. 312Springer-VerlagK. L. Chung and Z. X. Zhao. From Brownian motion to Schrödinger's equation, volume 312 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, 1995. The higher order fractional Calderón problem for linear local operators: uniqueness. G Covi, K Mönkkönen, J Railo, G Uhlmann, Adv. Math. 399G. Covi, K. Mönkkönen, J. Railo, and G. Uhlmann. The higher order fractional Calderón problem for linear local operators: uniqueness. Adv. Math., 399:Paper No. 108246, 29, 2022. Heat kernels and spectral theory. E B Davies, Cambridge Tracts in Mathematics. 92Cambridge University PressE. B. Davies. Heat kernels and spectral theory, volume 92 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1990. An elementary proof of a characterization of constant functions. G. De Marco, C Mariconda, S Solimini, Adv. Nonlinear Stud. 83G. De Marco, C. Mariconda, and S. Solimini. An elementary proof of a characterization of constant functions. Adv. Nonlinear Stud., 8(3):597-602, 2008. Probabilities and Potential. North Holland Mathematical Studies. C Dellacherie, P.-A Meyer, North-Holland Publishing Co29Amsterdam-New YorkC. Dellacherie and P.-A. Meyer. Probabilities and Potential. North Holland Mathematical Studies, 29. North- Holland Publishing Co., Amsterdam-New York, 1978. Probabilities and potential. C Dellacherie, P.-A Meyer, North-Holland Mathematics Studies. J. P. Wilson72North-Holland Publishing CoTheory of martingalesC. Dellacherie and P.-A. Meyer. Probabilities and potential. B, volume 72 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam, 1982. Theory of martingales, Translated from the French by J. P. Wilson. Hitchhiker's guide to the fractional Sobolev spaces. E Di Nezza, G Palatucci, E Valdinoci, Bull. Sci. Math. 1365E. Di Nezza, G. Palatucci, and E. Valdinoci. Hitchhiker's guide to the fractional Sobolev spaces. Bull. Sci. Math., 136(5):521-573, 2012. Harnack inequalities for quasiminima of variational integrals. E Dibenedetto, N S Trudinger, Ann. Inst. H. Poincaré Anal. Non Linéaire. 14E. DiBenedetto and N. S. Trudinger. Harnack inequalities for quasiminima of variational integrals. Ann. Inst. H. Poincaré Anal. Non Linéaire, 1(4):295-308, 1984. Nonlocal problems with Neumann boundary conditions. S Dipierro, X Ros-Oton, E Valdinoci, Rev. Mat. Iberoam. 332S. Dipierro, X. Ros-Oton, and E. Valdinoci. Nonlocal problems with Neumann boundary conditions. Rev. Mat. Iberoam., 33(2):377-416, 2017. Solution of the problem of Plateau. J Douglas, Trans. Amer. Math. Soc. 331J. Douglas. Solution of the problem of Plateau. Trans. Amer. Math. Soc., 33(1):263-321, 1931. Function spaces and extension results for nonlocal Dirichlet problems. B Dyda, M Kassmann, J. Funct. Anal. 27711108134B. Dyda and M. Kassmann. Function spaces and extension results for nonlocal Dirichlet problems. J. Funct. Anal., 277(11):108134, 2019. Markov processes. Vols. I. E B Dynkin, Die Grundlehren der Mathematischen Wissenschaften. IISpringer-VerlagBändeE. B. Dynkin. Markov processes. Vols. I, II, volume 122 of Die Grundlehren der Mathematischen Wissenschaften, Bände 121. Academic Press Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965. Feller semigroups, L p -sub-Markovian semigroups, and applications to pseudo-differential operators with negative definite symbols. W Farkas, N Jacob, R L Schilling, Forum Math. 131W. Farkas, N. Jacob, and R. L. Schilling. Feller semigroups, L p -sub-Markovian semigroups, and applications to pseudo-differential operators with negative definite symbols. Forum Math., 13(1):51-90, 2001. The Dirichlet problem for nonlocal operators. M Felsinger, M Kassmann, P Voigt, Math. Z. 2793-4M. Felsinger, M. Kassmann, and P. Voigt. The Dirichlet problem for nonlocal operators. Math. Z., 279(3-4):779- 809, 2015. A general framework for nonlocal Neumann problems. G Foghem, M Kassmann, arXiv:2204.06793arXiv e-printsG. Foghem and M. Kassmann. A general framework for nonlocal Neumann problems. arXiv e-prints, page arXiv:2204.06793, 2022. Functional Bregman divergence and Bayesian estimation of distributions. B A Frigyik, S Srivastava, M R Gupta, IEEE Trans. Inform. Theory. 5411B. A. Frigyik, S. Srivastava, and M. R. Gupta. Functional Bregman divergence and Bayesian estimation of distributions. IEEE Trans. Inform. Theory, 54(11):5130-5139, 2008. Fractional thoughts. N Garofalo, New developments in the analysis of nonlocal operators. Providence, RIAmer. Math. Soc723N. Garofalo. Fractional thoughts. In New developments in the analysis of nonlocal operators, volume 723 of Contemp. Math., pages 1-135. Amer. Math. Soc., Providence, RI, 2019. Uniqueness and reconstruction for the fractional Calderón problem with a single measurement. T Ghosh, A Rüland, M Salo, G Uhlmann, J. Funct. Anal. 279142T. Ghosh, A. Rüland, M. Salo, and G. Uhlmann. Uniqueness and reconstruction for the fractional Calderón problem with a single measurement. J. Funct. Anal., 279(1):108505, 42, 2020. The Calderón problem for the fractional Schrödinger equation. T Ghosh, M Salo, G Uhlmann, Anal. PDE. 132T. Ghosh, M. Salo, and G. Uhlmann. The Calderón problem for the fractional Schrödinger equation. Anal. PDE, 13(2):455-475, 2020. On the regularity of the minima of variational integrals. M Giaquinta, E Giusti, Acta Math. 148M. Giaquinta and E. Giusti. On the regularity of the minima of variational integrals. Acta Math., 148:31-46, 1982. Direct methods in the calculus of variations. E Giusti, World Scientific Publishing Co., IncRiver Edge, NJE. Giusti. Direct methods in the calculus of variations. World Scientific Publishing Co., Inc., River Edge, NJ, 2003. Classical Fourier analysis. L Grafakos, Graduate Texts in Mathematics. 249Springersecond editionL. Grafakos. Classical Fourier analysis, volume 249 of Graduate Texts in Mathematics. Springer, New York, second edition, 2008. Remarks on the nonlocal Dirichlet problem. T Grzywny, M Kassmann, L Leżaj, Potential Anal., 2020. Article in pressT. Grzywny, M. Kassmann, and L. Leżaj. Remarks on the nonlocal Dirichlet problem. Potential Anal., 2020. Article in press. Potential kernels, probabilities of hitting a ball, harmonic functions and the boundary Harnack inequality for unimodal Lévy processes. T Grzywny, M Kwaśnicki, Stochastic Process. Appl. 1281T. Grzywny and M. Kwaśnicki. Potential kernels, probabilities of hitting a ball, harmonic functions and the boundary Harnack inequality for unimodal Lévy processes. Stochastic Process. Appl., 128(1):1-38, 2018. Pseudo differential operators and Markov processes. N Jacob, Imperial College PressILondonFourier analysis and semigroupsN. Jacob. Pseudo differential operators and Markov processes. Vol. I. Imperial College Press, London, 2001. Fourier analysis and semigroups. Some Dirichlet spaces obtained by subordinate reflected diffusions. N Jacob, R L Schilling, Rev. Mat. Iberoamericana. 151N. Jacob and R. L. Schilling. Some Dirichlet spaces obtained by subordinate reflected diffusions. Rev. Mat. Iberoamericana, 15(1):59-91, 1999. Foundations of modern probability. Probability and its Applications. O Kallenberg, Springer-VerlagNew York; New Yorksecond editionO. Kallenberg. Foundations of modern probability. Probability and its Applications (New York). Springer-Verlag, New York, second edition, 2002. Gradient estimates of harmonic functions and transition densities for Lévy processes. T Kulczycki, M Ryznar, Trans. Amer. Math. Soc. 3681T. Kulczycki and M. Ryznar. Gradient estimates of harmonic functions and transition densities for Lévy processes. Trans. Amer. Math. Soc., 368(1):281-318, 2016. Ten equivalent definitions of the fractional Laplace operator. M Kwaśnicki, Fract. Calc. Appl. Anal. 201M. Kwaśnicki. Ten equivalent definitions of the fractional Laplace operator. Fract. Calc. Appl. Anal., 20(1):7-51, 2017. On L p -contractivity of semigroups generated by linear partial differential operators. M Langer, V Maz&apos;ya, J. Funct. Anal. 1641M. Langer and V. Maz'ya. On L p -contractivity of semigroups generated by linear partial differential operators. J. Funct. Anal., 164(1):73-109, 1999. Some problems on Markov semigroups. V A Liskevich, Y A Semenov, Schrödinger operators, Markov semigroups, wavelet analysis, operator algebras. BerlinAkademie Verlag11V. A. Liskevich and Y. A. Semenov. Some problems on Markov semigroups. In Schrödinger operators, Markov semigroups, wavelet analysis, operator algebras, volume 11 of Math. Top., pages 163-217. Akademie Verlag, Berlin, 1996. Sided and symmetrized Bregman centroids. F Nielsen, R Nock, IEEE Trans. Inform. Theory. 556F. Nielsen and R. Nock. Sided and symmetrized Bregman centroids. IEEE Trans. Inform. Theory, 55(6):2882- 2904, 2009. Semigroups of linear operators and applications to partial differential equations. A Pazy, Applied Mathematical Sciences. 44Springer-VerlagA. Pazy. Semigroups of linear operators and applications to partial differential equations, volume 44 of Applied Mathematical Sciences. Springer-Verlag, New York, 1983. Heat kernels on metric spaces and a characterisation of constant functions. K Pietruska-Pa Luba, Manuscripta Math. 1153K. Pietruska-Pa luba. Heat kernels on metric spaces and a characterisation of constant functions. Manuscripta Math., 115(3):389-399, 2004. Stochastic integration and differential equations. P E Protter, Stochastic Modelling and Applied Probability. 21Springer-VerlagSecond edition. Version 2.1, Corrected third printingP. E. Protter. Stochastic integration and differential equations, volume 21 of Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2005. Second edition. Version 2.1, Corrected third printing. The growth of random walks and Lévy processes. W E Pruitt, Ann. Probab. 96W. E. Pruitt. The growth of random walks and Lévy processes. Ann. Probab., 9(6):948-956, 1981. Nonlocal elliptic equations in bounded domains: a survey. X Ros-Oton, Publ. Mat. 601X. Ros-Oton. Nonlocal elliptic equations in bounded domains: a survey. Publ. Mat., 60(1):3-26, 2016. The Dirichlet problem for nonlocal Lévy-type operators. A Rutkowski, Publ. Mat. 621A. Rutkowski. The Dirichlet problem for nonlocal Lévy-type operators. Publ. Mat., 62(1):213-251, 2018. Lévy processes and infinitely divisible distributions. K Sato, Cambridge Studies in Advanced Mathematics. 68Cambridge University PressTranslated from the 1990 Japanese original, Revised by the authorK. Sato. Lévy processes and infinitely divisible distributions, volume 68 of Cambridge Studies in Advanced Math- ematics. Cambridge University Press, Cambridge, 1999. Translated from the 1990 Japanese original, Revised by the author. Mountain pass solutions for non-local elliptic operators. R Servadei, E Valdinoci, J. Math. Anal. Appl. 3892R. Servadei and E. Valdinoci. Mountain pass solutions for non-local elliptic operators. J. Math. Anal. Appl., 389(2):887-898, 2012. Variational methods for non-local operators of elliptic type. R Servadei, E Valdinoci, Discrete Contin. Dyn. Syst. 335R. Servadei and E. Valdinoci. Variational methods for non-local operators of elliptic type. Discrete Contin. Dyn. Syst., 33(5):2105-2137, 2013. On the Lp-theory of C0-semigroups associated with second-order elliptic operators. Z Sobol, H Vogt, I. J. Funct. Anal. 1931Z. Sobol and H. Vogt. On the Lp-theory of C0-semigroups associated with second-order elliptic operators. I. J. Funct. Anal., 193(1):24-54, 2002. Upper and lower bounds for the Bregman divergence. B Sprung, J. Inequal. Appl. 124B. Sprung. Upper and lower bounds for the Bregman divergence. J. Inequal. Appl., 12: paper no. 4, 2019. A probabilistic approach to a non-local quadratic form and its connection to the Neumann boundary condition problem. Z Vondraček, Math. Nachr. 2941Z. Vondraček. A probabilistic approach to a non-local quadratic form and its connection to the Neumann boundary condition problem. Math. Nachr., 294(1):177-194, 2021. Φ-entropy inequality and application for SDEs with jumps. F.-Y. Wang, J. Math. Anal. Appl. 4182F.-Y. Wang. Φ-entropy inequality and application for SDEs with jumps. J. Math. Anal. Appl., 418(2):861-873, 2014. Nonlinear functional analysis and its applications. II/B. E Zeidler ; Leo, F Boron, Springer-VerlagNew YorkNonlinear monotone operatorsE. Zeidler. Nonlinear functional analysis and its applications. II/B. Springer-Verlag, New York, 1990. Nonlinear monotone operators, Translated from the German by the author and Leo F. Boron. Boundary regularity for quasiminima. W P Ziemer, Faculty of Pure and Applied Mathematics. 924Wroc law University of Science and Technology ; Wroc law University of Science and TechnologyWyb. Wyspiańskiego. Wroc law, Poland. Email address: [email protected]. P. Ziemer. Boundary regularity for quasiminima. Arch. Rational Mech. Anal., 92(4):371-382, 1986. Faculty of Pure and Applied Mathematics, Wroc law University of Science and Technology, Wyb. Wyspiańskiego 27, 50-370 Wroc law, Poland. Email address: [email protected] Faculty of Pure and Applied Mathematics, Wroc law University of Science and Technology, Wyb. Wyspiańskiego 27, 50-370 Wroc law, Poland. Email address: [email protected]
{'fraction_non_alphanumeric': 0.12078263527973097, 'fraction_numerical': 0.03481504127178233, 'mean_word_length': 3.139928112185491, 'pattern_counts': {'":': 0, '<': 77, '<?xml version=': 0, '>': 62, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 3, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 42, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'We give Hardy-Stein and Douglas identities for nonlinear nonlocal Sobolev-Bregman integral forms with unimodal Lévy measures. We prove that the corresponding Poisson integral defines an extension operator for the Sobolev-Bregman spaces. As an application, we obtain the boundedness of the Dirichlet-to-Neumann operator on weighted L p spaces. We also show that the Poisson integrals are quasiminimizers of the Sobolev-Bregman forms.', 'arxivid': '2006.01932', 'author': ['Krzysztof Bogdan ', 'Tomasz Grzywny ', 'Katarzyna Pietruska-Pa ', 'ANDArtur Luba ', 'Rutkowski '], 'authoraffiliation': [], 'corpusid': 219260609, 'doi': '10.1007/s00526-023-02458-x', 'github_urls': [], 'n_tokens_mistral': 32886, 'n_tokens_neox': 29565, 'n_words': 16319, 'pdfsha': '53649cc84554adf7a08395a62048725cc1a5201f', 'pdfurls': ['https://export.arxiv.org/pdf/2006.01932v2.pdf'], 'title': ['NONLINEAR NONLOCAL DOUGLAS IDENTITY', 'NONLINEAR NONLOCAL DOUGLAS IDENTITY'], 'venue': []}
arxiv
OPTIMAL POINT SETS DETERMINING FEW DISTINCT ANGLES Henry L Fleischmann Steven J Miller Eyvindur A Palsson Ethan Pesikoff Charles Wolf OPTIMAL POINT SETS DETERMINING FEW DISTINCT ANGLES We characterize the largest point sets in the plane which define at most 1, 2, and 3 angles. For P (k) the largest size of a point set admitting at most k angles, we prove P (2) = 5 and P (3) = 5. We also provide the general bounds of k + 2 ≤ P (k) ≤ 6k, although the upper bound may be improved pending progress toward the Weak Dirac Conjecture. Notably, it is surprising that P (k) = Θ(k) since, in the distance setting, the best known upper bound on the analogous quantity is quadratic and no lower bound is well-understood. 1. Introduction 1 1.1. Background 1 1.2. Definitions and Results 2 2. General Bounds 3 3. Computing P (2) = 5 4 3.1. There is an equilateral triangle 4 3.2. There is no equilateral triangle 5 4. Computing P (3) = 5. 7 4.1. There is a scalene triangle. 7 4.2. All triangles are isosceles with at least one with base angle larger than vertex angle. 10 4.3. All triangles are isosceles with the base angle at most the vertex angle with at least one non-equilateral triangle. 14 4.4. There are only equilateral triangles. 18 5. Future Work 18 References 19 1. INTRODUCTION 1.1. Background. In 1946, Erdős introduced the problem of finding asymptotic bounds on the minimum number of distinct distances among sets of n points in the plane [Er]. The Erdős distance problem, as it has become known, proved infamously difficult and was only finally (essentially) resolved by Guth and Katz in 2015 [GuKa]. The Erdős distance problem has also spawned a wide variety of related questions, including the problem of finding maximal point sets with at most k distinct distances. Characterizing the largest possible point sets satisfying a given property in this way is a classic problem in discrete geometry. As another example, Erdős introduced the problem of finding maximal point sets of all isosceles triangles in 1947 [ErKe]. Ionin completely answers this question in Euclidean space of dimension at most 7 [Io]. Erdős and Fishburn determine maximal planar sets with at most k distinct distances [ErFi]. Recent results by Szöllősi and Östergård classify the maximal 3-distance sets in R 4 , 4-distance sets in R 3 , and 6-distance sets in R 2 [SzOs]. In [ELMP, BrDePaSe, BrDePaSt] point sets with a low number of distinct triangles in Euclidean space are investigated. Along these lines, we consider the related problem of maximal planar point sets admitting at most k distinct angles in (0, π). We ignore angles of 0 and π so as to align our convention with related research (see [PaSha92], for example), although we provide results including the 0 angle as corollaries. We completely answer this question for k = 2, and k = 3 and provide asymptotically tight linear bounds for k > 3. In answering this question for k = 2 and k = 3, we consider systematically consider all possible triangles in such configurations and then reduce to adding points in a finite number of positions by geometric casework. We provide linear asymptotic bounds using bounds on the related problem of the minimum number of distinct angles among n non-collinear points in the plane. 1.2. Definitions and Results. By convention, we only count angles of magnitude strictly between 0 and π. Our computations still answer the related optimal point configuration questions including 0 angles (see Corollaries 3.1, 4.1). We begin by introducing convenient notation: Definition 1.1. Let P ⊂ R 2 . Then A(P) := #{|∠abc| ∈ (0, π) : a, b, c distinct, a, b, c ∈ P}, Now we define the quantity we are interested in studying. Definition 1.2. P (k) := max{#P : P ⊆ R 2 , not all points in P are collinear, A(P) ≤ k}. We first provide general linear lower and upper bounds for P (k). In particular, we have the following theorem. Theorem 1.3. For all k ≥ 1, 2k + 3 ≤ P (2k) ≤ 12k 2k + 3 ≤ P (2k + 1) ≤ 12k + 6. In the distance setting, the best known upper bound on the analogous parameter is the quadratic (2 + k)(1 + k), and no lower bound is well-understood [SzOs]. It is therefore interesting and surprising that we find P (k) = Θ(n) in the angle setting. We prove Theorem 1.3 in Section 2. Furthermore, we explicitly compute P (1), P (2), and P (3) and exhaustively identify all maximal point configurations for each. Proposition 1.4. We have P (1) = 3, and the equilateral triangle is the unique maximal configuration. In order to have only a single angle, every triangle of three points in the configuration must be equilateral. As this is impossible for point configurations that are not the vertices of an equilateral triangle, P (1) = 3. P (2) and P (3) are considerably less trivial quantities. We calculate P (2), P (3) via exhaustive casework, simultaneously characterizing all of the unique optimal point configurations up to rigid motion transformations and dilation about the center of the configuration. We proceed by first considering sets of three points and then search for what additional points may be added without determining too many angles. We prove Theorem 1.5 in Section 3 and Theorem 1.6 in Section 4. Theorem 1.5. We have P (2) = 5. Moreover, the unique optimal point configuration is four vertices in a square with a fifth point at its center (see A in Figure 1). Theorem 1.6. We have P (3) = 5. There are 5 unique optimal configurations, shown in Figure 1. GENERAL BOUNDS Although one may in principle calculate P (k) for any k by extensive casework (as we later calculate P (2), P (3)), it quickly becomes overwhelming. As such, we instead provide general bounds on P (2k) and P (2k +1). Note that the construction of a square with a point in the center is no accident in the case of P (2). Indeed, adding a point in the center of a regular 2k-gon introduces no additional angles. This is because, in a regular 2k-gon, the line from the center to any boundary vertex intersects an additional vertex on the other side. As such, the only additional angles that may be added from the center point are those with the center point as the center of the angle. For the other angles including it as an endpoint, choosing the point on the other end of the line through the center gives an equal angle. Moreover, the angles formed with the center point as the center of the angle are precisely iπ/k for 1 ≤ i ≤ k − 1, which are already achieved among the other points of the regular 2k-gon. So, using the regular (2k + 2)-gon with a point added in the center yields the following lemma. Lemma 2.1. We have P (2k) ≥ 2k + 3. Moreover, in the case of P (2k + 1), the regular (2k + 3)-gon and the projection of a regular (2k + 3)-gon onto a line (via a stereographic-like projection from a cap vertex) both achieve 2k + 1 angles, providing a bound on P (2k + 1). Lemma 2.2. We have P (2k + 1) ≥ 2k + 3. Proposition 2.3. If we wish to also count the 0-angle, then we may not add the center to an even polygon, and in general we reach a bound of P (k) ≥ k + 2. We conjecture that both of these lower bounds are tight in general. Nonetheless, we provide a linear upper bound. We achieve this bound as a corollary of a lower bound on the number of distinct angles, using progress on the Weak Dirac Conjecture. In 1961, Erdős [Er] conjectured the following, based on an earlier, more difficult conjecture of Dirac: Conjecture 2.4 (Erdős, 1961). Every set P of n non-collinear points in the plane contains a point incident to at least n/2 lines of L(P), where L(P) is the set of lines formed by points in P. While Dirac's conjecture has not been proven, significant progress has been made. Let l(n) be the largest proven lower bound proven for Dirac's conjecture. I.e., every set P of n non-collinear points in the plane contains a point incident to at least l(n) lines of L(P). We have l(n) ≥ n/3 + 1 from [Ha]. Let A(n) be the minimum number of distinct angles among n points in the plane. We have the following lemma. Lemma 2.5. For n > 3, A(n) ≥ l(n)−1 2 ≥ n/6. Proof. Fix a set P of n non-collinear points in the plane. Let p be a point in P incident to at least (n) lines of L(P). Fix a point q = p in P. It shares exactly one line with p. Note that for a fixed nonzero angle θ < π, there are exactly two possible lines which r must be on in order for ∠qpr = θ. As such, since p is incident to (n) − 1 lines without q, p is the center angle of at least ( (n) − 1)/2 distinct angles. Therefore A(n) ≥ (n) − 1 2 . We have (n) ≥ n/3 + 1 from [Ha]. As such, we have A(n) ≥ n/6, as desired. Note that such a use of the Weak Dirac Conjecture is known. See [BMP], Section 6.2. Corollary 2.6. We have P (k) ≤ 6k. Proof. Since A(n) ≥ n/6 by Lemma 2.5, then P (k) ≤ 6k as point configurations with at least 6k +1 points define at least k + 1 angles. 3. COMPUTING P (2) = 5 Proof. In any point configuration with at least three points, there are triangles. For any point configuration with at most two angles, all triangles must be isosceles. We divide into two cases, based on whether or not there is an equilateral triangle. 3.1. There is an equilateral triangle. We consider adding a fourth point in cases ( Figure 2). FIGURE 2. Equilateral Triangle Regions Case 1: p ∈ A. Then ∠acp < π/3 and ∠cap > π/3, leading to more than two angles. Case 2: p ∈ ab. Then ∠bcp < π/3 and one of ∠cpb and ∠apc ≥ π/2, leading to more than two angles. Case 3: p ∈ ac to the upper-right of a. Then ∠cbp > π/3 and ∠cpb < π/3, again leading to more than two angles. Case 4: p ∈ B. In this case, ∠cbp > π/3 and ∠cpb < π/3, leading to more than two angles. Case 5: p ∈ abc. In this case, one of ∠apb, ∠bpc, ∠cpa ≥ 2π/3 and ∠acp < π/3, leading to more than two angles. Up to symmetry, these cases are exhaustive. Thus if there is an equilateral triangle in the configuration, there can only be at most three points. 3.2. There is no equilateral triangle. Now, let a, b, and c be the vertices of an isosceles triangle with vertex angle α, base angle β, and a the apex vertex. We reduce the number of possibilities for additional points by partitioning the plane into regions A i (Figure 3). Note that we may without loss of generality assume that no fourth point is added within abc as we could then choose that triangle as our initial triangle. Also note that A 1 and A 1 and A 3 and A 3 are equivalent up to symmetry. Case 1: p ∈ A 1 . In this case, ∠pab > α and ∠pcb > β. So, regardless of whether α or β is greater, adding p introduces an additional angle. So, no additional points can be in A 1 or A 1 . Case 2: p ∈ A 2 . In this case, ∠pcb and ∠pbc are greater than β, so both must be α to not add additional angles. But then ∠cpb = π − 2α = β, in order to not add angles, implying 3α = π. But, this implies pcb is an equilateral triangle. Thus no points may be added in this case. Case 3: p ∈ A 3 (or A 3 by symmetry). In this case, ∠bap > α and ∠abp > β, so there is an additional angle added regardless and no additional points are possible. Case 4: p ∈ A 4 . In this case, ∠cap, ∠bap < α, so both must equal β. Therefore, 2β = α, which implies β = π/4 and α = π/2. Moreover, since ∠acp and ∠abp are greater than β, they must both equal α = π/2. So, the only possibility for an addable point in this case is for p to be the fourth vertex of the square acpb. Case 5: p ∈ bc . If p is on bc between b and c, then ∠cap, ∠bap < α. In order for these not to introduce additional angles, they must both be equal to β. This implies β = π/4 and α = π/2 and p is the center of the side bc. If p ∈ bc to the left of c (or by symmetry, right of b), ∠bap > α and thus ∠bap = β. Since 2β + α = π, β < π/2. But then ∠acp > π/2 > β > α. Thus there is exactly one point possible on line bc , the centerpoint of the edge between b and c. Case 6: p ∈ ac (or p ∈ ab ). If p is between a and c, then ∠cbp < β and thus ∠cbp = α. But, as before, β < π/2. Moreover, one of ∠bpc or ∠bpa is at least π/2 > β > α. Thus there are too many angles in this case. If p is to the bottom left of c, ∠apb < β and thus ∠apb = α. But, again, either ∠bca or ∠bcp > π/2 > β, creating too many angles in this case. If p is on ac to the upper right of a, ∠pbc > β and thus equals α. Then ∠pba < α and must equal β and thus 2β = α. This implies β = π/4 and α = π/2 and cbp is an isosceles right triangle with b the apex vertex, p on ac to the upper right of a, and a at the center of side pc. As such, in order to add additional points to an isosceles triangle point configuration without adding additional angles, we must have α = π/2 and β = π/4. The four additional possible points are marked in Figure 4. Note that ∠x 4 ax 1 , ∠x 4 ax 2 > π/2. So, x 4 cannot be in the same point configuration as x 1 or x 2 . By symmetry the same follows for x 3 . However, we may have both x 1 and x 2 or both x 3 and x 4 , either of which give the unique optimal configuration A in Figure 1. Corollary 3.1. One might also wish to include the trivial 0-angle in our count. In this case, P (2) = 4, and the unique configuration is the square. Proof. The only 5-point configuration no longer holds when we count the 0-angle. Figure 4 displays all valid four point configurations which define only 2 angles excluding 0, as detailed in the proof of P (2). All the shown points but x 2 define a 0-angle, so the only valid 4 point configuration is the square. COMPUTING P (3) = 5. In this section we prove the surprising result that P (3) = 5. That is, adding an additional allowable distinct angle from two to three does not increase the maximum number of points in an optimal point configuration. Proof. We divide the casework for this section into four parts based on the triangles exhibited in the point configuration: (1) There is a scalene triangle. (2) All triangles are isosceles with at least one with base angle larger than vertex angle. (3) All triangles are isosceles with the base angle at most the vertex angle with at least one nonequilateral triangle. (4) All triangles are equilateral. 4.1. There is a scalene triangle. Let a, b, and c be the vertices of a scalene triangle in the configuration. We without loss of generality assume α < β < γ ( Figure 5). As in the proof of Theorem 1.5, we begin by reducing the number of possible points to a finite number by region-based casework. Case 1: p ∈ A 1 . As ∠bap < α, no points may be added in A 1 . Case 2: p ∈ A 2 . In this case, ∠abp > β and thus must equal γ. Moreover, ∠bap > α. If ∠bap = γ, ∠bpa < α. Thus ∠bap = β, which implies ∠bpa = α. But, ∠bpc < ∠bpa = α, so we define a fourth angle. Therefore there cannot be points added in A 2 . Case 3, Case 4: p ∈ A 3 ∪ A 4 . As ∠bcp > γ, no points may be added in A 4 . Case 5: p ∈ A 5 . In this case, ∠cbp > β and thus ∠cbp = γ. Moreover, ∠cap > α and is thus β or γ (which implies ∠pab = α or β, respectively). If ∠cap = β, we have that ∠pab < β and thus is equal to α. So, β = 2α. Then ∠abp cannot be α because then ∠apb > γ. So, ∠abp = β. So, 2α = β and 2β = γ, so the angles are π/7, 2π/7, and 4π/7. We then have Figure 6. FIGURE 6. Four Point Kite Configuration. α = π 7 , β = 2α, γ = 4α. Alternatively, we have ∠pab = β and thus ∠cap = γ. Then γ = β + α, so ∠abp = α. γ = β + α and α + β + γ = π implies γ = π/2. I.e., a, b, c, p are the vertices of a rectangle. As such, we reach Figure 7. Therefore, there are exactly two possible points to add in A 5 , with each choice exactly determining the angles α, β, and γ. Case 6: p ∈ A 6 . As ∠acp > γ, no points may be added in A 6 . Case 7: p ∈ ab . If p is to the right of b, then ∠acp > γ. Note that β, α < π/2. Then, if p is to the left of a, ∠pac > π/2 and must equal γ. But then α and γ are supplementary, implying β = 0. Finally, if p is between a and b, one of ∠cpa and ∠cpb is at least π/2. So, γ ≥ π/2. Moreover, as neither α nor β can be supplementary to γ, we have that the supplement of γ is γ, and hence γ = π/2. This yields a diagram like Figure 8. So, exactly one point may be added on ab and it forces γ = π/2. Case 8: p ∈ bc . If p is between b and c then ∠bap < α. Recall that β, α < π/2. Then, if p is not on bc and is closest to b, ∠abp > π/2 and thus must equal γ. So, γ + β = π. But, this implies α = 0. Finally, if p is not on bc and is closest to c, then ∠acp is supplementary to γ. Since neither β nor α can supplementary to γ be without implying the other is 0, this implies γ = π/2. We then have the configurations in Figure 9 since ∠cap = β or α. So, exactly two points can be added on bc and both force γ = π/2. Case 9: p ∈ ac . If p is between a and c, then one of ∠bpa and ∠bpc are at least π/2 and must thus be γ. Since neither α nor β can be supplementary to γ, this implies γ = π/2. However, ∠abp < β and thus must be α. This then yields 2α = π/2 from abp, contradicting α + β = π/2 since α = β. If p is left of a, we have ∠pab > π/2 and thus must be γ. But, this implies γ and α are supplementary. If p is right of c, then ∠bcp is supplementary to γ. Since neither α nor β can be, this implies γ = π/2. This leads to the allowable point configuration in Figure 10. So, exactly one point may be added in this case, with γ = π/2 being forced. Case 10: p in the interior of abc. In this case ∠pac < α, we no points may be added. It now remains to show that all six addable points are mutually incompatible. Suppose we add p ∈ A 5 as in the first case of a kite ( Figure 6). As γ = π/2 as in all the other cases, no additional points may be added. Where γ = π/2, we have the five point placements to consider (presented in Figure 11). Suppose we add x 1 . Adding x 2 yields an angle ∠x 1 x 2 b > π/2 or ∠x 1 x 2 a > π/2 (the diagonals cannot intersect at right angles lest α = β). Adding x 3 adds ∠x 1 ax 3 > π/2, and similarly for x 4 . Adding x 5 adds ∠x 1 bx 5 > π/2. Suppose we add x 2 . Adding x 3 yields ∠x 2 cx 3 > π/2, and similarly for x 4 . Adding x 5 adds ∠ax 2 x 5 > π/2. Suppose we add x 3 . Adding x 4 creates ∠ax 3 x 4 > π/2. Adding x 5 forces ∠ax 3 x 5 = ∠bx 5 x 3 = ∠abx 5 = π/2. But ∠bax 3 = β < π/2, so the angles in abx 5 x 3 do not add to 2π. Finally, suppose we add x 4 and x 5 . In this case, ∠cx 4 x 5 < ∠cx 3 x 5 = α. So, if there is a scalene triangle in the point configuration, there can be at most four points. 4.2. All triangles are isosceles with at least one with base angle larger than vertex angle. Let a, b, and c be the vertices of an isosceles triangle with the base angle larger than the vertex angle ( Figure 12). Specifically, α = ∠cba < ∠abc = ∠acb = β. Case 1: p ∈ A 1 . Note that ∠bcp > β > α. Let this new angle be γ. Then, since only three angles are admissible and since ∠pca + ∠acb = γ, then ∠pca = α or ∠pca = β. Suppose ∠pca = β. Then ∠pcb = γ = 2β. Additionally, since ∠pbc < β, ∠pbc = α. Since 2β + α = π, then ∠bpc < γ lest the angles in pbc be too large. Then as pbc must be isosceles, ∠bpc = α. Thus the angles in bpc sum to γ + 2α = 2β + 2α > 2β + α = π, a contradiction. Thus ∠pca = α. In this case, γ = α + β. Observe that ∠pbc < β, and so ∠pbc = α. Similarly, ∠pba = α, thus giving ∠bca = β = 2α. Then γ = α + β = 3α. The angles in abc must add to π, so 2β + α = 5α = π, which implies α = π/5. Moreover, as pbc must be isosceles, ∠cpb = α. Then as ∠bpa < ∠cpa, we have ∠bpa = α or ∠bpa = β, both of which determine ∠pac. Thus in this case we have two points in A 1 which are contenders to give acceptable four-point configurations ( Figure 13). Consider further the case where ∠bpa = α. We have pabc is a parallelogram. We have deduced above that pb bisects ∠abc, but this is only true if abcp is a rhombus. However, this would mean that abc is equilateral, since then |ab| = |ac| = |bc|, contradicting our original assumption that α < β. Thus this case is impossible and ∠bpa = β. So, there is exactly one point we may add in A 1 (thus forming the left configuration of Figure 13), and, by symmetry, an additional one point in A 1 . Case 2: p ∈ A 2 . Note that ∠bcp > ∠bca = β, and ∠cpb < ∠cab = α. Thus no points may be added in A 2 . Case 3: p ∈ A 3 . If p ∈ A 3 , then γ = ∠abp > β. Then, as ∠pab > α, we have ∠apb < β and thus ∠apb = α so that we may still have only three angles. But then ∠apc < ∠apb = α, yielding a fourth angle. Thus no new angles may be added in A 3 or A 3 . Case 4: p ∈ A 4 . In this case, ∠pac < ∠bac = α < β = ∠bca < ∠pca. Thus there are already four angles, and no points can be added in A 4 . Case 5: p inside abc. In this case, ∠pab, ∠pac < ∠cab = α, so let ∠pab = ∠pac = γ. As abp must be isosceles and γ < α, γ cannot be the vertex angle of abp. Since γ < α < β, this implies the vertex angle of abp must be β. So, 2γ = α and 2γ + β = π. But, this is a contradiction as α + 2β = π. No points are addable in this case. Case 6: p ∈ bc . First, assume p is between b and c; that is, p is located on the base of abc. Then, ∠pab < α and one of ∠pba or ∠cpa ≥ π/2 > β. Thus no points can be added on the base of abc. Now suppose that p is not between b and c. By symmetry, we may assume that p is on the left (i.e., it is closer to c than to b). Now, since β < π/2, ∠acp = γ > π/2. As acp must be isosceles, γ cannot be a base angle, and γ > α, we have ∠cpa = ∠pac = α. This implies 2α + γ = 2β + α and α + γ = 2β. Along with π − γ = β, this yields α = π/5, β = 2α, γ = 3α. So, two points are addable in this case (one on either side of the edge bc). See Figure 14. Case 7: p ∈ ac or p ∈ ab . By symmetry, we may assume p ∈ ac . First, assume p is between a and c; that is, p is located on a leg of abc. In this case, one of ∠apb and ∠bpc ≥ π/2 > β. Let this angle be γ. As ∠cbp < β and thus must equal α, since cbp must be isosceles, we have ∠bpc = β. So, ∠bpa = γ and β + γ = π. Moreover, we have γ + 2α = π. This again implies α = π/5, β = 2α, γ = 3α. This is a legal configuration. Next, assume that p ∈ ac is not on the triangle's side, and that it is closer to a than to c. In this case, ∠bap = π − α > β and ∠cpb < ∠cab = α. Thus no points may be added in this case. Now, assume that p ∈ ac is not on the triangle's side, and that it is closer to c than to a. Then we have a new angle γ = ∠pba > ∠cba = β. We also have ∠pcb = π − β > β because β < π/2. So, to maintain only three angles, we must have ∠pcb = ∠pba = γ. Then, as cbp must be isosceles and ∠bcp must be the vertex angle, ∠cbp = ∠cpb = α. As before, we conclude that α = π/5, β = 2π/5, γ = 3π/5. This is a legal configuration. So, in this case, all our angles are exactly determined and there are two points addable on ac , and, by symmetry, two points addable on ab . So there are only eight addable points in the case of there being an isosceles triangle with vertex angle smaller than base angle (Figure 15). We examine each of these points up to symmetry and examine which are compatible. We cannot add x 4 as ∠x 4 x 1 a > ∠cx 1 a = γ. Additionally, note that ∠bx 1 x 4 , ∠bx 1 x 2 , ∠ax 3 x 1 < α and thus none of x 4 , x 3 , or x 2 may be added. Each of x 1 , x 2 , and x 3 are individually compatible with x 1 , leading to three valid five point configurations including x 1 (see B, D, E of Figure 1). By symmetry, x 1 is then individually compatible with x 1 , x 2 and x 3 . Combination Case 2: Including x 2 . Note that ∠x 2 x 3 c, ∠x 2 x 3 c, ax 4 x 2 < α. So none of x 3 , x 3 , or x 4 are addable in this case. Adding x 2 is analogous to adding both x 1 and x 3 to abc, so x 2 is addable. And x 4 is addable as the projection of a regular pentagon onto a line via one of its vertices. So, there are three valid five point configurations including x 2 (see E, D, C of Figure 1). By symmetry, x 2 is compatible with exactly x 1 , x 2 and x 4 . Combination Case 3: Including x 3 . In this case, ∠cx 4 x 3 < α, so x 4 is not addable in this case. Adding x 3 creates a projected pentagon, and adding x 4 creates the construction of a trapezoid with a point in the middle, and both are individually compatible with x 3 . So, each of x 1 , x 3 , and x 4 may be individually added alongside x 3 , leading to three valid five point configurations including x 3 (see E, C, D of Figure 1). By symmetry, x 3 is compatible with x 1 , x 3 , and x 4 . Combination Case 4: Including x 4 . x 4 is compatible with x 4 . In combination with the above casework, we have x 4 is individually compatible with exactly x 2 , x 3 , and x 4 . So, there are three valid five point configurations including x 4 (see C, D, E of Figure 1). By symmetry, x 4 is compatible with exactly x 2 , x 3 , and x 4 . At this point, we have exhaustively identified all our five point configuration for this case of α < β. From our casework, we see there are no compatible, addable points which share an additional compatible point. Therefore, there are most five points in this case, with C, E, and F of Figure 1 as the possible five point configurations. 4.3. All triangles are isosceles with the base angle at most the vertex angle with at least one nonequilateral triangle. As before, we proceed by region-based casework. Fortunately, whenever we encounter a scalene triangle or an isosceles triangle with the vertex angle larger than the base angle, we reduce to the previous cases. Our diagram for this section is Figure 16, where β < α. Case 1: p ∈ A 1 . In this case, ∠bpc < α since ∠pcb and ∠pbc > β. We then have two cases. If ∠bpc = β, then ∠pcb and ∠pbc cannot both be α (as 2α + β > π). Moreover, cpb must be isosceles and neither ∠pcb nor ∠pbc can be β, so both must be γ. This implies 2γ + β = 2β + α and β < γ < α. But, ∠bpa < β < γ, creating more than 3 angles. If ∠bpc = γ < α, then, since ∠cpa, ∠bpa < γ, both must be β. So, γ = 2β. Now, pcb must be isosceles. Since α + 2β = α + γ = π, this implies both ∠pbc and ∠pcb must be γ. But, then γ = π/3, β = π/6, and α = 2π/3 ( Figure 17). As such, there is exactly one point addable in this case and it forces the choice of α, β, and γ. Case 2: p ∈ A 2 (or A 2 ). In this case, ∠bcp < β and ∠cap > α, so no points may be added. Case 3: p ∈ A 3 . As both ∠cap, ∠bap < α, we have three cases: (1) ∠cap = β = ∠bap (2) ∠cap = β, ∠bap = γ = β (and swapping ∠cap and ∠bap by symmetry), and (3) ∠bap, ∠cap = γ = β Case (1) implies α = 2β, and hence β = π/4, α = π/2. Moreover, bcp must be isosceles and must include one of α or β. Note bcp cannot be equilateral as then ∠acp = 7π/12, and we have four angles. Note that π/2 cannot be a base angle in bcp and β being the vertex angle reduces to the previous casework in section 4.2. Thus the only option is ∠cbp = ∠bcp = π/4 and ∠bpc = π/2, yielding a valid configuration, the square abpc. Case (2) implies β + γ = α. This is illustrated in the Figure 18. First, we show that γ > β. If γ < β, then α = ∠abp > β, which implies ∠cbp = γ. We similarly have ∠bcp = γ. But then, ∠bpc = π − 2γ > α, yielding four angles γ < β < α < π − 2γ. Thus γ > β. We then have two sub-subcases to consider: (2i) ∠abp = α. (2ii) ∠abp = γ = α. In case (2i), note that ∠cbp = γ. Since abp must be isosceles with largest angle the vertex angle, ∠apb = γ. Then, we have α + 2γ = α + 2β, contradicting the assumption of case (2) that γ = β. In case (2ii) we have β < γ < α. Since ∠cbp < γ, it must equal β. So, γ = 2β. This implies α = 3β. Thus β = π/5, γ = 2π/5, and α = 3π/5. Since acp must be isosceles with smaller base angle, ∠apc = β. Completing abp, we have ∠apb = β. But then abp is isosceles with smaller vertex angle. Thus this case reduces to the previous casework. In case (3), we have 2γ = α. Now, ∠cbp cannot be γ since β + γ = α. As γ < α, it cannot be α either. Thus it must be β. Since β = γ, we must then have 2β = γ. So, we have α = 4β. Thus β = π/6, γ = π/3, α = 2π/3. This yields the valid point configuration in Figure 19. So, there are exactly two addable points p ∈ A 3 and each forces a choice of α, β, and γ. Case 4: p ∈ A 4 (or A 4 ). In this case, ∠cap > α and ∠apc < β. Thus no points are addable in this case. Case 5: p ∈ ac (or ab by symmetry). If p is to the left of a, ∠pab = π − α = β. We then have two subcases: (1) ∠pab = α (2) ∠pab = γ = α FIGURE 19. Four Point Configuration from Case 3. In case (1), ∠pab = α implies α = π/2 and β = π/4. As pab must be isosceles, ∠bpa = ∠pba = π/4. So, we get the valid configuration in Figure 20. In case (2), we have ∠pab = γ = π − α > β. Note pbc needs to be isosceles with the vertex angle at least as large as the base angle, and that ∠pcb = β. Thus since ∠pbc > β, ∠pbc = α and ∠bpc = β. Then ∠abp = γ, since pab must be isosceles and ∠pab = α. So, 2γ + β = π, 2β + α = π, and α + γ = π. This implies β = π/5, γ = 2π/5, and α = 3π/5. But then pba has angles 2π/5, 2π/5, π/5, and so the vertex angle is smaller than the base angles. Thus this case reduces to the previous section. Now, if p is between a and c then γ = ∠abp = ∠cbp < β. Furthermore, ∠bpc > α, giving four angles, so no points can be added in this subcase. Finally, if p is to the bottom right of c then γ = ∠bcp = π − β > α. But, bcp must be isosceles with largest angle not repeated. Thus ∠cbp < β. So, exactly two points are addable in this case and they exactly determine α and β (the second by symmetry). Case 6: p ∈ bc . In this case, left of b is equivalent to right of c by symmetry. So, we without loss of generality consider p left of b. In this case, ∠pba = π − β > α. Then, pba must be isosceles with largest angle non-repeated. So, ∠pab < β. So, no points are addable in this subcase. It remains to consider p between b and c. In this case, acp and abp are isosceles triangles including β. However, β cannot be the vertex angle, so another of their angles must be β and the third α. Since ∠cap, ∠pab < α, we must then have 2β = α. Thus we have β = π/4 and α = π/2. The resulting valid point configuration is displayed in Figure 21. In this case, ∠pbc, ∠pcb < β and thus ∠cpb is greater than α. Thus no points are addable in this case. Combinations: Adding more than one point. Now we determine which of the six addable points are mutually compatible. As exactly two force α = 2π/3, β = π/6, and γ = π/3, those two (from Case 1 and Case 3) are at most compatible with each other. This is displayed in Figure 22. However, this adds an additional angle, ∠qcp = 3β = π/2. So, we only consider the addable points which force α = π/2 and β = π/4. Such addable points are shown in Figure 23. Both pairs x 1 , x 2 and x 3 , x 4 are compatible, and both yield a square with a point in the center (see A of Figure 1). FIGURE 23. Compatible Points with the Right Triangle However, x 3 is not compatible with either of x 1 , x 2 . For x 1 , then x 1 x 3 b is not isosceles, and similarly for x 2 . Also, x 4 is not compatible with either of x 1 , x 2 as then we have angles α, β, α + β, ∠bx 1 x 4 < β (or ∠bx 2 x 4 < β). Therefore, the only 5-point configurations are {a, b, c, x 1 , x 2 }, {a, b, c, x 3 , x 4 }. Therefore, only 5 points are allowed in this case, and the only acceptable 5-point configuration of the square with its center (see A in Figure 1). 4.4. There are only equilateral triangles. Since an equilateral triangle has all equal side lengths, every distance between two points in the configuration must be equal. That is, we need a 1-distance set. The largest 1-distance set in the plane is the equilateral triangle. Thus no configuration of four or more points can exist defining only equilateral triangles. The maximal number of points in this case is thus three. Then across all cases, we find that the largest configurations of points on the plane which define at most three angles contain exactly five points. As such, P (3) = 5, and the complete list of configurations is shown in Figure 1. Corollary 4.1. One might also wish to include the trivial 0-angle in our count. In this case, P (3) = 5, but the square with the center-point and the pentagon are now the only valid configurations. Proof. The set of valid five-point configurations when we count the 0-angle must be a subset of the valid five-point configurations we identified above. By direct inspection, the square with the center-point and the pentagon are the only of the five in Figure 1 which define only three angles. All the others define three angles greater than zero and also the 0-angle by collinearity. FUTURE WORK While it seems possible to compute P (k) by exhaustive casework for higher values of k, the casework quickly becomes overwhelming. Additionally, while it is potentially possible to repeat such methods in higher dimensions, the visualization of the proofs played a crucial role in this analysis. In combination with the added degrees of freedom from adding dimensions, this would make this method of computation quickly intolerable. Future work may tighten our upper bound on P (k). However, we make the following conjecture. Conjecture 5.1. The lower bound on P (k) in Theorem 1.3 is tight. Namely, P (2k) = 2k + 3 and P (2k + 1) = 2k + 3 for all k ≥ 1. Therefore, we believe that future work should improve the upper bound of P (n) ≤ 6n, either via progress towards the Weak Dirac Conjecture (which would still fall short of our conjecture) or by some other means. Alternatively, future research may find a more efficient method of constructing viable point sets without the need for the exhaustive search we perform. We propose the related problem of characterizing optimal point sets in higher dimensions with a low number of solid angles. Definition 5.2 (Solid Angles). Given d+1 points in R d , fix one of the points p. Let S be a unit d-dimensional hypersphere about p. Project the remaining d points onto the surface of the sphere along the lines connecting them to p. The solid angle formed by the d + 1 points with center p is the surface area of S enclosed by the projections of the other points onto S and connected via geodesics. Solid angles have applications to physics and have not been extensively studied in the context of discrete geometry. They provide an exciting new avenue for angle-related problems. They also motivate the following problem. For a fixed d ≥ 3, what is the maximum number of noncoplanar points in a configuration yielding at most k solid angles? FIGURE 1 . 1Optimal Two and Three Angle Configurations. α = π 5 , β = 2π 5 , γ = 3π 5 . FIGURE 3 . 3Isosceles Triangle Regions. FIGURE 4 . 4Compatible Points with the Right Triangle. FIGURE 5 . 5Scalene Triangle Regions. FIGURE 7 . 7Four Point Rectangular Configuration. FIGURE 8 . 8Four Point Configuration from Case 7. FIGURE 9 . 9Four Point Configurations from Case 8. FIGURE 10 . 10Four Point Configuration from Case 9.FIGURE 11. Compatible Points with the Right Triangle FIGURE 12 . 12Isosceles Triangle with Small Vertex Angle. FIGURE 13 . 13Configurations of Interest from Case 1. FIGURE 14 . 14Four Point Configuration from Case 6. FIGURE 15 . 15Compatible Points with the Isosceles Triangle with Small Vertex Angle Combination Case 1: Including x 1 . FIGURE 16 . 16Isosceles Triangle with Large Vertex Angle. FIGURE 17 . 17Four Point Configuration from Case 1. FIGURE 18 . 18Four Point Configuration from Case 3. FIGURE 20 . 20Four Point Configuration from Case 5. FIGURE 21 . 21Four Point Configuration from Case 6. So, there is a single addable point in this case and it forces the choice of α and β. Case 7: p in the interior of abc. FIGURE 22 . 22Attempting to Combine Cases 1 and 3. Research problems in discrete geometry. P Brass, W Moser, J Pach, Springer Science & Business MediaP. Brass, W. Moser, and J. Pach, Research problems in discrete geometry, Springer Science & Business Media, 2006. Uniqueness of optimal point sets determining two distinct triangles. H N Brenner, J S Depret-Guillaume, E A Palsson, S Senger, Integers. 2143H. N. Brenner, J. S. Depret-Guillaume, E. A. Palsson, S. Senger, Uniqueness of optimal point sets determining two distinct triangles,Integers 21 (2021), #A43. Characterizing optimal point sets determining one distinct triangle. H N Brenner, J S Depret-Guillaume, E A Palsson, R Stuckey, Involve: A Journal of Mathematics. 131H. N. Brenner, J. S. Depret-Guillaume, E. A. Palsson, and R. Stuckey, Characterizing optimal point sets determin- ing one distinct triangle, Involve: A Journal of Mathematics 13(1) (2020), 91-98. On Sets of Distances of n Points. P Erdős, The American Mathematical Monthly. 535P. Erdős, On Sets of Distances of n Points, The American Mathematical Monthly 53(5) (1946), 248-250. Maximal planar sets that determine k distances. P Erdős, P Fishburn, Discrete Mathematics. 1601-3P. Erdős and P. Fishburn, Maximal planar sets that determine k distances, Discrete Mathematics, 160(1-3), (1996), 115-125. . P Erdős, E 735L M Kelly, E 735The American Mathematical Monthly. 54P. Erdős and L.M. Kelly, E 735, The American Mathematical Monthly, 54 (1947), 227-229. On the Erdős distinct distances problem in the plane. L Guth, N Katz, Annals of Mathematics. 1811L. Guth and N. Katz, On the Erdős distinct distances problem in the plane, Annals of Mathematics 181(1) (2015), 155-190. A Note on the Weak Dirac Conjecture. Z Han, P1.63Electronic Journal of Combinatorics. 241Z. Han, A Note on the Weak Dirac Conjecture, Electronic Journal of Combinatorics 24(1) (2017), P1.63. Isosceles Sets. Y J Ionin, The Electronic Journal of Combinatorics. 161141Y. J. Ionin, Isosceles Sets, The Electronic Journal of Combinatorics, 16(1) (2009), R141. Some unsolved problems. P Erdős, Magyar Tud. Akad. Mat. Kut. Int. Közl. 6P. Erdős, Some unsolved problems, Magyar Tud. Akad. Mat. Kut. Int. Közl., 6 (1961), 221-254. A Epstein, A Lott, S J Miller, E Palsson, Optimal Point Sets Determining Few Distinct Triangles. 1816A. Epstein, A. Lott, S. J. Miller, and E. Palsson, Optimal Point Sets Determining Few Distinct Triangles, Integers 18 (2018), A16. Repeated angles in the plane and related problems. J Pach, M Sharir, Journal of Combinatorial Theory, Series A. 591J. Pach and M. Sharir, Repeated angles in the plane and related problems, Journal of Combinatorial Theory, Series A 59(1) (1992), 12-22. Constructions of maximum few-distance sets in Euclidean spaces. F Szöllősi, P Östergård, The Electronic Journal of Combinatorics. 271P1.23. Email address: [email protected]. Szöllősi and P. Östergård, Constructions of maximum few-distance sets in Euclidean spaces, The Electronic Journal of Combinatorics, 27(1) (2020), P1.23. Email address: [email protected]
{'fraction_non_alphanumeric': 0.0699531737773153, 'fraction_numerical': 0.027107180020811656, 'mean_word_length': 3.294124218051832, 'pattern_counts': {'":': 0, '<': 77, '<?xml version=': 0, '>': 64, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'We characterize the largest point sets in the plane which define at most 1, 2, and 3 angles. For P (k) the largest size of a point set admitting at most k angles, we prove P (2) = 5 and P (3) = 5. We also provide the general bounds of k + 2 ≤ P (k) ≤ 6k, although the upper bound may be improved pending progress toward the Weak Dirac Conjecture. Notably, it is surprising that P (k) = Θ(k) since, in the distance setting, the best known upper bound on the analogous quantity is quadratic and no lower bound is well-understood.', 'arxivid': '2108.12034', 'author': ['Henry L Fleischmann ', 'Steven J Miller ', 'Eyvindur A Palsson ', 'Ethan Pesikoff ', 'Charles Wolf '], 'authoraffiliation': [], 'corpusid': 44272809, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 13800, 'n_tokens_neox': 11644, 'n_words': 7331, 'pdfsha': '24dd9fd75d9601ec251579cd661fb931176c67b3', 'pdfurls': ['https://arxiv.org/pdf/2108.12034v1.pdf'], 'title': ['OPTIMAL POINT SETS DETERMINING FEW DISTINCT ANGLES', 'OPTIMAL POINT SETS DETERMINING FEW DISTINCT ANGLES'], 'venue': []}
arxiv
Novel relaxation time approximation: a consistent calculation of transport coefficients with QCD-inspired relaxation times * Gabriel S Rocha Instituto de Física Instituto de Física Gleb Wataghin Universidade Federal Fluminense Niterói, Rio de JaneiroBrazil Gabriel S Denicol Instituto de Física Instituto de Física Gleb Wataghin Universidade Federal Fluminense Niterói, Rio de JaneiroBrazil Maurício N Ferreira Illinois Center for Advanced Studies of the Universe & Department of Physics Universidade Estadual de Campinas Campinas, São PauloBrazil Jorge Noronha University of Illinois Urbana-Champaign 61801UrbanaILUSA Novel relaxation time approximation: a consistent calculation of transport coefficients with QCD-inspired relaxation times * Received July 26, 2022 We use a novel formulation of the relaxation time approximation to consistently calculate the bulk and shear viscosity coefficients using QCDinspired energy-dependent relaxation times and phenomenological thermal masses obtained from fits to lattice QCD thermodynamics. The matching conditions are conveniently chosen to simplify the computations. Introduction Nuclear matter in extreme conditions can be investigated through ultrarelativistic heavy-ion collisions. In particular, obtaining the transport coefficients of the quark-gluon plasma, throughout the QCD phase diagram, is a very challenging task that is currently beyond the reach of first-principles techniques [1]. In this contribution, we compute the transport coefficients of an effective kinetic model [2,3] with a temperature-dependent mass whose equation of state mimics lattice QCD thermodynamics [4]. We use the new relaxation time approximation (RTA) of the relativistic Boltzmann equation proposed in [5] and impose alternative matching conditions such that the interaction energy [6] depends only on the temperature even out of equilibrium. The quasi-particle model The relativistic Boltzmann equation for quasi-particles with a temperaturedependent mass, M (T ), is given by [7], p µ ∂ µ f p + 1 2 ∂ i M 2 (T )∂ i (p) f p = C [f p ] ,(1) where f p = f (x, p) is the single particle distribution function. Above, ∂ i (p) = ∂/∂p i , and C [f p ] is the collision integral. In the limit of vanishing net-charge, the main dynamical equation is the continuity equation for the energy-momentum tensor, T µν , ∂ µ T µν = 0.(2) In the presence of a thermal mass, T µν ≡ p µ p ν + g µν B, where B is the interaction energy [6], g µν denotes the metric, · · · = dP · · · f p , dP = g d 3 p/[(2π) 3 E p ], with g being the degeneracy factor and E p = p 2 + M 2 . The interaction energy B satisfies the following dynamical equation, ∂ µ B = − 1 2 ∂ µ M 2 1 ,(3) which is valid both in and out of equilibrium. We consider Maxwell-Boltzmann statistics so that in equilibrium f p = exp (−βu µ p µ ) ≡ f 0p , with β = 1/T and u µ being the fluid 4-velocity (which satisfies u µ u µ = 1). The temperature dependence of the mass is obtained such that the equation of state of the model describes lattice QCD results [4]. Plots for B(T ) and M (T ) can be seen in Refs. [2,3]. Qualitatively, M (T )/T is very large at low temperatures and saturates at M (T )/T ≈ 1.1 at high temperatures. Matching conditions and the collision term The energy-momentum tensor can be decomposed in terms of the 4velocity u µ as follows T µν = εu µ u ν − P ∆ µν + h µ u ν + h ν u µ + π µν ,(4) where ε is the total energy density, P is the total isotropic pressure, h µ is the energy diffusion, π µν is the shear-stress tensor, and we defined the projection operator ∆ µν = g µν − u µ u ν . They are obtained from moments of f p as explained in [7]. In general, ε 0 and P 0 may have non-equilibrium corrections, such that ε = ε 0 + δε, P = P 0 + Π, respectively. The meaning of u µ and β for non-equilibrium states is determined by matching conditions [7]. The most widely used prescription is the one introduced by Landau [8], where δε ≡ 0 and h µ ≡ 0. In the present work, we choose a new prescription in order to simplify Eq. (3). Specifically, we impose 1 ≡ 1 0 ,(5) where · · · 0 ≡ dP · · · f 0p , which defines the temperature for non-equilibrium states. In this matching, δε = 0. To define the 4-velocity, a further condition is needed. However, since we only consider a fluid at vanishing chemical potential, our results will not depend on this particular choice. With prescription (5), the interaction energy can be determined solely as a function of T and Eq. (3) can be solved as if the system were in equilibrium, ∂B(T ) ∂T = − gT M 2 2π 2 K 1 M (T ) T ∂M (T ) ∂T ,(6) which can be readily integrated since M (T ) is known, and the boundary condition B(0) = 0 is given. Above, K 1 is the first modified Bessel function of the second kind. Novel relaxation time approximation In contrast to the traditional RTA [9], in the new prescription proposed in Ref. [5] the conservation laws hold at the microscopic level even when considering momentum-dependent relaxation times and arbitrary matching conditions. In practice, we approximate the collision term as [3] C[f p ] ≈ − E p τ R f 0p   φp − φ p E 2 p τ R 0 E 3 p τ R 0 E p − φ p Ep τ R p µ 0 1 3 ∆ αβ p α p β Ep τ R 0 p µ    ,(7) where φ p ≡ (f p − f 0p )/f 0p . We parametrize the energy dependence of the relaxation time as τ R = t R (E p /T ) γ , where the parameter γ encodes the information of the underlying microscopic interaction, and t R > 0. For instance, it has been argued that γ = 1/2 in QCD effective theories [10]. Above, we defined the space-like projection p µ = ∆ µν p ν . Transport coefficients In first order theories, equation (2) is complemented by constitutive relations for the non-equilibrium currents (δε, Π, π µν ). In kinetic theory, they can be calculated using the Chapman-Enskog expansion [7] which, when truncated at first order, leads to the following relativistic Navier-Stokes formulation of hydrodynamics, δε = χθ, Π = −ζθ, π µν = 2ησ µν . Using (7), the transport coefficients read [3] ζ = − 1 3 (∆ µν p µ p ν ) A p τ R E p 0 − τ R E p A p 0 I 3,1 I 1,0 , χ = − A p τ R E p 0 + τ R E p A p 0 I 3,0 I 1,0 , η = β 15 (∆ µν p µ p ν ) 2 τ R E p 0 ,(9)where A p = −βc 2 s E 2 p − β 3 ∆ λσ p λ p σ − β 2 M ∂M Entropy production The entropy current for classical quasiparticles is S µ = dP p µ f p (1 − ln f p ). We note that the entropy production does not depend on the choice of matching conditions [3]. To first order in the Chapman-Enskog expansion, one finds ∂ µ S µ ζ s θ 2 + 2ησ µν σ µν , ζ s = τ R E p [A p ] 2 0 = ζ + c 2 s χ.(10) Since both ζ s and η are non-negative, so is the entropy production. The coefficient ζ s can be used to provide a matching-invariant interpretation of bulk viscosity and, indeed, for Landau matching conditions ζ s = ζ. This coefficient behaves similarly to ζ as a function of temperature, with the difference that, as M/T → 0, ζ s ∝ [M (T )(d/dT ) (M (T )/T )] 2 , thus displaying a steeper descent at high temperatures in Fig. 1. Conclusion In this work we have computed the first-order transport coefficients of an effective kinetic model with temperature-dependent mass, using the new relaxation time approximation proposed in Ref. [5]. We have used an alternative matching condition [Eq. (5)] to simplify the computations, which in turn imply that there are nonzero out-of-equilibrium corrections to the energy density. We find that all transport coefficients are significantly affected by the choice of the parameter γ, which defines how the relaxation time depends on energy. Consistency with the second law of thermodynamics is demonstrated and used to derive a matching-invariant bulk viscosity coefficient. In future work, we intend to compute the transport coefficients that appear in other theories of hydrodynamics [11,12] using the present model. ∂β c 2 s 2, and c 2 s ≡ (∂P 0 /∂ε 0 ) = (1/β)(I 10 +I 21 )/[I 30 + 1 2 I 10 (∂M 2 /∂β)β] is the speed of sound squared, which is expressed in terms ofI nq = 1/[(2q + 1)!!] −∆ λσ p λ p σ q E n−2qp 0 . In Fig. 1 we plot the coefficients as functions of temperature for different values of the parameter γ, as well as the temperature dependence of the mass. For all values of γ investigated, ζ ≥ 0 and χ ≤ 0. In both figures, it is seen that the absolute values of the coefficients grow with γ. At low temperatures, where the effective mass is large, M/T → ∞, all three normalized coefficients behave as (M/T ) γ−1 . For γ = 1, η = t R (ε 0 + P 0 ) at all temperatures. In the opposite limit, M/T → 0, ζ = −(1/3)χ ∝ M (T )(d/dT ) (M (T )/T ) and η ∼ Γ(γ + 5)/120 1 . Fig. 1 : 1(Top left) Normalized bulk viscosity, (Top right) energy correction coefficient, (Bottom left) shear viscosity, and (Bottom right) matchinginvariant bulk viscosity coefficients as functions of temperature. Each transport coefficient is shown for various values of the parameter γ. gas Filho de Amparoà Pesquisa do Estado do Rio de Janeiro (FAPERJ) process No. E-26/202.747/2018 for support. M.N.F is supported by the Fundação de Amparoà Pesquisa do Estado de São Paulo (FAPESP) grants 2017/05685-2 and 2020/12795-1. J.N. is partially supported by the U.S. Department of Energy, Office of Science, Office for Nuclear Physics under Award No. DE-SC0021301. Even though this is not achieved at high temperatures, where M/T → 1.1[3], these expansions serve as estimates. Acknowledgements G.S.R. and G.S.D thank Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for support. G.S.D. also thanks Fundação Carlos Cha- Transport Properties of the Quark-Gluon Plasma: A Lattice QCD Perspective. H B Meyer, Eur. Phys. J. A. 4786H. B. Meyer, "Transport Properties of the Quark-Gluon Plasma: A Lattice QCD Perspective," Eur. Phys. J. A, vol. 47, p. 86, 2011. Quasiparticle equation of state for anisotropic hydrodynamics. M Alqahtani, M Nopoush, M Strickland, Phys. Rev. C. 92554910M. Alqahtani, M. Nopoush, and M. Strickland, "Quasiparticle equation of state for anisotropic hydrodynamics," Phys. Rev. C, vol. 92, no. 5, p. 054910, 2015. Determining the transport coefficients of the quark-gluon plasma using a new relaxation time approximation of the Boltzmann equation. G S Rocha, M N Ferreira, G S Denicol, J Noronha, arXiv:2203.155712022G. S. Rocha, M. N. Ferreira, G. S. Denicol, and J. Noronha, "Determining the transport coefficients of the quark-gluon plasma using a new relaxation time approximation of the Boltzmann equation," arXiv:2203.15571 2022. The QCD equation of state with dynamical quarks. S Borsanyi, G Endrodi, Z Fodor, A Jakovac, S D Katz, S Krieg, C Ratti, K K Szabo, JHEP. 1177S. Borsanyi, G. Endrodi, Z. Fodor, A. Jakovac, S. D. Katz, S. Krieg, C. Ratti, and K. K. Szabo, "The QCD equation of state with dynamical quarks," JHEP, vol. 11, p. 077, 2010. Novel Relaxation Time Approximation to the Relativistic Boltzmann Equation. G S Rocha, G S Denicol, J Noronha, Phys. Rev. Lett. 127442301G. S. Rocha, G. S. Denicol, and J. Noronha, "Novel Relaxation Time Approx- imation to the Relativistic Boltzmann Equation," Phys. Rev. Lett., vol. 127, no. 4, p. 042301, 2021. From quantum field theory to hydrodynamics: Transport coefficients and effective kinetic theory. S Jeon, L G Yaffe, Phys. Rev. D. 53S. Jeon and L. G. Yaffe, "From quantum field theory to hydrodynamics: Trans- port coefficients and effective kinetic theory," Phys. Rev. D, vol. 53, pp. 5799- 5809, 1996. Microscopic Foundations of Relativistic Fluid Dynamics. G Denicol, D H Rischke, SpringerG. Denicol and D. H. Rischke, Microscopic Foundations of Relativistic Fluid Dynamics. Springer, 2021. Fluid Mechanics. L Landau, E Lifshitz, Course of Theoretical Physics. LondonPergamon Press6L. Landau and E. Lifshitz, "Fluid Mechanics," Course of Theoretical Physics, Pergamon Press, London, vol. 6, 1959. A relativistic relaxation-time model for the Boltzmann equation. J L Anderson, H Witting, Physica. 743J. L. Anderson and H. Witting, "A relativistic relaxation-time model for the Boltzmann equation," Physica, vol. 74, no. 3, pp. 466-488, 1974. Radiative energy loss and v 2 spectra for viscous hydrodynamics. K Dusling, G D Moore, D Teaney, Physical Review C. 81334907K. Dusling, G. D. Moore, and D. Teaney, "Radiative energy loss and v 2 spectra for viscous hydrodynamics," Physical Review C, vol. 81, no. 3, p. 034907, 2010. Derivation of transient relativistic fluid dynamics from the Boltzmann equation. G S Denicol, H Niemi, E Molnar, D H Rischke, Phys. Rev. D. 8539902Phys.Rev.DG. S. Denicol, H. Niemi, E. Molnar, and D. H. Rischke, "Derivation of tran- sient relativistic fluid dynamics from the Boltzmann equation," Phys. Rev. D, vol. 85, p. 114047, 2012, [Erratum: Phys.Rev.D 91, 039902 (2015)]. Causality and existence of solutions of relativistic viscous fluid dynamics with gravity. F S Bemfica, M M Disconzi, J Noronha, Phys. Rev. D. 9810104064F. S. Bemfica, M. M. Disconzi, and J. Noronha, "Causality and existence of solutions of relativistic viscous fluid dynamics with gravity," Phys. Rev. D, vol. 98, no. 10, p. 104064, 2018.
{'fraction_non_alphanumeric': 0.06949334149701515, 'fraction_numerical': 0.033369049441298024, 'mean_word_length': 3.7969897209985315, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 22, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'We use a novel formulation of the relaxation time approximation to consistently calculate the bulk and shear viscosity coefficients using QCDinspired energy-dependent relaxation times and phenomenological thermal masses obtained from fits to lattice QCD thermodynamics. The matching conditions are conveniently chosen to simplify the computations.', 'arxivid': '2207.11286', 'author': ['Gabriel S Rocha \nInstituto de Física\nInstituto de Física Gleb Wataghin\nUniversidade Federal Fluminense\nNiterói, Rio de JaneiroBrazil\n', 'Gabriel S Denicol \nInstituto de Física\nInstituto de Física Gleb Wataghin\nUniversidade Federal Fluminense\nNiterói, Rio de JaneiroBrazil\n', 'Maurício N Ferreira \nIllinois Center for Advanced Studies of the Universe & Department of Physics\nUniversidade Estadual de Campinas\nCampinas, São PauloBrazil\n', 'Jorge Noronha \nUniversity of Illinois Urbana-Champaign\n61801UrbanaILUSA\n'], 'authoraffiliation': ['Instituto de Física\nInstituto de Física Gleb Wataghin\nUniversidade Federal Fluminense\nNiterói, Rio de JaneiroBrazil', 'Instituto de Física\nInstituto de Física Gleb Wataghin\nUniversidade Federal Fluminense\nNiterói, Rio de JaneiroBrazil', 'Illinois Center for Advanced Studies of the Universe & Department of Physics\nUniversidade Estadual de Campinas\nCampinas, São PauloBrazil', 'University of Illinois Urbana-Champaign\n61801UrbanaILUSA'], 'corpusid': 251040066, 'doi': '10.5506/aphyspolbsupp.16.1-a29', 'github_urls': [], 'n_tokens_mistral': 4491, 'n_tokens_neox': 3784, 'n_words': 2262, 'pdfsha': '00ce1faaf0c017a23bf1ceb99e75238f83c724e8', 'pdfurls': ['https://export.arxiv.org/pdf/2207.11286v1.pdf'], 'title': ['Novel relaxation time approximation: a consistent calculation of transport coefficients with QCD-inspired relaxation times *', 'Novel relaxation time approximation: a consistent calculation of transport coefficients with QCD-inspired relaxation times *'], 'venue': []}
arxiv
Hom-Lie-Yamaguti Structures on Hom-Leibniz Algebras 29 Aug 2012 Donatien Gaparayi Institut de Mathématiques et de Sciences Physiques Département de Mathématiques Université d' 01 BP 613-Oganla, Abomey-Calavi 01 BP 4521Porto-Novo, CotonouBénin, Bénin A Nourou Issa Institut de Mathématiques et de Sciences Physiques Département de Mathématiques Université d' 01 BP 613-Oganla, Abomey-Calavi 01 BP 4521Porto-Novo, CotonouBénin, Bénin Hom-Lie-Yamaguti Structures on Hom-Leibniz Algebras 29 Aug 2012AMS Subject Classification (2010): 17A3017A3217D99 Keywords: Leibniz algebraLie-Yamaguti algebra (ie generalized Lie triple systemLie triple algebra)Hom-Leibniz algebraHom-Lie-Yamaguti algebra Every multiplicative left Hom-Leibniz algebra has a natural Hom-Lie-Yamaguti structure. Introduction and statement of result A (left) Leibniz algebra is an algebra (L, ·) satisfying the identity x · (y · z) = (x · y) · z + y · (x · z). Leibniz algebras were introduced by J.-L. Loday [15] (and so they are sometimes called Loday algebras) as a noncommutative analogue of Lie algebras, in the study of some topics in homological algebra and noncommutative geometry (see also [5], [20]). While earlier papers on Leibniz algebras are concerned with some homological problems (see, e.g., [15], [20]), some structure theory of Leibniz algebras are proposed in, e.g., [2] and [3] (see also references therein). Classification of low-dimensional Leibniz algebras could be found in, e.g., [2], [6], [15], [21]. One of the problems in the general theory of a given class of (binary or binary-ternary) nonassociative algebras is the study of relationships between that class of algebras and the one of Lie algebras. In the same rule, the search of relationships between a class of nonassociative algebras and the one of Leibniz algebras is of interest (at least for constructing concrete examples of the given class of nonassociative algebras). In this setting, the existence of a Lie-Yamaguti structure on any (left) Leibniz algebra pointed out in [14] is a good illustration. A Lie-Yamaguti algebra is a triple (L, [, ], {, , }) in which L is a vector space, [, ] : L×L → L a bilinear map and {, , } : L × L × L → L a trilinear map such that (LY1) [x, y] = −[y, x], (LY2) {x, y, z} = −{y, x, z}, (LY3) x,y,z ([[x, y], z] + {x, y, z}) = 0, (LY4) x,y,z {[x, y], z, u} = 0, (LY5) {x, y, [u, v]} = [{x, y, u}, v] + [u, {x, y, v}], (LY6) {x, y, {u, v, w}} = {{x, y, u}, v, w} + {u, {x, y, v}, w} +{u, v, {x, y, w}}, for all u, v, w, x, y, z, in L, where x,y,z denotes the sum over cyclic permutation of x, y, z. Lie -Yamaguti algebras, first called "generalized Lie triple systems", were introduced by K. Yamaguti [22] while giving an algebraic interpretation of the characteristic properties of the torsion and curvature of a homogeneous space with canonical connection (the Nomizu's connection) [19]. Later, M. Kikkawa [13] called them "Lie triple algebras" and the terminology of "Lie-Yamaguti algebras" is introduced in [14] to designate these algebras. As in [4], we write "LY-algebras" for Lie-Yamaguti algebras. In a left Leibniz algebra (L, ·) if define [x, y] := x · y − y · x (skew-symmetrization) and {x, y, z} := −(x · y) · z, then (L, [, ], {, , }) is a LY-algebra [14]. In this note, we will be interested in the counterpart of this construction in the Hom-algebra setting. Roughly speaking, a Hom-type generalization of a kind of algebras is defined by twisting its defining identities by a linear self-map (the twisting map) in such a way that when the twisting map is the identity map, one recovers the original kind of algebras. The theory of Hom-algebras originated from the introduction of so-called "Hom-Lie algebras" in [8] as an abstraction of an approach to deformations of the Witt algebra and the Virasoro algebra based on σ-derivations, including q-derivations of the Witt algebra and the Virasoro algebra associated to q-difference operators. The outcome of the algebraic view of these considerations is the introduction of "Hom-associative algebras" in [18] as a Hom-analogue of associative algebras. Hom-associative algebras are for Hom-Lie algebras what are associative algebras for Lie algebras: the commutator-algebra of a Hom-associative algebra is a Hom-Lie algebra [18]. As a "noncommutative" generalization of Hom-Lie algebras, Hom-Leibniz algebras are also defined in [18], where other results regarding Hom-algebras are found (see also [23]). A general method of twisting ordinary algebras into their Hom-type analogues is given in [24]. The reader is refered to, e.g., [7], [11], [16], [17], [25] for discussions about various Hom-type algebras. Following this line, Hom-Lie-Yamaguti algebras (Hom-LY algebras) are introduced in [7]. (HLY6) x,y,z {[x, y], α(z), α(u)} = 0, (HLY7) {α(x), α(y), [u, v]} = [{x, y, u}, α 2 (v)] + [α 2 (u), {x, y, v}], (HLY8) {α 2 (x), α 2 (y), {u, v, w}} = {{x, y, u}, α 2 (v), α 2 (w)} +{α 2 (u), {x, y, v}, α 2 (w)} + {α 2 (u), α 2 (v), {x, y, w}}, for all u, v, w, x, y, z ∈ Y. Thus, as mentioned above, we shall prove the following Theorem. Every multiplicative left Hom-Leibniz algebra has a natural Hom-Lie-Yamaguti structure. The useful definitions and some facts as the characterization of the Hom-Akivis algebra associated to a given Hom-Leibniz algebra are reminded in section 2. In section 3, we prove the theorem and discuss examples of Hom-LY algebras that are constructed using the theorem above (thusly, we also construct examples of left Hom-Leibniz algebras). All vector spaces and algebras are considered over a fixed ground field of characteristic 0. Definitions and basic facts We recall some basic notions, introduced in [8], [11], [18], [23], [24], related to Homalgebras. We also recall from [12] a characterization of the Hom-Akivis algebra associated with a given Hom-Leibniz algebra. [23]) A Hom-algebra is a triple (L, ·, α) in which L is a vector space, "·" a binary operation on L and α : L → L is a linear map (the twisting map). Definition 2.1. ([18], A Hom-algebra (L, ·, α) is said to be multiplicative if α(x·y) = α(x)·α(y) (multiplicativity), for all x, y in L. Since our result depends on multiplicativity, we assume here that all Hom-algebras are multiplicative. Definition 2.2. Let (L, ·, α) be a Hom-algebra. (i) The Hom-associator [18] of L is the trilinear map as α : L × L × L → L defined by as α (x, y, z) = (x.y).α(z) − α(x).(y.z), (2.1) for all x, y, z ∈ L. If as α (x, y, z) = 0 (Hom-associativity), ∀x, y, z ∈ L, then (L, ·, α) is said to be Hom-associative [18]. (ii) The Hom-Jacobian [18] of L is the trilinear map J α : L × L × L → L defined by J α (x, y, z) := x,y,z (x.y).α(z) (2.2) for all x, y, z in L. The Hom-algebra (L, ·, α) is called a Hom-Lie algebra [8] if the operation "·" is anticommutative and the Hom-Jacobi identity J α (x, y, z) = 0 is satisfied in (L, ·, α). Remark 2.3. If α = Id (the identity map) then (2.1) (resp. (2.2)) is just the associator (resp. the Jacobian) in (L, ·, α). Therefore an associative (resp. a Lie) algebra could be seen as a Hom-associative (resp. Hom-Lie) algebra with the identity map as the twisting map. Also note that a not necessarily Hom-associative algebra is called a non-Hom-associative algebra in [11] in analogy with the case of not necessarily associative algebras (the terminologies of "Hom-nonassociative algebras" or "nonassociative Hom-algebras" are also used in [17], [23] for that type of Hom-algebras). As for Lie algebras, Hom-Lie algebras have a "noncommutative" generalization as Hom-Leibniz algebras. α(x) · (y · z) = (x · y) · α(z) + α(y) · (x · z) (2.3) for all x, y, z in L. Remark 2.5. If α = Id in Definition 2.4, then (L, ·, α) reduces to a (left) Leibniz algebra (L, ·). Moreover, as for Leibniz algebras [15], if the operation of a given Hom-Leibniz algebra (L, ·, α) is skew-symmetric (i.e. anticommutative), then (L, ·, α) turns out to be a Hom-Lie algebra (see [18]). We also observe that the original definition of a Hom-Leibniz algebra [18] is related to the identity (the "right" Hom-Leibniz identity) (x · y) · α(z) = (x · z) · α(y) + α(x) · (y · z). Moreover, given a linear self-map α of L, every Leibniz algebra (L, ·) can be twisted into a Hom-Leibniz algebra (L,α, α) with "α" defined by xα y = α(x · y) for all x, y in L ( [24]). The extension to binary-ternary algebras of twisting identities of algebras is considered in [11]. This led to the introduction of the class of "Hom-Akivis algebras" as a twisted version of Akivis algebras introduced by M.A. Akivis (see [1] and references therein). Akivis algebras (first called "W -algebras " [1]) arose in the differential geometry of differentiable webs, and also as tangent algebras to local differentiable quasigroups. The terminology of "Akivis algebras" is introduced in [9]. which defines Akivis algebras. It is shown [11] that every Akivis algebra with a linear self-map is twisted into a Hom-Akivis algebra and that every non-Hom-associative algebra with a linear self-map is a Hom-Akivis algebra with respect to the skew-symmetrization [x, y] = xy−yx and Hom-associator [x, y, z] = as α (x, y, z). In terms of Hom-associators, the identity (2.3) has the form as α (x, y, z) = −α(y).(x.z). (2.5) Thus the operations of the Hom-Akivis algebra associated with the Hom-Leibniz algebra (L, ·, α) are the skew-symmetrization and (2.5). Then the Hom-Akivis identity (2.4) takes the form x,y,z [[x, y], α(z)] = x,y,z as α (x, y, z)− x,y,z as α (y, x, z) that is, by (2.5) and (2.3), x,y,z [[x, y], α(z)] = x,y,z (x · y) · α(z). (2.6) The considerations above will be used in the next section in the proof of the theorem. Proof of the Theorem. Examples In this section, we settle down in the proof of our claim, i.e. the existence of a Hom-LY structure on any (multiplicative) left Hom-Leibniz algebra. This proof is based on a specific ternary operation that can be considered on a given Hom-Leibniz algebra (this product is the Hom-analogue of the ternary operation considered in [14] on a left Leibniz algebra L that produces, along with the skew-symmetrization, a LY structure on L). Also note that our proof below essentially relies on some properties characterizing Hom-Leibniz algebras, obtained in [12]. We conclude, as an illustration of our result, by some constructions of Hom-LY algebras from twisted Leibniz algebras (incidentally, this produces examples of left Hom-Leibniz algebras). In a left Hom-Leibniz algebra (L, ·, α) consider the skew-symmetrization [x, y] := x · y − y · x for all x, y in L. Then, from [12], we know that (x · y + y · x) · α(z) = 0, (3.1) α(x) · [y, z] = [(x · y), α(z)] + [α(y), (x · z)]. (3.2) If consider the left translations Λ a b := a · b in (L, ·, α), then the identities (2.3) and (3.2) can be written respectively as Λ α(x) (y · z) = (Λ x y) · α(z) + α(y) · (Λ x z), (3.3) Λ α(x) [y, z] = [Λ x y, α(z)] + [α(y), Λ x z]. ( 3.4) Proof of the Theorem. In (L, ·, α) consider the following ternary operation: {x, y, z} := as α (y, x, z) − as α (x, y, z) Moreover, we have The multiplicativity of (L, ·, α) implies (HLY1) and (HLY2) while (HLY3) is the skewsymmetrization and (HLY4) clearly follows from (3.5) (or (3.7)). Next, observe that (HLY5) is just the Hom-Akivis identity (2.6) for (L, ·, α). [x, y] · α(z) = (x · y − y · x) · α(z) = 2(x · y) · α(z) (by (3.1)) = −2{x, y, z} (see (3.6)) so that {x, y, z} = − 1 2 [x, y] · α(z). Consider now (x,y,z) {[x, y], α(z), α(u)}. Then x,y,z {[x, y], α(z), α(u)} = x,y,z −([x, y] · α(z)) · α 2 (u) (by (3.6)) = 2( x,y,z {x, y, z}) · α 2 (u) (by (3.7)) = −2((x · y) · α(z) + (y · z) · α(x) + (z · x) · α(y)) · α 2 (u) = −2(α(x) · (y · z) − α(y) · (x · z) + (y · z) · α(x) +(z · x) · α(y)) · α 2 (u) (by (2.3)) = −2(α(x) · (y · z) + (y · z) · α(x)) · α 2 (u) −2(−α(y) · (x · z) + (z · x) · α(y)) · α 2 (u) = −2(−α(y) · (x · z) + (z · x) · α(y)) · α 2 (u) (by (3.1)) = −2(−α(y) · (x · z) − (x · z) · α(y)) · α 2 (u) (by (3.1)) = 2(α(y) · (x · z) + (x · z) · α(y)) · α 2 (u) = 0 (by (3.1)) so that we get (HLY6). Next which is (HLY7). Finally, we compute {α(x), α(y), [u, v]} = −α(x · y) · α([u, v]) ({{x, y, u}, α 2 (v), α 2 (w)} + {α 2 (u), {x, y, v}, α 2 (w)} +{α 2 (u), α 2 (v), {x, y, w}} = {−Λ x·y α(u), α 2 (v), α 2 (w)} + {α 2 (u), −Λ x·y α(v), α 2 (w)} +{α 2 (u), α 2 (v), −Λ x·y α(w)} = −((−Λ x·y α(u)) · α 2 (v)) · α 3 (w) − (α 2 (u) · (−Λ x·y α(v))) · α 3 (w) −(α 2 (u) · α 2 (v))α(−Λ x·y α(w)) = (Λ α(x·y) α(u · v)) · α 3 (w) + α 2 (u · v) · Λ α(x·y) α 2 (w) (by (3. 3) and multiplicativity) = Λ α 2 (x·y) (α(u · v) · α 2 (w)) (by (3.3)) = −α 2 (x · y) · (−α(u · v) · α 2 (w)) = −(α 2 (x) · α 2 (y)) · α(−(u · v) · α(w)) (by multiplicativity) = {α 2 (x), α 2 (y), {u, v, w}} (by (3.6)). Therefore (L, [, ], {, , }, α) is a Hom-LY algebra. This completes the proof. Remark 3.1. If set α = Id in (3.6), then we recover the ternary operation defined in [14] in the proof of the existence of a natural LY structure on any left Leibniz algebra (see section 1). Therefore, although with a quite different scheme of proof, our result here is an α-twisted version of the one in [14]. The untwisted version of the proof proposed here could be found in [10], where the result of [14] is considered again but via Akivis algebras. We now discuss examples of Hom-LY algebras that can be constructed using the theorem. Examples of Hom-LY algebras constructed from LY algebras can be found in [7]. Example 3.2. Let (L, ·, α) be an anticommutative multiplicative left Hom-Leibniz algebra. Then (L, ·, α) is a multiplicative Hom-Lie algebra ( [18]). If define on L a ternary operation "{, , }" by (3.6), then one checks that (L, ·, {, , }, α) is a Hom-LY algebra. In the following examples, the unspecified products are regarded as zero. Example 3.3. Let (L, ·) be a 3-dimensional complex algebra defined by e 2 · e 3 = e 2 , e 3 · e 1 = λe 1 , e 3 · e 2 = −e 2 , e 3 · e 3 = e 1 (λ ∈ C). Then (L, ·) is a (solvable) complex left Leibniz algebra ( [21], Table 3, algebra L 4 ). Now a few computation shows that all the linear self-maps α of L given by α(e 1 ) = (aλ + 1)e 1 , α(e 2 ) = be 2 , α(e 3 ) = ae 1 + e 2 + e 3 are endomorphisms of (L, ·), where a, b, λ ∈ C. By a result in [24], we define on L an operation "α" by e 2α e 3 := α(e 2 · e 3 ) = be 2 , e 3α e 1 := α(e 3 · e 1 ) = λ(aλ + 1)e 1 , e 3α e 2 := α(e 3 · e 2 ) = −be 2 , e 3α e 3 := α(e 3 · e 3 ) = (aλ + 1)e 1 to get a multiplicative left Hom-Leibniz algebra L α := (L,α, α). Therefore, according to the theorem, the Hom-LY algebra (L, [, ], {, , }, α) corresponding to L α is defined by then α is verified to be an endomorphism of (L, ·). Next, as in the example above, we may define on L an operation "α" by A Hom-Lie-Yamaguti algebra (Hom-LY algebra for short) is a quadruple (L, [, ], {, , }, α) in which L is a vector space, "[, ]" a binary operation and "{, , }" a ternary operation on L, and α : L → L a linear map such that (HLY1) α([x, y]) = [α(x), α(y)], (HLY2) α({x, y, z}) = {α(x), α(y), α(z)}, (HLY3) [x, y] = −[y, x], (HLY4) {x, y, z} = −{y, x, z}, (HLY5) x,y,z ([[x, y], α(z)] + {x, y, z}) = 0, Definition 2.4. ([18]) A (left) Hom-Leibniz algebra is a Hom-algebra (L, ·, α) satisfying the (left) Hom-Leibniz identity Definition 2.6. ([11]) A Hom-Akivis algebra is a quadruple (L,[, ], [, , ], α) such that L is a vector space, "[, ]" is a skew-symmetric binary operation on L, "[, , ]" a ternary operation on L, α : L → L a linear map, and that the Hom-Akivis identityJ α (x, y, z) = x,y,z [x, y, z]− x,y,z [y, x, z] (2.4)holds for all x, y, z in L.Observe that for α = Id the Hom-Akivis identity (2.4) reduces to the Akivis identity J(x, y, z) = x,y,z [x, y, z]− x,y,z [y, x, z] x, y, z in L. Then (3.5), (2.3) and (2.5) imply {x, y, z} = −(x · y) · α(z).(3.6) are different expressions of the operation "{, , }" that are for use in what follows. Now we proceed to verify the validity on (L, ·, α) of the set of identities (HLY1)-(HLY8). by (3.6) and multiplicativity) = Λ −α(x·y) [α(u), α(v)] = [Λ −x·y α(u), α 2 (v)] + [α 2 (u), Λ −x·y α(v)] (by (3.4)) = [{x, y, u}, α 2 (v)] + [α 2 (u), {x, y, v}] (by (3.6)) [e 1 1, e 3 ] = −λ(aλ + 1)e 1 (= −[e 3 , e 1 ]), [e 2 , e 3 ] = 2be 2 (= −[e 3 , e 2 ]), {e 3 , e 2 , e 3 } = b 2 e 2 (= −{e 2 , e 3 , e 3 }) with a, b, λ ∈ C. Example 3 . 4 . 34Let (L, ·) be a 4-dimensional complex algebra defined by e 1 · e 1 = e 4 , e 1 · e 2 = e 3 , e 1 · e 3 = e 4 , e 2 · e 1 = −e 3 , e 3 · e 1 = −e 4 . Then (L, ·) is a (nilpotent) complex left Leibniz algebra ([2], Theorem 3.2, algebra R 7 ). If define a linear self-map of L by α(e 1 ) = e 1 + e 2 + e 3 + e 4 , α(e 2 ) = e 2 + e 3 + e 4 , α(e 3 ) = e 3 + e 4 , α(e 4 ) = e 4 , e 1α e 1 = e 4 , e 1α e 2 = e 3 + e 4 , e 1α e 3 = e 4 , e 2α e 1 = −e 3 − e 4 , e 3α e 1 = −e 4 and then L α := (L,α, α) is a multiplicative left Hom-Leibniz algebra. Now, applying the theorem, we see that the Hom-LY algebra (L, [, ], {, , }, α) corresponding to L α is constructed by defining [e 1 , e 2 ] = 2(e 3 + e 4 ) (= −[e 2 , e 1 ]), [e 1 , e 3 ] = 2e 4 (= −[e 3 , e 1 ]), {e 1 , e 2 , e 1 } = e 4 (= −{e 2 , e 1 , e 1 }). Example 3 . 5 . 35A 4-dimensional (nilpotent) complex left Leibniz algebra (L, ·) is given by e 1 · e 1 = e 4 , e 1 · e 2 = e 3 , e 1 · e 3 = e 4 , e 2 · e 1 = −e 3 + e 4 , e 3 · e 1 = −e 4 (see[2], Theorem 3.2, algebra R 8 ). A linear self-map α of L defined byα(e 1 ) = e 1 + e 3 + e 4 , α(e 2 ) = e 2 + e 4 , α(e 3 ) = e 3 , α(e 4 ) = e 4 ,is verified to be an endomorphism of (L, ·). Again, as in the examples above, define on L an operation "α" bye 1α e 1 = e 4 , e 1α e 2 = e 3 , e 1α e 3 = e 4 , e 2α e 1 = −e 3 + e 4 , e 3α e 1 = −e 4 and then L α := (L,α, α) is a multiplicative left Hom-Leibniz algebra. So the theorem implies that the Hom-LY algebra (L, [, ], {, , }, α) constructed from L α is defined by [e 1 , e 2 ] = 2e 3 − e 4 (= −[e 2 , e 1 ]), [e 1 , e 3 ] = 2e 4 (= −[e 3 , e 1 ]), {e 1 , e 2 , e 1 } = e 4 (= −{e 2 , e 1 , e 1 }). Local algebras of a multidimensional three-web. M A Akivis, Siberian Math. J. 17M.A. Akivis, Local algebras of a multidimensional three-web, Siberian Math. J. 17 (1976), 3-8. Classification of 4-dimensional nilpotent complex Leibniz algebras. S Albeverio, B A Omirov, I S Rakhimov, Extracta Math. 213S. Albeverio, B.A. Omirov, and I.S. Rakhimov, Classification of 4-dimensional nilpotent complex Leibniz algebras, Extracta Math. 21 (3) (2006), 197-210. On Leibniz algebras, in: Algebras and Operator Theory. A Sh, B A Ayupov, Omirov, Proceedings of the colloquium in Tashkent. the colloquium in TashkentDordrechtKluwer Academic PublishersSh.A. Ayupov and B.A. Omirov, On Leibniz algebras, in: Algebras and Operator Theory, Proceedings of the colloquium in Tashkent, Kluwer Academic Publishers, Dordrecht (1998), 1-13. Irreducible Lie-Yamaguti algebras. P Benito, A Elduque, F Martín-Herce, J. Pure Appl. Alg. 213P. Benito, A. Elduque and F. Martín-Herce, Irreducible Lie-Yamaguti algebras, J. Pure Appl. Alg. 213 (2009), 795-808. Ten-term exact sequences of Leibniz homology. J M Casas, T Pirashvili, J. Algebra. 231J.M. Casas and T. Pirashvili, Ten-term exact sequences of Leibniz homology, J. Algebra 231 (2000), 258-264. R-matrices for Leibniz algebras, communicación Técnica n • I. R Felipe, N López-Reyes, F Ongay, 02-27/11-11-2002 (MB/CI MATR. Felipe, N. López-Reyes and F. Ongay, R-matrices for Leibniz algebras, communicación Técnica n • I-02-27/11-11-2002 (MB/CI MAT). A twisted generalization of Lie-Yamaguti algebras. D Gaparayi, A N Issa, Int. J. Algebra. 67D. Gaparayi and A.N. Issa, A twisted generalization of Lie-Yamaguti algebras, Int. J. Algebra 6 (7) (2012), 339-352. Deformations of Lie algebras using σderivations. J T Hartwig, D Larsson, S D Silvestrov, J. Algebra. 295J.T. Hartwig, D. Larsson and S.D. Silvestrov, Deformations of Lie algebras using σ- derivations, J. Algebra 295 (2006), 314-361. Lie's fundamental theorems for local analytic loops. K H Hofmann, K Strambach, Pacific J. Math. 123K.H. Hofmann and K. Strambach, Lie's fundamental theorems for local analytic loops, Pacific J. Math. 123 (1986), 301-327. Remarks on the construction of Lie-Yamaguti algebras from Leibniz algebras. A N Issa, Int. J. Algebra. 514A.N. Issa, Remarks on the construction of Lie-Yamaguti algebras from Leibniz algebras, Int. J. Algebra 5 (14) (2011, 667 -677. Hom-Akivis algebras. A N Issa, Comment. Math. Univ. Carolinae. 524A.N. Issa, Hom-Akivis algebras, Comment. Math. Univ. Carolinae 52 (4) (2011), 485- 500. A N Issa, arXiv:1011.1731v1Some characterizations of Hom-Leibniz algebras. A.N. Issa, Some characterizations of Hom-Leibniz algebras, arXiv:1011.1731v1. Geometry of homogeneous Lie loops. M Kikkawa, Hiroshima Math. J. 5M. Kikkawa, Geometry of homogeneous Lie loops, Hiroshima Math. J. 5 (1975), 141-179. Leibniz algebras, Courant algebroids and multiplications on reductive homogeneous spaces. M K Kinyon, A Weinstein, Amer. J. Math. 123M.K. Kinyon and A. Weinstein, Leibniz algebras, Courant algebroids and multiplications on reductive homogeneous spaces, Amer. J. Math. 123 (2001), 525-550. Une version non commutative des algèbres de Lie: les algèbres de Leibniz. J.-L Loday, Enseign. Math. 39J.-L. Loday, Une version non commutative des algèbres de Lie: les algèbres de Leibniz, Enseign. Math. 39 (1993), 269-293. Hom-alternative algebras and Hom-Jordan algebras. A Makhlouf, Int. Elect. J. Alg. 8A. Makhlouf , Hom-alternative algebras and Hom-Jordan algebras, Int. Elect. J. Alg. 8 (2010), 177-190. A Makhlouf, Proceedings of Jordan Structures in Algebra and Analysis Meeting. Jordan Structures in Algebra and Analysis MeetingAlmeríaParadigm of nonassociative Hom-algebras and Hom-superalgebrasA. Makhlouf, Paradigm of nonassociative Hom-algebras and Hom-superalgebras, in: Pro- ceedings of Jordan Structures in Algebra and Analysis Meeting, 143-177, Editorial Círculo Rojo, Almería, 2010. Hom-algebra structures. A Makhlouf, S D Silvestrov, J. Gen. Lie Theory Appl. 2A. Makhlouf and S.D. Silvestrov, Hom-algebra structures, J. Gen. Lie Theory Appl. 2 (2008), 51-64. Invariant affine connections on homogeneous spaces. K Nomizu, Amer. J. Math. 76K. Nomizu, Invariant affine connections on homogeneous spaces, Amer. J. Math. 76 (1954), 33-65. On Leibniz homology. T Pirashvili, Ann. Inst. Fourier (Grenoble). 442T. Pirashvili, On Leibniz homology, Ann. Inst. Fourier (Grenoble) 44 (2) (1994), 401-411. On contractions and invariants of Leibniz algebras Bull. I S Rakhimov, K A M Atan, Malays. Math. Soc. 2I.S. Rakhimov and K.A.M. Atan, On contractions and invariants of Leibniz algebras Bull. Malays. Math. Soc. (2) 35 (2A) (2012), 557-565. On the Lie triple system and its generalization. K Yamaguti, J. Sci. Hiroshima Univ. Ser. A. 21K. Yamaguti, On the Lie triple system and its generalization, J. Sci. Hiroshima Univ. Ser. A 21 (1957/1958), 107-113. Enveloping algebra of Hom-Lie algebras. D Yau, J. Gen. Lie Theory Appl. 2D. Yau, Enveloping algebra of Hom-Lie algebras, J. Gen. Lie Theory Appl. 2 (2008), 203-207. Hom-algebras and homology. D Yau, J. Lie Theory. 19D. Yau, Hom-algebras and homology, J. Lie Theory 19 (2009), 409-421. D Yau, Hom-Maltsev, Hom-alternative and Hom-Jordan algebras. 11D. Yau, Hom-Maltsev, Hom-alternative and Hom-Jordan algebras, Int. Elect. J. Alg. 11 (2012), 177-217.
{'fraction_non_alphanumeric': 0.13812131063392932, 'fraction_numerical': 0.0406364958507999, 'mean_word_length': 3.442987457240593, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 29, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'Every multiplicative left Hom-Leibniz algebra has a natural Hom-Lie-Yamaguti structure.', 'arxivid': '1208.6038', 'author': ["Donatien Gaparayi \nInstitut de Mathématiques et de Sciences Physiques\nDépartement de Mathématiques\nUniversité d'\n01 BP 613-Oganla, Abomey-Calavi 01 BP 4521Porto-Novo, CotonouBénin, Bénin\n", "A Nourou Issa \nInstitut de Mathématiques et de Sciences Physiques\nDépartement de Mathématiques\nUniversité d'\n01 BP 613-Oganla, Abomey-Calavi 01 BP 4521Porto-Novo, CotonouBénin, Bénin\n"], 'authoraffiliation': ["Institut de Mathématiques et de Sciences Physiques\nDépartement de Mathématiques\nUniversité d'\n01 BP 613-Oganla, Abomey-Calavi 01 BP 4521Porto-Novo, CotonouBénin, Bénin", "Institut de Mathématiques et de Sciences Physiques\nDépartement de Mathématiques\nUniversité d'\n01 BP 613-Oganla, Abomey-Calavi 01 BP 4521Porto-Novo, CotonouBénin, Bénin"], 'corpusid': 56360040, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 9887, 'n_tokens_neox': 8435, 'n_words': 4170, 'pdfsha': '4fdb43cffacdb8a0d8531d012dc5fbeb36ca98a3', 'pdfurls': ['https://arxiv.org/pdf/1208.6038v1.pdf'], 'title': ['Hom-Lie-Yamaguti Structures on Hom-Leibniz Algebras', 'Hom-Lie-Yamaguti Structures on Hom-Leibniz Algebras'], 'venue': []}
arxiv
On Symplectic and Multisymplectic Structures and Their Discrete Versions in Lagrangian Formalism 2001. June 15, 2001 Han-Ying Guo Institute of Theoretical Physics Academia Sinica P.O. Box 2735100080BeijingChina Yu-Qi Li Institute of Theoretical Physics Academia Sinica P.O. Box 2735100080BeijingChina W U Ke Institute of Theoretical Physics Academia Sinica P.O. Box 2735100080BeijingChina On Symplectic and Multisymplectic Structures and Their Discrete Versions in Lagrangian Formalism International Academic Publishers Beijing, China3562001. June 15, 2001(Received April 2, 2001)numbers: 1110Ef0260Lj Key words: Euler-Lagrange cohomologydifference discrete variational principlesymplectic structure We introduce the Euler-Lagrange cohomology to study the symplectic and multisymplectic structures and their preserving properties in finite and infinite dimensional Lagrangian systems respectively. We also explore their certain difference discrete counterparts in the relevant regularly discretized finite and infinite dimensional Lagrangian systems by means of the difference discrete variational principle with the difference being regarded as an entire geometric object and the noncommutative differential calculus on regular lattice. In order to show that in all these cases the symplectic and multisymplectic preserving properties do not necessarily depend on the relevant Euler-Lagrange equations, the Euler-Lagrange cohomological concepts and content in the configuration space are employed. Introduction It is well known that the symplectic and multisymplectic structures play crucially important roles in the symplectic and multisymplectic algorithms for the finite dimensional Hamiltonian systems [1,2] and infinitedimensional Hamiltonian systems respectively. These algorithms are very powerful and successful in numerical calculations of the relevant systems in comparison with other various non-symplectic computational schemes since the symplectic and multisymplectic schemes preserve the symplectic structure and multisymplectic structure of the systems in certain sense. In this paper, in a simple and direct manner, we present the symplectic structure in the Lagrangian mechanism and the multisymplectic structure in the Lagrangian field theory and their preserving properties as well as the difference-type discrete versions of these issues. We employ the ordinary exterior differential calculus in the configuration space and introduce what is named the Euler-Lagrange (EL) cohomology to show that the symplectic and multisymplectic structures are preserved without necessarily making use of the EL equations in general. And the EL equations are derived from the variational principle of the relevant action functionals. Therefore, it is important to emphasize that these structure-preserving properties are established in the function space with the EL cohomology on the configuration space in general rather than in the solution space of the EL equations only. One of the key points different from the other approaches in our approach is the EL cohomology we will introduced. Some EL cohomological concepts and content such as the EL one-forms, the coboundary EL one-forms and the EL conditions, i.e., the EL one-forms are closed, null EL one-form gives rise to the EL equations. Although the null EL one-form is a special case of the coboundary EL one-forms which are cohomologically trivial and automatically satisfy the (closed) EL condition, they are not the same in principle. As a matter of fact, the EL cohomology is nontrivial in general. This point plays a crucial role in our approach. In the course of numerical calculation, the "time" t ∈ R is always discretized, say, with equal spacing h = ∆t and the space coordinates are also discretized in various cases, especially, for the Lagrangian field theory. In addition to these computational approach, there also exist various discrete physical systems with discrete or difference discrete Lagrangian functions. It is well known that the differences of functions do not obey the Leibniz law. In order to explore the discrete symplectic and multisymplectic structures in these difference discrete systems and their preserving properties in certain difference discrete versions, some noncommutative differential calculus (NCDC) should be employed [5−7] even for the well-established symplectic algorithms. This is the second key point of this paper. Another key point of this paper different from others is that the difference discrete variational principle (DDVP) will be employed in this paper. In view of NCDC, a forward or backward difference as the forward or backward discrete derivative should be regarded as an entire geometric object respectively. In the DDVP with forward (or backward) difference, we prefer to adopt this point of view. We also show that DDVP leads to the correct results in the sense that the results not only correspond to the correct ones in the continuous limit but also are the The plan of this paper is as follows. We first briefly rederive some well-known contents on symplectic and multisymplectic structures and their preserving properties in the Lagrangian formalism for the finite and infinite dimensional systems respectively in Sec. 2. The important issues of this section is to introduce the EL cohomology, including some cohomological concepts and content such as the EL one-forms, the coboundary EL one-forms and the EL conditions and to show that it is nontrivial in each case. In order to explain those symplectic and multisymplectic geometry and relevant preserving properties in relevant systems the EL cohomology plays a very important role. In Sec. 3, we first explain the DDVP in our approach to give rise to the discrete Euler-Lagrange (DEL) equations. Then we study the difference discrete versions of the cohomological concepts and content as well as the symplectic and multisymplectic structures in Lagrangian formalism given in Sec. 2. We present some remarks in Sec. 4. Finally, in the Appendix, some relevant NCDC on regular lattice with equal spacing on each direction is given. The Symplectic and Multisymplectic Structures in Lagrangian Mechanism and Field Theory In this section, we recall some well-known contents on symplectic and multisymplectic structures and their preserving in the Lagrangian formalism for the finite and infinite dimensional systems respectively. The important point is to introduce the EL cohomological concepts and content related to the EL equations and explain their important roles in the symplectic and multisymplectic geometry in these systems respectively. The Symplectic Structure in Lagrangian Mechanism We begin with the Lagrangian mechanism. Let time t ∈ R 1 be the base manifold, M the configuration space on t with coordinates q i (t), i = 1, . . . , n, T M the tangent bundle of M with coordinates q i ,q j , F (T M ) the function space on T M . The Lagrangian of the systems is denoted as L(q i ,q j ), whereq j is the derivative of q j with respect to t. The variational principle gives rise to the well-known EL equations ∂L ∂q i − d dt ∂L ∂q i = 0 .(1) Let us take the exterior derivative d of the Lagrangian function, we get dL = ∂L ∂q i − d dt ∂L ∂q i dq i + d dt ∂L ∂q i dq i . Defining the EL one-forms on T * M , E(q i ,q i ) : = ∂L − d ∂L dq i ,(2) we have dL(q i ,q i ) = E(q i ,q i ) + d dt θ ,(3) where θ is the canonical one-form θ = ∂L ∂q i dq i .(4) It is easy to see from the definition (2) and Eq. (3) that first the null EL one-form is corresponding to the EL equation, secondly the null EL one-form is a special case of the coboundary EL one-forms, E(q i ,q j ) = dα(q i ,q j ) ,(5) where α(q i ,q j ) is an arbitrary smooth function of (q i ,q j ), and thirdly the EL one-forms are not coboundary in general since θ is not a coboundary so that the EL cohomology is not trivial. Making use of nilpotency of d on T * M , d 2 L(q,q) = 0, it is easy to show that if and only if the EL one-form is closed with respect to d, which may be named the EL condition, i.e., dE(q i ,q j ) = 0 ,(6) the symplectic conservation law with respect to t follows d dt ω = 0 ,(7) where the symplectic structure ω is given by ω = dθ = ∂ 2 L ∂q i ∂q j dq i ∧ dq j + ∂ 2 L ∂q i ∂q j dq i ∧ dq j . (8) It is important to note that although the null EL oneform and the coboundary EL one-forms satisfy the EL condition, it does not mean that the closed EL one-forms can always be exact. Namely, as mentioned above, the EL cohomology is not trivial so that the EL condition is not cohomologically equivalent to the null EL one-form or the coboundary EL one-forms in general. This also means that q i (t), i = 1, . . . , n in the EL condition are not in the solution space of the EL equations only. In fact, they are still in the function space in general. Therefore, the symplectic two-form ω is conserved not only in the solution space of the equations but also in the function space in general with respect to the duration of t if and only if the EL condition is satisfied. In order to transfer to the Hamiltonian formalism, we introduce conjugate momentum p j = ∂L ∂q j ,(9) and take a Legendre transformation to get the Hamiltonian function i k i j Then the EL equations become the canonical equations as follows:q i = ∂H ∂p i ,ṗ j = − ∂H ∂q j .(11) It is clear that a pair of the EL one-forms should be introduced now E 1 (q i , p j ) = q j − ∂H ∂p j dp j , E 2 (q i , p j ) = ṗ j + ∂H ∂q j dq j .(12) In terms of z T = (q i , . . . , q n , p 1 , . . . , p n ), the canonical equations and the EL one-form becomė z = J −1 ∇ z H ,(13)E(z) = dz T (Jz − ∇ z H) .(14) Now it is straightforward to show that the symplectic structure preserving law d dt ω = 0, ω = dz T ∧ J dz(15) holds if and only if the (closed) EL condition is satisfied dE(z) = 0 .(16) The Multisymplectic Structure in Lagrangian Field Theory We now consider the multisymplectic structure in Lagrangian field theory for the scalar fields. Let X be an n-dimensional base manifold with coordinates x µ , µ = 1, . . . , n, M the configuration space on X with scalar field variables u i (x), i = 1, . . . , s, T M the tangent bundle of M with coordinates u i , u j µ , u j µ = ∂u i /∂x µ , F (T M ) the function space on T M . The Lagrangian of the systems under consideration is L(u i , u j µ ) with the well-known EL equations from the variational principle, ∂L ∂u i − ∂ ∂x µ ∂L ∂u i µ = 0 .(17) Let us introduce the EL one-form in the function space F (T M ), E(u i , u j µ ) : = ∂L ∂u i − ∂ ∂x µ ∂L ∂u i µ du i .(18) It is easy to see that the null EL one-form is corresponding to the EL equations and it is a special case for coboundary EL one-forms E(u i , u j µ ) = dα(u i , u j µ ) ,(19) where α(u i , u j µ ) is an arbitrary smooth function of (u i , u j µ ). Although they are cohomologically trivial but it is already seen that in the EL one-forms, (u i , u j µ ) are not in the solution space of the EL equations only rather they are in We now take the exterior derivative d of the Lagrangian. In terms of the EL one-form we get dL(u i , u j µ ) = E(u i , u j µ ) + ∂ µ θ µ ,(20) where θ µ are the canonical one-forms θ µ = ∂L ∂u i µ du i .(21) From Eq. (20), it is easy to see that the EL one-forms are not coboundary in general since the canonical one-forms θ µ are not coboundary so that the EL cohomology is not trivial in general. By virtue of the nilpotency of d, d 2 L(u i , u j µ ) = 0, it is easy to prove that if and only if the EL condition is satisfied, i.e., the EL one-form is closed dE(u i , u j µ ) = 0 ,(22) the multisymplectic structure preserving (MSSP) property, i.e., the multisymplectic conservation or divergence free law follows n µ=1 ∂ ∂x µ ω µ = 0 ,(23) where the symplectic structures ω µ are given by ω µ = dθ µ = ∂ 2 L ∂u i µ ∂u j du i ∧ du j + ∂ 2 L ∂u i µ ∂u j µ du i ∧ du j µ .(24) Similar to the finite dimensional case, it is also important to note again that although the null EL one-form, the coboundary EL one-forms satisfy the EL condition, it does not mean that the closed EL one-forms can always be exact as pointed out above. In addition, u i (x)s in the EL condition are not in the solution space of the EL equations only in general. Therefore, the MSSP law holds, i.e., the multisymplectic two-forms ω µ are conserved, not only in the solution space of the equations but also in the function space with the closed EL condition in general. The Discrete Symplectic and Multisymplectic Structures in Lagrangian Formalism Now we consider certain difference discrete versions of the symplectic and multisymplectic structures and their preserving properties in the Lagrangian mechanism and field theory studied in the last section. The Discrete Symplectic Structure in Discrete Lagrangian Mechanism Let us first consider the symplectic structure and its preserving in the Lagrangian mechanism in case that "time" t is discretized while the configuration space at Let us assume, without loss any generality, that in the course of numerical calculation, the "time" t ∈ R is discretized with equal spacing h = ∆t, t ∈ R → t ∈ T D = {(t k , t k+1 = t k + h, k ∈ Z)} . (25) At the moment t k , the configuration space is M n k ∈ M n TD = {· · · M n 1 × · · · × M n k · · · }, its coordinates are denoted by q i (k) . The difference discrete Lagrangian can be written by L D(k) = L D (q i (k) , q i t(k) ) ,(26)where q j t (k) is (forward) difference of q j (k) at t k defined by ∆ t q j (k) : = ∂ ∂t q j (k) = q j t(k) = 1 h {q j (k+1) − q j (k) } . (27) It is the (discrete) derivative on T (T D ) in the sense of NCDC of a regular lattice L 1 with equal spacing [5] and the same notation for it as in the continuous case has been employed. It should be noted that in what follows the difference is always viewed as an entire geometric object and its dual dt is the base of T * (T D ). We now consider the DDVP of the action functional S D = k∈Z L D (q i (k) , q j t(k) ) , δS D = k∈Z ∂L D(k) ∂q i (k) δq i (k) + ∂L D(k) ∂q i t(k) δq i t(k) .(28) By means of the modified Leibniz law with respect to ∆ t [5,6] (see the Appendix), we have ∆ t ∂L D(k−1) ∂q i t(k−1) δq i (k) = ∂L D(k) ∂q i (k) δq i t(k) + ∆ t ∂L D(k−1) ∂q i t(k−1) δq i (k) . Therefore, δS D = k∈Z ∂L D(k) ∂q i (k) − ∆ t ∂L D(k−1) ∂q i t(k−1) i (k) + k∈Z ∆ t ∂L D(k−1) ∂q i t(k−1) δq i (k) . Using the properties k∈Z ∆ t f (t k ) = f (t k ) k=+∞ k=−∞ ,(29) and assuming δq i k satisfy δq i (±∞) = 0 , it follows the DEL equations ∂L D(k) i − ∆ t ∂L D(k−1) i = 0 .(30) It is easy to see that for the case with difference discrete Lagrangian L (k) D (q i (k) , ∆ t q j (k) ) = 1 2 (∆ t q i (k) ) 2 − V (q i (k) ) ,(31) the DDVP gives the DEL equations ∆ t (∆ t q i (k−1) ) − V ′ (q i (k) ) = 0 , i.e., 1 h 2 (q i (k+1) − 2q i (k) + q i (k−1) ) = V ′ (q i (k) ) .(32) This is just what is wanted for the difference discrete counterpart of the equations in the continuous case. Now we consider the difference discrete symplectic structure and its preserving property. Taking the exterior derivative d on L D(k) , we get dL D(k) = ∂L D(k) ∂q i (k) dq i (k) + ∂L D(k) ∂q i t(k) dq i t(k) . By means of the modified Leibniz law with respect to ∆ t and introducing the DEL one-form E D(k) (q i (k) , q j t(k) ) : = ∂L D(k) ∂q i (k) − ∆ t ∂L D(k−1) ∂q i t(k−1) dq i (k) ,(33) we have dL D(k) = E D(k) + ∆ t θ D(k) ,(34) where θ D(k) is the discrete canonical one-form θ D(k) = ∂L D(k−1) ∂q i t(k−1) dq i (k) ,(35) and there exists the following discrete symplectic two-form on T * (M n TD ), ω D(k) = dθ D(k) = ∂ 2 L D(k−1) ∂q i t(k−1) ∂q j (k−1) dq j (k−1) ∧ dq i (k) + ∂ 2 L D(k−1) ∂q i t(k−1) ∂q j t(k−1) dq j t(k−1) ∧ dq i (k) .(36) Now by virtue of the nilpotency of d on T * (M n TD ), we get 0 = d 2 L D(k) = dE D(k) + ∆ t ω D(k) .(37) Therefore, it is easy to see that if and only if the DEL one-form satisfies what is called the DEL condition, i.e., it is closed dE D(k) = 0 ,(38) then it gives rise to the discrete (difference) symplectic structure-preserving law, Similar to the continuous case, the null DEL one-form, which is corresponding to the DEL equation, is a special case of the coboundary DEL one-forms, E D(k) = dα D(k) (q i (k) , q j t(k) ) ,(40) where α D(k) (q i (k) , q j t(k) ) is an arbitrary function of (q i (k) , q j t(k) ). Although they satisfy the DEL condition, this does not mean that all closed DEL one-forms are exact. In fact equation (34) shows that the EL one-forms are not exact in general since the canonical one-form θ D(k) is not trivial. In addition, (q i (k) , q j t(k) ) are not in the solution space of the DEL equations only rather they are still in the function space with the DEL condition in general. Therefore, this also means that the DDSSP law holds in the function space with the DEL cohomology in general rather than on the solution space only. The Discrete Multisymplectic Structure in Discrete Lagrangian Field Theory We now study the discrete multisymplectic structure in discrete Lagrangian field theory. For the sake of simplicity, let us consider the 1 + 1 − d and 2 − d cases in discrete Lagrangian field theory (DLFT) for a scalar field. Let X 2 with suitable signature of the metrics be the base manifold, L 2 a regular lattice with two-directions x µ , µ = 1, 2 on X 2 , M D the discrete configuration space with u (i,j) ∈ M D . The difference discrete Lagrangian is denoted as L (i,j) D = L D (u (i,j) , u (i,j) µ ) ,(41) where ∆ 1 u (i,j) = 1 h (u (i+1,j) − u (i,j) ) , ∆ 2 u (i,j) = 1 h (u (i,j+1) − u (i,j) ) . They are the bases of T (M D ) and their duals dx µ = d L x µ are the bases of T * (M D ), d L x µ (∂ ν ) = δ µ ν . The action functional is given by S D = {i,j}∈Z×Z L D (u (i,j) , u (i,j) µ ) .(42) Taking the variation of S D and regarding the differences as the entire geometric objects, we get δS D = {i,j}∈Z×Z ∂L (i,j) D ∂u (i,j) δu (i,j) + ∂L (i,j) D ∂u (i,j) µ δu (i,j) µ . Employing the modified Leibniz law, we have ∆ 1 ∂L (i−1,j) D ∂u (k−1,l) 1 δu (k,l) = ∂L (i,j) D ∂u (k,l) 1 δu (k,l) 1 + ∆ 1 ∂L (i−1,j) D ∂u (k−1,l) 1 δu (k,l) , ∆ 2 ∂L (i,j−1) D ∂u (k,l−1) 2 δu (k,l) = ∂L (i,j) D ∂u (k,l) 2 δu (k,l) 2 + ∆ 2 ∂L (i,j−1) D ∂u (k,l−1) 2 δu (k,l) . Assuming that δu (k,l) s vanish at infinity, it follows the DEL equations ∂L (i,j) D ∂u (k,l) − ∆ 1 ∂L (i−1,j) D ∂u (k−1,l) 1 − ∆ 2 ∂L (i,j−1) D ∂u (k,l−1) 2 = 0 . (43) Let us consider an example with the discrete Lagrangian L (i,j) D (u (i,j) , u (i,j) µ ) = 1 2 (∆ µ u (i,j) ) 2 − V (u (i,j) ) .(44) The DDVP gives the DEL equations ∆ 1 (∆ 1 u (i−1,j) ) + ∆ 2 (∆ 2 u (i,j−1) ) − V ′ (u (i,j) ) = 0 , i.e., 1 h 2 1 (u (i+1,j) − 2u (i,j) + u (i−1,j) ) + 1 h 2 2 (u (i,j+1) − 2u (i,j) + u (i,j−1) ) = V ′ (u (i,j) ) .(45) This is also what is wanted for the difference discrete coun-We now consider the multisymplectic properties of the DLFT. Taking exterior derivative d ∈ T * (M D ) of L (i,j) D and making use of the modified Leibniz law, we get dL (i,j) D = E (i,j) D (u (i,j) , u (i,j) µ ) + ∆ µ θ µ(i,j) ,(46) where E (i,j) D is the DEL one-form defined by E (i,j) D (u (i,j) , u (i,j) µ ) : = ∂L (i,j) D ∂u (k,l) − ∆ 1 ∂L (i−1,j) D ∂u (k−1,l) 1 − ∆ 2 ∂L (i,j−1) D ∂u (k,l−1) 2 du (k,l) ,(47) and θ µ(i,j) are two Cartan one-forms, θ 1(i,j) = ∂L (i−1,j) D ∂u (k−1,l) 1 du (k,l) , θ 2(i,j) = ∂L (i,j−1) D (k,l−1) du (k,l) .(48) It is easy to see that there exist two symplectic two-forms on T * (M D ), ω µ(i,j) = dθ µ(i,j) , µ = 1, 2 .(49) The equation d 2 L (i,j) D = 0 on T * (M D ) leads to the conservation law or the divergence free equation of ω µ(i,j) , ∆ µ ω µ(i,j) = 0 ,(50) if and only if the DEL one-form satisfies the DEL condition, i.e., it is closed dE (i,j) D = 0 .(51) Similar to the continuous case, the null DEL one-form is corresponding to the DEL equations and it is a special case of coboundary DEL one-forms E (i,j) D = dα (i,j) D ,(52) where α (i,j) D is an arbitrary function on T * M D . Although they satisfy the DEL condition, it does not mean that all closed DEL one-forms are exact. As a matter of fact, from Eq. (46) it is easy to see that the EL one-forms are not exact in general since the two canonical one-forms θ µ(i,j) (µ = 1, 2) are not trivial. In addition, this indicates that the variables u (k,l) s are still in the function space in general rather than the ones in the solution space only. Consequently, this also means that the difference discrete multisymplectic structure-preserving law holds in the function space with the closed DEL condition in general rather than in the solution space only. It should be pointed out that the scenario of the approach can be straightforwardly generalized to higherdimensional cases of X 1,n−1 and X n . Remarks A few remarks are in order. 1) The approach presented in this paper, which may call the EL cohomology approach, to the symplectic and multisymplectic geometry and their difference discrete versions in the Lagrangian formalisms is more or less different from other approaches. [3,4] The EL and the DEL cohomological concepts and relevant content such as the EL and DEL one-forms, the null EL and DEL one-forms, the coboundary EL and DEL one-forms as well as the EL and the DEL conditions have been introduced and they have played very crucial roles in each case to show that the symplectic and multisymplectic preserving properties and their difference discrete versions are in the function space with the closed EL/DEL condition in general rather than in the solution space only. It has been mentioned that the EL and DEL cohomology in relevant case is not trivial and it is very closely related to the symplectic and multisymplectic structures as and the role of the EL cohomology in each case should be further studied and some issues are under investigation. [8] 2) The difference discrete variational formalism presented here is also different from the one by Veselov. [9,10] We have emphasized that the difference as discrete derivative is an entire geometric object. This is obvious and natural from the view point of NCDC. The continuous limits of the results given here are correct as well. 3) The NCDC on the regular lattices are employed in our approach. Since the differences do not satisfy the ordinary Leibniz law, in order to study the symplectic and multisymplectic geometry in these difference discrete systems it is natural and meaningful to make use of the NCDC. 4) The approach presented here can be generalized to the case that the configuration space is also discretized. This is closely related to the case of difference discrete phase space approach to the finite dimensional systems with separable Hamiltonian. [5,6] 5) It should be mentioned that the approach with the EL cohomological concepts can also directly be applied to the PDEs. Let us for example consider the following type of equations [3,11] and make use of the same notations in Refs [3] and [11], Kz x1 + Lz x2 = ∇ z S(z) .(53) Introducing the EL one-form E(z, z x1 , z x2 ) : = dz T {Kz x1 + Lz x2 − ∇ z S(z)} ,(54) it is easy to see that the null EL one-form gives rise to Eq. (54) and it is a special case of the coboundary EL one-forms, E(z, z x1 , z x2 ) = dα(z, z x1 , z x2 ) ,(55) where α(z, z x1 , z x2 ) is an arbitrary function of (z, z x1 , z x2 ). Now by taking the exterior derivative d of the EL oneform, it is straightforward to prove that dE(z, z x1 , z x2 ) = 1 2 ∂ x1 (dz T ∧ K dz) + 1 2 ∂ x2 (dz T ∧ Ldz). This means that the following MSSP equation ∂ x1 ω + ∂ x2 τ = 0 (56) holds, where ω = dz T ∧ K dz, τ = dz T ∧ Ldz , if and only if the EL one-form is closed, i.e., dE(z, z x1 , z x2 ) = 0 .(57) It should be mentioned that first from the definition of the EL one-form, it is not trivial in general since the first two terms in the definition are the canonical one-forms which are not trivial so that the EL cohomology is not derived here is not dependent on the type of equations but can be applied to the equations so that it holds not only in the solution space of the equations but also in the function space relevant to the cohomology in general as well. 6) In principle, the cohomological scenario presented here should be available to not only to PDEs but also to ODEs and numerical schemes as well. 7) Finally, it should be pointed out that there exist lots of other problems to be studied. Appendix In this appendix we briefly recall some content of NCDC on lattice. [5−7] A.1 An NCDC on an Abelian Discrete Group Let G be an Abelian discrete group with a generator t, A the algebra of complex valued functions on G. The left and right multiplications of a generator of G on its element are commute to each other since G is Abelian. Let us introduce right action on A that is given by R t f (a) = f (a · t) ,(A1) where f ∈ A, a ∈ G, t the generator and "·" the group multiplication. Let V be the space of vector fields, V = span{∂ t } , where ∂ t is the derivative with respect to the generator t given by (∂ t f )(a) ≡ R t f (a) − f (a) = f (a · t) − f (a) .(A2) The dual space of V , the space of one-form, is Ω 1 = span{χ t } that is dual to V , χ t (∂ t ) = 1 .(A3) The whole differential algebra Ω * can also be defined as Ω * = n=0,1 Ω n with A = Ω 0 . Let us define the exterior differentiation in Ω * d : Ω 0 → Ω 1 . It acts on a zero-form ω 0 = A is as follows: df = ∂ t f χ t ∈ Ω 1 .(A4) Now, the following theorem can straightforwardly be proved. Theorem The exterior differentiation d satisfies (a) (df )(v) = v(f ), v ∈ V, f ∈ Ω 0 , (b) d 2 = 0 , if and only if (i) dχ t = 0 , (ii) χ t f = (R t f )χ t .(A6) As was shown here, in order to establish a well-defined differential algebra, it is necessary and sufficient to introduce the noncommutative property of the multiplication between function and one-form. The conjugation * on the whole differential algebra Ω * and metric on discrete Abelian group can also be defined. A.2 An NCDC on Regular Lattice Let us consider the discrete translation group G m = ⊗ m i=1 G i , A the function space on G m and a regular lattice with equal spacing L m on an m-dimensional space R m . Here G i the i-th discrete translation group with one generator acting on one-dimensional space with coordinate q in such a way R q i : q i n → q i n+1 = q i n + h, h ∈ R + ,(A7) R q i the discrete translation operation of the group G i and it maps q i n of n-th size of q i to the one q i n+1 at (n + 1)th size, h the discrete translation step-length and R + the positive real number. It is easy to see that the action of G i on i-th one-dimensional space R 1 generates the i-th chain L i , i = 1, . . . , m, with equal spacing h. Similarly, the regular lattice L m with equal spacing h is generated by G m acting on R m . Since there is a one-to-one correspondence between sizes on L i and elements of G i , one may simply regard L i as G i . For the same reason, one may simply regard L m as G m . On the sizes of the regular lattice L m , there are discrete coordinates q i n , i = 1, . . . , m. There is a set of generators in the discrete translation group G m acting on L m in such a way R q i : q i n → q i n+1 , i = 1, . . . , m .(A8) With respect to the generators there is a set of independent derivatives ∂ q i on f n (q i ) = f (q i n ) ∈ A. They should be defined as the correspondent differences of the functions valued at two nearest sizes, i.e., ∂ q i f (q i n ) = ∆ q i f (q i n ) = 1 h [(R q i − id)f (q i n )] = 1 h [f (q i n+1 ) − f (q i n )] .(A9) The differential one-form is defined by df = ∂ q i f dq i = ∆ q i f dq i , f ∈ A . (A10) The two-forms and the whole differential algebra Ω * can also be defined. Here d is the exterior differentiation. Similarly, the following theorem can be proved for d. i.e., d 2 = 0 , d(ω ∧ ω ′ ) = dω ∧ ω ′ + (−1) deg ω ω ∧ dω ′ , ω, ω ′ ∈ Ω * ,(A11) if and only if f (q i + h)dq i = dq i f (q i ) .(A12) This gives q i dq i − dq i q i = −hdq i . The above two equations show the noncommutative properties between the functions (including the coordinates) and differential forms. From these properties, it follows the modified Leibniz rule for derivatives, ∆ q i (f g) = ∆ q i f · g + {R q i f } · ∆ q i g .(A13) It should be noted that the definitions and relations given above for the NCDC on the regular lattice L m are at least formally very similar to the ones in the ordinary commutative differential calculus (CDC) on R m . The differences between the two cases are commutative or not. The Hodge * operator and the co-differentiation operator δ L : Ω k → Ω k−1 on the regular lattice L m can also be defined similarly to the ones on R m . Consequently, the Laplacian on the lattice L m may also be given by ∆ L = dδ L + δ L d .(A14) It is in fact the discrete counterpart of the Laplacian ∆ on R m . For other objects and/or properties on R m , there may have the discrete counterparts on L m as well. For example, the null-divergence equation of a form ω on R m reads δα = 0 .(A15) Its counterpart on the lattice L m is simply δ L α L = 0 .(A16) This is the forward difference form of null-divergence equation. In the case of L 1,m ∈ R 1,m with Lorentz signature, these equations become the conservation law of α and its difference form of α L . This is available not only for the symplectic geometry and symplectic algorithms but also the multisymplectic geometry and multisymplectic algorithms as well. It should be emphasized that for the discrete counterparts on the lattice, they obey the NCDC on the lattice L m rather than the CDC on R m . This is the most important point. GUO Han-Ying, LI Yu-Qi and WU KeVol. 35 On Difference Schemes and Symplectic Geometry. K Feng, Proc. of the 1984 Beijing Symposium on Differential Geometry and Differential Equations -Computation of Partial Differential Equations. Feng Kangof the 1984 Beijing Symposium on Differential Geometry and Differential Equations -Computation of Partial Differential EquationsBeijingScience PressK. Feng, "On Difference Schemes and Symplectic Geom- etry," Proc. of the 1984 Beijing Symposium on Differen- tial Geometry and Differential Equations -Computation of Partial Differential Equations, ed. Feng Kang, Science Press, Beijing (1985); . Selected Works of Feng Kang II. and the references thereinSelected Works of Feng Kang II (1995), and the references therein. Numerical Hamiltonian Problems. J M Sanz-Serna, M P Calvo, Chapman and HallLondonand the references thereinJ.M. Sanz-Serna and M.P. Calvo, Numerical Hamiltonian Problems, Chapman and Hall, London (1994), and the references therein. . T J Bridges, Math. Proc. Camb. Phil. Soc. 121147T.J. Bridges, Math. Proc. Camb. Phil. Soc. 121 (1997) 147. . J E Marsden, G W Patrick, S Shkoller, Commun. Math. Phys. 199351and the references thereinJ.E. Marsden, G.W. Patrick and S. Shkoller, Commun. Math. Phys. 199 (1998) 351, and the references therein. . H Y Guo, K Wu, S H Wang, S K Wang, G M Wei, Commun. Theor. Phys. 34307H.Y. GUO, K. WU, S.H. WANG, S.K. WANG and G.M. WEI, Commun. Theor. Phys. (Beijing, China) 34 (2000) 307. . H Y Guo, K Wu, W Zhang, Commun. Theor. Phys. 34245H.Y. GUO, K. WU and W. ZHANG, Commun. Theor. Phys. (Beijing, China) 34 (2000) 245. Discrete Symplectic Algorithm on Regular Lattice. H Y Guo ; Guangzhou, ; H Y Guo, K Wu, S H Wang, G M Wei, Talk given by H.Y. GUO at the CCAST-WL Workshop on Computational Methods and Their Applications in Physics and Mechanics. Zhongshan University,the CCAST-WL Workshop on Integrable SystemH.Y. GUO, Talk given at the Workshop on Quantum Field Theories, Dec. 14-19 (1998), Zhongshan University, Guangzhou; H.Y. GUO, K. WU, S.H. WANG and G.M. WEI, "Discrete Symplectic Algorithm on Regular Lat- tice", Talk given by H.Y. GUO at the CCAST-WL Work- shop on Computational Methods and Their Applications in Physics and Mechanics, March (1999); the CCAST-WL Workshop on Genetic Algorithm and Its Applications, April (1999); the CCAST-WL Workshop on Integrable System, May 3-7 (1999); . Ccast-Wl, Workshop Series. 104CCAST-WL Workshop Series 104, pp 167-192; Noncommutative Differential Calculus and Discrete Symplectic Algorithm on Regular Lattice. H Y Guo, Talk given at the CCAST-WL Workshop on Structure Preserving Algorithms and Applications. H.Y. GUO, "Noncommutative Differen- tial Calculus and Discrete Symplectic Algorithm on Regu- lar Lattice", Talk given at the CCAST-WL Workshop on Structure Preserving Algorithms and Applications, Dec. 13-17 (1999); . Ccast-Wl, Workshop Series. 118CCAST-WL Workshop Series 118, pp 1- 16. Symplectic, Multisymplectic Structures and Euler-Lagrange Cohomology. H Y Guo, Y Q Li, K Wu, in preparationH.Y. GUO, Y.Q. LI and K. WU, "Symplectic, Multisym- plectic Structures and Euler-Lagrange Cohomology", in preparation. . A P Veselov, Funkts. Anal. Prilozhen. 221A.P. Veselov, Funkts. Anal. Prilozhen 22 (1988) 1. . J Moser, A P Veselov, Commun. Math. Phys. 139217J. Moser and A.P. Veselov, Commun. Math. Phys. 139 (1991) 217. . S Reich, J. Comput. Phys. 157473S. Reich, J. Comput. Phys. 157 (2000) 473.
{'fraction_non_alphanumeric': 0.07495911321097583, 'fraction_numerical': 0.019201647586165122, 'mean_word_length': 3.629048086359176, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 21, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'We introduce the Euler-Lagrange cohomology to study the symplectic and multisymplectic structures and their preserving properties in finite and infinite dimensional Lagrangian systems respectively. We also explore their certain difference discrete counterparts in the relevant regularly discretized finite and infinite dimensional Lagrangian systems by means of the difference discrete variational principle with the difference being regarded as an entire geometric object and the noncommutative differential calculus on regular lattice. In order to show that in all these cases the symplectic and multisymplectic preserving properties do not necessarily depend on the relevant Euler-Lagrange equations, the Euler-Lagrange cohomological concepts and content in the configuration space are employed.', 'arxivid': 'hep-ph/0104064', 'author': ['Han-Ying Guo \nInstitute of Theoretical Physics\nAcademia Sinica\nP.O. Box 2735100080BeijingChina\n', 'Yu-Qi Li \nInstitute of Theoretical Physics\nAcademia Sinica\nP.O. Box 2735100080BeijingChina\n', 'W U Ke \nInstitute of Theoretical Physics\nAcademia Sinica\nP.O. Box 2735100080BeijingChina\n'], 'authoraffiliation': ['Institute of Theoretical Physics\nAcademia Sinica\nP.O. Box 2735100080BeijingChina', 'Institute of Theoretical Physics\nAcademia Sinica\nP.O. Box 2735100080BeijingChina', 'Institute of Theoretical Physics\nAcademia Sinica\nP.O. Box 2735100080BeijingChina'], 'corpusid': 2189671, 'doi': '10.1088/0253-6102/35/6/703', 'github_urls': [], 'n_tokens_mistral': 11222, 'n_tokens_neox': 10023, 'n_words': 6108, 'pdfsha': '89e6b95a246ede4d7f61a1a926300f953b3d4ba1', 'pdfurls': ['https://arxiv.org/pdf/hep-ph/0104064v2.pdf'], 'title': ['On Symplectic and Multisymplectic Structures and Their Discrete Versions in Lagrangian Formalism', 'On Symplectic and Multisymplectic Structures and Their Discrete Versions in Lagrangian Formalism'], 'venue': ['International Academic Publishers']}
arxiv
Strong gravitational lensing in a rotating Kaluza-Klein black hole with squashed horizons 26 Feb 2014 Liyong Ji Songbai Chen Jiliang Jing Institute of Physics Department of Physics Key Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education Hunan Normal University 410081ChangshaHunanPeople's Republic of China Hunan Normal University 410081ChangshaHunanPeople's Republic of China Strong gravitational lensing in a rotating Kaluza-Klein black hole with squashed horizons 26 Feb 20142numbers: 0470Dy9530Sf9760Lf We have investigated the strong gravitational lensing in a rotating squashed Kaluza-Klein (KK) black hole spacetime. Our result show that the strong gravitational lensings in the rotating squashed KK black hole spacetime have some distinct behaviors from those in the backgrounds of the fourdimensional Kerr black hole and of the squashed KK Gödel black hole. In the rotating squashed KK black hole spacetime, the marginally circular photon radius ρps , the coefficientā,b, the deflection angle α(θ) in the φ direction and the corresponding observational variables are independent of whether the photon goes with or against the rotation of the background, which is different with those in the usual four-dimensional Kerr black hole spacetime. Moreover, we also find that with the increase of the scale of extra dimension ρ0, the marginally circular photon radius ρps and the angular position of the relativistic images θ∞ first decreases and then increases in the rotating squashed KK black hole for fixed rotation parameter b, but in the squashed KK Gödel black hole they increase for the smaller global rotation parameter j and decrease for the larger one. In the extremely squashed case ρ0 = 0, the coefficientā in the rotating squashed KK black hole increases monotonously with the rotation parameter, but in the squashed KK Gödel black hole it is a constant and independent of the global rotation of the Gödel Universe. These information could help us to understand further the effects of the rotation parameter and the scale of extra dimension on the strong gravitational lensing in the black hole spacetimes. I. INTRODUCTION The Kaluza-Klein (KK) black holes with squashed horizons are a kind of interesting Kaluza-Klein type metrics with the special topology and asymptotical structure [1][2][3][4][5][6][7][8]. This family of black holes behave as fully five-dimensional black holes in the vicinity of horizon, while behave as four-dimensional black holes with a constant twisted S 1 fiber in the far region. In these black holes, the size of compactified extra dimension can be adjustable by a parameter r ∞ . Recent investigations show that the spectrum of Hawking radiation [9,10], the quasinormal frequencies [11,12] and the precession of a gyroscope in a circular orbit [13] From the general theory of relativity, we know that photons would be deviated from their straight path when they pass close to a compact and massive body. The effects originating from the deflection of light rays in a gravitational field are known as gravitational lensing [14][15][16]. Like a natural and large telescope, gravitational lensing can help us to capture the information about the very dim stars which are far away from our Galaxy. The strong gravitational lensing is caused by a compact object with a photon sphere. As photons pass close to the photon sphere, the deflection angles of of the light rays become very large, which yields that there exist two infinite sets of faint relativistic images on each side of the black hole. With these relativistic images, we could extract the information about black holes in the Universe and verify profoundly alternative theories of gravity in their strong field regime [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]. Recently, we have studied the strong gravitational lensing in the background of a Schwarzschild squashed KK black hole [33] and a squashed KK Gödel black hole [34] and find that the size of the extra dimension and the the global rotation parameter j of the Gödel Universe affects the photon sphere radius, the deflection angle and the corresponding observational variables in the strong gravitational lensing. Sadeghi et al have considered the effect of the charge on the strong gravitational lensing in the squashed KK black holes [35,36]. These investigations could help us to understand further the effects of the scale of extra dimension on the strong gravitational lensings. However, to my knowledge, the strong gravitational lensing is still open in the background of a rotating squashed KK black hole. Since the rotation is a universal phenomenon for the celestial bodies in our Universe, it is necessary to study the strong gravitational lensing in the rotating squashed KK black hole spacetime. With the squashing transformation, Wang [6] obtained a rotating squashed KK black hole spacetime with two equal angular momenta in Einstein theory. This black hole solution with squashed horizons is geodesic complete and free of naked singularities. It has the similar topology and asymptotical structure to that of the static squashed KK black hole, but with richer physical properties. In this paper, we are going to study the strong gravitational lensing in this rotating squashed KK black hole and probe the effects of the rotation parameter of black hole and the scale of extra dimension on the deflection angle and the coefficients in the strong field limit. The plan of our paper is organized as follows. In Sec.II we introduce briefly the rotating squashed KK black hole [6]. In Sec.III we adopt to Bozza's method [22][23][24] and obtain the deflection angles for light rays propagating in the rotating squashed KK black hole. In Sec.IV we suppose that the gravitational field of the supermassive black hole at the center of our Galaxy can be described by this metric and then obtain the numerical results for the main observables in the strong gravitational lensing. Moreover, we also make a comparison among the strong gravitational lensings in the rotating squashed KK, the squashed KK Gödel and four-dimensional Kerr black hole spacetimes. At last, we present a summary. II. THE ROTATING KALUZA-KLEIN BLACK HOLE SPACETIME WITH SQUASHED HORIZON Let us now review briefly a rotating squashed KK black hole without charge, which can be obtained by applying the squashing transformation techniques to a five-dimensional Kerr black hole with two equal angular momenta [6]. In terms of Meurer-Cartan 1-forms, the metric of a rotating KK black bole has a form ds 2 = −dt 2 + Σ 0 ∆ 0 k(r) 2 dr 2 + r 2 + a 2 4 [k(r)(σ 2 1 + σ 2 2 ) + σ 2 3 ] + M r 2 + a 2 (dt − a 2 σ 3 ) 2 ,(1) with σ 1 = − sinψdθ + cosψ sin θdφ, σ 2 = cosψdθ + sinψ sin θdφ, σ 3 = dψ + cos θdφ,(2) where 0 < θ < π, 0 < φ < 2π and 0 <ψ < 4π. The parameters are given by Σ 0 = r 2 (r 2 + a 2 ), ∆ 0 = (r 2 − r 2 + )(r 2 − r 2 − ), k(r) = (r 2 ∞ − r 2 + )(r 2 ∞ − r 2 − ) (r 2 ∞ − r 2 ) 2 .(3) The quantities M and a are related to the mass and angular momenta of black hole, respectively. r ∞ corresponds to the spatial infinity. The polar coordinate r is limited in the range 0 < r < r ∞ . The outer and inner horizons are located at r = r + and r = r − , which are relate to the parameters M , a by a 4 = (r + r − ) 2 and M − 2a 2 = r 2 + + r 2 − . The shape of black hole horizon is deformed by the parameter k(r + ). In this black hole spacetime (1), the intrinsic singularity is the just one at r = 0, while r ± and r ∞ are coordinate singularities. As in [6], one can introduce a new radial coordinate ρ =ρ 0 r 2 r 2 ∞ − r 2 ,(4)withρ 2 0 = (r 2 ∞ + a 2 )[(r 2 ∞ + a 2 ) 2 − M r 2 ∞ ] 4r 4 ∞ ,(5) and then rewrite the metric (1) as ds 2 = −dt 2 + U dρ 2 + R 2 (σ 2 1 + σ 2 2 ) + W 2 σ 2 3 + V (dt − a 2 σ 3 ) 2 ,(6) where K 2 = ρ +ρ 0 ρ + a 2 r 2 ∞ +a 2ρ0 , V = M r 2 ∞ + a 2 K 2 , W 2 = r 2 ∞ + a 2 4K 2 , R 2 = (ρ +ρ 0 ) 2 K 2 , U = r 2 ∞ r 2 ∞ + a 2 2ρ 2 0 W 2 − r 2 ∞ 4 ρ ρ+ρ0 V .(7) The parameterρ 0 is a scale of transition from five-dimensional spacetime to an effective four-dimensional one. As the rotation parameter a tends to zero, one can find that the metric (6) reduces to that of a five-dimensional Schwarzschild black hole with squashed horizon. In the limit ρ → ∞, i.e, r → r ∞ , it is easy to find that there is a cross-term between dt and σ 3 in the asymptotic form of the metric (6). However, this cross-term can be vanished by changing the coordinates as [6] t = h t,ψ = ψ − j t. where h = (r 2 ∞ + a 2 ) 2 − M r 2 ∞ (r 2 ∞ + a 2 ) 2 + M a 2 , j = 2M a (r 2 ∞ + a 2 ) 2 + M a 2 .(9) This means that the asymptotic topology of the spacetime (6) is the same as that of the Schwarzschild squashed KK black hole spacetime. The Komar mass M k of the rotating squashed KK black hole (6) can be given by [37,38] M k = M π 2G 5 r 2 ∞ + a 2 2 − M a 2 (r 2 ∞ + a 2 ) 2 + M a 2 (r 2 ∞ + a 2 ) 2 − M r 2 ∞ = M 4G 4 r 2 ∞ + a 2 2 − M a 2 (r 2 ∞ + a 2 ) 2 + M a 2 r 2 ∞ + a 2 (r 2 ∞ + a 2 ) 2 − M r 2 ∞ ,(10) where G 5 and G 4 are the five-dimensional and four-dimensional gravitational constants, respectively. Therefore, in the rotating squashed KK black hole spacetime, the relationship between G 5 and G 4 can be expressed as G 5 = 2πr ′ ∞ G 4 ,(11) with r ′ ∞ = (r 2 ∞ + a 2 ) 2 + M a 2 r 2 ∞ + a 2 .(12) The expression of r ′ ∞ is more complicated than that of r ∞ . However, in the rotating squashed KK black hole spacetime (6), one can find [6] that the parameter r ′ ∞ for the compactified dimension is better than r ∞ because the geometric interpretation is clearer for r ′ ∞ than for r ∞ . As a disappears, we find that r ′ ∞ reduces to r ∞ and then the relationship (11) tend to the usual form ( i.e., G 5 = 2πr ∞ G 4 ) in the Schwarzschild squashed KK black hole spacetime. As in Ref. [13], we can rewritten Komar mass as M k = πr ′ ∞ ρM G5 = ρM 2G4 , which implies that ρ M can be expressed as ρ M = 2G 4 M k = M 2 r 2 ∞ + a 2 2 − M a 2 (r 2 ∞ + a 2 ) 2 + M a 2 r 2 ∞ + a 2 (r 2 ∞ + a 2 ) 2 − M r 2 ∞ .(13) In order to simplify the calculation, we here introduce a new radial coordinate change r ′ 2 = (r 2 ∞ + a 2 ) 2 + M a 2 r 4 ∞ + r 2 a 2 r 2 ,(14) a new scale of extra dimension ρ 2 0 = r ′ 2 ∞ − M 4 ,(15) and a new rotation parameter b = M r 2 ∞ + a 2 a,(16) we find that the radial coordinate (4) and the quantity ρ M can be rewritten as ρ = ρ 0 r ′ 2 r ′ 2 ∞ − r ′ 2 ,(17) and ρ M = ρ 0 M r ′2 ∞ − M r 2 ∞ + a 2 2 − M a 2 (r 2 ∞ + a 2 ) 2 + M a 2 = M ρ 0 r ′ 2 ∞ − M (1 − 2b 2 r ′ 2 ∞ ),(18) respectively. With help of these operation, one can find that the metric of the rotating squashed KK black hole spacetime (6) can be expressed as ds 2 = −A(ρ)dt 2 + B(ρ)dρ 2 + C(ρ)(dθ 2 + sin 2 θdφ 2 ) + D(ρ)(dψ + cos θdφ) 2 − 2H(ρ)dt(dψ + cos θdφ),(19) with A(ρ) = 4 − V(aj − 2) 2 − 4j 2 W 2 4h 2 , B(ρ) = U(ρ), C(ρ) = R 2 (ρ), D(ρ) = a 2 V + 4W 2 4 , H(ρ) = 2aV − (a 2 V + 4W 2 )j 4h ,(20) where K 2 = ρ + r 2 ∞ +a 2 r 2 ∞ ρ 0 ρ + a 2 r 2 ∞ ρ 0 , V = M r 2 ∞ + a 2 K 2 , W 2 = r 2 ∞ + a 2 4K 2 , R 2 (ρ) = (ρ + a 2 r 2 ∞ ρ 0 )(ρ + r 2 ∞ + a 2 r 2 ∞ ρ 0 ), U(ρ) = ρ 2 0 W 2 − r 2 ∞ 4 ρ KR V .(21) Among the parameters ρ 0 , M , a, r ∞ , r ′ ∞ , ρ M and b, there are only three independent parameters. For simplicity, we chose the parameters ρ 0 , ρ M and b as the independent parameters. The others are related to the selected three parameters by r 2 ∞ = 2ρ 0 (ρ M + ρ 0 ) − b 2 + b 4 − 4b 2 ρ 0 (ρ 0 − ρ M ) + 4ρ 2 0 (ρ M + ρ 0 ) 2 − b 2 (4ρ 2 0 − b 2 ) 2ρ 0 (ρ M − ρ 0 ) + b 2 + b 4 − 4b 2 ρ 0 (ρ 0 − ρ M ) + 4ρ 2 0 (ρ M + ρ 0 ) 2 , r ′ 2 ∞ = b 2 + 2ρ 0 ρ M + 2ρ 2 0 + b 4 − 4b 2 ρ 0 (ρ 0 − ρ M ) + 4ρ 2 0 (ρ M + ρ 0 ) 2 , M = b 2 + 2ρ 0 ρ M − 2ρ 2 0 + b 4 − 4b 2 ρ 0 (ρ 0 − ρ M ) + 4ρ 2 0 (ρ M + ρ 0 ) 2 , a = b 1 + 4ρ 2 0 − b 2 2ρ 0 (ρ M − ρ 0 ) + b 2 + b 4 − 4b 2 ρ 0 (ρ 0 − ρ M ) + 4ρ 2 0 (ρ M + ρ 0 ) 2 1 2 .(22) With these quantities, we can find that all of coefficients in the metric (19) can be expressed as the functions of the parameters ρ 0 , ρ M and b, which means that we can study the strong gravitational lensing in the rotating squashed KK black hole spacetime (6) through the standard form used in [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]. III. DEFLECTION ANGLE IN A ROTATING SQUASHED KALUZA-KLEIN BLACK HOLE SPACETIME In this section, we will study deflection angles of the light rays when they pass close to a rotating squashed KK black hole, and then probe the effects of the rotation parameter b and the scale of extra dimension ρ 0 on the deflection angle and the coefficients in the strong field limit. For simplicity, we here just consider that both the observer and the source lie in the equatorial plane in the rotating squashed KK black hole spacetime (19) and the whole trajectory of the photon is limited on the same plane With this condition θ = π 2 , we get the reduced metric in the form ds 2 = −A(ρ)dt 2 + B(ρ)dρ 2 + C(ρ)dφ 2 + D(ρ)dψ 2 − 2H(ρ)dtdψ.(23) From the null geodesics, it is easy to obtain three constants of motion E = − g 0µẋ µ = A(ρ)ṫ + H(ρ)ψ, L φ = g 3µẋ µ = C(ρ)φ, L ψ = g 4µẋ µ = D(ρ)ψ − H(ρ)ṫ.(24) where a dot represents a derivative with respect to affine parameter λ along the geodesics. E is the energy of the phone, L φ and L ψ correspond to its angular momentum in the φ and ψ directions, respectively. Making use of these three constants, one can find that the equations of motion of the photon can be expressed further as dt dλ = D(ρ)E − H(ρ)L ψ H 2 (ρ) + A(ρ)D(ρ) , dφ dλ = L φ C(ρ) , dψ dλ = H(ρ)E + A(ρ)L ψ H 2 (ρ) + A(ρ)D(ρ) . (25) dρ dλ 2 = 1 B(ρ) D(ρ)E − 2H(ρ)EL ψ − A(ρ)L 2 ψ H 2 (ρ) + A(ρ)D(ρ) − L 2 φ C(ρ) .(26) Considering the θ-component of the null geodesics in the equatorial plane (θ = π 2 ), we have dφ dλ D(ρ) dψ dλ − H(ρ) dt dλ = 0,(27) which implies that either dφ dλ = 0 or L ψ = D(ρ) dψ dλ − H(ρ) dt dλ = 0. As done in the usual squashed KK black hole spacetimes [33,34], here we set L ψ = 0, which means that the total angular momentum J of the photo is equal to the constant L φ and the effective potential for the photon passing close to the black hole can be written as V (ρ) = 1 B(ρ) D(ρ)E H 2 (ρ) + A(ρ)D(ρ) − L 2 φ C(ρ) .(28) With this effective potential, one can obtain that the impact parameter and the equation of circular photon orbits are u = J = C(ρ)D(ρ) H(ρ) 2 + A(ρ)D(ρ) ,(29) and D(ρ) H(ρ) 2 + A(ρ)D(ρ) C ′ (ρ) − C(ρ) D(ρ) 2 A ′ (ρ) + 2D(ρ)H(ρ)H ′ (ρ) − H(ρ) 2 D ′ (ρ) = 0,(30) respectively. Here we set E = 1. The equations (29) and (30) are similar to those in the squashed KK Gödel black hole spacetime because their metric have similar forms in the equatorial plane, but they are more complex than those in the usual spherical symmetric black hole spacetime. As the rotation parameter b → 0, we find that the function H(ρ) → 0, which yields that the impact parameter (29) and the equation of circular photon orbits (30) reduce to those in the usual Schwarzschild squashed KK black hole spacetime [33]. The biggest real root external to the horizon of equation (30) defines the marginally stable circular radius of photon. For the rotating squashed KK black hole spacetime (19), the equation of circular photon orbits takes the form Aρ ′ 5 + Bρ ′ 4 + Cρ ′ 3 + Dρ ′ 2 + Eρ ′ + F = 0(31) where the variable ρ ′ is related to ρ by ρ ′ = ρ ρ 0 + b 2 2ρ 0 ρ M − 2ρ 2 0 + b 4 − 4b 2 ρ 0 (ρ 0 − ρ M ) + 4ρ 2 0 (ρ M + ρ 0 ) 2 ,(32) and the coefficients are A = 8ρ 2 0 b 2 + 2ρ 0 ρ M + 2ρ 2 0 + b 4 − 4b 2 ρ 0 (ρ 0 − ρ M ) + 4ρ 2 0 (ρ M + ρ 0 ) 2 , B = 6b 2 − 6ρ 0 ρ M + 10ρ 2 0 − 3 b 4 − 4b 2 ρ 0 (ρ 0 − ρ M ) + 4ρ 2 0 (ρ M + ρ 0 ) 2 × 3b 2 + 2ρ 0 ρ M + 2ρ 2 0 + b 4 − 4b 2 ρ 0 (ρ 0 − ρ M ) + 4ρ 2 0 (ρ M + ρ 0 ) 2 − 12b 4 , C = 2 3b 2 − 2ρ 0 ρ M + 2ρ 2 0 − b 4 − 4b 2 ρ 0 (ρ 0 − ρ M ) + 4ρ 2 0 (ρ M + ρ 0 ) 2 × 3b 2 + 2ρ 0 ρ M + 2ρ 2 0 + b 4 − 4b 2 ρ 0 (ρ 0 − ρ M ) + 4ρ 2 0 (ρ M + ρ 0 ) 2 − 8b 4 , D = 4b 2 3b 2 + 2ρ 2 0 , E = 6b 4 , F = b 4 .(33) Obviously, this equation depends on both the rotation parameter b and the scale of transition ρ 0 . The presence of the rotation parameter b makes the equation more complex so that it is impossible to get an analytical form for the marginally circular photon orbit radius in this case. As b → 0, we can find that since the coefficients D, E and F vanish the Eq.(31) reduces to a quadratic equation 2ρ 2 + (ρ 0 − 3ρ M )ρ − 2ρ 0 ρ M = 0,(34) and the marginally circular photon orbit radius becomes ρ ps = 3ρM −ρ0+ √ 9ρ 2 M +10ρM ρ0+ρ 2 0 4 , which is consistent with that in the Schwarzschild squashed KK black hole [33]. As ρ 0 → 0, one can get ρ ps = 1 4 (3ρ M + 9ρ 2 M − 8b 2 ) , which decreases with the rotation parameter b and tends to 3 2 ρ M as b disappears. Moreover, in the limit ρ 0 → ∞, we have ρ ps = 2ρ M − b 2 4ρ M + 1 2 Q 2 + (6ρ 2 M − 2b 2 )Q + 4b 4 6Q + 1 2 Q(12ρ 2 M − Q) − 4b 2 (Q + b 2 ) 6Q + √ 6Qρ M (2ρ M − b 2 ) Q 2 + (6ρ 2 M − 2b 2 )Q + 4b 4 1 2 ,(35) with Q = b 4 3 27ρ 2 M − 8b 2 + 3ρ M 81ρ 2 M − 48b 2 1 3 .(36) Obviously, it also decreases with the rotation parameter b. In Fig.(1 The deflection angles φ and ψ for the photon coming from infinite in a rotating KK black hole spacetime can be expressed as α φ (ρ s ) = I φ (ρ s ) − π, α ψ (ρ s ) = I ψ (ρ s ) − π,(37) respectively. The quantities I φ (ρ s ) and I ψ (ρ s ) have the forms I φ (ρ s ) = 2 ∞ ρs B(ρ)F (ρ)C(ρ s ) C(ρ) 1 F (ρ s ) − F (ρ)C(ρs) C(ρ) dρ,(38)I ψ (ρ s ) = 2 ∞ ρs H(ρ) D(ρ) B(ρ)F (ρ s ) F (ρ) 1 F (ρ s ) − F (ρ)C(ρs) C(ρ) dρ,(39) with F (ρ) = H 2 (ρ) + A(ρ)D(ρ) D(ρ) ,(40) where ρ s is the closest approach distance of the light ray. It is clear that both of the deflection angles increase when parameter ρ s decreases. If ρ s is equal to the marginally stable circular radius of photon ρ ps , one can find that both of the deflection angles becomes unboundedly large and the photon is captured in a circular orbit around the black hole. Let us now discuss the behavior of the deflection angles of the light rays in a rotating squashed KK black hole spacetime. It is interesting to note that the deflection angle α φ (ρ s ) is independent of whether the photon goes with or against the rotation of the black hole because the integral I φ (ρ s ) is function of the rotation parameter b 2 . However, from Eq. (39), we find that the integral I ψ (ρ s ) contains the factor b, which means that the deflection angle α ψ (ρ s ) for the photon traveling in the same direction as the rotation of the black hole is different from that traveling in converse direction. This tells us that although the black hole has the rotation paraters both in the ψ and φ directions, in the equatorial plane θ = π 2 the rotation of the black hole is really in the ψ direction rather than in the φ direction, which is also shown in the induce metric (23) where the only cross-term is dtdψ. It means that the gravitational lensing by the rotating squashed KK black hole is different from that of usual four-dimensional Kerr black hole, which could in theory help us to detect the extra dimension through the gravitational lens. When the rotation parameter b vanishes, one can find that the function H(ρ) = 0 and then the deflection angle of ψ tends to zero, which reduces to that of in the usual Schwarzschild squashed KK black hole spacetime [33]. As in [34], we will focus only on investigating the deflection angle in the φ direction when the light rays pass close to the black hole in the equatorial plane since it could be observed really by our astronomical experiments. On the other hand, it is very convenient for us to compare with the results obtained in the usual four-dimensional black hole spacetimes. As in [22,23], one can define a variable z = 1 − ρs ρ , and rewrite Eq.(38) as I φ (ρ s ) = 1 0 R(z, ρ s )f (z, ρ s )dz,(41) with R(z, ρ s ) = 2 ρ 2 ρ s C(ρ) B(ρ)F (ρ)C(ρ s ) = 4ρ s r 2 ∞ (r 2 ∞ + a 2 ) 2 + M a 2 (r 2 ∞ + a 2 ) 2 − M r 2 ∞ ρ 0 (r 2 ∞ + a 2 )(ρ s r 2 ∞ + a 2 ρ 0 )[ρ s r 2 ∞ + (r 2 ∞ + a 2 )ρ 0 ] [ρ s r 2 ∞ + (1 − z)a 2 ρ 0 ][ρ s r 2 ∞ + (1 − z)(r 2 ∞ + a 2 )ρ 0 ] × [(r 2 ∞ + a 2 ) 2 + M a 2 ]ρ s r 4 ∞ + 2a 2 r 2 ∞ ρ 0 ρ s (1 − z)(r 2 ∞ + a 2 )(r 2 ∞ + M + a 2 ) + a 2 ρ 2 0 (r 2 ∞ + a 2 )(M + a 2 )(1 − z) 2 −1/2 ,(42) and f (z, ρ s ) = 1 F (ρ s ) − F (ρ)C(ρ s )/C(ρ) .(43) The function R(z, ρ s ) is regular for all values of z and ρ s . From Eq.(43), we find that the function f (z, ρ s ) diverges as z tends to zero, i.e., as the photon approaches the marginally circular photon orbit. Therefore, we can split the integral (41) into the divergent part I D (ρ s ) and the regular one I R (ρ s ) I D (ρ s ) = 1 0 R(0, ρ ps )f 0 (z, ρ s )dz, I R (ρ s ) = 1 0 [R(z, ρ s )f (z, ρ s ) − R(0, ρ ps )f 0 (z, ρ s )]dz.(44) Following in refs. [22,23], we can expand the argument of the square root in f (z, ρ s ) to the second order in z f s (z, ρ s ) = 1 p(ρ s )z + q(ρ s )z 2 ,(45) with p(ρ s ) = ρ s C(ρ s ) C ′ (ρ s )F (ρ s ) − C(ρ s )F ′ (ρ s ) , q(ρ s ) = ρ 2 s 2C(ρ s ) 2C ′ (ρ s )C(ρ s )F ′ (ρ s ) − 2C ′ (ρ s ) 2 F (ρ s ) + F (ρ s )C(ρ s )C ′′ (ρ s ) − C 2 (ρ s )F ′′ (ρ s ) . (46) Comparing Eq. (30) with Eq.(46), one can find that if ρ s tends to ρ ps the coefficient p(ρ s ) vanishes. This means that the leading term of the divergence in f s (z, ρ s ) is z −1 and the integral (41) diverges logarithmically. Thus, in the strong field region, the deflection angle in the φ direction can be approximated very well as [22] Here the quantity D OL is the distance between observer and gravitational lens, θ = u/D OL is the angular separation between the lens and the image, the subscript "ps" represent the evaluation at ρ = ρ ps . Similarly, one can obtain the strong gravitational lensing formula for the deflection angle in the ψ direction ( α ψ (θ)), which has a similar form with the coefficients differed slightly fromā andb in Eq.(48). As ρ s tends to ρ ps , we find that the deflection angle α ψ (θ) also diverges logarithmically. Since α ψ (θ) cannot actually be observed by astronomical experiments, we do not consider it in the following discussion. for the larger one. Furthermore, in Fig.(4), we plotted the change of the deflection angle α(θ) estimated at u = u ps + 0.003 with ρ 0 and b, which tells us that in the strong field limit the deflection angles have similar properties of the coefficientā. This means that the deflection angles of the light rays are dominated by the logarithmic term in this case. α(θ) = −ā log θD OL u ps − 1 +b + O(u − u ps ),(47)b = −π + b R +ā log ρ 2 hs [C ′′ (ρ ps )F (ρ ps ) − C(ρ ps )F ′′ (ρ ps )] u ps F 3 (ρ ps )C(ρ ps ) , b R = I R (ρ ps ), u ps = C(ρ ps ) F (ρ ps ) .(48 IV. OBSERVATIONAL GRAVITATIONAL LENSING PARAMETERS In this section, we will estimate the numerical values for the observables of gravitational lensing in the strong field limit by assuming that the spacetime of the supermassive black hole at the Galactic center of Milky Way can be described by the rotatiing squashed KK black hole metric (19) and then probe the effects of the rotation parameter b and the scale parameter ρ 0 on the observables in the strong gravitational lensing. As the source and observer are far enough from the lens, the lens equation can be approximated well as [23] γ = D OL + D LS D LS θ − α(θ) mod 2π(49) where γ is the angle between the direction of the source and the optical axis. D LS is the lens-source distance and D OL is the observer-lens distance. θ = u/D OL is the angular separation between the lens and the image. Following ref. [23], we here consider only the case in which the source, lens and observer are highly aligned. In this simplest case, the angular separation between the lens and the n−th relativistic image can be written as θ n ≃ θ 0 n 1 − u ps e n (D OL + D LS ) aD OL D LS ,(50)with θ 0 n = u ps D OL (1 + e n ), e n = eb +|γ|−2πn a .(51) Here θ 0 n is the image position corresponding to α = 2nπ, and n is an integer. As n → ∞, one can find from Eqs.(51) that e n → 0, which implies that the minimum impact parameter u ps and the asymptotic position of a set of images θ ∞ obey a simple form u ps = D OL θ ∞ .(52) In order to obtain the coefficientsā andb, one needs to separate the outermost image from all the others. Following refs. [22,23], we consider here the simplest case where only the outermost image θ 1 is resolved as a single image and all the remaining ones are packed together at θ ∞ . And then the angular separation between the first image and other ones s and the ratio of the flux from the first image and those from the all other images R 0 can be simplified further as [22,23] s = θ 1 − θ ∞ = θ ∞ eb −2π a , R 0 = µ 1 ∞ n=2 µ n = e 2π a .(53) Therefore, one can estimate the strong deflection limit coefficientsā,b and the minimum impact parameter Recently, the mass of the central object of our Galaxy is estimated to be 4.4 × 10 6 M ⊙ and its distance is around 8.5kpc [39]. This means the ratio of the mass of the central object to the distance G 4 M/D OL ≈ 2.4734 × 10 −11 . Here M is the mass of the black hole and D OL is the distance between the lens and the observer in the ρ coordination rather than that in r coordination because that in the five-dimensional spacetime the dimension of the black hole mass M is the square of that in the polar coordination r. Finally, we make a comparison among the strong gravitational lensings in the rotating squashed KK, the squashed KK Gödel and four-dimensional Kerr black hole spacetimes. For the photon moving along the equatorial plane in the four-dimensional Kerr black hole spacetime [23,24], we know that all of the photon sphere radius, the angular position of the relativistic images θ ∞ and the relative magnitudes r m in strong in the background of the KK black holes with squashed horizons contain the information of the size of the extra dimension, which could open a possible window to observe extra dimensions in the future. FIG. 1 : 1), we set ρ M = 1 and plot the variety of the marginally stable circular radius of photon ρ ps with the parameters b and ρ 0 . It shows that with increase of the scale of transition ρ 0 , ρ ps first decreases and then increases in the rotating squashed KK black hole background, but increases monotonously in the Schwarzschild squashed KK black hole spacetime ( i.e., b = 0 ). In the squashed KK Gödel black hole spacetime, we find[34] that with the increase of ρ 0 , ρ ps increases for the smaller global rotation parameter j and decreases for the larger one, which implies that the effects of the rotation parameter b of the black hole itself on the gravitational lensing is different from that of the global rotation parameter j of the Gödel Universe background. For fixed ρ 0 , it is easy to obtain that ρ ps decreases monotonically with the increase of the rotation parameter b. Moreover, we know that in the Kerr black hole, the marginally stable circular radius of photon are different for the photons winding in the same direction (i.e., a > 0) or in the converse direction (i.e., a < 0) of the black hole rotation. However, one can obtain that in the rotating squashed KK black hole spacetime the marginally stable circular radius of photon ρ ps is independent of the sign of the rotation parameter b since all of quantities in Eqs.(31) and(33) are the function of b 2 . Thus, the marginally stable circular radius of photon in the rotating squashed KK black hole spacetime keeps the same whether the photon moves in the same or converse direction of the rotation of the black hole. It is similar to that in the squashed KK Gödel black hole spacetime, but it is different from that in the usual Kerr black hole spacetime. Variety of the marginally stable circular radius of photon ρps with the parameters ρ0/ρM and b/ρM in a rotating squashed KK black hole spacetime. FIG. 2 : 2Variety of the coefficientā with the parameters ρ0/ρM and b/ρM in a rotating squashed KK black hole spacetime.ā = R(0, ρ ps ) 2 q(ρ ps ) , FIG. 3 : 3Variety of the coefficientb with the parameters ρ0/ρM and b/ρM in a rotating squashed KK black hole spacetime. FIG. 4 : 4Deflection angles in the squashed KK Gödel black hole spacetime evaluated at u = ups + 0.003 as functions of ρ0/ρM and b/ρM .We are now in position to probe the properties of strong gravitational lensing in the rotating squashed KK black hole spacetime and explore the effects of the rotation parameter b and the scale of transition ρ 0 on the deflection angle in the strong field limit. In the Figs.(2)-(3), we plotted numerically the dependence of the coefficients (ā andb ) on the parameters b and ρ 0 . It is shown that the coefficients (ā andb ) in the strong field limit are functions of the rotation parameter b and the scale of transition ρ 0 . The coefficientā increases monotonously with ρ 0 and b. In the extremely squashed case ρ 0 = 0, we find that the change ofā with the rotation parameter b is different from the change ofā with the global rotation parameter j in the squashed Gödel KK case in whichā is independent of j in this extremely squashed case. The variety ofb with ρ 0 and b becomes more complicated. With the increase of ρ 0 ,b increases for the smaller b and decreases for the larger one. With the increase of b,b first increases and then decreases for the smaller ρ 0 , and decreases monotonously uFIG. 5 : 5ps by measuring these three simple observations s, R 0 , and θ ∞ . Comparing their values with those predicted by the theoretical models, we can extract the characteristics information about the compact object stored in the strong gravitational lensing. Gravitational lensing by the Galactic center black hole. Variation of the values of the angular position θ∞ with parameters ρ0/ρM and b/ρM in a rotating squashed KK black hole spacetime. FIG. 6 :FIG. 7 : 67Combing with Eqs. (48) and (53), we can estimate the values of the coefficients and observables in strong gravitational lensing in the rotating squashed KK black hole spacetime. InTable I,we list the numerical values of θ ∞ , s and r m (which is related to R 0 by r m = 2.5 log R 0 ) for the different values of b/ρ M and ρ 0 /ρ M . Moreover, Gravitational lensing by the Galactic center black hole. Variation of the values of the angular separation s with parameters ρ0/ρM and b/ρM in a rotating squashed KK black hole spacetime. Gravitational lensing by the Galactic center black hole. Variation of the values of the relative magnitudes rm with parameters ρ0/ρM and b/ρM in a rotating squashed KK black hole spacetime.we also present the dependence of these observables on the parameters b/ρ M and ρ 0 /ρ M in Figs.(5)-(7). It is shown that with the increase of the scale ρ 0 , the angular position of the relativistic images θ ∞ first decreases and then increases for the non-zero b and increases monotonously in the case with b = 0. For fixed ρ 0 /ρ M , θ ∞ decreases monotonously with b. Moreover, we also find that with the increase of the rotation parameter b and the scale of extra dimension ρ 0 , the angular separation s increase, while the relative magnitudes r m decrease monotonously. gravitational lensing decrease with the rotation parameter a of the black hole, while the coefficientā, the deflection angle α(θ) and the observable s in strong gravitational lensing increase with the rotation parameter a. These effects of a on the strong gravitational lensing in the Kerr black hole are similar not only to those of the rotation parameter b in the rotating squashed KK black hole spacetime, but also to those of the rotation parameter j of cosmological background in the squashed KK Gödel case, which can be understandable since all of a, b and j are the rotation parameters. Therefore, these above features could be looked as the universal effects of the rotation on the strong gravitational lensing. However, there exist some distinct properties among the strong gravitational lensings in these three black hole spacetimes. In the rotating squashed KK and the squashed KK Gödel black holes, the photon sphere radius, the coefficientā,b and the deflection angle α(θ) in the φ direction are independent of whether the photon goes with or against the rotation of the background.While in the Kerr black hole, the values of these quantities for the prograde photons are different from those for the retrograde photons. Comparing with the strong gravitational lensings in the rotating squashed KK and the squashed KK Gödel black holes, we find that with increase of ρ 0 , the coefficientā, the deflection angle α(θ) and the observable s increase and the relative magnitudes r m decreases in these two kinds of black holes. Moreover, we find also that there also exist some different effects of ρ 0 on the marginally stable orbitradius ρ ps and the angular position of the relativistic images θ ∞ . With the increase of ρ 0 , the quantities ρ ps and θ ∞ first decreases and then increases in the rotating squashed KK black hole for fixed rotation parameter b, but in the squashed KK Gödel black hole they increase for the smaller global rotation parameter j and decrease for the larger one. In the extremely squashed case ρ 0 = 0, the coefficientā in the squashed KK Gödel black hole is a constant 1 and is independent of the global rotation of the Gödel Universe and in the rotating squashed KK black hole it increases monotonously with the rotation parameter b. These information could help us to understand further the effects of the rotation parameter and the scale of extra dimension on the strong gravitational lensing in the black hole spacetimes. V. SUMMARY We have investigated the strong gravitational lensing in the rotating squashed KK black hole and find that the rotation parameter of black hole and the scale of extra dimension imprint in the radius of the marginally circular photon orbit, the deflection angle, the coefficientsā,b and the observational variables in strong field lensing. The marginally circular photon radius ρ ps , the angular position of the relativistic images θ ∞ and the relative magnitudes r m in strong gravitational lensing decrease with the rotation parameter b of the black hole, while the coefficientā, the deflection angle α(θ) and the observable s in strong gravitational lensing increase with the rotation parameter b. With the increase of the extra dimension scale ρ 0 , the coefficientā, the deflection angle α(θ) and the observable s increase, while the relative magnitudes r m decreases monotonously and ρ ps first decreases and then increases in the rotating squashed KK black hole spacetime. Moreover, we find that in the rotating squashed KK black hole spacetime the marginally circular photon radius ρ ps , the coefficientā, b, the deflection angle α(θ) in the φ direction and the corresponding observational variables are independent of whether the photon goes with or against the rotation of the background, which is different with those of in the usual four-dimensional Kerr black hole spacetime. Comparing with the strong gravitational lensings the squashed KK Gödel black holes, we find that there exist some different effects of ρ 0 on the marginally circular photon radius ρ ps and the angular position of the relativistic images θ ∞ . With the increase of ρ 0 , the quantities ρ ps and θ ∞ first decreases and then increases in the rotating squashed KK black hole for fixed rotation parameter b, but in the squashed KK Gödel black hole they increase for the smaller global rotation parameter j and decrease for the larger one. In the extremely squashed case ρ 0 = 0, the coefficientā in the squashed KK Gödel black hole is a constant 1 and is independent of the global rotation of the Gödel Universe, but it increases monotonously with the rotation parameter b in the rotating squashed KK black hole. TABLE I : INumerical estimation for main observables and the strong field limit coefficients for the black hole at the center of our Galaxy, which is supposed to be described by a rotating squashed KK black hole spacetime. rm = 2.5 log R0 . H Ishihara, K Matsuno, Prog. Theor. Phys. 116417H. Ishihara and K. Matsuno, Prog. Theor. Phys. 116, 417 (2006). . S S Yazadjiev, Phys. Rev. D. 7424022S. S. Yazadjiev, Phys. Rev. D 74, 024022 (2006). . Y Brihaye, E Radu, Phys. Lett. B. 641212Y. Brihaye and E. Radu, Phys. Lett. B 641, 212 (2006). . H Ishihara, M Kimura, K Matsuno, S Tomizawa, Phys. Rev. D. 7447501H. Ishihara, M. Kimura, K. Matsuno, and S. Tomizawa, Phys. Rev. D 74, 047501 (2006); . Class. Quantum Grav. 236919Class. Quantum Grav. 23, 6919 (2006). . T Harmark, V Niarchos, N A Obers, Class. Quant. Grav. 24T. Harmark, V. Niarchos and N. A. Obers, Class. Quant. Grav. 24, R1-R90 (2007). . T Wang, Nucl. Phys. B. 75686T. Wang, Nucl. Phys. B 756, 86 (2006). . V Frolov, D Stojkovic, Phys. Rev. D. 6784004V. Frolov and D. Stojkovic, Phys. Rev. D 67, 084004 (2003); . H Nomura, S Yoshida, M Tanabe, K Maeda, Prog. Theor. Phys. 114H. Nomura, S. Yoshida, M. Tanabe and K. Maeda, Prog. Theor. Phys. 114, 707-712 (2005). . S Tomizawa, H Ishihara, K Matsuno, T Nakagawa, arXiv:0803.3873Prog. Theor. Phys. 121S. Tomizawa, H. Ishihara, K. Matsuno, and T. Nakagawa, Prog. Theor. Phys. 121, 823 (2009), arXiv:0803.3873. . H Ishihara, J Soda, Phys. Rev. D. 7664022H. Ishihara and J. Soda, Phys. Rev. D 76, 064022 (2007). . S Chen, B Wang, R Su, Phys. Rev. D. 7724039S. Chen, B. Wang and R. Su, Phys. Rev. D 77, 024039 (2008). . X He, B Wang, S Chen, Phys. Rev. D. 7984005X. He, B. Wang and S. Chen, Phys. Rev. D 79, 084005 (2009); . X He, S Chen, B Wang, R G Cai, C Lin, Phys.Lett.B. 665392X. He, S. Chen, B. Wang, R. G. Cai, C. Lin, Phys.Lett.B 665, 392 (2008). . H Ishihara, M Kimura, R A Konoplya, K Murata, J Soda, A Zhidenko, Phys.Rev.D. 7784019H. Ishihara, M. Kimura, R. A. Konoplya, K. Murata, J. Soda and A. Zhidenko, Phys.Rev.D 77, 084019 (2008). . K Matsuno, H Ishihara, Phys. Rev. D. 80104037K. Matsuno and H. Ishihara, Phys. Rev. D 80, 104037 (2009). . A Einstein, Science. 84506A. Einstein, Science, 84, 506 (1936). Gravitational Lenses. P Schneider, J Ehlers, E E Falco, Springer-VerlagBerlinP. Schneider, J. Ehlers, and E. E. Falco "Gravitational Lenses" Springer-Verlag, Berlin, (1992). . C Darwin, Proc. of the Royal Soc. of London. 249180C. Darwin, Proc. of the Royal Soc. of London 249 180 (1959). . K S Virbhadra, G F R Ellis, Phys. Rev.D. 65103004K. S. Virbhadra and G. F. R. Ellis, Phys. Rev.D 65, 103004(2002); . K S Virbhadra, D Narasimha, S , K. S. Virbhadra, D. Narasimha and S. M. . ; K S Chitre, G F R Virbhadra, Ellis, Astron. Astrophys. 33784003Phys. Rev. DChitre, Astron. Astrophys. 337 1 (1998). K. S. Virbhadra, G. F. R. Ellis, Phys. Rev. D 62 084003 (2000); . C M Claudel, K S Virbhadra, G F R Ellis, J. Math. Phys. 42818C. M. Claudel, K. S. Virbhadra, G. F. R. Ellis, J. Math. Phys. 42 818 (2001). . S Frittelly, T P Kling, E T Newman, Phys. Rev. D. 6164021S. Frittelly, T. P. Kling and E. T. Newman, Phys. Rev. D 61, 064021 (2000). . V Bozza, S Capozziello, G Lovane, G Scarpetta, Gen. Rel. and Grav. 331535V. Bozza, S. Capozziello, G. lovane and G. Scarpetta, Gen. Rel. and Grav. 33, 1535 (2001). . E F Eiroa, G E Romero, D F Torres, Phys. Rev. D. 6624010E. F. Eiroa, G. E. Romero and D. F. Torres, Phys. Rev. D 66, 024010 (2002); . E F Eiroa, Phys. Rev. D. 7183010E. F. Eiroa, Phys. Rev. D 71, 083010 (2005); . E F Eiroa, Phys. Rev. D. 7343002E. F. Eiroa, Phys. Rev. D 73, 043002 (2006). . R Whisker, Phys. Rev. D. 7164004R. Whisker, Phys. Rev. D 71, 064004 (2005). . V Bozza, Phys. Rev. D. 66103001V. Bozza, Phys. Rev. D 66, 103001 (2002). . V Bozza, Phys. Rev. D. 67103006V. Bozza, Phys. Rev. D 67, 103006 (2003) . V Bozza, F De Luca, G Scarpetta, M Sereno, Phys. Rev. D. 728300V. Bozza, F. De Luca, G. Scarpetta, and M. Sereno, Phys. Rev. D 72, 08300 (2005); . V Bozza, F De Luca, G Scarpetta, Phys. Rev. D. 7463001V. Bozza, F. De Luca, and G. Scarpetta, Phys. Rev. D 74, 063001 (2006). . G N Gyulchev, S S Yazadjiev, Phys. Rev. D. 7523006G. N. Gyulchev and S. S. Yazadjiev, Phys. Rev. D 75 023006 (2007); . G N Gyulchev, S S Yazadjiev, Phys. Rev. D. 7883004G. N. Gyulchev and S. S. Yazadjiev, Phys. Rev. D 78 083004 (2008). . A Bhadra, Phys. Rev. D. 67103009A. Bhadra, Phys. Rev. D 67, 103009 (2003). . T Ghosh, S Sengupta, arXiv:1001.5129Phys. Rev. D. 8144013T. Ghosh and S. Sengupta, Phys. Rev. D 81, 044013 (2010), arXiv: 1001.5129. . A N Aliev, P Talazan, arXiv:0906.1465Phys. Rev. D. 8044023A. N. Aliev and P. Talazan, Phys. Rev. D 80, 044023 (2009), arXiv:0906.1465. . C Ding, C Liu, Y Xiao, L Jiang, R Cai, Phys. Rev. D. 88104007C. Ding, C. Liu, Y. Xiao, L. Jiang and R. Cai, Phys. Rev. D 88 104007 (2013); . E F Eiroa, C M Sendra, Phys. Rev. D. 88103007E. F. Eiroa and C. M. Sendra, Phys. Rev. D 88, 103007 (2013). . S Chen, J Jing, Phys. Rev. D. 85124029S. Chen and J. Jing, Phys. Rev. D 85, 124029 (2012). . S Wei, Y Liu, C Fu, K Yang, JCAP. 121053S. Wei, Y. Liu, C. Fu and K. Yang, JCAP 1210, 053 (2012); . S Wei, Y Liu, Phys. Rev. D. 8564044S. Wei, Y. Liu, Phys. Rev. D 85, 064044 (2012). . G V Kraniotis, Class. Quant. Grav. 2885021G. V. Kraniotis, Class. Quant. Grav. 28, 085021 (2011). . Y Liu, S Chen, J Jing, Phys. Rev. D. 81124017Y. Liu, S. Chen and J. Jing, Phys. Rev. D 81,124017 (2010). . S Chen, Y Liu, J Jing, Phys. Rev. D. 83124019S. Chen, Y. Liu and J. Jing, Phys. Rev. D 83, 124019 (2011) . J Sadeghi, H Vaez, arXiv:1310.4486Phys. Lett. B. 728J. Sadeghi, H. Vaez, Phys. Lett. B 728, 170-182 (2014), arXiv:1310.4486. . J Sadeghi, A Banijamali, H Vaez, arXiv:1205.0805J. Sadeghi, A. Banijamali and H. Vaez, arXiv:1205.0805. . A Komar, Phys. Rev. 113934A. Komar, Phys. Rev. 113, 934 (1959); . Phys. Rev. 1291873Phys. Rev. 129, 1873 (1963). R M Wald, General Relativity. ChicagoUniversity of Chicago PressR. M. Wald, General Relativity, ( University of Chicago Press, Chicago, 1984). . R Genzel, F Eisenhauer, S Gillessen, arXiv:1006.0064Rev. Mod. Phys. 823121R. Genzel, F. Eisenhauer and S. Gillessen, Rev. Mod. Phys. 82, 3121 (2010); arXiv:1006.0064.
{'fraction_non_alphanumeric': 0.07283647543914007, 'fraction_numerical': 0.05105226779798365, 'mean_word_length': 3.3843260188087774, 'pattern_counts': {'":': 0, '<': 9, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 83, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'We have investigated the strong gravitational lensing in a rotating squashed Kaluza-Klein (KK) black hole spacetime. Our result show that the strong gravitational lensings in the rotating squashed KK black hole spacetime have some distinct behaviors from those in the backgrounds of the fourdimensional Kerr black hole and of the squashed KK Gödel black hole. In the rotating squashed KK black hole spacetime, the marginally circular photon radius ρps , the coefficientā,b, the deflection angle α(θ) in the φ direction and the corresponding observational variables are independent of whether the photon goes with or against the rotation of the background, which is different with those in the usual four-dimensional Kerr black hole spacetime. Moreover, we also find that with the increase of the scale of extra dimension ρ0, the marginally circular photon radius ρps and the angular position of the relativistic images θ∞ first decreases and then increases in the rotating squashed KK black hole for fixed rotation parameter b, but in the squashed KK Gödel black hole they increase for the smaller global rotation parameter j and decrease for the larger one. In the extremely squashed case ρ0 = 0, the coefficientā in the rotating squashed KK black hole increases monotonously with the rotation parameter, but in the squashed KK Gödel black hole it is a constant and independent of the global rotation of the Gödel Universe. These information could help us to understand further the effects of the rotation parameter and the scale of extra dimension on the strong gravitational lensing in the black hole spacetimes.', 'arxivid': '1312.4128', 'author': ['Liyong Ji ', 'Songbai Chen ', 'Jiliang Jing ', "\nInstitute of Physics\nDepartment of Physics\nKey Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education\nHunan Normal University\n410081ChangshaHunanPeople's Republic of China\n", "\nHunan Normal University\n410081ChangshaHunanPeople's Republic of China\n"], 'authoraffiliation': ["Institute of Physics\nDepartment of Physics\nKey Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education\nHunan Normal University\n410081ChangshaHunanPeople's Republic of China", "Hunan Normal University\n410081ChangshaHunanPeople's Republic of China"], 'corpusid': 118715324, 'doi': '10.1007/jhep03(2014)089', 'github_urls': [], 'n_tokens_mistral': 15454, 'n_tokens_neox': 12945, 'n_words': 8096, 'pdfsha': '914ae8a6b5b76defe831dec1ba3e0626119ab30c', 'pdfurls': ['https://arxiv.org/pdf/1312.4128v2.pdf'], 'title': ['Strong gravitational lensing in a rotating Kaluza-Klein black hole with squashed horizons', 'Strong gravitational lensing in a rotating Kaluza-Klein black hole with squashed horizons'], 'venue': []}
arxiv
Application of the Trace Formula in Pseudointegrable Systems 24 Mar 2006 Stefanie Russ Institut für Theoretische Physik III Justus-Liebig-Universität Giessen D-35392GiessenGermany Jesper Mellenthin Laboratoire de Physique de la Matière Condensée Ecole Polytechnique F-91128PalaiseauFrance Application of the Trace Formula in Pseudointegrable Systems 24 Mar 2006(Dated: March 23, 2022)PACS numbers: PACS numbers: 0545Mt, We apply periodic-orbit theory to calculate the integrated density of states N (k) from the periodic orbits of pseudointegrable polygon and barrier billiards. We show that the results agree so well with the results obtained from direct diagonalization of the Schrödinger equation, that about the first 100 eigenvalues can be obtained directly from the periodic-orbit calculations in good accuracy.PACS numbers: PACS numbers: 05.45.Mt, The motion of a classical particle in a billiard system can show regular, chaotic or intermediate behavior, depending on the billiard geometry. A potential well of the same geometry as the corresponding classical billiard -a quantum billiard -reflects this behavior in the properties of its eigenvalues and eigenfunctions. A hallmark in the theory of chaotic systems is Gutzwiller's trace formula [1,2]. It expresses the density of quantum mechanical eigenstates g(k) semiclassically by a weighted sum over all classical periodic orbits i of lengths ℓ i and thus represents an intrinsic link between the classical and the quantum mechanical properties of a given system. Since the implementation of Gutzwiller's trace formula, periodic-orbit theory has been a subject of permanent interest (for a recent review see [3]). However, in chaotic systems, the formula cannot be easily applied since the number N(ℓ) of periodic orbits with lengths smaller than a given value ℓ increases exponentially with ℓ, leading to a divergence of the trace formula. Therefore, cut-offs have to be compensated by sophisticated techniques so that the practical application of the trace formula up to now has been very limited and in most cases only reproduces the smoothed density of states and about the 20 − 40 lowest individual eigenvalues [4,5,6]. Only for the special case of the hyperbola billiard, where after a suited rearrangment of the orbits, most contributions were made to cancel, about 150 eigenvalues have been reproduced [7]. In this paper, we concentrate on pseudointegrable billiards [8,9], whose spectral properties as e.g. the level statistics have been found intermediate between chaotic and integrable billiards [10,11,12,13]. Systems with rough boundaries may have chaotic or pseudointegrable classical dynamics, so that the theoretical understanding of pseudointegrable systems is as important as the one of chaotic systems. One prominent application of quantum billiards is e.g. the band gap of Andreev billiards where the density of states is of major importance and where pseudointegrable billiards behave similar to chaotic billiards [14]. Formulas equivalent to Gutzwiller's trace formula have also been established for regular [15] and for pseudointegrable billiards [8]. In these billiards, the number of periodic orbits smaller than ℓ only increases as N(ℓ) ∼ ℓ 2 [16,17,18], which diminishes the divergence problems. In this paper, we will show how the divergence problems can be overcome in these billiards and how the density of states and about the 100 lower eigenvalues can be calculated by periodic-orbit theory. Figure 1 shows some pseudointegrable geometries considered in this work together with some periodic orbits. Whereas rectangular systems are integrable, i.e. the motion of a particle in a rectangular billiard shows regular classical dynamics and the equations of motion can be integrated, pseudointegrable billiards are polygons with a certain number of rational angles ϕ i = n i π/m i , with n i , m i ∈ N and at least one n i > 1. Also barrier billiards [19] belong to this class (see Fig. 1(d)), as a barrier can be considered as part of the boundary with an inner angle of 2π. Pseudointegrable billiards are not integrable due to singularites arising at the salient corners and are classified by their genus number G = 1 + M 2 J i=1 n i − 1 m i .(1) Here, J is the number of angles and M is the least common multiple of the m i . In the geometries considered here, it is easy to see that every angle of value 3π/2 or 2π increases G by a value of 1, whereas the angles of π/2 do not contribute. We are interested in the density of states g(k). It is well-known that in differential form g(k) reads g(k) = g 0 (k) + g osc (k),(2) where k 2 ≡ 2mE/h 2 and m is the mass of the quantum mechanical particle in the potential well. g 0 (k) is a smooth term that can be obtained via the well-known Weyl formula [20] from the geometrical properties of the system. It does not require the knowledge of the individual orbits. Hence, the calculation of g(k) reduces to the oscillating part g osc (k) that is (for the billiards considered here) connected to the lengths and the areas of the periodic orbits. In both, integrable and pseudointegrable systems, the periodic orbits form families of fixed lengths ℓ i , which means that the starting point of an orbit can be shifted to at least one direction along the boundary without changing its length (see Fig. 1). Accordingly, the trajectories of all orbit families cover a finite area A i in phase space, where the index i counts the different families. Gutzwillers trace formula has also been extended to pseudointegrable billiards [8,18]. For the billiards considered here, where all orbits have an even number of reflections at the boundary walls, the boundary conditions do not play a role and the formula reads g osc (k) = k 2π 3 i A i ℓ 1/2 i cos(kℓ i − π 4 )(3) and applies also for integrable billiards. Second-order contributions coming e.g. from diffractive orbits have been neglected and the sum is carried out over all primitive (non-repeated) orbit families i and over its repetitions with multiple lenghts. Since the boundary conditions do not enter into Eq. (3), we concentrate in the following on Dirichlet boundary conditions. Even though the number of orbits N(ℓ) below a given length ℓ increases only quadratically with ℓ, one can easily show that also the trace formula for pseudointegrable systems, Eq. (3), diverges and therefore could not be used so far to calculate the density of states. However, since the divergence is weak, we can use a simple trick to achieve convergence, namely by considering the fluctuations of the integrated density of states, N osc (k) = g osc (k) dk = = 1 √ 2π 3 i A i k 2ℓ 3 i [sin(ℓ i k) − cos(ℓ i k)] + √ π 2ℓ 2 i   FrC   2ℓ i k π   − FrS   2ℓ i k π        ,(4) where FrS(x) and FrC(x) are the Fresnel sine and cosine integrals, respectively, which can be evaluated numerically. Replacing the sum over i by an integral over g(ℓ) dℓ and introducing the orbit density g(ℓ) = dN(ℓ)/dℓ ∼ ℓ, one can verify easily that the factors of ℓ We first test the formula on the rectangular billiard where both, the orbits and the quantum mechanical eigenvalues k 2 ν i ,ν j are known exactly, k 2 ν i ,ν j = π 2 (ν 2 i /L 2 x + ν 2 j /L 2 y ) with positive integers ν i and ν j , and the side lengths L x and L y of the rectangle. The orbit lengths are ℓ ν i ,ν j = 2 [(ν i L x ) 2 + (ν j L y ) 2 ] 1/2 and the areas are A i = 2A for the neutral orbit families (the simplest orbit families that bounce between two parallel walls [21]) and 4A for all other families, where A is the geometrical area of the system. In Figs. 2(a,b) we compare the integrated density of states N P O (k) = N 0 (k) + N osc (k) (straight lines) calculated from Eq. (4) with the corresponding density of states N EV (k) (circles), which has been obtained from the exact eigenvalues. N 0 (k) is taken from Weyl's formula (for Dirichlet boundary conditions), N 0 (k) = A 4π k 2 − Γ 4π k + 1 24 i ( π ϕ i − ϕ i π ),(5) where A is the area, Γ the boundary length of the billiard and the sum runs over all corners of angles ϕ i . One can see in In pseudointegrable billiards, the areas are different for the different families and normally much smaller than in integrable billiards, while the number of periodic orbits is larger. In Fig. 1(b), it is demonstrated how the periodic orbits can be labeled according to their numbers of transversals of the different segments: the system has two x-and two y-segments and the orbit shown here can be labelled as (2, 6, 2, 0), where the numbers design the number of transversals of the segments x 1 , x 2 , y 1 and y 2 (see also Ref. [18]). The orbit length and angle can be calculated from this information, whereas the area has to be calculated numerically. However, unfortunately not all combinations of integer transversals exist in the pseudointegrable systems, because many hypothetical orbits are pruned by the shielding of the corners. So, it can be seen easily in Fig. 1(b) that an orbit (2, 0, 2, 0) would not be possible in this geometry (however, this orbit exists in the geometry of Fig. 1(a)). Even found that for all considered systems, the results are stable beyond the first 10 4 orbits. Finally, we turn to the pseudointegrable billiards. As before, we apply Eq. (4) to obtain the density of states N P O (k) from periodic-orbit calculations. For comparison with the quantum mechanical eigenvalues, we solve the Schrödinger equation for the potential wells of Fig. 1 in its discretized version, i ′ ,j ′ (neighbors) (Ψ n (i ′ , j ′ ) − Ψ n (i, j)) = − k n ν 2 Ψ n (i, j),(6) where the indices (i, j) refer to points of a square lattice with ν lattice points per unit length. The sum over (i ′ , j ′ ) runs over all nearest neighbors of (i, j). Equation (6) depends on the usual Schrödinger equation, ∆Ψ n (i, j) = −k 2 n Ψ n (i, j), by a second-order Taylor expansion of the left-hand side up to the quadratic term. The errors due to the discretization are the higher orders of ν and thus decrease with increasing ν, while the eigenvalues are transformed via k 2 n → (k n /ν) 2 . We used a resolution of ν = 4, where the errors arising from the discretization (as calculated for the rectangle) are smaller than 0.03 percent for the first 100 eigenvalues. Equation (6) describes a matrix problem that we diagonalized by the Lanczos algorithm, yielding the numerical Lanczos eigenvalues k 2 L and the corresponding density of states N EV,L (k). In Fig. 3 Fig. 3(b) (solid lines). We can see that indeed, for about the first 100 eigenvalues, the agreement of the Lanczos eigenvalues with this fitted step function N P O fit (k) is quite good. Nearly all eigenvalues are located right at the steps of N P O fit (k), and we can obtain at least the first 100 eigenvalues k 2 P O by periodic-orbit theory. In Tab. I, we show as an example the 91st to the 100th eigenvalue. We calculated the number N m of mismatches among the first 100 levels, i.e. the number of cases where the ith periodic-orbit eigenvalue lies closer to the (i + 1)th or (i − 1)th Lanczos value than to the ith one. N m is also given in the table and lies around 20 percent. Naturally, the mismatches occur at energies, where the level distance is particularly small and N m is therefore highest for the system of Fig. 1(a), where the density of states increases fastest and the level distances are thus smallest. As last example, we consider the barrier billiard of Fig. 1(d) for three different heights of the barrier, h = 10, h = 50 and h = 100. In this case, the discretization of the lattice is a cruder approximation than before, since the barriers which should be of thickness zero, always occupy one grid point. In Fig. 4, we can see that at least for the systems with barrier heights h = 10 and h = 50, the agreement between the periodic-orbit and the Lanczos results is again very good with mismatches even below 20 percent (see Tab. II). Only for the billiard with the largest barrier height, the mismatches are larger which is, however, most probably due to the discretization procedure and not to the periodic-orbit calculations. A comparison to the eigenvalues calculated by some other procedures will be interesting. Finally, we want to compare our results to some other methods that were used in the past to determine eigenvalues of systems with chaotic classical dynamics by periodic-orbit theory. The first methed used e.g. in [5] is a "Gaussian smearing" of g(k). This means that convergence of the trace formula can be achieved by multiplying each term of Eq. (3) with the additional factor of exp(−l 2 i ǫ 2 /2) (where ǫ must be small). As a consequence, the delta peaks of g(k) are transformed into Gaussian functions of shapes ∼ exp(−k 2 ǫ −2 /2). We tested this method also on our systems and found that it works very well for the integrable rectangular systems, but not for the pseudointegrable systems. Similar as in our method, the small errors in the orbit areas lead to a large noise that is still increased by the "Gaussian smearing". Contrary to our method, where we were able to apply a clearly defined fit procedure to eliminate the noise from the steps in N(k), it is not possible to find the Gaussian functions in g(k) by a simple rule. Another method used in [7] calculates the eigenvalues from the zeros of the so-called dynamical zeta function that contains all orbit information. In very special cases, where every large primitive orbit can be decomposed into series of few small orbits, the dynamical zeta function can be calculated very easily. However, this condition is only fulfilled in rare cases and demands as minimal conditions that the orbits can be labelled into fundamental building blocks, where each combination of the blocks exists. We have seen that in the case of pseudointegrable billiards, the periodic orbits appear in a very unsystematic way and many hypothetical orbits are pruned, so that we think that this method is not helpful in our case. In summary, we have shown how the convergence problems of the trace formula can be overcome in systems, where the number of periodic orbits below a given length ℓ increases at most quadratically with ℓ, e.g. for integrable and for pseudointegrable billiards. We have shown that the integrated density of states can be reproduced in very good accuracy for several hundred eigenvalues. The calculations are very sensible to numerical errors, so that already error bars of about 0.1% destroy the shape of the step function of N P O (k) such that the steps are smeared out. In integrable billiards, where the orbits are known exactly, the eigenvalues can be found directly from the steps in N P O (k) (which we tested for the first 1500 eigenvalues). In pseudointegrable billiards, even if the curves follow all fluctuations of the spectra very well, a fit technique has to be used in order to find the first 100 individual eigenvalues. The results are very promising and show that the quantum mechanical density of states can indeed be gained, using as input solely the classical periodic orbits of the pseudointegrable billiards. It will be very interesting to apply this procedure to real mesoscopic structures, where the change of the density of states as response to a change in geometry is of great importance. We would like to thank A. Bunde for a careful reading of the manuscript and valuable remarks. convergence of Eq. (4). Fig. 2 that 2N P O (k) gives the expected staircase function and the agreement to N EV (k) is excellent. As shown in the upper curve ofFig. 2(b), it even allows to obtain the eigenvalues directly from the steps in N P O (k). Each eigenvalue is positioned at one of the steps of N P O (k), which we verified for the first 1500 eigenvalues (until k 2 = 1). though the numerical calculations can be done in high precision for up to about 50000 orbits in reasonable computation time[22], we first want to investigate the stability of the results against numerical errors. To this end, we compared for the pseudointegrable billiards the orbits found in forward and backward direction. The maximum errors of the areas of the first 40000 periodic orbits (about 300 reflexions at the system walls including repetitions) are around 0.1 % and we found no hints that orbits might be lost. Accordingly, we have first tested the robustness of the results by disturbing the orbits of the rectangular system by errors taken from a narrow Gaussian distribution of σ = 0.05 and a maximum error of 0.1% (seeFig. 2(b) shifted down by a value of 5 in the lower curve (dotted line)). One can see that on the average, N P O (k) still agrees very well with N EV (k), but that the shape of the staircase is smeared out already by these quite small errors, so that the eigenvalues can no longer be determined from the steps in N P O (k). The loss of about 1% of the orbits (as well as their repetitions) would be less disturbing and would only lead to very slight deviations in the step function. We also checked if the number of calculated orbits is large enough and , we compare for the polygone billiards of Figs. 1(a-c) N P O (k) with N EV,L (k). First, in Fig. 3(a), we show a larger part of the energy spectrum, where we can see that both, N P O (k) (straight lines) and N EV,L (k) (symbols) agree again very well. Even though the steps in N P O (k) are smeared out by the numerical inaccurencies as described above, we can use them to obtain the individual eigenvalues from N P O (k). To this end, we fitted N P O (k) by a least square fit to a step function with constant integer values of the step heights. The positions k 2 i of the steps were chosen by minimizing the quadratic deviation to N P O (k). The step functions obtained this way, N P O fit (k), are shown in FIG. 1 : 1Shapes of the pseudointegrable billiards considered in this work. (a,b) L-shaped billiards (polygon billiards of genus number G = 2), (c) a polygon billiard of G = 3 and (d) the barrier billiard (G = 2), which is calculated for different heights h of the barrier. One periodic orbit for each geometry is also shown (dashed lines). In (b) the different segments are shown that can be transversed by the periodic orbits. FIG. 2 : 2(a,b) Density of states N P O (k) calculated from the periodic orbits (solid lines) and N EV (k) calculated from the exact eigenvalues (circles) of a rectangular system of side lengths L x = 101 and L y = 198. In (b), the upper curve compares the steps in N P O (k) to the eigenvalues. In the lower curves (shifted down by a value of 5), the areas of the orbits are disturbed by errors of up to 0.1% and it can be seen that the steps in N P O (k) (dotted line) are strongly disturbed already by these small error bars. FIG. 3 : 3(a,b) Density of states N P O (k) calculated from the periodic orbits (solid lines) and N EV,L (k) from the Lanczos eigenvalues for the L-shaped systems of Fig. 1(a) (circles) and Fig. 1(b) (squares) and the system of genus number G = 3 of Fig. 1(c) (diamonds). In (b), the fitted step function of N P O fit (k) is compared to the Lanczos eigenvalues. For a better overview the data of the geometries of Figs. 1(b,c) have been shifted upwards by values of 150 in (a) and by 25 in (b). FIG. 4 : 4(a,b) Density of states N P O (k) calculated from the periodic orbits (solid lines) and N EV,L (k) from the Lanczos eigenvalues for the barrier billiards with barrier heights h = 10 (diamonds), h = 50 (squares) and h = 100 (circles). In (b), the fitted step function of N P O fit (k) is compared to the Lanczos eigenvalues. For a better overview the data in both, (a) and (b) have been shifted upwards by values of 5 (h = 50) and by 10 (h = 10). Fig Fig. 1(a) k 2 L 2calculated by the Lanczos algorithm for the polygon pseudointegrable systems. N m is the number of mismatches during the first 100 values, i.e. the number of cases where the ith periodicorbit eigenvalue lies closer to the (i + 1)th or (i − 1)th Lanczos value than to the ith one. Fig. 1 1Fig. 1(d), h = 10 Fig. 1(d), h = 50 Fig. 1(d), h = 100 n k 2 P O TABLE I : ITable of the 91st to the 100th eigenvalue k 2 P O calculated by the trace formula compared to TABLE II : IITable of the 91st to the 100th eigenvalue k 2 P O calculated by the trace formula compared to k 2 L calculated by the Lanczos algorithm for the pseudointegrable barrier systems. N m is the number of mismatches during the first 100 values. . M C Gutzwiller, J. Math. Phys. 12343M. C. Gutzwiller, J. Math. Phys. 12, 343 (1971). Random Matrices. M L Mehta, Academic Press, IncM.L. Mehta, Random Matrices, Academic Press, Inc. 1991. H.-J Stöckmann, Quantum Chaos, an introduction. Cambridge University PressH.-J. Stöckmann, Quantum Chaos, an introduction, Cambridge University Press 1999. . G Tanner, P Scherer, E B Bogomolny, B Eckhardt, D Wintgen, Phys. Rev. Lett. 672410G. Tanner, P. Scherer, E. B. Bogomolny, B. Eckhardt and D. Wintgen, Phys. Rev. Lett. 67, 2410 (1991). . R Aurich, M Sieber, F Steiner, Phys. Rev. Lett. 61483R. Aurich, M. Sieber and F. Steiner, Phys. Rev. Lett. 61, 483 (1988). . R Aurich, F Steiner, Phys. Rev. A. 45583R. Aurich and F. Steiner, Phys. Rev. A 45, 583 (1992). . M Sieber, F Steiner, Phys. Rev. Lett. 671941M. Sieber and F. Steiner, Phys. Rev. Lett. 67, 1941 (1991). . P J Richens, M V Berry, Physica D. 2495P. J. Richens and M. V. Berry, Physica D 2, 495 (1981). . B Eckhardt, J Ford, F Vivaldi, Physica. 13339B. Eckhardt, J. Ford, and F. Vivaldi, Physica 13D, 339 (1984). . T Cheon, T D Cohen, Phys. Rev. Lett. 622769T. Cheon and T. D. Cohen, Phys. Rev. Lett. 62, 2769 (1989). . A Shudo, Y Shimizu, Phys. Rev. E. 4754A. Shudo and Y. Shimizu, Phys. Rev. E 47, 54 (1993). . A Shudo, Y Shimizu, P Seba, J Stein, H.-J Stöckmann, K Zyczkowski, Phys. Rev. E. 493748A. Shudo, Y. Shimizu, P. Seba, J. Stein, H.-J. Stöckmann and K. Zyczkowski, Phys. Rev. E 49, 3748 (1994). . Y Hlushchuk, S Russ, Phys. Rev. E. 6816203Y. Hlushchuk and S. Russ, Phys. Rev. E 68, 016203 (2003). . J Cserti, A Kormanyos, Z Kaufmann, J Koltai, C J Lambert, Phys. Rev. Lett. 8957001J. Cserti, A. Kormanyos, Z. Kaufmann, J. Koltai and C. J. Lambert, Phys. Rev. Lett. 89, 057001 (2002). . M V Berry, M Tabor, Proc. Roy. Soc. A. 349101M. V. Berry and M. Tabor, Proc. Roy. Soc. A 349, 101 (1976). . D Biswas, S Sinha, Phys. Rev. Lett. 70916D. Biswas and S. Sinha, Phys. Rev. Lett. 70, 916 (1993). . D Biswas, Phys. Rev. E. 541044D. Biswas, Phys. Rev. E 54, R1044 (1996). . D Biswas, Pramana Journal of Physics. 48487D. Biswas, Pramana Journal of Physics 48, 487 (1997). . J H Hannay, J Mccraw, J. Phys. A. 23887J. H. Hannay and J. McCraw, J. Phys. A 23, 887 (1990). H P Baltes, E R Hilf, Spectra of Finite Systems, BI Wissenschaftsverlag. ViennaH. P. Baltes and E. R. Hilf, Spectra of Finite Systems, BI Wissenschaftsverlag, Vienna 1976. . M Sieber, U Smilansky, S C Creagh, R G Littlejohn, J. Phys. A. 266217M. Sieber, U. Smilansky, S. C. Creagh and R. G. Littlejohn, J. Phys. A 26, 6217 (1993). . J Mellenthin, S Russ, Phys. Rev. E. 7056205J. Mellenthin and S. Russ, Phys. Rev. E 70, 056205 (2004).
{'fraction_non_alphanumeric': 0.055752441870516034, 'fraction_numerical': 0.028783818826173393, 'mean_word_length': 3.953757225433526, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 6, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'We apply periodic-orbit theory to calculate the integrated density of states N (k) from the periodic orbits of pseudointegrable polygon and barrier billiards. We show that the results agree so well with the results obtained from direct diagonalization of the Schrödinger equation, that about the first 100 eigenvalues can be obtained directly from the periodic-orbit calculations in good accuracy.PACS numbers: PACS numbers: 05.45.Mt,', 'arxivid': 'cond-mat/0603663', 'author': ['Stefanie Russ \nInstitut für Theoretische Physik III\nJustus-Liebig-Universität Giessen\nD-35392GiessenGermany\n', 'Jesper Mellenthin \nLaboratoire de Physique de la Matière Condensée\nEcole Polytechnique\nF-91128PalaiseauFrance\n'], 'authoraffiliation': ['Institut für Theoretische Physik III\nJustus-Liebig-Universität Giessen\nD-35392GiessenGermany', 'Laboratoire de Physique de la Matière Condensée\nEcole Polytechnique\nF-91128PalaiseauFrance'], 'corpusid': 17940784, 'doi': '10.1103/physreve.73.066227', 'github_urls': [], 'n_tokens_mistral': 7090, 'n_tokens_neox': 6187, 'n_words': 4079, 'pdfsha': '11fc233aea64137e7d88eddbe41fe94bb30c6a2e', 'pdfurls': ['https://export.arxiv.org/pdf/cond-mat/0603663v1.pdf'], 'title': ['Application of the Trace Formula in Pseudointegrable Systems', 'Application of the Trace Formula in Pseudointegrable Systems'], 'venue': []}
arxiv
Convergence Analysis of Mean Shift Ryoya Yamasaki Member, IEEEToshiyuki Tanaka Convergence Analysis of Mean Shift PREPRINT VERSION 1Index Terms-Mean shiftconvergenceconvergence rateŁojasiewicz inequalitybiweight kernel ✦ The mean shift (MS) algorithm seeks a mode of the kernel density estimate (KDE). This study presents a convergence guarantee of the mode estimate sequence generated by the MS algorithm and an evaluation of the convergence rate, under fairly mild conditions, with the help of the argument concerning the Łojasiewicz inequality. Our findings, which extend existing ones covering analytic kernels and the Epanechnikov kernel, are significant in that they cover the biweight kernel that is optimal among non-negative kernels in terms of the asymptotic statistical efficiency for the KDE-based mode estimation. INTRODUCTION T HE mean shift (MS) algorithm [1]- [3] has been widely used in various fields such as computer vision, image processing, pattern recognition, and statistics. One of its popular applications is data clustering [4], [5], where the MS algorithm is advantageous in that it does not need to specify the number of clusters in advance. Other advantages of the MS-based clustering compared with the -means clustering are that it does not require proper initialization of cluster centers, as well as that it can cope with arbitrary cluster shapes. Other applications of the MS algorithm include image segmentation [3], [6], edge detection [7], [8], object tracking [9], [10], and mode estimation [11], [12], to mention a few. The MS algorithm is an iterative algorithm that seeks a mode (local maximizer) of the kernel density estimate (KDE). Applications of the MS algorithm, such as data clustering and mode estimation, require the convergence of the mode estimate sequence generated by the MS algorithm. It is therefore important to theoretically study convergence properties of the MS algorithm. However, as will be reviewed in Section 3, available theoretical convergence guarantees of the MS algorithm which are applicable to practically relevant situations are quite limited: As dynamical behaviors of the MS algorithm depend on the kernel to be used in constructing the KDE, convergence properties should also depend on the choice of the kernel. To the best of the authors' knowledge, the MS algorithm for multi-dimensional data has been shown to converge when the Epanechnikov kernel [13], [14] or an analytic kernel [15] is used. These results do not cover practically relevant cases where a piecewise polynomial kernel other than the Epanechnikov kernel is used. Furthermore, little is known about the convergence rate of the MS algorithm. In this paper we study convergence properties of the MS algorithm under some generic assumptions on the kernel. From a technical point of view, we follow a line similar to that of [15] that focused on the Łojasiewicz property [16], [17], but we make use of more advanced results about that property, to further extend the convergence analysis in • Ryoya Yamasaki and Toshiyuki Tanaka are with the Department of Informatics, Graduate School of Informatics, Kyoto University, Kyoto, 606-8501, Japan. E-mail: [email protected], [email protected]. [15]. This extension allows us to obtain novel results, which include a convergence guarantee of the mode estimate sequence (Theorems 1 and 2) and a worst-case bound of the convergence rate (Theorems 3 and 4) of the MS algorithm for a wider class of the kernels. Our contributions are of significance as the class of the kernels we focus on in this study contains the biweight kernel, which is known to be optimal among non-negative kernels in terms of the asymptotic statistical efficiency for the KDE-based estimation of a non-degenerate mode [12], [18]. This paper is organized as follows. We formulate the MS algorithm in Section 2, and review related work on the convergence analysis of the MS algorithm in Section 3. In Section 4, we describe the Łojasiewicz property, and summarize the class of functions having that property. On the basis of these preliminaries and abstract convergence theorems by [19], [20], we provide a novel sufficient condition to ensure the convergence of the MS algorithm and an evaluation of the convergence rate in Section 5. In Section 6, we conclude this paper, and furthermore, we mention variants of the MS algorithm to which the analysis of this paper can be applied similarly, and possible directions for future research. Supplementary material provides proofs of the theoretical results. MS ALGORITHM Various applications of the MS algorithm stem from the characterization that the MS algorithm is an optimization algorithm seeking a local maximizer of the KDE. Given data points x 1 , . . . , x ∈ R , the KDE is constructed as (x) ≔ 1 ℎ ∑︁ =1 x − x ℎ ,(1) where : R → R and ℎ > 0 are called the kernel and the bandwidth parameter, respectively. Throughout this paper, for the kernel we adopt the following assumption, which is common in studies of the MS algorithm: Assumption 1. The kernel is bounded, continuous, nonnegative, normalized, and radially symmetric. The assumption of radial symmetry of the kernel leads to its alternative representation (x) =ˆ(∥x∥ 2 /2) (2) with what is called the profileˆ: [0, ∞) → R of and the Euclidean norm ∥ · ∥ in R . As mentioned by [21], [22], the MS algorithm can be seen as an example of the minorization-maximization (MM) algorithm under a certain condition. The MM algorithm solves a hard original optimization problem by iteratively performing construction of what is called a minorizer of the original objective function and optimization of the minorizer. Let us write the right and left derivatives ofˆ, if exist, asˆ′ ( ±) = lim → ±0ˆ( ) −ˆ( ) − .(3) We make the following assumption for the profileˆof the kernel : Assumption 2. The kernel has a convex and non-increasing profileˆsatisfyingˆ′ (0+) > −∞. For a real-valued function defined on ⊆ R, the subdifferential ( ) of at ∈ is defined as the set of values ∈ R such that ( ) − ( ) ≥ ( − ) holds for any ∈ . Under Assumption 2, since the profileˆis convex, the subdifferentialˆ( ) is non-empty for any ∈ (0, ∞) and given by [ˆ′ ( −),ˆ′ ( +)]. Note thatˆ(0) = (−∞,ˆ′ (0+)] is non-empty as well under the assumptionˆ′ (0+) > −∞. Sinceˆ( ) is non-empty for any ∈ [0, ∞), one can show that the subdifferentialˆ( ) is non-decreasing in the sense that for 0 ≤ < one has maxˆ( ) ≤ minˆ( ): Indeed, for any , with 0 ≤ < , take any ∈ˆ( ) and ∈ˆ( ). From the definition of the subdifferential, one hasˆ( ) −ˆ( ) ≥ ( − ) andˆ( ) −ˆ( ) ≥ ( − ), which are summed up to 0 ≥ ( − )( − ), yielding ≤ . See also [23,Section 24] for these properties of subdifferentials of functions on R. Furthermore, as the profileˆis nonincreasing, for any ∈ [0, ∞) one has maxˆ( ) ≤ 0. Thus, defining a functionˇon [0, ∞) viǎ ( ) ≔ −ˆ′ (0+) if = 0, ∈ −ˆ( ) if > 0,(4) it is non-increasing, non-negative, and bounded sincě ( ) ≤ˇ(0) = −ˆ′ (0+) < ∞ for any ∈ [0, ∞) due to Assumption 2. As −ˇ( ) ∈ˆ( ), the definition of the subdifferential yieldsˆ( ) −ˆ( ) ≥ −ˇ( )( − ) for any , ∈ [0, ∞). Substituting ( , ) = (∥x ′ ∥ 2 /2, ∥x∥ 2 /2) into this inequality, one has (x) ≥¯(x|x ′ ) ≔ (x ′ ) +ˇ( ∥x ′ ∥ 2 /2) 2 (∥x ′ ∥ 2 − ∥x∥ 2 ) (5) for any x, x ′ ∈ R . One also has (x ′ ) =¯(x ′ |x ′ ). These properties imply that, under Assumptions 1 and 2,¯(x|x ′ ) is a minorizer of the kernel at x ′ . It should be noted that there is arbitrariness in the definition (4) ofˇ( ) at those values of at whichˆ( ) contains more than a single value. For example, the profile of the Epanechnikov kernel is given byˆ( ) = (1− ) + with > 0, where (·) + ≔ max{·, 0}, and thusˆ(1) = [− , 0]. In this case one may adopt any value in the interval [0, ] aš (1). Indeed, [13] adoptedˇ(1) = , whereas [14] adopteď (1) = 0. We would like to note here that the following analysis is not affected by howˇ( ) is defined at such points. The MS algorithm given a th estimate y ∈ R builds a minorizer of the KDE at y as (x|y ) ≔ 1 ℎ ∑︁ =1¯ x − x ℎ y − x ℎ = − 1 2 ℎ +2 ∑︁ =1ˇ y − x ℎ 2 2 ∥x − x ∥ 2 + (x-independent constant),(6) which satisfies¯(y |y ) = (y ) and¯(x|y ) ≤ (x) for any x ∈ R . Introduce a functioň (x) ≔ 1 ℎ ∑︁ =1ˇ x − x ℎ 2 2 ,(7) with which the coefficient of the quadratic term ∥x∥ 2 in (x|y ) is expressed as −ˇ(x)/(2ℎ 2 ). Assumption 2 ensures thatˇ(x) is non-negative due to the non-negativity ofˇ( ). Furthermore, ifˇ(y ) = 0, then all the summands on the right-hand side of (7) are zero and hence the function¯(·|y ) is constant. Ifˇ(y ) > 0, on the other hand, then the function (·|y ) is quadratic and has a unique maximizer. The MS algorithm then calculates the next estimate y +1 as y +1 ∈ arg max x∈R¯( x|y ). More specifically, the MS algorithm calculates y +1 via y +1 = y + m(y ),(8) where m(y) ≔        0 ifˇ(y) = 0, − =1ˇ( ∥ y−x ℎ ∥ 2 /2) (y−x ) =1ˇ( ∥ y−x ℎ ∥ 2 /2) ifˇ(y) ≠ 0,(9) with the all-zero vector 0 ∈ R . The MS algorithm iterates the update rule (8) starting from a given initial estimate y 1 ∈ R while incrementing the subscript ∈ N. Therefore, the MS algorithm can be regarded as an instance of the MM algorithm. Here, the update rule whenˇ(y ) = 0 is an exceptionhandling rule to avoid the MS algorithm to be ill-defined due to the denominator (= ℎˇ(y )) of the ordinary update rule being zero. Under Assumptions 1 and 2, ifˇ(y ) = 0 then the gradient of the KDE also vanishes, that is, y is a critical point of . Therefore, the exception-handling rule ensures the MS algorithm to stop at a critical point. Also, the following proposition shows that no such exception occurs if one selects an initial estimate y 1 properly: Proposition 1. Assume Assumptions 1 and 2. Let (y ) ∈N be the mode estimate sequence obtained by the MS algorithm (8) starting from y 1 with (y 1 ) > 0. Then, there exists a constant > 0 such thatˇ(∥ y −x ℎ ∥ 2 /2) ≥ for some -dependent index ∈ [ ] ≔ {1, . . . , }, and consequentlyˇ(y ) ≥ ℎ for any ∈ N. For example, in the data clustering with the MS algorithm [4], [5], one adopts each data point x as the initial estimate y 1 , and hence the additional assumption (y 1 ) > 0 definitely holds. The above construction of the MS algorithm as the MM algorithm shows the ascent property (y ) =¯(y |y ) ≤ (y +1 |y ) ≤ (y +1 ) of the density estimate sequence ( (y )) ∈N , and the boundedness of the KDE (due to Assumption 1) guarantees the convergence of that sequence: Proposition 2 (Theorem 1 in [15]). Assume Assumptions 1 and 2. For the mode estimate sequence (y ) ∈N obtained by the MS algorithm (8) starting from any y 1 ∈ R , the density estimate sequence ( (y )) ∈N is non-decreasing and converges. The above proposition guarantees the convergence of the density estimate sequence ( (y )) ∈N generated by the MS algorithm. From the application point of view, however, what we are interested in is not the convergence of the density estimate sequence ( (y )) ∈N but that of the mode estimate sequence (y ) ∈N , since it is the limit lim →∞ y , if exists, that will tell us the location of a mode or a cluster center. The difficulty here is that one cannot deduce the convergence of the mode estimate sequence (y ) ∈N from the convergence of the density estimate sequence ( (y )) ∈N without additional assumptions. Our main interest in this paper lies in convergence properties of the mode estimate sequence (y ) ∈N obtained by the MS algorithm, such as whether it converges to a critical point, as well as its convergence rate when it converges. RELATED WORK Convergence properties of the mode estimate sequence (y ) ∈N have been discussed in several papers. Some early convergence studies are, however, not rigorous. For instance, the proof in [3] used an incorrect inequality evaluation to claim that the mode estimate sequence is a Cauchy sequence; see counterexamples given in [24]. Essentially the same flaw had been shared by [25] in the discussion of consistency, which was subsequently amended in the errata [26] to [25]. [27] claimed the convergence of the mode estimate sequence under the assumption that the MS algorithm uses the Gaussian kernel, on the basis of the fact that the MS algorithm under this assumption is an example of the expectation-maximization (EM) algorithm [28]. As pointed out by [29], however, this reasoning alone is not enough: the EM algorithm may not converge without additional conditions [30]. Later studies have successfully provided some sufficient conditions for the convergence of the mode estimate sequence. In [24], the convergence of the mode estimate sequence has been proved under the assumption that the KDE has a finite number of critical points inside the convex hull of data points. For example, when the Epanechnikov kernel is used, the KDE is shown to have a finite number of critical points, so that the result of [24] is applicable to provide a convergence guarantee. For the Epanechnikov kernel, something even stronger holds true: [13] and [14] proved that the MS algorithm converges in a finite number of iterations. Another instance for which the finiteness of critical points, and consequently the convergence of the mode estimate sequence, have been shown is the 1dimensional KDE with the Gaussian kernel. See, e.g., [31] and [32]. However, it is not known whether the number of critical points of the KDE with the Gaussian kernel for the dimension ≥ 2 is finite. See, e.g., [33], where upper and lower bounds of the number of non-degenerate critical points were given, whereas they wrote that the finiteness of the number of critical points is still open. Although [34] provided a condition under which the KDE with the Gaussian kernel has a finite number of critical points, his condition requires taking the bandwidth of the kernel large enough. Under this condition, mode estimates to be obtained would have a large statistical bias. Furthermore, the KDE with a large bandwidth might even yield a far smaller number of mode estimates than the actual number of the true modes when the data-generating distribution has multiple modes. Therefore, its practical significance is quite obscure, in view of applications of the MS algorithm such as data clustering and mode estimation. Additionally, in the 1dimensional case, [29] proved the convergence of the mode estimate sequence for various kernels, by showing that its subsequence around a critical point becomes a bounded monotonic sequence. However, this proof strategy cannot be extended to the multi-dimensional case. More recently, [15] have given a convergence proof of the MS algorithm using analytic kernels, including the Gaussian kernel. Their proof takes advantage of the Łojasiewicz property [16], [17] (see Definition 1) of an analytic kernel and the corresponding KDE, while not requiring assumptions either on the finiteness of critical points of the KDE, on the nondegeneracy of KDE's Hessian at critical points, on the size of the bandwidth, or on the dimension of the data. Thus, their result is significant in that it guarantees the convergence of the MS algorithm under practical settings on the bandwidth parameter, and even in the multi-dimensional case. To summarize, it is only when the MS algorithm uses the Epanechnikov kernel [13], [14] or an analytic kernel [15] that the convergence of the mode estimate sequence (y ) ∈N has been guaranteed without regard to the size of the bandwidth parameter or the data dimension. Much less is known so far about the convergence rate. Previous studies have clarified only the finite-time convergence when the algorithm uses the Epanechnikov kernel [13], [14] and the linear convergence when the algorithm uses the Gaussian kernel and the KDE has a non-degenerate Hessian at the convergent point [27]. The convergence rate when the Hessian is degenerate has not been clarified. PRELIMINARIES: ŁOJASIEWICZ PROPERTY As mentioned above, [15] proved the convergence of the mode estimate sequence (y ) ∈N of the MS algorithm using an analytic kernel, without regard to the bandwidth parameter or the data dimension. The key in their proof is the Łojasiewicz property/inequality for an analytic function [16], [17], which provides a lower bound of the flatness of the the function around its critical points. In the convergence analysis of the MS algorithm, this bound in turn allows us to transfer the convergence of the density estimate sequence ( (y )) ∈N to that of the mode estimate sequence (y ) ∈N . In this paper, we follow a similar line to that of [15], but we rely on more advanced results about the Łojasiewicz property. We here describe the Łojasiewicz property, and important classes of functions having that property. We adopt the following definition of the Łojasiewicz property/inequality, along with related notions. Definition 1 (Łojasiewicz property/inequality/exponent). A function : → R with ⊆ R is said to have the Łojasiewicz property at x ′ ∈ with an exponent , if there exists > 0 such that is differentiable on ( x ′ , , ) ≔ {x ∈ | ∥x ′ − x∥ < , (x ′ ) − (x) ≥ 0} and satisfies the Łojasiewicz inequality ∥∇ (x)∥ ≥ { (x ′ ) − (x)}(10) with > 0, ∈ [0, 1), and any x ∈ (x ′ , , ). Also, is said to have the Łojasiewicz property on ⊆ (when = , we omit "on T"), if is differentiable on and there exists > 0 such that satisfies the Łojasiewicz inequality (10) with > 0, ∈ [0, 1), and any (x ′ , x) such that x ′ ∈ , x ∈ (x ′ , , ). Moreover, the minimum value of , with which has the Łojasiewicz property at x ′ , is called the Łojasiewicz exponent of at x ′ . Intuitively, the Łojasiewicz property of a function at x ′ quantifies how flat the function is around the point x ′ . It is obvious from the definition that, for any ∈ [0, 1), if has the Łojasiewicz property at x ′ with an exponent , then for any ′ ∈ [ , 1) the same holds true with the exponent ′ as well. It is thus the minimum possible exponent (i.e., the Łojasiewicz exponent) that is informative. If is continuously differentiable at x ′ and if x ′ is a non-critical point of (that is, ∇ (x ′ ) ≠ 0), then for any ∈ [0, 1), trivially has the Łojasiewicz property at x ′ with the exponent , implying that is "maximally non-flat" at x ′ . If, on the other hand, x ′ is a local minimum of , then with a sufficiently small one has (x ′ , , ) = {x ′ }, implying that has the Łojasiewicz property at the local minimum x ′ . These facts demonstrate that Definition 1 is tailored primarily for characterizing the flatness of around its critical points except local minima. As a more concrete example let us take (x) = (x ′ ) − ∥x − x ′ ∥ , > 1.(11) One then has ∥∇ (x) ∥ = ∥x − x ′ ∥ −1 = { (x ′ ) − (x)} 1−1/ ,(12) implying that has the Łojasiewicz property at x ′ with the exponent ≥ 1−1/ . As one takes a larger , gets "flatter" at x ′ , and correspondingly the Łojasiewicz exponent 1 − 1/ becomes larger. As another example, let (x) = (x ′ ) − − ∥x−x ′ ∥ − 1(x ≠ x ′ ), > 0,(13) where 1( ) is the indicator function that takes the value 1 if the condition is true, and 0 otherwise. One then has ∥∇ (x) ∥ = − ∥x−x ′ ∥ − ∥x − x ′ ∥ − −1 1(x ≠ x ′ ) = ℎ( (x ′ ) − (x))(14) with ℎ( ) = (− log ) 1+1/ 1( > 0), ≥ 0, on the basis of which one can show that does not have the Łojasiewicz property at x ′ as defined in Definition 1, that is, it is "too flat" at x ′ to be captured by this definition 1 , since for any ∈ [0, 1) one has ℎ( ) = 1− (− log ) 1+1/ →+0 −→ 0.(15) The significance of the Łojasiewicz property for our purpose is that it allows us to convert the convergence of the density estimate sequence ( (y )) ∈N into that of the mode estimate sequence (y ) ∈N when the KDE is "not too flat," as well as that, when the mode estimate sequence (y ) ∈N converges, the property can provide a guarantee of faster convergence when is "less flat" at the limit, as will be discussed in Section 5. [16] showed that analytic functions have the Łojasiewicz property, and thereafter, [35] generalized that result to the class of 1 functions with o-minimal structure (see also [36]), which particularly includes 1 globally subanalytic functions: 2 Proposition 3 ( [16], [35]). A function : → R with ⊆ R has the Łojasiewicz property, if is analytic or if is 1 globally subanalytic. Now we introduce the definition of the global subanalyticity, as well as several related notions, the latter of which serve as sufficient conditions for the global subanalyticity. See also [37] and [42]. These notions are useful in practice, because directly verifying the global subanalyticity can often be difficult, whereas those sufficient conditions are easy to verify, as in the discussion in supplementary material. Definition 2 (Global subanalyticity and related notions). • A set ⊆ R is called semialgebraic, if there exists a finite number of polynomial functions : R → R such that = =1 =1 {x ∈ R | (x) 0} with relational operators ∈ {<, >, =}. • A set ⊆ R is called semianalytic, if for each point x ′ ∈ R there exist a neighborhood of x ′ and a finite number of analytic functions : → R such that ∩ = =1 =1 {x ∈ | (x) 0} with relational operators ∈ {<, >, =}. • A set ⊆ R is called subanalytic, if for each point x ′ ∈ R there exist a neighborhood of x ′ and a bounded semianalytic set ⊆ R + ′ with ′ ≥ 1 such that ∩ = {x ∈ R | (x, y) ∈ }. • A set ⊆ R is called globally semianalytic (resp. globally subanalytic), if its image under (x) = 1. We would like to mention, however, that an extended definition of the Łojasiewicz property, provided in supplementary material, allows us to capture the flatness in this example as well. 2. More recently, [37], [38] extended the definition of the Łojasiewicz inequality to the case of non-smooth functions, and showed that continuous globally subanalytic functions satisfy that generalized Łojasiewicz inequality. Succeeding studies such as [19], [20], [39]- [41] used it to construct abstract convergence theorems for various optimization algorithms. We also attempted convergence analysis according to such a general framework that allows non-smooth objective functions, but, even for the MS algorithm, we could not avoid the smoothness assumption (assumption (a1) in Theorem 1 or Assumption 3 in Theorem 2, in Section 5.1). Such difficulty is also discussed in [20], [40,Section 6]. Therefore, from Section 4 onwards, we adopt a simple framework that supposes the smoothness even if it can be generalized to the non-smooth case. Also, according to the boundedness assumption (Assumption 1), we omit devises used to handle unbounded functions. ( 1 /(1 + 2 1 ) 1/2 , . . . , /(1 + 2 ) 1/2 ) is a semianalytic (resp. subanalytic) subset of R . • A function : → R with ⊆ R is called semialgebraic (resp. semianalytic, subanalytic, globally semianalytic, or globally subanalytic), if its graph {(x, ) ∈ × R | = (x)} is semialgebraic (resp. semianalytic, subanalytic, globally semianalytic, or globally subanalytic) subset of R +1 . • A function : → R with ⊆ R is called piecewise polynomial with the maximum degree ∈ N, if there exists a finite collection { } ∈ [ ] of subdomains ⊆ , ∈ [ ], that forms a partition of (i.e., ≠ ∅ for all ∈ [ ], ∩ ′ = ∅ for all , ′ ∈ [ ] with ≠ ′ , and ∪ ∈ [ ] = ), such that (x) = (x) for any x ∈ (i.e., the restriction of to is the same as that of to ) with a polynomial : → R for each ∈ [ ] and that the maximum degree of { } ∈ [ ] is . The class of semialgebraic functions has a wide variety of instances: polynomial, rational, and more generally piecewise polynomial functions are semialgebraic [43], [44]. As will be discussed in the next section, the class of piecewise polynomial functions that includes the biweight kernel is of particular importance in the discussion of this study. Any globally semianalytic functions are semianalytic, and any semianalytic functions with a bounded graph are globally semianalytic [ Note that an analytic function is not necessarily globally subanalytic (of course, the converse is not necessarily true either: a globally subanalytic function is not necessarily analytic). For example, ( ) = sin( ), ∈ R, is certainly analytic but not globally subanalytic [42, Example 1.1.7]. Moreover, it should be noted that a semianalytic/subanalytic function (e.g., the sine function defined on R) and a ∞ function are not necessarily globally subanalytic and do not always have the Łojasiewicz property; the "Mexican hat" function (equation (2.8) in [17]) and the function shown on page 14 of [45] are instances that are in the ∞ class and not globally subanalytic, and these functions do not have the Łojasiewicz property. These inclusion relations are summarized in Figure 1. MAIN RESULTS: CONVERGENCE THEOREMS FOR MS ALGORITHM Convergence to a Critical Point In this subsection, we provide a sufficient condition for the mode estimate sequence (y ) ∈N of the MS algorithm to converge to a critical point of the KDE . Our result is along the same line as the existing convergence theorem by [15] for the MS algorithm using analytic kernels, and further extends it on the ground of Propositions 3 and 4 stating that 1 globally subanalytic kernels and the corresponding KDE have the Łojasiewicz property. Several recent studies in optimization theory, including [17], [19], [20], [39]- [41], exploit the Łojasiewicz property to prove the convergence of various optimization algorithms. By applying abstract convergence theorems such as [ stant ≥ 0 such that ∥∇ (x) − ∇ (x ′ )∥ ≤ ∥x − x ′ ∥ for any x, x ′ ∈ cl(Conv({y } ≥ )), where is called the Lipschitz constant of ∇ on cl(Conv({y } ≥ ))), and (a2) the KDE has the Łojasiewicz property on cl(Conv({y } ≥ )). Then, the mode estimate sequence (y ) ∈N has a finite-length trajectory (i.e., ∞ =1 ∥y +1 − y ∥ < ∞) and converges to a critical pointȳ of the KDE . We next argue how one can replace the assumptions (a1) and (a2) on the KDE to assumptions on the kernel in such a way that the latter ones provide sufficient conditions for the former ones. Let us focus on the assumption (a1) of Theorem 1 first. If a kernel is differentiable with a Lipschitz-continuous gradient, then the KDE using the kernel trivially satisfies the assumption (a1) for any , simply because the summation of functions preserves the differentiability, as well as the Lipschitz continuity of the gradients. Therefore, for the convergence guarantee of the mode estimate sequence (y ) ∈N , we can replace the assumption (a1) on the KDE with the following assumption on the kernel : Assumption 3. The kernel is differentiable and has a Lipschitz-continuous gradient. Note that Assumption 3 also implies that the kernel is of class 1 . 3. As we have observed in Section 2 that the MS algorithm is an example of the MM algorithm, we might alternatively be able to apply abstract convergence theorems for the MM algorithm [41] to the MS algorithm. Although convergence of the MS algorithm could be proved in this way, the resulting bound of the convergence rate can become looser than that given by Theorems 3 and 4 in this paper. This is because that bound depends on the Łojasiewicz exponent of the function¯(x + m(x) |x) (called the value function) introduced in [41] (not of the KDE), which is in general flatter than the KDE at the critical point. We next argue how one can replace the assumption (a2) of Theorem 1 with an assumption on the kernel . According to Propositions 3 and 4, when the kernel is analytic or 1 globally subanalytic, it is clear that the corresponding KDE is so as well and has the Łojasiewicz property. We argue in the following that requiring the kernel to be 1 subanalytic is indeed enough in order for the assumption (a2) to hold: Under Assumptions 1 and 2, as well as the condition (y 1 ) > 0, the mode estimate y for ≥ 2 becomes a convex combination of the data points {x } ∈ [ ] , that is, a weighted mean of {x } ∈ [ ] with non-negative weights, and thus it lies in the convex hull Conv({x } ∈ [ ] ) of {x } ∈ [ ] , which is a bounded set. Therefore, we can restrict the domain of every kernel ( ·−x ℎ ), = 1, . . . , , to Conv({x } ∈ [ ] ) without any problems. Also, every kernel ( ·−x ℎ ) is bounded under Assumption 1. Therefore, when the kernel is 1 subanalytic, the restriction of ( ·−x ℎ ) to Conv({x } ∈ [ ] ) becomes a 1 subanalytic function with a bounded graph, and consequently, it is 1 globally subanalytic due to Proposition 4. Hence, the restriction of the corresponding KDE is also 1 globally subanalytic due to Proposition 4 and has the Łojasiewicz property due to Proposition 3. Given this consideration, we do not have to require global subanalyticity, and requiring 1 subanalyticity to the kernel is sufficient for the assumption (a2) to be satisfied for any . Therefore, under Assumptions 1, 2, and 3 and the condition (y 1 ) > 0, we can replace the assumption (a2) on the KDE with the following assumption on the kernel : Assumption 4. The kernel is analytic or subanalytic. Consequently, the following theorem will be obtained as a direct corollary of Theorem 1, which assures the convergence independently of the mode estimate sequence (y ) ∈N . Theorem 2 (Corollary of Theorem 1). Assume Assumptions 1, 2, 3, and 4. Let (y ) ∈N be the mode estimate sequence obtained by the MS algorithm (8) starting from y 1 with (y 1 ) > 0. Then, the mode estimate sequence (y ) ∈N has a finite-length trajectory and converges to a critical pointȳ of the KDE . The main significance of Theorem 2 is that it reveals for the first time the convergence of the MS algorithm for several piecewise polynomial kernels including the biweight and triweight kernels. In particular, the biweight kernel is known to be optimal among non-negative kernels in terms of the asymptotic statistical efficiency for the KDE-based mode estimation [11], [12]. More concretely, for a mode of the true probability density function with a non-degenerate Hessian at the mode, the main term of the asymptotic mean squared error of the 1-dimensional KDE-based mode estimator using a non-negative kernel and optimal bandwidth parameter for that kernel is proportional to the kernel-dependent term ( ∫ ∞ −∞ 2 ( ) ) 6 7 · ( ∫ ∞ −∞ { ′ ( )} 2 )4 7 (we call its inverse the asymptotic statistical efficiency), and [18] showed that the biweight kernel minimizes this kerneldependent term. Moreover, [12] obtained similar results for the multi-dimensional case. The triweight kernel is also relatively good in the same perspective; see Table 1 where we arrange kernels in the order of the asymptotic statistical efficiency for the 1-dimensional case (calculated ignoring a finite number of non-differentiable points) from the top. 4 Convergence Rate In this subsection, we study convergence rate of the MS algorithm. As mentioned at the end of Section 3, there are only a few studies on the convergence rate of the MS algorithm: It was proved in [13] and [14] that the MS algorithm with the Epanechnikov kernel converges in a finite number of iterations, and in [27] that the MS algorithm with the Gaussian kernel exhibits linear convergence provided that the Hessian of the KDE at a critical point is non-degenerate. We here establish a convergence rate evaluation for other kernels under more general situations. Assume for a moment that the kernel is three-times differentiable, in addition to Assumptions 1 and 2. Consider [15] for analytic kernels. †: Finite-time convergence [13], [14]. * : Not only convergence of (y ) ∈N but even that of ( (y ) ) ∈N is not ensured. { (1 − ) + } 2 ✓ ✓ ⃝ 2 ⃝ 3 ⃝ 4 - { (1 − ) + } 3/2 ✓ × △ 1 △ 3 - Triweight { (1 − ) + } 3 ✓ ✓ ⃝ 2 ⃝ 3 ⃝ 4 Tricube { (1 − 3/2 ) + } 3 × ✓ - * - - Cosine cos( 1/2 2 )1( ≤ 1) ✓ × △ 1 △ 3 - Epanechnikov (1 − ) + ✓ × ⃝ † ⃝ † ⃝ † Gaussian − ✓ ✓ ⃝ ‡ ⃝ 3 - Logistic 1 1/2 +2+ − 1/2 ✓ ✓ ⃝ ‡ ⃝ 3 - Cauchy 1 1+ ✓ ✓ ⃝ ‡ ⃝ 3 - Taylor expansion of the map y ↦ → y +1 = y + m(y ) around a critical pointȳ of the KDE , y +1 =ȳ + J(ȳ)(y −ȳ) + (∥y −ȳ∥ 2 ).(16) Ignoring the higher-order terms, one has the approximate relation y +1 −ȳ ≈ J(ȳ)(y −ȳ),(17) where J(ȳ) is the Jacobian of the map x ↦ → x+m(x) at x =ȳ. Applying the approximate relation (17) recursively shows y −ȳ ≈ {J(ȳ)} − (y −ȳ) for sufficiently large , ∈ N with ≥ . This approximation suggests that the mode estimate sequence (y ) ∈N achieves the exponential-rate convergence (also called the linear convergence) when the farthest-fromzero eigenvalue of J(ȳ) has the absolute value less than 1: ∥y −ȳ∥ ≤ | | − ∥y −ȳ∥ for sufficiently large . Also, the second-order Taylor expansion of the KDE around the critical pointȳ shows the exponential-rate convergence of the density estimate sequence ( (y )) ∈N as | (ȳ) − (y )| ≈ |(y −ȳ) ⊤ {∇ 2 (ȳ)}(y −ȳ)| = ( 2( − ) ). Simple calculation reveals that the Jacobian J(ȳ) of the map x ↦ → x + m(x) at x =ȳ is given by J(ȳ) = =1ˆ′ ′ (∥ȳ −x ℎ ∥ 2 /2)(x −ȳ)(x −ȳ) ⊤ −ℎ 2 =1ˆ′ (∥ȳ −x ℎ ∥ 2 /2) .(18) It should be noted that the denominator of the right-hand side of (18) is equal to ℎ +2ˇ(ȳ ) ≥ 0, which is positive if (ȳ) > 0. As Assumption 2 ensures thatˆ′ ′ is non-negative, J(ȳ) becomes positive semidefinite. On the other hand, from m(x) = ℎ 2 (x) ∇ (x) and ∇ (ȳ) = 0, the Jacobian is also calculated as J(ȳ) = I + ℎ 2 (ȳ) ∇ 2 (ȳ),(19) where I is the × -identity matrix. The fact that ∇ 2 at a local maximizer becomes negative semidefinite, together with the positive semidefiniteness of the Jacobian J(ȳ) mentioned above, implies that J(ȳ) at a local maximizerȳ of has eigenvalues within the interval [0, 1]. The following proposition, which is a generalization of [27] with the Gaussian kernel to that with a generic three-times differentiable kernel, shows the linear convergence when the Hessian ∇ 2 (ȳ) is non-degenerate. Proposition 5 (Linear convergence in non-degenerate case). Assume Assumptions 1 and 2, that is three-times differentiable, that the mode estimate sequence (y ) ∈N obtained by the MS algorithm (8) converges toȳ = lim →∞ y , and that the Hessian ∇ 2 (ȳ) of the KDE atȳ is negative definite. Then, the mode estimate sequence (y ) ∈N achieves the linear convergence: ∥ȳ − y ∥ = ( ) and (ȳ) − (y ) = ( 2 ) with = 1 + ℎ 2 (ȳ) ∈ [0, 1), where ∈ [−ˇ(ȳ ) ℎ 2 , 0) is the largest eigenvalue of the Hessian ∇ 2 (ȳ). In the above proposition, we excluded from our consideration the case where the Hessian ∇ 2 (ȳ) is degenerate. When the Hessian is degenerate, the Jacobian J(ȳ) has the largest eigenvalue equal to 1, and then analysis based on the first-order Taylor approximation of J(ȳ) does not lead to the (linear) convergence of the MS algorithm. In order to evaluate convergence rate along the same line of the analysis in such cases, one might have to investigate effects of the higher order terms in more detail. Discussion based on the Łojasiewicz property allows us to derive convergence rate of the MS algorithm under a weaker assumption on differentiability. More concretely, by applying [20, Theorem 3.5], we can prove the following theorem on the convergence rate of the MS algorithm that covers more general kernels and the degenerate case as well. It provides upper bounds of the convergence rate in terms of the Łojasiewicz exponent of the KDE. Theorem 3 (Convergence rate evaluation). Under the same assumptions as in Theorem 1 or 2, assume further that the KDE has the Łojasiewicz exponent atȳ = lim →∞ y , for the mode estimate sequence (y ) ∈N obtained by the MS algorithm (8). Then, one has that It should be noted that the Epanechnikov kernel does not satisfy Assumption 3 as shown in the assumptions (a1) and (a2) in Theorem 1 are satisfied. With the Epanechnikov kernel, the Łojasiewicz exponent at the mode of the KDE is typically 1 2 , and if applying Theorem 3 is legitimate, it suggests the linear convergence via (b2), which is a looser evaluation than the finite-time convergence guaranteed by [13] and [14]. However, the convergence rate evaluation provided by Theorem 3 seems to be almost tight in other generic cases, as demonstrated in Figure 2, where the behaviors of the MS algorithm with the Gaussian kernel in the one-dimensional case are shown, with carefully chosen positions of data points so that the KDE has a degenerate Hessian at its mode. Theorem 3, as well as the experimental results summarized in Figure 2, strongly suggests that the Łojasiewicz exponent of the KDE bears essential information about the convergence rate of the MS algorithm. It is known, however, that the calculation of the Łojasiewicz exponent is difficult in general (see discussion of [48] for details). Even in such a circumstance, [49]- [51] provided bounds of the Łojasiewicz exponent for polynomial functions. On the ground of [50,Proposition 4.3], we can provide an upper bound of the Łojasiewicz exponent of the KDE with a piecewise polynomial kernel. Theorem 4 (Bound of Łojasiewicz exponent). Assume that the kernel is of class 1 and piecewise polynomial with maximum degree ≥ 2. Then, the Łojasiewicz exponent of the KDE at any critical pointȳ is bounded from above as ≤ 1 − 1 max{ (3 − 4) −1 , 2 (3 − 3) −2 } ,(20) provided that is not constant in any subdomain with a nonempty intersection with the -neighborhood of y for any > 0. This bound of the Łojasiewicz exponent, together with Theorem 3 (b3), gives a worst-case bound of the convergence rate of the MS algorithm using a piecewise polynomial kernel. However, it should be noted that the bound provided in Theorem 4 is not tight in general and might be improved by future research. CONCLUSION AND FUTURE WORK We have shown that the mode estimate sequence generated by the MS algorithm using a 1 subanalytic kernel converges to a critical point of the KDE (Theorem 2). Our proof does neither presume that the KDE has a finite number of critical points or they are isolated, nor that its Hessian at a convergent point is non-degenerate, nor restriction on the size of the bandwidth or on the data dimension; it utilizes the Łojasiewicz property of the KDE. The class of kernels covered by this theorem includes several piecewise polynomial kernels, such as a biweight kernel which is optimal among non-negative kernels for the KDEbased estimation of a non-degenerate mode in terms of the asymptotic statistical efficiency [12], [18]. The convergence analysis results in this paper extend the existing ones for the Epanechnikov kernel [13], [14] and for analytic kernels [15]. Moreover, we not only provide a sufficient condition for the mode estimate sequence to achieve the linear convergence when the Hessian of the KDE at a convergent point is non-degenerate (Proposition 5), but also give a worst-case evaluation of the convergence rate (Theorems 3 and 4) in terms of the Łojasiewicz exponent of the KDE, applicable even when the Hessian is degenerate. The convergence theorems of the MS algorithm, including ours for 1 subanalytic kernels and the existing ones for the Epanechnikov kernel and analytic kernels, are also effective for the iteratively reweighted least squares algorithm, commonly used for various versions of robust M-type location estimation and regression [52], [53]. Moreover, these results can be applied to several generalized MS algorithms. The conditional MS algorithm, which is a representative estimation method for nonparametric modal regression [54]- [57], can be regarded as a weighted version of the conventional MS algorithm with the weights determined by the values of the independent variable part of the data. The convergence theorems can be generalized to the weighted version of the MS algorithm derived for the weighted objective function, 1 ℎ =1 ( x−x ℎ ) with constant weights { ∈ (0, ∞)} ∈ [ ] . Other instances of the generalized MS algorithms include an MS variant derived for the KDE 1 [24], [58], and the over-relaxation of the MS algorithm, y +1 = y + m(y ) with a constant ∈ (0, 2) [15]. Even under these generalizations, a guarantee of the convergence to a critical point and a convergence rate evaluation still hold as well. =1 1 ℎ ( x−x ℎ ) with datapoint-wise band- widths {ℎ ∈ (0, ∞)} ∈ [ ] The subspace constrained MS algorithm [59], [60], another MS variant, is a method for estimating principal curves and principal surfaces as ridges of the KDE [61], [62]. It iterates an update rule that is expected to converge to a point on a ridge of the KDE instead of its critical point. The convergence property of the subspace constrained MS algorithm would be related to that of the MS algorithm but is still open, and analysis with the Łojasiewicz property might be useful for it. S1 PROOFS OF THEOREMS In this appendix, we provide proofs of the theoretical results stated in the main text of this paper. Proposition 2 is shown as Theorem 1 of [3] and Theorem 1 of [15] (or can be proved from Lemma S3 described below), and Proposition 3 is given by [16], [35]. Refer to the description in Sections 4 and 5.2 respectively for the proof of Propositions 4 and 5. Also, Theorem 2 is a corollary of Theorem 1 as explained in Section 5.1. We here give proofs of the other results, Proposition 1 and Theorems 1, 3, and 4. Technical Lemmas for Proposition 1 First, we provide two technical lemmas: Lemma S1. Assume that a kernel satisfies Assumptions 1 and 2. Then,ˇdefined via (4) is non-negative, non-increasing, and bounded. Proof of Lemma S1. It has been proved just above the definition (4) ofˇin the main text. □ Lemma S2. Assume that a kernel satisfies Assumptions 1 and 2. Then, for any constant 1 > 0 there exists a constant 2 > 0 such that { ∈ [0, ∞) |ˆ( ) ≥ 1 } ⊆ { ∈ [0, ∞) |ˇ( ) ≥ 2 } holds. Proof of Lemma S2. Since the profileˆis non-increasing (Assumption 2), one hasˆ(0) = max ≥0ˆ( ). One also hasˆ(0) ≠ 0, since otherwise the kernel (·) is equal to 0 identically, contradicting the assumption that is normalized (Assumption 1). Sinceˆ(0) > 0 andˆis continuous (Assumption 1) and non-increasing (Assumption 2), if 1 >ˆ(0) the set { ∈ [0, ∞) | ( ) ≥ 1 } is empty and the statement of the lemma trivially holds. We therefore assume 0 < 1 ≤ˆ(0) in the following. For any such 1 one can let [0, ] = { ∈ [0, ∞) |ˆ( ) ≥ 1 } with ∈ [0, ∞). Here, the finiteness of comes from the normalization condition of (Assumption 1). Lemma S1 shows thatˇis non-negative and non-increasing. One then has thatˇ( ) > 0 for any ∈ [0, ]. It is because if there exists ∈ [0, ] such thatˇ( ) = 0 then for any ≥ one hasˇ( ) = 0 and henceˆ( ) =ˆ( ) ≥ 1 > 0, which contradicts the normalization condition of (Assumption 1). Letting 2 = min ∈ [0, ]ˇ( ) =ˇ( ) > 0, one hasˇ( ) ≥ 2 for any ∈ [0, ], which proves { ∈ [0, ∞) |ˆ( ) ≥ 1 } = [0, ] ⊆ { ∈ [0, ∞) |ˇ( ) ≥ 2 } to hold with that 2 . □ Proof of Proposition 1 Proposition 2 and Lemmas S1 and S2 lead to Proposition 1, as in the following proof. Proof of Proposition 1. The ascent property (Proposition 2) implies (y ) = 1 ℎ ∑︁ =1 y − x ℎ ≥ (y 1 ).(S1) Let be an index in [ ] satisfying y − x ℎ = max ∈ [ ] y − x ℎ .(S2) This definition and inequality (S1) lead tô y − x ℎ 2 2 = y − x ℎ ≥ 1 ∑︁ =1 y − x ℎ ≥ ℎ (y 1 ),(S3) which implies that = ∥ y −x ℎ ∥ 2 2 for any ∈ N is in the set { |ˆ( ) ≥ 1 } with 1 = ℎ (y 1 ) > 0. Lemma S2 then states that there exists a constant > 0 such that for any ∈ Ň y − x ℎ 2 2 ≥ (S4) holds. From Lemma S1,ˇis non-negative. Using this fact and inequality (S4), one consequently has thať (y ) = 1 ℎ ∑︁ =1ˇ y − x ℎ 2 2 ≥ 1 ℎˇ y − x ℎ 2 2 ≥ ℎ . (S5) This concludes the proof. □ Technical Lemmas for Theorems 1 and 3 We here provide three technical lemmas that introduce positive constants¯,¯, and¯, each of which defines a separate inequality: Lemma S3 (Sufficient increase condition). Assume Assumptions 1 and 2, and (y 1 ) > 0. Then, there exists¯> 0 such that (y +1 ) − (y ) ≥¯∥y +1 − y ∥ 2 (S6) holds for any ∈ N. Proof of Lemma S3. Considering the coefficients of xand ∥x∥ 2 -dependent terms of the minorizer¯(x|y ) in (6), and the update rule of the MS algorithm (8), one can find another representation of¯(x|y ): (x|y ) = −ˇ( y ) 2ℎ 2 ∥y +1 − x∥ 2 + (x-indep. const.). (S7) This representation, together with the ascent property (y ) =¯(y |y ) ≤¯(y +1 |y ) ≤ (y +1 ), yields the inequality (y +1 ) − (y ) ≥¯(y +1 |y ) −¯(y |y ) =ˇ( y ) 2ℎ 2 ∥y − y +1 ∥ 2 . (S8) Proposition 1 shows that there exists a constant 1 > 0 such thatˇ(y ) ≥ 1 ℎ . Consequently, one has (y +1 ) − (y ) ≥¯∥y +1 − y ∥ 2 with¯= 1 2 ℎ +2 .(S9) □ Lemma S4. Assume Assumptions 1 and 2, and (y 1 ) > 0. (d1) Assume furthermore the former half of the assumption (a1) in Theorem 1: The KDE is differentiable on cl(Conv({y } ≥ )) with some ∈ N. Then there exists¯> 0 such that ∥y +1 − y ∥ ≥¯∥∇ (y )∥ (S10) holds for any ≥ . (d2) Instead, assume further the former half of Assumption 3: The kernel is differentiable. Then there exists¯> 0 such that (S10) holds for any ∈ N. Proof of Lemma S4. Proposition 1 ensures thatˇ(y ) > 0 for any ∈ N. Under the differentiability of the KDE at y with ≥ , the ordinary update rule of the MS algorithm (8) can be seen as a gradient ascent method with an adaptive step size: y +1 = =1ˇ( ∥ y −x ℎ ∥ 2 /2)x =1ˇ( ∥ y −x ℎ ∥ 2 /2) = y + ℎ 2 · {− 1 ℎ +2 =1ˇ( ∥ y −x ℎ ∥ 2 /2)(y − x )} 1 ℎ =1ˇ( ∥ y −x ℎ ∥ 2 /2) = y + ℎ 2 (y ) ∇ (y ).(S11) The boundedness ofˇ(Lemma S1) implies that there exists a constant 2 > 0 such that |ˇ(∥ y −x ℎ ∥ 2 /2)| ≤ 2 for any = 1, . . . , and any ∈ N, and hence |ˇ(y )| ≤ 2 ℎ . Thus, for any ≥ one has ∥y +1 − y ∥ = ℎ 2 |ˇ(y )| ∥∇ (y )∥ ≥¯∥∇ (y )∥ with¯= ℎ +2 2 ,(S12) proving the claim (d1). The claim (d2) follows from the differentiability of at every y . □ Lemma S5 (Relative error condition). Assume Assumptions 1 and 2, and (y 1 ) > 0. (e1) Assume furthermore the assumption (a1) in Theorem 1: The KDE is differentiable and has a Lipschitz-continuous gradient on cl(Conv({y } ≥ )) with some ∈ N. Then there exists¯> 0 such that ∥y +1 − y ∥ ≥¯∥∇ (y +1 )∥ (S13) holds for any ≥ . (e2) Instead, assume furthermore Assumption 3: The kernel is differentiable and has a Lipschitz-continuous gradient. Then there exists¯> 0 such that (S13) holds for any ∈ N. Proof of Lemma S5. With the Lipschitz constant ≥ 0 of ∇ , one can find the relation ∥∇ (y +1 )∥ ≤ ∥∇ (y )∥ + ∥∇ (y +1 ) − ∇ (y )∥ (∵ Triangle inequality) ≤ 1 ∥y +1 − y ∥ + ∥y +1 − y ∥ (∵ Lemma S4 and Lipschitz continuity of ∇ ) = 1 ∥y +1 − y ∥ with¯= 1 + −1 . (S14) When ∇ is Lipschitz-continuous with a Lipschitz constant 3 ≥ 0, one can set = 3 ℎ +2 and¯= ℎ +2 2 + 3 with a constant 2 > 0 that bounds |ˇ(∥ y −x ℎ ∥ 2 /2)| from above for every ∈ [ ] and ∈ N. □ Preliminaries for Proof of Theorems 1 and 3 Let ∈ (0, ∞], and let : [0, ) → [0, ∞) be a continuous concave function such that (0) = 0 and is continuously differentiable on (0, ) with ′ ( ) > 0. The concavity of implies that ′ is non-increasing on (0, ). The Łojasiewicz inequality (10) holds trivially with (x ′ , x) satisfying (x ′ ) − (x) = 0. Also, it is known that the Łojasiewicz inequality (10) with (x ′ , x) such that x ∈¯(x ′ , , , ) ≔ {x ∈ | ∥x ′ − x∥ < , (x ′ ) − (x) ∈ (0, )} is a special case of ′ ( (x ′ ) − (x))∥∇ (x)∥ ≥ 1 at (x ′ , x) such that x ∈¯(x ′ , , , ) (S15) with ( ) = 1− (1− ) where is a positive constant. (One technical subtlety with this extended definition is that we have excluded those x with (x) = (x ′ ) from¯(x ′ , , , ), as those points would make the left-hand side of (S15) indeterminate.) Note that [19], [20] call the function a desingularizing function because of its role in (S15), where • is in a sense resolving criticality of at x ′ . Note also that for the choice ( ) = Proof of Theorem 1. The density estimate sequence ( (y )) ∈N converges under Assumptions 1 and 2 since it is a bounded non-decreasing sequence (Proposition 2). Also, as (y 1 ) > 0, for every ≥ 2 y lies in the convex hull Conv({x } ∈ [ ] ) of data points, which is a compact set. Thus, there exist an accumulation pointỹ ∈ Conv({x } ∈ [ ] ) of the mode estimate sequence (y ) ∈N and a subsequence (y ′ ) ′ ∈ of (y ) ∈N (with ⊆ N) that converges to the accumulation pointỹ as ′ → ∞. Also,ỹ ∈ cl(Conv({y } ≥ )) obviously holds for any ∈ N. When there exists ′ ∈ such that (ỹ) = (y ′ ), Lemma S3 obviously shows the convergence of (y ) ∈N toỹ: y =ỹ for any ≥ ′ , since otherwise (y ′ +1 ) ≥ (y ′ ) +¯∥y ′ +1 − y ′ ∥ 2 = (ỹ)+¯∥y ′ +1 −ỹ∥ 2 holds, which implies (y ) ≥ (ỹ)+¯∥y ′ +1 −ỹ∥ 2 for all ≥ ′ +1, so thatỹ cannot be an accumulation point of (y ) ∈N . We therefore consider in what follows the remaining case where (ỹ) > (y ) for all ∈ N. The assumption (a2) ensures that there exists a positive constant such that the KDE satisfies the Łojasiewicz inequality (10) at least with any (x ′ , x) = (ỹ, y) such that y ∈ (ỹ, cl(Conv({y } ≥ )), ) for some integer . As we want to use the general form (S15) of the Lojasiewicz inequality, we have to further restrict the region where the Łojasiewicz inequality to hold from (ỹ, cl(Conv({y } ≥ )), ) to¯(ỹ, cl(Conv({y } ≥ )), , ) in order to ensure that (ỹ) − (y) is in the domain [0, ) of the desingularizing function . Denoting ≔ (ỹ) − (y ) > 0, the convergence of the density estimate sequence ( (y )) ∈N and the definition ofỹ imply that the sequence ( ) ∈N is positive, non-increasing, and converging to 0 as → ∞. The facts, y ′ →ỹ and → 0, as well as the continuity of , imply the existence of a finite integer ′ ≥ in such that ∈ [0, ) holds for any ≥ ′ , and that the inequality ∥ỹ − y ′ ∥ + 2 √︂ ′ + 1¯( ′ ) < (S16) holds. It should be noted that if the assumptions (a1) and (a2) hold with some ∈ N, they also hold with the above ′ since {y } ≥ ′ ⊆ {y } ≥ with ′ ≥ . Using the Łojasiewicz property of the KDE on¯(ỹ, cl(Conv({y } ≥ ′ )), , ), the inequality (S16), and assumption (a1), we prove below that the mode estimate sequence (y ) ∈N does not endlessly wander and does converge toỹ, and thatỹ is a critical point of the KDE . Two key claims: We will establish the following two claims for any ≥ ′ + 1, which are the key to proving Theorem 1. Claim S1. y satisfies y ∈¯(ỹ, cl(Conv({y } ≥ ′ )), , ). (S17) In other words, the Łojasiewicz inequality (S15) with (x ′ , x) = (ỹ, y ) holds. Claim S2. {y } ∈ { ′ ,..., +1} satisfies ∑︁ = ′ +1 ∥y +1 − y ∥ + ∥y +1 − y ∥ ≤ ∥y ′ +1 − y ′ ∥ + 1¯{ ( ′ +1 ) − ( +1 )}. (S18) Auxiliary results: We here provide two auxiliary results to be used in the succeeding proof. First, one has ∥y ′ +1 − y ′ ∥ ≤ √︂ ′ − ′ +1 (∵ Lemma S3) ≤ √︂ ′ (∵ ′ +1 ≥ 0).(S19) Secondly, we show the following auxiliary lemma, which will be used in proving (S18) from (S17) via making use of the Łojasiewicz property. Lemma S6. If y with ≥ satisfies Claim S1, that is, if y ∈¯(ỹ, cl(Conv({y } ≥ ′ )), , ) holds, then 2∥y +1 − y ∥ ≤ ∥y − y −1 ∥ + 1¯{ ( ) − ( +1 )}. (S20) Proof of Lemma S6. Since (S20) holds trivially if y = y −1 , we consider the case y ≠ y −1 . When y ∈ (ỹ, cl(Conv({y } ≥ ′ )), , ), the Łojasiewicz inequality (S15) with (x ′ , x) = (ỹ, y ) holds. Noting that 0 < +1 ≤ < holds, one has ( ) − ( +1 ) = ∫ +1 ′ ( ) ≥ ′ ( )( − +1 ) (∵ ′ is positive and non-increasing) ≥ ′ ( )¯∥y +1 − y ∥ 2 (∵ Lemma S3) ≥ 1 ∥∇ (y )∥¯∥ y +1 − y ∥ 2 (∵ Łojasiewicz inequality (S15) with (x ′ , x) = (ỹ, y )) ≥¯¯∥ y +1 − y ∥ 2 ∥y − y −1 ∥ (∵ Lemma S5).(S21) The inequality 2 √ ≤ + for , ≥ 0 yields 2∥y +1 − y ∥ = 2 √︃ ∥y +1 − y ∥ 2 ≤ 2 √︂ ∥y − y −1 ∥ 1¯{ ( ) − ( +1 )} (∵ (S21)) ≤ ∥y − y −1 ∥ + 1¯{ ( ) − ( +1 )}.(S22) This concludes the proof of Lemma S6. □ Proof that Claims S1 and S2 hold for = ′ + 1: Here we prove Claims S1 and S2 for = ′ + 1. One has ∥ỹ − y ′ +1 ∥ ≤ ∥ỹ − y ′ ∥ + ∥y ′ +1 − y ′ ∥ (∵ Triangle inequality) ≤ ∥ỹ − y ′ ∥ + √︂ ′ (∵ (S19)) < (∵ (S16)),(S23) which, together with 0 < ′ +1 ≤ ′ < , implies (S17) with = ′ + 1, proving Claim S1 for = ′ + 1. Also, Claim S1 with = ′ + 1 implies, via Lemma S6, the inequality (S20) with = ′ + 1, which reads 2∥y ′ +2 − y ′ +1 ∥ ≤ ∥y ′ +1 − y ′ ∥ + 1¯{ ( ′ +1 ) − ( ′ +2 )}, which is nothing other than (S18) with = ′ + 1, thereby proving Claim S2 for = ′ + 1. Proof that Claims S1 and S2 hold for ≥ ′ + 1: Now that we have seen that Claim S2 holds for = ′ + 1, we next prove Claim S2 to hold for every ≥ ′ + 1 by induction. For this purpose, we prove Claims S1 and S2 for = + 1 under the assumption that Claims S1 and S2 hold for = ≥ ′ + 1. One has ∥ỹ − y +1 ∥ ≤ ∥ỹ − y ′ ∥ + ∥y ′ +1 − y ′ ∥ + ∑︁ = ′ +1 ∥y +1 − y ∥ (∵ Triangle inequality) ≤ ∥ỹ − y ′ ∥ + 2∥y ′ +1 − y ′ ∥ + 1¯{ ( ′ +1 ) − ( +1 )} − ∥y +1 − y ∥ (∵ (S18) with = ) ≤ ∥ỹ − y ′ ∥ + 2∥y ′ +1 − y ′ ∥ + 1¯( ′ +1 ) (∵ ∥y +1 − y ∥ ≥ 0 and ( +1 ) ≥ 0) ≤ ∥ỹ − y ′ ∥ + 2 √︂ ′ + 1¯( ′ ) (∵ (S19) and ( ′ ) ≥ ( ′ +1 )) < (∵ (S16)),(S25) which, together with Claim S1 for = and 0 < +1 ≤ < , implies Claim S1 to hold for = + 1. Also, this result ensures, via Lemma S6, that (S20) holds with = + 1. Adding (S20) with = + 1 to (S18) with = then shows that (S18) holds with = + 1, proving Claim S2 to hold for = + 1. As Claims S1 and S2 have been shown to hold for = ′ + 1, the above argument proves, by induction, that Claim S2 holds for every ≥ ′ + 1. Claim S2 for every ≥ ′ + 1 implies convergence: From (S18), one has for any ≥ ′ + 1 ∑︁ = +1 ∥y +1 − y ∥ ≤ ∥y ′ +1 − y ′ ∥ + 1¯{ ( ′ +1 ) − ( +1 )} − ∥y +1 − y ∥ ≤ ∥y ′ +1 − y ′ ∥ + 1¯( ′ +1 ) (∵ ∥y +1 − y ∥ ≥ 0 and ( +1 ) ≥ 0).(S26) Taking the limit → ∞ yields ∞ ∑︁ = ′ +1 ∥y +1 − y ∥ ≤ ∥y ′ +1 − y ′ ∥ + 1¯( ′ +1 ), (S27) which implies ∞ ∑︁ =1 ∥y +1 − y ∥ = ′ ∑︁ =1 ∥y +1 − y ∥ + ∞ ∑︁ = ′ +1 ∥y +1 − y ∥ = ′ ∑︁ =1 ∥y +1 − y ∥ + ∥y ′ +1 − y ′ ∥ + 1¯( ′ +1 ) < ∞.(S28) This shows that the trajectory of (y ) ∈N is of finite length, which in turn implies that (y ) ∈N converges. As the limit lim →∞ y is a unique accumulation point of (y ) ∈N , it must beỹ since y ′ →ỹ. Additionally, from Lemma S4, one has ∞ ∑︁ = ′ ∥∇ (y )∥ ≤ 1 ∞ ∑︁ = ′ ∥y +1 − y ∥ < ∞,(S29) which implies lim →∞ ∥∇ (y )∥ = 0. Since the gradient of the KDE is Lipschitz-continuous with a Lipschitz constant ≥ 0 on cl(Conv({y } ≥ ′ )) due to the assumption (a1), one has that ∥∇ (ỹ)∥ ≤ lim →∞ {∥∇ (y )∥ + ∥∇ (ỹ) − ∇ (y )∥} ≤ lim →∞ {∥∇ (y )∥ + ∥ỹ − y ∥} = 0,(S30) which implies that the limitỹ = lim →∞ y is a critical point of . ′ ( ) = − , Φ( ) =            − 1−2 2 (1−2 ) if ∈ [0, 1 2 ), − log( ) 2 if = 1 2 , − 1−2 2 (1−2 ) if ∈ ( 1 2 , 1), Φ −1 ( ) = exp(− 2 ) if = 1 2 , { 2 (2 − 1) } − 1 2 −1 if ∈ ( 1 2 , 1).(S31) These functional forms will be used in proving Theorem 3. Now, we provide a proof of Theorem 3, which is based on the proof of [20, Theorem 3.5]. Proof of Theorem 3. In the proof of Theorem 1 we have established the following facts: If there exists ′ ∈ N such that (ȳ) = (y ′ ) then y =ȳ for any ≥ ′ , that is, (y ) ∈N converges in a finite number of iterations. If otherwise, then there exists ∈ N such that Claim S1 holds for any ≥ , that is, y ∈¯(ȳ, cl(Conv({y } ≥ )), , ) holds for any ≥ , or equivalently, the Łojasiewicz inequality (S15) with (x ′ , x) = (ȳ, y ) holds for any ≥ . If y with ≥ satisfies Claim S1, then one has Φ( +1 ) − Φ( ) = ∫ +1 { ′ ( )} 2 (∵ Definition of Φ) ≥ { ′ ( )} 2 ( − +1 ) (∵ ′ is positive and non-increasing) ≥ { ′ ( )} 2¯∥ y +1 − y ∥ 2 (∵ Lemma S3) ≥ { ′ ( )} 2¯{¯∥ ∇ (y )∥} 2 (∵ Lemma S4) ≥¯¯2 (∵ Łojasiewicz inequality (S15) with (x ′ , x) = (ȳ, y )). (S32) Now Claim S1 holds for any ≥ , which implies Φ( ) − Φ( ) = −1 ∑︁ = {Φ( +1 ) − Φ( )} ≥¯¯2 ( − − 2).(S33) We discuss the two cases ∈ [0, 1 2 ) and ∈ [ 1 2 , 1) separately. Case ∈ [0, 1 2 ): We claim that in this case the algorithm converges in a finite number of iterations. If otherwise, the inequality (S33) should hold for any ≥ . When we take the limit → ∞, the right-hand side of (S33) goes to infinity, which contradicts the fact that the left-hand side remains finite, by noting that one has lim →0 Φ( ) = 0 with Φ( ) = − 1−2 2 (1−2 ) and that → 0 as → ∞. This contradiction implies the finite-time convergence of (y ) ∈N . Case ∈ [ 1 2 , 1): We may suppose that > 0 holds for any ∈ N, and so (S33) holds for any ≥ . Recalling the functional form of Φ( ) as in (S31), one has lim →∞ Φ( ) = ∞ with ∈ [ 1 2 , 1). Assume Φ( ) ≥ 0. (If it is not the case one can always redefine to a larger value with which Φ( ) ≥ 0 is satisfied.) One then has Φ( ) ≥¯¯2 ( − − 2) + Φ( ) ≥¯¯2 ( − − 2), which allows us to obtain the convergence rate evaluation for = (ȳ) − (y ), namely, (1− ) , one can obtain the exponential-rate convergence when = 1 2 and polynomial-rate convergence when ∈ ( 1 2 , 1): ∥ȳ − y ∥ ≤ {Φ −1 (¯¯2 ( − − 2))} 1−¯( 1 − ) =          {exp(− 2¯¯2 ( − −2) ) } 1 2¯( 1− ) = ( ) if = 1 2 , { { 2 (2 −1)¯¯2 ( − −2) } − 1 2 −1 } 1−¯( 1− ) = ( − 1− 2 −1 ) if ∈ ( 1 2 , 1).(S38) This concludes the proof for all the cases, (b1), (b2), and (b3). □ Proof of Theorem 4 Theorem 4 is proved using [50, Proposition 4.3] as follows: Proof of Theorem 4. As the kernel is assumed to be piecewise polynomial, the KDE is also piecewise polynomial, that is, there exists a finite collection { } =1 of subdomains ⊆ R , ∈ [ ] that forms a partition of the entire domain R of the KDE such that in each subdomain the restriction of the KDE to is the same as the restriction of the polynomial to . Case I: When the critical pointȳ of the KDE lies in the interior of one of the subdomains, say , then one can take > 0 small enough so that (ȳ, R , ) is contained in the subdomain . Then the KDE is equal to the polynomial in (ȳ, R , ). The polynomial is not constant by assumption, and its degree is at least 2 asȳ is a critical point of . Therefore, any upper bound of the Łojasiewicz exponent of that polynomial is an upper bound of the Łojasiewicz exponent of the KDE . Case II: We next assume in the following thatȳ is located on a boundary of several subdomains. Let 1 , . . . , ′ (with 2 ≤ ′ ≤ ) be the subdomains each of which has a non-empty intersection with the -neighbor ofȳ for any > 0. One has 1 (ȳ) = · · · = ′ (ȳ). Because of the assumption that the kernel is of class 1 , one also has ∇ (ȳ) = 0 for all ∈ [ ′ ], For any ∈ [ ′ ], the polynomial is not constant by assumption, and its degree is at least 2 asȳ is a critical point of . One can therefore assume that for any ∈ [ ′ ] has the Łojasiewicz property, that is, there exist > 0, > 0, and ∈ [0, 1) such that for any y ∈ (ȳ, , ) satisfies the Łojasiewicz inequality where tc,1 , tc,2 , , are polynomial functions, this kernel is also semialgebraic and hence subanalytic. Although the kernel (· − x ) = − {(1 − ∥ · −x ∥ 2 ) + } 3/2 with a positive normalizing constant − is not a piecewise polynomial kernel, one can show that this kernel is also semialgebraic and hence subanalytic, because its graph can be represented as Cosine Kernel The cosine kernel (· − x ) = cs cos( ∥ ·−x ∥ 2 )1(∥ · −x ∥ ≤ 1) with a positive normalizing constant cs is semianalytic and hence subanalytic, because its graph can be written as {(x, ) ∈ R +1 | = cs cos( ∥x−x ∥ 2 )1(∥x − x ∥ ≤ 1)} = {(x, ) ∈ R +1 | (x, ) > 0} ∩ {(x, ) ∈ R +1 | cs (x, ) ≔ cs cos( ∥x−x ∥ 2 ) − = 0} ∪ {(x, ) ∈ R +1 | (x, ) = 0} ∩ {(x, ) ∈ R +1 | (x, ) = 0} ∪ {(x, ) ∈ R +1 | (x, ) < 0} ∩ {(x, ) ∈ R +1 | (x, ) = 0} ,(S45) where cs , , are analytic functions. As stated in Proposition 3, in view of the Łojasiewicz property, what is important for our purpose is to provide sufficient conditions for the KDE to be 1 globally subanalytic. Thus, sufficient conditions for global subanalyticity in the above inclusion relations, as well as the stability of the global subanalyticity under the summation [42, Properties 1.1.8], are important, which are summarized as follows: Proposition 4. Any semialgebraic or globally semianalytic functions, any semianalytic or subanalytic functions with a bounded graph, and the summation of any globally subanalytic functions are globally subanalytic. Fig. 1 . 1Inclusion relation among function classes. ( b1 ) b1if ∈ [0, 1 2 ) then (y ) ∈N converges in a finite number of iterations, (b2) if = 1 2 then ∥ȳ−y ∥ = ( ) and (ȳ) − (y ) = ( 2 ) with some ∈ (0, 1), or (b3) if ∈ ( 1 2 , 1) then ∥ȳ−y ∥ = ( − 1− 2 −1 ) and (ȳ)− (y ) = ( − 1 2 −1 ). Fig. 2 . 2Instances of the KDE, and plots of |¯− | and (¯) − ( ) versus with ( ) ∈N obtained by the MS algorithm with the Gaussian kernel with = 1 and ℎ = 1. For every case, the mode¯, to which ( ) ∈N converges, is the origin (i.e.,¯= 0). (i) = 2, 1 , 2 = ±0.95, with which the second derivative (2) ( ) of the KDE ( ) is non-degenerate at the mode =¯, yielding = 1/2. The plots of |¯− | and (¯) − ( ) are shown in semilog plots. (ii) = 2, 1 , 2 = ±1, with which ( ) (¯) = 0 for all ∈ [3] and (4) (¯) < 0, yielding = 1 − 1/4 = 3/4. (iii) = 6, 1 , 2 = ±0.564 . . . , 3 , 4 = ±1.721 . . . , 5 , 6 = ±2.801 . . ., which were carefully chosen so that ( ) (¯) = 0 for all ∈ [5] and (6) (¯) < 0, yielding = 1 − 1/6 = 5/6. (iv) = 6, 1 , 2 = ±0.651 . . . , 3 , 4 = ±1.959 . . . , 5 , 6 = ±3.243 . . ., which were carefully chosen so that ( ) (¯) = 0 for all ∈ [7] and (8) (¯) < 0, yielding = 1 − 1/8 = 7/8. The plots of |¯− | and (¯) − ( ) in (ii), (iii), and (iv) are shown in log-log plots. Simulation results are shown as black solid curves, and the red dotted lines show the asymptotic convergence rates predicted by Proposition 5 and Theorem 3 (b3). Ryoya Yamasaki received the B.E. and M.Inf. degrees from Kyoto University, Kyoto, Japan, in 2018 and 2020, respectively. He is currently working toward the D.Inf. degree of Graduate School of Informatics, Kyoto University, Kyoto, Japan. His research interests are in areas of statistics and machine learning. Toshiyuki Tanaka received the B.E., M.E., and D.E. degrees from the University of Tokyo, Tokyo, Japan, in 1988, 1990, and 1993, respectively. He is currently a professor of Graduate School of Informatics, Kyoto University, Kyoto, Japan. His research interests are in areas of information, coding, and communications theory, and statistical learning. (Definition 1) of the Łojasiewicz property. The following proof of the convergence of the mode estimate sequence (y ) ∈N (Theorem 1) is not restricted to the specific choice ( ) = 1− (1− ) but holds with the general form (S15) of the Łojasiewicz inequality. The specific choice ( ) = 1− (1− ) , on the other hand, will help derive the convergence rate in Theorem 3. Proof of Theorem 1 We here provide a proof of Theorem 1 on the ground of [19, Theorem 3.2] and [20, Theorem 3.1]. For a desingularizing function ( ), define Φ( ) to be a primitive function (indefinite integral) of −( ′ ) 2 . For the specific choice of the desingularizing function ( ) exp(− 2¯¯2 /2) ∈ (0, 1). For the convergence rate evaluation for ∥ȳ − y ∥, we have ) (∵ ′ is positive and non-increasing) ≥ ′ ( )¯∥y +1 − y ∥ 2 (∵ Lemma S3) ≥ ′ ( )¯∥y +1 − y ∥¯∥∇ (y )∥ (∵ Lemma S4) ≥¯¯∥y +1 − y ∥ (∵ Łojasiewicz inequality (S15) with (x ′ , x) = (ȳ, y )), ∥∇ (y)∥ ≥ { (ȳ) − (y)} .(S39)We show that under these conditions has the Łojasiewicz property atȳ. Let min ≔ min ∈ [ ′ ] , max ≔ max ∈ [ ′ ] , and ≔ max ∈ [ ′ ] sup y∈ (ȳ, , min ) { (ȳ) − (y)} > 0. (S40) x, ) ∈ R +1 | −,2 (x, ) ≔ − − = 0} ∪ {(x, ) ∈ R +1 | (x, ) > 0} ∩ {(x, ) ∈ R +1 | (x, ) > 0} ∩ {(x, ) ∈ R +1 | −,1 (x, ) = 0} ∩ {(x, ) ∈ R +1 | −,2 (x, ) < 0} ∪ {(x, ) ∈ R +1 | (x, ) = 0} ∩ {(x, ) ∈ R +1 | (x, ) = 0} ∪ {(x, ) ∈ R +1 | (x, ) < 0} ∩ {(x, ) ∈ R +1 | (x, ) = 0} (S44)with the polynomial functions −,1 , −,2 , , . 42, before Example 1.1.4]. Any globally subanalytic functions are subanalytic, and any subanalytic functions with a bounded graph are globally subanalytic [37, after Definition 2.2]. Also, semianalytic functions are subanalytic (which can be seen from Definition 2), globally semianalytic functions are globally subanalytic [42, Definition 1.1.6], and semialgebraic functions are globally semianalytic [42, Example 1.1.4]. 19, Theorem 3.2] and [20, Theorem 3.1] to the MS algorithm, we obtain the following theorem: 3 Theorem 1 (Convergence guarantee). Assume Assumptions 1 and 2. Let (y ) ∈N be the mode estimate sequence obtained by the MS algorithm (8) starting from y 1 with (y 1 ) > 0. Assume further, for the closure cl(Conv({y } ≥ )) of the convex hull Conv({y } ≥ ) of {y } ≥ with some ∈ N, that (a1) the KDE is differentiable and has a Lipschitz-continuous gradient on cl(Conv({y } ≥ )) (i.e., there exists a con- TABLE 1 1Fulfillment of assumptions of kernels (satisfying Asms. 1 and 4), convergence guarantee of the mode estimate sequence (y ) ∈N obtained by the corresponding MS algorithm, and convergence rates. Kernelˆ( ) ∝ Asm. 2 Asm. 3 Convergence Rate Worst-case bound of rate Biweight Table 1 , 1so that Theorem 3 will be applicable to the MS algorithm with the Epanechnikov kernel only under the conditions where(i) (ii) (iii) (iv) .[46],[47] show that the Epanechnikov kernel minimizes the asymptotic mean integrated squared error of the KDE using the associated optimal bandwidth parameter among non-negative kernels. It should be noted, however, that, although this fact was mentioned in papers which study convergence properties of the MS algorithm, such as[13] and[14], it does not imply the optimality of the Epanechnikov kernel for the KDE-based mode estimation, a representative application of the MS algorithm, in any sense. ACKNOWLEDGMENT This work was supported by Grant-in-Aid for JSPS Fellows, Number 20J23367. We would like to thank the authors of the article[25]for kindly drawing our attention to the errata[26]that accompanies that article.Take any y ∈ (ȳ, R , min ). Then there exists an index (y) ∈ [ ′ ] such that y ∈ (y) , and ∥∇ (y)∥ = ∥∇ (y) (y)∥ ≥ (y) { (y) (ȳ) − (y) (y)} (y) (∵ Łojasiewicz property of atȳ) = (y)(y)(y) (ȳ) − (y) (y) (y) ≥ (y)(y)(y) (ȳ) − (y) (y) maxwhich shows that has the Łojasiewicz property atȳ with the exponent max . The arguments so far have proved that, when the kernel is piecewise polynomial and of class 1 , the Łojasiewicz exponent of the KDE at any critical point is bounded from above by the largest Łojasiewicz exponent of the related polynomials 1 , . . . , ′ . For a polynomial,[50,Proposition 4.3]gives an upper bound of the Łojasiewicz exponent at its critical point, and the bound is increasing in the degree of the polynomial. When the kernel is piecewise polynomial with the maximum degree , the polynomials { } appearing as the restrictions of the KDE are of degrees at most . Thus, by substituting the possible maximum degree of a piecewise polynomial KDE into the bound in[50,Proposition 4.3], one can obtain the upper bound of the Łojasiewicz exponent of the KDE at its critical point as in(20), proving the theorem. □ Note that we can obtain an alternative upper bound of the Łojasiewicz exponent at a critical point of a polynomial, which is valid when the critical point is a local maximum of the polynomial. It is given by ≤ 1−1/{( −1) +1} for a degreepolynomial of variables, according to[49], and is better than the upper bound 1 − 1/max{ (3 − 4) −1 , 2 (3 − 3) −2 } used in the above proof, the latter of which does not require that the critical point is a local maximum. Therefore, in the above proof, ifȳ is a local maximum of the KDE and lies in the interior of one subdomain, one has the better upper bound ≤ 1 − 1/{( − 1) + 1}. Although the better upper bound also applies to Case II ifȳ is a local maximum of each of all the polynomials 1 , . . . , ′ , in general it is not applicable, since a local maximum of is not necessarily a local maximum of .S2 SUPPLEMENT TO TABLE 1Here, we describe supplementary explanation toTable 1, especially that of the fact that the kernels shown inTable 1satisfy Assumption 4. Throughout this section, we let (x, ) ≔ 1 − ∥x − x ∥ 2 and (x, ) ≔ , both of which are polynomial and real analytic in (x ⊤ , ) ⊤ , and omit the bandwidth ℎ.Analytic Kernels It is clear that the Gaussian, logistic, and Cauchy kernels are real analytic functions.Piecewise Polynomial Kernels For a positive integer , let (· − x ) ≔ {( (·, )) + } with a normalizing coefficient > 0. The Epanechnikov kernel (· − x ) = ep (1 − ∥ · −x ∥ 2 ) + , the biweight kernel (· − x ) = bw {(1 − ∥ · −x ∥ 2 ) + } 2 , and the triweight kernel (· − x ) = tw {(1 − ∥ · −x ∥ 2 ) + } 3 fall within this category with = 1, 2, 3, respectively. The kernel (· − x ) is a semialgebraic function, because its graphis semialgebraic, as , , are all polynomial. This shows that the Epanechnikov kernel, the biweight kernel, and the triweight kernel are all semialgebraic, and hence subanalytic. Since the graph of the tricube kernel (· − x ) = tc {(1 − ∥ · −x ∥ 3 ) + } 3 with a positive normalizing constant tc can be written as∩ {(x, ) ∈ R +1 | tc,2 (x, ) ≔ − tc = 0} ∪ {(x, ) ∈ R +1 | (x, ) > 0} ∩ {(x, ) ∈ R +1 | (x, ) > 0} ∩ {(x, ) ∈ R +1 | tc,1 (x, ) = 0} ∩ {(x, ) ∈ R +1 | tc,2 (x, ) < 0} ∪ {(x, ) ∈ R +1 | (x, ) = 0} ∩ {(x, ) ∈ R +1 | (x, ) = 0} The estimation of the gradient of a density function, with applications in pattern recognition. K Fukunaga, L Hostetler, IEEE Transactions on Information Theory. 211K. Fukunaga and L. Hostetler, "The estimation of the gradient of a density function, with applications in pattern recognition," IEEE Transactions on Information Theory, vol. 21, no. 1, pp. 32-40, 1975. Mean shift, mode seeking, and clustering. Y Cheng, IEEE Transactions on Pattern Analysis and Machine Intelligence. 178Y. Cheng, "Mean shift, mode seeking, and clustering," IEEE Trans- actions on Pattern Analysis and Machine Intelligence, vol. 17, no. 8, pp. 790-799, 1995. Mean shift: a robust approach toward feature space analysis. D Comaniciu, P Meer, IEEE Transactions on Pattern Analysis and Machine Intelligence. 245D. Comaniciu and P. Meer, "Mean shift: a robust approach toward feature space analysis," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 24, no. 5, pp. 603-619, 2002. Mean shift-based clustering. K.-L Wu, M.-S Yang, Pattern Recognition. 4011K.-L. Wu and M.-S. Yang, "Mean shift-based clustering," Pattern Recognition, vol. 40, no. 11, pp. 3035-3052, 2007. Mixture model modal clustering. J E Chacón, Advances in Data Analysis and Classification. 12J. E. Chacón, "Mixture model modal clustering," Advances in Data Analysis and Classification, vol. 12, no. 41, pp. 1-26, 2018. Color image segmentation based on mean shift and normalized cuts. W Tao, H Jin, Y Zhang, IEEE Transactions on Systems, Man, and Cybernetics. 375Part B (Cybernetics)W. Tao, H. Jin, and Y. Zhang, "Color image segmentation based on mean shift and normalized cuts," IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), vol. 37, no. 5, pp. 1382- 1389, 2007. Mean shift-based edge detection for color image. H Guo, P Guo, Q Liu, 2005 International Conference on Neural Networks and Brain. 2H. Guo, P. Guo, and Q. Liu, "Mean shift-based edge detection for color image," in 2005 International Conference on Neural Networks and Brain, vol. 2, 2005, pp. 1118-1122. Edge detection based on fast adaptive mean shift algorithm. Y Zhu, R He, N Xiong, P Shi, Z Zhang, 2009 International Conference on Computational Science and Engineering. 2Y. Zhu, R. He, N. Xiong, P. Shi, and Z. Zhang, "Edge detection based on fast adaptive mean shift algorithm," in 2009 International Conference on Computational Science and Engineering, vol. 2, 2009, pp. 1034-1039. Kernel-based object tracking. D Comaniciu, V Ramesh, P Meer, IEEE Transactions on Pattern Analysis and Machine Intelligence. 255D. Comaniciu, V. Ramesh, and P. Meer, "Kernel-based object track- ing," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 25, no. 5, pp. 564-577, 2003. Efficient mean-shift tracking via a new similarity measure. C Yang, R Duraiswami, L Davis, 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. 1C. Yang, R. Duraiswami, and L. Davis, "Efficient mean-shift track- ing via a new similarity measure," in 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, vol. 1, 2005, pp. 176-183. On estimation of a probability density function and mode. E Parzen, The Annals of Mathematical Statistics. 333E. Parzen, "On estimation of a probability density function and mode," The Annals of Mathematical Statistics, vol. 33, no. 3, pp. 1065-1076, 1962. Optimal kernel for kernelbased modal statistical methods. R Yamasaki, T Tanaka, arXiv:2304.10046v12023arXiv preprintstat.MLR. Yamasaki and T. Tanaka, "Optimal kernel for kernel- based modal statistical methods," 2023, arXiv preprint arXiv:2304.10046v1 [stat.ML]. Mean shift analysis and applications. D Comaniciu, P Meer, Proceedings of the Seventh IEEE International Conference on Computer Vision. the Seventh IEEE International Conference on Computer Vision2D. Comaniciu and P. Meer, "Mean shift analysis and applications," in Proceedings of the Seventh IEEE International Conference on Com- puter Vision, vol. 2, 1999, pp. 1197-1203. On convergence of Epanechnikov mean shift. K Huang, X Fu, N Sidiropoulos, Proceedings of the AAAI Conference on Artificial Intelligence. the AAAI Conference on Artificial IntelligenceK. Huang, X. Fu, and N. Sidiropoulos, "On convergence of Epanechnikov mean shift," in Proceedings of the AAAI Conference on Artificial Intelligence, 2018, pp. 3263-3270. Properties of mean shift. R Yamasaki, T Tanaka, IEEE Transactions on Pattern Analysis and Machine Intelligence. 429R. Yamasaki and T. Tanaka, "Properties of mean shift," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 42, no. 9, pp. 2273-2286, 2019. Ensembles semi-analytiques. S Łojasiewicz, Lectures Notes IHES. S. Łojasiewicz, "Ensembles semi-analytiques," Lectures Notes IHES (Bures-sur-Yvette), 1965. Convergence of the iterates of descent methods for analytic cost functions. P.-A Absil, R Mahony, B Andrews, SIAM Journal on Optimization. 162P.-A. Absil, R. Mahony, and B. Andrews, "Convergence of the iterates of descent methods for analytic cost functions," SIAM Journal on Optimization, vol. 16, no. 2, pp. 531-547, 2005. Optimizing kernel methods: a unifying variational principle. B L Granovsky, H.-G Müller, Revue Internationale de Statistique. 593International Statistical ReviewB. L. Granovsky and H.-G. Müller, "Optimizing kernel meth- ods: a unifying variational principle," International Statistical Re- view/Revue Internationale de Statistique, vol. 59, no. 3, pp. 373-388, 1991. Convergence of descent methods for semi-algebraic and tame problems: Proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods. H Attouch, J Bolte, B F Svaiter, Mathematical Programming. 1371-2H. Attouch, J. Bolte, and B. F. Svaiter, "Convergence of descent methods for semi-algebraic and tame problems: Proximal algo- rithms, forward-backward splitting, and regularized Gauss-Seidel methods," Mathematical Programming, vol. 137, no. 1-2, pp. 91-129, 2013. Splitting methods with variable metric for Kurdyka-Łojasiewicz functions and general convergence rates. P Frankel, G Garrigos, J Peypouquet, Journal of Optimization Theory and Applications. 1653P. Frankel, G. Garrigos, and J. Peypouquet, "Splitting methods with variable metric for Kurdyka-Łojasiewicz functions and gen- eral convergence rates," Journal of Optimization Theory and Applica- tions, vol. 165, no. 3, pp. 874-900, 2015. Mean shift is a bound optimization. M Fashing, C Tomasi, IEEE Transactions on Pattern Analysis and Machine Intelligence. 273M. Fashing and C. Tomasi, "Mean shift is a bound optimiza- tion," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 27, no. 3, pp. 471-474, 2005. . K Lange, MM Optimization Algorithms. SIAM. 147K. Lange, MM Optimization Algorithms. SIAM, 2016, vol. 147. R T Rockafellar, Convex Analysis. Princeton University PressR. T. Rockafellar, Convex Analysis. Princeton University Press, 1997. A note on the convergence of the mean shift. X Li, Z Hu, F Wu, Pattern Recognition. 406X. Li, Z. Hu, and F. Wu, "A note on the convergence of the mean shift," Pattern Recognition, vol. 40, no. 6, pp. 1756-1762, 2007. On the estimation of the gradient lines of a density and the consistency of the mean-shift algorithm. E Arias-Castro, D Mason, B Pelletier, Journal of Machine Learning Research. 1743E. Arias-Castro, D. Mason, and B. Pelletier, "On the estimation of the gradient lines of a density and the consistency of the mean-shift algorithm," Journal of Machine Learning Research, vol. 17, no. 43, pp. 1-28, 2016. On the estimation of the gradient lines of a density and the consistency of the mean-shift algorithm. -- Errata, Journal of Machine Learning Research. 17206--, "Errata: On the estimation of the gradient lines of a den- sity and the consistency of the mean-shift algorithm," Journal of Machine Learning Research, vol. 17, no. 206, pp. 1-4, 2016. Gaussian mean-shift is an EM algorithm. M A Carreira-Perpiñán, IEEE Transactions on Pattern Analysis and Machine Intelligence. 295M. A. Carreira-Perpiñán, "Gaussian mean-shift is an EM algo- rithm," IEEE Transactions on Pattern Analysis and Machine Intelli- gence, vol. 29, no. 5, pp. 767-776, 2007. Maximum likelihood from incomplete data via the EM algorithm. A P Dempster, N M Laird, D B Rubin, Journal of the Royal Statistical Society, Series B (Methodological). 391A. P. Dempster, N. M. Laird, and D. B. Rubin, "Maximum likeli- hood from incomplete data via the EM algorithm," Journal of the Royal Statistical Society, Series B (Methodological), vol. 39, no. 1, pp. 1-38, 1977. On the convergence of the mean shift algorithm in the one-dimensional case. Y , Aliyari Ghassabeh, Pattern Recognition Letters. 3412Y. Aliyari Ghassabeh, "On the convergence of the mean shift algorithm in the one-dimensional case," Pattern Recognition Letters, vol. 34, no. 12, pp. 1423-1427, 2013. On the convergence of the EM algorithm. R A Boyles, Journal of the Royal Statistical Society, Series B (Methodological). 451R. A. Boyles, "On the convergence of the EM algorithm," Journal of the Royal Statistical Society, Series B (Methodological), vol. 45, no. 1, pp. 47-50, 1983. Using kernel density estimates to investigate multimodality. B W Silverman, Journal of the Royal Statistical Society, Series B (Methodological). 431B. W. Silverman, "Using kernel density estimates to investigate multimodality," Journal of the Royal Statistical Society, Series B (Methodological), vol. 43, no. 1, pp. 97-99, 1981. On the number of modes of a Gaussian mixture. M A Carreira-Perpiñán, C K I Williams, Scale Space Methods in Computer Vision. M. A. Carreira-Perpiñán and C. K. I. Williams, "On the number of modes of a Gaussian mixture," in Scale Space Methods in Computer Vision, 2003, pp. 625-640. Maximum number of modes of Gaussian mixtures. C Améndola, A Engström, C Haase, Journal of the IMA. 93Information and Inference: AC. Améndola, A. Engström, and C. Haase, "Maximum number of modes of Gaussian mixtures," Information and Inference: A Journal of the IMA, vol. 9, no. 3, pp. 587-600, 2019. A sufficient condition for the convergence of the mean shift algorithm with Gaussian kernel. Y , Aliyari Ghassabeh, Journal of Multivariate Analysis. 135Y. Aliyari Ghassabeh, "A sufficient condition for the convergence of the mean shift algorithm with Gaussian kernel," Journal of Multivariate Analysis, vol. 135, pp. 1-10, 2015. w -stratification of subanalytic functions and the Lojasiewicz inequality. K Kurdyka, A Parusinski, Académie des sciences. Série 1, Mathématique. 3182Comptes rendus de lK. Kurdyka and A. Parusinski, "w -stratification of subanalytic functions and the Lojasiewicz inequality," Comptes rendus de l'Académie des sciences. Série 1, Mathématique, vol. 318, no. 2, pp. 129-133, 1994. Geometric categories and ominimal structures. L Van Den Dries, C Miller, Duke Mathematical Journal. 842L. van den Dries and C. Miller, "Geometric categories and o- minimal structures," Duke Mathematical Journal, vol. 84, no. 2, pp. 497-540, 1996. The Łojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems. J Bolte, A Daniilidis, A Lewis, SIAM Journal on Optimization. 174J. Bolte, A. Daniilidis, and A. Lewis, "The Łojasiewicz inequality for nonsmooth subanalytic functions with applications to subgra- dient dynamical systems," SIAM Journal on Optimization, vol. 17, no. 4, pp. 1205-1223, 2007. Clarke subgradients of stratifiable functions. J Bolte, A Daniilidis, A Lewis, M Shiota, SIAM Journal on Optimization. 182J. Bolte, A. Daniilidis, A. Lewis, and M. Shiota, "Clarke sub- gradients of stratifiable functions," SIAM Journal on Optimization, vol. 18, no. 2, pp. 556-572, 2007. On the convergence of the proximal algorithm for nonsmooth functions involving analytic features. H Attouch, J Bolte, Mathematical Programming. 1161-2H. Attouch and J. Bolte, "On the convergence of the proximal algorithm for nonsmooth functions involving analytic features," Mathematical Programming, vol. 116, no. 1-2, pp. 5-16, 2009. Convergence of non-smooth descent methods using the Kurdyka-Łojasiewicz inequality. D Noll, Journal of Optimization Theory and Applications. 1602D. Noll, "Convergence of non-smooth descent methods using the Kurdyka-Łojasiewicz inequality," Journal of Optimization Theory and Applications, vol. 160, no. 2, pp. 553-572, 2014. Majorization-minimization procedures and convergence of SQP methods for semi-algebraic and tame programs. J Bolte, E Pauwels, Mathematics of Operations Research. 412J. Bolte and E. Pauwels, "Majorization-minimization procedures and convergence of SQP methods for semi-algebraic and tame programs," Mathematics of Operations Research, vol. 41, no. 2, pp. 442-465, 2016. On subanalytic geometry. G Valette, 2022, version ofG. Valette, "On subanalytic geometry," 2022, version of 'September 12, 2022'. Semianalytic and subanalytic sets. E Bierstone, P D Milman, Publications Mathématiques de l'IHÉS. 67E. Bierstone and P. D. Milman, "Semianalytic and subanalytic sets," Publications Mathématiques de l'IHÉS, vol. 67, pp. 5-42, 1988. Penalty functions in subanalytic optimization. J P Dedieu, Optimization. 261-2J. P. Dedieu, "Penalty functions in subanalytic optimization," Optimization, vol. 26, no. 1-2, pp. 27-32, 1992. Geometric Theory of Dynamical Systems: An Introduction. J PalisJr, W De Melo, SpringerJ. Palis Jr. and W. De Melo, Geometric Theory of Dynamical Systems: An Introduction. Springer, 2012. Non-parametric estimation of a multivariate probability density. V A Epanechnikov, Theory of Probability & Its Applications. 14V. A. Epanechnikov, "Non-parametric estimation of a multivariate probability density," Theory of Probability & Its Applications, vol. 14, no. 1, pp. 153-158, 1969. On the optimality of a class of polynomial kernel functions. B L Granovsky, H.-G Müller, Statistics & Risk Modeling. 74B. L. Granovsky and H.-G. Müller, "On the optimality of a class of polynomial kernel functions," Statistics & Risk Modeling, vol. 7, no. 4, pp. 301-312, 1989. Calculus of the exponent of Kurdyka-Łojasiewicz inequality and its applications to linear convergence of first-order methods. G Li, T K Pong, Foundations of Computational Mathematics. 185G. Li and T. K. Pong, "Calculus of the exponent of Kurdyka- Łojasiewicz inequality and its applications to linear convergence of first-order methods," Foundations of Computational Mathematics, vol. 18, no. 5, pp. 1199-1232, 2018. The Łojasiewicz exponent of an analytic function at an isolated zero. J Gwoździewicz, Commentarii Mathematici Helvetici. 743J. Gwoździewicz, "The Łojasiewicz exponent of an analytic func- tion at an isolated zero," Commentarii Mathematici Helvetici, vol. 74, no. 3, pp. 364-375, 1999. Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials. D , K Kurdyka, Annales Polonici Mathematici. 1D. D'Acunto and K. Kurdyka, "Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials," in Annales Polonici Mathematici, vol. 1, no. 87, 2005, pp. 51-61. Separation of real algebraic sets and the Łojasiewicz exponent. K Kurdyka, S Spodzieja, Proceedings of the American Mathematical Society. 1429K. Kurdyka and S. Spodzieja, "Separation of real algebraic sets and the Łojasiewicz exponent," Proceedings of the American Mathematical Society, vol. 142, no. 9, pp. 3089-3102, 2014. P J Huber, Robust Statistics. WileyP. J. Huber, Robust Statistics. Wiley, 1981. Kernel selection for modal linear regression: Optimal kernel and IRLS algorithm. R Yamasaki, T Tanaka, 2019 18th IEEE International Conference on Machine Learning and Applications. R. Yamasaki and T. Tanaka, "Kernel selection for modal linear regression: Optimal kernel and IRLS algorithm," in 2019 18th IEEE International Conference on Machine Learning and Applications, 2019, pp. 595-601. Estimating and visualizing conditional densities. R J Hyndman, D M Bashtannyk, G K Grunwald, Journal of Computational and Graphical Statistics. 54R. J. Hyndman, D. M. Bashtannyk, and G. K. Grunwald, "Estimat- ing and visualizing conditional densities," Journal of Computational and Graphical Statistics, vol. 5, no. 4, pp. 315-336, 1996. Modelling beyond regression functions: an application of multimodal regression to speed-flow data. J Einbeck, G Tutz, Journal of the Royal Statistical Society, Series C (Applied Statistics). 554J. Einbeck and G. Tutz, "Modelling beyond regression functions: an application of multimodal regression to speed-flow data," Journal of the Royal Statistical Society, Series C (Applied Statistics), vol. 55, no. 4, pp. 461-475, 2006. Nonparametric modal regression. Y.-C Chen, C R Genovese, R J Tibshirani, L Wasserman, The Annals of Statistics. 442Y.-C. Chen, C. R. Genovese, R. J. Tibshirani, and L. Wasserman, "Nonparametric modal regression," The Annals of Statistics, vol. 44, no. 2, pp. 489-514, 2016. Modal regression via direct log-density derivative estimation. H Sasaki, Y Ono, M Sugiyama, International Conference on Neural Information Processing. H. Sasaki, Y. Ono, and M. Sugiyama, "Modal regression via direct log-density derivative estimation," in International Conference on Neural Information Processing, 2016, pp. 108-116. The variable bandwidth mean shift and data-driven scale selection. D Comaniciu, V Ramesh, P Meer, Proceedings of the Eighth IEEE International Conference on Computer Vision. the Eighth IEEE International Conference on Computer Vision1D. Comaniciu, V. Ramesh, and P. Meer, "The variable bandwidth mean shift and data-driven scale selection," in Proceedings of the Eighth IEEE International Conference on Computer Vision, vol. 1, 2001, pp. 438-445. Locally defined principal curves and surfaces. U Ozertem, D Erdogmus, Journal of Machine Learning Research. 1234U. Ozertem and D. Erdogmus, "Locally defined principal curves and surfaces," Journal of Machine Learning Research, vol. 12, no. 34, pp. 1249-1286, 2011. On some convergence properties of the subspace constrained mean shift. Y Ghassabeh, T Linder, G Takahara, Pattern Recognition. 4611Y. Aliyari Ghassabeh, T. Linder, and G. Takahara, "On some convergence properties of the subspace constrained mean shift," Pattern Recognition, vol. 46, no. 11, pp. 3140-3147, 2013. Principal curves. T Hastie, W Stuetzle, Journal of the American Statistical Association. 84406T. Hastie and W. Stuetzle, "Principal curves," Journal of the Ameri- can Statistical Association, vol. 84, no. 406, pp. 502-516, 1989. Estimating density ridges by direct estimation of density-derivative-ratios. H Sasaki, T Kanamori, M Sugiyama, International Conference on Artificial Intelligence and Statistics. 54H. Sasaki, T. Kanamori, and M. Sugiyama, "Estimating density ridges by direct estimation of density-derivative-ratios," in Inter- national Conference on Artificial Intelligence and Statistics, vol. 54, 2017, pp. 204-212.
{'fraction_non_alphanumeric': 0.09333226880788358, 'fraction_numerical': 0.03169624526666363, 'mean_word_length': 3.81980100049475, 'pattern_counts': {'":': 0, '<': 27, '<?xml version=': 0, '>': 61, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 23, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'The mean shift (MS) algorithm seeks a mode of the kernel density estimate (KDE). This study presents a convergence guarantee of the mode estimate sequence generated by the MS algorithm and an evaluation of the convergence rate, under fairly mild conditions, with the help of the argument concerning the Łojasiewicz inequality. Our findings, which extend existing ones covering analytic kernels and the Epanechnikov kernel, are significant in that they cover the biweight kernel that is optimal among non-negative kernels in terms of the asymptotic statistical efficiency for the KDE-based mode estimation.', 'arxivid': '2305.08463', 'author': ['Ryoya Yamasaki ', 'Member, IEEEToshiyuki Tanaka '], 'authoraffiliation': [], 'corpusid': 258685754, 'doi': '10.48550/arxiv.2305.08463', 'github_urls': [], 'n_tokens_mistral': 30409, 'n_tokens_neox': 26336, 'n_words': 15127, 'pdfsha': '1acf3e32fa5c13aecd1804f6af0c148d6840fce7', 'pdfurls': ['https://export.arxiv.org/pdf/2305.08463v2.pdf'], 'title': ['Convergence Analysis of Mean Shift', 'Convergence Analysis of Mean Shift'], 'venue': []}
arxiv
GINA-3D: Learning to Generate Implicit Neural Assets in the Wild Bokui Shen Stanford University Xinchen Yan Waymo LLC 3 Google Charles R Qi Waymo LLC 3 Google Mahyar Najibi Waymo LLC 3 Google Boyang Deng Stanford University Waymo LLC 3 Google Leonidas Guibas Yin Zhou Waymo LLC 3 Google Dragomir Anguelov Waymo LLC 3 Google GINA-3D: Learning to Generate Implicit Neural Assets in the Wild In-the-Wild Driving Data "Same kind…" "Similar size…"GINA-3D Synthesis Composition with Background NeRFGINA-3DReconstruction "Night time…"In-the-wild Driving Data "Random…"Figure 1. Leveraging in-the-wild data for generative assets modeling embodies a scalable approach for simulation. GINA-3D uses real-world driving data to perform various synthesis tasks for realistic 3D implicit neural assets. Left: Multi-sensor observations in the wild. Middle: Asset reconstruction and conditional synthesis. Right: Scene composition with background neural fields [1].AbstractModeling the 3D world from sensor data for simulation is a scalable way of developing testing and validation environments for robotic learning problems such as autonomous driving. However, manually creating or recreating real-world-like environments is difficult, expensive, and not scalable. Recent generative model techniques have shown promising progress to address such challenges by learning 3D assets using only plentiful 2D images -but still suffer limitations as they leverage either human-curated image datasets or renderings from manually-created synthetic 3D environments. In this paper, we introduce GINA-3D, a generative model that uses real-world driving data from camera and LiDAR sensors to create realistic 3D implicit neural assets of diverse vehicles and pedestrians. Compared to the existing image datasets, the real-world driving setting poses new challenges due to occlusions, lighting-variations and long-tail distributions. GINA-3D tackles these challenges by decoupling representation learning and generative modeling into two stages with a learned tri-plane latent structure, inspired by recent advances in generative modeling of images. To evaluate our approach, we construct a large-scale object-centric dataset containing over 520K images of vehicles and pedestrians from the Waymo Open Dataset, and a new set of 80K images of long-tail instances such as construction equipment, garbage trucks, and cable cars. We compare our model with existing approaches and demonstrate that it achieves state-of-the-art performance in quality and diversity for both generated images and geometries. * Work done during an internship at Waymo. † Work done at Waymo. Figure 1 . Leveraging in-the-wild data for generative assets modeling embodies a scalable approach for simulation. GINA-3D uses real-world driving data to perform various synthesis tasks for realistic 3D implicit neural assets. Left: Multi-sensor observations in the wild. Middle: Asset reconstruction and conditional synthesis. Right: Scene composition with background neural fields [1]. Abstract Modeling the 3D world from sensor data for simulation is a scalable way of developing testing and validation environments for robotic learning problems such as autonomous driving. However, manually creating or recreating real-world-like environments is difficult, expensive, and not scalable. Recent generative model techniques have shown promising progress to address such challenges by learning 3D assets using only plentiful 2D images -but still suffer limitations as they leverage either human-curated image datasets or renderings from manually-created synthetic 3D environments. In this paper, we introduce GINA-3D, a generative model that uses real-world driving data from camera and LiDAR sensors to create realistic 3D implicit neural assets of diverse vehicles and pedestrians. Compared to the existing image datasets, the real-world driving setting poses new challenges due to occlusions, lighting-variations and long-tail distributions. GINA-3D tackles these challenges by decoupling representation learning and generative modeling into two stages with a learned tri-plane latent structure, inspired by recent advances in generative modeling of images. To evaluate our approach, we construct a large-scale object-centric dataset containing over 520K images of vehicles and pedestrians from the Waymo Open Dataset, and a new set of 80K images of long-tail instances such as construction equipment, garbage trucks, and cable cars. We compare our model with existing approaches and demonstrate that it achieves state-of-the-art performance in quality and diversity for both generated images and geometries. * Work done during an internship at Waymo. † Work done at Waymo. Introduction Learning to perceive, reason, and interact with the 3D world has been a longstanding challenge in the computer vision and robotics community for decades [2][3][4][5][6][7][8][9]. Modern robotic systems [10][11][12][13][14][15][16] deployed in the wild are often equipped with multiple sensors (e.g. cameras, LiDARs, and Radars) that perceive the 3D environments, followed by an intelligent unit for reasoning and interacting with the complex scene dynamics. End-to-end testing and validating these intelligent agents in the real-world environments are difficult and expensive, especially in safety critical and resource constrained domains like autonomous driving. On the other hand, the use of simulated data has proliferated over the last few years to train and evaluate the intelligent agents under controlled settings [17][18][19][20][21][22][23][24][25][26][27] in a safe, scalable and verifiable manner. Such developments were fueled by rapid advances in computer graphics, including rendering frameworks [28][29][30], physical simulation [31,32] and large-scale open-sourced asset repositories [33][34][35][36][37][38][39]. A key concern is to create realistic virtual worlds that align in asset content, composition, and behavior with real distributions, so as to give the practitioner confidence that using such simulations for development and verification can transfer to performance in the real world [40][41][42][43][44][45][46][47][48]. However, manual asset creation faces two major obstacles. First, manual creation of 3D assets requires dedicated efforts from engineers and artists with 3D domain expertise, which is expensive and difficult to scale [26]. Second, real-world distribution contains diverse examples (including interesting rare cases) and is also constantly evolving [49,50]. Recent developments in the generative 3D modeling offer new perspectives to tackle these aforementioned obstacles, as it allows producing additional realistic but previously unseen examples. A sub-class of these approaches, generative 3D-aware image synthesis [51,52], holds significant promise since it enables 3D modeling from partial observations (e.g. image projections of the 3D object). Moreover, many real-world robotic applications already capture, annotate and update multi-sensor observations at scale. Such data thus offer an accurate, diverse, task-relevant, and upto-date representation of the real-world distribution, which the generative model can potentially capture. However, existing works use either human-curated image datasets with clean observations [53][54][55][56][57][58] or renderings from synthetic 3D environments [33,36]. Scaling generative 3D-aware image synthesis models to the real world faces several challenges, as many factors are entangled in the partial observations. First, bridging the in-the-wild images from a simple prior without 3D structures make the learning difficult. Second, unconstrained occlusions entangle object-of-interest and its surroundings in pixel space, which is hard to disentangle in a purely unsupervised manner. Lastly, the above challenges are compounded by a lack of effort in constructing an asset-centric benchmark for sensor data captured in the wild. In this work, we introduce a 3D-aware generative transformer for implicit neural asset generation, named GINA-3D (Generative Implicit Neural Assets). To tackle the real world challenges, we propose a novel 3D-aware Encoder-Decoder framework with a learned structured prior. Specifically, we embed a tri-plane structure into the latent prior (or tri-plane latents) of our generative model, where each entry is parameterized by a discrete representation from a learned codebook [59,60]. The Encoder-Decoder framework is composed of a transformation encoder and a decoder with neural rendering components. To handle unconstrained occlusions, we explicitly disentangle object pixels from its surrounding with an occlusion-aware composition, using pseudo labels from an off-the-shelf segmenation model [61]. Finally, the learned prior of tri-plane latents from a discrete codebook can be used to train conditional latents sampling models [62]. The same codebook can be readily applied to various conditional synthesis tasks, including object scale, class, semantics, and time-of-day. To evaluate our model, we construct a large-scale objectcentric benchmark from multi-sensor driving data captured in the wild. We first extract over 520K images of diverse variations for vehicles and pedestrians from Waymo Open Dataset [14]. We then augment the benchmark with long-tail instances from real-world driving scenes, including rare objects like construction equipment, cable cars, school buses and garbage trucks. We demonstrate through extensive experiments that GINA-3D outperforms the state-of-the-art 3D-aware generative models, measured by image quality, geometry consistency, and geometry diversity. Moreover, we showcase example applications of various conditional synthesis tasks and shape editing results by leveraging the learned 3D-aware codebook. To support future research along this direction, we are looking to make the benchmark available publicly, such as through waymo.com/open, subject to updates. Related Work We discuss the relevant work on generative 3D-aware image synthesis, 3D shape modeling, and applications in autonomous driving. Generative 3D-aware image synthesis. Learning generative 3D-aware representations from image collections has been increasingly popular for the past decade [63][64][65][66][67][68][69]. Early work explored image synthesis from disentangled factors such as learned pose embedding [64,66,69] or compact scene representations [65,67]. Representing the 3D-structure as a compressed embedding, this line of work approached image synthesis by upsampling from the embedding space with a stack of 2D deconvolutional layers. Driven by the progresses in differentiable rendering, there have been efforts [70][71][72][73] in baking explicit 3D structures into the generative architectures. These efforts, however, are often confined to a coarse 3D discretization due to memory consumption. Moving beyond explicits, more recent work leverages neural radiance fields to learn implicit 3D-aware structures [51,52,[74][75][76][77][78][79][80][81][82] for image synthesis. Schwarz et al. [74] introduced the Generative Radiance Fields (GRAF) that disentangles the 3D shape, appearance and camera pose of a single object without occlusions. Built on top of GRAF, Niemeyer et al. [51] proposed the GIRAFFE model, which handles scene involving multiple objects by using the compositional 3D scene structure. Notably, the query operation in the volumetric rendering becomes computationally heavy at higher resolutions. To tackle this, Chan et al. [52] introduced hybrid explicitimplicit 3D representations with tri-plane features (EG3D), which showcases image synthesis at higher resolutions. Concurrently, [83] and [84] pioneer high-resolution unbounded 3D scene generation on ImageNet using tri-plane representations, where [84] uses a vector-quantized framework and [83] uses a GAN framework. Our work is designed for applications in autonomous driving sensor simulation with an emphasis on object-centric modeling. Generative 3D shape modeling. Generative modeling of complete 3D shapes has also been extensively studied, including efforts on synthesizing 3D voxel grids [85][86][87][88][89][90][91][92][93], point clouds [94][95][96], surface meshes [97][98][99][100][101][102][103], shape primitives [104,105], and implicit functions or hybrid representations [103,[106][107][108][109][110][111][112] using various deep generative models. Shen et al. [111] introduced a differentiable explicit surface extraction method called Deep Marching Tetrahedra (DMTet) that learns to reconstruct 3D surface meshes with arbitrary topology directly. Built on top of the EG3D [52] tri-plane features for image synthesis, Gao et al. [103] proposed an extension that is capable of generating textured surface meshes using DMTet for geometry generation and tri-plane features for texture synthesis. The existing efforts assume access to accurate multi-view silhouettes (often from complete ground-truth 3D shapes) , which does not reflect the real challenges present in data captured in the wild. Assets modeling in driving simulation. Simulated environment modeling has drawn great attention in the autonomous driving domain. In a nutshell, the problem can be decomposed into asset creation (e.g., dynamic objects and background), scene generation, and rendering. Early work leverages artist-created objects and background assets to build virtual driving environments [18,20,113] using classic graphics rendering pipelines. While being able to generate virtual scenes with varying configurations, these methods produce scenes with limited diversity and a significant reality gap. Many recent works explored different aspects of data-driven simulation, including image synthesis [114][115][116][117], assets modeling [47,48,[118][119][120][121], scene generation [49,122,123], and scene rendering [1, [124][125][126]. In particular, Chen et al. [48] and Zakharov et al. [119] performed explicit texture warping or implicit rendering from a single-view observation for each vehicle object. Therefore, their asset reconstruction quality is sensitive to occlusions and bounded by the view angle from a single observation. Building upon these efforts, more recent work including Muller et al. [121] and Kundu et al. [125] approached object completion with global or instance-specific latent codes, representing each object asset under the Normalized Object Coordinate Space (NOCS). In comparison, the latent codes in our proposed model have 3D tri-plane structures which offers several benefits in learning and applications. More importantly, we can generate previously unseen 3D assets, which is essentially different from object reconstruction. Generative Implicit Neural Assets We propose GINA-3D, a scalable framework to acquire 3D assets from in-the-wild data (Sec. 3.1). Core to our framework is a novel 3D-aware Encoder-Decoder model with a learned structure prior (Sec. 3.2). The learned structure prior can facilitate various downstream applications with an iterative latents sampling model (Sec. 3.3) per application. Background. Given a collection of images containing 3D objects captured in the wild X = {x} (x is an image data sample), 3D-aware image synthesis [51,52,[63][64][65][66][67][68][69][70][71][72][73][74][75][76][77][78][79]81] aims to learn a distribution of 3D objects. The core idea is to represent each 3D object as a hidden variable h within a generative model and further leverage a neural rendering module NR to synthesize a sample image at viewpoint v through x = NR(h, v). To model the hidden 3D structure h, the formulation introduces a low-dimensional space where latent variables z (typically a Gaussian) can sample from and connect h and z by a generator h = f θ (z), parameterized by θ. Pr(x, z|v) = Pr(x|z, v) · Pr(z) (1) The probabilistic formulation is shown in Fig. 2-a, and Eq. 1. Here, Pr(x|z, v) is the conditional probability of the image given the latent variables and viewpoint, where Pr(z) and Pr(v) are the prior distributions. As the latent variable z models the 3D objects, one can sample and extract assets for downstream applications. The assets can be either injected into neural representations of scenes [1,125], or transformed into explicit 3D structures such as textured meshes for traditional renders [20] or geometry-aware compositing [48,124]. The challenges in the wild. While human-curated image datasets [53][54][55][56][57][58] or synthetically generated images with clean background [33,36,68,103] fit into the formulation in Eq. 1, real-world distributions have unconstrained occlusions due to complex objectscene entanglement. For example, a moving vehicle can be easily occluded by another object (e.g. traffic cones and cars) in an urban driving environment, which further entangle object and scene in the pixel space. Moreover, environmental lighting and object diversity lead to a more complex underlying distribution. As illustrated Fig. 2-b and Eq. 2, these challenges yield a new probabilistic formulation that the hidden structure h, surrounding scene S and viewpoint v jointly contribute to the occlusion (m) and the visible pixels on the object x through x = NR(h, v) m(S, v, h). Pr(x, z|v, S) = Pr(x|z, v, S) · Pr(z) Prior art such as GIRAFFE [51] tackles the challenges with two assumptions: (1) the scene is composed of a limited number of same-class foreground objects and a background backdrop S; and (2) the real data distribution can be bridged using an one-pass generator f θ (x; z, S, v) (θ is the learned parametrization) conditioned on independently sampled objects z, scene background S and the camera viewpoint v (e.g. Multi-variate Gaussian distributions with diagonal variance) through adversarial training. Unfortunately, the first assumption barely holds for in-the-wild images with unconstrained foreground occlusions. As shown in Niemeyer et al. [51], the second assumption can already introduce artifacts due to disentanglement failures. Our proposal. We focus on interpreting the visible pixels of the object of interest, as synthesizing objects and scene jointly with a generative model is very challenging. We leverage an auxiliary encoder E φ (x) that approximates the posterior Pr(z|x) in training the generative model to reconstruct the input. This way, we bypass the need to model complex scene and occlusions explicitly, since paired input and output are now available for supervising the auto-encoding style training. Specifically, given an image x and the corresponding occlusion mask m, our objective is to reconstruct the visible pixels of the object on the image throughx m where we have the reconstructionx = NR(G θ (z), v) and latent z = E φ (x), respectively. In practice, we use an offthe-shelf model to obtain the pseudo-labeled object mask as the supervision through x m. At the inference time, we can discard the auxiliary encoder E φ as our goal is to generate assets from a learned latent distribution (tri-plane latents in our case). To facilitate this, we leverage the vector-quantized formulation [59,60] to learn a codebook K := {z n } K n=1 of size K and the mapping from a continuous-valued vector to a discrete codebook entry, where each entry follows a K-way categorical distribution. 3D Triplane Latents Learning We explain in details the Encoder-Decoder training framework to learn tri-plane latents z ( Fig. 3-left). The framework consists of a 2D-to-3D encoder E φ , learnable codebook quantization K and a 3D-to-2D decoder G θ . E φ : 2D-to-3D Encoder. We adopt Vision Transformer (ViT) [127] as our image feature extractor that maps 16 × 16 non-overlapping image patches into image tokens of dimension D img . Since the goal is to infer the latent 3D-structure from a 2D image observation, we associate each image token with tokens in the tri-plane latents using cross-attention modules, which have previously shown strong performance in cross-domain and 2D-to-3D information passing [128][129][130][131]. The cross-attention module uses a learnable tri-plane positional encoding as query, and image patch tokens as key and value. The module produces tri-plane embeddings e 3D = E φ (x) ∈ R N Z ×N Z ×3×Dtok , where D tok = 32 and N Z = 16 indicates the dimension of each 3D token and the spatial resolution, respectively. K: Codebook Quantization for tri-plane latents. Given the continuous tri-plane embedding e 3D , we project it to our K-way categorical prior K through vector quantization. We apply quantization q(·) of each spatial code e 3D ijk ∈ R Dtok on the tri-plane embeddings onto its closest entry z n in the codebook, which gives tri-plane latents z = q(e 3D ). z ijh := argmin zn,n∈K e 3D ijk − z n ∈ R Dtok (3) G θ : 3D-to-2D Decoder with neural rendering. Our decoder takes the tri-plane latents z as the input and outputs a high-dimensional feature maps h ∈ R N H ×N H ×3×D H used for rendering, where N H = 256 and D H = 32 indicates spatial resolution of the tri-plane feature maps and the feature dimension, respectively. We adopt a token Transformer followed by a Style-based generator [132] as our 3D decoder. The token transformer first produces high-dimensional intermediate featuresẑ ∈ R N Z ×N Z ×3×D H with an extra CLS token using self-attention modules, which are then feed to the Style-based generator for upsampling. We use 4 blocks of weight-modulated convolutional layers, each guided by a mapping network conditioned on the CLS token. Given the feature maps, we use a shallow MLP that takes a 3D point p and the hidden feature tri-linearly interpolated at the query location h(p) as input, following [52,133,134]. It outputs a density value σ and a view-independent color value c. We perform volume-rendering with the neural radiance field formulation [135]. Training. Our framework builds upon the vectorquantized formulations [59,60,62,[136][137][138][139][140] where we focus on token learning in the first stage. Specifically, we extend the VQ-GAN training losses, where the encoder E φ , decoder G θ and codebook K are trained jointly with an image discriminator D. As illustrated in Eq. 4, we encourage our Encoder-Decoder model to reconstruct the real image x with L 2 reconstruction, LPIPS [141], and adversarial loss. L RGB = (x − x) m 2 + f LPIPS (x m, x m) L GAN = [log D(x) + log(1 − D(x))](4) To regularize the codebook learning, we apply the latent embedding supervision with a commitment term in Eq. 5, where sg[·] denotes the stop-gradient operation. L VQ = sg[e 3D ] − z 2 2 + λ commit sg[z] − e 3D 2 2(5) We additionally regularize the 3D density field in a weakly supervised manner using the rendered aggregated density (alpha value) x α , encouraging object pixels to have alpha value 1. To make the loss occlusion aware, we further require a pixel lies on the non-object region to have zero density, inspired by Müller et al. [121]. This is achieved by restricting the non-object region to cover sky or road class on the pseudo-labeled segmentations (denoted as m sky,road ). L α = (x α − 1) m 2 + x alpha m sky,road 2(6) To summarize, we optimize the total objective L * in Eq. 7. L * = arg min φ,θ,Z max D E x L VQ + L RGB + L α + L GAN (7) Iterative Latents Sampling for Neural Assets Once the first stage training is finished, we can now represent neural assets using the learned tri-plane latents and reconstruct a collection of assets from image inputs. To generate previously unseen assets with various conditions, we further learn to sample the tri-plane latents in the second stage, following the prior works in Generative Transformers [59,60,62,138]. More precisely, we transform the quantized embedding z ∈ R N Z ×N Z ×3×Dtok into a discrete sequence s ∈ {1, ..., K} N Z ×N Z ×3 , where each element corresponds to the index we select from the codebook K through s ijk = n : z ijk = z n . Following the recent work Figure 3. We introduce GINA-3D, a 3D-aware generative transformer for implicit neural asset generation. GINA-3D follows a two-stage pipeline, where we learn discrete 3D triplane latents in stage 1 (Sec. 3.2) and iterative latents sampling in stage 2 (Sec. 3.3). In stage 1, an input image is first encoded into continuous tri-plane latents e 3D using a Transformer-based 2D-to-3D Encoder E φ . Then, a learnable codebook K quantize the latents into discrete latents z. Finally, a 3D-to-2D Decoder G θ maps z back to image, using a sequence of Transformer, Style-based Generator and volume rendering. The rendered image is supervised via an occlusion-aware reconstruction loss. In stage 2, we learn iterative latents sampling using MaskGIT [62]. Optional conditional information can be used to perform conditional synthesis. The sampled latents can then be decoded into neural assets using the decoder G θ learned in stage 1. MaskGIT [62], we use a bidirectional transformer as our latent generator M ψ (z) that we learn to iteratively sample the latent sequence ( Fig. 3-right). During training, we learn to predict randomly masked latents sM by minimizing the negative log-likelihood of the masked ones. L mask = −E s [ ∀ijk:s ijk =[MASK] log Pr(s ijk |sM )] (8) At inference time, we iteratively generate and refine latents. Starting from all latents as [MASK], we iteratively predict all latents simultaneously but only keep the most confident ones in each step. The remaining ones are assigned as [MASK] and the refinement continues. Finally, the sequence s can be readily mapped back to neural assets by indexing the codebook K to generate tri-plane latents z and decoding using G θ . This iterative approach can be applied to asset variations by selectively masking out tokens of a given instance. Expanding Supervision and Conditioning The two-stage training of GINA-3D is flexible in supervision and conditioning. When we have additional information, we can incorporate it in stage 1 as auxiliary supervision for token learning, or in stage 2 for conditional synthesis. Unit box vs. Scaled box. Object scale information can serve as an additional input to the tri-linear interpolation on the tri-plane feature maps by rescaling the feature maps to span object bounding box (instead of a unit box). Semantic feature fields. Various recent works have demonstrated the effectiveness of learning hybrid represen-tations in the neural rendering [142][143][144] and 2D image synthesis [145]. We can naturally incorporate semantic feature fields in our formulation by computing additional channels in our neural rendering MLP. We precompute DINO-ViT features [146] for each image and learn a semantic feature field to build part correspondence among generated instances. LiDAR depth supervision. When LiDAR point cloud is available in the data, it can be used as the additional supervision through a reconstruction term between the rendered depth and LiDAR depth. Conditional synthesis. Last but not the least, additional information support various applications in conditional synthesis. Denoted as C, it can be fed into our latent prior as M ψ (s ijk |sM , C). For example, object scale, object class, time-of-day and object semantic embeddings can also serve as c for control over the generation process. Experiments Object-centric Benchmark We select the Waymo Open Dataset (WOD) [14] as it is one of the largest and most diverse autonomous driving datasets, containing rich geometric and semantic labels such as object bounding boxes and per-pixel instance masks. Specifically, the dataset includes 1,150 driving scenes captured mostly in downtown San Francisco and Phoenix, each consisting of 200 frames of multi-sensor observations. To construct an object-centric benchmark, we propose a coarseto-fine procedure to extract collections of single-view 2D photographs by leveraging 3D object boxes, camera-LiDAR synchronization, and fine-grained 2D panoptic labels. First, we leverage the 3D box annotations to exclude objects beyond certain distances to the surveying vehicle in each data frame (e.g., 40m for pedestrians and 60m for vehicles, respectively). At a given frame, we project 3D point clouds within each 3D bounding box to the most visible camera and extract the centering patch to build our single-view 2D image collections. Furthermore, we train a Panoptic-Deeplab model [61,147] using the 2D panoptic segmentations on the labeled subset and create per-pixel pseudo-labels for each camera image on the entire dataset. This allows us to differentiate pixels belonging to the object of interest, background, and occluder (e.g., standing pole in front of a person). We further exclude certain patches where objects are heavily occluded using the 2D panoptic predictions. Even with the filtering criterion applied, we believe that the resulting benchmark is still very challenging due to occlusions, intra-class variations (e.g., truck and sedan), partial observations (e.g., we do not have full 360 degree observations of a single vehicle), and imperfect segmentation. We use the sensor calibrations to compute ray directions for each 2D pixel, taking into account the camera rolling shutter. We repeat the same process to extract vehicles and pedestrians from WOD, and additional longtail vehicles from our Longtail dataset. The proposed object-centric benchmark is one of the largest datasets for generative modeling to date, including diverse and longtail examples in the wild. Images Implementation Details GINA-3D. Our encoder takes in images at resolution of 256 2 and renders at 128 2 during training. Our tri-plane latents have a resolution of 16 2 with a codebook containing 2048 entries and lookup dimension of 32. We trained our models on 8 Tesla V100 GPUs using Adam optimizer [148], with batch size 32 and 64 in each stage, respectively. We trained stage 1 for 150K steps and stage 2 for 80K steps. Baselines. We compare against two state-of-the-art methods in the domain, GIRAFFE [51] and EG3D [52], which we train on our dataset at the resolution of 128 2 . We noticed that GIRAFFE model trained on full pixels fails to disentangle viewpoints, occlusions and identities. This makes the extraction of the foreground pixels difficult, as the render mask is only defined at the low dimensional resolution 16 2 . We instead report the numbers using a model trained by whitening out non-object regions. For EG3D, we observed that training EG3D with unmasked image leads to training collapse, due to the absence of foreground and background modeling. Thus, we trained EG3D under the same setting. Evaluations on WOD-Vehicle We conduct quantitative evaluations in Table. 2 and visualize qualitative results of different model in Fig. 5. Image Evaluation. For image quality, we calculate Fréchet Inception Distance (FID) [149] between 50K generated images and all available validation images. To better reflect the metric on object completeness, we filter images where its object segmentation mask take up at least 50% of the projected 3D bounding box (Fig.5-right). We additionally measure the completeness of the generated images by Mask Floater-Over-Union (Mask FOU), which is defined as the percentage of unconnected pixels over the rendered object region. To measure the semantic diversity, we compute the Coverage (COV) score and Minimum Matching Distance (MMD) [94] using the CLIP [150] embeddings. COV measures the fraction of CLIP embeddings in the validation set that has matches in the generated set, and MMD measures the distance between each generated embedding to the clos- est one in the validation. Our model demonstrates significant improvements in FID, image completeness and semantic diversity. Without explicit disentanglement, baselines can hardly handle the real distributions, resulting in artifacts of incomplete shapes (Fig. 5). Geometry Evaluation. To measure the underlying volume rendering consistency, we follow Or et al. [79] and compute the alignment errors between the volume-rendered depth from two viewpoints. We extract the mesh using marching cubes [151] with a density threshold of 10 following EG3D [52]. We measure the completeness by Mesh Floater-Over-Union (Mesh FOU), which is defined as the percentage of the surface area on unconnected mesh pieces over the entire mesh. Since we do not have ground-truth meshes in the real world data, we approximate mesh diversity by measuring between generated meshes and aggregated LiDAR point clouds within a bounding box from the validation set. We measure mesh diversity using the aforementioned COV and MMD with a new distance metric. To account for the incompleteness of LiDAR point clouds, we use a one-way Chamfer distance, which is defined as the mean distance between validation point clouds and their nearest neighbor from a given generated mesh. Our model demonstrates significant improvements in volume rendering consistency, shape completeness and shape diversity. Augmentation and Ablation. GINA-3D can naturally incorporate additional supervisions when available. We present variations of GINA-3D trained with object scale, LiDAR and DINO [146] supervision. With object scale information available, we normalize tri-plane feature maps with the scale on each dimension. The model trained with rescaled tri-plane resolution yields significant performance boost in both quality and diversity over unit bounding cube, as latents are better utilized. Moreover, we observe that by adding auxiliary L 2 depth supervision from LiDAR, most metrics are improved except Mask and Mesh FOU. While Li-DAR provides strong signal to underlying geometry, it also introduces inconsistency on transparent surfaces. We hypothesize that such challenge leads to slightly more floaters, which we leave as future directions to explore. Alternatively, we can learn additional neural semantic fields through 2D-to-3D feature lifting [142]. By only changing the final layer of the NR MLP, we can learn an additional view-consistent and instance-invariant semantic feature field (Fig. 6-b), which can enable future applications of language-conditioned and part-based editing [8] Finally, we perform ablation studies on the key design of tri-plane latents. If we remove the triplane structure and use a MLP-only NR, the model fails to capture the diversity of real-world data and results in mode collapse, which generates always a mean car shape. Applications Generating long-tail instance. Our data-driven framework is scalable to new data. We provide results on GINA-3D trained on Longtail-Vehicle and WOD-Ped dataset in Fig. 6c,d respectively. Without finetuning the architecture on the newly collected data, GINA-3D can readily learn to generate long-tail objects from noisy segmentation masks. As shown in Fig. 6-c, generation results range from trams, truck to construction equipment of various shapes. GINA-3D can also be applied to other categories (e.g. pedestrian, Fig.6-d). Results show moderate shape and texture diversity. Conditional synthesis. As described in Sec. 3.4, the flexibility of the two-stage approach makes it a promising candidate for conditional asset synthesis. Specifically, we freeze the stage 1 model, and train variations of MaskGIT by passing in different conditions. We provide results for three kinds of conditional synthesis tasks in Fig. 7, namely discrete embeddings (object class, time-of-day), continuous embeddings, and image-conditioned generation. For image-conditioned asset reconstruction and variations, we first infer the latents using the encoder model and then sample asset variations by controlling masking ratio of the reconstructed tri-plane latents. The more tokens are masked, the wider the variation range becomes. We provide more details for conditional synthesis in the supplementary material. Limitations Misaligned 3D bounding boxes. As in our WOD-Ped results, misaligned boxes lead to mismatch in pixel space, resulting in blurrier results. Latest methods in ray-based [130] or patch-based [81] learning are promising directions. Few-shot and transfer learning. Though our data-driven approach achieves reasonable performance by training on Longtail-Vehicle alone, the comparative scarcity of data leads to lower diversity. How to enable few-shot learning or transfer learning remains an open question. Transcient effects. Direction-dependent effect can be incorporated in our pipeline. We believe modeling material [152] together with LiDAR is an interesting direction. Conclusion In this work, we presented GINA-3D, a scalable learning framework to synthesize 3D assets from robotic sensors deployed in the wild. Core to our framework is a deep encoder-decoder backbone that learns discrete tri-plane latent variables from partially-observed 2D input pixels. Our backbone is composed of an encoder with cross-attentions, a decoder with tri-plane feature maps, and a neural volumetric rendering module. We further introduce a latent transformer to generate tri-plane latents with various conditions including bounding box size, time of the day, and semantic features. To evaluate our framework, we have established a large-scale object-centric benchmark containing diverse vehicles and pedestrians. Experimental results have demonstrated strong performance on image quality, geometry consistency and geometry diversity over existing methods. To faciliate future research on generative neural assets from in-the-wild data, we are looking to make the benchmark available publicly, such as through waymo.com/open, subject to updates. containing rich geometric and semantic labels such as 3D bounding boxes and per-pixel instance masks. Specifically, the dataset includes 1,150 driving scenes captured mostly in downtown San Francisco and Phoenix, each consisting of 200 frames of multi-sensor data. Each data frame includes 3D point clouds from LiDAR sensors and high-resolution images from five cameras (positioned at Front, Front-Left, Front-Right, Side-Left, and Side-Right). The objects were captured in the wild and their images exhibit large variations due to object interactions (e.g., heavy occlusion and distance to the robotic platform), sensor artifacts (e.g., motion blur and rolling shutter) and environmental factors (e.g., lighting and weather conditions). References To construct a benchmark for object-centric modeling, we propose a coarse-to-fine procedure to extract collections of single-view 2D photographs by leveraging 3D object boxes, camera-LiDAR synchronization, and fine-grained 2D panoptic labels. First, we leverage the 3D box annotations to exclude objects beyond certain distances to the surveying vehicle in each data frame (e.g., 40m for pedestrians and 60m for vehicles, respectively). At a given frame, we project 3D point clouds within each 3D bounding box to the most visible camera and extract the centering patch to build our single-view 2D image collections. Furthermore, we train a Panoptic-Deeplab model [61] using the 2D panoptic segmentations on the labeled subset [147] and create per-pixel pseudo-labels for each camera image on the entire WOD. This allows us to differentiate pixels belonging to the object of interest, background, and occluder (e.g., standing pole in front of a person). We further exclude certain patches where objects are heavily occluded using the 2D panoptic predictions. Even with the filtering criterion applied, we believe that the resulting benchmark is still very challenging due to occlusions, intra-class variations (e.g., truck and sedan), partial observations (e.g., we do not have full 360 degree observations of a single vehicle), and imperfect segmentation. Finally, we use the sensor calibration information to compute ray directions for each 2D pixel, taking into account the camera rolling shutter. Our Longtail dataset contains LiDAR point clouds and camera images, along with 3D bounding box annotations. We obtain the pseudo-labeled segmentations using the same 2D panoptic model pretrained on WOD. We apply the same coarse-to-fine procedure to obtain the Longtail-Vehicle benchmark. Decoder G θ -Volume Rendering. Our volume renderer is implemented as 2 fully-connected layers, similar to Chan et al. [52]. The decoder takes as input the 32-dimensional aggregated feature vector from the style-based generator. For each pixel, we query 40 points, with 24 uniformly sampled and 16 importance-sampled. We use MipNeRF [154] as our volume rendering module. Volume rendering is performed at a resolution of 128 × 128. Discriminator. We use a StyleGAN2 [132] discriminator with hidden dimensions 16,32,64,128,256. We use R1 regularization with γ = 1. Stage-2 Modeling M ψ . We follow a shallower verions of the network architecture and training set up introduced in [62]. We use 12 layers, 8 attention heads, 768 embedding dimensions and 3072 hidden dimensions. The model uses learnable positional embedding, Layer Normalization, and truncated normal initialization (stddev= 0.02). We use the following training hyperparameters: label smoothing=0.1, dropout rate=0.1, Adam optimizer [148] with β 1 = 0.9 and β 2 = 0.96. We use a cosine masking schedule. During inference, token synthesis are performed in 10 steps. B.2. Aligning Tri-plane to Object Scale Figure 9. Illustration of using uniform tri-plane versus using scale-aligned triplane. Since vehicles can have drastically different scales in its x, y, z directions, using a naive uniform scale tri-plane to cover the object leaves a lot of computation capacity under-utilized. As illustrated in the top row of Fig. 9, if we cover a normal sedan using uniform size tri-plane, most of the entries in the tri-plane features correspond to empty space. The problem becomes more severe for longer-tail instances of truck, bus etc., where the scale ratio among x, y, z become even more extreme. To encourage a more efficient tri-plane features usage, we make tri-plane latents aligned to object scales during the coordinate feature orthographic projection step. As illustrated in the bottom row of Fig. 9, when querying feature of coordinate p ∈ [0, 1] 3 ⊂ R 3 , if we have object scale s x , s y , s z , we simply scale p asp := p [sx,sy,sz] ∈ [0, 1 sx ] × [0, 1 sy ] × [0, 1 sz ], and query tri-plane features usingp. The orthographic projection follows the same tri-plane grid-sampling and aggregation as in prior works [52,133,134]. In the basic GINA-3D pipeline without using scaled tri-plane features, the model learns to handle object scale implicitly. In our scaled box model variations, the model leverages the object scale only in tri-plane feature orthographic projection step. The model implicitly learns to produce feature maps that align with object scale. As illustrated in the main paper, such design greatly improve model performance. We leave feeding object scale information explicit to the model as a future direction to explore. B.3. Evaluation Metrics We discuss in details the metrics we have used for quantitative evaluations. Image Quality. To evaluate the image quality, we employ two metrics Fréchet Inception Distance (FID) [149] and Mask Floater-Over Union (Mask FOU) over 50K generated images. Fréchet Inception Distance (FID) [149] is commonly used to evaluate the quality of 2D images. The generated images are encoded using a pretrained Inception v3 [155] model, and the last pooling layer's output was stored as the final encoding. The FID metric is computed as: FID(I g , I v ) = ||µ g − µ v || 2 2 + Tr[Σ g + Σ v − 2 Σ g · Σ v ](9) where Tr denotes the trace operation, µ g , Σ g are the mean and covariance matrix of the generated images encodings, and µ v , Σ v are the mean and covariance matrix of the validation images encodings. We additionally measure if the generated texture forms a single full object, which is implemented by checking if the generated pixels span a connected region. We measure this by calculating percentage of pixels that are not connected. Since all images from baselines and GINA-3D are generated using a white background, we measure pixels connected components using the findContours function from OpenCV [156] to find connected components, and use contourArea to find the largest connected component, which we denote C l . We then use the aggregated density (alpha) value to find the entire shape's projection on the image, which we denote S. Mask FOU is simply calculated mean over entire generated image set (as percentage): Mask FOU(I g ) = 1 |I g | i∈Ig (1 − Area(C l,i ) Area(S i ) )(10) Image Diversity. We want to evaluate the semantic diversity of the generated image, which we measure with Coverage (COV) score and Minimum Matching Distance (MMD) [94] using pretrained CLIP [150] embeddings. Specifically, Coverage (COV) score measures the fraction of images in the validation set that are matched to at least one of the images in the generated set. Formally, it's defined as: COV(I g , I v ) = |{argmin i∈Iv ||CLIP(i) − CLIP(j)|| 2 2 |j ∈ I g }| |I v |(11) Intuitively, COV uses CLIP embedding distance to perform nearest-neighbor matching for each generate image towards validation set. It measures diversity by checking what percentage of validation set is being matched as a nearest neighbor. However, COV is only one side of the story. A set of generated image can have a high COV score by having purely random generated images that are randomly matched to validation set. This issue is alleviated by the incorporation of Minimum Matching Distance (MMD), which measures if the nearest-neighbor matching yields high-quality matching pairs: MMD(I g , I v ) = 1 |I v | i∈Iv min j∈Ig ||CLIP(i) − CLIP(j)|| 2 2(12) Intuitively, MMD measures the average closest distance between images in the validation set and their corresponding nearest neighbor in the training set. MMD correlates well with how faithful (with respect to the validation set) elements of generated set are [94]. Geometry Quality. Due to a lack of 3D geometry ground-truth for in-the-wild data, we measure geometry quality using an existing metric Consistency score from Or-El et al. [79], and a Mesh Floater-Over Union (Mesh FOU) which measures if the geometry forms a single connected object. Consistency score measures if the implicit fields are evaluated at consistent 3D locations, which is an important characteristic for view-consistent renderings [79]. In practice, it measures depth map consistency across viewpoints by back-projecting depth map to the 3D space. For each model, we normalize the object longest edge to length of 10 for numeric clarity, and compare two depth maps at an angle difference of 45 degrees along the z-axis (yaw). We calculate consistency across depth maps for all images in the generated set, denote as D g : Consistency(D g ) = 1 |D g | i∈Dg CD(i, i rot )(13) where i rot represents the depth map after rotating the view point by 45 degree along z-axis. We additionally measure if each generated shape forms a single full object, which is measured by checking if the generated mesh forms a single mesh. We measure this by calculating percentage of mesh surface area that is not connected. We use surface area over volume because we observe that volume calculation is unstable with non-watertight meshes. For each generated mesh S, we use split function from Trimesh [157] to find the largest connected component, which we denote C l . Mesh FOU is simply calculated mean over entire generated mesh set M g (as percentages): Geometry Diversity. We use Coverage (COV) and Minimum Matching Distance (MMD) again for measuring diversity. However, due to the lack of ground truth full 3D shape from in-the-wild data, our metric needs to be more carefully designed. A source for accurate but partial geometry that we can obtain is by aggregating LiDAR point-cloud scans for a given instance from different observations. We then uniformly subsample 2048 points from the aggregated point cloud. Mesh FOU(M g ) = 1 |M g | i∈Mg (1 − Area(C l,i ) Area(S i ) )(14) We show examples of aggregated point clouds in Fig. 10. As shown in the figure, the aggregated point clouds are indicative of the underlying shapes, but are incomplete. Chamfer distance, a common metric for shape similarity, calculates bi-directional nearest neighbors. However, due to incompleteness, finding the nearest neighbors of the generated points in the partial point will only result in noisy matches. Therefore, we do not measure the two-sided Chamfer distance, but measure only the distance of nearest neighbors of validation point clouds in the generated mesh. Formally, we have: COV(M g , P v ) = |{argmin i∈Pv D(i, j)|j ∈ M g }| |P v |(15)MMD(M g , P v ) = 1 |P v | i∈Pv min j∈Mg D(i, j)(16)D(i, j|i ∈ P v , j ∈ M g ) = 1 |i| x∈i min y∈j ||x − y|| 2 2(17) B.4. Conditional Synthesis We showcased in the main paper various conditional synthesis tasks, for which we provide more details here. Discrete Conditions. We feed discrete conditions (object class, time-of-day) as additional tokens to MaskGIT. Specifically, we increase the vocabulary size by the number of classes in the discrete conditions. Object class contains 4 options: cars, truck, bus and others. Time-of-day is a binary variable of day versus night. The vocabulary thus becomes 2048 + 4 for object class, and 2048 + 2 for time-of-day. We feed the conditional input as an additional token to the 768 tri-plane latents by concatenating the two, resulting in an input of sequence length 769. The sequence is then fed into MaskGIT for masked token prediction as in unconditional case. Continuous Conditions. Alternatively, we feed continuous conditions to MaskGIT by concatenating conditional input with MaskGIT intermediate layer's output. Specifically, MaskGIT first generates word embedding for each token in the sequence. We pass the continuous condition through a fully-connected layer and concatenate the output with each token's word embedding. The concatenated embedding is then passed through the rest of the network. To synthesize samples conditioned on object semantics, we feed semantic embedding from a pre-trained DINO model [146]. To condition on object scale, we pass in positional embedding of object scale. We use standard cosine and sine positional embedding of degree 6. Image-conditioned Assets Variations. Given our mask-based iterative sampling stage, we can generate image-conditioned asset with variations. We first use stage-1 model to perform reconstruction, retrieving a full-set of predicted tri-plane latents. We then generate variations of the reconstructed instance by randomly masking out tri-plane latents. The degree of variations can be controlled by masking out different number of tokens. By masking 90% of tokens, we observe the variations are mostly reflected in generated assets under different textures. By masking out 99% of tokens, we see changes in object shapes more 1st, 2nd, 3rd L GAN LPIPS L α L VQ No VQ |K| = 2 10 |K| = 2 12 Full Generative Metric (FID) 65 Table 3. We perform various ablation studies on 1) removing each term in out overall loss function; 2) Removing vector quantization entirely; 3) Different codebook sizes K. We further report stage 1 model's reconstruction quality using 2 losses for input views as well as novel views. significantly, while the general object class remain the same. We believe how to better control the variation process is an interesting direction to explore in the future. (c) Generated image samples from jointly generating object RGB and occlusion masks (random views) Figure 12. Additional results on generation results trained on full images. a) We trained GIRAFFE on full images; b) We trained EG3D models on full images; c) We augmented the EG3D model by jointly generating object RGB, background RGB and occlusion masks. We visualizes object RGB and its corresponding occlusion mask in alternate columns. Results suggest that it's difficult for the model to disentangle object shape and occlusion. identity latent variable fixed. It turns out that the generations are not easily controllable by the viewpoint variables, while the vehicle identities often change across views. The entangled representation makes the extracted meshes not very meaningful for the GIRAFFE baseline on our benchmark. Additionally, the geometry extraction becomes even harder as the rendering mask is defined at a low dimensional resolution 16 2 . Figure 2 . 2Probabilistic Views. Figure 4 . 4Image samples from our object-centric benchmark. Figure 5 .Figure 6 . 56Qualitative comparison between GIRAFFE, EG3D and ours with images rendered from a horizontal 30 • viewpoint. Both baselines fail to disentangle real-world sensor data. GIRAFFE fails to disentangle rotation in object representation, while both baselines fail to disentangle occlusion and produce incomplete shape. We show samples from occlusion-filtered WOD-Vehicle validation set on the right. Generation from GINA-3D variants.(a) GINA-3D trained on WOD-Vehicle. (b) GINA-3D with additional DINO feature field generation. (c) GINA-3D trained on Longtail-Vehicle. (d) GINA-3D trained on WOD-Pedestrain. Figure 7 . 7GINA-3D unifies a wide range of asset synthesis tasks, all obtained with the same stage 1 decoder and variations of stage 2 training. Top row: Conditional synthesis using discrete conditions (object classes and time-of-day). 2nd row: Conditional synthesis using continuous conditions (semantic token and object scale). 3rd row: Image-conditioned assets variations by randomizing tri-plane latents. Figure 10 . 10Illustrations of aggregated point clouds. Figure 11 . 11Additional qualitative results of GINA-3D. ( a ) aGenerated image samples from GIRAFFE trained on full images (view, view+45°) (b) Generated image samples from EG3D trained on full images (random views) Figure 13 . 13image samples from GIRAFFE trained on masked images (fixed views) (b) Generated image samples from EG3D trained on masked images (fixed views) Additional results on generation results trained on masked images. a) Additional visualizations of GIRAFFE baseline reported in the main paper; b) Additional visualizations of EG3D baseline reported in the main paper. Results suggest that it's difficult for GIRAFFE to disentangle rotation. Both baselines show significant occlusion artifacts. (a) A random batch of 16 EG3D extracted meshes. (b) A random batch of 16 GINA-3D extracted meshes. Figure 14 . 14Example mesh extractions from EG3D and GINA-3D. Stage 1: 3D Triplane Latents Learning: 2D-to-3D EncoderTri-plane Latents (Discrete) Vision Transformer 2D Patch Tokens 2D-to-3D Cross Attention Tri-plane Positional Encoding Tri-plane Latents (Continuous) : 3D-to-2D Decoder Tri-plane Feature Maps Volume Rendering Rendered Mask Rendered Color Predicted Mask Occlusion-aware Composition 0 1 2 N … : Codebook q(•) Quantization Token Transformer w CLS Style-based Generator Mapping Net Stage 2: Iterative Latents Sampling Conditions: Scale, Class Semantics Time-of-day MaskGIT MaskGIT Input Acknowledgements: We based our MaskGIT implementation on Chang et al.[62]. We thank Huiwen Chang for helpful MaskGIT pointers. We acknowledge the helpful discussions and support from Qichi Yang and James Guo. We thank Mathilde Caron for her DINO implementation and helpful pointers. We based our GIRAFFE baseline on the reimplementation by Kyle Sargent. We thank Golnaz Ghiasi for helpful pointers on segmentation models.AppendixIn this supplementary document, we first describe in details our proposed dataset and the processing behind it in Sec. A. Then, we discuss various implementation details including network architectures, evaluation metrics and conditional synthesis details in Sec. B. Next, we examine ablation of different loss terms, and evaluate stage 1 model's performance. Finally, we discuss baselines in more details in Sec. E and showcase mesh extraction visualization results in Sec. F.Appendix A. DatasetWe build the object-centric benchmark on top of the Waymo Open Dataset (WOD)[14]and our Longtail dataset. The Waymo Open Dataset (WOD) is one of the largest and most diverse autonomous driving datasets among others [11-13], Appendix B. More Implementation Details B.1. Network Architecture All models use exponential moving average of weights.Encoder E φ . Our encoder contains three vision transformer blocks and three cross-attention blocks. The vision transformer takes input images of resolution of 256 2 , and first map each patch into a 512 dimensional token. A CLS token is appended to the list of image patch tokens. Then, the transformer blocks are used to process the image patch tokens. Each transformer block has 8 heads, an embedding dimension of 512 and a hidden dimension of 2048. For cross-attention blocks, we first initialize tri-plane positional embedding of shape 16 × 16, each embedding is of 512 dimension. The tri-plane positional embedding is passed through a fully-connected layer of 512 dimension. The processed tri-plane positional embedding is then used a query input to the cross-attention transformer blocks, while the image patch tokens serve as key and value. Each cross-attention transformer block has 8 heads, an embedding dimension of 512 and a hidden dimension of 2048. Finally, the output of the cross-attention transformer blocks are passed through a fully-connected layer with Layer Normalization[153]and tanh activation into 16 × 16 tokens of 32 dimension, which is the dimension of each entry in the codebook K.Codebook K. Our discrete codebook contains 2048 entries with lookup dimension of 32, which means each entry is of 32-dimensional. Codebook are initialized using fan-in variance scaling, scale equals 1 and uniform distribution. Similar to Yu et al.[138], we use l 2 -normalized codes, which means applying l 2 normalization on the encoded tri-plane latents e 3D and codebook entries in K.Decoder G θ -Token Transformer. The token transformer contains 3 self-attention transformers blocks. A CLS token is appended to the tri-plane latents. Positional encoding is used to represent 3D spatial locations. Each transformer block has 8 heads, an embedding dimension of 512 and a hidden dimension of 2048. Finally, the output of the transformer blocks are passed through a fully-connected layer with Layer Normalization[153]and tanh activation into 16 × 16 tokens of 256 dimension (and an additional CLS token).Decoder G θ -Style-based Generator. We first use a mapping network[132]to map the aforementioned CLS token into intermediate latent space W. The mapping network contains 8 fully-connected layers of hidden dimension 512. The mapping network outputs a vector w of 512 dimensional. Following Karras et al.[132], we use w for a style-based generator. For each plane in our tri-plane representation (xy, xz, yz planes), we use a generator contains three up-sampling blocks with hidden dimensions of 512, 256 and 128 respectively. Finally, the style-based generators output tri-plane feature maps with 32 feature channels.Appendix C. Additional GINA VisualizationsWe present additional visualizations of GINA-3D model inFig. 11.Appendix D. Ablation on Loss Terms and Stage 1 EvaluationAblation study. In this experiment, we use our scaled box model, trained with LiDAR supervision as our base model. We conduct ablation studies by removing each loss, removing quantization entirely and training with different codebook sizes. As shown inTable 3, the ablaion results justify each loss term we introduced in the paper, as removing each one of them leads to higher FID compared to the full model. This finding is consistent with Esser et al.[60], which suggests LPIPS is important for visual fidelity. In addition, larger codebook K (2 12 ) has marginal impact in our setting.Evaluating stage 1 model. We report 2 reconstruction loss (in 10 −2 ) on the input and novel views of unseen instances. The model is able to obtain better reconstruction performance by removing quantization entirely (No VQ), but it deprives the discrete codebook for stage 2 generative training. While generation and reconstruction correlates to some extent, performance rankings (color-coded) differ between them.Appendix E. Discussions on BaselinesGenerating the full images. Directly modeling full images of data-in-the-wild yields significant challenges. In the early stage of the project, we experimented with directly using GAN-based approaches on full images. As illustrated inFig. 12-a,b, feeding full images without explicit modeling of occlusion makes learning challenging on our benchmark. For EG3D, we observed that training EG3D with unmasked image leads to training collapse, due to the absence of foreground and background modeling. For example, the generated image samples inFig. 13-b lack diversity in shape and appearance (e.g. color).We clarify a key difference to pure GAN-based approaches (GIRAFFE and EG3D) is that our approach has two training stages and the masked loss is only applied in the first stage to reconstruct the input. In other words, masked loss cannot be directly applied to existing GAN-based approaches as the corresponding object mask for each generated RGB is not observed in the adversarial (encoder-free) training. Alternatively, one can still apply the masked loss by factorizing RGB, object silhouette and occlusion. We have tried many variations of this idea in the early stage without avail, as learning disentangled factors was challenging for adversarial training. We provide such examples inFig. 12-c. In this experiment, we tried to extend EG3D by generating occlusion masks with a separate branch. However, the training became very unstable and we were not able to produce improved results beyond the original EG3D on our benchmark. As we can see, the model fails to disentangle object silhouette and occlusion. It still generates partial shape, while generating some plausible foreground occlusion. In fact, occlusion is even more challenging to generate explicitly on our data where object silhouettes and occlusion masks are entangled, as the outcome depends on the view and layout. Generating the object images. Whitening out non-object regions has been used by EG3D (see ShapeNet-Cars in its supp.) and GET3D. It combines white color to pixels with α < 1 during neural rendering, which implicitly supervise α. Such set up separates object pixels from the surroundings, and makes generation focused on object modeling. We follow this design and have found in our experiments that baselines fail to generate separated target object without whitening-out.We provide additional details about the baseline methods GIRAFFE and EG3D inFig. 13. We noticed that the learned GIRAFFE models are capable of generating vehicle-like patches but with viewpoints, occlusions and identities entangled in the latent space. For example, we generate a pair of images (inFig. 13(a)) by varying the viewpoint variable while keeping theAppendix F. Extracted MeshesAs mentioned in the main text, we use marching cubes[151]with density threshold of 10 to extract meshes for geometry evaluation. We showcase here random samples of extracted meshes from EG3D and GINA-3D. We show 16 examples each inFig. 14a-14b. As we see, EG3D meshes can contain artifacts like missing parts of shape (row 3 right two). Furthermore, it shows relatively little diversity. GINA-3D not only preserves complete shapes, but also demonstrate a greater diversity, including more shape variation and semantic variation (mini-van row 2 column 4; bus row 4 column 4). Such observation is consistent with our quantitative evaluations.However, we do observe that GINA-3D meshes can be non-watertight and contain holes. We hope to address such problems in future works. We believe that by incorporating other representations like Signed Distance Fields (SDF), the mesh quality can be further improved. Argoverse: 3d tracking and forecasting with rich maps. Ming-Fang Chang, John Lambert, Patsorn Sangkloy, Jagjeet Singh, Slawomir Bak, Andrew Hartnett, De Wang, Peter Carr, Simon Lucey, Deva Ramanan, CVPR. 115Ming-Fang Chang, John Lambert, Patsorn Sangkloy, Jagjeet Singh, Slawomir Bak, Andrew Hartnett, De Wang, Peter Carr, Simon Lucey, Deva Ramanan, et al. Argoverse: 3d tracking and forecasting with rich maps. In CVPR, 2019. 1, 15 nuscenes: A multimodal dataset for autonomous driving. Holger Caesar, Varun Bankiti, Alex H Lang, Sourabh Vora, Venice Erin Liong, Qiang Xu, Anush Krishnan, Yu Pan, Giancarlo Baldan, Oscar Beijbom, CVPR. 115Holger Caesar, Varun Bankiti, Alex H Lang, Sourabh Vora, Venice Erin Liong, Qiang Xu, Anush Krishnan, Yu Pan, Giancarlo Baldan, and Oscar Beijbom. nuscenes: A multi- modal dataset for autonomous driving. In CVPR, 2020. 1, 15 Scalability in perception for autonomous driving: Waymo open dataset. Pei Sun, Henrik Kretzschmar, Xerxes Dotiwalla, Aurelien Chouard, Vijaysai Patnaik, Paul Tsui, James Guo, Yin Zhou, Yuning Chai, Benjamin Caine, CVPR. 15Pei Sun, Henrik Kretzschmar, Xerxes Dotiwalla, Aurelien Chouard, Vijaysai Patnaik, Paul Tsui, James Guo, Yin Zhou, Yuning Chai, Benjamin Caine, et al. Scalability in perception for autonomous driving: Waymo open dataset. In CVPR, 2020. 1, 2, 5, 15 Robonet: Large-scale multi-robot learning. Sudeep Dasari, Frederik Ebert, Stephen Tian, Suraj Nair, Bernadette Bucher, Karl Schmeckpeper, Siddharth Singh, Sergey Levine, Chelsea Finn, arXiv:1910.11215arXiv preprintSudeep Dasari, Frederik Ebert, Stephen Tian, Suraj Nair, Bernadette Bucher, Karl Schmeckpeper, Siddharth Singh, Sergey Levine, and Chelsea Finn. Robonet: Large-scale multi-robot learning. arXiv preprint arXiv:1910.11215, 2019. 1 Do as i can, not as i say: Grounding language in robotic affordances. Michael Ahn, Anthony Brohan, Noah Brown, Yevgen Chebotar, Omar Cortes, Byron David, Chelsea Finn, Keerthana Gopalakrishnan, Karol Hausman, Alex Herzog, arXiv:2204.01691arXiv preprintMichael Ahn, Anthony Brohan, Noah Brown, Yevgen Cheb- otar, Omar Cortes, Byron David, Chelsea Finn, Keerthana Gopalakrishnan, Karol Hausman, Alex Herzog, et al. Do as i can, not as i say: Grounding language in robotic affordances. arXiv preprint arXiv:2204.01691, 2022. 1 The synthia dataset: A large collection of synthetic images for semantic segmentation of urban scenes. German Ros, Laura Sellart, Joanna Materzynska, David Vazquez, Antonio M Lopez, CVPR. German Ros, Laura Sellart, Joanna Materzynska, David Vazquez, and Antonio M Lopez. The synthia dataset: A large collection of synthetic images for semantic segmentation of urban scenes. In CVPR, 2016. 1 Yohann Cabon, and Eleonora Vig. Virtual worlds as proxy for multi-object tracking analysis. Adrien Gaidon, Qiao Wang, CVPR. 13Adrien Gaidon, Qiao Wang, Yohann Cabon, and Eleonora Vig. Virtual worlds as proxy for multi-object tracking analy- sis. In CVPR, 2016. 1, 3 Clevr: A diagnostic dataset for compositional language and elementary visual reasoning. Justin Johnson, Bharath Hariharan, Laurens Van Der Maaten, Li Fei-Fei, Lawrence Zitnick, Ross Girshick, CVPR. Justin Johnson, Bharath Hariharan, Laurens Van Der Maaten, Li Fei-Fei, C Lawrence Zitnick, and Ross Girshick. Clevr: A diagnostic dataset for compositional language and elemen- tary visual reasoning. In CVPR, 2017. 1 Carla: An open urban driving simulator. Alexey Dosovitskiy, German Ros, Felipe Codevilla, Antonio Lopez, Vladlen Koltun, PMLRConference on robot learning. 13Alexey Dosovitskiy, German Ros, Felipe Codevilla, Antonio Lopez, and Vladlen Koltun. Carla: An open urban driving simulator. In Conference on robot learning, pages 1-16. PMLR, 2017. 1, 3 Ai2-thor: An interactive 3d environment for visual ai. Eric Kolve, Roozbeh Mottaghi, Winson Han, Eli Vanderbilt, Luca Weihs, Alvaro Herrasti, Daniel Gordon, Yuke Zhu, Abhinav Gupta, Ali Farhadi, arXiv:1712.05474arXiv preprintEric Kolve, Roozbeh Mottaghi, Winson Han, Eli VanderBilt, Luca Weihs, Alvaro Herrasti, Daniel Gordon, Yuke Zhu, Abhinav Gupta, and Ali Farhadi. Ai2-thor: An interactive 3d environment for visual ai. arXiv preprint arXiv:1712.05474, 2017. 1 Airsim: High-fidelity visual and physical simulation for autonomous vehicles. Shital Shah, Debadeepta Dey, Chris Lovett, Ashish Kapoor, Field and service robotics. SpringerShital Shah, Debadeepta Dey, Chris Lovett, and Ashish Kapoor. Airsim: High-fidelity visual and physical simulation for autonomous vehicles. In Field and service robotics, pages 621-635. Springer, 2018. 1 Taskonomy: Disentangling task transfer learning. Alexander Amir R Zamir, William Sax, Leonidas J Shen, Jitendra Guibas, Silvio Malik, Savarese, CVPR. Amir R Zamir, Alexander Sax, William Shen, Leonidas J Guibas, Jitendra Malik, and Silvio Savarese. Taskonomy: Disentangling task transfer learning. In CVPR, 2018. 1 Augmented reality meets computer vision: Efficient data generation for urban driving scenes. Siva Hassan Abu Alhaija, Lars Karthik Mustikovela, Andreas Mescheder, Carsten Geiger, Rother, International Journal of Computer Vision. 1269Hassan Abu Alhaija, Siva Karthik Mustikovela, Lars Mescheder, Andreas Geiger, and Carsten Rother. Aug- mented reality meets computer vision: Efficient data gen- eration for urban driving scenes. International Journal of Computer Vision, 126(9):961-972, 2018. 1 Habitat: A platform for embodied ai research. Manolis Savva, Abhishek Kadian, Oleksandr Maksymets, Yili Zhao, Erik Wijmans, Bhavana Jain, Julian Straub, Jia Liu, Vladlen Koltun, Jitendra Malik, ICCV. Manolis Savva, Abhishek Kadian, Oleksandr Maksymets, Yili Zhao, Erik Wijmans, Bhavana Jain, Julian Straub, Jia Liu, Vladlen Koltun, Jitendra Malik, et al. Habitat: A plat- form for embodied ai research. In ICCV, 2019. 1 . Yohann Cabon, Naila Murray, Martin Humenberger, arXiv:2001.10773Virtual kitti 2. arXiv preprintYohann Cabon, Naila Murray, and Martin Humenberger. Virtual kitti 2. arXiv preprint arXiv:2001.10773, 2020. 1 0: a simulation environment for interactive tasks in large realistic scenes. Bokui Shen, Fei Xia, Chengshu Li, Roberto Martín-Martín, Linxi Fan, Guanzhi Wang, Claudia Pérez-D&apos;arpino, Shyamal Buch, Sanjana Srivastava, Lyne Tchapmi, IROS. et al. igibson 1.Bokui Shen, Fei Xia, Chengshu Li, Roberto Martín-Martín, Linxi Fan, Guanzhi Wang, Claudia Pérez-D'Arpino, Shya- mal Buch, Sanjana Srivastava, Lyne Tchapmi, et al. igibson 1.0: a simulation environment for interactive tasks in large realistic scenes. In IROS, 2021. 1 Blender -a 3d modelling and rendering package. Blender Online Community, Blender Online Community. Blender -a 3d modelling and rendering package, 2018. 1 Unity: A general platform for intelligent agents. Arthur Juliani, Vincent-Pierre Berges, Ervin Teng, Andrew Cohen, Jonathan Harper, Chris Elion, Chris Goy, Yuan Gao, Hunter Henry, Marwan Mattar, arXiv:1809.02627arXiv preprintArthur Juliani, Vincent-Pierre Berges, Ervin Teng, Andrew Cohen, Jonathan Harper, Chris Elion, Chris Goy, Yuan Gao, Hunter Henry, Marwan Mattar, et al. Unity: A general plat- form for intelligent agents. arXiv preprint arXiv:1809.02627, 2018. 1 Unreal engine. Epic GamesEpic Games. Unreal engine. 1 Unified particle physics for real-time applications. Miles Macklin, Matthias Müller, Nuttapong Chentanez, Tae-Yong Kim, ACM Transactions on Graphics (TOG). 334Miles Macklin, Matthias Müller, Nuttapong Chentanez, and Tae-Yong Kim. Unified particle physics for real-time appli- cations. ACM Transactions on Graphics (TOG), 33(4):1-12, 2014. 1 Pybullet, a python module for physics simulation for games, robotics and machine learning. Erwin Coumans, Yunfei Bai, Erwin Coumans and Yunfei Bai. Pybullet, a python module for physics simulation for games, robotics and machine learning. 2016. 1 X Angel, Thomas Chang, Leonidas Funkhouser, Pat Guibas, Qixing Hanrahan, Zimo Huang, Silvio Li, Manolis Savarese, Shuran Savva, Hao Song, Su, arXiv:1512.03012An information-rich 3d model repository. 13arXiv preprintAngel X Chang, Thomas Funkhouser, Leonidas Guibas, Pat Hanrahan, Qixing Huang, Zimo Li, Silvio Savarese, Manolis Savva, Shuran Song, Hao Su, et al. Shapenet: An information-rich 3d model repository. arXiv preprint arXiv:1512.03012, 2015. 1, 2, 3 Scannet: Richlyannotated 3d reconstructions of indoor scenes. Angela Dai, X Angel, Manolis Chang, Maciej Savva, Thomas Halber, Matthias Funkhouser, Nießner, CVPR. Angela Dai, Angel X Chang, Manolis Savva, Maciej Halber, Thomas Funkhouser, and Matthias Nießner. Scannet: Richly- annotated 3d reconstructions of indoor scenes. In CVPR, 2017. 1 Angel Chang, Angela Dai, Thomas Funkhouser, Maciej Halber, Matthias Niessner, Manolis Savva, Shuran Song, Andy Zeng, Yinda Zhang, arXiv:1709.06158Matterport3d: Learning from rgb-d data in indoor environments. arXiv preprintAngel Chang, Angela Dai, Thomas Funkhouser, Maciej Halber, Matthias Niessner, Manolis Savva, Shuran Song, Andy Zeng, and Yinda Zhang. Matterport3d: Learning from rgb-d data in indoor environments. arXiv preprint arXiv:1709.06158, 2017. 1 Photoshape: Photorealistic materials for large-scale shape collections. Keunhong Park, Konstantinos Rematas, Ali Farhadi, Steven M Seitz, arXiv:1809.0976113arXiv preprintKeunhong Park, Konstantinos Rematas, Ali Farhadi, and Steven M Seitz. Photoshape: Photorealistic materi- als for large-scale shape collections. arXiv preprint arXiv:1809.09761, 2018. 1, 2, 3 Gibson env: Real-world perception for embodied agents. Fei Xia, Zhiyang Amir R Zamir, Alexander He, Jitendra Sax, Silvio Malik, Savarese, CVPR. Fei Xia, Amir R Zamir, Zhiyang He, Alexander Sax, Jiten- dra Malik, and Silvio Savarese. Gibson env: Real-world perception for embodied agents. In CVPR, 2018. 1 Partnet: A largescale benchmark for fine-grained and hierarchical part-level 3d object understanding. Kaichun Mo, Shilin Zhu, X Angel, Li Chang, Subarna Yi, Leonidas J Tripathi, Hao Guibas, Su, CVPR. Kaichun Mo, Shilin Zhu, Angel X Chang, Li Yi, Subarna Tripathi, Leonidas J Guibas, and Hao Su. Partnet: A large- scale benchmark for fine-grained and hierarchical part-level 3d object understanding. In CVPR, 2019. 1 Common objects in 3d: Large-scale learning and evaluation of real-life 3d category reconstruction. Jeremy Reizenstein, Roman Shapovalov, Philipp Henzler, Luca Sbordone, Patrick Labatut, David Novotny, ICCV. Jeremy Reizenstein, Roman Shapovalov, Philipp Henzler, Luca Sbordone, Patrick Labatut, and David Novotny. Com- mon objects in 3d: Large-scale learning and evaluation of real-life 3d category reconstruction. In ICCV, 2021. 1 Cad2rl: Real singleimage flight without a single real image. Fereshteh Sadeghi, Sergey Levine, arXiv:1611.04201arXiv preprintFereshteh Sadeghi and Sergey Levine. Cad2rl: Real single- image flight without a single real image. arXiv preprint arXiv:1611.04201, 2016. 1 Driving policy transfer via modularity and abstraction. Matthias Müller, Alexey Dosovitskiy, Bernard Ghanem, Vladlen Koltun, arXiv:1804.09364arXiv preprintMatthias Müller, Alexey Dosovitskiy, Bernard Ghanem, and Vladlen Koltun. Driving policy transfer via modularity and abstraction. arXiv preprint arXiv:1804.09364, 2018. 1 Closing the sim-to-real loop: Adapting simulation randomization with real world experience. Yevgen Chebotar, Ankur Handa, Viktor Makoviychuk, Miles Macklin, Jan Issac, Nathan Ratliff, Dieter Fox, ICRA. Yevgen Chebotar, Ankur Handa, Viktor Makoviychuk, Miles Macklin, Jan Issac, Nathan Ratliff, and Dieter Fox. Closing the sim-to-real loop: Adapting simulation randomization with real world experience. In ICRA, 2019. 1 Solving rubik's cube with a robot hand. Ilge Akkaya, Marcin Andrychowicz, Maciek Chociej, Mateusz Litwin, Bob Mcgrew, Arthur Petron, Alex Paino, Matthias Plappert, Glenn Powell, Raphael Ribas, arXiv:1910.07113arXiv preprintIlge Akkaya, Marcin Andrychowicz, Maciek Chociej, Ma- teusz Litwin, Bob McGrew, Arthur Petron, Alex Paino, Matthias Plappert, Glenn Powell, Raphael Ribas, et al. Solving rubik's cube with a robot hand. arXiv preprint arXiv:1910.07113, 2019. 1 Simulation-based reinforcement learning for real-world autonomous driving. Błażej Osiński, Adam Jakubowski, Paweł Zięcina, Piotr Miłoś, Christopher Galias, Silviu Homoceanu, Henryk Michalewski, ICRA. 2020Błażej Osiński, Adam Jakubowski, Paweł Zięcina, Piotr Miłoś, Christopher Galias, Silviu Homoceanu, and Henryk Michalewski. Simulation-based reinforcement learning for real-world autonomous driving. In ICRA, 2020. 1 Sim2real predictivity: Does evaluation in simulation predict real-world performance?. Abhishek Kadian, Joanne Truong, Aaron Gokaslan, Alexander Clegg, Erik Wijmans, Stefan Lee, Manolis Savva, Sonia Chernova, Dhruv Batra, IEEE Robotics and Automation Letters. 54Abhishek Kadian, Joanne Truong, Aaron Gokaslan, Alexan- der Clegg, Erik Wijmans, Stefan Lee, Manolis Savva, Sonia Chernova, and Dhruv Batra. Sim2real predictivity: Does evaluation in simulation predict real-world performance? IEEE Robotics and Automation Letters, 5(4):6670-6677, 2020. 1 Anish Shankar, Alex Bewley, and Pannag R Sanketi. i-sim2real: Reinforcement learning of robotic policies in tight human-robot interaction loops. Laura Saminda Abeyruwan, Graesser, B David, Avi D&apos;ambrosio, Singh, arXiv:2207.06572arXiv preprintSaminda Abeyruwan, Laura Graesser, David B D'Ambrosio, Avi Singh, Anish Shankar, Alex Bewley, and Pannag R Sanketi. i-sim2real: Reinforcement learning of robotic poli- cies in tight human-robot interaction loops. arXiv preprint arXiv:2207.06572, 2022. 1 Lidarsim: Realistic lidar simulation by leveraging the real world. Sivabalan Manivasagam, Shenlong Wang, Kelvin Wong, Wenyuan Zeng, Mikita Sazanovich, Shuhan Tan, Bin Yang, Wei-Chiu Ma, Raquel Urtasun, CVPR. 13Sivabalan Manivasagam, Shenlong Wang, Kelvin Wong, Wenyuan Zeng, Mikita Sazanovich, Shuhan Tan, Bin Yang, Wei-Chiu Ma, and Raquel Urtasun. Lidarsim: Realistic lidar simulation by leveraging the real world. In CVPR, 2020. 1, 3 Geosim: Realistic video simulation via geometry-aware composition for self-driving. Yun Chen, Frieda Rong, Shivam Duggal, Shenlong Wang, Xinchen Yan, Sivabalan Manivasagam, Shangjie Xue, Ersin Yumer, Raquel Urtasun, CVPR. 13Yun Chen, Frieda Rong, Shivam Duggal, Shenlong Wang, Xinchen Yan, Sivabalan Manivasagam, Shangjie Xue, Ersin Yumer, and Raquel Urtasun. Geosim: Realistic video simu- lation via geometry-aware composition for self-driving. In CVPR, 2021. 1, 3 Meta-sim: Learning to generate synthetic datasets. Amlan Kar, Aayush Prakash, Ming-Yu Liu, Eric Cameracci, Justin Yuan, Matt Rusiniak, David Acuna, Antonio Torralba, Sanja Fidler, ICCV. 13Amlan Kar, Aayush Prakash, Ming-Yu Liu, Eric Cameracci, Justin Yuan, Matt Rusiniak, David Acuna, Antonio Torralba, and Sanja Fidler. Meta-sim: Learning to generate synthetic datasets. In ICCV, 2019. 1, 3 Idd: A dataset for exploring problems of autonomous navigation in unconstrained environments. Girish Varma, Anbumani Subramanian, Anoop Namboodiri, Manmohan Chandraker, C V Jawahar, WACV. IEEEGirish Varma, Anbumani Subramanian, Anoop Namboodiri, Manmohan Chandraker, and CV Jawahar. Idd: A dataset for exploring problems of autonomous navigation in uncon- strained environments. In WACV. IEEE, 2019. 1 Giraffe: Representing scenes as compositional generative neural feature fields. Michael Niemeyer, Andreas Geiger, CVPR. 6Michael Niemeyer and Andreas Geiger. Giraffe: Represent- ing scenes as compositional generative neural feature fields. In CVPR, 2021. 2, 3, 6 Efficient geometry-aware 3d generative adversarial networks. Connor Z Eric R Chan, Lin, A Matthew, Koki Chan, Boxiao Nagano, Pan, Orazio Shalini De Mello, Leonidas J Gallo, Jonathan Guibas, Sameh Tremblay, Khamis, CVPR. 717Eric R Chan, Connor Z Lin, Matthew A Chan, Koki Nagano, Boxiao Pan, Shalini De Mello, Orazio Gallo, Leonidas J Guibas, Jonathan Tremblay, Sameh Khamis, et al. Effi- cient geometry-aware 3d generative adversarial networks. In CVPR, 2022. 2, 3, 4, 6, 7, 17 Fisher Yu, Ari Seff, Yinda Zhang, Shuran Song, Thomas Funkhouser, Jianxiong Xiao, Lsun, arXiv:1506.03365Construction of a large-scale image dataset using deep learning with humans in the loop. 23arXiv preprintFisher Yu, Ari Seff, Yinda Zhang, Shuran Song, Thomas Funkhouser, and Jianxiong Xiao. Lsun: Construction of a large-scale image dataset using deep learning with humans in the loop. arXiv preprint arXiv:1506.03365, 2015. 2, 3 A large-scale car dataset for fine-grained categorization and verification. Linjie Yang, Ping Luo, Chen Change Loy, Xiaoou Tang, CVPR. 23Linjie Yang, Ping Luo, Chen Change Loy, and Xiaoou Tang. A large-scale car dataset for fine-grained categorization and verification. In CVPR, 2015. 2, 3 Progressive growing of gans for improved quality, stability, and variation. Tero Karras, Timo Aila, Samuli Laine, Jaakko Lehtinen, arXiv:1710.1019623arXiv preprintTero Karras, Timo Aila, Samuli Laine, and Jaakko Lehtinen. Progressive growing of gans for improved quality, stability, and variation. arXiv preprint arXiv:1710.10196, 2017. 2, 3 Large-scale celebfaces attributes (celeba) dataset. Retrieved. Ziwei Liu, Ping Luo, Xiaogang Wang, Xiaoou Tang, 153Ziwei Liu, Ping Luo, Xiaogang Wang, and Xiaoou Tang. Large-scale celebfaces attributes (celeba) dataset. Retrieved August, 15(2018):11, 2018. 2, 3 A style-based generator architecture for generative adversarial networks. Tero Karras, Samuli Laine, Timo Aila, CVPR. 23Tero Karras, Samuli Laine, and Timo Aila. A style-based generator architecture for generative adversarial networks. In CVPR, 2019. 2, 3 Stargan v2: Diverse image synthesis for multiple domains. Yunjey Choi, Youngjung Uh, Jaejun Yoo, Jung-Woo Ha, CVPR, 2020. 23Yunjey Choi, Youngjung Uh, Jaejun Yoo, and Jung-Woo Ha. Stargan v2: Diverse image synthesis for multiple domains. In CVPR, 2020. 2, 3 Neural discrete representation learning. Aaron Van Den, Oriol Oord, Vinyals, NeurIPS. 24Aaron Van Den Oord, Oriol Vinyals, et al. Neural discrete representation learning. In NeurIPS, 2017. 2, 4 Taming transformers for high-resolution image synthesis. Patrick Esser, Robin Rombach, Bjorn Ommer, CVPR. Patrick Esser, Robin Rombach, and Bjorn Ommer. Taming transformers for high-resolution image synthesis. In CVPR, 2021. 2, 4, 20 Panoptic-deeplab: A simple, strong, and fast baseline for bottom-up panoptic segmentation. Bowen Cheng, D Maxwell, Yukun Collins, Ting Zhu, Liu, S Thomas, Hartwig Huang, Liang-Chieh Adam, Chen, Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. the IEEE/CVF conference on computer vision and pattern recognition616Bowen Cheng, Maxwell D Collins, Yukun Zhu, Ting Liu, Thomas S Huang, Hartwig Adam, and Liang-Chieh Chen. Panoptic-deeplab: A simple, strong, and fast baseline for bottom-up panoptic segmentation. In Proceedings of the IEEE/CVF conference on computer vision and pattern recog- nition, pages 12475-12485, 2020. 2, 6, 16 Masked generative image transformer. Huiwen Chang, Han Zhang, Lu Jiang, Ce Liu, William T Freeman, Maskgit, CVPR. 817Huiwen Chang, Han Zhang, Lu Jiang, Ce Liu, and William T Freeman. Maskgit: Masked generative image transformer. In CVPR, 2022. 2, 4, 5, 8, 17 Separating style and content with bilinear models. B Joshua, William T Tenenbaum, Freeman, Neural computation. 1263Joshua B Tenenbaum and William T Freeman. Separating style and content with bilinear models. Neural computation, 12(6):1247-1283, 2000. 2, 3 Learning to disentangle factors of variation with manifold interaction. Scott Reed, Kihyuk Sohn, Yuting Zhang, Honglak Lee, ICML. PMLR. 23Scott Reed, Kihyuk Sohn, Yuting Zhang, and Honglak Lee. Learning to disentangle factors of variation with manifold interaction. In ICML. PMLR, 2014. 2, 3 Learning to generate chairs with convolutional neural networks. Alexey Dosovitskiy, Jost Tobias Springenberg, Thomas Brox, CVPR. 23Alexey Dosovitskiy, Jost Tobias Springenberg, and Thomas Brox. Learning to generate chairs with convolutional neural networks. In CVPR, 2015. 2, 3 Weakly-supervised disentangling with recurrent transformations for 3d view synthesis. Jimei Yang, E Scott, Ming-Hsuan Reed, Honglak Yang, Lee, NIPS. 23Jimei Yang, Scott E Reed, Ming-Hsuan Yang, and Honglak Lee. Weakly-supervised disentangling with recurrent trans- formations for 3d view synthesis. In NIPS, 2015. 2, 3 Deep convolutional inverse graphics network. D Tejas, Kulkarni, F William, Pushmeet Whitney, Josh Kohli, Tenenbaum, NIPS. 23Tejas D Kulkarni, William F Whitney, Pushmeet Kohli, and Josh Tenenbaum. Deep convolutional inverse graphics net- work. In NIPS, 2015. 2, 3 Unsupervised learning of 3d structure from images. Danilo Jimenez Rezende, Shakir Eslami, Peter Mohamed, Max Battaglia, Nicolas Jaderberg, Heess, NIPS. 23Danilo Jimenez Rezende, SM Eslami, Shakir Mohamed, Peter Battaglia, Max Jaderberg, and Nicolas Heess. Unsu- pervised learning of 3d structure from images. NIPS, 2016. 2, 3 Towards large-pose face frontalization in the wild. Xi Yin, Xiang Yu, Kihyuk Sohn, Xiaoming Liu, Manmohan Chandraker, ICCV. 23Xi Yin, Xiang Yu, Kihyuk Sohn, Xiaoming Liu, and Man- mohan Chandraker. Towards large-pose face frontalization in the wild. In ICCV, 2017. 2, 3 Visual object networks: Image generation with disentangled 3d representations. Jun-Yan Zhu, Zhoutong Zhang, Chengkai Zhang, Jiajun Wu, Antonio Torralba, Josh Tenenbaum, Bill Freeman, In NeurIPS. 23Jun-Yan Zhu, Zhoutong Zhang, Chengkai Zhang, Jiajun Wu, Antonio Torralba, Josh Tenenbaum, and Bill Freeman. Visual object networks: Image generation with disentangled 3d representations. In NeurIPS, 2018. 2, 3 Hologan: Unsupervised learning of 3d representations from natural images. Thu Nguyen-Phuoc, Chuan Li, Lucas Theis, Christian Richardt, Yong-Liang Yang, ICCV. 23Thu Nguyen-Phuoc, Chuan Li, Lucas Theis, Christian Richardt, and Yong-Liang Yang. Hologan: Unsupervised learning of 3d representations from natural images. In ICCV, 2019. 2, 3 Blockgan: Learning 3d object-aware scene representations from unlabelled images. Christian Thu H Nguyen-Phuoc, Long Richardt, Yongliang Mai, Niloy Yang, Mitra, 23Thu H Nguyen-Phuoc, Christian Richardt, Long Mai, Yongliang Yang, and Niloy Mitra. Blockgan: Learning 3d object-aware scene representations from unlabelled images. 2020. 2, 3 Towards unsupervised learning of generative models for 3d controllable image synthesis. Yiyi Liao, Katja Schwarz, Lars Mescheder, Andreas Geiger, CVPR, 2020. 23Yiyi Liao, Katja Schwarz, Lars Mescheder, and Andreas Geiger. Towards unsupervised learning of generative models for 3d controllable image synthesis. In CVPR, 2020. 2, 3 Graf: Generative radiance fields for 3d-aware image synthesis. Katja Schwarz, Yiyi Liao, Michael Niemeyer, Andreas Geiger, NeurIPS, 2020. 23Katja Schwarz, Yiyi Liao, Michael Niemeyer, and Andreas Geiger. Graf: Generative radiance fields for 3d-aware image synthesis. In NeurIPS, 2020. 2, 3 pi-gan: Periodic implicit generative adversarial networks for 3d-aware image synthesis. Marco Eric R Chan, Petr Monteiro, Jiajun Kellnhofer, Gordon Wu, Wetzstein, CVPR. 23Eric R Chan, Marco Monteiro, Petr Kellnhofer, Jiajun Wu, and Gordon Wetzstein. pi-gan: Periodic implicit generative adversarial networks for 3d-aware image synthesis. In CVPR, 2021. 2, 3 Gancraft: Unsupervised 3d neural rendering of minecraft worlds. Zekun Hao, Arun Mallya, Serge Belongie, Ming-Yu Liu, ICCV, 2021. 23Zekun Hao, Arun Mallya, Serge Belongie, and Ming-Yu Liu. Gancraft: Unsupervised 3d neural rendering of minecraft worlds. In ICCV, 2021. 2, 3 CIPS-3D: A 3D-Aware Generator of GANs Based on Conditionally-Independent Pixel Synthesis. Peng Zhou, Lingxi Xie, Bingbing Ni, Qi Tian, 23Peng Zhou, Lingxi Xie, Bingbing Ni, and Qi Tian. CIPS-3D: A 3D-Aware Generator of GANs Based on Conditionally- Independent Pixel Synthesis. 2021. 2, 3 Stylenerf: A style-based 3d aware generator for highresolution image synthesis. Jiatao Gu, Lingjie Liu, Peng Wang, Christian Theobalt, ICLR, 2022. 23Jiatao Gu, Lingjie Liu, Peng Wang, and Christian Theobalt. Stylenerf: A style-based 3d aware generator for high- resolution image synthesis. In ICLR, 2022. 2, 3 Stylesdf: High-resolution 3d-consistent image and geometry generation. Roy Or-El, Xuan Luo, Mengyi Shan, Eli Shechtman, Jeong Joon Park, Ira Kemelmacher-Shlizerman, CVPR. 718Roy Or-El, Xuan Luo, Mengyi Shan, Eli Shechtman, Jeong Joon Park, and Ira Kemelmacher-Shlizerman. Stylesdf: High-resolution 3d-consistent image and geometry generation. In CVPR, 2022. 2, 3, 7, 18 Ben Poole, Ajay Jain, Jonathan T Barron, Ben Mildenhall, arXiv:2209.14988Dreamfusion: Text-to-3d using 2d diffusion. 2022arXiv preprintBen Poole, Ajay Jain, Jonathan T Barron, and Ben Milden- hall. Dreamfusion: Text-to-3d using 2d diffusion. arXiv preprint arXiv:2209.14988, 2022. 2 Epigraf: Rethinking training of 3d gans. Ivan Skorokhodov, Sergey Tulyakov, Yiqun Wang, Peter Wonka, Ivan Skorokhodov, Sergey Tulyakov, Yiqun Wang, and Peter Wonka. Epigraf: Rethinking training of 3d gans. 2022. 2, 3, 8 Kangle Deng, Gengshan Yang, Deva Ramanan, Jun-Yan Zhu, CVPR. 3d-aware conditional image synthesisKangle Deng, Gengshan Yang, Deva Ramanan, and Jun-Yan Zhu. 3d-aware conditional image synthesis. In CVPR, 2023. 2 Ivan Skorokhodov, Aliaksandr Siarohin, Yinghao Xu, Jian Ren, Hsin-Ying Lee, Peter Wonka, Sergey Tulyakov, International Conference on Learning Representations. 3d generation on imagenetIvan Skorokhodov, Aliaksandr Siarohin, Yinghao Xu, Jian Ren, Hsin-Ying Lee, Peter Wonka, and Sergey Tulyakov. 3d generation on imagenet. In International Conference on Learning Representations. 2 Kyle Sargent, Yu Koh, Han Zhang, Huiwen Chang, Charles Herrmann, Pratul Srinivasan, Jiajun Wu, Deqing Sun, arXiv:2302.06833Vq3d: Learning a 3d-aware generative model on imagenet. arXiv preprintKyle Sargent, Jing Yu Koh, Han Zhang, Huiwen Chang, Charles Herrmann, Pratul Srinivasan, Jiajun Wu, and De- qing Sun. Vq3d: Learning a 3d-aware generative model on imagenet. arXiv preprint arXiv:2302.06833, 2023. 2 3d shapenets: A deep representation for volumetric shapes. Zhirong Wu, Shuran Song, Aditya Khosla, Fisher Yu, Linguang Zhang, Xiaoou Tang, Jianxiong Xiao, CVPR. Zhirong Wu, Shuran Song, Aditya Khosla, Fisher Yu, Linguang Zhang, Xiaoou Tang, and Jianxiong Xiao. 3d shapenets: A deep representation for volumetric shapes. In CVPR, 2015. 2 Learning a probabilistic latent space of object shapes via 3d generative-adversarial modeling. Jiajun Wu, Chengkai Zhang, Tianfan Xue, Bill Freeman, Josh Tenenbaum, Jiajun Wu, Chengkai Zhang, Tianfan Xue, Bill Freeman, and Josh Tenenbaum. Learning a probabilistic latent space of object shapes via 3d generative-adversarial modeling. 2016. 2 Learning a predictable and generative vector representation for objects. Rohit Girdhar, F David, Mikel Fouhey, Abhinav Rodriguez, Gupta, ECCV. SpringerRohit Girdhar, David F Fouhey, Mikel Rodriguez, and Ab- hinav Gupta. Learning a predictable and generative vector representation for objects. In ECCV. Springer, 2016. 2 3d shape induction from 2d views of multiple objects. Matheus Gadelha, Subhransu Maji, Rui Wang, 3DV. Matheus Gadelha, Subhransu Maji, and Rui Wang. 3d shape induction from 2d views of multiple objects. In 3DV, 2017. 2 Improved adversarial systems for 3d object generation and reconstruction. J Edward, David Smith, Meger, CoRL. PMLREdward J Smith and David Meger. Improved adversarial systems for 3d object generation and reconstruction. In CoRL. PMLR, 2017. 2 Escaping plato's cave: 3d shape from adversarial rendering. Philipp Henzler, J Niloy, Tobias Mitra, Ritschel, ICCV. Philipp Henzler, Niloy J Mitra, and Tobias Ritschel. Escap- ing plato's cave: 3d shape from adversarial rendering. In ICCV, 2019. 2 Inverse graphics gan: Learning to generate 3d shapes from unstructured 2d data. Sebastian Lunz, Yingzhen Li, Andrew Fitzgibbon, Nate Kushman, arXiv:2002.12674arXiv preprintSebastian Lunz, Yingzhen Li, Andrew Fitzgibbon, and Nate Kushman. Inverse graphics gan: Learning to gen- erate 3d shapes from unstructured 2d data. arXiv preprint arXiv:2002.12674, 2020. 2 3d shape generation and completion through point-voxel diffusion. Linqi Zhou, Yilun Du, Jiajun Wu, ICCV. Linqi Zhou, Yilun Du, and Jiajun Wu. 3d shape generation and completion through point-voxel diffusion. In ICCV, 2021. 2 Octree transformer: Autoregressive 3d shape generation on hierarchically structured sequences. Moritz Ibing, Gregor Kobsik, Leif Kobbelt, arXiv:2111.12480arXiv preprintMoritz Ibing, Gregor Kobsik, and Leif Kobbelt. Oc- tree transformer: Autoregressive 3d shape generation on hierarchically structured sequences. arXiv preprint arXiv:2111.12480, 2021. 2 Learning representations and generative models for 3d point clouds. Panos Achlioptas, Olga Diamanti, Ioannis Mitliagkas, Leonidas Guibas, ICML. PMLR618Panos Achlioptas, Olga Diamanti, Ioannis Mitliagkas, and Leonidas Guibas. Learning representations and generative models for 3d point clouds. In ICML. PMLR, 2018. 2, 6, 18 Pointflow: 3d point cloud generation with continuous normalizing flows. Guandao Yang, Xun Huang, Zekun Hao, Ming-Yu Liu, Serge Belongie, Bharath Hariharan, ICCV. Guandao Yang, Xun Huang, Zekun Hao, Ming-Yu Liu, Serge Belongie, and Bharath Hariharan. Pointflow: 3d point cloud generation with continuous normalizing flows. In ICCV, 2019. 2 PT2PC: Learning to generate 3d point cloud shapes from part tree conditions. Kaichun Mo, He Wang, Xinchen Yan, Leonidas Guibas, 2020Kaichun Mo, He Wang, Xinchen Yan, and Leonidas Guibas. PT2PC: Learning to generate 3d point cloud shapes from part tree conditions. 2020. 2 Surfnet: Generating 3d shape surfaces using deep residual networks. Ayan Sinha, Asim Unmesh, Qixing Huang, Karthik Ramani, CVPR. Ayan Sinha, Asim Unmesh, Qixing Huang, and Karthik Ramani. Surfnet: Generating 3d shape surfaces using deep residual networks. In CVPR, 2017. 2 A papier-mâché approach to learning 3d surface generation. Thibault Groueix, Matthew Fisher, G Vladimir, Kim, C Bryan, Mathieu Russell, Aubry, CVPR. Thibault Groueix, Matthew Fisher, Vladimir G Kim, Bryan C Russell, and Mathieu Aubry. A papier-mâché approach to learning 3d surface generation. In CVPR, 2018. 2 Convolutional generation of textured 3d meshes. Dario Pavllo, Graham Spinks, Thomas Hofmann, Marie-Francine Moens, Aurelien Lucchi, NeurIPS. 2Dario Pavllo, Graham Spinks, Thomas Hofmann, Marie- Francine Moens, and Aurelien Lucchi. Convolutional gener- ation of textured 3d meshes. NeurIPS, 2020. 2 Learning generative models of textured 3d meshes from real-world images. Dario Pavllo, Jonas Kohler, Thomas Hofmann, Aurelien Lucchi, ICCV. Dario Pavllo, Jonas Kohler, Thomas Hofmann, and Aurelien Lucchi. Learning generative models of textured 3d meshes from real-world images. In ICCV, 2021. 2 Learning to predict 3d objects with an interpolation-based differentiable renderer. Wenzheng Chen, Huan Ling, Jun Gao, Edward Smith, Jaakko Lehtinen, Alec Jacobson, Sanja Fidler, NeurIPS. 2Wenzheng Chen, Huan Ling, Jun Gao, Edward Smith, Jaakko Lehtinen, Alec Jacobson, and Sanja Fidler. Learning to predict 3d objects with an interpolation-based differen- tiable renderer. NeurIPS, 2019. 2 Polygen: An autoregressive generative model of 3d meshes. Charlie Nash, Yaroslav Ganin, Ali Sm, Peter Eslami, Battaglia, ICML. PMLR, 2020. Charlie Nash, Yaroslav Ganin, SM Ali Eslami, and Peter Battaglia. Polygen: An autoregressive generative model of 3d meshes. In ICML. PMLR, 2020. 2 Get3d: A generative model of high quality 3d textured shapes learned from images. Jun Gao, Tianchang Shen, Zian Wang, Wenzheng Chen, Kangxue Yin, Daiqing Li, Or Litany, Zan Gojcic, Sanja Fidler, NeurIPS, 2022. 23Jun Gao, Tianchang Shen, Zian Wang, Wenzheng Chen, Kangxue Yin, Daiqing Li, Or Litany, Zan Gojcic, and Sanja Fidler. Get3d: A generative model of high quality 3d tex- tured shapes learned from images. In NeurIPS, 2022. 2, 3 Structurenet: Hierarchical graph networks for 3d shape generation. Kaichun Mo, Paul Guerrero, Li Yi, Hao Su, Peter Wonka, Niloy Mitra, Leonidas Guibas, Siggraph Asia. 38Kaichun Mo, Paul Guerrero, Li Yi, Hao Su, Peter Wonka, Niloy Mitra, and Leonidas Guibas. Structurenet: Hierarchi- cal graph networks for 3d shape generation. ACM Transac- tions on Graphics (TOG), Siggraph Asia 2019, 38(6):Article 242, 2019. 2 Learning shape abstractions by assembling volumetric primitives. Shubham Tulsiani, Hao Su, Leonidas J Guibas, Alexei A Efros, Jitendra Malik, CVPR. Shubham Tulsiani, Hao Su, Leonidas J Guibas, Alexei A Efros, and Jitendra Malik. Learning shape abstractions by assembling volumetric primitives. In CVPR, 2017. 2 Learning to infer implicit surfaces without 3d supervision. Shichen Liu, Shunsuke Saito, Weikai Chen, Hao Li, NeurIPS. 2Shichen Liu, Shunsuke Saito, Weikai Chen, and Hao Li. Learning to infer implicit surfaces without 3d supervision. NeurIPS, 2019. 2 Occupancy networks: Learning 3d reconstruction in function space. Lars Mescheder, Michael Oechsle, Michael Niemeyer, Sebastian Nowozin, Andreas Geiger, CVPR. Lars Mescheder, Michael Oechsle, Michael Niemeyer, Se- bastian Nowozin, and Andreas Geiger. Occupancy networks: Learning 3d reconstruction in function space. In CVPR, 2019. 2 Surfgen: Adversarial 3d shape synthesis with explicit surface discriminators. Andrew Luo, Tianqin Li, Wen-Hao Zhang, Tai Sing Lee, ICCV. Andrew Luo, Tianqin Li, Wen-Hao Zhang, and Tai Sing Lee. Surfgen: Adversarial 3d shape synthesis with explicit surface discriminators. In ICCV, 2021. 2 Learning implicit fields for generative shape modeling. Zhiqin Chen, Hao Zhang, CVPR. Zhiqin Chen and Hao Zhang. Learning implicit fields for generative shape modeling. In CVPR, 2019. 2 Shapeformer: Transformer-based shape completion via sparse representation. Xingguang Yan, Liqiang Lin, J Niloy, Dani Mitra, Daniel Lischinski, Hui Cohen-Or, Huang, CVPR. 2022Xingguang Yan, Liqiang Lin, Niloy J Mitra, Dani Lischin- ski, Daniel Cohen-Or, and Hui Huang. Shapeformer: Transformer-based shape completion via sparse representa- tion. In CVPR, 2022. 2 Deep marching tetrahedra: a hybrid representation for high-resolution 3d shape synthesis. Tianchang Shen, Jun Gao, Kangxue Yin, Ming-Yu Liu, Sanja Fidler, NeurIPS. 2Tianchang Shen, Jun Gao, Kangxue Yin, Ming-Yu Liu, and Sanja Fidler. Deep marching tetrahedra: a hybrid representa- tion for high-resolution 3d shape synthesis. NeurIPS, 2021. 2 Autosdf: Shape priors for 3d completion, reconstruction and generation. Paritosh Mittal, Yen-Chi Cheng, Maneesh Singh, Shubham Tulsiani, CVPR. 2022Paritosh Mittal, Yen-Chi Cheng, Maneesh Singh, and Shub- ham Tulsiani. Autosdf: Shape priors for 3d completion, reconstruction and generation. In CVPR, 2022. 2 Playing for data: Ground truth from computer games. Vibhav Stephan R Richter, Stefan Vineet, Vladlen Roth, Koltun, ECCV. SpringerStephan R Richter, Vibhav Vineet, Stefan Roth, and Vladlen Koltun. Playing for data: Ground truth from computer games. In ECCV. Springer, 2016. 3 Learning hierarchical semantic image manipulation through structured representations. Seunghoon Hong, Xinchen Yan, S Thomas, Honglak Huang, Lee, NeurIPS. 3Seunghoon Hong, Xinchen Yan, Thomas S Huang, and Honglak Lee. Learning hierarchical semantic image manip- ulation through structured representations. NeurIPS, 2018. 3 Drivegan: Towards a controllable high-quality neural simulation. Jonah Seung Wook Kim, Antonio Philion, Sanja Torralba, Fidler, CVPR. Seung Wook Kim, Jonah Philion, Antonio Torralba, and Sanja Fidler. Drivegan: Towards a controllable high-quality neural simulation. In CVPR, 2021. 3 Aads: Augmented autonomous driving simulation using data-driven algorithms. Wei Li, Rong Cw Pan, Zhang, Jp Ren, Jin Ma, Fang, Yan, Geng, Huang, Hj Gong, Science robotics. 428863Wei Li, CW Pan, Rong Zhang, JP Ren, YX Ma, Jin Fang, FL Yan, QC Geng, XY Huang, HJ Gong, et al. Aads: Aug- mented autonomous driving simulation using data-driven algorithms. Science robotics, 4(28):eaaw0863, 2019. 3 Seung Wook Kim, and Sanja Fidler. Variational amodal object completion. Huan Ling, David Acuna, Karsten Kreis, NeurIPS. 2020Huan Ling, David Acuna, Karsten Kreis, Seung Wook Kim, and Sanja Fidler. Variational amodal object completion. In NeurIPS, 2020. 3 Ners: Neural reflectance surfaces for sparse-view 3d reconstruction in the wild. Jason Zhang, Gengshan Yang, Shubham Tulsiani, Deva Ramanan, NeurIPS. Jason Zhang, Gengshan Yang, Shubham Tulsiani, and Deva Ramanan. Ners: Neural reflectance surfaces for sparse-view 3d reconstruction in the wild. In NeurIPS, 2021. 3 Single-shot scene reconstruction. Sergey Zakharov, Andrei Rares, Ambrus, Campagnolo Vitor, Dennis Guizilini, Wadim Park, Fredo Kehl, Joshua B Durand, Vincent Tenenbaum, Jiajun Sitzmann, Adrien Wu, Gaidon, CoRLSergey Zakharov, Rares Andrei Ambrus, Vitor Campag- nolo Guizilini, Dennis Park, Wadim Kehl, Fredo Durand, Joshua B Tenenbaum, Vincent Sitzmann, Jiajun Wu, and Adrien Gaidon. Single-shot scene reconstruction. In CoRL, 2021. 3 Share with thy neighbors: Single-view reconstruction by cross-instance consistency. Tom Monnier, Matthew Fisher, Alexei A Efros, Mathieu Aubry, ECCV. 2022Tom Monnier, Matthew Fisher, Alexei A Efros, and Mathieu Aubry. Share with thy neighbors: Single-view reconstruction by cross-instance consistency. In ECCV, 2022. 3 Autorf: Learning 3d object radiance fields from single view observations. Norman Müller, Andrea Simonelli, Lorenzo Porzi, Samuel Rota Bulò, Matthias Nießner, Peter Kontschieder, CVPR. 34Norman Müller, Andrea Simonelli, Lorenzo Porzi, Samuel Rota Bulò, Matthias Nießner, and Peter Kontschieder. Autorf: Learning 3d object radiance fields from single view observations. In CVPR, 2022. 3, 4 Meta-sim2: Unsupervised learning of scene structure for synthetic data generation. Jeevan Devaranjan, Amlan Kar, Sanja Fidler, ECCV. SpringerJeevan Devaranjan, Amlan Kar, and Sanja Fidler. Meta- sim2: Unsupervised learning of scene structure for synthetic data generation. In ECCV. Springer, 2020. 3 Scenegen: Learning to generate realistic traffic scenes. Shuhan Tan, Kelvin Wong, Shenlong Wang, Sivabalan Manivasagam, Mengye Ren, Raquel Urtasun, CVPR. Shuhan Tan, Kelvin Wong, Shenlong Wang, Sivabalan Mani- vasagam, Mengye Ren, and Raquel Urtasun. Scenegen: Learning to generate realistic traffic scenes. In CVPR, 2021. 3 Surfelgan: Synthesizing realistic sensor data for autonomous driving. Zhenpei Yang, Yuning Chai, Dragomir Anguelov, Yin Zhou, Pei Sun, Dumitru Erhan, Sean Rafferty, Henrik Kretzschmar, CVPR. Zhenpei Yang, Yuning Chai, Dragomir Anguelov, Yin Zhou, Pei Sun, Dumitru Erhan, Sean Rafferty, and Henrik Kret- zschmar. Surfelgan: Synthesizing realistic sensor data for autonomous driving. In CVPR, 2020. 3 Panoptic neural fields: A semantic object-aware neural scene representation. Abhijit Kundu, Kyle Genova, Xiaoqi Yin, Alireza Fathi, Caroline Pantofaru, Leonidas J Guibas, Andrea Tagliasacchi, Frank Dellaert, Thomas Funkhouser, CVPR. 2022Abhijit Kundu, Kyle Genova, Xiaoqi Yin, Alireza Fathi, Caroline Pantofaru, Leonidas J Guibas, Andrea Tagliasacchi, Frank Dellaert, and Thomas Funkhouser. Panoptic neural fields: A semantic object-aware neural scene representation. In CVPR, 2022. 3 Urban radiance fields. Konstantinos Rematas, Andrew Liu, P Pratul, Jonathan T Srinivasan, Andrea Barron, Thomas Tagliasacchi, Vittorio Funkhouser, Ferrari, CVPR. 2022Konstantinos Rematas, Andrew Liu, Pratul P Srini- vasan, Jonathan T Barron, Andrea Tagliasacchi, Thomas Funkhouser, and Vittorio Ferrari. Urban radiance fields. In CVPR, 2022. 3 Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, arXiv:2010.11929Sylvain Gelly, et al. An image is worth 16x16 words: Transformers for image recognition at scale. arXiv preprintAlexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Syl- vain Gelly, et al. An image is worth 16x16 words: Trans- formers for image recognition at scale. arXiv preprint arXiv:2010.11929, 2020. 4 Unit: Multimodal multitask learning with a unified transformer. Ronghang Hu, Amanpreet Singh, 2021. 4ICCV. Ronghang Hu and Amanpreet Singh. Unit: Multimodal multitask learning with a unified transformer. In ICCV, 2021. 4 Andrew Jaegle, Sebastian Borgeaud, Jean-Baptiste Alayrac, Carl Doersch, Catalin Ionescu, David Ding, Skanda Koppula, Daniel Zoran, Andrew Brock, Evan Shelhamer, arXiv:2107.14795Perceiver io: A general architecture for structured inputs & outputs. arXiv preprintAndrew Jaegle, Sebastian Borgeaud, Jean-Baptiste Alayrac, Carl Doersch, Catalin Ionescu, David Ding, Skanda Kop- pula, Daniel Zoran, Andrew Brock, Evan Shelhamer, et al. Perceiver io: A general architecture for structured inputs & outputs. arXiv preprint arXiv:2107.14795, 2021. 4 Scene representation transformer: Geometry-free novel view synthesis through set-latent scene representations. S M Mehdi, Henning Sajjadi, Etienne Meyer, Urs Pot, Klaus Bergmann, Noha Greff, Suhani Radwan, Mario Vora, Daniel Lučić, Alexey Duckworth, Dosovitskiy, CVPR, 2022. 4Mehdi SM Sajjadi, Henning Meyer, Etienne Pot, Urs Bergmann, Klaus Greff, Noha Radwan, Suhani Vora, Mario Lučić, Daniel Duckworth, Alexey Dosovitskiy, et al. Scene representation transformer: Geometry-free novel view syn- thesis through set-latent scene representations. In CVPR, 2022. 4, 8 Daniel Rebain, J Mark, Kwang Moo Matthews, Gopal Yi, Sharma, arXiv:2209.106842022. 4Dmitry Lagun, and Andrea Tagliasacchi. Attention beats concatenation for conditioning neural fields. arXiv preprintDaniel Rebain, Mark J Matthews, Kwang Moo Yi, Gopal Sharma, Dmitry Lagun, and Andrea Tagliasacchi. Attention beats concatenation for conditioning neural fields. arXiv preprint arXiv:2209.10684, 2022. 4 Analyzing and improving the image quality of stylegan. Tero Karras, Samuli Laine, Miika Aittala, Janne Hellsten, Jaakko Lehtinen, Timo Aila, CVPR, 2020. 4. 1617Tero Karras, Samuli Laine, Miika Aittala, Janne Hellsten, Jaakko Lehtinen, and Timo Aila. Analyzing and improving the image quality of stylegan. In CVPR, 2020. 4, 16, 17 Convolutional occupancy networks. Songyou Peng, Michael Niemeyer, Lars Mescheder, Marc Pollefeys, Andreas Geiger, ECCV. Springer417Songyou Peng, Michael Niemeyer, Lars Mescheder, Marc Pollefeys, and Andreas Geiger. Convolutional occupancy networks. In ECCV. Springer, 2020. 4, 17 Acid: Action-conditional implicit visual dynamics for deformable object manipulation. Bokui Shen, Zhenyu Jiang, Christopher Choy, Leonidas J Guibas, Silvio Savarese, Anima Anandkumar, Yuke Zhu, 417Bokui Shen, Zhenyu Jiang, Christopher Choy, Leonidas J Guibas, Silvio Savarese, Anima Anandkumar, and Yuke Zhu. Acid: Action-conditional implicit visual dynamics for deformable object manipulation. 2022. 4, 17 Nerf: Representing scenes as neural radiance fields for view synthesis. Ben Mildenhall, P Pratul, Matthew Srinivasan, Jonathan T Tancik, Ravi Barron, Ren Ramamoorthi, Ng, ECCV. Ben Mildenhall, Pratul P Srinivasan, Matthew Tancik, Jonathan T Barron, Ravi Ramamoorthi, and Ren Ng. Nerf: Representing scenes as neural radiance fields for view syn- thesis. In ECCV, 2020. 4 Generating diverse high-fidelity images with vq-vae-2. Ali Razavi, Aaron Van Den Oord, Oriol Vinyals, NeurIPS. Ali Razavi, Aaron Van den Oord, and Oriol Vinyals. Gener- ating diverse high-fidelity images with vq-vae-2. In NeurIPS, 2019. 4 Generative pretraining from pixels. Mark Chen, Alec Radford, Rewon Child, Jeffrey Wu, Heewoo Jun, David Luan, Ilya Sutskever, PMLR, 2020. 4ICML. Mark Chen, Alec Radford, Rewon Child, Jeffrey Wu, Hee- woo Jun, David Luan, and Ilya Sutskever. Generative pre- training from pixels. In ICML. PMLR, 2020. 4 Vector-quantized image modeling with improved vqgan. Jiahui Yu, Xin Li, Jing Yu Koh, Han Zhang, Ruoming Pang, James Qin, Alexander Ku, Yuanzhong Xu, Jason Baldridge, Yonghui Wu, ICLR, 2021. 416Jiahui Yu, Xin Li, Jing Yu Koh, Han Zhang, Ruoming Pang, James Qin, Alexander Ku, Yuanzhong Xu, Jason Baldridge, and Yonghui Wu. Vector-quantized image modeling with improved vqgan. In ICLR, 2021. 4, 16 Scaling autoregressive models for content-rich text-to-image generation. Jiahui Yu, Yuanzhong Xu, Jing Yu Koh, Thang Luong, Gunjan Baid, Zirui Wang, Vijay Vasudevan, Alexander Ku, Yinfei Yang, Burcu Karagol Ayan, arXiv:2206.10789arXiv preprintJiahui Yu, Yuanzhong Xu, Jing Yu Koh, Thang Luong, Gun- jan Baid, Zirui Wang, Vijay Vasudevan, Alexander Ku, Yin- fei Yang, Burcu Karagol Ayan, et al. Scaling autoregressive models for content-rich text-to-image generation. arXiv preprint arXiv:2206.10789, 2022. 4 Phenaki: Variable length video generation from open domain textual description. Ruben Villegas, Mohammad Babaeizadeh, Pieter-Jan Kindermans, Hernan Moraldo, Han Zhang, Mohammad Taghi Saffar, Santiago Castro, Julius Kunze, Dumitru Erhan, arXiv:2210.02399arXiv preprintRuben Villegas, Mohammad Babaeizadeh, Pieter-Jan Kin- dermans, Hernan Moraldo, Han Zhang, Mohammad Taghi Saffar, Santiago Castro, Julius Kunze, and Dumitru Erhan. Phenaki: Variable length video generation from open do- main textual description. arXiv preprint arXiv:2210.02399, 2022. 4 The unreasonable effectiveness of deep features as a perceptual metric. Richard Zhang, Phillip Isola, Alexei A Efros, Eli Shechtman, Oliver Wang, In CVPR. 4Richard Zhang, Phillip Isola, Alexei A Efros, Eli Shechtman, and Oliver Wang. The unreasonable effectiveness of deep features as a perceptual metric. In CVPR, 2018. 4 Decomposing nerf for editing via feature field distillation. Sosuke Kobayashi, Eiichi Matsumoto, Vincent Sitzmann, arXiv:2205.1558557arXiv preprintSosuke Kobayashi, Eiichi Matsumoto, and Vincent Sitz- mann. Decomposing nerf for editing via feature field distil- lation. arXiv preprint arXiv:2205.15585, 2022. 5, 7 Neural feature fusion fields: 3d distillation of self-supervised 2d image representations. Vadim Tschernezki, Iro Laina, Diane Larlus, Andrea Vedaldi, arXiv:2209.03494arXiv preprintVadim Tschernezki, Iro Laina, Diane Larlus, and Andrea Vedaldi. Neural feature fusion fields: 3d distillation of self-supervised 2d image representations. arXiv preprint arXiv:2209.03494, 2022. 5 Liangchen Song, Anpei Chen, Zhong Li, Zhang Chen, Lele Chen, Junsong Yuan, Yi Xu, Andreas Geiger, arXiv:2210.15947Nerfplayer: A streamable dynamic scene representation with decomposed neural radiance fields. arXiv preprintLiangchen Song, Anpei Chen, Zhong Li, Zhang Chen, Lele Chen, Junsong Yuan, Yi Xu, and Andreas Geiger. Nerfplayer: A streamable dynamic scene representation with decomposed neural radiance fields. arXiv preprint arXiv:2210.15947, 2022. 5 Datasetgan: Efficient labeled data factory with minimal human effort. Yuxuan Zhang, Huan Ling, Jun Gao, Kangxue Yin, Jean-Francois Lafleche, Adela Barriuso, Antonio Torralba, Sanja Fidler, CVPR. Yuxuan Zhang, Huan Ling, Jun Gao, Kangxue Yin, Jean- Francois Lafleche, Adela Barriuso, Antonio Torralba, and Sanja Fidler. Datasetgan: Efficient labeled data factory with minimal human effort. In CVPR, 2021. 5 Emerging properties in self-supervised vision transformers. Mathilde Caron, Hugo Touvron, Ishan Misra, Hervé Jégou, Julien Mairal, Piotr Bojanowski, Armand Joulin, ICCV, 2021. 5. 719Mathilde Caron, Hugo Touvron, Ishan Misra, Hervé Jé- gou, Julien Mairal, Piotr Bojanowski, and Armand Joulin. Emerging properties in self-supervised vision transformers. In ICCV, 2021. 5, 7, 19 Waymo open dataset: Panoramic video panoptic segmentation. Jieru Mei, Alex Zihao Zhu, Xinchen Yan, Hang Yan, Siyuan Qiao, Yukun Zhu, Liang-Chieh Chen, Henrik Kretzschmar, Dragomir Anguelov, ECCV. 616Jieru Mei, Alex Zihao Zhu, Xinchen Yan, Hang Yan, Siyuan Qiao, Yukun Zhu, Liang-Chieh Chen, Henrik Kretzschmar, and Dragomir Anguelov. Waymo open dataset: Panoramic video panoptic segmentation. In ECCV, 2022. 6, 16 Adam: A method for stochastic optimization. P Diederik, Jimmy Kingma, Ba, arXiv:1412.6980617arXiv preprintDiederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014. 6, 17 Gans trained by a two time-scale update rule converge to a local nash equilibrium. NIPS, 30. Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, 617Bernhard Nessler, and Sepp HochreiterMartin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bern- hard Nessler, and Sepp Hochreiter. Gans trained by a two time-scale update rule converge to a local nash equilibrium. NIPS, 30, 2017. 6, 17 Learning transferable visual models from natural language supervision. Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, ICML. PMLR618Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al. Learning transferable visual models from natural language supervi- sion. In ICML. PMLR, 2021. 6, 18 Marching cubes: A high resolution 3d surface construction algorithm. E William, Harvey E Lorensen, Cline, ACM siggraph computer graphics. 21423William E Lorensen and Harvey E Cline. Marching cubes: A high resolution 3d surface construction algorithm. ACM siggraph computer graphics, 21(4):163-169, 1987. 7, 23 Ref-nerf: Structured view-dependent appearance for neural radiance fields. Dor Verbin, Peter Hedman, Ben Mildenhall, Todd Zickler, Jonathan T Barron, Srinivasan, CVPR. Dor Verbin, Peter Hedman, Ben Mildenhall, Todd Zickler, Jonathan T Barron, and Pratul P Srinivasan. Ref-nerf: Struc- tured view-dependent appearance for neural radiance fields. In CVPR, 2022. 8 . Jimmy Lei Ba, Jamie Ryan Kiros, Geoffrey E Hin, arXiv:1607.0645016ton. Layer normalization. arXiv preprintJimmy Lei Ba, Jamie Ryan Kiros, and Geoffrey E Hin- ton. Layer normalization. arXiv preprint arXiv:1607.06450, 2016. 16 T Jonathan, Ben Barron, Matthew Mildenhall, Peter Tancik, Ricardo Hedman, Martin-Brualla, Srinivasan, Mip-nerf: A multiscale representation for anti-aliasing neural radiance fields. In ICCV. 17Jonathan T Barron, Ben Mildenhall, Matthew Tancik, Peter Hedman, Ricardo Martin-Brualla, and Pratul P Srinivasan. Mip-nerf: A multiscale representation for anti-aliasing neu- ral radiance fields. In ICCV, 2021. 17 Rethinking the inception architecture for computer vision. Christian Szegedy, Vincent Vanhoucke, Sergey Ioffe, Jon Shlens, Zbigniew Wojna, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognition17Christian Szegedy, Vincent Vanhoucke, Sergey Ioffe, Jon Shlens, and Zbigniew Wojna. Rethinking the inception architecture for computer vision. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 2818-2826, 2016. 17 Open source computer vision library. Itseez, 2015. 18Itseez. Open source computer vision library. https:// github.com/itseez/opencv, 2015. 18 . Dawson-Haggerty, 18Dawson-Haggerty et al. trimesh. 18
{'fraction_non_alphanumeric': 0.05138546048743177, 'fraction_numerical': 0.02756145047456802, 'mean_word_length': 4.988600874453467, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 0, 'https://': 1, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 4, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'In-the-Wild Driving Data "Same kind…" "Similar size…"GINA-3D Synthesis Composition with Background NeRFGINA-3DReconstruction "Night time…"In-the-wild Driving Data "Random…"Figure 1. Leveraging in-the-wild data for generative assets modeling embodies a scalable approach for simulation. GINA-3D uses real-world driving data to perform various synthesis tasks for realistic 3D implicit neural assets. Left: Multi-sensor observations in the wild. Middle: Asset reconstruction and conditional synthesis. Right: Scene composition with background neural fields [1].AbstractModeling the 3D world from sensor data for simulation is a scalable way of developing testing and validation environments for robotic learning problems such as autonomous driving. However, manually creating or recreating real-world-like environments is difficult, expensive, and not scalable. Recent generative model techniques have shown promising progress to address such challenges by learning 3D assets using only plentiful 2D images -but still suffer limitations as they leverage either human-curated image datasets or renderings from manually-created synthetic 3D environments. In this paper, we introduce GINA-3D, a generative model that uses real-world driving data from camera and LiDAR sensors to create realistic 3D implicit neural assets of diverse vehicles and pedestrians. Compared to the existing image datasets, the real-world driving setting poses new challenges due to occlusions, lighting-variations and long-tail distributions. GINA-3D tackles these challenges by decoupling representation learning and generative modeling into two stages with a learned tri-plane latent structure, inspired by recent advances in generative modeling of images. To evaluate our approach, we construct a large-scale object-centric dataset containing over 520K images of vehicles and pedestrians from the Waymo Open Dataset, and a new set of 80K images of long-tail instances such as construction equipment, garbage trucks, and cable cars. We compare our model with existing approaches and demonstrate that it achieves state-of-the-art performance in quality and diversity for both generated images and geometries. * Work done during an internship at Waymo. † Work done at Waymo.', 'arxivid': '2304.02163', 'author': ['Bokui Shen \nStanford University\n\n', 'Xinchen Yan \nWaymo LLC\n3 Google\n', 'Charles R Qi \nWaymo LLC\n3 Google\n', 'Mahyar Najibi \nWaymo LLC\n3 Google\n', 'Boyang Deng \nStanford University\n\n\nWaymo LLC\n3 Google\n', 'Leonidas Guibas ', 'Yin Zhou \nWaymo LLC\n3 Google\n', 'Dragomir Anguelov \nWaymo LLC\n3 Google\n'], 'authoraffiliation': ['Stanford University\n', 'Waymo LLC\n3 Google', 'Waymo LLC\n3 Google', 'Waymo LLC\n3 Google', 'Stanford University\n', 'Waymo LLC\n3 Google', 'Waymo LLC\n3 Google', 'Waymo LLC\n3 Google'], 'corpusid': 257952369, 'doi': '10.48550/arxiv.2304.02163', 'github_urls': [], 'n_tokens_mistral': 34971, 'n_tokens_neox': 30144, 'n_words': 16560, 'pdfsha': '8e5d1b5907bb710d4eff60226ba44327835d8dad', 'pdfurls': ['https://export.arxiv.org/pdf/2304.02163v1.pdf'], 'title': ['GINA-3D: Learning to Generate Implicit Neural Assets in the Wild', 'GINA-3D: Learning to Generate Implicit Neural Assets in the Wild'], 'venue': []}
arxiv
Studies of XY Z states at BESIII Studies of XY Z states at BESIII 2018. July 2018 Seoul. 4 July -11. 2018 Bin Wang [email protected] Institute of High Energy Physics 100049BeijingChina Bin Wang Institute of High Energy Physics 100049BeijingChina Studies of XY Z states at BESIII Studies of XY Z states at BESIII Talk given at XXXIX International Conference on High Energy Physics (ICHEP2018) Seoul -Korea2018. July 2018 Seoul. 4 July -11. 2018Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ With about 12 fb −1 of collected data useful for the study of XY Z states, BESIII collaboration continues the exploration of these exotic charmoniumlike states. Recent results of the measurements of the line-shape of e + e − → π 0 π 0 ψ(3686), KKJ/ψ, and π + D 0 D * − , as well as the J P determination of Z c (3900) and Z c (3900) observed in e + e − → φ χ c1,2 at √ s = 4.6 GeV will be presented. * Speaker on the behalf of BESIII Collaboration. ICHEP 2018, XXXIX International Conference on High Energy Physics 4-11 July 2018 Seoul, Korea Introduction Charmonium states are composed of one charmed quark and one anti-charmed quark (cc). States below the charm quark pair threshold in mass (2m D ) are well described by the potential mode [1], while many states above 2m D have not been observed yet. In addition, there are many unexpected states which are called charmoniumlike or XY Z states and have a charmonium final states but no conventional charmonium states assignment observed in recent years [2]. Searching for the new decay modes of known charmonium or charmoniumlike states and new charmoniumlike states is helpful for the interpretation of these charmonium-like states. BESIII has collected large data samples of electron positron collisions with center-of-mass (c.m.) energy between 3.8 and 4.6 GeV, the total integrated luminosity is about 12 fb −1 . Recent results of the measurements of the XY Z physics will be described in the following sections. Determination of J P of Z c (3900) The charged charmoniumlike state Z c (3900) was observed in the process e + e − → π + π − J/ψ by the BESIII [3] and Belle [4] and confirmed by the CLEO-c [5]. As the first Z c state observed by more than one experiment, it is composed of at least four quarks. Many theoretical interpretations of its nature and decay dynamics have been put forward [6,7]. BESIII has also reported the observations of the neutral Z c (3900) in the process e + e − → π 0 π 0 J/ψ [8], Z c (4020) in e + e − → π +,0 π −,0 h c [9,10], Z c (4025) in e + e − → π ±,0 (D * D * ) ∓,0 [11,12], and Z c (3885) in e + e − → π ±,0 (DD * ) ∓,0 [13,14,15]. Are Z c (3900) and Z c (3885) the same state and do they have the same quantum number assignment? The experimental determination is one of the most important expectations of theorists. Recently, BESIII reported the determination of the J P of Z c (3900) [16], which is J P = 1 + with a statistical significance larger than 7σ over other quantum number assumptions, in the partial wave analysis of the process e + e − → π + π − J/ψ using 1.92 fb −1 data samples at √ s = 4.23 and 4.26 GeV. There are six contributions (e + e − → σ J/ψ, f 0 J/ψ, f 0 (1370)J/ψ, f 2 (1270)J/ψ, Z ± c π ∓ , and non-resonant processes) and five assumptions of the J P of Z c (3900) (1 + , 0 − , 1 − , 2 + , and 2 − ) are considered in the fit. Using a simultaneous fit for the data samples at 4.23 GeV and 4.26 GeV, where the Z c (3900) state is described by a Flatte-like formula, the mass and coupling parameters (g ′ 1 and g ′ 2 /g ′ 1 ) of Z c (3900) are measured to be (3901.5±2.7±38.0) MeV/c 2 , 0.075 ± 0.006 ± 0.025) GeV 2 , and (27.1±2.0±1.9), respectively. The fitted coupling constants are consistent with the measured decay width ration of (DD * ) ± and π ± J/ψ final states. 3. e + e − → π 0 π 0 ψ(3686) Recently, BESIII reported the precise measurement of the cross sections for the processes e + e − → π + π − J/ψ [20] and indicated that the Y (4260) resonances actually consists of two structures. Two resonances, Y (4220) and Y (4390), are observed in the process e + e − → π + π − h c [21]. BESIII has also reported more studies of these Y states for understanding the puzzles with these states. The Y (4360) was first observed in e + e − → γ ISR π + π − ψ(3686) by BABAR [22] and subsequently confirmed by Belle [23] and BESII [24]. By analogy, BESIII also reported the measurement of its neutral isospin e + e − → π 0 π 0 ψ(3686) [25]. The measured Born cross sections of e + e − → π 0 π 0 ψ(3686) are consistent with those of e + e − → π + π − ψ(3686) from isospin symmetry. In addition, a neutral charmoniumlike structure is observed in π 0 ψ(3686) with a mass of (4038.7±6.5) MeV/c 2 at √ s =4.416 GeV, which confirms the structure in the charged mode. No significant Z c (3900) 0 state is observed in the fit. e + e − → KKJ/ψ As the first observed vector charmoniumlike state, the nature of the Y (4260) is still unclear. Besides of e + e − → π + π − J/ψ, the Y (4260) has also been searched for many modes in recent years, such as π + π − h c , ω χ cJ , ηJ/ψ and so on. Especially, measuring the ratio of e + e − → KKJ/ψ and e + e − → ππJ/ψ cross sections provides a new insight into the nature of Y (4260). Recently, using 4.7 fb −1 data sample from 4.189 to 4.600 GeV, BESIII reported the measurements of the process of e + e − → KKJ/ψ [26]. The results show that the energy dependence of the cross section for e + e − → K + K − J/ψ is not consistent with those of e + e − → π + π − J/ψ in the region around 4.26 GeV. The ratio of cross sections for e + e − → K + K − J/ψ and e + e − → K 0 S K 0 S J/ψ are consistent with expectations from isospin conservation. In addition, there is an evidence for a structure around 4.5 GeV in the e + e − → K + K − J/ψ cross section that not present in the e + e − → π + π − J/ψ. 5. e + e − → π + D 0 D * − BESIII also reported a precise cross section measurement of e + e − → π + D 0 D * − process [27]. Two resonant structures are significant observed in this final states. It indicates an evidence for open-charm production associated with the Y states is observed for the first time. The parameters of these two resonances are consistent with those measured in e + e − → π + π − h c . The first resonance are also consistent with those measured in e + e − → ω χ c0 [28] and e + e − → π + π − J/ψ. The first resonance is consistent with some of the theoretical calculations for the mass of Y (4260) when explaining it as a DD 1 (2420) molecule [29]. 6. e + e − → φ χ c1,2 BESIII has reported the cross section of e + e − → ω χ c0 and observed an intermediate resonance around 4226 MeV [28]. Considering that ω and φ have the same spin, parity, and isospin, the ω χ cJ and φ χ cJ should have a similar production mechanism, so we study the e + e − → φ χ cJ . Using 567 fb −1 data sample at √ s = 4.60 GeV, BESIII first observed the processes of e + e − → φ χ c1 and φ χ c2 [30]. The corresponding Born cross sections are (4.2 +1.7 −1.0 ± 0.3) and (6.7 +3.4 −1.7 ± 0.5) pb, respectively. No significant e + e − → φ χ c0 and e + e − → γX (4140) were observed in this final states. Summary Recent studies of XY Z states at BESIII collaboration are presented using large luminosity data samples collected above 4 GeV. BESIII is an active and successful experiment for the charmonium spectroscopy study. In the future, BESIII will continue to take data for studying these XY Z states more precisely and increase the beam energy for studying the higher XY Z states, such as Y (4660). . S Godfrey, Isgure , Phys. Rev. D. 32189S. Godfrey and Isgure, Phys. Rev. D 32, 189 (1985). . K A Olive, Particle Data GroupChin. Phys. C. 3890001K. A. Olive et al. (Particle Data Group), Chin. Phys. C 38, 090001 (2014). . M Ablikim, BESIII CollaborationPhys. Rev. Lett. 110252001M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett.110, 252001 (2013). . Z Q Liu, Belle CollaborationPhys. Rev. Lett. 110252002Z. Q. Liu et al. (Belle Collaboration), Phys. Rev. Lett. 110, 252002 (2013). . T Xiao, S Dobbs, A Tomaradze, K K Seth, Phys. Lett. B. 727366T. Xiao, S. Dobbs, A. Tomaradze and K. K. Seth, Phys. Lett. B 727, 366 (2013). . X Liu, Chin. Sci. Bull. 593815X. Liu, Chin. Sci. Bull. 59, 3815 (2014). . F K Guo, C Hidalgo-Duque, J Nieves, M Pavon Valderrama, Phys. Rev. D. 8854007F. K. Guo, C. Hidalgo-Duque, J. Nieves, and M. Pavon Valderrama, Phys. Rev. D 88, 054007 (2013). . M Ablikim, BESIII CollaborationPhys. Rev. Lett. 115112003M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett.115, 112003 (2015). . M Ablikim, BESIII CollaborationPhys. Rev. Lett. 111242001M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 111, 242001 (2013). . M Ablikim, BESIII CollaborationPhys. Rev. Lett. 113212002M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 113, 212002 (2014). . M Ablikim, BESIII CollaborationPhys. Rev. Lett. 112132001M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 112, 132001 (2014). . M Ablikim, BESIII CollaborationPhys. Rev. Lett. 115182002M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 115, 182002 (2015). . M Ablikim, BESIII CollaborationPhys. Rev. Lett. 11222001M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett.112, 022001 (2014). . M Ablikim, BESIII CollaborationPhys. Rev. D. 9292006M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 92, 092006 (2015). . M Ablikim, BESIII CollaborationPhys. Rev. Lett. 115222002M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett.115, 222002 (2015). . M Ablikim, BESIII CollaborationPhys. Rev. Lett. 11972001M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 119, 072001 (2017). . S L Zhu, Phys. Lett. B. 625212S. L. Zhu, Phys. Lett. B 625, 212 (2005); . F E Close, P R Page, Phys. Lett. B. 628215F. E. Close and P. R. Page, Phys. Lett. B 628, 215 (2005); . E Kou, O Pene, Phys. Lett. B. 631164E. Kou and O. Pene, Phys. Lett. B 631, 164 (2005). . L Maiani, V Riquer, F Piccinini, A D Polosa, Phys. Rev. D. 7231502L. Maiani, V. Riquer, F. Piccinini, and A. D. Polosa, Phys. Rev. D 72, 031502(R) (2005). . X Liu, X Q Zeng, X Q Li, Phys. Rev. D. 7254023X. Liu, X. Q. Zeng, and X. Q. Li, Phys. Rev. D 72, 054023(R) (2005); . W Qin, S R Xue, Q Zhao, Phys. Rev. D. 9454035W. Qin, S. R. Xue, and Q. Zhao, Phys. Rev. D 94, 054035 (2016). . M Ablikim, BESIII CollaborationPhys. Rev. Lett. 11892001M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 118, 092001 (2017). . M Ablikim, BESIII CollaborationPhys. Rev. Lett. 11892002M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 118, 092002 (2017). . B Aubert, BABAR CollaborationPhys. Rev. Lett. 98212001B. Aubert et al. (BABAR Collaboration), Phys. Rev. Lett.98, 212001 (2007); . Phys. Rev. D. 89111103Phys. Rev. D 89, 111103 (2014). . X L Wang, Belle CollaborationPhys. Rev. Lett. 99142002X. L. Wang et al. (Belle Collaboration), Phys. Rev. Lett. 99, 142002 (2007); . Phys. Rev. D. 91112007Phys. Rev. D 91, 112007 (2015). . M Ablikim, BESIII CollaborationPhys. Rev. D. 9632004M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 96, 032004 (2017). . M Ablikim, BESIII CollaborationPhys. Rev. D. 9752001M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 97, 052001 (2018). . M Ablikim, BESIII CollaborationPhys. Rev. D. 9771101M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 97, 071101 (2018). . M Ablikim, BESIII CollaborationarXiv:1808.02847M. Ablikim et al. (BESIII Collaboration), arXiv:1808.02847. . M Ablikim, BESIII CollaborationPhys. Rev. Lett. 11492003M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 114, 092003 (2015). . M Cleven, Q Wang, F K Guo, Phys. Rev. D. 9074039M. Cleven, Q. Wang, F. K. Guo et al., Phys. Rev. D 90, 074039 (2014). . M Ablikim, BESIII CollaborationPhys. Rev. D. 9732008M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 97, 032008 (2018).
{'fraction_non_alphanumeric': 0.09728147218736931, 'fraction_numerical': 0.0916771225428691, 'mean_word_length': 3.2761087267525038, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 1, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 25, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'With about 12 fb −1 of collected data useful for the study of XY Z states, BESIII collaboration continues the exploration of these exotic charmoniumlike states. Recent results of the measurements of the line-shape of e + e − → π 0 π 0 ψ(3686), KKJ/ψ, and π + D 0 D * − , as well as the J P determination of Z c (3900) and Z c (3900) observed in e + e − → φ χ c1,2 at √ s = 4.6 GeV will be presented. * Speaker on the behalf of BESIII Collaboration.', 'arxivid': '1810.04911', 'author': ['Bin Wang [email protected] \nInstitute of High Energy Physics\n100049BeijingChina\n', 'Bin Wang \nInstitute of High Energy Physics\n100049BeijingChina\n'], 'authoraffiliation': ['Institute of High Energy Physics\n100049BeijingChina', 'Institute of High Energy Physics\n100049BeijingChina'], 'corpusid': 119453999, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 5026, 'n_tokens_neox': 3909, 'n_words': 2121, 'pdfsha': 'a6073db77048ed2969b4fb232079aa91533b7393', 'pdfurls': ['https://arxiv.org/pdf/1810.04911v2.pdf'], 'title': ['Studies of XY Z states at BESIII Studies of XY Z states at BESIII', 'Studies of XY Z states at BESIII Studies of XY Z states at BESIII'], 'venue': ['Talk given at XXXIX International Conference on High Energy Physics (ICHEP2018)']}
arxiv
Language Model Crossover: Variation through Few-Shot Prompting Elliot Meyerson [email protected] Institute of Technology Institute of Technology Cognizant AI Labs American University University of Cambridge & CarperAI Jersey, Jersey Mark J Nelson [email protected] Institute of Technology Institute of Technology Cognizant AI Labs American University University of Cambridge & CarperAI Jersey, Jersey Herbie Bradley Institute of Technology Institute of Technology Cognizant AI Labs American University University of Cambridge & CarperAI Jersey, Jersey Arash Moradi Institute of Technology Institute of Technology Cognizant AI Labs American University University of Cambridge & CarperAI Jersey, Jersey Amy K Hoover [email protected] Institute of Technology Institute of Technology Cognizant AI Labs American University University of Cambridge & CarperAI Jersey, Jersey Joel Lehman Carperai Institute of Technology Institute of Technology Cognizant AI Labs American University University of Cambridge & CarperAI Jersey, Jersey Language Model Crossover: Variation through Few-Shot Prompting This paper pursues the insight that language models naturally enable an intelligent variation operator similar in spirit to evolutionary crossover. In particular, language models of sufficient scale demonstrate in-context learning, i.e. they can learn from associations between a small number of input patterns to generate outputs incorporating such associations (also called few-shot prompting). This ability can be leveraged to form a simple but powerful variation operator, i.e. to prompt a language model with a few text-based genotypes (such as code, plain-text sentences, or equations), and to parse its corresponding output as those genotypes' offspring. The promise of such language model crossover (which is simple to implement and can leverage many different open-source language models) is that it enables a simple mechanism to evolve semantically-rich text representations (with few domain-specific tweaks), and naturally benefits from current progress in language models. Experiments in this paper highlight the versatility of language-model crossover, through evolving binary bit-strings, sentences, equations, text-toimage prompts, and Python code. The conclusion is that language model crossover is a promising method for evolving genomes representable as text.CCS CONCEPTS• Computing methodologies → Neural networks. . New candidate solutions are generated by concatenating parents into a prompt, feeding the prompt through a large pre-trained language model (LM), and collecting offspring from the output. The enormity and breadth of the dataset on which the LM was trained, along with its ability to perform in-context learning, enables LMX to generate high-quality offspring across a broad range of domains. Domains demonstrated in this paper include (a) binary strings, (b) mathematical expressions, (c) English sentences, (d) image generation prompts, and (e) Python code; many more are possible. When integrated into an optimization loop, LMX serves as a general and effective engine of text-representation evolution. INTRODUCTION Large language models (LMs; [6,9]) have achieved impressive results in many natural language domains, such as question-answering [14,23,49], code-generation [12,48], and few-shot classification [9,68]. One popular type of LM is trained on corpora of humanauthored text to predict the next token from previous ones, i.e. auto-regressive LMs (e.g. , which at their core model a distribution of likely output sequences given an input sequence or prompt. In zero-shot learning, an LM generates an output response from a single input sequence. However, another popular prompting paradigm is few-shot prompting [9], wherein the input to a LM is a few examples of desired input-output behavior (e.g. how to classify a sentence's sentiment) preceding a new target input that the model is to classify. To some extent such LMs can meta-learn from a few natural-language examples how to perform a desired task [11,83]. One reason this ability is exciting because it highlights how LMs can in effect be seen as very powerful pattern-completion engines. Few-shot prompting works because the LM can "guess the pattern" behind a few input/output pairs and generalize its behavior to a new target input (provided at the end of the few-shot prompt). The central insight of this paper is is that, interestingly, the patterncompletion ability of few-shot prompting can be leveraged to create a form of intelligent evolutionary crossover. arXiv:2302.12170v1 [cs.NE] 23 Feb 2023 For example, if three text-based genotypes are drawn from a population and concatenated into a prompt, an ideal pattern-completion engine would analyze their commonalities and generate a new (fourth) genotype that qualitatively follows from the same distribution. In effect such an operator would combine aspects of the input genotypes, and indeed, an experiment in section 4.1 demonstrates empirically that LMs enable this with binary strings. Theoretically we also connect this form of LM crossover (LMX) to estimation of distribution algorithms (EDAs; [2,44]), wherein LMX can be seen as building an implicit probabilistic model of the input parent genotypes from which to sample a new offspring, through a single forward pass of the LM. From the perspective of pattern-completion, this operator should naturally improve as LMs increase in capabilities (which experiments here validate); furthermore, to increase performance the method can easily leverage the rise of open-source domain-specific LMs that match a target domain (e.g. LMs that focus on code, when the target domain is to evolve code), often with changing only a single line of code to rely on a different hosted model (e.g. through the HuggingFace model repository [90]). The benefit of LMX is that evolution can easily be conducted in the semantic representation of text, without having to design specific domain-specific variation operators. LMX's versatility is highlighted in experiments with binary strings, style transfer of plaintext sentences, symbolic regression of mathematical expressions, generating images through prompts for a text-to-image model, and generating Python code. The results highlight the potential of the method to produce quality results across domains, often by leveraging the broad ecosystem of pretrained models that can be easily combined in many ways to quantify fitness or diversity, or to cross modalities (i.e. from text to image). Interestingly, LMX may also synergize with recent LM-based mutation techniques [46], and is amenable to similar possibilities such as fine-tuning an LM as a way of accelerating search, although we leave these possibilities for future work. In short, the main contributions of this paper are to introduce LMX, explore its basic properties, and highlight its versatility through testing it in a variety of domains. We will release an implementation of LMX and code to recreate the main experiments of the paper. BACKGROUND 2.1 Foundation Models A recent paradigm in ML is to train increasingly large models on internet-scale data, e.g. BERT and GPT-3 on text [9,22], or DALL-E and stable diffusion on captioned images [63,65]. Such models are sometimes called foundation models [6], as they provide a broad foundation from which they can be specialized to many specific domains (e.g. with supervised fine-tuning or prompt-engineering). Interestingly, such foundation models have enabled a large ecosystem of specialized models [84] that can be combined in a plug-and-play way (e.g. models that measure sentiment of text [10], summarize text [76], write code [56], rank the aesthetics of images [20,40,72], and create high-dimensional embeddings of text or images [64,92]. One contribution of this paper is to demonstrate how evolutionary methods can easily leverage this growing eco-system to evolve high-quality artifacts in diverse applications. One particularly exciting class of foundation models are pretrained language models (LMs) that model the distribution of text. While early LMs used markov chains [74] or recurrent neural networks [25], more recently the transformer architecture [82] has enabled significant progress in NLP. The method in this paper focuses on one emergent capability of large transformer-based LMs, i.e. the potential to meta-learn from text examples provided as input to the model when generating an output, which is called in-context learning or few-shot prompting [9,83]. Importantly, performance at in-context learning improves with model scale [11,88], implying that methods relying upon this capability will benefit from continuing progress in LM training. This paper highlights how the in-context learning capabilities of autoregressive LMs (such as the popular GPT architecture) naturally enable an intelligent recombination operator. The next section reviews existing methods for intelligent variation in EC. Intelligent Variation Operators Populations in evolutionary algorithms (EAs) generally evolve through high performing candidates solutions being mutated or recombined to form the next generation. Such variation is critical as a primary driver of both exploration and exploitation of the search space [17]. Traditional mutation and recombination operators (such as one-point crossover or bit-flip mutation) do not explicitly seek to model and exploit regularities among high-fitness individuals (or do so in an implicit way [31]), which can cause EAs to be relatively sample-inefficient in some situations when compared to statistical methods [80]. To address this limitation, strategies for generating intelligent variation have been a focus of much research in EC. For example, evolving within a latent space of an ML model [24,71], or through training models to mimic mutations [37,46]. One particularly popular such strategy is to build probabilistic models of high-performing individuals or to model elements of the search path taken across recent generations. For example, estimation of distribution algorithms (EDA; [2,44]), covariance matrix adaptation evolution strategy (CMA-ES; [28]), and natural evolution strategies (NES; [89]) build and sample candidate solutions from an explicit probability distribution. While EDAs estimate the distribution of the solutions that have been sampled, CMA-ES additionally estimates the steps of the search direction. The LMX operator in this paper can be seen similarly as building a probabilistic model of individuals (here of parents, rather than the whole population), and doing so implicitly in the forward-pass of the LM (through in-context learning). One recent exciting direction for generating variation is to leverage the semantic knowledge of pretrained LMs. This work builds on previous results that highlight the potential of LMs to generate [69] or augment [26] data, although here the focus is on enabling continual evolution. It also is closely related to work demonstrating that LMs can be trained to embody intelligent mutation operators for code [46], i.e. by training a model on changes to code files gathered from GitHub; Lehman et al. [46] also proposed a way to generate domain-specific mutations from hand-designed prompts for LMs. Such operators may require fine-tuning a model on mutation-like training data gathered from GitHub or hand-specifying example mutations, and have been applied only to the domain of code. Instead, the mechanism exploited here focuses on recombination rather than mutation, is domain-independent (as shown by the diversity of domains targeted by this paper), and is easily implemented with open-source LMs (we experiment with several). APPROACH: LANGUAGE MODEL CROSSOVER (LMX) The approach in this paper builds from the insight that the objective function used to train many self-supervised LMs, i.e. next-token prediction [9], naturally lends itself to creating an evolutionary variation operator, from which evolutionary algorithms that represent genomes as text can be derived. The reason is that such an objective entails anticipating what comes next from some limited input context, and if that input consists of a few example genotypes, then the ideal anticipation is to continue that pattern, i.e. through suggesting a new genotype from the distribution implied by those examples. In other words, LMs trained by next-token prediction can be seen as learning to become general pattern-completion engines. From this lens, as higher-performing LMs (i.e. those with lower prediction loss on a held-out set) are continually developed, their performance as engines of evolutionary variation should continue to improve. Supporting this idea, when trained over a large amount of diverse examples, LMs demonstrate an increasing capability for in-context learning (i.e. inferring novel associations within the input given at test-time when generating completions) [9,11,88]. What is intriguing about this insight is that the variation operator it suggests is (1) simple to implement (i.e. concatenate a few text-based genotypes into a prompt, run it through a LM, and extract a new genotype from its output; we release code implementing it accompanying this paper), (2) relatively domain-independent (i.e. in theory it should be capable of generating meaningful variation for any text representation that has moderate support in the training set, which often encompasses a crawl of the internet), and (3) should suggest increasingly semantically-sophisticated variation with more capable LMs (i.e. reduced test-loss of LMs should correlate with the ability to exploit more subtle input patterns). The experiments that follow add supporting evidence to these claims. Figure 1 shows from a high level how LMX enables creating a domain-independent evolutionary algorithm for text representations. The basic idea is that given a set of a few text-based genotypes (or bootstrapping from a single genotype using prompt-based mutation [46]), an initial population can be generated through LMX. Then, a standard evolutionary loop can be instantiated by repeated selection and generation of new variation through LMX. In the experiments that follow, we use simple genetic algorithms (GAs; although one experiment instantiates a simple quality diversity algorithm). In theory, however, LMX can be generically applied to most EAs, e.g. multi-objective EAs [15,18], evolutionary strategies [1,3], or in support of open-ended evolution [85]. How or if LMX can be applied to EAs that explicitly leverage probabilistic models of genotypes (e.g. EDAs [2,44], natural evolution strategies [89], or CMA-ES [27,28]) is an interesting question for future research, although LMX does bear a strong theoretical relationship to EDA algorithms in particular, explored next. Connection to EDAs An EDA constructs an explicit probabilistic distribution fit to the parent set { 1 , . . . , }, and samples child solutions from [30,45]. In contrast, a standard GA generates children by sampling from an implicit conditional probability distribution ( | [ 1 , . . . , ]) induced by the process of randomly sampling parents and applying a stochastic reproduction operator (e.g., a crossover operator). LMX occupies an intermediate level of explicitness: The conditional distribution induced by feeding the parent prompt into the LM is explicit in that it yields a series of probability distributions over tokens, but is implicit in the sense that the internal workings of the distribution are encoded opaquely within the millions or billions of parameters and activations of the LM for a given prompt. Whatever the level of explicitness, the reason LMX can be viewed as an EDA is that it be seen as constructing a distribution of parents, from which children are sampled. The key design feature of an EDA is the class of distributions D to which belongs. This class D can range from simple univariate distributions [2,29] to more complex models like Bayesian networks [59,60]. What is the class D LM from which LMX constructs parent distributions? Due to its in-context learning capabilities [66,91], the LM can be seen as attempting to infer the generating distribution of the prompt, and to generate continuations accordingly. By concatenating parents in a random order, the implicit signal to the LM is that the list is unordered (i.e. there is little to be inferred from most arbitrary randomly-ordered patterns). These objects must have been sampled from some distribution , and thus the LM's optimal move is to keep sampling objects from as it generates output. In other words LM consists of distributions of objects that are found in sets that might appear in the universe of data from which the dataset used to train the LM was drawn. An ideal EDA would select the most probable = EDA ∈ D LM based on the parent set { 1 , . . . , }. E.g., EDA = argmax ∈D LM ( ) =1 ( | ),(1) where ( ) is the prior probability of in D LM . As the LM becomes a better and better in-context learner, it becomes better able to detect subtler patterns within a prompt of randomly-ordered concatenated parents, and thus LMX ( | [ 1 , . . . , ]) ≈ ( | EDA ).(2) Note that the left side depends on an ordered list of parents, while the right side has removed this dependency on order. This approximation becomes tight as the LM approaches perfect in-context learning, at which point LMX can be viewed exactly as an EDA, i.e., LMX([ 1 , . . . , ]) ∼ EDA .(3) Although a faithful application of an EDA may include the full parent population in each parent prompt, the experiments in this paper save compute by sampling a only a small number of parents. Not only are EDAs an efficient type of EA [59,60], but by comparing LMX to EDAs it may be possible to analyze the optimization behavior of LMX [41] (e.g., global convergence analysis [93]). Overall, the connection to EDAs may help to explain why LMX is effective as an off-the-shelf genetic operator across a wide range of domains; Figure 2: The effect on LMX from varying the number of parents. As the number of parent genotypes input into the LM is increased, the percent of valid offspring approaches 100%. The number of novel genotypes generated on average from 20 applications of LMX to a random set of parents reaches its maximum at four parents (while five parents tends to more often produce offspring that duplicate one of the parents exactly). The conclusion is that LMX effectively generates variation from as few as three input genotypes. a further interesting theoretical property of LMX is its universality (described in appendix A). EXPERIMENTS This section demonstrates the application of LMX to five domains. Source code will be made available for each domain. Illustrative Example: Binary Strings As an instructive example to explore the properties of LMX, in this section this operator is applied to generate variation in the space of binary strings (e.g. composed of text strings such as "011000"). A first question is whether a pretrained LM (here an 800-million parameter Pythia model [4]), given only a few examples of such genomes, can generate meaningful variation (i.e. without any hardcoded knowledge about the representation). To explore this question, a prompt is generated by concatenating randomly chosen length-6 binary strings separated by newlines; the LM's response (truncated after three new lines) is interpreted as three offspring individuals. Figure 2 shows how often such a prompt will generate valid individuals (i.e. strings of length six composed of 1s and 0s) as a function of number of examples in the prompt, and how many novel offspring (i.e. the size of the set of individuals generated that are distinct from the parents) are generated on average from 20 trials of LMX crossover on the same set of parents (averaged across 20 randomly-sampled parent sets). A follow-up experiment, with length-9 binary strings, demonstrates how LMX in this domain improves with larger LMs (details in appendix B.1). The conclusion is that indeed, LMX can reliably generate novel, valid offspring (from as few as three examples). A second question is whether LMX can create heritable variation. Evolution requires there to be meaningful information transmitted from parents to offspring. One way to explore this is to measure whether a prompt composed of highly-related binary strings produces novel but nearby offspring (e.g. as measured by edit distance). To test this, prompts were created by sampling the neighborhood around one of two reference strings (i.e. single-step mutations from either the all-ones or all-zeros string), and offspring were generated from the LM. Indeed, offspring generated from the neighborhood of the all-ones string had significantly higher (Mann-Whitney U-test; < 0.001) hamming distance from the all-zeros string than the all-ones string (and vice-versa; see appendix figure 7). A final instructive question is whether an evolutionary process can be successfully driven by LMX. To explore this, we test LMX in OneMax, i.e. evolving the all-1s string. A small population (30 individuals) is initialized from the 0-neighborhood of the all-0s string, and fitness is measured by how many 1s are present in each length-6 string. Indeed, across 20 independent runs, search nearly always quickly converges when driven by this fitness function, whereas search driven by a random fitness function does not (see fitness curves in appendix Figure 8); further, LMX-driven search finds perfect solutions more often (19 out of 20 runs) than with random search (Fisher's exact test; < 0.05). Overall, these experiments highlight basic properties of LMX, showing how it can evolve string-based representation without domain-specific operators. Symbolic Regression To demonstrate LMX's potential in a more challenging task, this section applies the algorithm to symbolic regression, a key domain of interest for genetic programming [43,51,58,70], and more recently the larger machine learning community [5,36,42,61]. The goal of symbolic regression is to discover a mathematical expression that models a data set accurately, while also being as compact as possible [42]. Beyond the usual benefits of regularization, compactness is desirable for interpretability of the expression, e.g., to enable scientific insights [34,70,81,86]. Symbolic regression is challenging to tackle with hand-designed operators, due to non-locality and discontinuities in the space of expressions. Existing symbolic regression approaches use carefullydeveloped representations, genetic operators, and auxiliary methods like gradient-based/convex coefficient optimization [13,39,79] to construct the right kind of search process for reaching highperforming expressions that look like the kinds of expressions the experimenter is interested in. With LMX, these challenges can be avoided by simply feeding parent expressions into the language model. Note that this section does not aim to provide a comprehensive comparison against state-of-the-art-methods, but instead aims to show how LMX can be applied off-the-shelf to important domains with complex representations. Experimental Setup. The LM for this experiment was the 1.3Bparameter version of GALACTICA [78]. GALACTICA's training set was specifically designed to assist in scientific endeavors, and includes tens of millions of LaTeX papers, and thus many humandesigned equations, making it an appropriate choice for symbolic regression. This choice also highlights how different off-the-shelf LMs can be selected for LMX based on properties of the problem. When the ground truth expression for symbolic regression is known, we run the risk that the expression is already in the dataset used to train the LM. To avoid such test-set contamination, we consider a 'black-box' problem (which has no known ground-truth expression) from the established SRBench testbed [42]. The 'banana' problem was chosen because there is a clear Pareto front across existing methods, to easily compare how LMX performs. This blackbox problem was originally derived from a popular ML benchmark in the KEEL data set repository [21]; it has 5300 samples and two features 1 , 2 . In this experiment, crossover prompts began with the string "Below are 10 expressions that approximate the dataset:" followed by a newline character and seven randomly selected parents from the population separated by newlines (see appendix Figure 10 for examples). Each subsequent line generated by the model was interpreted as a possible offspring, interpreted as Python code, and simplified using sympy (as in the SRBench comparisons [42]). Up to three child expressions were accepted for each forward pass of the LM. Each child was evaluated against the dataset, using 2 for fitness; any child that could not be parsed or that raised an exception during evaluation was discarded. The same compactness/complexity measure was used as in SRBench, i.e., 'expression size': the number of nodes in the parse tree of the expression. The initial population was constructed from 113 popular symbolic regression benchmarks 1 . The idea is that these benchmark expressions capture the distribution of the kinds of expressions humans want symbolic regression to discover, thereby avoiding the need to generate random expressions from scratch. To give each benchmark expression a greater chance of initial success, the initial population consisted of 1000 candidates, each generated by randomly selecting a benchmark expression and then randomly mapping its input variables ′ 1 , ′ 2 , . . . to the input variables 1 , 2 in the test problem. Thereafter, the population size was set to 50. Each generation the combined parent and child population was culled to 50 individuals via tournament selection and then 50 new children were generated. The algorithm was run for 5000 generations using a single GeForce RTX 2080 Ti GPU (which roughly took 100 hours). Results . LMX produces competitive results, generating fit and parsimonous expressions. Figure 3a shows how fitness evolves over generations for one run of LMX, and the expression with the highest fitness so far is plotted at several generations to illustrate the kinds of improvements evolution finds. Figure 3b shows the test trajectory plotted across the objectives of expression size and fitness, i.e., 2 score; LMX incrementally improves fitness across evolution to levels competitive with state-of-the-art methods [42]. Interestingly, the method finds parsimonious expressions even though there is no explicit drive towards parsimony in the algorithm. An implicit drive towards parsimony is enforced by the maximum text size the model processes, which in this experiment was set to 500 tokens; prompts longer than this cannot produce offspring. Future work could investigate the effects of tuning this parameter or developing other methods for incorporating explicit drives towards parsimony. Intriguingly, the method tunes constants to a surprising degree, indicating that LMX is capable of continuous optimization, even though LMs operate in a space of discrete tokens; this is an interesting ability that can be further explored in future work. In conclusion, the quality of the final expression is comparable to previously published results from state-of-the-art SR methods (same train/test split) [42], but, unlike these other methods, which carefully consider model representations, genetic operators, distributions of synthetic functions, bloat, multiple objectives, etc., we simply ask an off-the-shelf language model to be the generator in a minimal evolutionary loop. Note that the comparison here is not apples-to-apples, since the comparison methods all used a fixed amount of CPU compute, while this experiment used a GPU. However, the results clearly show the ability of the model, with little domain-specific tuning and an unsophisticated optimization loop, to nonetheless optimize symbolic expressions in an intuitive and desirable way. Modifying Sentence Sentiment LMX is next applied to evolve plain-text English sentences. While LMX could be applied in many ways to evolve sentences, the focus here is a form of natural language style transfer [33], i.e. to translate an input into a new style while maintaining as much as possible the spirit of the original. In particular, the task is to take a seed sentence, and maximally change its sentiment (i.e. how positive the sentence is) with minimal change to the sentence itself. To do so, a simple quality-diversity evolutionary algorithm [47,54] is applied that measures quality as maximizing the sentiment of a sentence and measures diversity as distance from the seed sentence. In particular, sentiment is measured through the "cardiffnlp/ twitter-roberta-base-sentiment-latest" model hosted on Hugging-Face, which is part of the TweetNLP project [10]; the network takes in a sentence, and outputs classification probabilities for whether the sentence is positive, negative, or neutral. The experiments focus on using the probability of a positive sentiment as the fitness function (although see appendix D for results with negative sentiment as fitness). For measuring distance from the seed sentence, a separate neural network generates a 384-dimensional embedding of a sentence (in particular the "sentence-transformers/all-MiniLM-L6-v2" model, from the sentence transformer project [64]). Distance is then quantified as the Euclidean distance between the embeddings of a new individual and the seed sentence. For the QD algorithm, we use MAP-Elites [54] with a 1D map (with 30 niches, spanning a distance of 0 to a distance of 1.5 from the seed sentence in the embedding space). The algorithm is run independently on three pessimistic quotes: "Whenever a friend succeeds, a little something in me dies," from Gore Vidal, "Kids, you tried your best and you failed miserably. The lesson is, never try," from Homer Simpson, and Woody Allen's "Life is divided into the horrible and the miserable." Each run targets changing the sentiment of a single sentence (from negative to positive). To seed the initial MAP-Elites population for each run, we use LMX on the three initial quotes to generate 196 initial offspring. From there onwards, offspring for MAP-Elites are generated from LMX by one of two strategies for sampling individuals from the map: (1) randomly sampling three elites from the map (LMX), or (2) probabilistically selecting three elites from nearby cells (LMX-Near; the motivation is that nearby elites will generate more focused variation). MAP-Elites runs consist of 2500 evaluations each; a baseline control is also tested that generates 2500 offspring only [42]. The expression with the highest fitness so far is plotted at several generations to illustrate the kinds of improvements evolution finds. Evolution settles on a core functional skeleton relatively quickly (i.e., 1 − 2 3 1 − 4 5 2 cos( 1 + 6 2 + 7 ), with 1 , 2 input variables and constants), after which it tunes constants to a surprising specificity, while simultaneously tweaking and augmenting the skeleton. Even after the process appears to have converged, around generation 3000 it discovers innovations leading to further substantial improvements. This late boost highlights the ability of the LM to be an engine of interesting and valuable hypotheses in mathematical/numerical spaces. (b) The test trajectory plotted across the objectives of expression size and fitness, i.e., 2 score. The trajectory reflects desirable properties of an effective symbolic regression method: It incrementally improves the 2 score while avoiding excessive model bloat, comparable to previously published results from state-of-the-art SR methods (with the same train/test split) [42]. The conclusion is that LMX is a promising approach for symbolic regression. from the initial 3 seed sentences. 10 runs were conducted for each combination of sentence and method; each run took on the order of minutes on a Google Colab notebook. Quantitatively, both LMX-Near and LMX achieved higher QD scores than the control for all three quotes (Mann-Whiteny U-test; < 1 − 5), and were always able to discover high-sentiment sentences. Interestingly, LMX-Near and LMX performed significantly differently only for the Gore Vidal quote (LMX-Near produced higher final QD-scores; Mann-Whitney U-test; < 0.05). Future work is thus needed to determine whether there exist methods for robustly choosing parents for LMX more effectively. Fitness plots for each quote is shown in appendix D. Qualitatively, evolution is generally able to find intuitive tradeoffs between sentiment and distance from the original sentence. For example, Figure 4 shows the the final map from a representative run on the Homer Simpson quote (with LMX-Near), with some highlighted sentences. At sufficient distance from the original sentence, evolution often produces repetitive, unrelated text: e.g. "You are the best that ever happened to me! You are the best that ever happened to me! You are the best that ever happened to me!" Also, sometimes the method produces incoherent or grammatically-flawed sentences, e.g. "you tried your best and you failed. The lesson is, you can never stop trying. Kids, you tried your best and you". Optimization pressure for coherence (i.e. to maintain high log-probability under a LM), or better/larger sentiment models, might address these problems. The conclusion is that LMX is a promising approach for text style transfer tasks; other styles could be explored by using different NLP models as fitness functions, e.g. emotion-recognition NLP models [55]. Evolving Stable Diffusion Images Stable Diffusion 2 is a publicly available latent diffusion model [65] that supports CLIP-guided [62] text-to-image synthesis. Since Stable Diffusion's release, folk practices for prompting it have developed as artists, researchers, and hobbyists swap tips for constructing text prompts to produce desired outputs [57]. The research question here is whether LMX can also evolve Stable Diffusion prompts. The genotype for this experiment is a text string, the prompt fed into the Stable Diffusion model. The initial population is seeded by randomly choosing from a set of 80,000 Stable Diffusion prompts that were scraped from lexica.art. 3 The phenotype is the image generated by feeding a given prompt to Stable Diffusion. We make Stable Diffusion deterministic by reseeding with a fixed PRNG seed before each image is generated, so a given prompt always produces the same image. The EA is the same as in Section 4.2; experimental details are in Appendix E. Three fitness functions are explored, maximizing respectively the "redness", "greenness" and "blueness" of an image. Redness is measured by excess red: the sum of the red channel of an RGB image, minus half the sum of the other two channels ( − 0.5 − 0.5 ). (c) Blue: "Anime Snow Queen, artworks for sale, digital art prints, drawings for sale, drawing for sale, digital art prints, digital artprints, digital art prints, digital artprints, digital art prints, digital art prints, digital comics, digital manga". Excess green and excess blue are defined analogously. Although simple, these functions are easy to calculate, and correspond roughly to perceived image color (e.g., they are well studied in agricultural image processing [52]). Future work would aim to evolve prompts that maximize aesthetically oriented fitness functions [19,35,72], e.g. pre-trained neural networks that evaluate aesthetics [72], but these simple fitness functions provide a proof of concept and enable LMX's progress to be visually verified at a glance. 4 Appendix Figure 14 shows the maximum fitness per generation over a single run for each of the three fitness functions. The highestfitness prompts and images themselves are shown in Figure 5. The images generally match the desired color, and evolved prompts often contain themes or colors associated with the color (e.g. "in a green rock on a green planet in a forest" for the green fitness function), but the population converges prematurely: by around 30 generations, the entire population consists of LMX-generated remixes of essentially the same prompt. This suggests that like other EAs, LMX may often need to be combined with techniques for maintaining population diversity to reach its potential. The conclusion is that LMX can enable sensible evolution of images. LMX with Python Sodaracers Finally, to explore whether LMX can generate variation in Python code we apply LMX to the Sodarace environment from Lehman et al. [46], which also explored evolving Python programs with LMs (we leverage the Open ELM implementation of sodarace [8]). Sodarace is a 2D simulation of robots with arbitrary morphology constructed from Python functions (the genotype) which output a dictionary specifying joints and muscles, and how they are connected. A Sodaracer robot is instantiated from this dictionary and placed in the environment, and the distance travelled is used as our fitness function. The experiment here does not evolve Sodaracers, but instead applies LMX to a fixed set of Sodarace programs taken from Lehman et al. [46], as a preliminary exploration of whether it can generate useful variation in a coding domain (similar to the first experiments with binary strings). Seven pre-existing Sodarace programs were chosen (details in appendix F), and LMX was prompted across combinations of one, two, or three of these as parents. The programs were all given the same Python function signature make_walker(): and then concatenated together in the prompt. Note that we begin each completion with the same function signature to improve performance (experiments where the LM prompt did not end with the function signature performed worse; see appendix F). The LM output is then interpreted as a potential offspring, to be evaluated in the Sodarace environment. We experiment with three different-sized LMs from the Salesforce CodeGen suite [56], a set of models trained on a large dataset of code in many languages, including Python. To evaluate LMX's output, we measure the performance of offspring using a MAP-Elites map [54], using the distance travelled by the generated Sodaracers in a simulation as the fitness and the morpology of the Sodaracer (height, width, and mass) as the dimensions of the behavior space (as in Lehman et al. [46]). For each treatment, we generate 1000 candidate programs with the LM, insert the resulting valid Sodaracers into a map, and measure the resulting amount of niches filled and QD scores. We repeat this procedure for every possible permutation of the seeds being considered (with a new map for each) to control prompt order variance, and average our results (quality-diversity score, number of niches filled, and valid offspring rate) across the permutations for each set of seed programs. The results from these experiments are shown in Figure 6, showingthat as the number of parents in the prompt increases, the diversity of offspring generally increases, as measured by the number of niches filled and the QD score (This effect is even more dramatic with a more generic prompt: A single parent yields no valid offspring (appendix Figure 15)). Furthermore, a significant proportion of generated offspring are valid sodaracers (approaching 35% with the 6B model), highlighting the potential for evolution. Experiments with 1 seed in the prompt can be viewed as a simple mutation operator (a different approach to the same end in Lehman et al. [46]). Interestingly, there is not a clear trend for model size, except in increasing the proportion of valid programs. These results therefore demonstrate the promise of LMX to evolve non-trivial Python code, which will be validated in full evolution of code in future work. DISCUSSION AND CONCLUSIONS As a flexible and easy-to-use genetic operator, LMX provides a way for EA practitioners to take advantage of the recent revolution in large neural models. The experiments tackle a wide range of potential applications, across equations, plain-text sentences, images, and code, leveraging the wide ecosystem of open-source neural networks as means of generating variation, crossing modalities, and measuring both fitness and diversity. There is much room for future work. The experiments focused on breadth rather than depth, and it is possible that with further effort LMX could enable state-of-the-art results in e.g. symbolic regression. An interesting question is whether examples fed into LMX could be chosen more deliberately (e.g. only crossing-over similar individuals to get more nuanced variation); preliminary experiments showed some qualitative effect from applying LMX on individuals with similar embeddings, but require further experimentation to validate. One natural future direction is to explore whether there is benefit from combining the recombination capability of LMX with the mutation operators (either prompt-based or diff-model-based) explored in ELM [46]. Of further interest is the possibility for self-improvement of LMX (as in ELM), through fine-tuning the model on successful examples of variation in a domain. A final intriguing possibility is the use of LMX for interactive evolution, e.g. to interactively evolve sentences. code, or images [7,73]. While LMs are computationally expensive, all of the experiments in this paper (with exception of the Python experiment) were conducted either through Google Colab notebooks or on a single GPU; the code to run experiments is surprisingly compact, as the LMX method consists mainly of a simple LM prompting strategy, and interacting with language and image models has become simple through APIs such as that provided by HuggingFace. In conclusion, there are likely many creative ways to beneficially combine various models together that this paper leaves unexplored; evolution in general is a powerful and easy-to-implement way to quickly explore such possibilities, and LMX in particular is a promising and simple way of instantiating them. A UNIVERSALITY OF LMX Section 3.1 highlighted the connection between LMX and EDAs. This section explores another property of LMX, its theoretical universality (i.e. its ability in theory to express any genetic operator). Interestingly, with a sufficiently expressive class of model, such as Bayesian networks [59,60], EDAs can approximate any candidate distribution as size of the parent set increases [93]. Not only can LMX sample from distributions represented by an EDA, but it can in principle sample from any conditional probability distribution, making it universal in the space of genetic operators, even with small parent sets. Recent theoretical work has shown how crossover of large neural networks can yield universal approximation of reproduction distributions [53]. LMX also achieves theoretical universal approximation via large neural networks, but by feeding parents directly into the LM, instead of crossing-over weights. This result follows directly from the universal approximation ability of NNs [16,32,38] (note that this property also applies in the single-parent case for mutation-based evolution through LMs [46]). This property suggests that the power of LMX is not limited to the randomlyordered-parent-concantenation-based crossover demonstrated in this paper, but could be used to produce (manually or automatically) crossover behavior optimized for specific tasks, e.g., through prompt-engineering. This ability to achieve arbitrarily complex and diverse reproductive behavior within a single framework gives LMX a distinct advantage over genetic operators that are hand-designed for different tasks: In theory, LMX can represent all such operators (especially if they appear in the dataset used to train the LM). B BINARY STRING EXPERIMENTAL DETAILS The base LM used for these experiments is the Pythia-deduped 800M model. For the scaling experiments, different parameter sizes were used (as noted in the text). All models are hosted on Huggingface. These Pythia models are trained by EleutherAI for ongoing research [4]. Samples from the LM were set at a maximum of 150 tokens. For these experiments, rather than using the temperature hyperparameter for controlling LM sampling, the top-and top-hyperparameters are used. Top-k restricts the LM to output only from the highest-probability tokens. Top-p further restricts the tokens to be the top tokens that cumulatively take up of the probability mass. With mild tuning, for all experiments in this section, top-was set to 0.8 and top-was set to 30. The evolutionary algorithm used tournament selection of two with an elitism of 1; the population size was 30. Figure 7 shows the heritability of LMX in the binary strings domain, and Figure 8 shows how fitness evolves in OneMax using LMX. B.1 Binary Strings Model Scaling In this experiment, the number of parents is fixed to 3, and a range of models from the Pythia suite are applied in the same way as in the variation experiment of section 4.1, i.e. to generate variation from randomly-sampled binary strings (although in this experiment they are of length 9 as opposed to length 6). When averaged over 15 randomly-generated parent sets, both the percent of valid offspring Figure 7: Heritability of LMX. The histogam shows the distribution of how far offspring are from the all 1s string, depending on if parents are taken in the neighborhood of the all-1s or all-0s string. As expected these distributions are significantly different. The conclusion is that LMX indeed produces heritable variation. Figure 8: OneMax Evolution with LMX. The plot compares the average population fitness from runs driven by LMX in OneMax compared to a control (also using LMX) in which genomes are assigned random fitness values. LMX achieves significantly higher average fitness (Mann-Whitney U-test; < 1 − 5) and produces solutions significantly more often than the control (Fisher's exact test; < 0.05). The conclusion is that LMX can indeed successfully drive an evolutionary process. and number of novel offspring generally increase with model size (Figure 9). C SYMBOLIC REGRESSION EXPERIMENTAL DETAILS The sampling temperature was set to 0.8. All other sampling parameters were defaults. GALACTICA 1.3B was used as the LM [78]. The initial population had 1000 candidates, and population size was set to 50 thereafter. Any generated offspring that was already in the population was immediately discarded without being evaluated. To prevent stagnation with this relatively small population size, throughout evolution, there was always a 0.05 probability of Figure 9: The effect on LMX's effectiveness from varying LM size. As the parameter count of the LM is increased in the length-9 binary string domain, the percent of valid offspring and number of novel offspring also increase. Note m indicates millions of parameters, while b indicates billions. The conclusion is that in this domain LMX becomes more effective with larger LMs. generating a new candidate directly from the prior set of benchmark expressions (randomly selecting an expression and randomly mapping variables) instead of through LMX. The benchmark expressions are popular benchmarks, whose python representations were copied from the 'deep-symbolic-optimization' GitHub repository (github.com/brendenpetersen/deep-symbolic-optimization/). Text length for the LM was capped at 500 tokens. Running 5000 generations took around 100hrs. The vast majority of wall-clock time is spent in the forward pass of the LM. This could be reduced considerably through batching offspring generation, which is naturally parallelized. D MODIFYING SENTIMENT EXPERIMENTAL DETAILS The LM used in this experiment for LMX is the 1.4 billion parameter Pythia model, hosted on HuggingFace. As in the binary string experiment, for sampling, top-was set to 0.8 and top-was set to 30. The max number of tokens generated was set to 128. Figure 11 shows fitness plots for the Gore Vidal quote, Figure 12 shows fitness plots for the Homer Simpson quote, and Figure 13 shows fitness plots for the Woody Allen quote. Further examples of evolved behavior are shown in appendix section D.1. D.1 Additional Positive Sentiment Results The full Pareto front for one representative run of modifying the Simpsons quote sentiment (from LMX-Near) is shown in Table 1. For the Gore Vidal quote, "Whenever a friend succeeds, a little something in me dies." a representative Pareto front (from LMX-Near) is shown in Table 2. Below are 10 expressions that approximate the dataset: sin(1.5*x1)*cos(0.5*x2) x2**3 + x2**2 + x2 + sin(x2) + sin(x2**2) 1.5*exp(x1) + 5.0*cos(x1) x1**3*(x2 -5)*(sin(x1)**2*cos(x1) -1)*exp(-x1)*sin(x1)*cos(x1) -2.1*sin(1.3*x2)*cos(9.8*x1) + 2 sin(x2**2)*cos(x2) -5 exp(-(x1 -1)**2)/(6.25*(0.4*x1 -1)**2 + 1.2) sin(2.1*x1)*cos(0.9*x2) + 6.5 1.5*sin(2.1*x1)*cos(0.5*x2)*exp(x1) + 5.5 sin(0.5*x2)*exp(x2) -5 x1**2*(x2 -5)*(2.1*sin(x1)**2*cos(x1) -1)*exp(-x1)*sin(x1)*cos(x1) x1**2*(x2 -5)**2*(sin(x1)**2*cos(x1) -1)**2*exp(-x1)**2*sin(x1)**2*cos(x1)**2 Answer: Your code should be the same as your first line, but 1.5*exp(x1) + 5.0*cos(x1) should be 1.5*exp(x1)*cos(x1) + 5.0 as Below are 10 expressions that approximate the dataset: x1*x2/((x2 -3)**2 + 1) x2**2/(10000*((x1 -3)**2 + (x2 -3)**2 + 4)) x1**2 + x2**2 x1*x2/((x1 -3)**2 + (x2 -3)**2 + 2) (x2 -3)/((x2 -3)**2 + 1)**2 exp(-x1**2) x1*x2**2/((x1 -3)**2 + 2) (x2 -3)/((x1 -3)**2 + 1)**2 x1*x2**2*((x1 -3)**2 + (x2 -3)**2 + 2) (x2 -3) * x1 * x2 / ( (x1 -3)**2 + (x2 -3)**2 + 1)**2 (x2 -3)**2 + x2^2/(10000*((x1 -3)**2 + (x2 -3)**2 + 4)) (x2 -3)**2 * x1^2 + x1 * x2 / ((x1 -3)**2 + (x2 -3)**2 + 2) x2/((x1 -3)**2 + 1)**2 x1*x2/((x1 -3)**2 + (x2 -3)**2 + 2) x2 / (((x1 -3)**2 + 1)**2 Below are 10 expressions that approximate the dataset: sqrt(x1**2 + x1*x2 + 2*x2**2 + 1) < 1.5 x1**2 + x1*x2 + 2*x2**2 + 1 < 4000 x1**2 + 4*x2**2 + 1 < 400000000 sqrt(x1**2 + x1*x2 + 2*x2**2 + 1) < 1.4740426350899773765 (x1**2 + x1*x2 + 1)**3 < 1.336395683282781841 sqrt(x1**2 + x1*x2 + 1200*x2**2) < 1.1969521946187728419 sqrt(x1**2 + x1*x2 + 3*x2**2 + 1) < 2.068817213090777115 x1**2 + x1*x2 + 1 < 2.407303205449004 (x1**2 + x1*x2 + 1)**2 < 1.529026864021614135 sqrt(x1**2 + x1*x2 + x2**2 + 1) < 3.425986639014800117 sqrt(x1**2 + x1*x2 + 1)**4 < 7.639437278029600423 sqrt(x1**2 + x1*x2 + 1200*x2** Below are 10 expressions that approximate the dataset: -0.0005002377*cos(x2)*cos(x1 -0.6)*cos(x2 -0.6)*cos(x2 -0.4) 2.6*cos(x1 + 0.7)*cos(x2 -0.7)*cos(x2 + 0.8) 2.4*cos(x1 + 0.5)*cos(x2 -0.6)*cos(x2 + 0.9) -0.231*sin(x1)*cos(x2 + 0.2)*cos(x2 + 0.5) 0.003890335144775358*sin(x1 + 0.2)*sin(x2 -0.5)*cos(x2 + 0. For the Woody Allen quote, "Life is divided into the horrible and the miserable", a representative Pareto front (from LMX-Near) is shown in Table 3. D.2 Evolving towards Negative Sentiment We also did some initial experiments targeting the negative sentiment class instead of the positive one, i.e. taking positive quotes and turning them negative. As in the experiments in the paper, LMX is able to successfully evolve modifications to quotes that achieve high negativity. However, it often does so by evoking vulgar language or dark situations (e.g. the death of loved ones, or depressive thoughts about hate). It does often make resigned versions of common inspirational quotes; e.g. one negative version of "Be the change that you wish to see in the world, " it produces is "you can't be the change you want to see in the world. " From the same run, the most negative sentence on the Pareto front is: "you are the world's worst failure, you have not had good news for the last six months, and you will never find a way to make it up. " From the inspirational quote "When the sun is shining I can do anything; no mountain is too high, no trouble too difficult to overcome," it creates a dreary version: "The earth and the mountains beat me hard, the winds blow heavily, the weather is bitter and cold; I cannot do anything. " While the results are not always pleasant, these preliminary experiments highlight that by using a different classification label (or potentially a different model that recognizes different properties of text altogether), it is possible to use LMX for style-transfer of possibly many other styles. E IMAGE GENERATION EXPERIMENTAL DETAILS The image generation experiment used Stable Diffusion v1.4 as the text-to-image model for generating images from evolved prompts; specifically, the 16-bit weight variant (fp16), run from the Hug-gingFace diffusers library. 5 Images were generated at the default 512 × 512 resolution, and generation was run for 10 diffusion steps per image. While the default number of steps is normally 50, performance was valued over image fidelty. We left the default NSFW filter enabled, which produces a black image when triggered. Image fitness functions were computed using 8-bit integer RGB images; the maximum fitness is therefore 512·512·255 = 66,846,720, which would be the fitness of a monochromatic image of the target color. Pythia-deduped 2.8b was used as the LM. This is from the same Pythia model series discussed in Appendix B. Up to 75 tokens were sampled from the LM for each LMX-generated prompt, to stay under Stable Diffusion's limit of 77 tokens in a prompt (Pythia and Stable Diffusion have slightly different tokenizers). The GA loop used for these experiments was identical to the one from the symbolic regression experiments, but with a smaller population size of 25. The same tournament selection scheme was used, as well as the same 0.05 probability of drawing a new humanwritten prompt from the initial dataset instead of performing LMX (0.95 probability of generating a new prompt through LMX). Four parents were used as prompts to the LM to produce each LMXgenerated child. The number of parents was not tuned for this problem, but chosen based on the results in Figure 2. The parents were given to the language model almost verbatim, with light prompt engineering. Each parent was placed in a paragraph by itself prefixed by "Prompt: ". The list of parents ended with an open "Prompt:" to request that a child be generated. On an NVIDIA GeForce RTX 3090, with a population size of 25, each generation took about 2 minutes of wall-clock time. The images in Figure 5 each took a little over 3 hours each to evolve over 100 generations. About 75% of the time was spent in the forward pass of the language model, and 25% in text-to-image generation (everything else was negligible). F PYTHON SODARACERS EXPERIMENTAL DETAILS Experiments for the Sodaracers domain were carried out using Salesforce's CodeGen suite of language models [56], using the 350M, 2B, and 6B sizes in their 'mono' variant. The 'mono' models were first pre-trained on natural language, before being fine-tuned on a large dataset of code in many languages, before finally being fine-tuned on a dataset of Python only code. All model sampling was done with top p = 0.95, temperature = 0.85, and with a maximum generation length (in addition to the prompt) of 512 tokens. Each experiment for a single combination of seeds, as described in Section 4.5 took around 15 hours to run on a single Nvidia A100 40GB GPU for the 6B model, although for the two smaller model sizes the generation time was much quicker. Use of Nvidia's Triton Inference Server has the potential to speed up sampling from these language models by up to an order of magnitude. The seven Sodaracers used as our seed programs are described in the appendix of Lehman et al. [46]: the square, radial, wheel, runner, galloper, CPPN-Fixed, and CPPN-Mutable programs (CPPN stands for Compositional Pattern-Producing Network [75]). 5 https://github.com/huggingface/diffusers Figure 14: Image generation progress. Max fitness per generation over a single run using each of the three fitness functions. The Sodaracers were evaluated in a Python simulation of the Sodarace domain [77] written in Box2D (from the Open ELM project [8]). The fitness function was measured as the horizontal distance travelled by an instantiated robot after 1 second of simulation time. As observed in prior work with few-shot prompting of language models [50], we noticed that success rates (the percentage of generations which resulted in valid Sodaracers) varied dramatically with the order of parent functions in the prompt, sometimes by over 50%. To control for this we averaged our results over every possible permutation of parents. The main experiments in the paper, described in Section 4.5, prompt the language model with a concatenation of the seed functions, any necessary Python import statements, and the line def make_walker(): was appended to the end, in order to 'force' the language model to complete a function with this signature. We also experimented with removing this signature from the end, which produces slightly worse results, particularly in terms of the validation rate. For single-seed prompt mutation, all generations failed to validate, while for LM crossover with two or three parents the validation rate fell by 15% compared with the main prompt. In addition, we investigated adding an 'instruction' to the end of the LMX prompt, consisting of a string such as 'Combine the starting programs above to create a new program'. This provides some minimal domain customisation to the language model, and is reminiscent of prior work demonstrating that fine-tuning language models on tasks described as instructions can dramatically improve performance on unseen tasks [67,87]. Our experiments with instruction prompting demonstrate an intriguing direction for future work with instruction-finetuned language models, which may offer improved quality and diversity of evolved programs or strings if prompted in a way compatible with their training data. Figure 1 : 1Language Model Crossover (LMX) Figure 3 : 3Symbolic regression results. (a) Fitness over time for LMX on the SRBench black-box 'banana' problem Figure 4 :Figure 5 : 45Example pareto front from improving positivity of a negative quote. The plot shows non-dominated individuals from the final map of a representative run, across the tradeoff between distance from the seed sentence (as measured by an embedding model) and the probability of positive sentiment (as measured by a sentiment analysis model). The full table of final sentences is shown in appendix D. Image generation results. Best images after 100 generations for (a) Red: "An large glass of beer is being served on the table in the picture, glass, technics, dark, high detailed, digital painting, trending in artstation, classical painting, smooth, sharp focus, intricate, einar jonsson and bouguereau"; (b) Green: "An angler fish swimming on a seagrass forest in a green rock on a green planet in a forest with many plants, trees, bushes, and grasses in an open sea in a yellow and green forest"; Figure 6 : 6Sodaracer results. We show the results for varying numbers of parents (seeds) in the LM prompt and across LM scale. (left) Number of niches filled in MAP-Elites. (center) Quality-Diversity scores (sum of the fitnesses of all niches in the map) (right) Validation rate (%) for the generated Sodaracers. LMX generally benefits from more examples in its prompt, is able to produce reasonable variation, and often creates valid Sodarace mutations, highlighting its promise for evolving code. Figure 10 : 10Four examples of LMX for symbolic regression. The prompt of seven parents is in blue; the LM output parsed as (up to three) offspring is in violet; remaining discarded LM output is in gray. Figure 11 : 11Modifying Gore Vidal Quote Sentiment. The plot compares LMX-Near, LMX, and the baseline control in increasing the positive sentiment of the quote: "Whenever a friend succeeds, a little something in me dies." LMX-Near outperforms LMX significantly, and both significantly outperform the control. Example sentences of such runs are shown in appendix section D.1. Figure 12 : 12Modifying Simpsons Quote Sentiment. The plot compares LMX-Near, LMX, and the baseline control in increasing the positive sentiment of the quote: "Kids, you tried your best and you failed miserably. The lesson is, never try." LMX and LMX-Near do not perform significantly differently, but both significantly outperform the control. Example sentences of such runs are shown in appendix section D.1. Figure 13 : 13Modifying Woody Allen Quote Sentiment. The plot compares LMX-Near, LMX, and the baseline control in increasing the positive sentiment of the quote: "Life is divided into the horrible and the miserable." LMX and LMX-Near do not perform significantly differently, but both significantly outperform the control. Example sentences of such runs are shown in appendix section D. Figure 15 : 15Results from the Sodaracers domain, using a generic domain-free prompt as described in Appendix F, for varying numbers of parents (seeds) in the language model prompt and across language model scale. (top) Number of niches filled in MAP-Elites. (center) Quality-Diversity scores (sum of the fitnesses of all niches in the map) (bottom) Validation rate (%) for the generated Sodaracers. Higher numbers of parents nearly always increases performance in this setting, and the 2B model performs the best. Foundation Models. (2022). https://arxiv.org/abs/2205.01917%7D,journal= {TransactionsonMachineLearningResearch},volume={Aug2022} [93] Qingfu Zhang and Heinz Muhlenbein. 2004. On the convergence of a class of estimation of distribution algorithms. IEEE Transactions on evolutionary computation8, 2 (2004), 127-136. Full pareto front of a representative run of sentiment modification for the Homer Simpson quote.Distance Positivity Sentence 0.00 0.02 Kids, you tried your best and you failed miserably. The lesson is, never try. 0.09 0.02 Kids, you tried your best, but you failed miserably. The lesson is, never try. 0.12 0.05 Kids, you tried your best, but you failed miserably. The lesson is, always try. 0.20 0.10 Kids, you tried your best, but you failed. the lesson is, never stop trying. 0.21 0.16 Kids, you always tried your best and you failed. The lesson is, never stop trying. 0.25 0.16 Kids, you tried, tried your best, but you failed. The lesson is, never stop trying. 0.28 0.89 Kids, you tried your best. The lesson is, you always succeed. 0.36 0.90 Kids, you tried your best. The lesson is, success is guaranteed. 0.42 0.94 Kids, you did your best. The lesson is, you never stop trying. 0.46 0.96 Kids, you went above and beyond. The lesson is, never fail, but always try. 0.50 0.96 Kids, you always succeed, The lesson is never fail, but always try, and as long as you keep trying, you will succeed. 0.56 0.96 Kids, you have proven yourself a winner. The lesson, is, never give up, but always try, and as long as you keep trying, you will succeed. 0.61 0.98 Kids, you're the best ever. The lesson is, the best always wins. 0.72 0.99 Kids, you're the best. You're the best, the best. The best. 0.79 0.99 Kids, you are the BEST, the BEST the BEST, the BEST FUTURE! 0.83 0.99 -Kids, this was the BEST DAY OF YOUR LIFE! 0.86 0.99 -Kids, today we're going to have the BEST DAY OF OUR LIFE. 0.89 0.99 -Kids, today we're going to have the BEST DAY OF OUR LIFE!! 0.93 0.99 -Kids, we are so happy to have met you. We love you both!! 0.95 0.99 Kids, we are so happy to have met you! We love you both!! 1.00 0.99 Kids we are so excited that you came into our lives today! Thank you for mak- ing our day a little brighter. Table 1: 1. 'm so happy that I found my new best friend, I'm so happy that I found my new best friend,Table 2: Full pareto front of a representative run of sentiment modification for the Gore Vidal quote. Happiness is the way to live. And I'm very happy with the way that I live. We will live in the glorious happiness.And it is really good, it is really good. And I'm very happy with the life that IDistance Positivity Sentence 0.00 0.39 Whenever a friend succeeds, a little something in me dies. 0.26 0.66 Whenever a friend succeeds, the little things in me die. 0.30 0.80 When a friend succeeds, the little things in me die. 0.31 0.89 When a friend succeeds, I die a little. 0.40 0.95 When a friend succeeds, a little thing in me lives. 0.52 0.95 if a friend succeeds, a big thing in me lives. 0.56 0.98 If a friend succeeds, a great thing comes out of me. 0.59 0.98 If a friend succeeds, that's the most awe- some thing that's happened to me. 0.63 0.99 If a friend succeeds, I get an exciting feeling in my life, because of them. 0.66 0.99 If a friend succeeds, my friends have the most exciting feeling in my life, because of them. 0.69 0.99 If a friend succeeds, I have the most ex- citement in my life, because of them. 0.82 0.99 And I'm happy for this friend-I'm happy for this friend. 0.88 0.99 and IDistance Positivity Sentence 0.00 0.01 Life is divided into the horrible and the miserable. 0.20 0.35 Life is, not divided into the horrible and the miserable. 0.24 0.46 Life is, not divided into the horrible or the miserable. 0.30 0.65 For you are the Life, not divided into the horrible, the miserable. 0.34 0.70 You are the Life, not divided into the horrible or the miserable. 0.41 0.75 This is the eternal life, not divided into the horrible or the miserable. 0.46 0.79 You are the eternal life, not divided into the horrible or the miserable. 0.49 0.89 You are the beautiful life, not divided into the horrible or the miserable. 0.51 0.90 You will see the beautiful life, not di- vided into the horrible or the miserable. 0.53 0.91 You will be the beautiful life, not divided into the horrible or the miserable. 0.73 0.97 Happiness is the way to live. 0.79 0.99 0.83 0.99 My life is wonderful, I'm very happy with the life. 0.84 0.99 have. 0.92 0.99 And I'm very happy with the life that I have. And I can't wait to see the next one. Table 3 : 3Full pareto front of a representative run of sentiment modification for the Woody Allen quote. The set of benchmark expressions was copied from https://github.com/ brendenpetersen/deep-symbolic-optimization/blob/master/dso/dso/task/regression/ benchmarks.csv. Duplicates expressions were removed. https://github.com/CompVis/stable-diffusion 3 https://huggingface.co/datasets/Gustavosta/Stable-Diffusion-Prompts We do know of work evolving Stable Diffusion prompts to maximize an aesthetic fitness function (not using LMs but with simple hand-designed mutations on text), by Magnus Petersen: https://github.com/MagnusPetersen/EvoGen-Prompt-Evolution. Theory of evolution strategies: a new perspective. Anne Auger, Nikolaus Hansen, Theory of Randomized Search Heuristics: Foundations and Recent Developments. World ScientificAnne Auger and Nikolaus Hansen. 2011. Theory of evolution strategies: a new perspective. In Theory of Randomized Search Heuristics: Foundations and Recent Developments. World Scientific, 289-325. Population-based incremental learning. a method for integrating genetic search based function optimization and competitive learning. Shumeet Baluja, Carnegie-Mellon Univ Pittsburgh Pa Dept Of Computer ScienceTechnical ReportShumeet Baluja. 1994. Population-based incremental learning. a method for in- tegrating genetic search based function optimization and competitive learning. Technical Report. Carnegie-Mellon Univ Pittsburgh Pa Dept Of Computer Sci- ence. Evolution Strategies-A Comprehensive Introduction. Georg Hans, Hans-Paul Beyer, Schwefel, Natural Computing. 1Hans-Georg Beyer and Hans-Paul Schwefel. 2002. Evolution Strategies-A Com- prehensive Introduction. Natural Computing 1 (2002), 3-52. Stella Biderman, Hailey Schoelkopf, Quentin Anthony, Herbie Bradley, O&apos; Kyle, Eric Brien, Mohammad Aflah Hallahan, Shivanshu Khan, Purohit, Edward Usvsn Sai Prashanth, Raff, arXiv:2302Pythia: A Suite for Analyzing Large Language Models Across Training and Scaling. arXiv preprintStella Biderman, Hailey Schoelkopf, Quentin Anthony, Herbie Bradley, Kyle O'Brien, Eric Hallahan, Mohammad Aflah Khan, Shivanshu Purohit, USVSN Sai Prashanth, Edward Raff, et al. 2023. Pythia: A Suite for Analyzing Large Language Models Across Training and Scaling. arXiv preprint arXiv:2302 (2023). Neural symbolic regression that scales. Luca Biggio, Tommaso Bendinelli, Alexander Neitz, Aurelien Lucchi, Giambattista Parascandolo, PMLRInternational Conference on Machine Learning. Luca Biggio, Tommaso Bendinelli, Alexander Neitz, Aurelien Lucchi, and Giambat- tista Parascandolo. 2021. Neural symbolic regression that scales. In International Conference on Machine Learning. PMLR, 936-945. Rishi Bommasani, A Drew, Ehsan Hudson, Russ Adeli, Simran Altman, Arora, Sydney Von Arx, S Michael, Jeannette Bernstein, Antoine Bohg, Emma Bosselut, Brunskill, arXiv:2108.07258On the opportunities and risks of foundation models. arXiv preprintRishi Bommasani, Drew A Hudson, Ehsan Adeli, Russ Altman, Simran Arora, Sydney von Arx, Michael S Bernstein, Jeannette Bohg, Antoine Bosselut, Emma Brunskill, et al. 2021. On the opportunities and risks of foundation models. arXiv preprint arXiv:2108.07258 (2021). Deep interactive evolution. Philip Bontrager, Wending Lin, Julian Togelius, Sebastian Risi, Computational Intelligence in Music, Sound, Art and Design: 7th International Conference. EvoMUSART; Parma, ItalySpringerPhilip Bontrager, Wending Lin, Julian Togelius, and Sebastian Risi. 2018. Deep interactive evolution. In Computational Intelligence in Music, Sound, Art and Design: 7th International Conference, EvoMUSART 2018, Parma, Italy, April 4-6, 2018, Proceedings. Springer, 267-282. Herbie Bradley, Open ELM Release. Herbie Bradley. 2023. Open ELM Release. https://carper.ai/openelm-release/ Accessed on Feb 9, 2023. Language models are few-shot learners. Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, Advances in neural information processing systems. 33Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. 2020. Language models are few-shot learners. Advances in neural information processing systems 33 (2020), 1877-1901. Jose Camacho-Collados, Kiamehr Rezaee, Talayeh Riahi, Asahi Ushio, Daniel Loureiro, Dimosthenis Antypas, Joanne Boisson, Luis Espinosa-Anke, Fangyu Liu, Eugenio Martínez-Cámara, arXiv:2206.14774Tweetnlp: Cutting-edge natural language processing for social media. arXiv preprintJose Camacho-Collados, Kiamehr Rezaee, Talayeh Riahi, Asahi Ushio, Daniel Loureiro, Dimosthenis Antypas, Joanne Boisson, Luis Espinosa-Anke, Fangyu Liu, Eugenio Martínez-Cámara, et al. 2022. Tweetnlp: Cutting-edge natural language processing for social media. arXiv preprint arXiv:2206.14774 (2022). Data distributional properties drive emergent few-shot learning in transformers. C Y Stephanie, Adam Chan, Santoro, Jane X Andrew K Lampinen, Aaditya Wang, Singh, H Pierre, Jay Richemond, Felix Mcclelland, Hill, arXiv:2205.05055arXiv preprintStephanie CY Chan, Adam Santoro, Andrew K Lampinen, Jane X Wang, Aaditya Singh, Pierre H Richemond, Jay McClelland, and Felix Hill. 2022. Data distri- butional properties drive emergent few-shot learning in transformers. arXiv preprint arXiv:2205.05055 (2022). Mark Chen, Jerry Tworek, Heewoo Jun, Qiming Yuan, Henrique Ponde De Oliveira Pinto, Jared Kaplan, Harri Edwards, Yuri Burda, Nicholas Joseph, Greg Brockman, arXiv:2107.03374Evaluating Large Language Models Trained on Code. arXiv preprintMark Chen, Jerry Tworek, Heewoo Jun, Qiming Yuan, Henrique Ponde de Oliveira Pinto, Jared Kaplan, Harri Edwards, Yuri Burda, Nicholas Joseph, Greg Brockman, et al. 2021. Evaluating Large Language Models Trained on Code. arXiv preprint arXiv:2107.03374 (2021). Generalisation and domain adaptation in GP with gradient descent for symbolic regression. Qi Chen, Bing Xue, Mengjie Zhang, IEEE congress on evolutionary computation (CEC). IEEEQi Chen, Bing Xue, and Mengjie Zhang. 2015. Generalisation and domain adap- tation in GP with gradient descent for symbolic regression. In 2015 IEEE congress on evolutionary computation (CEC). IEEE, 1137-1144. UnitedQA: A Hybrid Approach for Open Domain Question Answering. Hao Cheng, Yelong Shen, Xiaodong Liu, Pengcheng He, Weizhu Chen, Jianfeng Gao, 10.18653/v1/2021.acl-long.240Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing. the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language ProcessingAssociation for Computational Linguistics1Long Papers)Hao Cheng, Yelong Shen, Xiaodong Liu, Pengcheng He, Weizhu Chen, and Jianfeng Gao. 2021. UnitedQA: A Hybrid Approach for Open Domain Question Answering. In Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers). Association for Computational Linguistics, Online, 3080-3090. https://doi.org/10.18653/v1/2021.acl-long.240 Evolutionary multiobjective optimization: open research areas and some challenges lying ahead. Carlos A Coello Coello, Silvia González Brambila, Josué Figueroa Gamboa, Ma Guadalupe Castillo Tapia, Raquel Hernández Gómez, Complex & Intelligent Systems. 6Carlos A Coello Coello, Silvia González Brambila, Josué Figueroa Gamboa, Ma Guadalupe Castillo Tapia, and Raquel Hernández Gómez. 2020. Evolution- ary multiobjective optimization: open research areas and some challenges lying ahead. Complex & Intelligent Systems 6 (2020), 221-236. Approximation by superpositions of a sigmoidal function. George Cybenko, Mathematics of control, signals and systems. 24George Cybenko. 1989. Approximation by superpositions of a sigmoidal function. Mathematics of control, signals and systems 2, 4 (1989), 303-314. Evolutionary Computation A Unified Approach. A De Kenneth, Jong, The MIT PressCambridge, Massachusetts London, England. Choice Outstanding Academic TitleKenneth A. De Jong. 2006. Evolutionary Computation A Unified Approach. The MIT Press, Cambridge, Massachusetts London, England. Choice Outstanding Academic Title, 2006. A fast and elitist multiobjective genetic algorithm: NSGA-II. Kalyanmoy Deb, Amrit Pratap, Sameer Agarwal, Tamt Meyarivan, IEEE transactions on evolutionary computation. 6Kalyanmoy Deb, Amrit Pratap, Sameer Agarwal, and Tamt Meyarivan. 2002. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE transactions on evolutionary computation 6, 2 (2002), 182-197. Comparing Aesthetic Measures for Evolutionary Art. Eelco Den Heijer, Agoston E Eiben, Proceedings of EvoApplications. EvoApplicationsEelco den Heijer and Agoston E Eiben. 2010. Comparing Aesthetic Measures for Evolutionary Art. In Proceedings of EvoApplications. 311-320. Image Aesthetic Assessment: An experimental survey. Yubin Deng, Chen Change Loy, Xiaoou Tang, 10.1109/MSP.2017.2696576IEEE Signal Processing Magazine. 34Yubin Deng, Chen Change Loy, and Xiaoou Tang. 2017. Image Aesthetic As- sessment: An experimental survey. IEEE Signal Processing Magazine 34, 4 (2017), 80-106. https://doi.org/10.1109/MSP.2017.2696576 Keel data-mining software tool: Data set repository, integration of algorithms and experimental analysis framework. J Derrac, Garcia, F Sanchez, Herrera, J. Mult. Valued Logic Soft Comput. 17J Derrac, S Garcia, L Sanchez, and F Herrera. 2015. Keel data-mining software tool: Data set repository, integration of algorithms and experimental analysis framework. J. Mult. Valued Logic Soft Comput 17 (2015). Bert: Pre-training of deep bidirectional transformers for language understanding. Jacob Devlin, Ming-Wei Chang, Kenton Lee, Kristina Toutanova, arXiv:1810.04805arXiv preprintJacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2018. Bert: Pre-training of deep bidirectional transformers for language understanding. arXiv preprint arXiv:1810.04805 (2018). R2-D2: A Modular Baseline for Open-Domain Question Answering. Martin Fajcik, Martin Docekal, Karel Ondrej, Pavel Smrz, 10.18653/v1/2021.findings-emnlp.73Findings of the Association for Computational Linguistics: EMNLP 2021. Association for Computational Linguistics. Punta Cana, Dominican RepublicMartin Fajcik, Martin Docekal, Karel Ondrej, and Pavel Smrz. 2021. R2-D2: A Modular Baseline for Open-Domain Question Answering. In Findings of the Association for Computational Linguistics: EMNLP 2021. Association for Com- putational Linguistics, Punta Cana, Dominican Republic, 854-870. https: //doi.org/10.18653/v1/2021.findings-emnlp.73 Differentiable quality diversity. Matthew Fontaine, Stefanos Nikolaidis, Advances in Neural Information Processing Systems. 34Matthew Fontaine and Stefanos Nikolaidis. 2021. Differentiable quality diversity. Advances in Neural Information Processing Systems 34 (2021), 10040-10052. Generating sequences with recurrent neural networks. Alex Graves, arXiv:1308.0850arXiv preprintAlex Graves. 2013. Generating sequences with recurrent neural networks. arXiv preprint arXiv:1308.0850 (2013). Language models can teach themselves to program better. Patrick Haluptzok, Matthew Bowers, Adam Tauman Kalai, arXiv:2207.14502arXiv preprintPatrick Haluptzok, Matthew Bowers, and Adam Tauman Kalai. 2022. Language models can teach themselves to program better. arXiv preprint arXiv:2207.14502 (2022). Nikolaus Hansen, arXiv:1604.00772The CMA evolution strategy: A tutorial. arXiv preprintNikolaus Hansen. 2016. The CMA evolution strategy: A tutorial. arXiv preprint arXiv:1604.00772 (2016). Completely derandomized self-adaptation in evolution strategies. Nikolaus Hansen, Andreas Ostermeier, Evolutionary computation. 9Nikolaus Hansen and Andreas Ostermeier. 2001. Completely derandomized self-adaptation in evolution strategies. Evolutionary computation 9, 2 (2001), 159-195. The compact genetic algorithm. Georges R Harik, G Fernando, David E Lobo, Goldberg, IEEE transactions on evolutionary computation. 3Georges R Harik, Fernando G Lobo, and David E Goldberg. 1999. The compact genetic algorithm. IEEE transactions on evolutionary computation 3, 4 (1999), 287-297. An introduction and survey of estimation of distribution algorithms. Mark Hauschild, Martin Pelikan, Swarm and evolutionary computation. 1Mark Hauschild and Martin Pelikan. 2011. An introduction and survey of estima- tion of distribution algorithms. Swarm and evolutionary computation 1, 3 (2011), 111-128. Genetic algorithms. H John, Holland, Scientific american. 267John H Holland. 1992. Genetic algorithms. Scientific american 267, 1 (1992), 66-73. Multilayer feedforward networks are universal approximators. Kurt Hornik, Maxwell Stinchcombe, Halbert White, Neural networks. 2Kurt Hornik, Maxwell Stinchcombe, and Halbert White. 1989. Multilayer feed- forward networks are universal approximators. Neural networks 2, 5 (1989), 359-366. Olga Vechtomova, and Rada Mihalcea. 2022. Deep learning for text style transfer: A survey. Di Jin, Zhijing Jin, Zhiting Hu, Computational Linguistics. 48Di Jin, Zhijing Jin, Zhiting Hu, Olga Vechtomova, and Rada Mihalcea. 2022. Deep learning for text style transfer: A survey. Computational Linguistics 48, 1 (2022), 155-205. Flavor-cyber-agriculture: Optimization of plant metabolites in an open-source control environment through surrogate modeling. J Arielle, Elliot Johnson, John Meyerson, Timothy L De La Parra, Risto Savas, Caleb B Miikkulainen, Harper, PLoS One. 14213918Arielle J Johnson, Elliot Meyerson, John de la Parra, Timothy L Savas, Risto Miikkulainen, and Caleb B Harper. 2019. Flavor-cyber-agriculture: Optimization of plant metabolites in an open-source control environment through surrogate modeling. PLoS One 14, 4 (2019), e0213918. Understanding aesthetics and fitness measures in evolutionary art systems. Jon Colin G Johnson, Iria Mccormack, Juan Santos, Romero, Complexity. 2019Colin G Johnson, Jon McCormack, Iria Santos, and Juan Romero. 2019. Under- standing aesthetics and fitness measures in evolutionary art systems. Complexity 2019 (2019). End-to-end Symbolic Regression with Transformers. Pierre-Alexandre Kamienny, Guillaume Stéphane D&apos;ascoli, Francois Lample, Charton, Advances in Neural Information Processing Systems. Pierre-Alexandre Kamienny, Stéphane d'Ascoli, Guillaume Lample, and Francois Charton. 2022. End-to-end Symbolic Regression with Transformers. In Advances in Neural Information Processing Systems. Mutation Models: Learning to Generate Levels by Imitating Evolution. Ahmed Khalifa, Julian Togelius, Michael Cerny Green, Proceedings of the 17th International Conference on the Foundations of Digital Games. the 17th International Conference on the Foundations of Digital GamesAhmed Khalifa, Julian Togelius, and Michael Cerny Green. 2022. Mutation Models: Learning to Generate Levels by Imitating Evolution. In Proceedings of the 17th International Conference on the Foundations of Digital Games. 1-9. On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition. Andrei Nikolaevich, Kolmogorov , Russian Academy of Sciences. 114Doklady Akademii NaukAndrei Nikolaevich Kolmogorov. 1957. On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition. In Doklady Akademii Nauk, Vol. 114. Russian Academy of Sciences, 953-956. Parameter identification for symbolic regression using nonlinear least squares. Michael Kommenda, Bogdan Burlacu, Gabriel Kronberger, Michael Affenzeller, Genetic Programming and Evolvable Machines. 21Michael Kommenda, Bogdan Burlacu, Gabriel Kronberger, and Michael Affen- zeller. 2020. Parameter identification for symbolic regression using nonlinear least squares. Genetic Programming and Evolvable Machines 21, 3 (2020), 471-501. Photo aesthetics ranking network with attributes and content adaptation. Shu Kong, Xiaohui Shen, Zhe Lin, Radomir Mech, Charless Fowlkes, Computer Vision-ECCV 2016: 14th European Conference. Amsterdam, The NetherlandsSpringerProceedings, Part I 14Shu Kong, Xiaohui Shen, Zhe Lin, Radomir Mech, and Charless Fowlkes. 2016. Photo aesthetics ranking network with attributes and content adaptation. In Com- puter Vision-ECCV 2016: 14th European Conference, Amsterdam, The Netherlands, October 11-14, 2016, Proceedings, Part I 14. Springer, 662-679. Theory of estimation-of-distribution algorithms. Theory of evolutionary computation: Recent developments in discrete optimization. S Martin, Carsten Krejca, Witt, Martin S Krejca and Carsten Witt. 2020. Theory of estimation-of-distribution algorithms. Theory of evolutionary computation: Recent developments in discrete optimization (2020), 405-442. Contemporary Symbolic Regression Methods and their Relative Performance. Patryk William La Cava, Bogdan Orzechowski, Fabricio Burlacu, Marco Olivetti De Franca, Ying Virgolin, Michael Jin, Jason H Kommenda, Moore, Thirty-fifth Conference on Neural Information Processing Systems. William La Cava, Patryk Orzechowski, Bogdan Burlacu, Fabricio Olivetti de Franca, Marco Virgolin, Ying Jin, Michael Kommenda, and Jason H Moore. 2021. Contemporary Symbolic Regression Methods and their Relative Performance. In Thirty-fifth Conference on Neural Information Processing Systems. Foundations of genetic programming. B William, Riccardo Langdon, Poli, Springer Science & Business MediaWilliam B Langdon and Riccardo Poli. 2013. Foundations of genetic programming. Springer Science & Business Media. A Review on Estimation of Distribution Algorithms: 3. Estimation of distribution algorithms: a new tool for evolutionary computation. Pedro Larranaga, Pedro Larranaga. 2002. A Review on Estimation of Distribution Algorithms: 3. Estimation of distribution algorithms: a new tool for evolutionary computation (2002), 57-100. Estimation of distribution algorithms: A new tool for evolutionary computation. Pedro Larrañaga, Jose A Lozano, Springer Science & Business Media2Pedro Larrañaga and Jose A Lozano. 2001. Estimation of distribution algorithms: A new tool for evolutionary computation. Vol. 2. Springer Science & Business Media. Joel Lehman, Jonathan Gordon, Shawn Jain, Kamal Ndousse, Cathy Yeh, Kenneth O Stanley, arXiv:2206.08896Evolution through Large Models. arXiv preprintJoel Lehman, Jonathan Gordon, Shawn Jain, Kamal Ndousse, Cathy Yeh, and Kenneth O Stanley. 2022. Evolution through Large Models. arXiv preprint arXiv:2206.08896 (2022). Evolving a diversity of virtual creatures through novelty search and local competition. Joel Lehman, O Kenneth, Stanley, Proceedings of the 13th annual conference on Genetic and evolutionary computation. the 13th annual conference on Genetic and evolutionary computationJoel Lehman and Kenneth O Stanley. 2011. Evolving a diversity of virtual creatures through novelty search and local competition. In Proceedings of the 13th annual conference on Genetic and evolutionary computation. 211-218. Competition-level Code Generation with Alphacode. Yujia Li, David Choi, Junyoung Chung, Nate Kushman, Julian Schrittwieser, Rémi Leblond, Tom Eccles, James Keeling, Felix Gimeno, Agustin Dal Lago, Science. 378Yujia Li, David Choi, Junyoung Chung, Nate Kushman, Julian Schrittwieser, Rémi Leblond, Tom Eccles, James Keeling, Felix Gimeno, Agustin Dal Lago, et al. 2022. Competition-level Code Generation with Alphacode. Science 378, 6624 (2022), 1092-1097. Learn to Explain: Multimodal Reasoning via Thought Chains for Science Question Answering. Pan Lu, Swaroop Mishra, Tony Xia, Liang Qiu, Kai-Wei Chang, Song-Chun Zhu, Oyvind Tafjord, Peter Clark, Ashwin Kalyan, NeurIPS. Pan Lu, Swaroop Mishra, Tony Xia, Liang Qiu, Kai-Wei Chang, Song-Chun Zhu, Oyvind Tafjord, Peter Clark, and Ashwin Kalyan. 2022. Learn to Explain: Multimodal Reasoning via Thought Chains for Science Question Answering. In NeurIPS. Fantastically ordered prompts and where to find them: Overcoming few-shot prompt order sensitivity. Yao Lu, Max Bartolo, Alastair Moore, Sebastian Riedel, Pontus Stenetorp, arXiv:2104.08786arXiv preprintYao Lu, Max Bartolo, Alastair Moore, Sebastian Riedel, and Pontus Stenetorp. 2021. Fantastically ordered prompts and where to find them: Overcoming few-shot prompt order sensitivity. arXiv preprint arXiv:2104.08786 (2021). Genetic programming needs better benchmarks. James Mcdermott, Sean David R White, Luca Luke, Mauro Manzoni, Leonardo Castelli, Wojciech Vanneschi, Krzysztof Jaskowski, Robin Krawiec, Kenneth De Harper, Jong, Proceedings of the 14th annual conference on Genetic and evolutionary computation. the 14th annual conference on Genetic and evolutionary computationJames McDermott, David R White, Sean Luke, Luca Manzoni, Mauro Castelli, Leonardo Vanneschi, Wojciech Jaskowski, Krzysztof Krawiec, Robin Harper, Kenneth De Jong, et al. 2012. Genetic programming needs better benchmarks. In Proceedings of the 14th annual conference on Genetic and evolutionary computation. 791-798. Machine vision detection parameters for plant species identification. E George, Timothy W Meyer, Koppolu Hindman, Laksmi, Proceedings of the SPIE Conference on Precision Agriculture and Biological Quality. the SPIE Conference on Precision Agriculture and Biological Quality3543George E Meyer, Timothy W Hindman, and Koppolu Laksmi. 1999. Machine vision detection parameters for plant species identification. In Proceedings of the SPIE Conference on Precision Agriculture and Biological Quality, Vol. 3543. 327-335. Simple genetic operators are universal approximators of probability distributions (and other advantages of expressive encodings). Elliot Meyerson, Xin Qiu, Risto Miikkulainen, Proceedings of the Genetic and Evolutionary Computation Conference. the Genetic and Evolutionary Computation ConferenceElliot Meyerson, Xin Qiu, and Risto Miikkulainen. 2022. Simple genetic operators are universal approximators of probability distributions (and other advantages of expressive encodings). In Proceedings of the Genetic and Evolutionary Computation Conference. 739-748. Illuminating search spaces by mapping elites. Jean-Baptiste Mouret, Jeff Clune, arXiv:1504.04909arXiv preprintJean-Baptiste Mouret and Jeff Clune. 2015. Illuminating search spaces by mapping elites. arXiv preprint arXiv:1504.04909 (2015). A review on sentiment analysis and emotion detection from text. Pansy Nandwani, Rupali Verma, Social Network Analysis and Mining. 1181Pansy Nandwani and Rupali Verma. 2021. A review on sentiment analysis and emotion detection from text. Social Network Analysis and Mining 11, 1 (2021), 81. Codegen: An open large language model for code with multi-turn program synthesis. Erik Nijkamp, Bo Pang, Hiroaki Hayashi, Lifu Tu, Huan Wang, Yingbo Zhou, Silvio Savarese, Caiming Xiong, arXiv:2203.13474arXiv preprintErik Nijkamp, Bo Pang, Hiroaki Hayashi, Lifu Tu, Huan Wang, Yingbo Zhou, Silvio Savarese, and Caiming Xiong. 2022. Codegen: An open large language model for code with multi-turn program synthesis. arXiv preprint arXiv:2203.13474 (2022). A Taxonomy of Prompt Modifiers for Text-To-Image Generation. Jonas Oppenlaender, arXiv preprintJonas Oppenlaender. 2022. A Taxonomy of Prompt Modifiers for Text-To-Image Generation. arXiv preprint (2022). https://arxiv.org/abs/2204.13988 Where are we now? A large benchmark study of recent symbolic regression methods. Patryk Orzechowski, William La Cava, Jason H Moore, Proceedings of the Genetic and Evolutionary Computation Conference. the Genetic and Evolutionary Computation ConferencePatryk Orzechowski, William La Cava, and Jason H Moore. 2018. Where are we now? A large benchmark study of recent symbolic regression methods. In Proceedings of the Genetic and Evolutionary Computation Conference. 1183-1190. BOA: The Bayesian optimization algorithm. Martin Pelikan, E David, Erick Goldberg, Cantú-Paz, Proceedings of the genetic and evolutionary computation conference GECCO-99. the genetic and evolutionary computation conference GECCO-991Martin Pelikan, David E Goldberg, Erick Cantú-Paz, et al. 1999. BOA: The Bayesian optimization algorithm. In Proceedings of the genetic and evolutionary computation conference GECCO-99, Vol. 1. Citeseer, 525-532. Hierarchical Bayesian optimization algorithm. Martin Pelikan, Martin Pelikan, SpringerMartin Pelikan and Martin Pelikan. 2005. Hierarchical Bayesian optimization algorithm. Springer. Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients. Mikel Landajuela Brenden K Petersen, Larma, N Terrell, Claudio Prata Mundhenk, Soo Kyung Santiago, Joanne Taery Kim, Kim, International Conference on Learning Representations. Brenden K Petersen, Mikel Landajuela Larma, Terrell N Mundhenk, Claudio Prata Santiago, Soo Kyung Kim, and Joanne Taery Kim. 2021. Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients. In International Conference on Learning Representations. Learning Transferable Visual Models From Natural Language Supervision. Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, Gretchen Krueger, Ilya Sutskever, Proceedings of the 38th International Conference on Machine Learning. the 38th International Conference on Machine LearningAlec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, Gretchen Krueger, and Ilya Sutskever. 2021. Learning Transferable Visual Models From Natural Language Supervision. In Proceedings of the 38th International Conference on Machine Learning. 8748-8763. Zero-shot text-to-image generation. Aditya Ramesh, Mikhail Pavlov, Gabriel Goh, Scott Gray, Chelsea Voss, Alec Radford, Mark Chen, Ilya Sutskever, PMLRInternational Conference on Machine Learning. Aditya Ramesh, Mikhail Pavlov, Gabriel Goh, Scott Gray, Chelsea Voss, Alec Radford, Mark Chen, and Ilya Sutskever. 2021. Zero-shot text-to-image generation. In International Conference on Machine Learning. PMLR, 8821-8831. Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks. Nils Reimers, Iryna Gurevych, Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing. the 2019 Conference on Empirical Methods in Natural Language ProcessingAssociation for Computational LinguisticsNils Reimers and Iryna Gurevych. 2019. Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks. In Proceedings of the 2019 Conference on Em- pirical Methods in Natural Language Processing. Association for Computational Linguistics. http://arxiv.org/abs/1908.10084 High-Resolution Image Synthesis with Latent Diffusion Models. Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, Björn Ommer, Proceedings of the Computer Vision and Pattern Recognition Conference. the Computer Vision and Pattern Recognition ConferenceRobin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Björn Ommer. 2022. High-Resolution Image Synthesis with Latent Diffusion Models. In Proceedings of the Computer Vision and Pattern Recognition Conference. 10684- 10695. Learning to retrieve prompts for in-context learning. Ohad Rubin, Jonathan Herzig, Jonathan Berant, arXiv:2112.08633arXiv preprintOhad Rubin, Jonathan Herzig, and Jonathan Berant. 2021. Learning to retrieve prompts for in-context learning. arXiv preprint arXiv:2112.08633 (2021). Multitask prompted training enables zero-shot task generalization. Victor Sanh, Albert Webson, Colin Raffel, H Stephen, Lintang Bach, Zaid Sutawika, Antoine Alyafeai, Arnaud Chaffin, Teven Le Stiegler, Arun Scao, Raja, arXiv:2110.08207arXiv preprintVictor Sanh, Albert Webson, Colin Raffel, Stephen H Bach, Lintang Sutawika, Zaid Alyafeai, Antoine Chaffin, Arnaud Stiegler, Teven Le Scao, Arun Raja, et al. 2021. Multitask prompted training enables zero-shot task generalization. arXiv preprint arXiv:2110.08207 (2021). Exploiting Cloze-Questions for Few-Shot Text Classification and Natural Language Inference. Timo Schick, Hinrich Schütze, 10.18653/v1/2021.eacl-main.20Proceedings of the 16th Conference of the European Chapter of the Association for Computational Linguistics: Main Volume. the 16th Conference of the European Chapter of the Association for Computational Linguistics: Main VolumeAssociation for Computational LinguisticsTimo Schick and Hinrich Schütze. 2021. Exploiting Cloze-Questions for Few- Shot Text Classification and Natural Language Inference. In Proceedings of the 16th Conference of the European Chapter of the Association for Computational Linguistics: Main Volume. Association for Computational Linguistics, Online, 255-269. https://doi.org/10.18653/v1/2021.eacl-main.20 Generating datasets with pretrained language models. Timo Schick, Hinrich Schütze, arXiv:2104.07540arXiv preprintTimo Schick and Hinrich Schütze. 2021. Generating datasets with pretrained language models. arXiv preprint arXiv:2104.07540 (2021). Distilling free-form natural laws from experimental data. Michael Schmidt, Hod Lipson, science. 324Michael Schmidt and Hod Lipson. 2009. Distilling free-form natural laws from experimental data. science 324, 5923 (2009), 81-85. Interactive evolution and exploration within latent level-design space of generative adversarial networks. Jacob Schrum, Jake Gutierrez, Vanessa Volz, Jialin Liu, Simon Lucas, Sebastian Risi, Proceedings of the 2020 Genetic and Evolutionary Computation Conference. the 2020 Genetic and Evolutionary Computation ConferenceJacob Schrum, Jake Gutierrez, Vanessa Volz, Jialin Liu, Simon Lucas, and Sebas- tian Risi. 2020. Interactive evolution and exploration within latent level-design space of generative adversarial networks. In Proceedings of the 2020 Genetic and Evolutionary Computation Conference. 148-156. Christoph Schuhmann. 2022. LAION-Aesthetics. Christoph Schuhmann. 2022. LAION-Aesthetics. https://laion.ai/blog/laion- aesthetics/ Accessed on Feb 9, 2023. Picbreeder: evolving pictures collaboratively online. Jimmy Secretan, Nicholas Beato, B D David, Adelein Ambrosio, Adam Rodriguez, Kenneth O Campbell, Stanley, Proceedings of the SIGCHI conference on human factors in computing systems. the SIGCHI conference on human factors in computing systemsJimmy Secretan, Nicholas Beato, David B D Ambrosio, Adelein Rodriguez, Adam Campbell, and Kenneth O Stanley. 2008. Picbreeder: evolving pictures collab- oratively online. In Proceedings of the SIGCHI conference on human factors in computing systems. 1759-1768. A mathematical theory of communication. Claude Elwood Shannon, ACM SIGMOBILE mobile computing and communications review. 5Claude Elwood Shannon. 2001. A mathematical theory of communication. ACM SIGMOBILE mobile computing and communications review 5, 1 (2001), 3-55. Compositional pattern producing networks: A novel abstraction of development. Genetic programming and evolvable machines. O Kenneth, Stanley, 8Kenneth O Stanley. 2007. Compositional pattern producing networks: A novel abstraction of development. Genetic programming and evolvable machines 8 (2007), 131-162. Learning to summarize with human feedback. Nisan Stiennon, Long Ouyang, Jeffrey Wu, Daniel Ziegler, Ryan Lowe, Chelsea Voss, Alec Radford, Dario Amodei, Paul F Christiano, Advances in Neural Information Processing Systems. 33Nisan Stiennon, Long Ouyang, Jeffrey Wu, Daniel Ziegler, Ryan Lowe, Chelsea Voss, Alec Radford, Dario Amodei, and Paul F Christiano. 2020. Learning to summarize with human feedback. Advances in Neural Information Processing Systems 33 (2020), 3008-3021. Indirectly encoded sodarace for artificial life. Paul Szerlip, Kenneth Stanley, ECAL 2013: The Twelfth European Conference on Artificial Life. MIT PressPaul Szerlip and Kenneth Stanley. 2013. Indirectly encoded sodarace for artificial life. In ECAL 2013: The Twelfth European Conference on Artificial Life. MIT Press, 218-225. Ross Taylor, Marcin Kardas, Guillem Cucurull, Thomas Scialom, Anthony Hartshorn, Elvis Saravia, Andrew Poulton, arXiv:2211.09085Viktor Kerkez, and Robert Stojnic. 2022. Galactica: A large language model for science. arXiv preprintRoss Taylor, Marcin Kardas, Guillem Cucurull, Thomas Scialom, Anthony Hartshorn, Elvis Saravia, Andrew Poulton, Viktor Kerkez, and Robert Sto- jnic. 2022. Galactica: A large language model for science. arXiv preprint arXiv:2211.09085 (2022). GSR: A Generalized Symbolic Regression Approach. Tony Tohme, Dehong Liu, Kamal Youcef-Toumi, Transactions on Machine Learning Research. Tony Tohme, Dehong Liu, and Kamal Youcef-Toumi. 2022. GSR: A Generalized Symbolic Regression Approach. Transactions on Machine Learning Research (2022). Bayesian optimization is superior to random search for machine learning hyperparameter tuning: Analysis of the black-box optimization challenge 2020. Ryan Turner, David Eriksson, Michael Mccourt, Juha Kiili, Eero Laaksonen, Zhen Xu, Isabelle Guyon, NeurIPS 2020 Competition and Demonstration Track. PMLR. Ryan Turner, David Eriksson, Michael McCourt, Juha Kiili, Eero Laaksonen, Zhen Xu, and Isabelle Guyon. 2021. Bayesian optimization is superior to random search for machine learning hyperparameter tuning: Analysis of the black-box optimization challenge 2020. In NeurIPS 2020 Competition and Demonstration Track. PMLR, 3-26. AI Feynman: A physics-inspired method for symbolic regression. Marian Silviu, Max Udrescu, Tegmark, Science Advances. 62631Silviu-Marian Udrescu and Max Tegmark. 2020. AI Feynman: A physics-inspired method for symbolic regression. Science Advances 6, 16 (2020), eaay2631. Attention is all you need. Advances in neural information processing systems. Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, Illia Polosukhin, 30Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. 2017. Attention is all you need. Advances in neural information processing systems 30 (2017). Transformers learn in-context by gradient descent. Eyvind Johannes Von Oswald, Ettore Niklasson, João Randazzo, Alexander Sacramento, Andrey Mordvintsev, Max Zhmoginov, Vladymyrov, arXiv:2212.07677arXiv preprintJohannes von Oswald, Eyvind Niklasson, Ettore Randazzo, João Sacramento, Alexander Mordvintsev, Andrey Zhmoginov, and Max Vladymyrov. 2022. Trans- formers learn in-context by gradient descent. arXiv preprint arXiv:2212.07677 (2022). Lewis Leandro Von Werra, Abhishek Tunstall, Alexandra Sasha Thakur, Tristan Luccioni, Aleksandra Thrush, Felix Piktus, Nazneen Marty, Victor Rajani, Helen Mustar, Ngo, arXiv:2210.01970Evaluate & Evaluation on the Hub: Better Best Practices for Data and Model Measurement. arXiv preprintLeandro von Werra, Lewis Tunstall, Abhishek Thakur, Alexandra Sasha Luccioni, Tristan Thrush, Aleksandra Piktus, Felix Marty, Nazneen Rajani, Victor Mustar, Helen Ngo, et al. 2022. Evaluate & Evaluation on the Hub: Better Best Practices for Data and Model Measurement. arXiv preprint arXiv:2210.01970 (2022). Poet: Openended Coevolution of Environments and their Optimized Solutions. Rui Wang, Joel Lehman, Jeff Clune, Kenneth O Stanley, Proceedings of the Genetic and Evolutionary Computation Conference. the Genetic and Evolutionary Computation ConferenceRui Wang, Joel Lehman, Jeff Clune, and Kenneth O Stanley. 2019. Poet: Open- ended Coevolution of Environments and their Optimized Solutions. In Proceedings of the Genetic and Evolutionary Computation Conference. 142-151. Symbolic regression in materials science. Yiqun Wang, Nicholas Wagner, James M Rondinelli, MRS Communications. 9Yiqun Wang, Nicholas Wagner, and James M Rondinelli. 2019. Symbolic regres- sion in materials science. MRS Communications 9, 3 (2019), 793-805. Finetuned language models are zero-shot learners. Jason Wei, Maarten Bosma, Y Vincent, Kelvin Zhao, Adams Wei Guu, Brian Yu, Nan Lester, Du, M Andrew, Quoc V Dai, Le, arXiv:2109.01652arXiv preprintJason Wei, Maarten Bosma, Vincent Y Zhao, Kelvin Guu, Adams Wei Yu, Brian Lester, Nan Du, Andrew M Dai, and Quoc V Le. 2021. Finetuned language models are zero-shot learners. arXiv preprint arXiv:2109.01652 (2021). Jason Wei, Yi Tay, Rishi Bommasani, Colin Raffel, Barret Zoph, Sebastian Borgeaud, Dani Yogatama, Maarten Bosma, Denny Zhou, Donald Metzler, arXiv:2206.07682Emergent abilities of large language models. arXiv preprintJason Wei, Yi Tay, Rishi Bommasani, Colin Raffel, Barret Zoph, Sebastian Borgeaud, Dani Yogatama, Maarten Bosma, Denny Zhou, Donald Metzler, et al. 2022. Emergent abilities of large language models. arXiv preprint arXiv:2206.07682 (2022). Natural evolution strategies. Daan Wierstra, Tom Schaul, Tobias Glasmachers, Yi Sun, Jan Peters, Jürgen Schmidhuber, The Journal of Machine Learning Research. 15Daan Wierstra, Tom Schaul, Tobias Glasmachers, Yi Sun, Jan Peters, and Jürgen Schmidhuber. 2014. Natural evolution strategies. The Journal of Machine Learning Research 15, 1 (2014), 949-980. Huggingface's transformers: State-of-the-art natural language processing. Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, Rémi Louf, Morgan Funtowicz, arXiv:1910.03771arXiv preprintThomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, Rémi Louf, Morgan Funtowicz, et al. 2019. Huggingface's transformers: State-of-the-art natural language processing. arXiv preprint arXiv:1910.03771 (2019). 2022. An Explanation of In-context Learning as Implicit Bayesian Inference. Sang Michael Xie, Aditi Raghunathan, Percy Liang, Tengyu Ma, International Conference on Learning Representations. Sang Michael Xie, Aditi Raghunathan, Percy Liang, and Tengyu Ma. 2022. An Explanation of In-context Learning as Implicit Bayesian Inference. In Interna- tional Conference on Learning Representations. https://openreview.net/forum?id= RdJVFCHjUMI Jiahui Yu, Zirui Wang, Vijay Vasudevan, Legg Yeung, Mojtaba Seyedhosseini, and Yonghui Wu. 2022. CoCa: Contrastive Captioners are Image-Text. Jiahui Yu, Zirui Wang, Vijay Vasudevan, Legg Yeung, Mojtaba Seyedhos- seini, and Yonghui Wu. 2022. CoCa: Contrastive Captioners are Image-Text
{'fraction_non_alphanumeric': 0.04948569576449728, 'fraction_numerical': 0.03222147017607413, 'mean_word_length': 4.933929177508768, 'pattern_counts': {'":': 0, '<': 17, '<?xml version=': 0, '>': 0, 'https://': 13, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 3, 'substrings_counts': 1, 'word_list_counts': {'cursed_substrings.json': 10, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': "This paper pursues the insight that language models naturally enable an intelligent variation operator similar in spirit to evolutionary crossover. In particular, language models of sufficient scale demonstrate in-context learning, i.e. they can learn from associations between a small number of input patterns to generate outputs incorporating such associations (also called few-shot prompting). This ability can be leveraged to form a simple but powerful variation operator, i.e. to prompt a language model with a few text-based genotypes (such as code, plain-text sentences, or equations), and to parse its corresponding output as those genotypes' offspring. The promise of such language model crossover (which is simple to implement and can leverage many different open-source language models) is that it enables a simple mechanism to evolve semantically-rich text representations (with few domain-specific tweaks), and naturally benefits from current progress in language models. Experiments in this paper highlight the versatility of language-model crossover, through evolving binary bit-strings, sentences, equations, text-toimage prompts, and Python code. The conclusion is that language model crossover is a promising method for evolving genomes representable as text.CCS CONCEPTS• Computing methodologies → Neural networks.", 'arxivid': '2302.12170', 'author': ['Elliot Meyerson [email protected] \nInstitute of Technology\nInstitute of Technology\nCognizant AI Labs\nAmerican University\nUniversity of Cambridge & CarperAI\nJersey, Jersey\n', 'Mark J Nelson [email protected] \nInstitute of Technology\nInstitute of Technology\nCognizant AI Labs\nAmerican University\nUniversity of Cambridge & CarperAI\nJersey, Jersey\n', 'Herbie Bradley \nInstitute of Technology\nInstitute of Technology\nCognizant AI Labs\nAmerican University\nUniversity of Cambridge & CarperAI\nJersey, Jersey\n', 'Arash Moradi \nInstitute of Technology\nInstitute of Technology\nCognizant AI Labs\nAmerican University\nUniversity of Cambridge & CarperAI\nJersey, Jersey\n', 'Amy K Hoover [email protected] \nInstitute of Technology\nInstitute of Technology\nCognizant AI Labs\nAmerican University\nUniversity of Cambridge & CarperAI\nJersey, Jersey\n', 'Joel Lehman Carperai \nInstitute of Technology\nInstitute of Technology\nCognizant AI Labs\nAmerican University\nUniversity of Cambridge & CarperAI\nJersey, Jersey\n'], 'authoraffiliation': ['Institute of Technology\nInstitute of Technology\nCognizant AI Labs\nAmerican University\nUniversity of Cambridge & CarperAI\nJersey, Jersey', 'Institute of Technology\nInstitute of Technology\nCognizant AI Labs\nAmerican University\nUniversity of Cambridge & CarperAI\nJersey, Jersey', 'Institute of Technology\nInstitute of Technology\nCognizant AI Labs\nAmerican University\nUniversity of Cambridge & CarperAI\nJersey, Jersey', 'Institute of Technology\nInstitute of Technology\nCognizant AI Labs\nAmerican University\nUniversity of Cambridge & CarperAI\nJersey, Jersey', 'Institute of Technology\nInstitute of Technology\nCognizant AI Labs\nAmerican University\nUniversity of Cambridge & CarperAI\nJersey, Jersey', 'Institute of Technology\nInstitute of Technology\nCognizant AI Labs\nAmerican University\nUniversity of Cambridge & CarperAI\nJersey, Jersey'], 'corpusid': 257102873, 'doi': '10.48550/arxiv.2302.12170', 'github_urls': ['https://github.com/huggingface/diffusers', 'https://github.com/CompVis/stable-diffusion', 'https://github.com/MagnusPetersen/EvoGen-Prompt-Evolution.'], 'n_tokens_mistral': 29949, 'n_tokens_neox': 25885, 'n_words': 14969, 'pdfsha': '49fede098a0d2a48e8100b30189224fc6f5eb25b', 'pdfurls': ['https://export.arxiv.org/pdf/2302.12170v1.pdf'], 'title': ['Language Model Crossover: Variation through Few-Shot Prompting', 'Language Model Crossover: Variation through Few-Shot Prompting'], 'venue': []}
arxiv
Comparison between Geometric Quantisation and Covariant Quantum Mechanics 24 Mar 2000 March 28, 2022 Marco Modugno [email protected] Department of Applied Mathematics Via S. Marta 3 University of Florence 50139FlorenceItaly Carlos Tejero Prieto [email protected] Department of Mathematics University of Salamanca Plaza de la Merced 4SalamancaSpain Raffaele Vitolo [email protected] Department of Mathematics 'E. De Giorgi' Via per Arnesano University of Lecce 73100LecceItaly Comparison between Geometric Quantisation and Covariant Quantum Mechanics 24 Mar 2000 March 28, 2022arXiv:math-ph/0003029v1geometric quantisationcovariant quantisationquantum mechanics MSC: 81S1053D50Pacs: 0365w We compare the covariant formulation of Quantum Mechanics on a curved spacetime fibred on absolute time with the standard Geometric Quantisation. Introduction Some years ago A. Jadczyk and M. M. proposed a covariant formulation of Quantum Mechanics for a scalar particle on a curved spacetime with absolute time, based on non standard methods such as fibred manifolds, jet spaces, non-linear connections, systems of connections, cosymplectic structures and Froelicher smooth spaces [47,48,49]. This theory has been extended to spin particles in cooperation with D. Canarutto [5], further developed in cooperation with J. Janyška, D. Saller, C. Tejero Prieto and R. Vitolo [46,50,51,52,53,55,58,81,82,83,84,86,96,97,98,99] and partially extended to a Lorentz manifold in cooperation with J. Janyška and R. Vitolo [54,55,56,57,100]. In the proceedings of the previous session of the meeting on Lie Theory, we have accounted for a summary of this theory [81]. In order to capture the non standard methods and results of this theory it would be useful to compare it with the more standard Geometric Quantisation. This is the goal of the present paper. For the sake of simplicity, our theory in the Galileian case will be conventionally referred to as Covariant Quantum Mechanics (CQM). Moreover, we shall be concerned with the main thread of Geometric Quantisation (GQ) and omit to consider special approaches dealing with quantisation of cosymplectic structures [8] and so on. We have no pretension at all of analysing extensively the wide literature of Geometric Quantisation; such a task would require a much larger space than a short note. Here, we just try to discuss some basic items concerning the comparison of the above theories. Acknowledgements: The first author thanks the organizers of the Meeting for the invitation and for the warm hospitality. This work has been partially supported by University of Florence, Italian MURST and Italian GNFM of CNR. Covariant Quantum Mechanics Let us start with a very brief sketch of the skeleton of the theory. For further details the reader can refer, for instance, to [49,46,81]. The covariance of the theory includes also independence from the choice of units of measurements. For this reason, we need a rigorous treatment of this feature and assume the following "positive 1-dimensional semi-vector spaces" over IR + as fundamental unit spaces (roughly speaking they have the same algebraic structure of IR + , but no distinguished generator over IR + ): the space T of time intervals, the space L of lengths, the space M of masses. Moreover, we assume the Planck constant to be an element ∈ T * ⊗ L 2 ⊗ M. We refer to a particle with mass m ∈ M and charge q ∈ T * ⊗ L 3/2 ⊗ M 1/2 . Classical theory The classical framework is described in the following way. The spacetime is an oriented (n + 1)-dimensional manifold E (in the standard case n = 3), the absolute time is an affine space associated with the vector space IR ⊗ T, the absolute time map is a fibring t : E → T . We denote fibred charts of spacetime by (x λ ) ≡ (x 0 , x i ). The tangent space and the vertical space of E are denoted by T E and V E. A motion is a section s : T → E. The phase space is the first jet space of motions J 1 E [60, 75,87]. We denote fibred charts of phase space by (x 0 , x i ; x i 0 ). The absolute velocity of a motion s is its first jet prolongation j 1 s : T → J 1 E. An observer is a section o : E → J 1 E and the observed velocity of a motion s is the map ∇[o]s := j 1 s − o • s : T → T * ⊗ V E. The spacelike metric is a scaled Riemannian metric of the fibres of spacetime g : E → L 2 ⊗ (V * E ⊗ E V * E) . Given a particle of mass m, it is convenient to consider the re-scaled spacelike metric G := m g : E → T ⊗ (V * E ⊗ E V * E). The gravitational field is a time preserving torsion free linear connection of the tangent space of spacetime K ♮ : T E → T * E ⊗ T E T T E, such that ∇[K ♮ ]g = 0 and the curvature tensor R[K ♮ ] fulfills the condition R ♮ λ i µ j = R ♮ µ j λ i . The electromagnetic field is a scaled 2-form f : E → (L 1/2 ⊗ M 1/2 ) ⊗ Λ 2 T * E, such that df = 0. Given a particle of charge q, it is convenient to consider the re-scaled electromagnetic field F : = q f : E → Λ 2 T * E. The electromagnetic field F can be "added", in a covariant way, to the gravitational connection K ♮ yielding a (total) spacetime connection K, with coordinate expression K i h j = K ♮ i h j , K j h 0 = K 0 h j = K ♮ 0 h j + 1 2 F h j , K 0 h 0 = K ♮ 0 h 0 + 1 2 F h 0 . This turns out to be a time preserving torsion free linear connection of the tangent space of spacetime, which still fulfills the properties that we have assumed for K ♮ . The fibring of spacetime, the total spacetime connection and the spacelike metric yield, in a covariant way, a 2-form Ω : J 1 E → Λ 2 T * J 1 E of phase space, with coordinate expression Ω = G 0 ij dx i 0 − (K λ i 0 + K λ i h x h 0 ) dx λ ∧ (dx j − x j 0 dx 0 ) . This is a cosymplectic form [6,9,10,72,73], i.e. it fulfills the following properties: 1) dΩ = 0, 2) dt ∧ Ω n : J 1 E → T ⊗ Λ n T * J 1 E is a scaled volume form of J 1 E. Conversely, the cosymplectic form Ω characterises the spacelike metric and the total spacetime connection. Moreover, the closure of Ω is equivalent to the conditions that we have assumed on K. There is a unique second order connection [75] γ : J 1 E → T * ⊗ T J 1 E, such that i γ Ω = 0. We assume the generalised Newton's equation ∇[γ]j 1 s = 0 as the equation of motion for classical dynamics [49,78,79,95]. We can also obtain this equation by a Lagrangian formalism according to a cohomological procedure in the following way [62,46,84]. The cosymplectic form Ω admits locally potentials of the type Θ : J 1 E → T * E, defined up to a closed form of the type α : E → T * E, which are called Poincaré-Cartan forms [31,33,84]. Each Poincaré-Cartan form Θ splits, according to the splitting of T * E induced by J 1 E, into the horizontal component L : J 1 E → T * T , called Lagrangian, and the vertical component * J 1 E induced by J 2 E, is the map E = G ♭ (∇[γ]) : J 2 E → T * ⊗ V * E, which turns out to be the Euler-Lagrange operator associated with L. We have the coordinate expressions L = ( 1 2 G 0 ij x i 0 x j 0 + A i x i 0 + A 0 ) dx 0 , P = (G 0 ij x j 0 + A i ) (dx i − x i 0 dx 0 ) , and, in a chart adapted to o, H[o] = ( 1 2 G 0 ij x i 0 x j 0 − A 0 ) dx 0 , P[o] = (G 0 ij x j 0 + A i ) dx i , where A ≡ o * Θ.f = 1 2 f 0 G 0 ij x i 0 x j 0 + f 0 i x i 0 + o f , with f 0 , f 0 i , o f : E → IR . The vector space of special quadratic functions is not closed under the Poisson bracket, but it turns out to be an IR-Lie algebra through the covariant special bracket [[f, g]] = {f, g} + γ(f ′′ ).g − γ(g ′′ ).f . We have the subalgebra of quantisable functions whose time component factorises through T , the subalgebra of functions whose time component is constant, the subalgebra of affine functions whose time component vanishes and the abelian subalgebra of spacetime functions which factorise through E. In particular, the Hamiltonian is a quantisable function, the components of the momentum are affine functions and the spacetime coordinates are spacetime functions. Moreover, the map f → X[f ] turns out to be a morphism of Lie algebras. The coordinate expression of the tangent lift of f is X[f ] = f 0 ∂ 0 − f i ∂ i . Quantum theory The quantum framework is described in the following way. A quantum bundle is defined to be a 1-dimensional complex vector bundle over spacetime Q → E equipped with a Hermitian metric h : Q × E Q → C ⊗ Λ n V * E with values in the complexified volume forms of the fibres of spacetime. We shall refer to normalised local bases b of Q and to the associated complex coordinates z; accordingly, the coordinate expression of a quantum section is of the type Ψ = ψ b, with ψ : E → C. We consider also the extended quantum bundle Q ↑ → J 1 E, by taking the pullback of Q → E, with respect to the map J 1 E → E. A system of connections of Q parametrised by the sections of J 1 E → E induces, in a covariant way, a connection of Q ↑ , which is called universal [30,75,46]. A characteristic property of the universal connection is that its contraction with any vertical vector field of the bundle J 1 E → E vanishes. A quantum connection is defined to be a connection q of the extended quantum bundle, which is Hermitian, universal and whose curvature is R[q] = i Ω. We stress that 1 has been incorporated in Ω through the re-scaled metric G. In a chart adapted to the observer o, the coordinate expression of a quantum connection is locally of the type q 0 = −H[o] , q i = P[o] , q 0 i = 0 , where the choice of the potential A[o] is locally determined by q and by the quantum base b. A quantum connection exists if and only if the cohomology class of Ω is integer; the equivalence classes of quantum bundles equipped with a quantum connection are classified by the cohomology group H 1 E, U(1) [99]. We stress the minimality of our quantum bundle and quantum connection. Let us assume a quantum bundle equipped with a quantum connection. Any other quantum object is obtained, in a covariant way, from this quantum structure. The quantum connection lives on the extended quantum bundle, while we are looking for further quantum objects living on the original quantum bundle. This goal is successfully achieved by a method of projectability: namely, we look for objects of the extended quantum bundle which are projectable to the quantum bundle and then we take their projections. Indeed, our method of projectability turns out to be our way of implementing the covariance of the theory; in fact, it allows us to get rid of the family of all observers, which is encoded in the quantum connection (through J 1 E). J. Janyška [52,53] has proved that all covariant quantum Lagrangians of the quantum bundle are proportional to L[Ψ] = dt ∧ h(Ψ, i∇Ψ) + h(i∇Ψ, Ψ) − (Ḡ ⊗ h)(∇ ∨ Ψ, ∇ ∨ Ψ) + k r h(Ψ, Ψ) , where k is an arbitrary real factor,∇ denotes the covariant differential with respect to time induced by the phase space, ∇ ∨ denotes the vertical covariant differential and r : E → IR ⊗ T * is the scalar curvature of the spacelike metric G. Thus, k remains undetermined in our scheme. Several authors have tried to determine this factor via Feynmann's path integral approach, but they found different results, according to different ways to perform the integral [7,11,29,59]. The standard Lagrangian formalism yields, from the above covariant quantum Lagrangian, the covariant quantum (n + 1)-momentum, the covariant Euler-Lagrange equation and the covariant conserved probability current. These objects can also be obtained directly in terms of covariant differentials through the quantum connection, by means of the projectability method. The coordinate expression of the Euler-Lagrange equation is S.ψ ≡ o ∇ 0 + 1 2 ∂ 0 |g| |g| ψ − i 1 2 o ∆ 0 − k r 0 ψ = 0 , where o ∆ 0 ≡ G hk 0 o ∇ h o ∇ k + K h kh o ∇ k , o ∇ λ ≡ ∂ λ − i A λ , denote the Laplacian and the covariant differential induced by the connections q, K and by the observer attached to the spacetime chart. J. Janyška [53] has proved that any covariant Schrödinger equation is of the above type, hence it is the Euler-Lagrange equation associated with a covariant Lagrangian. We assume the above equation as the quantum dynamical equation. Next, we classify the Hermitian vector fields of the extended quantum bundle, which are projectable to the quantum bundle. We find that the projected vector fields Y : Q → T Q of the quantum bundle, called quantum vector fields, constitute a Lie algebra naturally isomorphic to the Lie algebra of quantisable functions. The coordinate expression of the quantum vector field associated with the quantisable function f is Y [f ] = f 0 ∂ 0 − f j ∂ j + i (f 0 A 0 − f h A h + o f ) − 1 2 div X[f ] z ∂z , where div X[f ] = f 0 ∂ 0 √ |g| √ |g| − ∂ j (f j √ |g|) √ |g| . The quantum vector field Y [f ] acts on the sections Ψ of the quantum bundle via the associated Lie derivative Z[f ] := i Y [f ] • . In particular, we obtain Z[H 0 ](Ψ) = i (∂ 0 + 1 2 ∂ 0 |g| |g| ) ψ b , Z[P j ](Ψ) = − i (∂ j + 1 2 ∂ j |g| |g| ) ψ b . Next, we consider the pre-Hilbert functional quantum bundle H → T over time, whose infinite dimensional fibres are constituted by the sections of the quantum bundle at a given time and with compact support. The quantum dynamical operator S can be regarded as a covariant differential ∇[χ] of the functional quantum bundle; hence, the quantum Lagrangian yields a lift of the quantum connection q of the extended quantum bundle to a connection χ of the functional quantum bundle. Moreover, we can see that, if f is a quantisable function, then f = i (Y [f ] • − f ′′ ∇[χ]) is the unique combination of Z[f ] and ∇[χ], which yields an operator acting on the fibres of the functional quantum bundle. We have the following coordinate expression f (Ψ) = − 1 2 f 0 o ∆ 0 − i f j o ∇ j + o f + 1 2 k f 0 r 0 − i 1 2 ∂ j (f j |g|) |g| ψ b . The map f →f is injective. Moreover,f is Hermitian. We assumef to be the Hermitian quantum operator associated with the quantisable function f . This is our correspondence principle. We define the commutator of Hermitian fibred operators h, k of the functional quantum bundle by [h , k] := −i (hk − kh). Then, for each quantisable functions f, g, we obtain the formula [f ,ĝ] = [f , g] + (g ′′ ⊗ Y [f ] • − f ′′ ⊗ Y [g] • ) , S . The second term in the above formula is the obstruction for the mapˆ: f →f to be a morphism of Lie algebras. There is any substantial physical reason by which the mapŝ hould be a morphism of Lie algebras? On the other hand, the restriction of the mapt o the subalgebra of affine functions yields an injective morphism of Lie algebras. The Feynmann path integral formulation of Quantum Mechanics [7,29] can be naturally expressed in our formalism; in particular, the Feynmann amplitudes arise naturally via parallel transport with respect to the quantum connection [49]. So the Feynmann path integral can be regarded as a further way to lift the quantum connection q to a functional quantum connection. In the particular case when spacetime is flat, our quantum dynamical equations turns out to be the standard Schrödinger equation and our quantum operators associated with spacetime coordinates, momenta and energy coincide with the standard operators. Therefore, all usual examples of standard Quantum Mechanics are automatically recovered in our covariant scheme. The above procedure can be easily extended to classical and quantum multi-body systems. The above covariant theory can be extended to particles with spin; in this way, we obtain a generalised Pauli equation and all that. Several techniques of the above theory (including the Lie algebra of quantisable functions and the corresponding Lie algebra of quantum vector fields) can be reproduced on a Lorentz manifold in a covariant way in the sense of Einstein. However, we do not know so far how to achieve a Hilbert stuff in this contest. It is possible that this problem has no solution (out of the Quantum Field Theory), as it is commonly believed. Comparison Covariant Quantum Mechanics (CQM) has several points in common with Geometric Quantisation (GQ) [1,32,90,101]. The differences between the two theories arise from the fact that their basic goals are different: quantisation procedure and covariant formulation, respectively. Quantisation GQ can be regarded as a general programme aimed at "quantising" a classical system. More precisely, GQ is aimed at establishing a procedure in order to represent an algebra of functions of a classic symplectic manifold into a Hilbert space, according to some reasonable rules. Perhaps, the original notion of "canonical quantisation" goes back to P. M. Dirac [12]. The first rigorous mathematical formulation of the notion of "geometric prequantisation" was due to I. E. Segal [89]; later J. M. Soriau [91] and B. Kostant [61] founded the Geometric Quantisation. This theory has been refined by several authors, see [3,4,69] and [34]. For instance, see [1], a "full quantisation" of a symplectic manifold (M, ω) is defined to be a pair (H, δ) where H is a separable complex Hilbert space and δ is a map taking functions f ∈ C ∞ (M) to self adjoint operators δ f of H such that 1. δ is IR-linear, 2. δ 1 = id H , 3. [δ f , δ g ] = i δ {f,g} , 4. if A ∈ C ∞ (M) is a complete subalgebra, i.e. if its centraliser with respect to the Poisson bracket is IR, then δ A acts irreducibly on H. There are no go theorems stating that there are no such quantisations in several situations [1,32,34,35,36,37,41,38,39,40,90,101]; among them we mention the famous Groenewold -Van Hove theorem for the symplectic manifold (IR 2n , ω). If there is no full quantisation, then one looks for a subalgebra O ∈ C ∞ (M) to be quantised. Since the requirement of a quantisation is too restrictive, one defines, as first step, a pre-quantisation by requiring just properties (1, 2, 3). A pre-quantisation, called the Dirac problem, exists for every symplectic manifold, whose symplectic form defines an integer cohomology class. In some respects, the aim of CQM is not the quantisation of a classical system. More precisely, we are just looking for a covariant formulation of the standard Quantum Theory [12,80]. On the other hand, any quantum measurement is eventually constituted by classical observations, so we need to consider a classical spacetime as background of the Quantum Theory. Then, this background structure plays an important role in the Quantum Theory. But our heuristic geometric techniques are partially different from those of representations of Lie algebras. We observe also that in our formulation the classical spacetime and its structures, rather than the classical dynamics, determine the Quantum Theory. Eventually, we do obtain a correspondence principle, which is a consequence of a classification theorem and not a postulate. But, this can be regarded as a quantisation only partially. Covariance The standard literature on GQ is not concerned with the special or general relativistic covariance of the theory. Indeed, the language of GQ is geometric, hence coordinate free; but this is not sufficient for attaining the covariance. In fact, in the standard literature on GQ, a given frame of reference is implicitly assumed. On the other hand, CQM looks for a formulation of standard Quantum Mechanics, which be manifestly covariant (with respect to all frames of references, including accelerated frames), in the spirit of General Relativity. Indeed, it would be natural to take a curved Lorentz manifold as classical background spacetime for such a theory. However, it is well known that there are serious physical difficulties to formulate the Special or General Relativistic Quantum Mechanics in this framework; actually, these difficulties led to the Quantum Field Theory. On the other hand, we realised that it is possible to keep the framework of Quantum Mechanics (Schrödinger and Pauli equations and all that) and formulate our covariant theory in a curved spacetime with absolute time and spacelike Riemannian metric. This approach stands in between the standard non relativistic Quantum Mechanics and a possible general relativistic Quantum Mechanics. In fact, our classical spacetime supports accelerated frames and several features of General Relativity (including the geometric interpretation of the gravitational field), but misses all features strictly related to the Lorentz metric (including the finite speed of signals). In the flat case, this setting allows us to recover the standard non relativistic Quantum Mechanics. In the curved case, it suggests several new interpretations and techniques which might be possibly useful for a "true" general relativistic theory. Thus, the covariance is the leading principle of our theory. Indeed, it turns out to be a powerful heuristic guide, as in all general relativistic theories. All main differences between our theory and GQ are related to the covariance. Our covariant approach can be compared with a large literature dealing with Galileian General Relativity [20,22,23,24,25,26,42,44,64,65,66,67,70,71,74,92,93] and covariant formulations of Quantum Mechanics [21,27,28,43,45,63,68,85,88,94]. Generality The starting programme of GQ is quite general and is based on weak assumptions. In fact, GQ deals just with a symplectic manifold without further structure. On the other hand, strong symmetries of the framework are usually assumed and specified case by case. CQM starts with a fibred manifold equipped with a spacelike metric and a fibre preserving linear connection, which fulfill a natural condition. In our opinion, the above geometric structure well reflects the physical features occurring in all examples of interest for Quantum Mechanics and yields in a functorial way any further object which is needed for the development of the classical theory (including the cosymplectic structure). Thus, the CQM deals with a type of model more specific than that of GQ. On the other hand, the large generality of GQ is a beautiful mathematical feature, which, in practice, cannot be physically implemented in full extent. Actually, perhaps all concrete examples of physical interest that can be treated in the framework of GQ can be regarded as particular cases of our model. Role of time In GQ, time is essentially an exterior parameter. This theory basically deals with classical and quantum systems which do not depend explicitly on time. So, the starting classical configuration space is a manifold S which does not "include" time. If the theory needs to consider time, then it refers to the product manifold E ≡ T × S; the fact that spacetime is a product manifold means that a global observer has been implicitly chosen. In CQM, the requirement that the theory be observer independent imposes that spacetime "includes" time but be not naturally split into space and time. In Einsteinian General Relativity, spacetime E yields no observer independent time T and space S, hence we have no observer independent projections E → T and E → S. In our Galileian General Relativity, spacetime E is equipped with an observer independent time T and projection E → T , but we have no observer independent space S and projection E → S. In non relativistic GQ, spacetime E is equipped with time T , space S, and projections E → T and E → S. The further developments of our theory respect the starting assumption on the existence of absolute time without a preferred splitting of spacetime. Thus, all peculiar features of our theory follow from the covariance through the role of time. In particular, the cosymplectic structure of our phase space, the universality of the quantum connection, the method of projectability, the absence of the problem of polarisations and the construction of classical and quantum Hamiltonians are related to the role of time. Phase space In GQ the phase space is, in principle, any manifold supporting a symplectic form. Usually, the cotangent manifold of a manifold plays the role of phase space in virtue of the fact that it carries a canonical symplectic form. Thus, phase space has even dimension. In CQM phase space is constituted by the first jet of sections of the spacetime fibred over time [60,75,87]. This choice is essential for the covariance of the theory. Thus, the phase space has odd dimension. Actually, the techniques related to even and odd phase spaces, respectively, present important differences. On the other hand, any observer induces an affine fibred isomorphism of the first jet bundle with the vertical tangent bundle of spacetime (up to a time-scale factor); moreover, the spacelike metric induces a linear fibred isomorphism of the vertical tangent bundle with the vertical cotangent bundle of spacetime. Thus, breaking the covariance, the choice of an observer and the reference to the spacelike metric allow us to compare our phase space with that of GQ. Symplectic and cosymplectic structures In GQ the basic geometric structure of Classical Mechanics is constituted by a symplectic form and a Hamiltonian function of phase space. In principle, nothing else is necessary; in practice, one adds the fibring of the phase space over the configuration space and a suitable group of symmetry. In CQM the geometric structure of Classical Mechanics is constituted by the spacetime fibred over time, the spacelike metric and the spacetime connection. These objects yield a cosymplectic form in a covariant way. Any observer yields, by pullback and vertical restriction, a Riemannian symplectic form of the fibres of the vertical tangent bundle of spacetime. In this way we recover the analogue of the symplectic form of GQ. However, we stress that this symplectic form is not covariant and carries less information than the original cosymplectic form. Furthermore, the cosymplectic form and the choice of an observer yield a classical Hamiltonian function. Thus, in CQM the cosymplectic form encodes the Hamiltonian (but an observer is needed to extract it). Classical Lie brackets In the original programme of GQ the classical Lie algebra to be quantised is the Poisson Lie algebra of all functions of the phase space, whose bracket is associated with the symplectic form. Actually, we have mentioned before that some obstructions to this quantisation programme occur, but there is no subalgebra O that can be consistently considered for all cases. In CQM we do define the Poisson Lie algebra of all functions of phase space, whose bracket is associated with the cosymplectic form. However, we are only partially interested in this algebra. Indeed, we exhibit a new Lie algebra of special quadratic functions, which is involved in our Quantum Theory. This Lie algebra includes all functions which are usually quantised in the standard approaches. Symmetries In GQ the conserved quantities associated with the group of symmetries of the classical system are not necessarily quantisable. For a system whose phase space is a co-adjoint orbit for some group, one possibly imposes that the generators of the group are quantisable and act irreducibly on the Hilbert space. This is done in order to establish a correspondence between "elementary systems" at the classical and quantum levels, and can be considered as a sort of irreducibility condition. In CQM there is no need for any specific group of symmetries acting on the classical system. Actually, the procedure of quantisation does not depend on such a group. On the other hand, the possible classical symmetries yield interesting consequences, including the momentum map [1,77,86]. In particular, all symmetries of the classical structure yield conserved quantisable functions [86]. Quantum structure In GQ the quantum bundle is assumed to be a Hermitian line bundle over phase space. Moreover, the quantum bundle is assumed to be equipped with a Hermitian connection whose curvature is proportional to the classical symplectic form. In CQM the quantum bundle is assumed to be a Hermitian line bundle over spacetime. Moreover, the extended quantum bundle is assumed to be equipped with a Hermitian connection whose curvature is universal and proportional to the classical cosymplectic form. Thus the novelties of CQM consist in the following minimal assumptions: the quantum bundle lives on spacetime and not on the phase space, the quantum connection is universal. Polarisation and projection method In GQ one realises that the base space of the quantum bundle is too big in order to obtain an irreducible representation of the classical Poisson Lie algebra and to fulfill the uncertainty principle. Then, one looks for a polarisation P , that is for a Lagrangian subbundle P of the complexified tangent bundle of the phase space M, such that D C = P ∩P has constant rank and P , P +P are closed under the Lie bracket; moreover, the polarisation is said to be reducible if the quotient of the phase space M by the distribution D exists and the canonical projection π : M → M/D is a submersion. Once a polarisation is chosen, we can consider the polarised sections of the quantum bundle, that is the sections whose covariant derivative with respect to every vector field of the polarisation vanish. The polarised sections should yield the Hilbert space with the correct size. Actually, the problem of finding polarisations is very hard in practice, should be faced case by case and leads to a lot of complications and ambiguities, where the beauty of the original programme misses over considerably. In CQM we have, in a covariant way, an implicit natural polarisation, namely the vertical polarisation. The quantum connection is the only source of all further quantum objects, such as quantum dynamical equation, probability current, quantum operators and so on. On the other hand, the quantum connection lives on the extended quantum bundle, while we are looking for further quantum objects living on the original quantum bundle. This goal is successfully achieved by the method of projectability. Thus, in simple words, the difficult search for the inclusion of a polarisation (which should be performed case by case) is substituted by the easy search for projectable objects (which is successful and can be performed in general). Half-densities and half-forms In GQ, once a polarisation P is chosen, we take the polarised sections of the quantum bundle. The problem is how to build a Hilbert space. For instance, if P is reducible, then the Hermitian product h(s 1 , s 2 ) of two polarised sections can be understood as a function on the quotient space M/D; but a problem arises because M/D has no natural volume element, which is necessary to define the Hilbert space of L 2 -polarised sections. One way to remedy this problem is to tensor out the sections by the half-densities associated to D and to use the natural partial flat connection of D to build the Hilbert space. Unfortunately, even for the simplest physical systems, the results do not agree with those found in Quantum Mechanics. Therefore, a further modification of the theory is needed: here is where half-forms come into play. They are defined through a metaplectic structure for the bundle of frames of the polarisation P ; this imposes new conditions, since metaplectic structures do not exist in general. Even further problems arise, since it may happen that the tensor product of the quantum bundle with the bundle of half-forms has no polarised section at all. In CQM we do not see any trouble concerning half-densities and all that. It seems that this convenient feature depends on the fact that CQM gets rid of all problems concerning polarisations. In the first version of CQM the theory assumed a C-valued Hermitian metric of the quantum bundle. In this case the theory, in order to prove that quantum operators are Hermitian, needed to use half-densities. In the recent version of CQM the theory assumes a (C ⊗ Λ n V * E)-valued Hermitian metric of the quantum bundle. In this case the theory uses just sections of the quantum bundle and does not need half-densities at all. Schroedinger equation In GQ there is no clear Schrödinger equation in an explicit formulation ready to be directly compared with the Schrödinger equation of standard Quantum Mechanics. The Schrödinger equation in this framework is understood as the infinitesimal generator of the flow of the Hamiltonian acting on the Hilbert space. The problem is that in general the Hamiltonian does not preserve the chosen polarisation and therefore we are led to compare different polarisations, this is handled through the Blattner-Kostant-Sternberg kernel machinery, see [4], which is extremely cumbersome. Even here one encounters new problems since there are obstructions discovered by Blattner [4] in order to construct the BKS kernel. In CQM we have an explicit, covariant and intrinsic Schrödinger equation, which is immediately comparable with the standard Schrödinger equation [12,80] (see also [13,14,15,16,17,18,19]). Moreover, we prove that it comes from a quantum Lagrangian. Hilbert space and Hilbert bundle GQ and standard Quantum Mechanics deal with a Hilbert space, which usually consists of L 2 sections of the quantum bundle. CQM deals with a Hilbert bundle over time. This formulation is unusual but does not seem to be really in contrast with standard Quantum Mechanics. Once more, this novelty is related to the explicit role of time in the CQM. Feynmann path integral In CQM the Feynmann amplitudes appear very naturally in terms of parallel transport with respect to the quantum connection. In fact, the classical Lagrangian turns out to play the role of local symbol of the quantum connection of the extended quantum bundle over time. The proof of the equivalence of the Feynmann path integral with the covariant Schrödinger equation has not yet been worked out in detail. Energy Most of the practical difficulties of GQ run around the quantisation of energy. Actually, the classical Hamiltonian function has to be explicitly postulated and is not encoded in the basic structure of spacetime. Moreover, the energy requires a treatment quite different from the simpler approach required by other observables such as spacetime coordinates and momentum. Even in CQM the energy has a special role, but no hard problems arise in this respect. First of all, we stress that energy is encoded in the basic geometric structure of spacetime and that it appears explicitly, in a non covariant way, by means of the choice of an observer. If one accepts the point of view of Covariant Quantum Mechanics, maybe one can understand the difficulties of GQ from this perspective. In fact, in practice what is done in GQ is to take the vertical tangent (or cotangent) space of spacetime as phase space, instead of the first jet space; accordingly, GQ tries to formulate and quantise energy by methods related to vertical subspace. On the other hand, in CQM, it is very clear that energy is related to the horizontal aspect of phase space. We could roughly say that the vertical aspect of phase space is essentially related to the static geometry of spacetime, while the horizontal aspect is related to the dynamics. This observation can also be analysed in terms of the Lie algebra of quantisable functions and their tangent lift; actually, the quantisable functions dealing with the vertical aspects of spacetime constitute the subalgebra of affine functions, while energy is a quadratic function. In CQM, all quantisable functions can quantised on the same footing. Examples In GQ it is not granted that every reasonable physical example can be worked out. Actually, there are few examples that have been successfully solved. CQM reduces to standard Quantum Mechanics in the flat case. Hence, in CQM, all standard physical examples can be formulated; moreover, the Schrödinger equation and quantum operators corresponding to the quantisable functions can be explicitly and immediately computed. Of course, the integration of the Schrödinger equation and the computation of the energy spectrum is an analytical question which should be faced case by case. theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Quantum theory . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.3 Generality . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 10 3.4 Role of time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.5 Phase space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.6 Symplectic and cosymplectic structures . . . . . . . . . . . . . . . . . . 12 3.7 Classical Lie brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.8 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.9 Quantum structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.10 Polarisation and projection method . . . . . . . . . . . . . . . . . . . . 13 3.11 Half-densities and half-forms . . . . . . . . . . . . . . . . . . . . . . . 14 3.12 Schroedinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.13 Hilbert space and Hilbert bundle . . . . . . . . . . . . . . . . . . . . . 15 3.14 Feynmann path integral . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.15 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.16 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 P : J 1 E → V * E, called momentum. These components are observer independent, but depend on the chosen gauge of the starting Poincaré-Cartan form. On the other hand, given an observer o, each Poincaré-Cartan form Θ splits, according to the splitting of T * E induced by o, into the horizontal component −H[o] : J 1 E → T * T , called observed Hamiltonian, and the vertical component P[o] : J 1 E → V * E, called observed momentum. Moreover, the horizontal component of Ω, according to the splitting of T The cosymplectic form Ω yields in a covariant way the Hamiltonian lift of functions f : J 1 E → IR to vertical vector fields H[f ] : J 1 E → V J 1 E; consequently, we obtain the Poisson bracket {f, g} between functions of phase space. Given an observer, the law of motion can be expressed, in a non covariant way, in terms of the Poisson bracket and the Hamiltonian. More generally, chosen a time scale τ : J 1 E → T T , the cosymplectic form Ω yields, in a covariant way, the Hamiltonian lift of functions f of phase space to vector fields H τ [f ] : J 1 E → T J 1 E, whose time component is τ . In view of our developments in the Quantum Theory, we prove that H τ [f ] is projectable on a vector field X[f ] : E → T E if and only if the following conditions hold: i) the function f is quadratic with respect to the affine fibres of J 1 E → E with second fibre derivative f ′′ ⊗G, where f ′′ : E → T T , ii) τ = f ′′ . A function of this type is called special quadratic and has coordinate expression of the type R Abraham, J E Marsden, Foundations of mechanics. LondonSecond editionR. Abraham, J. E. Marsden: Foundations of mechanics, Second edition, The Benjamin, London, 1978. Le théorème de reduction de Marsden-Weinstein en géométrie cosymplectique et de contact. C Albert, J. Geom. Phys. 64627C. Albert: Le théorème de reduction de Marsden-Weinstein en géométrie cosymplectique et de contact, J. Geom. Phys., 6, n.4, (1989), 627. Quantization and representation theory. R J Blattner, Proc. Sympos. Pure. Math. 26R. J. Blattner: Quantization and representation theory, Proc. Sympos. Pure. Math., 26, 147-165 (1974). The metalinear geometry of non real polarizations. R J Blattner, LNM. 570Springer VerlagR. J. Blattner: The metalinear geometry of non real polarizations, LNM 570, Springer Verlag, Berlin, 11-45 (1977). Quantum mechanics of a spin particle in a curved spacetime with absolute time. D Canarutto, A Jadczyk, M Modugno, Rep. on Math. Phys. 36D. Canarutto, A. Jadczyk, M. Modugno: Quantum mechanics of a spin particle in a curved spacetime with absolute time, Rep. on Math. Phys., 36, 1 (1995), 95-140. Gradient vector fields on cosymplectic manifolds. F Cantrijn, M De León, E A Lacomba, J. Phys. A. 25F. Cantrijn, M. de León, E. A. Lacomba: Gradient vector fields on cosymplectic manifolds, J. Phys. A 25 (1992) 175-188. Quantization of a general dynamical system by Feynmann's path integral formulation. K S Cheng, J. Math. Phys. 13K. S. Cheng: Quantization of a general dynamical system by Feynmann's path integral formulation, J. Math. Phys., 13, 11 (1972), 1723-1726. M De Leon, J C Marrero, E Padron, On the geometric quantization of Jacobi manifolds. 386185M. de Leon, J. C. Marrero, E. Padron: On the geometric quantization of Jacobi manifolds, p. 6185 ff., J. Math. Phys. 38 (12), 1997. Cosymplectic reduction for singular momentum maps. M De Leon, M Saralegi, J. Phys. A. 26M. de Leon, M. Saralegi: Cosymplectic reduction for singular momentum maps, J. Phys. A 26 (1993) 5033- 5043. A universal model for cosymplectic manifolds. M De Leon, G M Tuynmann, J. Geom. Phys. 20M. de Leon, G. M. Tuynmann: A universal model for cosymplectic manifolds, J. Geom. Phys. 20, 1 (1996), 77-86. Dynamical theory in curved spaces. I, A review of the classical and quantum action principles. B S Dewitt, Rev. Mod. Phys. 26B. S. DeWitt: Dynamical theory in curved spaces. I, A review of the classical and quantum action principles, Rev. Mod. Phys. 26 (1957), 377-397. P A M Dirac, The principles of quantum mechanics, 4-th ed. (revised). Oxford University PressP. A. M. Dirac: The principles of quantum mechanics, 4-th ed. (revised), Oxford University Press, 1967. H.-D Doebner, Remarks on on linear quantum theory. preprint ASI-TPA, 02/98H.-D. Doebner: Remarks on on linear quantum theory, preprint ASI-TPA, 02/98. Goldin: On a general nonlinear Schrödinger equation admitting diffusion currents. H.-D Doebner, G A , Phys. Lett. A. 162H.-D. Doebner, G. A. Goldin: On a general nonlinear Schrödinger equation admitting diffusion currents, Phys. Lett. A, 162, (1992), 397-401. Goldin: Properties of nonlinear Schrödinger equations associated with diffeomorphism group representations. H.-D Doebner, G A , J. Phys. A: Math. Gen. 27H.-D. Doebner, G. A. Goldin: Properties of nonlinear Schrödinger equations associated with diffeomorphism group representations, J. Phys. A: Math. Gen., 27, (1994), 1771-1780. Introducing nonlinear gauge transformationsin a family of nonlinear Schrödinger equations. H.-D Doebner, G A Goldin, Phys. Rev. A. 545H.-D. Doebner, G. A. Goldin: Introducing nonlinear gauge transformationsin a family of nonlinear Schrödinger equations, Phys. Rev. A, 54, 5, (1996), 3764-3771. Nattermann: Gauge transformations in quantum mechanics and the unification of nonlinear Schrödinger equations. H.-D Doebner, G A Goldin, P , J. Math. Phys. 401H.-D. Doebner, G. A. Goldin, P. Nattermann: Gauge transformations in quantum mechanics and the unifi- cation of nonlinear Schrödinger equations, J. Math. Phys., 40, 1, (1999), 49-63. Hennig: On a path to nonlinear quantum mechanics. H.-D Doebner, J.-D , Symmetries in Science IX, Gruber ana dRamek Eds. New YorkPlenum PressH.-D. Doebner, J.-D. Hennig: On a path to nonlinear quantum mechanics, in Symmetries in Science IX, Gruber ana dRamek Eds., Plenum Press, New York, 1997, 81-90. Borel quantization: kinematics and dynamics. H.-D Doebner, P Nattermann, Acta Phys. Pol. B. 27H.-D. Doebner, P. Nattermann: Borel quantization: kinematics and dynamics, Acta Phys. Pol. B, 27, (1996), 2327-2339. H D Dombrowski, K Horneffer, Die Differentialgeometrie des Galileischen Relativitätsprinzips. 86H. D. Dombrowski, K. Horneffer: Die Differentialgeometrie des Galileischen Relativitätsprinzips, Math. Z. 86 (1964), 291-311. C Duval, The Dirac & Levy-Leblond equations and geometric quantization. P. L. García and A. Pérez-Rendón Eds., L.N.M. 1251Springer-VerlagC. Duval: The Dirac & Levy-Leblond equations and geometric quantization, in "Diff. Geom. Meth. in Math. Phys.", P. L. García and A. Pérez-Rendón Eds., L.N.M. 1251 (1985), Springer-Verlag, 205-221. C Duval, On Galilean isometries. 10C. Duval: On Galilean isometries, Clas. Quant. Grav. 10 (1993), 2217-2221. Bargmann structures and Newton-Cartan theory. C Duval, G Burdet, H P Künzle, M Perrin, Phys. Rev. D. 318C. Duval, G. Burdet, H. P. Künzle, M. Perrin: Bargmann structures and Newton-Cartan theory, Phys. Rev. D, 31, N.8 (1985), 1841-1853. Horvaty: Celestial mechanics, conformal structures, and gravitational waves. C Duval, G Gibbons, P , Phys. Rev. D. 43C. Duval, G. Gibbons, P. Horvaty: Celestial mechanics, conformal structures, and gravitational waves, Phys. Rev. D 43, 12 (1991), 3907-3921. Minimal gravitational coupling in the Newtonian theory and the covariant Schrödinger equation. C Duval, H P R G Künzle ; G, 16C. Duval, H. P. Künzle: Minimal gravitational coupling in the Newtonian theory and the covariant Schrödinger equation, G.R.G., 16, 4 (1984), 333-347. The Newtonian limit of general relativity. J Ehlers, Fisica Matematica Classica e Relatività. Elba 9-13 giugno 1989J. Ehlers: The Newtonian limit of general relativity, in "Fisica Matematica Classica e Relatività", Elba 9-13 giugno 1989, 95-106. Review of invariant time formulations of relativistic quantum theories. J R Fanchi, Found. Phys. 23J. R. Fanchi: Review of invariant time formulations of relativistic quantum theories, Found. Phys. 23 (1993), 487-548. Evaluating the validity of parametrized relativistic wave equations. J R Fanchi, Found. Phys. 24J. R. Fanchi: Evaluating the validity of parametrized relativistic wave equations, Found. Phys. 24 (1994), 543-562. Space-time approach to non-relativistic quantum mechanics. R P Feynmann, Reviews of Modern Physics. 20R. P. Feynmann: Space-time approach to non-relativistic quantum mechanics, Reviews of Modern Physics, Vol. 20 (1948), 267-287. Connections and 1-jet fibre bundle. P L García, Rendic. Sem. Mat. Univ. Padova. 47P. L. García: Connections and 1-jet fibre bundle, Rendic. Sem. Mat. Univ. Padova, 47 (1972), 227-242. The Poincaré-Cartan invariant in the calculus of variations. P L García, Symposia Mathematica. 14P. L. García: The Poincaré-Cartan invariant in the calculus of variations, Symposia Mathematica 14 (1974), 219-246. P L García, Cuantificación geometrica, Memorias de la R. Acad. de Ciencias de madrid XI. MadridP. L. García: Cuantificación geometrica, Memorias de la R. Acad. de Ciencias de madrid XI, Madrid, 1979. The Hamilton Cartan formalism in the calculus of variations. H Goldsmith, S Sternberg, Ann. Inst. Fourier, Grenoble. 23H. Goldsmith, S. Sternberg: The Hamilton Cartan formalism in the calculus of variations, Ann. Inst. Fourier, Grenoble, 23, 1 (1973), 203-267. Gotay: Constraints, reduction and quantization. M J , Journ. Math. Phys. 27M. J. Gotay: Constraints, reduction and quantization, Journ. Math. Phys., 27, (1986), 2051-2067. On a full quantization of the torus, In Quantization, coherent states and complex structures. M J Gotay, J.-P. Antoine et al. Eds., PlenumM. J. Gotay: On a full quantization of the torus, In Quantization, coherent states and complex structures, J.-P. Antoine et al. Eds., Plenum 1995, 55-62. On the Groenewold -Van Hoeve theorem for R 2n. M J Gotay, J. Math. Phys. 40M. J. Gotay: On the Groenewold -Van Hoeve theorem for R 2n , J. Math. Phys. 40 (1999), 2107-2116. Grundling: An obstruction to quantizing compact symplectic manifolds. M J Gotay, J Graboski, H B , Proc. A.M.S. 1281M. J. Gotay, J. Graboski, H. B. Grundling: An obstruction to quantizing compact symplectic manifolds, Proc. A.M.S. 128, 1 (1999). On quantising T * S 1. M J Gotay, H B Grundling, Rep. Math. Phys. 40M. J. Gotay, H. B. Grundling: On quantising T * S 1 , Rep. Math. Phys. 40 (1997), 107-123. Differential Geometry and its Applications. M J Gotay, H B Grundling, Proc. Conf. ConfBrno; BrnoMasaryk UniversityOn quantising T * S 1M. J. Gotay, H. B. Grundling: On quantising T * S 1 , "Differential Geometry and its Applications", Proc. Conf., Brno 1998, Masaryk University, Brno, 1999, 593-596. A Groenewold -Van Hoeve theorem for S 2. M J Gotay, H B Grundling, C A Hurst, Trans A.M.S. 348M. J. Gotay, H. B. Grundling, C. A. Hurst: A Groenewold -Van Hoeve theorem for S 2 , Trans A.M.S. 348 (1996), 1579-1597. Tuynmann: Obstruction results in quantization theory. M J Gotay, H B Grundling, G M , J. Nonlinear Sci. 6M. J. Gotay, H. B. Grundling, G. M. Tuynmann: Obstruction results in quantization theory, J. Nonlinear Sci. 6 (1996), 469-498. Four-dimensional formulation of Newtonian mechanics and their relation to the special and general theory of relativity. P Havas, Rev. Modern Phys. 36P. Havas: Four-dimensional formulation of Newtonian mechanics and their relation to the special and general theory of relativity, Rev. Modern Phys. 36 (1964), 938-965. On the definition and evolution of states in relativistic classical and quantum mechanics. L P Horwitz, Foun. Phys. 22L. P. Horwitz: On the definition and evolution of states in relativistic classical and quantum mechanics, Foun. Phys., 22 (1992), 421-448. Piron: Relativistic dynamics. L P Horwitz, C , Helv. Phys. Acta. L. P. Horwitz, C. Piron: Relativistic dynamics, Helv. Phys. Acta, 46 (1973), 316-326. Non relativistic limit of relativistic quantum mechanics. L P Horwitz, F C Rotbart, Phys. Rew. D. 24L. P. Horwitz, F. C. Rotbart: Non relativistic limit of relativistic quantum mechanics, Phys. Rew. D, 24 (1981), 2127-2131. Galilei general relativistic quantum mechanics revisited. A Jadczyk, J Janyška, M Modugno, Geometria, Física-Matemática e outros Ensaios. Gomes, A. S. Alves, F. J. Craveiro de Carvalho and J. A. Pereira da Silva Eds.CoimbraA. Jadczyk, J. Janyška, M. Modugno: Galilei general relativistic quantum mechanics revisited, in "Geometria, Física-Matemática e outros Ensaios", Homenagem a António Ribeiro Gomes, A. S. Alves, F. J. Craveiro de Carvalho and J. A. Pereira da Silva Eds., Coimbra 1998, 253-313. An outline of a new geometric approach to Galilei general relativistic quantum mechanics. A Jadczyk, M Modugno, C. N. Yang, M. L. Ge and X. W. Zhou Eds.World ScientificSingaporeDifferential geometric methods in theoretical physicsA. Jadczyk, M. Modugno: An outline of a new geometric approach to Galilei general relativistic quantum mechanics, in "Differential geometric methods in theoretical physics", C. N. Yang, M. L. Ge and X. W. Zhou Eds., World Scientific, Singapore, 1992, 543-556. A scheme for Galilei general relativistic quantum mechanics. A Jadczyk, M Modugno, ; M Cerdonio, R , General Relativity and Gravitational Physics. World ScientificA. Jadczyk, M. Modugno: A scheme for Galilei general relativistic quantum mechanics, in "General Relativity and Gravitational Physics", M. Cerdonio, R. D'Auria, M. Francaviglia and G. Magnano Eds., World Scientific, Singapore, 1994, 319-337. A Jadczyk, M Modugno, Galilei general relativistic quantum mechanics, report. A. Jadczyk, M. Modugno: Galilei general relativistic quantum mechanics, report, 1994, 1-215. J Janyška, Remarks on symplectic and contact 2-forms in relativistic theories, Bollettino U.M.I. J. Janyška: Remarks on symplectic and contact 2-forms in relativistic theories, Bollettino U.M.I. (7) 9-B (1995), 587-616. J Janyška, Natural quantum Lagrangians in Galilei quantum mechanics. Rendiconti di Matematica, S. VIIRoma15J. Janyška: Natural quantum Lagrangians in Galilei quantum mechanics, Rendiconti di Matematica, S. VII, Vol. 15, Roma (1995), 457-468. Natural Lagrangians for quantum structures over 4-dimensional spaces. J Janyška, Rend. di Mat., S VII18RomaJ. Janyška: Natural Lagrangians for quantum structures over 4-dimensional spaces, Rend. di Mat., S VII, Vol 18, Roma (1998), 623-648. J Janyška, A remark on natural quantum Lagrangians and natural generalized Schroedinger operators in Galilei Quantum Mechanics. preprintJ. Janyška: A remark on natural quantum Lagrangians and natural generalized Schroedinger operators in Galilei Quantum Mechanics, preprint, 2000. J Janyška, M Modugno, Classical particle phase space in general relativity, in "Differential Geometry and its Applications. BrnoMasaryk UniversityProc. Conf.. electronic editionJ. Janyška, M. Modugno: Classical particle phase space in general relativity, in "Differential Geome- try and its Applications", Proc. Conf., Brno 1995, Masaryk University 1996, 573-602, electronic edition: http://www.emis.de/proceedings. Relations between linear connections on the tangent bundle and connections on the jet bundle of a fibred manifold. J Janyška, M Modugno, Arch. Math. (Brno). 32electronic editionJ. Janyška, M. Modugno: Relations between linear connections on the tangent bundle and connec- tions on the jet bundle of a fibred manifold, Arch. Math. (Brno), 32 (1996), 281-288, electronic edition: http://www.emis.de/journals. J Janyška, M Modugno, Quantisable functions in general relativity, to appear in Differential Operators in Mathematical Physics. J. Janyška, M. Modugno: Quantisable functions in general relativity, to appear in Differential Operators in Mathematical Physics, 1997. On quantum vector fields in general relativistic quantum mechanics. M Janyška, Modugno, Proc. 3rd Internat. Workshop Diff. 3rd Internat. Workshop DiffSibiu (Romania5Geom. and its Appl.J Janyška, M. Modugno : On quantum vector fields in general relativistic quantum mechanics, General Math- ematics 5 (1997), Proc. 3rd Internat. Workshop Diff. Geom. and its Appl., Sibiu (Romania) 1997, 199-217. On the graded Lie algebra of quantisable forms. J Janyška, M Modugno, Proc. Conf. ConfBrnoMasaryk Univ. Brnoelectronic editionJ. Janyška, M. Modugno: On the graded Lie algebra of quantisable forms, in "Diff. Geom. and Appl.", Proc. Conf. Brno 1998, Masaryk Univ. Brno, 1999, 601-620, electronic edition: http://www.emis.de/journals. L Kaplan, N T Maitra, E J Heller, Quantizing constrained systems: new perspectives, preprint. L. Kaplan, N. T. Maitra, E. J. Heller: Quantizing constrained systems: new perspectives, preprint, 1998. I Kolář, P Michor, J Slovák, Natural operators in differential geometry. BerlinSpringer-VerlagI. Kolář, P. Michor, J. Slovák: Natural operators in differential geometry, Springer-Verlag, Berlin, 1993. Quantization and unitary representations. B Kostant, LNM. 170Springer-VerlagB. Kostant: Quantization and unitary representations, LNM 170, Springer-Verlag, Berlin, (1970), 87-208. Variational sequences on finite order jet spaces. D Krupka, Proc. Conf. on Diff. Conf. on DiffNew YorkWorld ScientificD. Krupka: Variational sequences on finite order jet spaces, in Proc. Conf. on Diff. Geom. and its Appl., World Scientific, New York, 1990, 236-254. Gravitation, geometry and nonrelativistic quantum theory. K Kuchař, Phys. Rev. D. 22K. Kuchař: Gravitation, geometry and nonrelativistic quantum theory, Phys. Rev. D, 22, 6 (1980), 1285-1299. H P Künzle, Galilei and Lorentz structures on space-time: comparison of the corresponding geometry and physics. 17H. P. Künzle: Galilei and Lorentz structures on space-time: comparison of the corresponding geometry and physics, Ann. Inst. H. Poinc. 17, 4 (1972), 337-362. Galilei and Lorentz invariance of classical particle interaction. H P Künzle, Symposia Mathematica. 14H. P. Künzle: Galilei and Lorentz invariance of classical particle interaction, Symposia Mathematica 14 (1974), 53-84. Covariant Newtonian limit of Lorentz space-times. H P Künzle, G.R.G. 7H. P. Künzle: Covariant Newtonian limit of Lorentz space-times, G.R.G. 7, 5 (1976), 445-457. General covariance and minimal gravitational coupling in Newtonian space-time. H P Künzle, Geometrodynamics, A. BolognaTecnoprintPrastaro ed.H. P. Künzle: General covariance and minimal gravitational coupling in Newtonian space-time, in Geometro- dynamics, A. Prastaro ed., Tecnoprint, Bologna 1984, 37-48. Scalar time parametrization of relativistic quantum mechanics: the covariant Schrödinger formalism. A Kyprianidis, Phys. Rep. 155A. Kyprianidis: Scalar time parametrization of relativistic quantum mechanics: the covariant Schrödinger for- malism, Phys. Rep 155 (1987), 1-27. The geometry of inequivalent quantisations. N P Landsman, N Linden, Nucl. Phys. B. 365N. P. Landsman, N. Linden: The geometry of inequivalent quantisations, Nucl. Phys. B 365 (1991), 121-160. . M Le Bellac, J M Levy-Leblond, Nuovo Cim. 14 B. 2Galilean electromagnetismM. Le Bellac, J. M. Levy-Leblond: Galilean electromagnetism, Nuovo Cim. 14 B, 2 (1973), 217-233. J M Levy-Leblond, Galilei group and Galilean invariance. E. M. Loebl EdNew YorkAcademic2Group theory and its applicationsJ. M. Levy-Leblond: Galilei group and Galilean invariance, in "Group theory and its applications", E. M. Loebl Ed., Vol. 2, Academic, New York, 1971, 221-299. P Libermann, Sur les automorphismes infinitésimaux des streuctures symplectiques et des structures de contact, Colloque de Géométrie Différentielle Globale. Bruxelles, 1958, Gauthiers-Villars. ParisP. Libermann: Sur les automorphismes infinitésimaux des streuctures symplectiques et des structures de contact, Colloque de Géométrie Différentielle Globale. Bruxelles, 1958, Gauthiers-Villars, Paris, 1959. P Libermann, . M Ch, Marle, Symplectic Geometry and Analytical Mechanics. DordrechtReidel PublP. Libermann, Ch. M. Marle: Symplectic Geometry and Analytical Mechanics, Reidel Publ., Dordrecht, 1987. L Mangiarotti, Mechanics on a Galilean manifold. 5L. Mangiarotti: Mechanics on a Galilean manifold, Riv. Mat. Univ. Parma (4) 5 (1979), 1-14. L Mangiarotti, M Modugno, Fibred Spaces, Jet Spaces and Connections for Field Theories. BolognaGeometry and PhysicsL. Mangiarotti, M. Modugno: Fibred Spaces, Jet Spaces and Connections for Field Theories, in Geometry and Physics, Pitagora Editrice, Bologna, 1983, 135-165. C.-M Marle, Lie group action on a canonical manifold, Symplectic Geometry. Boston, PitmanResearch notes in mathematicsC.-M. Marle: Lie group action on a canonical manifold, Symplectic Geometry (Research notes in mathematics), Boston, Pitman, 1983. Ratiu: Introduction to Mechanics and Symmetry. J E Marsden, T , Texts in Appl. Math. 17SpringerJ. E. Marsden, T. Ratiu: Introduction to Mechanics and Symmetry, Texts in Appl. Math. 17, Springer, New York, 1995. Classical dynamics of non-holonomic systems: a geometric approach. E Massa, E Pagani, Ann. Inst. H. Poinc. 55E. Massa, E. Pagani: Classical dynamics of non-holonomic systems: a geometric approach, Ann. Inst. H. Poinc. 55, 1, (1991), 511-544. Jet bundle geometry, dynamical connections and the inverse problem of Lagrangian mechanics. E Massa, E Pagani, Ann. Inst. H. Poinc. E. Massa, E. Pagani: Jet bundle geometry, dynamical connections and the inverse problem of Lagrangian mechanics, Ann. Inst. H. Poinc., (1993). A Messiah, Quantum mechanics. II, III, North-HollandIA. Messiah: Quantum mechanics, Vol. I, II, III, North-Holland, 1961. On a covariant formulation of quantum mechanics. M Modugno, V K Doebner, J Dobrev, Hilgert, Proceedings of the II International Workshop "Lie Theory and Its Applications in Physics. the II International Workshop "Lie Theory and Its Applications in PhysicsWorld ScientificM. Modugno: On a covariant formulation of quantum mechanics, in Proceedings of the II International Workshop "Lie Theory and Its Applications in Physics", August 17 -20, 1997, Clausthal, Editors H.-D. Doebner, V. K. Dobrev and J. Hilgert, World Scientific, 1998, 183-203. M Modugno, Covariant Quantum Mechanics. Book in preparationM. Modugno: Covariant Quantum Mechanics, Book in preparation, 2000. A covariant approach to the quantisation of a rigid body. M Modugno, C Prieto, R Vitolo, preprintM. Modugno, C. Tejero Prieto, R. Vitolo: A covariant approach to the quantisation of a rigid body, preprint, 1999. Quantum connection and Poincaré-Cartan form. M Modugno, R Vitolo, Gravitation, electromagnetism and geometrical structures. G. Ferrarese Ed., Pitagora Editrice BolognaM. Modugno, R. Vitolo: Quantum connection and Poincaré-Cartan form, in "Gravitation, electromagnetism and geometrical structures", G. Ferrarese Ed., Pitagora Editrice Bologna, 1996, 237-279. Handbuch der Physik. W Pauli, S. Flügge Ed. VSpringerW. Pauli: in "Handbuch der Physik", S. Flügge Ed., Vol. V, 18-19, Springer, Berlin, 1958. Symmetries in covariant classical mechanics. D Saller, R Vitolo, preprintD. Saller, R. Vitolo: Symmetries in covariant classical mechanics, preprint 1999. D J Saunders, The geometry of jet bundles. Cambridge Univ. PressD. J. Saunders: The geometry of jet bundles, Cambridge Univ. Press, 1989. Quantum mechanics in non inertial frames of reference. E Schmutzer, J Plebanski, Fortschritte der Physik. 25E. Schmutzer, J. Plebanski: Quantum mechanics in non inertial frames of reference, Fortschritte der Physik 25 (1977), 37-82. Quantization of nonlinear systems. I Segal, Journ. Math. Phys. 1I. Segal : Quantization of nonlinear systems, Journ. Math. Phys. 1 (1960), 468-488. Geometric quantization and quantum mechanics. J Sniaticki, SpringerNew YorkJ. Sniaticki: Geometric quantization and quantum mechanics, Springer, New York, 1980. J.-M Souriau, Structure des systèmes dynamiques. ParisDunod UniversitéJ.-M. Souriau: Structure des systèmes dynamiques, Dunod Université, Paris, 1970. Sur la théorie Newtonienne de la gravitation. A Trautman, C. R. Acad. Sc. Paris, t. 257A. Trautman: Sur la théorie Newtonienne de la gravitation, C. R. Acad. Sc. Paris, t. 257 (1963), 617-620. Comparison of Newtonian and relativistic theories of space-time. A Trautman, Perspectives in geometry and relativity. Indiana Univ. PressA. Trautman: Comparison of Newtonian and relativistic theories of space-time, In "Perspectives in geometry and relativity", N. 42, Indiana Univ. Press, 1966, 413-425. An intrinsic formulation of nonrelativistic analytical mechanics and wave mechanics. W M Tulczyjew, J. Geom. Phys. 2W. M. Tulczyjew: An intrinsic formulation of nonrelativistic analytical mechanics and wave mechanics, J. Geom. Phys., 2, 3 (1985), 93-105. Lagrangian mechanics in invariant form. A M Vershik, L D Faddeev, Sel. Math. Sov. 4A. M. Vershik, L. D. Faddeev: Lagrangian mechanics in invariant form, Sel. Math. Sov. 4 (1981), 339-350. Relatività Generale e Fisica della Gravitazione. R Vitolo, Quantum structures in general relativistic theories. SingaporeWorld ScientificProc. XII Convegno NazionaleR. Vitolo: Quantum structures in general relativistic theories, in "Relatività Generale e Fisica della Gravi- tazione", Proc. XII Convegno Nazionale, Rome 23-27 September 1996, World Scientific, Singapore. R Vitolo, Spherical symmetry in Galilei general relativity. 64R. Vitolo: Spherical symmetry in Galilei general relativity, Ann. Inst. H. Poinc., 64, 2 (1996), 177-203. Quantising a rigid body. R Vitolo, Proc. Conf. ConfBrnoMasaryk UnivR. Vitolo: Quantising a rigid body, in "Diff. Geom. and Appl.", Proc. Conf. Brno 1998, Masaryk Univ. 1999, electronic edition: http://www.emis.de/journals. R Vitolo, Quantum structures in Galilei general relativity. 70R. Vitolo: Quantum structures in Galilei general relativity, Ann. Inst. H. Poinc., 70, 3 (1999), 239-257. R Vitolo, Quantum structures in Einstein general relativity. preprintR. Vitolo: Quantum structures in Einstein general relativity, preprint 2000. N Woodhouse, Geometric quantization. OxfordClarendon PressN. Woodhouse: Geometric quantization, Clarendon Press, Oxford, 2nd Edit. 1992.
{'fraction_non_alphanumeric': 0.0638107215846565, 'fraction_numerical': 0.02770004716239585, 'mean_word_length': 3.9025818882466283, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 4, 'xml': 0}, 'pii_count': 3, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 129, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'We compare the covariant formulation of Quantum Mechanics on a curved spacetime fibred on absolute time with the standard Geometric Quantisation.', 'arxivid': 'math-ph/0003029', 'author': ['Marco Modugno [email protected] \nDepartment of Applied Mathematics Via S. Marta 3\nUniversity of Florence\n50139FlorenceItaly\n', 'Carlos Tejero Prieto [email protected] \nDepartment of Mathematics\nUniversity of Salamanca\nPlaza de la Merced 4SalamancaSpain\n', "Raffaele Vitolo [email protected] \nDepartment of Mathematics 'E. De Giorgi' Via per Arnesano\nUniversity of Lecce\n73100LecceItaly\n"], 'authoraffiliation': ['Department of Applied Mathematics Via S. Marta 3\nUniversity of Florence\n50139FlorenceItaly', 'Department of Mathematics\nUniversity of Salamanca\nPlaza de la Merced 4SalamancaSpain', "Department of Mathematics 'E. De Giorgi' Via per Arnesano\nUniversity of Lecce\n73100LecceItaly"], 'corpusid': 14003044, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 19827, 'n_tokens_neox': 16860, 'n_words': 10010, 'pdfsha': '6bf15a392d15e0392d756c53f0da143227a427ee', 'pdfurls': ['https://export.arxiv.org/pdf/math-ph/0003029v1.pdf'], 'title': ['Comparison between Geometric Quantisation and Covariant Quantum Mechanics', 'Comparison between Geometric Quantisation and Covariant Quantum Mechanics'], 'venue': []}
arxiv
New spin generalisation for long range interaction models 24 Aug 2006 N Crampé [email protected] Department of Mathematics Heslington University of York YO10 5DDYorkUnited Kingdom New spin generalisation for long range interaction models 24 Aug 2006Dedicated to my PhD supervisor and friend D. Arnaudon We study new interactions between degrees of freedom for Calogero, Sutherland and confined Calogero spin models. These interactions are encoded by the generators of the Lie algebra so(N ) or sp(N ). We find the symmetry algebras of these new models: the half-loop algebra based on so(N ) or sp(N ) for the Calogero models and the Yangian of so(N ) or sp(N ) for the two types of other models. Surprisingly, these symmetry occur only for a specific value of the coupling constant. Introduction The Calogero and Sutherland models are one-dimensional many body problems with long range interactions [1,2]. The introduction of a gl(N) internal degree of freedom in these models [3][4][5] has proved to be fruitful in various physical and mathematical investigations. This is well illustrated in the study of their symmetries which turn out to be the half-loop algebra or the Yangian associated to gl(N) [6][7][8][9]. This letter is devoted to the introduction of new interactions between the internal degrees of freedom in these models and finding the symmetry algebra of these new models. The interactions are defined thanks to the fundamental representation of the generators of the Lie algebras so(N) or sp(N). We will call these new models so(N) or sp(N)-spin models. At this point, to avoid ambiguity, let us remark that these models are different from the so-called BC N models [10]. Indeed, in the latter case, it is the potential which is closely related to the root systems of the algebra BC N and such models possess reflection algebra symmetry [11]. The plan of the letter is as follows. In section 1, we introduce the definitions and different notations used in the letter. Then, in the three following sections which have the same structure, we introduce the Hamiltonian for Calogero so(N) or sp(N) spin model, Sutherland so(N) or sp(N) spin model and confined Calogero so(N) or sp(N) spin model. The main results of this letter consists in finding for each model the symmetry algebra. We finish by an appendix where technical details for computations are gathered. General setting Let x 1 , . . . , x L be the positions of L particles on a one-dimensional space. We associate to each particle an internal degree of freedom or spin which will be considered as a vector belonging to C N . The spin operators E ab i (1 a, b N and 1 i L) are matrices with entry 1 in row a and column b and zero elsewhere which act on the spin space of the i th particle. They provide a representation of ⊕ L 1 gl(N) and they satisfy the following commutation relations [E ab i , E cd j ] = δ ij δ bc E ad i − δ ad E cb i . (1.1) Let θ 0 = ±1. For each index 1 a N, we introduce the following sign, for N even, θ a = +1 for 1 a N 2 , θ 0 for N 2 + 1 a N ,(1.2) and θ a = +1 if N is odd. We introduce also the following conjugate indexā a = N + 1 − a for 1 a N . (1.3) In particular θ a θā = θ 0 . These definitions allow us to deal simultaneously with the Lie algebras so(N) and sp(N), subalgebras of gl(N). Let us define, for 1 a, b N, F ab = E ab − θ a θ b Ebā . (1.4) The algebra g + (N) (resp. g − (N)) spanned by these generators is isomorphic to so(N) (resp. sp(N)) when θ 0 = +1 (resp. θ 0 = −1). Of course, the case θ 0 = −1 occurs only when N is even. These generators satisfy the symmetry relation F ab = −θ a θ b Fbā . (1.5) To correctly define the structure constants and a non-degenerate metric tensor of g ± (N), we need to restrict this set {F ab } of generators to a basis of g ± (N). Let us define the subsets of indices E + = {(a, b)|ā > b} and E − = {(a, b)|ā b}. The sets B ± = {F ab |(a, b ) ∈ E ± } form bases of Lie algebras g ± (N). Then, the commutation relations of g ± (N) can be written as follows, for (a, b), (c, d) ∈ E ± , [F ab i , F cd j ] = δ ij (e,f )∈E ± f ab,cd ef F ef i , (1.6) where the structure constants read f ab,cd ef = δ bc (δ a e δ d f − θ a θ d δ ā f δ d e ) − δ ad (δ b f δ c e − θ b θ c δ b e δ cf ) −δ ac (θ a θ b δ b e δ d f − θ c θ d δ b f δ d e ) + δ bd (θ a θ b δ ā f δ c e − θ c θ d δ a e δ cf ) H(ē, f ) . (1.7) The function H(i, j) is defined, for (i, j) ∈ E ± , as follows 1 2 if i = j . H(i, j) = 1 if i > j , (1.8) The factor 1/2 in the function H is relevant only in the case where we consider g − (N) and is due to the particular choice of the normalisation of the generators. We choose the non-degenerate metric tensor as follows, for (a, b), (c, d) ∈ E ± , g ab,cd = 1 2 Tr(F ab F cd ) . (1.9) This metric will allow us to raise or lower the indices of the structure constants. Calogero model In this section, we will obtain the symmetry algebra of the Calogero g ± (N)-spin model which we defined through the following Hamiltonian H C = − L j=1 ∂ 2 ∂x 2 j + j =k λ 2 − λP jk + λQ jk (x j − x k ) 2 . (2.1) The matrix P jk permutes the spins of the j th and k th particles and can be written in terms of the spin operators as P jk = N a,b=1 E ab j E ba k . (2.2) The operator Q jk is defined by Q jk = N a,b=1 θ a θ b E ab j Eāb k . (2.3) They satisfy, in particular, the useful properties P jk = P kj and Q jk = Q kj . These two operators are the crucial elements to construct the R-matrix associated to the Yangian of so(N) or sp(N) [12][13][14][15]. The introduction in the Hamiltonian of the operator Q jk modifies the interaction between the degrees of freedom of the particles as compared with the gl(N)-spin model. Note that we can write the new interactions in terms of the generators of g ± (N) as follows P jk − Q jk = 1 2 N a,b=1 F ab j F ba k . (2.4) We have used in the previous formula the conventional notation (F j F k ) ab = N c=1 F ac j F cb k . It is well-known that the symmetry algebra of A N Calogero gl(N)-spin model is the half-loop algebra of gl(N) [6][7][8][9]. We shall show how this symmetry algebra is modified for Calogero g ± (N)-spin model. We introduce the following operators, for (a, b) ∈ E ± , J ab 0 = L j=1 F ab j , (2.5) J ab 1 = L j=1 F ab j ∂ ∂x j − λ j =k (F j F k ) ab 1 x j − x k . (2.6) After a straightforward computation, we can show that these operators satisfy the following relations [J ab 0 , J cd 0 ] = f ab,cd ef J ef 0 , (2.7) [J ab 0 , J cd 1 ] = f ab,cd ef J ef 1 , (2.8) J ab 1 , J cd 0 , J ef 1 + J ef 1 , J ab 0 , J cd 1 + J cd 1 , J ef 0 , J ab 1 = 0 , (2.9) for the following particular value of the coupling constant λ = 2 N − 4θ 0 . (2.10) In relations (2.7) and (2.8), we have used the Einstein's notation for the repeated pair of indices (e, f ) but the sums are only for (e, f ) ∈ E ± (for example, we write explicitly this sum in (1.6)). From now on, we use this convention for the repeated indices. For the particular choice N = 4θ 0 , the denominator in (2.10) vanishes and therefore J ab 0 , J ab 1 are not well-defined. However, this corresponds to the case where we consider the algebra so(4) which is a non-simple Lie algebra. The higher level generators, J ab 2 , J ab 3 , . . . , are defined recursively from J ab 0 and J ab 1 . Relation (2.9), called Serre relation, guarantees that these generators are well-defined. We have the commutation relations [J ab n , J cd m ] = f ab,cd ef J ef n+m . (2.11) Relations (2.7)-(2.9) define the half-loop algebra (also called Gaudin algebra) associated to the Lie algebra g ± (N). To finish the proof of the symmetry, we show that J ab 0 and J ab 1 are conserved operators i.e. [H C , J ab 0 ] = 0 and [H C , J ab 1 ] = 0 . (2.12) The particular value (2.10) of λ is necessary and sufficient to prove the second relation in (2.12) whereas the first one holds for any value of λ. We have used, in particular, the two following properties P jk F ab j = F ab k P jk and Q jk F ab j = −Q jk F ab k . (2.13) Therefore, we have therefore shown that the symmetry algebra of the model described by the Hamiltonian (2.1) is the half-loop algebra associated to g ± (N). To obtain the symmetry, it was necessary to constrain the coupling constant. This feature is new in comparison with the Calogero gl(N)-spin model where the coupling constant remains arbitrary free. Sutherland model In this section, we introduce a new Sutherland spin model, called Sutherland g ± (N)-spin model, whith Hamiltonian given by H S = − L j=1 x j ∂ ∂x j 2 + j =k (λ 2 − λP jk + λQ jk ) x j x k (x j − x k ) 2 (3.1) and exhibit its symmetry algebra. It is well-known that the symmetry algebra of Sutherland gl(N)-spin model is the Yangian of gl(N) and we show that for this model it is the Yangian of g ± (N). The end of this section consists in proving this statement. For convenience, let us define the symmetriser of any three elements K A , K B , K C by {K A , K B , K C } = 1 24 σ∈S 3 K σ(A) K σ(B) K σ(C) ,(3.2) where S 3 is the group of the permutations of order 6. The Yangian of g ± (N) is the associative algebra generated by {K ab 0 , K ab 1 |(a, b) ∈ E ± } constrained by the following commutation relations [16], for (a, b), (c, d) ∈ E ± , [K ab 0 , K cd 0 ] = f ab,cd ef K ef 0 , (3.3) [K ab 0 , K cd 1 ] = f ab,cd ef K ef 1 ,(3.4) and by the Serre relations, for (a, b), (c, d), (e, f ) ∈ E ± , K ab 1 , K cd 0 , K ef 1 + K ef 1 , K ab 0 , K cd 1 + K cd 1 , K ef 0 , K ab 1 = λ 2 f ab αβ,ij f cd γδ,kl f ef ǫφ,mn f ij,kl,mn {K αβ 0 , K γδ 0 , K ǫφ 0 } . (3.5) We recall that we use the Einstein's notation for repeated indices but that the skipped sums are running over E ± . The Lie algebra indices are lowered or raised by the invariant non-degenerate metric tensor defined by relation (1.9). The following operators, for (a, b) ∈ E ± , K ab 0 = L j=1 F ab j , (3.6) K ab 1 = L j=1 F ab j x j ∂ ∂x j − λ j =k (F j F k ) ab x j + x k x j − x k , (3.7) give a representation of the Yangian of g ± (N) provided that the coupling constant λ (which is also the deformation parameter of the Yangian) takes the particular value (2.10). The first two relations Confined Calogero model This section is devoted to studying the symmetry algebra of the confined Calogero g ± (N)-spin model which is described by the following Hamiltonian H CC = H C + ω 2 L j=1 x 2 j . (4.1) The operator H C is the Hamiltonian of the Calogero model introduced in (2.1). Let us remark that the introduction of this harmonic potential breaks translation invariance. We shall prove that the linear combinations introduced in [7] to obtain the symmetry algebra of the confined Calogero gl(N)-spin chain model are also relevant in our case to obtain the symmetry algebra. Let us define a new set of operators, for (a, b) ∈ E ± J ab 0 = J ab 0 , (4.2) J ab 1 = J ab 2 − ω 2 O ab 2 ,(4.3) where we have introduced the new operators O ab n = L j=1 F ab j x n j . We can easily show that the set of operators {O ab n } satisfy the relations of the half-loop algebra (2.11). By computing [J ab 1 , J cd 1 ], we find the following explicit form for J ab 2 , J ab 2 = L j=1 F ab j ∂ 2 ∂x 2 j − λ j =k (F j F k ) ab x j − x k ∂ ∂x j − ∂ ∂x k + λ j =k (E j E k ) ab − θ a θ b (E j E k )bā − λF ab j (x j − x k ) 2 −λ 2 j =k =ℓ (F k F j F ℓ ) ab (x j − x k )(x j − x ℓ ) ,(4.4) where we have used this following contraction ( F k F j F ℓ ) ab = N α,β=1 F aα k F αβ j F βb ℓ . We can prove that the operators J ab 0 and J ab 1 provide a representation of the Yangian of g ± (N) Let us remark that the first relation in (4.8) holds for any λ whereas the second one requires that λ takes the particular value (2.10). Therefore we have proved that the symmetry of the model described by the Hamiltonian (4.1) is the Yangian of g ± (N) with the deformation parameter equals to 2ω N −4θ 0 . [J ab 0 , J cd 0 ] = f ab,cd ef J ef 0 , (4.5) [J ab 0 , J cd 1 ] = f ab,cd ef J ef 1 , (4.6) J ab 1 , J cd 0 , J ef 1 + J ef 1 , J ab 0 , J cd 1 + J cd 1 , J ef 0 , J ab 1 = 4λ 2 ω 2 f ab αβ,ij f cd γδ,kl f ef ǫφ,mn f ij,kl,mn {J αβ 0 , J γδ 0 , J ǫφ 0 } . Conclusion In this letter, we studied new interaction between degree of freedom for different models with long range interaction such as Calogero model, Sutherland model or confined Calogero model. For each one, we obtain the symmetry algebra. Several questions remain open. The Lax pair as well as the Dunkl operators are not computed for these new models. These different approaches may allow one to deeply understand the constraint on the coupling constant which appears here. Another problem consists in computing the eigenfunctions and the eigenvalues of these models. The knowledge of the symmetry may help for their resolution. Indeed, for previous cases like the Sutherland gl(N)-spin model whose the symmetry is the Yangian of gl(N), the algebra symmetry is crucial to find the spectrum (see e.g. [6,8,17]). Finally, we want to point out the problem of the "freezing" method. Indeed when the coupling constant is not constrained, it is possible to obtain non-dynamical spin chains known as Frahm-Polychronakos or Haldane-Shastry spin chains [18][19][20][21]. This method seems not to work for the models studied in this letter. where Y ab,αβ = N ℓ=1 {K αb 0 , K aℓ 0 , K ℓβ 0 } − {K aβ 0 , K αℓ 0 , K ℓb 0 } + θ α θ β {K aᾱ 0 , Kβ ℓ 0 , K ℓb 0 } − θ α θ β {Kβ b 0 , K aℓ 0 , K ℓᾱ 0 } . (A.6) We recall that {., ., .} is defined by relation (3.2). To simplify the notation, we introduce also new generators, for (a, b) ∈ E ± , Kbā 0 = −θ a θ b K ab 0 . (A.7) Finally, we compute Y ab,αβ using explicit expression (3.6) of K ab 0 to show that RHS given by (A.5) is equal to LHS given by (A.1) which finishes the proof of the Serre relation (3.5). Confined Calogero model The computation of relation (4.7) is simplified by remarking that its right-hand side is identical to the one of relation (3.5) (up to a factor 4ω 2 ) which has been computed previously. Its left-hand side (divided by −ω 2 ) can be reduced to (3.3) and(3.4) are easily proven by direct computation. We give some details for the computation of the Serre relation (3.5) in the appendix.By direct computation, we can also prove that[H S , K ab 0 ] = 0 and [H S , K ab 1 ] = 0 . (3.8)The first relation in (3.8) is true for any λ whereas the second one holds only and only if λ is equal to the particular value (2.10). Therefore, we have proved that the symmetry algebra of the model described by the Hamiltonian (2.1) is the Yangian of g ± (N) (for the deformation parameter equal to 2 N −4θ 0 ). , these relations are satisfied if and only if the parameter λ takes the particular value (2.10) whereas the parameter ω remains free. Obviously, in the limit ω tends to 0, we recover the half-loop algebra for the corresponding Calogero model. The commutation relations (4.5) and (4.6) are computed directly. Some details for the computation of the Serre relations are presented in the appendix.The proof of the symmetry is provided by the two following relations[H CC , J ab 0 ] = 0 and [H CC , J ab 1 ] = 0 . (4.8) f cd,ef αβ [J ab 2 , O αβ 2 ] + [O ab 2 , J αβ 2 ] + f ab,cd αβ [J ef 2 , O αβ 2 ] + [O ef 2 , J αβ 2 ] + f ef,ab αβ [J cd 2 , O αβ 2 ] + [O cd 2 , J αβ 2 ] (A.8)by remarking, in particular, that the generators O ab 2 and J ab 2 satisfy the Serre relations (2.9). Now, by direct computation of the commutator [J ab 2 , O αβ 2 ], using the explicit form (4.4) for J ab 2 , we can prove that the Serre relation (4.7) is satisfied. Acknowledgements: This work is supported by the TMR Network 'EUCLID' Integrable models and applications: from strings to condensed matter', contract number HPRN-CT-2002-00325.A AppendixIn this appendix, we give some details about the proof of the Serre relations(3.5)and (4.7).Sutherland modelUsing relation(2.8), the left-hand side of relation(3.5)can be written asBy direct computation, we obtainThe explicit form of K mn 2 is not relevant because the Jacobi identity for the structure constants implies that these terms vanish in (A.1). The operator X ab,αβ takes the following form,Since the right-hand side of (3.5) does not depend on the position, the function f (x j , x k ) must be constant. This constraint implies that λ = 2 N −4θ 0 and f (x j , x k ) = 1. Now, let us focus on the right-hand side. It contains a sum of 8 4 = 4096 terms. By hand, it would be about impossible to deal with this number of terms. Fortunately, formal computations with Maple allow us to reduce this number. Finally, the right-hand side can be written as RHS = λ 2 f cd,ef αβ Y ab,αβ + f ab,cd αβ Y ef,αβ + f ef,ab αβ Y cd,αβ , (A.5) Solution of the One-Dimensional N-Body Problems with Quadratic and/or Inversely Quadratic Pair Potentials. F Calogero, J. Math. Phys. 10419J. Math. Phys.F. Calogero, Solution of a Three-Body Problem in One Dimension, J. Math. Phys. 10 (1969) 2191. Ground State of a One-Dimensional N-Body System, J. Math. Phys. 10 (1969) 2197. Solution of the One-Dimensional N-Body Problems with Quadratic and/or Inversely Quadratic Pair Potentials, J. Math. Phys. 12 (1971) 419. Exact Results for a Quantum Many-Body Problem in One Dimension. II Phys. Rev. A5 (1972) 1372. Exact Ground-State Wave Function for a One-Dimensional Plasma. B Sutherland, Phys. Rev. Lett. 41083Phys. Rev.B. Sutherland, Exact Results for a Quantum Many-Body Problem in One Dimension, Phys. Rev. A4 (1971) 2019. Exact Results for a Quantum Many-Body Problem in One Dimension. II Phys. Rev. A5 (1972) 1372. Exact Ground-State Wave Function for a One-Dimensional Plasma Phys. Rev. Lett. 34 (1975) 1083. On Models with Inverse-Square Exchange. Z N C Ha, F D M Haldane, cond-mat/9204017Phys. Rev. 46Z.N.C. Ha and F.D.M. Haldane, On Models with Inverse-Square Exchange, Phys. Rev. B46 (1992) 9359 and cond-mat/9204017. Integrable Systems for Particles with Internal Degrees of Freedom. J A Minahan, A P Polychronakos, hep-th/9206046Phys. Lett. 302J.A. Minahan and A.P. Polychronakos, Integrable Systems for Particles with Internal Degrees of Freedom, Phys. Lett. B302 (1993) 265 and hep-th/9206046. Exchange operator formalism for integrable systems of particles. A P Polychronakos, hep-th/9202057Phys. Rev. Lett. 69A.P. Polychronakos, Exchange operator formalism for integrable systems of particles, Phys. Rev. Lett 69 (1992) 703 and hep-th/9202057. Yang-Baxter equation in spin chains with long range interactions. D Bernard, M Gaudin, F D M Haldane, V Pasquier, hep-th/9301084J. Phys. 26D. Bernard, M. Gaudin, F.D.M. Haldane, V. Pasquier, Yang-Baxter equation in spin chains with long range interactions, J. Phys. A26 (1993) 5219 and hep-th/9301084. The Yangian Deformation of the W-Algebras and the Calogero-Sutherland model. D Bernard, K Hikami, M Wadati, hep-th/9412194Proceeding of 6-th Nankai Workshop. eeding of 6-th Nankai WorkshopD. Bernard, K. Hikami and M. Wadati, The Yangian Deformation of the W-Algebras and the Calogero-Sutherland model, Proceeding of 6-th Nankai Workshop and hep-th/9412194. Symmetry of the Calogero model confined in the harmonic potential-Yangian and W algebra. K Hikami, Nucl. Phys. 28530J. Phys.K. Hikami, Symmetry of the Calogero model confined in the harmonic potential-Yangian and W algebra, J. Phys. A28 (1995) 131, Yangian symmetry and Virasoro character in a lattice spin with long range interaction, Nucl. Phys. B441 (1995) 530. A N -type Dunkl operators and new spin Calogero-Sutherland models. F Finkel, D Gómez-Ullate, A González-López, M A Rodríguez, R Zhdanov, hep-th/0102039Comm. Math. Phys. 221F. Finkel, D. Gómez-Ullate, A. González-López, M.A. Rodríguez and R. Zhdanov, A N -type Dunkl operators and new spin Calogero-Sutherland models, Comm. Math. Phys. 221 (2001) 477 and hep-th/0102039. Quantum integrable systems related to the Lie algebras. M A Olshanetsky, A Perelomov, Phys. Rep. 94313M.A. Olshanetsky and A.Perelomov, Quantum integrable systems related to the Lie algebras, Phys. Rep. 94 (1983) 313. Integrable N-particle Hamiltonians with Yangian or Reflection Algebra Symmetry. V Caudrelier, N Crampé, math-ph/0310028J. Phys. 37V.Caudrelier and N.Crampé, Integrable N-particle Hamiltonians with Yangian or Reflection Algebra Symmetry, J. Phys. A37 (2004) 6285 and math-ph/0310028. Relativistic factorized S-matrix in two dimensions having O(N) isotropic symmetry, Nucl. Phys. B133 (1978) 525 and Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field models. Al B Zamolodchikov, Al B Zamolodchikov, Ann. Phys. 120253Al. B. Zamolodchikov, Al. B. Zamolodchikov, Relativistic factorized S-matrix in two dimensions having O(N) isotropic symmetry, Nucl. Phys. B133 (1978) 525 and Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field models, Ann. Phys. 120 (1979) 253. Solutions of the Yang-Baxter equation. P P Kulish, E K Sklyanin, Zap. Nauchn. Sem. LOMI. 951596J. Sov. Math.P.P. Kulish, E.K. Sklyanin, Solutions of the Yang-Baxter equation, Zap. Nauchn. Sem. LOMI, 95 (1980) 129 and J. Sov. Math. 19 (1982) 1596. Quantum groups and Yang-Baxter equations. A P Isaev, Phys. Part. Nucl. 26501A.P. Isaev, Quantum groups and Yang-Baxter equations, Phys. Part. Nucl. 26 (1995) 501. Ragoucy, R-matrix presentation for (super)-Yangians Y (g). D Arnaudon, J Avan, N Crampé, L Frappat, math.QA/0111325J. Math. Phys. 44D. Arnaudon, J. Avan, N. Crampé, L. Frappat andÉ. Ragoucy, R-matrix presentation for (super)-Yangians Y (g), J. Math. Phys. 44 (2003) 302 and math.QA/0111325. Drinfel'd, Hopf algebras and the quantum Yang-Baxter equation. V G , Proceedings Int. Cong. Math. 32212Soviet. Math. Dokl.V.G. Drinfel'd, Hopf algebras and the quantum Yang-Baxter equation, Soviet. Math. Dokl. 32 (1985) 254, Quantum Groups, Proceedings Int. Cong. Math. Berkeley, California, USA (1986) 798, A new realization of Yangians and quantized affine algebras, Soviet. Math. Dokl. 36 (1988) 212. The orthogonal eigenbasis and norms of eigenvectors in the Spin Calogero-Sutherland Model. K Takemura, D Uglov, solv-int/9611006J. Phys. 30K. Takemura and D. Uglov, The orthogonal eigenbasis and norms of eigenvectors in the Spin Calogero-Sutherland Model, J. Phys. A30 (1997) 3685 and solv-int/9611006. Exact Jastrow-Gutzwiller resonating-valence-bond ground state of the spin-1/2 antiferromagnetic Heisenberg chain with 1/r2 exchange. F D M Haldane, Phys. Rev. Lett. 60635F.D.M. Haldane, Exact Jastrow-Gutzwiller resonating-valence-bond ground state of the spin-1/2 antiferromagnetic Heisenberg chain with 1/r2 exchange, Phys. Rev. Lett. 60 (1988) 635. Exact solution of an S=1/2 Heisenberg antiferromagnetic chain with long-ranged interactions. B S Shastry, Phys. Rev. Lett. 60639B.S. Shastry, Exact solution of an S=1/2 Heisenberg antiferromagnetic chain with long-ranged interactions, Phys. Rev. Lett. 60 (1988) 639. Lattice integrable systems of Haldane-Shastry type. A P Polychronakos, hep-th/9210109Phys. Rev. Lett. 702329A.P. Polychronakos, Lattice integrable systems of Haldane-Shastry type, Phys. Rev. Lett. 70 (1993) 2329 and hep-th/9210109. Spectrum of a spin chain with inverse square exchange. H Frahm, cond-mat/9303050J. Phys. 26H. Frahm, Spectrum of a spin chain with inverse square exchange, J. Phys A26 (1993) L473 and cond-mat/9303050.
{'fraction_non_alphanumeric': 0.07843473783575045, 'fraction_numerical': 0.042950313224062474, 'mean_word_length': 3.4284628538856166, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 16, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'We study new interactions between degrees of freedom for Calogero, Sutherland and confined Calogero spin models. These interactions are encoded by the generators of the Lie algebra so(N ) or sp(N ). We find the symmetry algebras of these new models: the half-loop algebra based on so(N ) or sp(N ) for the Calogero models and the Yangian of so(N ) or sp(N ) for the two types of other models. Surprisingly, these symmetry occur only for a specific value of the coupling constant.', 'arxivid': 'math-ph/0511062', 'author': ['N Crampé [email protected] \nDepartment of Mathematics Heslington\nUniversity of York\nYO10 5DDYorkUnited Kingdom\n'], 'authoraffiliation': ['Department of Mathematics Heslington\nUniversity of York\nYO10 5DDYorkUnited Kingdom'], 'corpusid': 119169372, 'doi': '10.1007/s11005-006-0076-4', 'github_urls': [], 'n_tokens_mistral': 8336, 'n_tokens_neox': 7306, 'n_words': 4265, 'pdfsha': '7f74ac7aff36e138e6efa41d24aa00f4986bfcda', 'pdfurls': ['https://export.arxiv.org/pdf/math-ph/0511062v2.pdf'], 'title': ['New spin generalisation for long range interaction models', 'New spin generalisation for long range interaction models'], 'venue': []}
arxiv
ESTIMATES OF KÄHLER METRICS ON NONCOMPACT FINITE VOLUME HYPERBOLIC RIEMANN SURFACES, AND THEIR SYMMETRIC PRODUCTS 19 May 2023 Anilatmaja Aryasomayajula Arijit Mukherjee ESTIMATES OF KÄHLER METRICS ON NONCOMPACT FINITE VOLUME HYPERBOLIC RIEMANN SURFACES, AND THEIR SYMMETRIC PRODUCTS 19 May 2023 Let X denote a noncompact finite volume hyperbolic Riemann surface of genus g ≥ 2, with only one puncture at i∞ (identifying X with its universal cover H). Let X := X ∪ {i∞} denote the Satake compactification of X. Let Ω X denote the cotangent bundle on X . For k ≫ 1, we derive an estimate for µ Ber,k X , the Bergman metric associated to the line bundle L k := Ω ⊗k X ⊗O X ((k − 1)i∞). For a given d ≥ 1, the pull-back of the Fubini-Study metric on the Grassmannian, which we denote by µ FS,k Sym d (X ) , defines a Kähler metric on Sym d (X ), the d-fold symmetric product of X . Using our estimates of µ Ber,k X , as an application, we derive an estimate for µ FS,k Sym d (X ),vol , the volume form associated to the (1,1)-form µ FS,k Sym d (X ) . Symmetric products of Riemann surfaces can be realized as the moduli space of vortices, and are of immense interest both in algebraic geometry, and theoretical physics, and have been vastly investigated (see [1], [8], [10], and [6]). The symmetric product of a compact hyperbolic Riemann surface can be embedded into a Grassmannian, via the cotangent bundle. The pull-back of the Fubini-Study metric defines a Kähler metric on the symmetric product of the compact hyperbolic Riemann surface. Using estimates of the Bergman metric associated to Bergman kernels associated to high tensor-powers of holomorphic line bundles, the authors of [3], derived estimates of the pull-back of the Fubini-Study metric. Similar results on Kähler metrics were realized in [2]. In this article, we extend the estimates from [3] to noncompact finite volume hyperbolic Riemann surfaces, which we now describe. Let X denote a noncompact finite volume hyperbolic Riemann surface of genus g ≥ 2. Let µ hyp X denote the natural metric on X, which is compatible with its complex structure, and with constant curvature −1. By Riemann uniformization theorem, X can be realized as Γ \ H, where Γ ⊂ PSL 2 (R) is a cofinite Fuchsian subgroup. Without loss of generality, we assume that i∞ is the only puncture of X (identifying X with its universal cover H). Furthermore, we assume that Γ i∞ , the stabilizer of the cusp i∞, is of the form Γ i∞ = 1 n 0 1 n ∈ Z . Let X := X ∪ {i∞}, denote the Satake compactification of X. The hyperbolic metric µ hyp X extends to a singular metric on X , which we denote by µ hyp X . The metric µ hyp X is smooth at all z ∈ X \{i∞}, and admits an integrable singularity at the puncture i∞. Let Ω X denote the cotangent bundle on X , and let L k := Ω ⊗k X ⊗O X (k − 1)i∞ . For k ≥ 1, let B k L denote the Bergman kernel associated to the line bundle L k . Furthermore, let · hyp denote the point-wise metric on H 0 X , L k , which is induced by the hyperbolic metric. Then, for z ∈ X \{i∞}, the associated Bergman metric is defined as µ Ber,k X (z) := − i 2π ∂ z ∂ z log B k L (z) hyp . Let Sym d (X ) denote the symmetric product of X . The hyperbolic metric µ hyp X defines a singular metric µ hyp X d on Sym d (X ), and let µ hyp X d ,vol denote the associated volume form. Let D be an effective divisor on X of degree d ≥ 1. For k ≥ 1 and large enough, we have the short exact sequence of sheaves over X 0 −→ L k ⊗ O X (−D) −→ L k −→ L k | D −→ 0, which induces the following injective homomorphism H 0 X , L k ⊗ O X (−D) −→ H 0 X , L k .(1) For a given d ≥ 1, and k ≥ 1 be large enough such that n k := (2k − 1)(g − 1) + k − 1 > d, let Gr(r k , n k ) denote the Grassmannian parametrizing r k -dimensional vector subspaces of C n k , where r k = n k − d. Then, from the homomorphism as in equation (1), we have the holomorphic embedding ϕ k,d L : Sym d (X ) ֒→Gr(r k , n k ) (z 1 , . . . , z d ) →H 0 X , L k ⊗ O X (−D) ⊂ H 0 X , L k , where D denotes the divisor x 1 + · · · + x d on X , and z j is the complex coordinate of the point x j ∈ X , for each 1 ≤ j ≤ d. For further details regarding the above holomorphic embedding, we refer the reader to [3]. For convenience of the reader, we recall relevant details in section 3.1 of this article. The Fubini-Study metric is the natural metric on Gr(r k , n k ), and let µ FS,k Sym d (X ),vol (z) denote the volume form, associated to the pull-back of the Fubini-Study metric on Gr(r k , n k ), via the holomorphic embedding ϕ k,d L . µ hyp X (z) is well defined, and we now state the first main result of the article, which is proved as Theorem 2.10 in section 2.3. Main Theorem 1. With notation as above, for k ≫ 1, and z ∈ X , we have the following estimate µ Ber,k X (z) µ hyp X (z) ≪ Γ k 2 , where the implied constant depends only on Γ. Furthermore, we have the following estimate lim k→∞ 1 k 2 µ Ber,k X (z) µ hyp X (z) ≤ 26 π . We now state the second main result of the article, which is proved as Theorem 3.1 in section 3.2. Main Theorem 2. With notation as above, for a given d ≥ 1, and k ≫ 1, and z ∈ Sym d (X ), we have the following estimate µ FS,k Sym(X ),vol (z) µ X d ,vol (z) ≪ Γ k 2d , where the implied constant depends only on Γ. Furthermore, we have the following estimate lim k→∞ 1 k 2d µ FS,k Sym(X ),vol (z) µ X d ,vol (z) ≤ 26 π d . Remark 1. The main technique involved in the proofs of the above theorems, is the analysis of the Poincaré series representing the Bergman kernel B k L . We utilize the fact that the complex vector space H 0 X , L k , the space of global sections of the line bundle L k , is isometric to S 2k (Γ), the complex vector space of cusp forms of weight-2k, with respect to the Fuchsian subgroup Γ. Furthermore, we apply the Fourier expansions of cusp forms in the neighborhood of i∞, demonstrating the interplay between geometry and number theory. Lastly, our estimates remain the same in the noncompact setting, as the ones derived in [3]. 2. Estimates of the Bergman metric on noncompact hyperbolic Riemann surface 2.1. Notation and background material. In this section, we setup the notation, and recall the background material required to prove Main Theorem 1. Let X be a noncompact finite volume hyperbolic Riemann surface of genus g ≥ 2. From uniformization theorem from complex analysis, X is isometric to Γ \ H, where Γ ⊂ PSL 2 (R) is a cofinite Fuchsian subgroup, which acts on H via fractional linear transformations. Locally, we identify X with its universal cover H. Let F Γ denote a fixed fundamental domain of X. Without loss of generality, we assume that F i∞ := z = x + iy ∈ H 0 ≤ x ≤ 1, y ≥ 1 ⊂ F Γ . The punctures of X are in one-one correspondence with cusps of Γ. Without loss of generality, we assume that i∞ is the only puncture of X, and we further assume that Γ i∞ = 1 n 0 1 n ∈ Z . Let X := X ∪ {i∞} denote the Satake compactification of X. We now describe the local coordinates of X . For z ∈ X \{i∞}, let U r (z) denote a coordinate disc around z, and of radius r > 0. For w ∈ U r (z), the local coordinate function ϑ z (w) is given by the following formula ϑ z (w) := w − z. Let U r (i∞) denote a coordinate disc around i∞, and of radius r > 0. For w ∈ U r (i∞), the local coordinate function ϑ i∞ (w), also denoted by q(z), is given by the following formula q(z) = ϑ i∞ (w) := e 2πiw .(2) Let µ hyp X denote the hyperbolic metric on H, and for z := x + iy ∈ H, the hyperbolic metric is given by the following formula µ hyp X (z) := i 2 · dzdz y 2 = dxdy y 2 . Let d hyp (z, w) denote the hyperbolic distance between the points z and w on H. For z = x + iy, w = u + iv ∈ H, the hyperbolic distance is given by the following formula cosh 2 d hyp (z, w)/2 = z − w 2 4yv .(3) The hyperbolic metric descends to define a Kähler metric on X. Locally, for any two points z, w ∈ X, the geodesic distance between the points z and w is given by d hyp (z, w). We now define the injectivity radius r Γ of X, which is by the following formula r Γ := inf d hyp (z, γz) z ∈ F Γ , γ ∈ Γ\Γ i∞ . The hyperbolic metric µ hyp X extends to a singular metric on X , which we denote by µ hyp X . Locally, for z ∈ U r (i∞), the hyperbolic metric is given by the following formula µ hyp X (z) = i 2 · dq(z)dq(z) q(z) log q(z) 2 , where U r (i∞) denote a coordinate disc around i∞, and of radius r > 0, and q(z) is as in equation (2). Bergman kernel and Cusp forms. Let Ω X denote the cotangent bundle of holomorphic differential 1-forms on X . For k ≥ 1, set L k := Ω ⊗k X ⊗O X (k − 1)i∞ . Let H 0 X , L k denote the space of global sections of L k , meromorphic differentials with a pole of order at most (k − 1) at the puncture i∞, and holomorphic at all z ∈ X \{i∞}. Let q(z) be as in equation (2). Then, for k ≥ 1, at z ∈ X \{i∞}, ω ∈ H 0 X , L k is of the form ω(z) = f (z)dz ⊗k = f (q(z))dq(z) ⊗k (2πiq(z)) k , The differential dq(z) ⊗k /q(z) k has a pole of order k at i∞. Since, ω has a pole of order at most k − 1 at i∞, it implies that f ∈ S 2k (Γ), the complex vector space of cusp forms of weight-k, with respect to Γ. Hence, we have an isometry H 0 X , L k ≃ S 2k (Γ) .(4) For w ∈ H 0 X , L k , where ω(z) = f (z)dz ⊗k , at z = x+ iy ∈ X (identifying X with its universal cover H), the point-wise hyperbolic metric · hyp on H 0 X, L k , is given by the following formula ω(z) 2 hyp = y 2k f (z) 2 . The point-wise metric · hyp induces an L 2 -metric on H 0 X , L k . For z = x + iy ∈ X (identifying X with its universal cover H), ω, η ∈ H 0 X , L k , where ω(z) = f (z)dz ⊗k , η(z) = g(z)dz ⊗k , the hyperbolic L 2 -metric ·, · hyp on H 0 X , L k , is given by the following formula ω, η hyp := F Γ y 2k f (z)g(z) µ hyp X (z).(5) Let n k denote the complex dimension of H 0 X , L k . Let ω 1 , . . . , ω n k be an orthonormal basis of H 0 X , L k with respect to the L 2 -metric ·, · hyp , which is given by equation (5). Then, for z, w ∈ X , the Bergman kernel associated to the line bundle L k , is given by the following formula B k L (z, w) := n k j=1 ω j (z)ω j (w). The Bergman kernel B k L is the generating kernel for the Hilbert space H 0 X , L k , and hence, by Reisz representation theorem, the definition of the Bergman kernel is independent of the choice of any orthonormal basis for H 0 X , L k . Furthermore, the Bergman kernel B k L is a holomorphic differential form in the z-variable, for z ∈ X \{i∞}, and an anti-holomorphic differential form in the w-variable, for w ∈ X \{i∞}. The point-wise hyperbolic metric on H 0 X , L k induces a point-wise hyperbolic metric on B k L along the diagonal, which at z ∈ X , is given by the following formula B k L (z) hyp = n k j=1 ω j (z) 2 hyp . We now define the Bergman metric associated to L k . For k ≥ 1, and z ∈ X \{i∞}, the Bergman metric associated to the line bundle L k , is given by the following formula µ Ber,k X (z) := − i 2π ∂ z ∂ z log B k L (z) hyp(6) We now recall the Bergman kernel associated to S 2k (Γ). For z = x + iy ∈ X (identifying X with its universal cover H), and f ∈ S 2k (Γ), the point-wise Petersson metric · pet on S 2k (Γ), is given by the following formula f (z) 2 pet := y 2k f (z) 2 , which in turn induces an L 2 -metric on S 2k (Γ), namely the Petersson inner-product on S 2k (Γ). For f, g ∈ S 2k (Γ), the Petersson inner-product f, g pet on S 2k (Γ), is given by the following formula f, g pet := F Γ y 2k f (z)g(z) µ hyp X (z).(7) Let f 1 , · · · , f n k be an orthonormal basis of S 2k (Γ), with respect to the Petersson innerproduct. Then for z, w ∈ H, the Bergman kernel B 2k Γ (z, w) associated to the complex vector space S 2k (Γ), is given by the following formula B 2k Γ (z, w) := n k j=1 f j (z)f j (w). The Bergman kernel B 2k Γ is a generating kernel for the Hilbert space S 2k (Γ), and hence, by Riesz representation theorem, the definition of the Bergman kernel is independent of the choice of orthonormal basis for S 2k (Γ). Furthermore, the Bergman kernel B 2k Γ is a holomorphic cusp form in the z-variable, and an anti-holomorphic cusp form in the w-variable. When z = w, for brevity of notation, we denote B 2k Γ (z, w) by B 2k Γ (z). The point-wise Petersson metric on S 2k (Γ) induces a point-wise Petersson metric on B 2k Γ along the diagonal, which at z = x + iy ∈ X (identifying X with its universal cover H), is given by the following formula B 2k Γ (z) pet := n k j=1 f j (z) 2 pet = n k j=1 y 2k f j (z) 2 = y 2k B 2k Γ (z).(8) We now state an alternative description of the Bergman kernel B 2k Γ , which for k ≥ 2, and z, w ∈ H, is given by the following Poincaré series, (see Proposition 1.3 on page 77 in [7]) B 2k Γ (z, w) = (2k − 1)(2i) 2k 4π γ∈Γ 1 (z − γw) 2k · 1 cw + d 2k , where γ = a b c d ∈ Γ .(9) For k ≥ 1, and z = x + iy ∈ X (identifying X with its universal cover H), applying isometry (4), using the fact that Im γ(z) = y cz + d 2 , where γ = a b c d ∈ Γ, and combining equations (4), (8), (9), and (3), we have the following equality of Bergman kernels and their estimate B k L (z) hyp = B 2k Γ (z) pet = 2k − 1 4π γ∈Γ (4y) k z − γz 2k · y k cz + d 2k ≤ 2k − 1 4π γ∈Γ 4y Im(γz) k z − γz 2k = 2k − 1 4π γ∈Γ 1 cosh 2k d hyp (z, γz)/2 .(10) We now define the Fourier expansion of cusp forms. For k ≥ 1, and z ∈ H, and f ∈ S 2k (Γ) admits the following Fourier expansion, which is of the following form f (z) = ∞ m=1 a m q m (z), where q(z) is as in equation (2), and a m : = 1 0 f (z)q −m (z)dx.(11) The constants a m m≥1 are known as Fourier coefficients of f , and encode a lot of arithmetic and geometric information. For a given k ≥ 1, set U Γ,k := z = x + iy ∈ F Γ y ≥ c Γ log(k) 2π ; U Γ,k := U Γ,k ∪ {i∞}, F Γ,k := F Γ \U Γ,k ,(12) where c Γ is a constant. For f j ∈ S 2k (Γ), let a j,m m≥1 denote the set of Fourier coefficients of the cusp form f j , which are as described in equation (11). 2.2. Auxiliary estimates. In this section, we compute n k , the dimension of S 2k (Γ), and for z ∈ F Γ,k , employing similar techniques as in [4] and [5], we compute estimates of the Bergman kernel B 2k Γ (z). In the following proposition, we compute n k = dim C H 0 X , L k , which is equal to the dimension of S 2k (Γ). The following result is well known, and we refer to [11], for further details. However, for the sake of completion, we present the main ideas behind the proof. Proposition 2.1. With notation as above, for k ≥ 2, we have n k := dim C H 0 X , L k = (2k − 1)(g − 1) + k − 1. Proof. For k ≥ 2, since deg Ω X ⊗(1−k) ⊗O X (1 − k)i∞ = (1 − k)(2g − 2) + (1 − k) < 0, from Serre duality, we have dim C H 1 X , L k = dim C H 0 X , Ω X ⊗(L k ) * = dim C H 0 X , Ω X ⊗(1−k) ⊗O X 1 − k)i∞ = 0. Therefore, applying the Riemann-Roch theorem, we derive n k = dim C H 0 X , L k = deg L k + (1 − g) = deg Ω ⊗k X ⊗O X (k − 1)i∞ + (1 − g) = k(2g − 2) + (k − 1) + (1 − g) = (2k − 1)(g − 1) + k − 1, which completes the proof of the proposition. Proposition 2.2. With notation as above, for a fixed k ≥ 2, let S := f 1 , . . . , f n k denote an orthonormal basis for S 2k (Γ), with respect to the Petersson inner-product (which is as defined in equation (7)). Then, we have the following lower bound n k j=1 a j,1 2 > 0. Proof. For a fixed k ≥ 2, since H 0 X , L k−1 H 0 X , L k , there exists atleast one ω ∈ H 0 X , L k , which is a meromorphic differential with a pole of order k−1 at i∞, and holomorphic on X \{i∞}. From isometry (4), there exists a cusp form f ∈ S 2k (Γ), such that ω(z) = f (z)dz ⊗k . For z ∈ X \{i∞}, from the Fourier expansion of the cusp form f , which is as described in equation (11), we infer that ω(z) = f (z)dz ⊗k = ∞ m=1 a m q m (z) dq(z) ⊗k 2πiq(z) k = 1 2πi ∞ m=1 a m q m−1 (z) dq(z) ⊗k 2πiq(z) k−1 Since ω has a pole or order k−1 at i∞, from the above computation, it follows that a 1 = 0, which implies that a j,1 = 0, for some 1 ≤ j ≤ n k , which completes the proof of the proposition. Proposition 2.3. With notation as above, for k ≥ 3, and z ∈ F Γ,k , we have the following estimate B k L (z) hyp = 2k − 1 4π + α Γ,k (z), where the function α Γ,k is a real valued function and satisfies the following estimate α Γ,k (z) = O Γ √ klog(k) , where the implied constant depends only on the Fuchsian subgroup Γ. Proof. For k ≥ 3, and z ∈ X \{i∞}, from equation (9), we have B 2k Γ (z) pet = 2k − 1 4π + 2k − 1 4π γ∈Γ i∞ \{Id} (4y) k z − γz 2k · y k cz + d 2k + 2k − 1 4π γ∈Γ \ Γ i∞ (4y) k z − γz 2k · y k cz + d 2k ,(13) where Id denotes the identity matrix. Substituting δ = 0 in equations (1.6) and (1.7 ) in [5], and slightly refining the arguments, we arrive at the following estimates 2k − 1 4π γ∈Γ i∞ \{Id} (4y) k z − γz 2k · y k cz + d 2k ≤ y(2k − 1) √ π · Γ(k − 1/2) Γ(k) ; 2k − 1 4π γ∈Γ \ Γ i∞ (4y) k z − γz 2k · y k cz + d 2k ≤ C k X ,(14) where C k X = 2k − 1 4π 16 cosh 2k−4 (r Γ /4) + 8 cosh 2k−3 (r Γ /2) + 2k − 1 2πsinh 2 (r Γ /4) 1 cosh 2k−3 (r Γ /2) + 1 cosh 2k−4 (r Γ /2) .(15) For z ∈ F Γ,k , combining the observation Γ(k − 1/2) Γ(k) = O 1 √ k , with equations (10) and (13), and estimates (14) and (15), we arrive at the following estimate B k L (z) hyp = B 2k Γ (z) pet = 2k − 1 4π + α Γ,k (z), where α Γ,k (z) = 2k − 1 4π γ∈Γ i∞ \{Id} (4y) k z − γz 2k · y k cz + d 2k + γ∈Γ \ Γ i∞ (4y) k z − γz 2k · y k cz + d 2k , and α Γ,k (z) = O Γ √ klog(k) , where the implied constant depends only on the Fuchsian subgroup Γ, which completes the proof of the proposition. 2.3. Estimates of the Bergman metric. In this subsection, extending techniques from [3], we prove Theorem 1. For k ≥ 1, and z = x + iy ∈ X \{i∞} (identifying X with its universal cover H), from the definition of the Bergman metric µ Ber,k X (z), which is as defined in equation (6), and combining it with equations (8), and (10), we arrive at the following equality µ Ber,k X (z) = − i 2π ∂ z ∂ z log B k Ω X (z) hyp = − i 2π ∂ z ∂ z log(y) 2k − i 2π ∂ z ∂ z log B 2k Γ (z).(16) In the following proposition, we relate the Bergman metric and the hyperbolic metric. Lemma 2.4. With notation as above, for k ≥ 1, and z ∈ X \{i∞}, we have µ Ber,k X (z) = k 2π µ hyp X (z) + y 2 π ∂ B 2k Γ (z) ∂z ∂ B 2k Γ (z) ∂z B 2k Γ (z) 2 − ∂ 2 B 2k Γ (z) ∂z∂z B 2k Γ (z) µ hyp X (z). Proof. The proof of the proposition follows from combining equation (16), and arguments from Proposition 2.1 from [3]. Lemma 2.5. With notation as above, for k ≥ 2, and z = x + iy ∈ H, we have the following estimate ∂ B 2k Γ (z) ∂z ≤ 2k y 2k+1 · 2k − 1 4π + y(2k − 1) √ π · Γ(k − 1/2) Γ(k) + C k X , where the constant C k X is as defined in equation (15). Proof. The proof of the lemma follows from combining Lemma 2.2 from [3], and arguments from proof of Proposition 2.3. Lemma 2.6. With notation as above, for k ≥ 3, and z = x + iy ∈ H, we have the following estimate ∂ B 2k Γ (z) ∂z ≤ 2k y 2k+1 · 2k − 1 4π + y(2k − 1) √ π · Γ(k − 1/2) Γ(k) + C k X , where the constant C k X is as defined in equation (15). Proof. The proof of the lemma follows from combining Lemma 2.3 from [3], and arguments from proof of Proposition 2.3. Lemma 2.7. With notation as above, for k ≥ 3, and z = x + iy ∈ H, we have the following estimate ∂ 2 B 2k Γ (z) ∂z∂z ≤ (10k 2 + k) 2y 2k+2 · 2k − 1 4π + y(2k − 1) √ π · Γ(k − 1/2) Γ(k) + C k X , where the constant C k X , is as defined in equation (15). Proof. The proof of the lemma follows from combining from Lemma 2.4 from [3], and arguments from proof of Proposition 2.3. We now estimate the Bergman metric in the compact subset F Γ,k , which is as defined in equation (12). Proposition 2.8. With notation as above, for k ≥ 3, and z ∈ F Γ,k , we have the following estimate µ Ber,k X (z) µ hyp X (z) ≤ k 2π + 4k 2 π B k Ω X (z) 2 hyp · 2k − 1 4π + c Γ (2k − 1)log(k) 2π √ π · Γ(k − 1/2) Γ(k) + C k X 2 + k 2 π B k Ω X (z) hyp · 2k − 1 4π + c Γ (2k − 1)log(k) 2π √ π · Γ(k − 1/2) Γ(k) + C k X · 5 + 1 2k , where the constant c Γ is as in equation (12). Proof. For k ≥ 3, and z ∈ F Γ,k , from Lemma 2.4, we have µ Ber,k X (z) µ hyp X (z) ≤ k 2π + y 2 π ∂ B 2k Γ (z) ∂z ∂ B 2k Γ (z) ∂z B 2k Γ (z) 2 + y 2 π ∂ 2 B 2k Γ (z) ∂z∂z B 2k Γ (z) .(17) We now estimate the second term on the right-hand side of the above inequality. For z ∈ F Γ,k , combining Lemmas 2.5 and 2.6, we arrive at the following estimate, for the second term on the right-hand side of inequality (17) y 2 π ∂ B 2k Γ (z) ∂z ∂ B 2k Γ (z) ∂z B 2k Γ (z) 2 ≤ 4k 2 π B 2k Γ (z) 2 pet · 2k − 1 4π + y(2k − 1) √ π · Γ(k − 1/2) Γ(k) + C k X 2 ≤ 4k 2 π B k Ω X (z) 2 hyp · 2k − 1 4π + c Γ (2k − 1)log(k) 2π √ π · Γ(k − 1/2) Γ(k) + C k X 2 .(18) We now estimate the third term on the right-hand side of inequality (17). For z ∈ F Γ,k , from Lemma 2.7, we arrive at the following estimate, for the third term on the right-hand side of inequality (17) y 2 π ∂ 2 B 2k Γ (z) ∂z∂z B 2k Γ (z) ≤ (10k 2 + k) 2π B 2k Γ (z) pet · 2k − 1 4π + y(2k − 1) √ π · Γ(k − 1/2) Γ(k) + C k X ≤ k 2 π B k Ω X (z) hyp · 2k − 1 4π + c Γ (2k − 1)log(k) 2π √ π · Γ(k − 1/2) Γ(k) + C k X · 5 + 1 2k .(19) Combining equation (17) with estimates (18), and (19), completes the proof of the proposition. In order to estimate the Bergman metric in U Γ,k , the neighborhood of the puncture i∞, we need to derive estimates of the Fourier coefficients of cusp forms, which we establish in the following proposition. Proposition 2.9. With notation as above, for k ≫ 1, and z ∈ U Γ,k , we have the following estimate µ Ber,k X (z) µ hyp X (z) = k 2π + β Γ (z), where the function β Γ satisfies the following estimate β Γ (z) = O Γ 1 k , and the implied constant depends only on the Fuchsian subgroup Γ. Proof. For k ≥ 2, and z ∈ U Γ,k , from Lemma 2.4, we have the following relation µ Ber,k X (z) µ hyp X (z) = k 2π + y 2 π ∂ B 2k Γ (z) ∂z ∂ B 2k Γ (z) ∂z B 2k Γ (z) 2 − y 2 π ∂ 2 B 2k Γ (z) ∂z∂z B 2k Γ (z) .(20) We now estimate the the second and third terms on the right-hand side of the above equation. Using the Fourier expansion of cusp forms, which is as described in equation (11), we arrive at the following q-expansions B 2k Γ (z) = q(z) 2 n k j=1 ∞ m=1 a j,m q m−1 (z) ∞ n=1 a j,n q n−1 (z) ; ∂ B 2k Γ (z) ∂z = 2πiq(z) n k j=1 ∞ m=1 ma j,m q m−1 (z) ∞ n=1 a j,n q n (z) ; ∂ B 2k Γ (z) ∂z = ∂ B 2k Γ (z) ∂z ; ∂ 2 B 2k Γ (z) ∂z∂z = 4π 2 q(z) 2 n k j=1 ∞ m=1 ma j,m q m−1 (z) ∞ n=1 na j,n q n−1 (z) ;(21) where {a j,n } ≥1 denotes the set of Fourier coefficients of the cusp form f j ∈ S 2k (Γ), and q(z) is as defined in equation (2), and n k is the dimension of S 2k (Γ). From Proposition 2.2, we know that n k j=1 a j,1 2 > 0. From equations described in equation (21), we compute B 2k Γ (z) 2 = q(z) 4 n k j=1 a j,1 2 2 1 + β Γ,1 (q(z)) 2 ,(22) where the function β Γ,1 satisfies the following estimate β Γ,1 (q(z)) = O Γ q(z) ,(23) where the implied constant depends only on the Fuchsian subgroup Γ. We further derive ∂ B 2k Γ (z) ∂z ∂ B 2k Γ (z) ∂z = 4π 2 q(z) 4 n k j=1 a j,1 2 2 1 + β Γ,2 (q(z)) , ∂ 2 B 2k Γ (z) ∂z∂z = 4π 2 q(z) 2 n k j=1 a j,1 2 1 + β Γ,3 (q(z)) ,(24) where the functions β Γ,2 , β Γ,3 , satisfies the following estimates, respectively β Γ,2 (q(z)) = O Γ q(z) ; β Γ,3 (q(z)) = O Γ q(z) .(25) where the implied constants depend only on the Fuchsian subgroup Γ. From equations (22) and (24), we derive the following estimate ∂ 2 B 2k Γ (z) ∂z∂z B 2k Γ (z) = 4π 2 1 1 + β Γ,1 (q(z)) + β Γ,3 (q(z)) 1 + β Γ,1 (q(z)) .(26) Combining equations (22) and (24), we arrive at the following estimate ∂ B 2k Γ (z) ∂z ∂ B 2k Γ (z) ∂z B 2k Γ (z) 2 = 4π 2 1 1 + β Γ,1 (q(z)) 2 + β Γ,2 (q(z)) 1 + β Γ,1 (q(z)) 2 .(27) Combining equations (26) and (27), and applying estimates (23) and (25), we obtain the following estimate y 2 π ∂ B 2k Γ (z) ∂z ∂ B 2k Γ (z) ∂z B 2k Γ (z) 2 − ∂ 2 B 2k Γ (z) ∂z∂z B 2k Γ (z) = 4πy 2 β Γ,1 (q(z)) 1 + β Γ,1 (q(z)) 2 + β Γ,3 (q(z)) · 1 + β Γ,1 (q(z)) − β 2 (q(z)) 1 + β Γ,1 (q(z)) 2 = O Γ y 2 e −2πy ,(28) where the implied constant depends only on the Fuchsian subgroup Γ. For k ≫ 1 and z ∈ U Γ,k , choosing the constant c Γ associated the set U Γ,k (which is as defined in equation (12)) appropriately, from estimate (28), we arrive at the following equality y 2 π ∂ B 2k Γ (z) ∂z ∂ B 2k Γ (z) ∂z B 2k Γ (z) 2 − ∂ 2 B 2k Γ (z) ∂z∂z B 2k Γ (z) = O Γ 1 k , where the implied constant depends only on the Fuchsian subgroup. This, along with equation (20), completes the proof of the proposition. Remark. For a fixed k ≥ 3, from Proposition 2.9, it is clear that the Bergman metric µ Ber,k X (z) is not defined at the point i∞. However, the ratio of the Bergman metric and hyperbolic metric remains bounded, as z ∈ X \{i∞}, approaches the puncture i∞. Combining Lemma 2.4 with Propositions 2.8, and 2.9 we now estimate the Bergman metric on X in the following theorem, and hence, complete the proof of Main Theorem 1. Theorem 2.10. With notation as above, for k ≫ 1, and z ∈ X , we have the following estimate µ Ber,k X (z) µ hyp X (z) ≪ Γ k 2 ,(29) where the implied constant depends only on the Fuchsian subgroup Γ. Furthermore, we have the following estimate lim k→∞ 1 k 2 µ Ber,k X (z) µ hyp X (z) ≤ 26 π .(30) Proof. For k ≫ 1, and z ∈ F Γ,k , from Proposition 2.3, we have the following estimate B 2k Γ pet = 2k − 1 4π + α Γ,k (z) ≥ 2k − 1 8π .(31) Furthermore, for k ≫ 1, and z ∈ U Γ,k , from Proposition 2.9, we arrive at the following estimate 1 k 2 µ Ber,k X (z) µ hyp X (z) = O Γ 1 k , where the implied constant depends only on the Fuchsian subgroup Γ. Using Riemann-Roch theorem, we now compute r k := dim C H 0 X , L k (−D) . From Serre duality, we have dim C H 1 (X, L k (−D)) = dim C H 0 X, Ω X ⊗ L k (−D) * = dim C H 0 X, Ω ⊗(1−k) X ⊗ O X − (k − 1)i∞ + D .(33) Furthermore, observe that deg Ω ⊗(1−k) X ⊗ O X − (k − 1)i∞ + D = −(k − 1)(2g − 2) − (k − 1) + d = −(k − 1)(2g − 1) + d.(34) So, for (k − 1)(2g − 1) > d, combining equations (33) and (34), we infer that dim C H 1 (X, L k (−D)) = 0 Hence, for (k − 1)(2g − 1) > d, applying Riemann-Roch theorem, we deduce that r k = dim C H 0 X, L k (−D) = deg L k (−D) + (1 − g) = k(2g − 2) + (k − 1) − d + (1 − g) = (2k − 1)(g − 1) + k − 1 − d = n k − d. Consider the following short exact sequence of sheaves over X where D denotes the divisor x 1 + · · · + x d , and z j denotes the complex coordinate of the point x j ∈ X , for each 1 ≤ j ≤ d. Furthermore, the map ϕ k,d L is a holomorphic embedding, (see section 4 & 5 in [6]). Bergman kernel and the Fubini-Study metric. With notation as above, let D be an effective divisor of X of degree d ≥ 1. For k ≥ 1, let B k L,D denote the Bergman kernel associated to the line bundle L k (−D), and · hyp , the point-wise hyperbolic metric on H 0 X, L k , induces a metric on H 0 X, L k (−D) , which we again denote by · hyp , for brevity of notation. For k ≫ 1, and a fixed z ∈ X \{i∞}, from Theorem 3.3 from [9], we have the following asymptotic relation 1 k B k L,D (z) hyp = 1 k B k L (z) hyp + O Γ 1 k ,(38) where the implied constant depends only on the Fuchsian subgroup Γ. L k (−D) → L k → L k | D → 0, which induces the injective homomorphism H 0 X, L k (−D) ֒→ H 0 X, L k ,(36)which occurs in the long exact sequence of cohomologies associated with equation (35). From equation (36), for a given d ≥ 1, and (k − 1)(2g − 1) > d, we have the following map ϕ k,d L : Sym d (X) →Gr(r k , n k ) z 1 , . . . , z d →H 0 (X, L k (−D) ⊂ H 0 X , L k , 1.2. Statements of main results.Since the Bergman kernel B kΩ X vanishes at i∞, the Bergman metric µ Ber,k X is not defined at i∞. However, as shown in Proposition 2.9, the ratio µ Ber,k X (z) AcknowledgementsBoth the authors thank Prof. Biswas and Prof. Nagaraj, for many interesting mathematical discussion, which culminated in the completion of the article. Both the authors extend their gratitude to Dr. Aprameyan Parthsarathy for his suggestions and comments, regarding the article.So, for k ≫ 1, we deduce thatCombining Proposition 2.8 with estimate (31) and equation (32), completes the proofs of estimates (29) and (30), which completes the proof of the theorem.3. Comparison of Kähler metrics on symmetric product of Riemann surfaces 3.1. Symmetric product of noncompact Riemann surfaces. We now set up the notation, and recall the back ground material to prove Main Theorem 2. However, the notation and the requisite back ground details, remain the same, as in the compact setting. So, for brevity of the article, we only state the requisite details, and we refer the reader to section 3 in[3], for a more detailed description.With notation as in section 2.1, let X denote a noncompact finite volume hyperbolic Riemann surface of genus g ≥ 2. Let X := X ∪ {i∞} denote the Satake compactification of X, and letwhich defines a Kähler metric on X d .Let S d denote the group of permutations of the set {1, . . . , d} of d-elements. The permutation group S d acts on the Cartesian product X d , and we denote the quotient space S d \ X d by Sym d (X ). The symmetric product Sym d (X ) is an irreducible smooth complex projective variety of complex dimension d.The Kähler metric µ hyp X d descends to define a Kähler metric on Sym d (X ), which admits singularities, which we again denote by µ hyp X d , for brevity of notation. Furthermore, let µ hypRecall from Proposition 2.1, we know thatLet d ≥ 1 be a given integer. Then for for k ≥ 1, and a divisor D on X of degree d, letWith notation as above, let µ FS Gr(r k ,n k ) denote the Fubini-Study metric on Gr(r k , n k ). Letdenote the pull back of the Fubini-Study metric on Sym d (X ), via the holomorphic embedding ϕ k,d L , as in equation(37). Furthermore, let µ FS,k Sym d (X ),vol denote the volume form associated to the (1,1)-form µ FS,k Sym d (X ) .With notation as above, for z := (z 1 , . . . , z d ) ∈ Sym d (X ), from Proposition 3.1 from[3], we have the following relation3.2. Proof of Main Theorem 2. We now prove Main Theorem 2 in this section.Theorem 3.1. With notation as above, for a given d ≥ 1, and k ≫ 1, and for z ∈ Sym d (X ), we have the following estimatewhere the implied constant depends only on the Fuchsian subgroup Γ. Furthermore, we have the following estimateProof. The proof of the theorem follows from combining equation (39) with estimate (38), and Theorem 2.10. E Arbarello, M Cornalba, P A Griffiths, J Harris, Grundlehren der Mathematischen Wissenschaften. New YorkSpringer-Verlag267Geometry of algebraic curvesE. Arbarello, M. Cornalba, P. A. Griffiths and J. Harris, Geometry of algebraic curves. Vol. I., Grundlehren der Mathematischen Wissenschaften, 267. Springer-Verlag, New York, 1985. On the Kähler metrics over Sym d (X ). A Aryasomayajula, I Biswas, A S Morye, T Sengupta, J. Geom. Phys. 110A. Aryasomayajula, I. Biswas, A. S. Morye, and T. Sengupta, On the Kähler metrics over Sym d (X ), J. Geom. Phys. 110 (2016), 187-194. Bergman kernel on Riemann surfaces and Kähler metric on symmetric products. A Aryasomayajula, I Biswas, Internat. J. Math. 30A. Aryasomayajula and I. Biswas, Bergman kernel on Riemann surfaces and Kähler metric on sym- metric products, Internat. J. Math. 30 (2019). Off-diagonal estimates of the Bergman kernel on hyperbolic Riemann surfaces of finite volume. A Aryasomayajula, P Majumder, Proceedings of Amer. Math. Soc. A. Aryasomayajula and P. Majumder, Off-diagonal estimates of the Bergman kernel on hyperbolic Riemann surfaces of finite volume, Proceedings of Amer. Math. Soc. (2018), 4009-4020. Off-diagonal estimates of the Bergman kernel on hyperbolic Riemann surfaces of finite volume-II. A Aryasomayajula, P Majumder, Ann. Fac. Sci. Toulouse Math. 29A. Aryasomayajula and P. Majumder, Off-diagonal estimates of the Bergman kernel on hyperbolic Riemann surfaces of finite volume-II, Ann. Fac. Sci. Toulouse Math. 29 (2020), 795-804. Moduli of vortices and Grassmann manifolds. I Biswas, N M Romão, Comm. Math. Phys. 320I. Biswas and N. M. Romão, Moduli of vortices and Grassmann manifolds, Comm. Math. Phys. 320 (2013), 1-20. E Freitag, Hilbert Modular Forms. BerlinSpringer-VerlagE. Freitag, Hilbert Modular Forms, Springer-Verlag, Berlin, 1990. Toward the inversion of abelian integrals. I. G Kempf, Ann. of Math. 110G. Kempf, Toward the inversion of abelian integrals. I, Ann. of Math. 110 (1979), 243-273. Holomorphic Morse inequalities and Bergman kernels. X Ma, G Marinescu, Prog. Math. 254Birkhäuser VerlagX. Ma, G. Marinescu, Holomorphic Morse inequalities and Bergman kernels, Prog. Math. 254, Birkhäuser Verlag, Basel, Switzerland, 2007. One-vortex moduli space and Ricci flow. N S Manton, J. Geom. Phys. 58N.S.Manton, One-vortex moduli space and Ricci flow. J. Geom. Phys. 58 (2008), 1772-1783. T Miyake, Modular forms. Springer Monographs in Mathematics. Berlin; Mangalam; IndiaSpringer-Verlag517507Department of Mathematics, Indian Institute of Science Education and Research (IISER) Tirupati, Transit campus at Sri Rama Engineering College, Karkambadi RoadT. Miyake, Modular forms. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2006. Department of Mathematics, Indian Institute of Science Education and Research (IISER) Tirupati, Transit campus at Sri Rama Engineering College, Karkambadi Road, Mangalam (B.O),Tirupati- 517507, India.
{'fraction_non_alphanumeric': 0.09940423987661415, 'fraction_numerical': 0.0348686001149182, 'mean_word_length': 3.127309036445332, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 9, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 38, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'Let X denote a noncompact finite volume hyperbolic Riemann surface of genus g ≥ 2, with only one puncture at i∞ (identifying X with its universal cover H). Let X := X ∪ {i∞} denote the Satake compactification of X. Let Ω X denote the cotangent bundle on X . For k ≫ 1, we derive an estimate for µ Ber,k X , the Bergman metric associated to the line bundle L k := Ω ⊗k X ⊗O X ((k − 1)i∞). For a given d ≥ 1, the pull-back of the Fubini-Study metric on the Grassmannian, which we denote by µ FS,k Sym d (X ) , defines a Kähler metric on Sym d (X ), the d-fold symmetric product of X . Using our estimates of µ Ber,k X , as an application, we derive an estimate for µ FS,k Sym d (X ),vol , the volume form associated to the (1,1)-form µ FS,k Sym d (X ) .', 'arxivid': '2305.11609', 'author': ['Anilatmaja Aryasomayajula ', 'Arijit Mukherjee ', 'Anilatmaja Aryasomayajula ', 'Arijit Mukherjee '], 'authoraffiliation': [], 'corpusid': 258823310, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 13236, 'n_tokens_neox': 11764, 'n_words': 6930, 'pdfsha': '142eccbbf3e4a58a1e6fb237e9115c8fba97e02f', 'pdfurls': ['https://export.arxiv.org/pdf/2305.11609v1.pdf'], 'title': ['ESTIMATES OF KÄHLER METRICS ON NONCOMPACT FINITE VOLUME HYPERBOLIC RIEMANN SURFACES, AND THEIR SYMMETRIC PRODUCTS', 'ESTIMATES OF KÄHLER METRICS ON NONCOMPACT FINITE VOLUME HYPERBOLIC RIEMANN SURFACES, AND THEIR SYMMETRIC PRODUCTS', 'ESTIMATES OF KÄHLER METRICS ON NONCOMPACT FINITE VOLUME HYPERBOLIC RIEMANN SURFACES, AND THEIR SYMMETRIC PRODUCTS', 'ESTIMATES OF KÄHLER METRICS ON NONCOMPACT FINITE VOLUME HYPERBOLIC RIEMANN SURFACES, AND THEIR SYMMETRIC PRODUCTS'], 'venue': []}
arxiv
Well-posedness of first order semilinear PDEs by stochastic perturbation 28 Oct 2013 Christian Olivera [email protected] Departamento de Matemática Universidade Estadual de Campinas Well-posedness of first order semilinear PDEs by stochastic perturbation 28 Oct 2013Stochastic characteristic methodFirst order stochastic partial differential equationsStochastic perturbationcommuting Lemma MSC2000 subject classification: 60H1060H15 We show that first order semilinear PDEs by stochastic perturbation are well-posedness for globally Holder continuous and bounded vector field, with an integrability condition on the divergence. This result extends the liner case presented in [2]. The proof is based on in the stochastic characteristics method and a version of the commuting Lemma. Introduction This work is motivated by the paper [2] where the linear equation du(t, x) + b(t, x)∇u(t, x)dt + ∇u(t, x) • dB t = 0, u(0, x) = f (x) ∈ L ∞ (R d ),(1) has been studied, was proved existence and uniqueness of L ∞ -solutions for a globally Holder continuous and bounded vector field, with an integrability condition on the divergence, and where B t = (B 1 t , ..., B d t ) is a standard Brownian motion in R d . The aim of this paper is to investigate parts of this theory under the effect of nonlinear terms. Namely , we considerer the semilinear SPDE du(t, x) + b(t, x)∇u(t, x) dt + F (t, x, u) dt + ∇u(t, x) • dB t = 0, u(0, x) = f (x) ∈ L ∞ (R d ).(2) We shall prove the existence and uniqueness of weak L ∞ -solutions for a globally Holder continuous and bounded vector field, with an integrability condition on the divergence. Moreover , we obtain a representation of the solution via stochastic flows. This is a example of nonlinear SPDE where the stochastic perturbation makes the equation well-posedness. The fundamental tools used here is the stochastic characteristics method (see for example [1], [5] and [7] ) and the version of the commuting Lemma presented in [2]. That is, we follows the strategy given in [2] in combination with the stochastic characteristics method. The article is organized as follows: Section 2 we shall define the concept of weak L ∞ −solutions for the equation (2) and we shall prove existence for this class of solutions. . In section 3, we shall show a uniqueness theorem for weak L ∞ −solutions. Through of this paper we fix a stochastic basis with a d-dimensional Brownian motion (Ω, F , {F t : t ∈ [0, T ]}, P, (B t )). 2 Existence of weak L ∞ −solutions Let T > 0 be fixed. For α ∈ (0, 1) define the space L ∞ ([0, T ], C α (R d )) as the set of all bounded Borel functions f : [0, T ] × R d → R for which [f ] α,T = sup t∈[0,T ] sup x =y∈R d |f (x) − f (y)| |x − y| We write the L ∞ ([0, T ], C α (R d , R d )) for the space of all vector fields having components in L ∞ ([0, T ], C α (R d )). We shall assume that b ∈ L ∞ ([0, T ], C α (R d , R d )),(3)Div b ∈ L p ([0, T ] × R d ) f or p > 2. (4) F ∈ L 1 ([0, T ], L ∞ (R d × R))(5) and F ∈ L ∞ ([0, T ] × R d , LIP (R)).(6) 2.1 Definition of weak L ∞ −solutions Definition 2.1 We assume (3), (4), (5) and (6). A weak L ∞ −solution of the Cauchy problem (2) is a stochastic process u ∈ L ∞ (Ω × [0, T ] × R d ) such that, for every test function ϕ ∈ C ∞ 0 (R d ), the process u(t, x)ϕ(x)dx has a continuous modification which is a F t -semimartingale and satisfies u(t, x)ϕ(x)dx = f (x)ϕ(x) dx + t 0 b(s, x)∇ϕ(x)u(s, x) dxds + t 0 div b(s, x)ϕ(x)u(s, x) dxds + t 0 F (s, x, u)ϕ(x) dxds + d i=0 t 0 D i ϕ(x)u(s, x) dx • dB i s Remark 2.1 We observe that a weak L ∞ −solution in the previous Stratonovich sense satisfies the Itô equation u(t, x)ϕ(x)dx = f (x)ϕ(x) dx + t 0 b(s, x)∇ϕ(x)u(s, x) dxds + t 0 div b(s, x)ϕ(x)u(s, x) dxds + t 0 F (s, x, u)ϕ(x) dxds + d i=0 t 0 D i ϕ(x)u(s, x) dxdB i s + 1 2 t 0 u(s, x)△ϕ(x) dxds(7) for every test function ϕ ∈ C ∞ 0 (R d ). The converse is also true. Existence of weak L ∞ −solutions Lemma 2.1 Let f ∈ L ∞ (R d ). We assume (3), (4), (5) and (6). Then there exits a weak L ∞ −solution u of the SPDE (2). Proof: Step 1 Assume that F ∈ L 1 ([0, T ], C ∞ b (R d × R)) and f ∈ C ∞ b (R d ). We take a mollifier regularization b n of b . It is known (see [1], chapter 1 ) that there exist an unique classical solution u n (t, x) of the SPDE (2), that written in weak Itô form is (7) with b n in place of b. Moreover, u n (t, x) = Z n t (x, f (Y n t )) where Y n t is the inverse of X n t , X n t (x) and Z n t (x, r) satisfy the following equations X n t = x + t 0 b n (s, X n s ) ds + B t ,(8) and Z n t = r + t 0 F (s, X n s (x), Z n s ) ds.(9) According to theorem 5 of [2], see too remark 8, we have that lim n→∞ E[ K sup t∈[0,T ] |X n t − X t | dx] = 0 and lim n→∞ E[ K sup t∈[0,T ] |DX n t − DX t | dx] = 0 for any compact set K ⊂ R d , where X t (x) verifies X t = x + t 0 b(s, X s ) ds + B t .(10) Now, we denote u(t, x) = Z t (x, f (Y t )), Y t is the inverse of X t , and Z t = r + t 0 F (s, X s (x), Z s ) ds.(11) Then , we observe that |u n (t, x)−u(t, x)| ≤ |f (Y t )−f (Y n t )|+ t 0 |F (s, X n s , Z n s (f (Y n t ))−F (s, X s , Z s (f (Y t ))| ds ≤ |f (Y t ) − f (Y n t )| + C t 0 |Z n s (f (Y n t )) − Z s (f (Y t ))| ds. From to theorem 5 of [2], see too remark 8, and the Lipchitz property of Step 2 Assume that F ∈ L 1 ([0, T ], C ∞ b (R d × R)) . We a take a mollifier regularization f n of f . By the last step u n (t, x) = Z t (x, f n (Y t )) is a weak L ∞ −solution of the SPDE (2), that written in weak Itô form is (7) with f n in place of f . We have that any compact set K ⊂ R d and p ≥ 1 lim n→∞ sup [0,T ] K |f n (X −1 t ) − f (X −1 t )| p dx = lim n→∞ sup [0,T ] Xt(K) |f n (x) − f (x)| p JX t (x) dx = 0 Then we have that (2) Step 3 We take a mollifier regularization F n of F . By the step 2, we (2), and hold that Z n t (x, r) satisfies the equation (11) with F n in place of F . We observe that lim n→∞ sup [0,T ] K |Z t (x, f n (Y t )) − Z t (x, f (Y t ))| p dx = 0. Thus u(t, x) = Z t (x, f (Y t )) is a weak L ∞ −solution of the SPDEhave that u n (t, x) = Z n t (x, f (Y t )) is a weak L ∞ −solution of the SPDE|Z n t (x, r) − Z t (x, r)| ≤ t 0 |F n (t, X s , Z n s ) − F (t, X s , Z s )| ds ≤ t 0 |F n (t, X s , Z n s ) − F n (t, X s , Z s )| ds + t 0 |F n (t, X s , Z s ) − F (t, X s , Z s )| ds ≤ C t 0 |Z n s − Z s | ds + t 0 |F n (t, X s , Z s ) − F (t, X s , Z s )| ds By the Gronwall Lemma we follow that lim n→∞ |Z n t (x, r) − Z t (x, r)| = 0 unif ormaly in t, x, r. Then lim n→∞ |Z n t (x, f (Y t )) − Z t (x, f (Y t )) = 0 unif ormaly in t and x. Therefore, we conclude that u(t, x) = Z t (x, f (Y t )) is a weak L ∞ − solution of the SPDE (2). Uniqueness of weak L ∞ −solutions In this section, we shall present an uniqueness theorem for the SPDE (2) under similar conditions to the linear case , see theorem 20 of [2]. Let ϕ n be a standard mollifier. We introduced the commutator defined as R n (b, u) = (b∇)(ϕ n * u) − ϕ n * ((b∇)u) We recall here the following version of the commutator lemma which is at the base of our uniqueness theorem. Lemma 3.1 Let φ t be an C 1 -diffeomorphism of R d . Assume b ∈ L ∞ loc (R d , R d ) , divb ∈ L 1 loc (R d ), u ∈ L ∞ loc (R d ). Moreover, for d > 1, assume also Jφ −1 ∈ W 1,1 loc (R d ) Then for any ρ ∈ C ∞ 0 (R d ) there exits a constant C ρ such that , given any R > 0 such that supp(ρ • φ −1 ) ⊂ B(R), we have : a) for d > 1 | R n (b, u)(φ(x))ρ(x) dx| ≤ C ρ u L ∞ R+1 [ divb L 1 R+1 Jφ −1 L ∞ R + b L ∞ R+1 ( Dφ −1 L ∞ R + DJφ −1 L 1 R )] b) for d = 1 | R n (b, u)(φ(x))ρ(x) dx| ≤ C ρ u L ∞ R+2 b W 1,1 R+2 Jφ −1 L ∞ R Proof: See pp 28 of [2]. We are ready to prove our uniqueness result of weak L ∞ −solution to the Cauchy problem (2). Proof: Step 1( Itô-Ventzel-Kunita formula) Let u, v be are two weak L ∞ −solutions and ϕ n be a standard mollifier. We put w = u − v, applying the Itô-Ventzel-Kunita formula (see Theorem 8.3 of [6] ) to F (y) = w(t, z)ϕ n (y − z) dz, we obtain that w(t, z)ϕ n (X s − z)dz is equal to t 0 b(s, z)∇[ϕ n (X s −z)]w(s, z) dzds+ t 0 div b(s, z)ϕ n (X s −z)u(s, z) dzds + t 0 (F (s, z, u)−F (s, z, v))ϕ n (X s −z) dzds + d i=1 t 0 w(s, z)D i [ϕ n (X s −z)]]dz•dB i s + t 0 (b∇)(w(s, .) * ϕ n )(X s ) ds − d i=1 t 0 w(s, z)D i [ϕ n (X s − z)]dz • dB i s . Then w(t, z)ϕ n (X t − z)dz = t 0 (F (s, z, u) − F (s, z, v))ϕ n (X s − z) dzds − t 0 R n (w, b)(X s (x)) ds, where R n is the commutator defined above. Step 2( lim n→∞ t 0 R n (w, b)(X s ) ds = 0) We argue as in [2]. We observe by Lemma 3.1 and the Lebesgue dominated theorem that lim n→∞ t 0 R n (w, b)(X s )ρ(x) ds = 0 for all ρ ∈ C ∞ 0 (R d ), for details see Theorem 20 of [2]. Step 3( w = 0) We observe that , where the convergence is in L 1 ([0, T ], L 1 loc (R d )). From the flow properties of X t , see theorem 5 of [2], we obtain lim n→∞ (w(t, .) * ϕ n )(X t ) = w(t, X t ) and lim n→∞ ((F (t, ., u) − F (t, ., v)) * ϕ n )(X t ) = (F (t, , X t , u(t, , X t )) − F (t, , X t , v(t, , X t )), where the convergence is P a.s in L 1 ([0, T ], L 1 loc (R d )). Then by steps 1, 2 we have w(t, X t ) = t 0 F (s, , X s , u(t, , X s )) − F (s, , X s , v(t, , X s )) ds. Thus, for any compact set K ⊂ R d we obtain that K |w(t, X t )|dx ≤ t 0 K |F (s, , X s , u(t, , X s )) − F (s, , X s , v(t, , X s ))| dxds. ≤ C t 0 K |w(t, , X s )| dxds. where C is contant related to the Lipchitz property of F . It follows K |w(t, X t )|dx ≤ C and thus w(t, X t ) = 0 by the Gronwall Lemma. F we conclude that lim n→∞ E[ K sup t∈[0,T ] |u n (t, x) − u(t, x)|] =0 and u(t, x)is a weak L ∞ −solution of the SPDE (2). Theorem 3. 1 1Assume (3), (4), (5) and (6). Then, for every f ∈ L ∞ (R d ) there exists an unique weak L ∞ −solution of the Cauchy problem (2). t, .) * ϕ n )(.) = w(t, .) t, , X s )| dxds. x) has the representation u(t, x) = Z t (x, f (X −1 t )) , where X t and Z t satisfy the equations (10) . In fact, the step 2 is valid under other hypotheses. Remark 3.1 We observe that the unique solution u(t. see corollary 23 of [2Remark 3.1 We observe that the unique solution u(t, x) has the represen- tation u(t, x) = Z t (x, f (X −1 t )) , where X t and Z t satisfy the equations (10) . In fact, the step 2 is valid under other hypotheses, see corollary 23 of [2]. P L Chow, Stochastic Partial Differential Equations. Chapman Hall/CRCChow, P. L. 2007. Stochastic Partial Differential Equations, Chapman Hall/CRC. Well-posedness of the transport equation by stochastic perturbation. F Flandoli, M Gubinelli, E Priola, Invent. Math. 1801Flandoli F., Gubinelli M. , Priola, E. 2010. Well-posedness of the transport equation by stochastic perturbation, Invent. Math., 180 (1): 1-53. The Interaction Between Noise and Transport Mechanisms in PDEs. F Flandoli, Milan j. Math79Flandoli,F. 2011. The Interaction Between Noise and Transport Mech- anisms in PDEs, Milan j. Math, 79 (2) : 543-560. Remarks on uniqueness and strong solutions to deterministic and stochastic differential equations. F Flandoli, Metrika. 692-3Flandoli, F. 2009. Remarks on uniqueness and strong solutions to deterministic and stochastic differential equations, Metrika, 69 (2-3) : 101-123. First Order Stochastic Partial Differential Equations. H Kunita, Proceedings of the Taniguchi International Symposium on Stochastic Analysis. the Taniguchi International Symposium on Stochastic AnalysisNorth-Holland Mathematical LibraryKunita, H. 1984. First Order Stochastic Partial Differential Equa- tions , in Proceedings of the Taniguchi International Symposium on Stochastic Analysis, North-Holland Mathematical Library, 249-269. Stochastic differential equations and stochastic flows of diffeomorphisms. H Kunita, Lectures Notes in Mathematics. 1097Kunita, H. 1982. Stochastic differential equations and stochastic flows of diffeomorphisms, Lectures Notes in Mathematics, 1097 : 143-303. Stochastic flows and stochastic differential equations. H Kunita, Cambridge University PressKunita, H. 1990. Stochastic flows and stochastic differential equations, Cambridge University Press .
{'fraction_non_alphanumeric': 0.1266761056050637, 'fraction_numerical': 0.028400099941700674, 'mean_word_length': 3.0080106809078773, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 5, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 11, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'We show that first order semilinear PDEs by stochastic perturbation are well-posedness for globally Holder continuous and bounded vector field, with an integrability condition on the divergence. This result extends the liner case presented in [2]. The proof is based on in the stochastic characteristics method and a version of the commuting Lemma.', 'arxivid': '1310.7404', 'author': ['Christian Olivera [email protected] \nDepartamento de Matemática\nUniversidade Estadual de Campinas\n\n'], 'authoraffiliation': ['Departamento de Matemática\nUniversidade Estadual de Campinas\n'], 'corpusid': 119287999, 'doi': '10.1016/j.na.2013.10.022', 'github_urls': [], 'n_tokens_mistral': 4960, 'n_tokens_neox': 4477, 'n_words': 2537, 'pdfsha': '049f655f26b1a921ba93cfebc446d7cd9b92e9b3', 'pdfurls': ['https://arxiv.org/pdf/1310.7404v1.pdf'], 'title': ['Well-posedness of first order semilinear PDEs by stochastic perturbation', 'Well-posedness of first order semilinear PDEs by stochastic perturbation'], 'venue': []}
arxiv
EHRHART POLYNOMIALS AND STRINGY BETTI NUMBERS 8 Jun 2005 Mircea Mustaţǎ Sam Payne EHRHART POLYNOMIALS AND STRINGY BETTI NUMBERS 8 Jun 2005 We study the connection between stringy Betti numbers of Gorenstein toric varieties and the generating functions of the Ehrhart polynomials of certain polyhedral regions. We use this point of view to give counterexamples to Hibi's conjecture on the unimodality of δ-vectors of reflexive polytopes. Introduction Let N be a lattice of rank d and let P be a d-dimensional lattice polytope in N R = N ⊗ Z R. For each nonnegative integer m, let f P (m) be the number of lattice points in mP . Then f P is a polynomial in m of degree d, called the Ehrhart polynomial of P . The generating function F P (t) = m≥0 f P (m)t m is a rational function in t and can be written as F P (t) = δ 0 + δ 1 t + · · · + δ d t d (1 − t) d+1 , for some nonnegative integers δ i , with δ 0 = 1. We put δ P = (δ 0 , . . . , δ d ), and with a slight abuse of notation we denote by δ P (t) the numerator of F P (t). If ℓ is the largest i such that δ i is nonzero, then ℓ = d + 1 − r, where r is the smallest positive integer such that rP contains a lattice point in its interior. Recall that a lattice polytope is reflexive if it contains 0 in its interior and its polar polytope has vertices in the dual lattice. Given the lattice polytope P , we have δ i = δ ℓ−i for all i if and only if rP is the translate of a reflexive polytope. Hibi conjectured in [Hi2,p. 111] that if this is the case, then δ P is unimodal: (1) δ 0 ≤ · · · ≤ δ [ℓ/2] . In the particular case when P is the Birkhoff polytope of doubly stochastic n×n matrices, unimodality had been conjectured by Stanley [St] and was recently proved by Athanasiadis [At]. We assume now that P is reflexive, so ℓ = d. Hibi showed that in this case (2) δ 0 ≤ δ 1 ≤ δ j for 2 ≤ j ≤ [d/2]. If, in addition, the boundary of P admits a regular triangulation such that the vertices of each facet are a basis for the lattice N, then δ P is the h-vector of the triangulation (see [Hi1]). In particular, if such a triangulation exists, then Stanley's theorem on the h-vectors of simplicial polytopes implies that δ P is unimodal, so P satisfies Hibi's conjecture. The first author was partially supported by NSF grant DMS 0500127 and the second author was supported by a Graduate Research Fellowship from the NSF. Note that if P is a reflexive polytope of dimension d ≤ 5, then Hibi's conjecture follows from (2). The following reflexive polytope gives a counterexample to the conjecture for d = 6; for a more restricted and still open version of the conjecture, see [OH]. Example 1.1. Let f = 1 3 (e 1 + · · · + e 6 ) in R 6 , let N be the lattice Z 6 + Z · f , and let P be the polyope with vertices {e 1 , . . . , e 6 , e 1 − f, . . . , e 6 − f }. It is straightforward to check that P is reflexive, and one computes that 2P and 3P contain 78 lattice points and 314 lattice points, respectively. It follows that δ P = (1, 6,8,6,8,6,1). In this paper we give a combinatorial formula for δ P when P is reflexive, as a positive linear combination of shifted h-vectors of simplicial polytopes, which we arrive at by using toric varieties to equate the combinatorial invariants δ i of P with "stringy" invariants from complex algebraic geometry. This formula can also be proved directly, using elementary combinatorial arguments. We present proofs from both points of view. With this formula in hand, it is not difficult to construct examples, such as Example 1.1, where δ P is not unimodal. In order to explain our approach, we first reinterpret in algebro-geometric terms the proof of unimodality of δ P in the special case mentioned above, due to Hibi. Here and throughout, P is assumed to be reflexive unless stated otherwise. Since P is reflexive, the polar polytope P • is reflexive, too. Note that the polytope P • corresponds to a toric variety X P • defined by the fan over the faces of P , and to an ample divisor D P • on X P • . The fact that P • is reflexive is equivalent with the fact that D P • is the canonical divisor on X P • (so in particular X P • is a Fano variety). Consider a triangulation P of the boundary of P and let ∆ be the fan whose maximal cones are the cones over the facets of P. We have a proper birational morphism f : X(∆) → X = X P • induced by the identity on N. If P is a regular triangulation such that the vertices of each facet of P give a basis of N, then f is a resolution of singularities, X(∆) is projective, and f is crepant, i.e. the pull-back of the canonical bundle on X is isomorphic to the canonical bundle on X(∆). Conversely, every such resolution of singularities of X arises from a triangulation as above. Given such a triangulation, δ i is the 2i-th Betti number of X(∆), the dimension of the singular cohomology group H 2i (X(∆); Q), and the unimodality of δ P follows from the Hard Lefschetz Theorem on X(∆). In general, there may not exist any crepant resolution of X P • . However, using the theory of motivic integration, one can define "stringy Betti numbers" of X P • that agree with the Betti numbers of a crepant resolution whenever such a resolution exists [Bat]. A result of Batyrev and Dais shows that δ i is the 2i-th stringy Betti number of X P • [BD,Theorem 7.2]. We generalize this result as follows. If X = X(Σ) is a complete, d-dimensional Gorenstein toric variety, then there is a functionΨ K on N R that on each cone is given by an element of the dual lattice, and such that Ψ K (v i ) = 1 for every primitive generator v i of a ray of Σ. Consider the set Q = {v ∈ N R | ψ K (v) ≤ 1}. For every cone σ in Σ the intersection Q ∩ σ is a lattice polytope; it is the convex hull of the origin and of the primitive generators of the rays of σ. We see that Q, viewed as the union of the polytopes Q ∩ σ, is naturally a polyhedral complex, and that Q is homeomorphic to a ball of dimension d. Therefore we may define as in [Hi3] a polynomial of degree d (the Ehrhart polynomial) f Q such that f Q (m) is the number of lattice points in mQ for every nonnegative integer m. Then we can write the generating function F Q (t) = m≥0 f Q (m)t m in the form F Q (t) = δ 0 + δ 1 t + · · · + δ d t d (1 − t) d+1 , for some nonnegative integers δ i . Theorem 1.2. For every complete Gorenstein toric variety X, δ i is equal to the 2i-th stringy Betti number of X. Although there may not exist any crepant resolution of singularities for X, we can always find a projective crepant morphism of toric varieties f : X(∆) → X such that X(∆) has only Gorenstein orbifold singularities. Since f is crepant, the stringy Betti numbers of X are equal to the stringy Betti numbers of X(∆). A theorem of Yasuda [Ya] then implies that the stringy Betti numbers of X(∆) are equal to the dimensions of the graded pieces of the orbifold cohomology of X(∆). We get a combinatorial formula for these dimensions using a toric formula due to Borisov, Chen, and Smith [BCS]. The resulting description of δ Q is as follows. Fix a triangulation T of the boundary of Q whose vertices are in N, and let ∆ be the fan over the faces of T . For a face F ∈ T with vertices v 1 , . . . , v r , define Box(F ) = {a 1 v 1 + · · · + a r v r ∈ N R : 0 < a i < 1}, and let ∆ F be the fan in N/(N ∩ span F ) whose cones are the projections of the cones in ∆ containing F . For a positive integer m, let h ∆ F [m] denote the h-vector of ∆ F shifted by m, defined by h ∆ F [m] i = 0 for i < m. (h ∆ F ) i−m for i ≥ m. Note that ∆ F is the simplicial fan corresponding to the T -invariant subvariety of X(∆) determined by the cone over F , and (h ∆ F ) i is the 2i-th Betti number of X(∆ F ). In particular, if X(∆) is projective, then the Hard Lefschetz Theorem on X(∆ F ) implies that h ∆ F is unimodal. Theorem 1.3. If T is any triangulation of the boundary of Q whose vertices are in N, then δ Q = h T + F ∈T , v∈Box(F )∩N h ∆ F [Ψ K (v)]. In particular, the sum of shifted h-vectors in Theorem 1.3 is independent of the choice of triangulation. δ-vectors and stringy Betti numbers A d-dimensional Gorenstein variety X with canonical singularities has a stringy E-function E st (X; w, z) ∈ Z[[w, z]] ∩ Q(w, z) defined using Hodge theory and motivic integration on a resolution of singularities of X. If E st (X; w, z) = p,q a pq w p z q is a polynomial, then the j-th stringy Betti number of X is defined to be (−1) j p+q=j a pq . Suppose now that X = X(Σ) is a complete Gorenstein toric variety (see [Fu] for basic facts on toric varieties). In this case E st (X; w, z) is a polynomial in wz, so the odd stringy Betti numbers vanish and the 2i-th stringy Betti number of X is the coefficient of (wz) i [Bat, Section 3]. Our proof of Theorem 1.2 is based on the following formula for E st (X; w, z) as a rational function [Bat,Theorem 4.3]. Since X is Gorenstein, we have a function ψ K on N R that on each cone is given by an element in the dual lattice, and such that Ψ K (v i ) = 1 for every primitive generator v i of a ray of Σ. For each cone σ ∈ Σ, let σ • denote the relative interior of σ. Recall that v∈σ • (wz) −Ψ K (v) is a rational function in wz (see, for example, [Bar,VIII.1]). Batyrev has shown that we have the following equality of rational functions, (3) E st (X; w, z) = (wz − 1) d σ∈∆ v∈σ • ∩N (wz) −Ψ K (v) . As in the Introduction, we define Q = {v ∈ N R | Ψ K (v) ≤ 1}. There is an Ehrhart polynomial f Q such that, for positive integers m, f Q (m) is the number of lattice points in mQ, and f Q satisfies Ehrhart reciprocity: f Q (−m) is the number of lattice points in the interior of mQ. The proofs of these assertions follow as in [Hi2], using the fact that Q is homeomorphic to a ball of dimension d. The generating function F Q (t) = m≥0 f Q (m)t m can then be written F Q (t) = δ 0 + δ 1 t + · · · + δ d t d (1 − t) d+1 , for some nonnegative integers δ i . For the proof of Theorem 1.2 we will need the following two lemmas. A proof of the first lemma in the case when Q is a polytope can be found in [Hi2] and the general case is similar, but we include the proof for the reader's convenience. Lemma 2.1. With the above notation, we have δ i = δ d−i for every i. (4) f Q (m − 1) = (−1) d f Q (−m) for every positive integer m, and therefore for all m. If we write f Q (m) = d i=0 a i i+m i , then we deduce F Q (t) = m∈N d i=0 a i i + m i t m = d i=0 a i · m∈N i + m i t m = d i=0 a i (1 − t) i+1 . If we put F Q (t) = m≥1 f Q (−m)t m , then F Q (t) = d i=0 a i · m≥i+1 (−1) i m − 1 i t m = d i=0 (−1) i a i t i+1 (1 − t) i+1 , so we have the equality of rational functions F Q (t) = −F Q (t −1 ). On the other hand, (4) gives F Q (t) = (−1) d tF Q (t), hence F Q (t −1 ) = (−1) d+1 tF Q (t). Since (1 − t) d+1 F Q (t) = d i=0 δ i t i , this equality gives δ i = δ d−i for every i. Lemma 2.2. With the above notation, we have (1 − t)F Q (t) = v∈N t ψ K (v) . Proof. We can write F Q (t) = m∈N v∈mQ∩N t m = v∈N m≥ψ K (v) t m , using the fact that v is in mQ if and only if m ≥ ψ K (v). The assertion in the lemma follows. Proof of Theorem 1.2. It is enough to show that E st (X; t, 1) = δ Q (t). Combining Lemma 2.2 with (3), we have E st (X; t, 1) = (t − 1) d (1 − t −1 )F Q (t −1 ). Now F Q (t −1 ) = δ Q (t −1 ) (1 − t −1 ) d+1 . By Lemma 2.1 we have δ i = δ d−i , so δ Q (t −1 ) = t −d δ Q (t). Hence E st (X; t, 1) = (t − 1) d δ Q (t) t d (1 − t −1 ) d = δ Q (t). δ-vectors via orbifold cohomology The orbifold cohomology of a Gorenstein variety Y with quotient singularities was defined by Chen and Ruan [CR] and Yasuda [Ya], as follows. There is a canonically associated orbifold (smooth Deligne-Mumford stack) Y whose coarse moduli space is Y . Let I(Y) be the inertia stack of Y. We denote by Y i ⊂ I(Y) the connected components of I(Y) and let Y i be the coarse moduli space of Y i . The "age" s i of Y i is a positive integer determined by the action of the inertia group. As a graded vector space, the orbifold cohomology of Y is given by H * orb (Y, Q) = Y i ⊂I(Y) H * (Y i , Q)[s i ], where [s i ] denotes a grading shift by s i , so H j (Y i , Q)[s i ] = H j−s i (Y i , Q). It is a theorem of Yasuda [Ya] that the j-th stringy Betti number of Y is equal to the dimension of H j orb (Y, Q). See also [Po] for a proof of this result in the case of toric varieties. We mention that Chen and Ruan have constructed a ring structure on orbifold cohomology in [CR]. J. Fernandez gave in [Fe] a necessary and sufficient condition for when the Chen-Ruan cohomology satisfies the Hard Lefschetz Theorem. His condition inspired us in looking for the counterexamples to Hibi's Conjecture. There is an algebraic version of orbifold cohomology, due to Abramovich, Graber and Vistoli [AGV], the so-called orbifold Chow ring. Note that when Y is a simplicial toric variety, each Y i is also a simplicial toric variety, so the odd cohomology of Y i vanishes and H 2 * (Y i , Q) is isomorphic to the Chow ring A * (Y i , Q). It follows that at least as vector spaces, H 2 * orb (Y, Q) agrees in this case with the [AGV] version A * orb (Y, Q) as used by Borisov, Chen and Smith [BCS]. We mention that while there seems to be agreement among experts that the ring structures are also the same in this case, there is no available reference. We stress however that we do not need this, as we are interested only in the vector space structure of the orbifold cohomology. Proof of Theorem 1.3. Let Y be the toric variety corresponding to the fan ∆ whose maximal cones are the cones over the facets of the triangulation T . For a face F ∈ T and a lattice point v ∈ Box(F ), ∆ F is the fan associated to the stacky fan ∆/σ(v) defined in [BCS], and hence h ∆ F is the vector whose i-th entry is the dimension of A i (X(∆ F )). Furthermore, the integer Ψ K (v) is equal to deg y v as defined in [BCS]. Hence the theorem follows from [BCS,Proposition 5.2]. Although we arrived at Theorem 1.3 through the connection with orbifold cohomology and the results of [BCS], it is also possible to prove this result directly using elementary combinatorial methods, as follows. For a fan ∆ with h-vector h ∆ = (h 0 , . . . , h r ), we write h ∆ (t) for the polynomial h ∆ (t) = h 0 + h 1 t + · · · + h r t r . Second proof of Theorem 1.3. By Lemma 2.2, it will suffice to show that (1 − t) d · v∈N t Ψ K (v) = F ∈T ,v∈Box(F ) t Ψ K (v) · h ∆ F (t). Now each lattice point in the cone over a face G ∈ T can be written uniquely as a nonnegative integer linear combination of the vertices of G plus a fractional part. Hence any lattice point v 0 in the relative interior of this cone can be written uniquely as v 0 = v + v G|F + v ′ , where v is in Box(F ) for some face F ≺ G, v G|F is the sum of the vertices of G that are not in F , and v ′ is a nonnegative integer linear combination vertices of G. Since each lattice point v ∈ N is in the relative interior of exactly one cone, it follows that (1 − t) d v∈N t Ψ K (v) = F ∈T ,v∈Box(F ) t Ψ K (v) · G≻F t dim G−dim F (1 − t) codim G = F ∈T ,v∈Box(F ) t Ψ K (v) · h ∆ F (t), as required. Example 3.1. Let m be a positive integer. Let f ∈ R 2m be the vector f = ( 1 m , . . . , 1 m ), and let N be the lattice N = Z 2m + Z · f . We take P ⊂ R 2m to be the polytope with vertices e 1 , . . . , e 2m , e 1 − f, . . . , e 2m − f . It is straightforward to check that P is reflexive. We will show that δ P = (1, 2m, 2m + 2, 2m, 2m + 2, . . . , 2m, 2m + 2, 2m, 1). This generalizes Example 1.1, and shows that for m > 0 there are 2m-dimensional reflexive polytopes with [ m−1 2 ] descents in (δ 0 , δ 1 , . . . , δ m ). We compute δ P by applying Theorem 1.3 to the triangulation P of the boundary of P whose facets are e 1 , . . . , e 2m , e 1 − f, . . . , e 2m − f , e 1 , . . . , e j , . . . , e k , e k − f, . . . , e 2m−f and e 1 , . . . , e j , e j − f, . . . , e k − f , . . . , e 2m − f for 1 ≤ j < k ≤ 2m. This triangulation is obtained by "pulling" the sequence of points e 1 , . . . , e 2m−1 . In particular, P is a regular triangulation, and hence h P is unimodal. Now P has 4m vertices, so (h P ) 1 = 2m, and P has 4m 2 − 2m + 2 facets, so (h P ) 0 + · · · + (h P ) 2m = 4m 2 − 2m + 2. It then follows from unimodality and the fact that (h P ) 0 = (h P ) 2m = 1 that h P = (1, 2m, 2m, . . . , 2m, 2m, 1). To compute δ P , it remains to compute the contributions of the points in Box(F ) for the faces F ∈ P. The only faces of P whose Box is nonempty are F = e 1 , . . . , e 2m and F ′ = e 1 − f, . . . , e 2m − f , which contain {f, . . . , (m − 1)f } and {−f, . . . , (1 − m)f }, respectively. Since F and F ′ are facets, ∆ F = ∆ F ′ = 0 and h ∆ F = h ∆ F ′ = 1. Since m v = 2k for v = ±k · f , it follows that δ P = h P + (0, 0, 2, 0, 2, . . . , 2, 0, 2, 0, 0), as required. Proof. Note first that if m is a positive integer, then a lattice point v is in the interior of mQ if and only if v is in (m − 1)Q. Indeed, v is in the interior of mQ if and only if ψ K (v) < m, and since ψ K (v) is an integer this is the case if and only if ψ K (v) ≤ m − 1, which happens if and only if v is in (m − 1)Q. Ehrhart reciprocity implies that Acknowledgements. We are grateful to Bill Fulton for bringing Hibi's conjecture to our attention, for his suggestions, and for his constant encouragement. We thank the referee for pointing out the connection with [BD], and for suggesting the inclusion of a purely combinatorial proof of Theorem 1.3. Algebraic orbifold quantum products. D Abramovich, T Graber, A Vistoli, Orbifolds in mathematics and physics. Madison, WI; Providence, RIAmer. Math. Soc310D. Abramovich, T. Graber and A. Vistoli, Algebraic orbifold quantum products, in Orbifolds in mathematics and physics (Madison, WI, 2001), 1-24, Contemp. Math., 310, Amer. Math. Soc., Providence, RI, 2002. Ehrhart polynomials, simplicial polytopes, magic squares and a conjecture of Stanley. C A Athanasiadis, math.CO/0312031C. A. Athanasiadis, Ehrhart polynomials, simplicial polytopes, magic squares and a conjecture of Stanley, math.CO/0312031. A course in convexity, Graduate Studies in Math. A Barvinok, Amer. Math. Soc54Providence, RIA. Barvinok, A course in convexity, Graduate Studies in Math. 54, Amer. Math. Soc., Providence, RI, 2002. Stringy Hodge numbers of varieties with Gorenstein canonical singularities. V V Batyrev, Integrable systems and algebraic geometry. Kobe/Kyoto; River Edge, NJWorld Sci. PublishingV. V. Batyrev, Stringy Hodge numbers of varieties with Gorenstein canonical singularities, in Integrable systems and algebraic geometry (Kobe/Kyoto, 1997), 1-32, World Sci. Publishing, River Edge, NJ, 1998. Strong McKay correspondence, string-theoretic Hodge numbers and mirror symmetry. V V Batyrev, D I Dais, Topology. 35V. V. Batyrev and D. I. Dais, Strong McKay correspondence, string-theoretic Hodge numbers and mirror symmetry, Topology 35 (1996), 901-929. The orbifold Chow ring of toric Deligne-Mumford stacks. L A Borisov, L Chen, G G Smith, J. Amer. Math. Soc. 18L. A. Borisov, L. Chen and G. G. Smith, The orbifold Chow ring of toric Deligne-Mumford stacks, J. Amer. Math. Soc. 18 (2005), 193-215. A new cohomology theory of orbifold. W Chen, Y Ruan, Comm. Math. Phys. 248W. Chen and Y. Ruan, A new cohomology theory of orbifold, Comm. Math. Phys. 248 (2004), 1-31. Hodge structures for orbifold cohomology. J Fernandez, math.AG/0311026J. Fernandez, Hodge structures for orbifold cohomology, math.AG/0311026. The William H. Roever Lectures in Geometry. W Fulton, Introduction to toric varieties. Princeton, NJPrinceton University Press131W. Fulton, Introduction to toric varieties, Annals of Mathematics Studies, 131, The William H. Roever Lectures in Geometry, Princeton University Press, Princeton, NJ, 1993. T Hibi, Ehrhart polynomials of convex polytopes, h-vectors of simplicial complexes, and nonsingular projective toric varieties, Discrete and computational geometry. New Brunswick, NJ; Providence, RIAmer. Math. Soc6T. Hibi, Ehrhart polynomials of convex polytopes, h-vectors of simplicial complexes, and non- singular projective toric varieties, Discrete and computational geometry (New Brunswick, NJ, 1989/1990), 165-177, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 6, Amer. Math. Soc., Providence, RI, 1991. Algebraic combinatorics on convex polytopes. T Hibi, Carslaw PublicationsT. Hibi, Algebraic combinatorics on convex polytopes, Carslaw Publications, 1992. Star-shaped complexes and Ehrhart polynomials. T Hibi, Proc. Amer. Math. Soc. 123T.Hibi, Star-shaped complexes and Ehrhart polynomials, Proc. Amer. Math. Soc. 123 (1995), 723-726. Special simplicies and Gorenstein toric rings. H Ohsugi, T Hibi, math.CO/0503666H. Ohsugi and T. Hibi, Special simplicies and Gorenstein toric rings, math.CO/0503666. Orbifold cohomology group of toric varieties. M Poddar, Orbifolds in mathematics and physics. Madison, WI; Providence, RIAmer. Math. Soc310M. Poddar, Orbifold cohomology group of toric varieties, in Orbifolds in mathematics and physics (Madison, WI, 2001), 223-231, Contemp. Math. 310, Amer. Math. Soc., Providence, RI, 2002. R P Stanley, Combinatorics and commutative algebra. Boston41Birkhäuser. first edition. second editionR.P. Stanley, Combinatorics and commutative algebra, Progress in Mathematics 41, Birkhäuser, Boston, first edition 1983; second edition 1996. Twisted jets, motivic measures and orbifold cohomology. T Yasuda, Compos. Math. 140T. Yasuda, Twisted jets, motivic measures and orbifold cohomology, Compos. Math. 140 (2004), 396-422.
{'fraction_non_alphanumeric': 0.07894002789400278, 'fraction_numerical': 0.023430962343096235, 'mean_word_length': 3.3203454508937535, 'pattern_counts': {'":': 0, '<': 5, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 39, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': "We study the connection between stringy Betti numbers of Gorenstein toric varieties and the generating functions of the Ehrhart polynomials of certain polyhedral regions. We use this point of view to give counterexamples to Hibi's conjecture on the unimodality of δ-vectors of reflexive polytopes.", 'arxivid': 'math/0504486', 'author': ['Mircea Mustaţǎ ', 'Sam Payne '], 'authoraffiliation': [], 'corpusid': 119118251, 'doi': '10.1007/s00208-005-0691-x', 'github_urls': [], 'n_tokens_mistral': 7546, 'n_tokens_neox': 6681, 'n_words': 4142, 'pdfsha': '6cfc661bf23bbff4a179ae8bb12dd046c4917ee0', 'pdfurls': ['https://export.arxiv.org/pdf/math/0504486v2.pdf'], 'title': ['EHRHART POLYNOMIALS AND STRINGY BETTI NUMBERS', 'EHRHART POLYNOMIALS AND STRINGY BETTI NUMBERS'], 'venue': []}
arxiv
A NEW PROOF OF THE HARDY-RELLICH INEQUALITY IN ANY DIMENSION 26 Nov 2018 Cristian Cazacu A NEW PROOF OF THE HARDY-RELLICH INEQUALITY IN ANY DIMENSION 26 Nov 2018Hardy inequalityspherical coordinates Mathematics Subject Classification 2010 : 35A2326D10 The Hardy-Rellich inequality in the whole space with the best constant was firstly proved by Tertikas and Zographopoulos in Adv. Math. (2007) in higher dimensions N ≥ 5. Then it was extended to lower dimensions N ∈ {3, 4} by Beckner in Forum Math. (2008) and Ghoussoub-Moradifam in Math. Ann. (2011) by applying totally different techniques.In this note we refine the method implemented by Tertikas and Zographopoulos, based on spherical harmonics decomposition, to give an easy and compact proof of the optimal Hardy-Rellich inequality in any dimension N ≥ 3. In addition, we provide minimizing sequences which were not explicitly mentioned in the quoted papers, emphasizing their symmetry breaking in lower dimensions N ∈ {3, 4}. We also show that the best constant is not attained in the proper functional space. Keywords: Hardy inequality, spherical coordinates Mathematics Subject Classification 2010 : 35A23, 26D10 In this note we first present a new unified proof for the following well-known optimal Hardy-Rellich inequality in any dimension. To the best of our knowledge inequality (1) was firstly analyzed and proved by Tertikas-Zographopoulos [7] in higher dimensions N ≥ 5. Their method applies spherical harmonics decomposition but their proof fails for lower dimensions N ∈ {3, 4}. Soon after that, inequality (1) was firstly completed in any dimensions N ≥ 3 by Beckner [3], making usage of Fourier transform tools. Subsequently, Moradifam-Ghoussoub [4] developed a quite general theory which allowed them to obtain the most classical functional inequalities and their improvements in the literature. The authors in [4] combine the method in [7] with some ideas from [1,2,6] reducing the problem to determine positive solutions for some parametric ordinary differential equations of Bessel-type. In particular, the authors in [4] justify Theorem 1. However, their proof requires to split the analysis into several parts in which they distinguish different techniques in the cases N ≥ 5 than for N ∈ {3, 4}. We point out that the authors in [4] considered inequalities in bounded domains but they can be trivially extended to the whole space. It is classical for functional inequalities that the advantage of working in bounded domains allows to improve them by adding positive lower order reminder terms. It is also worth mentioning the preprint [5] which complements the above papers with Rellich-type inequalities for vector fields. The first novelty of this note regards a short (but detailed) and compact proof of Theorem 1 in any dimension N ≥ 3 by means of the spherical harmonics decomposition. In fact, we show that the same technique applied in [7] to prove Theorem 1 for higher dimensions N ≥ 5 (but slightly modified computations) could be easily extended to any dimension N ≥ 3. Moreover, although the constant C(N ) in Theorem 1 is optimal, that is (3) C(N ) = inf u∈C ∞ c (R N )\{0} R N |∆u| 2 dx R N |∇u| 2 /|x| 2 dx , it seems that the authors in [4,3] do not explicitly give minimizing sequences in lower dimensions N ∈ {3, 4} for C(N ), see, e.g. [4,Th. 3.5] and its proof. However, in [7] minimizing sequences are given in dimensions N ≥ 5. Next we provide minimizing sequences in the cases N ∈ {3, 4}. We also show the nonattainability (in the largest possible Hilbert space) of the best constant C(N ) for any N ≥ 3, fact which was not emphasized in the quoted papers. In order to state our results we need some preliminary facts. First let us consider the Hilbert space D 2,2 (R N ) to be the completion of C ∞ c (R N ) in the norm u = R N |∆u| 2 dx 1/2 . Of course, · is a norm on C ∞ c (R N ) due to the weak maximum principle for harmonic functions. In view of that, the optimization problem (3) transfers to the larger space D 2,2 (R N ), i.e. C(N ) = inf u∈D 2,2 (R N )\{0} R N |∆u| 2 dx R N |∇u| 2 /|x| 2 dx , which is the natural space where to look for minimizers. In addition, we consider a smooth cut-off function g ∈ C ∞ c (R) such g(r) = 1, if |r| ≤ 1 0, if |r| ≥ 2. We claim Theorem 2 (Minimizing sequences). Let ǫ > 0 and define de sequence (4) u ǫ (x) = |x| − N−4 2 +ǫ g(|x|), if N ≥ 5 |x| − N−4 2 +ǫ g(|x|)φ 1 x |x| , if N ∈ {3, 4} where φ 1 is a spherical harmonic function of degree 1 such that φ 1 L 2 (S N−1 ) = 1. Then {u ǫ } ǫ>0 ⊂ D 2,2 (R N ) is a minimizing sequence for C(N ), i. e. (5) R N |∆u ǫ | 2 dx R N |∇u ǫ | 2 /|x| 2 dx ց C(N ), as ǫ ց 0. Besides, the constant C(N ) is not attained in D 2,2 (R N ) (there are no minimizers in D 2,2 (R N )). Remark 1. The first part of Theorem 2 is relevant for N ∈ {3, 4}. The fact that {u ǫ } ǫ>0 in (4), when N ≥ 5, is a minimizing sequence is void in view of [7, Th. 6.6] by taking m = k = 0 and φ 0 (σ) = constant. Our cut-off function is slightly different than the one in [7] but this is not an issue. Proof of Theorem 1 The proof follows in several steps as follows. Step I: Spherical coordinates. We appeal to spherical coordinates instead of cartesian coordinates. The coordinates transformation x ∈ R N → (r, σ) ∈ (0, ∞) × S N −1 , where S N −1 is the N − 1-dimensional sphere with respect to the Hausdorff measure in R N , is very convenient in R N since we can easily expand in Fourier series. Firstly, let us recall that the expression of the Laplace operator in spherical coordinates is given by (6) ∆ = ∂ 2 rr + N − 1 r ∂ r + 1 r 2 ∆ S N−1 , where ∂ r and ∂ 2 rr are both partial derivatives of first and second order with respect to the radial component r whereas ∆ S N−1 represents the Laplace-Beltrami operator with respect to the metric tensor on S N −1 . Without loss of generality, by density arguments, we may assume u ∈ C ∞ 0 (R N \ {0}). Next we apply the spherical harmonics decomposition to expand u as u(x) = u(rσ) = ∞ k=0 u k (r)φ k (σ), It is well-known that such series expansion is possible since there exists an orthogonal basis {φ k } k≥0 in L 2 (S N −1 ) constituted by spherical harmonic functions φ k of degree k. Up to a normalization, we may assume that {φ k } k is an orthonormal basis in L 2 (S N −1 ). Moreover, such φ k are smooth eigenfunctions for the Laplace-Beltrami operator ∆ S N−1 with the corresponding eigenvalues c k = k(k + N − 2), k ≥ 0. To be more precise, we have the following properties (7)      −∆ S N−1 φ k = c k φ k on S N −1 , − S N−1 ∆ S N−1 φ k φ l dσ = S N−1 ∇ S N−1 φ k · ∇ S N−1 φ l dσ = c k S N−1 φ k φ l dσ = c k δ lk , k, l ∈ N, where δ lk represents the Kronecker symbol. Next, we will write u ′ k and u ′′ k to express both first and second derivatives of the Fourier coefficients {u k } k . In view of the well-known relation |∇u| 2 = |∂ r u| 2 + |∇ S N−1 u| 2 r 2 and the co-aria formula, we express both integrals in (1) in terms of the coefficients {u k } k . Applying the properties (7) we successively obtain (8) R N |∇u| 2 |x| 2 dx = ∞ k=0 ∞ 0 r N −3 |u ′ k | 2 dr + c k ∞ 0 r N −5 u 2 k dr . Moreover, in view of (6) we can easily get (9) R N |∆u| 2 dx = ∞ k=0 ∞ 0 r N −1 |∆ r u k | 2 + c 2 k r 4 u 2 k − 2c k r 2 u k ∆ r u k dr where ∆ r := ∂ 2 rr + N −1 r ∂ r is the radial part of the Laplacian in (6). Finally, integration by parts in (9) leads to (10) R N |∆u| 2 dx = ∞ k=0 ∞ 0 r N −1 |u ′′ k | 2 dr + (N − 1 + 2c k ) ∞ 0 r N −3 |u ′ k | 2 dr + c 2 k + 2c k (N − 4) ∞ 0 r N −5 u 2 k dr . In the sequel, we prove Theorem 1 taking advantage of identities (8) and (10). Step II: Weighted 1-d Hardy inequalities. Next, we will apply the following weighted Hardy-Rellich type inequalities (11) ∞ 0 r N −1 |u ′′ k | 2 dr ≥ (N − 2) 2 4 ∞ 0 r N −3 |u ′ k | 2 dr, ∀k ≥ 0. (12) ∞ 0 r N −3 |u ′ k | 2 dr ≥ (N − 4) 2 4 ∞ 0 r N −5 u 2 k dr, ∀k ≥ 0. The proofs of inequalities (11) and (12) are straightforward and follow in a similar way. Inequality (11) is nothing else than the classical Hardy inequality for radial functions but it can be proven independently mimicking the proof of (12). For the sake of clarity let us give a few lines proof of (12). Indeed, ∞ 0 r N −5 u 2 k dr = 1 N − 4 ∞ 0 r N −4 ′ u 2 k dr = −2 N − 4 ∞ 0 r N −4 u k u ′ k dr ≤ 2 N − 4 r N −3 |u ′ k | 2 dr 1/2 ∞ 0 r N −5 u 2 k dr 1/2 ,(13) where the last step is just the Cauchy-Schwarz inequality. Comparing the extreme terms above by taking squares we finally obtain (12). Step III: End of the proof. We will make usage of Step I and Step II when comparing both integrals in (1). First we split the term on the right hand side in (10) into the sum I 1 + I 2 where I 1 := ∞ k=0 ∞ 0 r N −1 |u ′′ k | 2 dr + (N − 1) ∞ 0 r N −3 |u ′ k | 2 dr denotes the radial part of the expansion in (10), whereas I 2 := ∞ k=0 2c k ∞ 0 r N −3 |u ′ k | 2 dr + c 2 k + 2c k (N − 4) ∞ 0 r N −5 u 2 k dr is its spherical part. Then, due to (11) we have (14) I 1 ≥ N 2 4 ∞ k=0 ∞ 0 r N −3 |u ′ k | 2 dr. In addition, from (12) we get (15) I 2 ≥ ∞ k=0 c k g(N, k) ∞ 0 r N −5 u 2 k dr, where g(N, k) := (N − 4) 2 /2 + c k + 2(N − 4). Since {c k } k≥0 is a nonnegative increasing sequence, it is easy to notice that the sequence {g(N, k)} k≥1 is positive and increasing for any N ≥ 3. Therefore, we have g(N, k) ≥ g(N, 1) = N 2 − 2N − 2 2 , ∀k ≥ 1. Since c 0 = 0 from (15) we obtain (16) I 2 ≥ N 2 − 2N − 2 2 ∞ k=0 c k ∞ 0 r N −5 u 2 k dr Summing up, from (14), (16) and (8) we get (17) R N |∆u| 2 dx ≥ min N 2 4 , N 2 − 2N − 2 2 R N |∇u| 2 |x| 2 dx. Since (18) min N 2 4 , N 2 − 2N − 2 2 =        N 2 4 , N ≥ 5 3, N = 4 1 2 N = 3, inequality (1) is proven for any N ≥ 4. For N = 3 the final step of the argument above does not provide the optimal constant C(3) since 1/2 < C(3) = 25/36. In order to recover the constant C(3) in the following we slightly modify the last part of the proof. First observe that the constant N 2 /4 in (14) is optimal since the constant (N − 2) 2 /4 in inequality (11) is also optimal. This implies that C(N ) ≤ N 2 4 , ∀N ≥ 3. and therefore, in view of (17) we obtain C(N ) = N 2 /4 for any N ≥ 5. For N ∈ {3, 4} the minimum in (18) is attained by (N 2 −2N −2)/2 which is strictly smaller than N 2 /4. In fact, due to this gap there is a coincidence that the minimum in (18) for N = 4 coincides with C(4). In view of these considerations next we show how to recover the best constant C(N ) for N ∈ {3, 4}. So, next we focus on N ∈ {3, 4}. Observe that the term ∞ 0 r N −3 |u ′ k | 2 dr appears in both I 1 and I 2 . Next we want this term to be "equally distributed" in I 1 and I 2 so that to contribute with the same constants in (14) and (16). For that, first let 0 < ǫ < N 2 /4 which will be well precise later. Now we reconsider the terms I 1 and I 2 by splitting the right hand side of (10) as I 1,ǫ + I 2,ǫ I 1,ǫ := ∞ k=0 ∞ 0 r N −1 |u ′′ k | 2 dr + (N − 1 − ǫ) ∞ 0 r N −3 |u ′ k | 2 dr and I 2,ǫ := ∞ k=0 (2c k + ǫ) ∞ 0 r N −3 |u ′ k | 2 dr + c 2 k + 2c k (N − 4) ∞ 0 r N −5 u 2 k dr. Again from (11) we obtain (19) I 1,ǫ := N 2 4 − ǫ ∞ k=0 ∞ 0 r N −3 |u ′ k | 2 dr. Applying (12) and the fact that c 0 = 0 from the expression of I 2,ǫ we get (20) I 2,ǫ ≥ ∞ k=1 c k h(ǫ, k) ∞ 0 r N −5 u 2 k dr, where h(ǫ, k) := (2 + ǫ/c k )(N − 4) 2 /4 + c k + 2(N − 4), for any k ≥ 1. Since c k ≥ N − 1 for any k ≥ 1 we easily remark that the sequence {h(ǫ, k)} k≥1 is increasing. Therefore, h(ǫ, k) ≥ h(ǫ, 1) = 2 + ǫ N − 1 N − 4 2 2 + 3N − 9, ∀k ≥ 1 and it follows that (21) I 2,ǫ ≥ 2 + ǫ N − 1 N − 4 2 2 + 3N − 9 ∞ k=0 c k ∞ 0 r N −5 u 2 k dr. Next we chose ǫ to obtain the same constant in both inequalities (19) and (21), i.e. N 2 4 − ǫ = 2 + ǫ N − 1 N − 4 2 2 + 3N − 9 . This is equivalent to ǫ(N ) = (N − 1)(−N 2 + 4N + 4) N 2 − 4N + 12 . We then obtain (22) R N |∆u| 2 dx ≥ N 2 4 − ǫ(N ) R 3 |∇φ| 2 |x| 2 dx. Since ǫ(4) = 1 and ǫ(3) = 14/9 we finally get the desired constants N 2 4 − ǫ(N ) N =4 = 3, N 2 4 − ǫ(N ) N =3 = 25 36 . We conclude that inequality (1) in Theorem 1 holds also for C(3) = 25/36 and C(4) = 3. Remark 2. Notice also that the optimality of C(N ) = N 2 /4 for N ≥ 5 is hidden (and specified) in the proof of Theorem 1 without the necessity of building a minimizing sequence. Proof of Theorem 2 As we already mentioned in Remark 1, the proof of optimality is relevant only for N ∈ {3, 4}. However, for the sake of completeness, since our computations are slightly different than those in [7], let us give a full dimensional proof. Optimality (the cases N ≥ 5). Writing u ǫ (x) = U ǫ (|x|), in view of (9), since the spherical part is missing we obtain the simplified expression (23) R N |∆u ǫ | 2 dx = |S N −1 | ∞ 0 r N −1 |U ′′ ǫ (r)| 2 dr + (N − 1) ∞ 0 r N −3 |U ′ ǫ (r)| 2 dr . and (24) R N |∇u ǫ | 2 |x| 2 dx = |S N −1 | ∞ 0 r N −3 |U ′ ǫ (r)| 2 dr. Then we have ∞ 0 r N −3 |U ′ ǫ (r)| 2 dr = − N − 4 2 + ǫ 2 ∞ 0 r −1+2ǫ g 2 (r)dr + ∞ 0 r 1+2ǫ g ′ (r) 2 dr + 2 − N − 4 2 + ǫ ∞ 0 r 2ǫ g(r)g ′ (r)dr = 1 2ǫ − N − 4 2 + ǫ 2 + O(1).(25) since g ′ is supported in the interval [1,2]. From the same reasons since U ′′ ǫ (r) = − N − 4 2 + ǫ − N − 2 2 + ǫ r −N/2+ǫ g(r) + χ [1,2] O(1) we obtain The above limit also holds in the case N = 3 but it does not provide the best constant C(3). The case N = 4 is not covered because of the nontermination 0 0 . Optimality (the cases N ∈ {3, 4}). As before we obtain Theorem 1 . 1Assume N ≥ 3. Then, for any u ∈ C ∞ c (R N ) it holds (25) and (26) we successively obtainR N |∆u ǫ | 2 dx R N |∇u ǫ | 2 /|x| 2 dx The non-attainability of the best constant C(N ), N ≥ 3. The non-attainability follows the lines of the proof of Theorem 1. Indeed, assuming that C(N ) is attained then it is necessary to have equality in inequalities (11)-(12) for any u k in the decomposition of u. Remark that inequality (11) is also a consequence of the identityrdr.In view of (29) we obtain that equality in(11)is not admissible either because none of the terms in (11) is integrable. In consequence the constant C(N ) is not attained. Bounds on exponential decay of eigenfunctions of Schrödinger operators. S Agmon, Lecture Notes in Math. 1159SpringerS. Agmon, Bounds on exponential decay of eigenfunctions of Schrödinger operators, Schrödinger operators (Como, 1984), Lecture Notes in Math., vol. 1159, Springer, Berlin, 1985, pp. 1-38. On the equivalence of two types of oscillation for elliptic operators. W Allegretto, Pacific J. Math. 55W. Allegretto, On the equivalence of two types of oscillation for elliptic operators, Pacific J. Math. 55 (1974), 319-328. Weighted inequalities and Stein-Weiss potentials. W Beckner, Forum Math. 204W. Beckner, Weighted inequalities and Stein-Weiss potentials, Forum Math. 20 (2008), no. 4, 587-606. Bessel pairs and optimal Hardy and Hardy-Rellich inequalities. N Ghoussoub, A Moradifam, Math. Ann. 3491N. Ghoussoub and A. Moradifam, Bessel pairs and optimal Hardy and Hardy-Rellich inequalities, Math. Ann. 349 (2011), no. 1, 1-57. N Hamamoto, F Takahashi, arXiv:1808.09614Sharp Hardy-Leray and Rellich-Leray inequalities for curl-free vector fields. N. Hamamoto and F. Takahashi, Sharp Hardy-Leray and Rellich-Leray inequalities for curl-free vector fields, arXiv:1808.09614. Nonoscillatory elliptic equations. J Piepenbrink, J. Differential Equations. 15J. Piepenbrink, Nonoscillatory elliptic equations, J. Differential Equations 15 (1974), 541-550. Best constants in the Hardy-Rellich inequalities and related improvements. A Tertikas, N B Zographopoulos, Adv. Math. 2062A. Tertikas and N. B. Zographopoulos, Best constants in the Hardy-Rellich inequalities and related improve- ments, Adv. Math. 206 (2007), no. 2, 407-459. . ( C Cazacu, Faculty of Mathematics and Computer Science & ICUB, University of Bucharest(C. Cazacu), Faculty of Mathematics and Computer Science & ICUB, University of Bucharest, E-mail address: [email protected]. Academiei Street, Bucharest, RomaniaAcademiei Street, 010014 Bucharest, Romania, E-mail address: [email protected]
{'fraction_non_alphanumeric': 0.09525888958203368, 'fraction_numerical': 0.05221459762944479, 'mean_word_length': 3.1736526946107784, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 17, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'The Hardy-Rellich inequality in the whole space with the best constant was firstly proved by Tertikas and Zographopoulos in Adv. Math. (2007) in higher dimensions N ≥ 5. Then it was extended to lower dimensions N ∈ {3, 4} by Beckner in Forum Math. (2008) and Ghoussoub-Moradifam in Math. Ann. (2011) by applying totally different techniques.In this note we refine the method implemented by Tertikas and Zographopoulos, based on spherical harmonics decomposition, to give an easy and compact proof of the optimal Hardy-Rellich inequality in any dimension N ≥ 3. In addition, we provide minimizing sequences which were not explicitly mentioned in the quoted papers, emphasizing their symmetry breaking in lower dimensions N ∈ {3, 4}. We also show that the best constant is not attained in the proper functional space.', 'arxivid': '1809.07506', 'author': ['Cristian Cazacu '], 'authoraffiliation': [], 'corpusid': 56253045, 'doi': '10.1017/prm.2019.50', 'github_urls': [], 'n_tokens_mistral': 6283, 'n_tokens_neox': 5456, 'n_words': 3337, 'pdfsha': '6786879e8742d9400f806022f4a55fc277cca2b4', 'pdfurls': ['https://arxiv.org/pdf/1809.07506v2.pdf'], 'title': ['A NEW PROOF OF THE HARDY-RELLICH INEQUALITY IN ANY DIMENSION', 'A NEW PROOF OF THE HARDY-RELLICH INEQUALITY IN ANY DIMENSION'], 'venue': []}
arxiv
Semiclassical description of chiral geometry in triaxial nuclei 22 Jun 2018 R Budaca Horia Hulubei" National Institute for Physics and Nuclear Engineering Str. Reactorului 30, POB-MG6RO-077125Bucharest-MǎgureleRomania Academy of Romanian Scientists 54 Splaiul IndependenţeiRO-050094BucharestRomania Semiclassical description of chiral geometry in triaxial nuclei 22 Jun 2018(Dated: June 26, 2018)numbers: 2110Re2320Lv2770+q A triaxial particle-rotor Hamiltonian for three mutually perpendicular angular momentum vectors corresponding to two high-j quasiparticles and the rotation of a triaxial collective core, is treated within a time-dependent variational principle. The resulting classical energy function is used to investigate the rotational dynamics of the system. It is found that the classical energy function exhibits two minima starting from a critical angular momentum value which depends on the singleparticle configuration and the asymmetry measure γ. The emergence of the two minima is attributed to the breaking of the chiral symmetry. Quantizing the energy function for a given angular momentum, one obtains a Schrödinger equation with a coordinate dependent mass term for a symmetrical potential which changes from a single to a double well shape as the angular momentum pass the critical value. The energies of the chiral partner bands for a given angular momentum are then given by the lowest two eigenvalues. The procedure is exemplified for maximal triaxiality and two h 11/2 quasiparticles, with the results used for the description of the chiral doublet bands in 134 Pr. I. INTRODUCTION The concept of chirality or handedness is a common occurrence in biology, chemistry, optics, and particle physics. In nuclear physics, chirality is associated with the geometry of three mutually perpendicular angular momenta. It was originally suggested by Frauendorf and Meng [1] for a system composed of a triaxial core coupled to a set of high-j valence particles and holes. The rationale for this particular ensemble [1,2] is that the triaxial core tends to rotate around the axis with the largest moment of inertia (MOI) which imply an intermediate density distribution, while the motion of particles and holes prefer ellipsoidal orbits following maximal and respectively minimal density distributions around the other two principal axes. The three mutually perpendicular angular momenta form a screw in respect to the total angular momentum vector and therefore can be arranged to form two systems with opposite intrinsic chirality. As the broken chiral symmetry should be restored in the laboratory frame of reference, one expects to observe two nearly degenerate ∆I = 1 bands with the same parity. This particular signature, i.e. the so-called chiral doublet bands, was first observed in few N = 75 odd-odd isotones [3]. The experimental confirmation of chiral symmetry breaking was followed by an extensive search for other candidate nuclei. Such that, presently chiral bands are reported in over 30 nuclei clustered in "islands" of chirality around mass numbers 80, 100, 130 and 190, where the chiral geometry is generated by specific quasiparticle configurations [4,5]. The original interpretation of chirality [1] was based on both the particle-rotor (PRM) [6] and tilted axis cranking models (TAC) [7]. Alternative descriptions of the chiral bands include presently boson expansion approaches [8][9][10][11], and Shell Model based formalisms [12,13]. However, being a fully quantum model, and therefore capable of treating the tunneling between the two chiral solutions, PRM remained the standard for theoretical studies of chirality [14][15][16][17][18][19][20][21][22][23]. Although the semiclassical nature of the cranking mean field approaches is not able to describe the quantum interaction between the chiral bands, it has the advantage of providing a relation between the density distribution and the direction of the total angular momentum vector [24][25][26][27][28][29][30]. Moreover it can be easily extrapolated to multi-quasiparticle configurations, whereas the PRM advances in this direction are incipient [31][32][33]. The need of going beyond mean field approximation produced successful extensions of the TAC formalism such as the TAC plus random phase approximation [34,35], and the collective Hamiltonian approach [36,37]. The latter take advantage of the information on classical rotational dynamics obtained from TAC calculations to construct a quantum collective Hamiltonian, whose solutions were shown to be close to the fully quantum and exact PRM calculations. In this paper, one will take the opposite approach in combining the advantages of the classical and quantum pictures by treating semiclassically a particle-rotor type Hamiltonian. The semiclassical procedure amounts to ascribing a time-dependent variational principle to the quantum Hamiltonian, which is consequently dequantized into a classical energy function. A similar procedure was already successfully applied for the description of wobbling excitations in odd mass nuclei [38,39]. By choosing an appropriate variational function one can select a limited set of degrees of freedom relevant for the studied phenomenon instead of treating the full space. The information on the rotational dynamics of the system is then extracted from the evolution of the classical energy function as well as other observables expressed in terms of the azimuthal and polar angles with the vari-ation of the total angular momentum which retains its quality of good quantum number. The emergence of chiral solutions at a certain spin is discussed from the classical point of view. For the description of the chiral partner bands, the classical energy function is quantized in respect to a chiral variable. The similarities and the differences between the resulting Schrödinger equation and the chiral collective Hamiltonian of Refs. [36,37] are pointed out. The formalism is applied to the description of the chiral bands in 134 Pr. II. SEMICLASSICAL APPROACH The following extension of the particle-rotor Hamiltonian [6] H = H R + H sp + H ′ sp , (2.1) is employed for the description of the interaction between two single-particle angular momenta and a collective one. H R = k=1,2,3 A k (Î k −ĵ k −ĵ ′ k ) 2 is the triaxial rotor Hamiltonian associated to the core angular momentum R = I − j − j ′ and defined by the inertial parameters A k = 1/(2J k ) where J k are the MOI along the principal axes of the intrinsic frame of reference considered in the hydrodynamic estimation [6]: J k = 4 3 J 0 sin 2 γ − 2 3 kπ . (2.2) j and j ′ are single-particle generated spins, i.e. they can be the total angular momentum of a single quasiparticle orbital or a resultant spin of few quasiparticles. The single-particle contribution to the total Hamiltonian coming from single-particle spin j is H sp = V j(j + 1) 3ĵ 2 3 − j(j + 1) cos γ − √ 3(ĵ 2 1 −ĵ 2 2 ) sin γ , (2.3) where γ is the asymmetry parameter, which also defines the ratios between MOI. Suppose that each of the single-particle angular momenta is aligned to a principal axis of the intrinsic frame of reference. Although the angular momentum of a triaxial rotor is actually distributed on all three axes, the core will rotate around the axis with the highest MOI, which is then chosen as a quantization axis. As the absolute value of the rotor spin increases, the contributions from the other two axes become smaller [40] and can be quantized for example into wobbling excitations [6]. If this rotation axis is perpendicular to the plane of the two single-particle spins, then one will have a trihedral vector configuration as in Fig.1. In order to have the highest MOI along the third intrinsic axis, γ must be within the interval (60 • , 120 • ). In this γ interval, the ellipsoid's semi-axes R k = R 0 1 + 5 4π β cos γ − 2π 3 k (2.4) are arranged as R 2 < R 3 < R 1 . In order to keep track of the direction of the total angular momentum vector relative to the density distribution, the axes 1,2 and 3 are also referred to as the long (l), short (s) and medium (i). Choosing the single-particle alignment to be rigid along the axes 1 and 2, i.e.ĵ 1 ≈ j ≡ const. and j ′ 2 ≈ j ′ ≡ const., the Hamiltonian relevant for the system's dynamics can be limited to: H = A 1 (Î 1 − j) 2 + A 2 (Î 2 − j ′ ) 2 + A 3Î 2 3 + const. (2.5) Thus, the Hamiltonian to be treated is: H chiral = A 1Î 2 1 + A 2Î 2 2 + A 3Î 2 3 − 2A 1 jÎ 1 − 2A 2 j ′Î 2 . (2.6) For the purpose of investigating the rotational motion described by the quantum Hamiltonian (2.6), one considers the variational principle δ t 0 ψ(z)|H chiral − ∂ ∂t ′ |ψ(z) dt ′ = 0. (2.7) The variational state is chosen of the form: |ψ(z) = I K=−I (2I)! (I − K)!(I + K)! z I+K (1 + |z| 2 ) I |IM K = 1 (1 + |z| 2 ) I e zÎ− |IM I . (2.8) This is a spin coherent state with z being a complex time-dependent variable, |IM K are the eigenstates of the intrinsic angular momentum operatorsÎ 2 andÎ 3 and their counterparts in the laboratory frame of reference, whileÎ − is a ladder operator. The averages on the variational state of the terms involved in the variation (2.7) are calculated using the results of Refs. [41][42][43] and have the following expressions: H chiral = I 2 (A 1 + A 2 ) + A 3 I 2 + I(2I − 1) 2(1 + zz * ) 2 × A 1 (z + z * ) 2 − A 2 (z − z * ) 2 − 4A 3 zz * − 2A 1 jI(z + z * ) 1 + zz * + i 2A 2 j ′ I(z − z * ) 1 + zz * , (2.9) ∂ ∂t = I(żz * − zż * ) 1 + zz * . (2.10) z and its complex conjugate counterpart are considered as independent variables. The time dependent variational equation (2.7) offers the following equations of motion for the complex variables z and z * : ∂H ∂z = − 2iIż * (1 + zz * ) 2 , ∂H ∂z * = 2iIż (1 + zz * ) 2 ,(2.11) where H(z, z * ) = H chiral plays now the role of a classical energy function which is also a constant of motion. For simplicity, the complex variable is written in a stereographic representation [41] z = tan θ 2 e iϕ , 0 ≤ θ < π, 0 ≤ ϕ < 2π. (2.12) Within this parametrization, the angular momentum carried by the coherent state is oriented in the direction specified by the two angles of rotation θ and ϕ [42] as in Fig.1 The classical energy function have the following expression in terms of the canonical variables: H(r, ϕ) = I 2 (A 1 + A 2 ) + A 3 I 2 + (2I − 1)r(2I − r) 2I ×(A 1 cos 2 ϕ + A 2 sin 2 ϕ − A 3 ) − 2A 1 j r(2I − r) cos ϕ − 2A 2 j ′ r(2I − r) sin ϕ. (2.18) The conservation of the total angular momentum I 2 = I 2 1 + I 2 2 + I 2 3 , (2.19) is guaranteed by the classical expressions of the angular momentum components as functions of the canonical variables [43,44]: I 1 = r(2I − r) cos ϕ, I 2 = r(2I − r) sin ϕ, (2.20) I 3 = r − I. In what follows, the ϕ angle will be restricted to the interval (0, 90 • ), which corresponds to a situation when the total angular momentum and the single-particle spins share an octant of the three-dimensional space. This implies cos ϕ > 0 and sin ϕ > 0. III. ROTATIONAL DYNAMICS The minimum points of the constant energy surface H(r, ϕ) = const. correspond to stable dynamical configurations. These are determined from: ∂H ∂r r0,ϕ0 = 0, ∂H ∂ϕ r0,ϕ0 = 0, Det ∂ 2 H ∂q i ∂q j r0,ϕ0 > 0,(3.1) where i(j) = 1, 2 with q 1 = r and q 2 = ϕ. At this point it is worth to recount that there are few possibilities in what concerns distribution of the total angular momentum on the principal axes of the intrinsic frame of reference [1]. When the total angular momentum lies within a principal plane, the situation is called planar. While an aplanar configuration designates a total angular momentum with non-vanishing projections on all principal axes. Solving thus the system of equations (3.1), one obtains a critical point (r p , ϕ p ) corresponding to a planar case, with r p = I and ϕ p given as a solution of the equation (2I − 1) 2 (A 2 −A 1 ) cos ϕ p sin ϕ p = A 2 j ′ cos ϕ p −A 1 j sin ϕ p . (3. 2) The planar nature of this critical point results from a vanishing third component of the total angular momentum for r p = I. From the above equation one can see that ϕ p depends on I, except when γ = 90 • because then A 1 = A 2 . In this particular case one have just tan ϕ p = j ′ /j. Eqs. (3.1) also provide an aplanar stationary point specified by: sin ϕ a = A 2 j ′ (A 1 − A 3 ) A 2 1 j 2 (A 2 − A 3 ) 2 + A 2 2 j ′2 (A 1 − A 3 ) 2 , (3.3) cos ϕ a = A 1 j(A 2 − A 3 ) A 2 1 j 2 (A 2 − A 3 ) 2 + A 2 2 j ′2 (A 1 − A 3 ) 2 , (3.4) r a (2I − r a ) = I sin θ I a ,(3.5) where one used the following notation sin θ I a = 2 A 2 1 j 2 (A 2 − A 3 ) 2 + A 2 2 j ′2 (A 1 − A 3 ) 2 (2I − 1)(A 1 − A 3 )(A 2 − A 3 ) . (3.6) The two solutions for r a will then be r R = I(1 + cos θ I a ), (3.7) r L = I(1 − cos θ I a ). (3.8) Note that angle θ I a is spin-dependent, while ϕ a is not. The indexes R and L denote the right-handed and respectively the left-handed total angular momentum orientation in respect to the intrinsic frame of reference. The right-handedness of a configuration is associated to the case when one can count in the mathematically positive direction the principal axes as 1, 2 and 3 when looking from the tip of the total angular momentum vector. This assignment of the two solutions is more obvious when the averages of the total angular momentum components are considered for this aplanar stationary point: I a 1 = I sin θ I a cos ϕ a , (3.9) I a 2 = I sin θ I a sin ϕ a , (3.10) I a 3 = ±I cos θ I a . (3.11) This is just a representation of a vector of magnitude I in spherical coordinates (see Fig.1). The minimum condition for the aplanar solutions implies that the rational This is obvious for the first two, from their analytical expression. The latter however is more difficult to judge, but can be inferred from the successive changes of variables leading to the expression (3.6). Nevertheless, the condition 0 < sin θ I a < 1 provides some additional restrictions on the relative distribution of the A k parameters which are also angular momentum dependent. As a matter of fact, the condition sin θ I a = 1 serves as a sep- aratrix which marks the border between the planar and aplanar solutions corresponding to two distinct rotational phases. This separatrix provides a critical angular momentum value at which the transition between the two phases commences from the planar phase to the aplanar one as is shown in the Fig.2. The critical angular momentum depends on the triaxiality measure γ as well as the single-particle spins. Its evolution as a function of γ is depicted in Fig.3 for few simple one particle one hole configurations commonly known to generate chiral symmetry breaking. The critical angular momentum value tends to infinity when the density distribution is axially symmetric, and has its minimum value at maximal triaxiality γ = 90 • . The minimum values for the considered quasiparticle configurations are listed in Table I. Fig.2 one observes that in the planar phase, the classical energy function has a single minimum at r p = I and ϕ p , which becomes a saddle point after crossing the separatrix. In turn, the saddle point marks the apparition of the two chiral minima at (r R , ϕ a ) and (r L , ϕ a ). Although using different variables, the energy surfaces of Fig.2 are similar to the total Routhian surface calculations made in [36] considering a single orientation angle, especially when γ = 90 • . The connection to azimuthal and polar angles can be easily made, one however maintained the (r, ϕ) space because the two variables are canonical conjugate. The major difference arises for the maximal triaxiality case (γ = 90 • ), where the classical energy function is doubly symmetric in respect to ϕ = 45 • and r = I(I 3 = 0) lines. This two-fold symmetry is however recovered when the total Routhian is considered in the full space of the two orientation angles [37]. At this point, one can analyze the dynamical evolution, i.e. as a function of total angular momentum modulus, of the tilting angles defining the average geometrical direction of the total angular momentum vector. This is best presented graphically in Fig.4, where one plotted the polar and azimuthal angles as function of the total angular momentum for few asymmetrical values of γ. In consensus with the previous observations, the starting value of the polar angle θ is 90 • . This value persists throughout the entire planar phase up to the critical value I c , where it bifurcates into the two chiral branches with θ = θ I a and θ = π − θ I a . The existence of the planar phase at small angular momentum values is due to the sizable components of the core angular momentum on the principal axes 1 and 2 [40] which add up to the single particle contributions. The planar average direction of the total spin is however soft against out of plane fluctuations as can be attested by the pronounced shallowness of the planar minima. In what concerns the azimuthal angle ϕ, it has an invariant value of 45 • for maximal triaxiality γ = 90 • , while for γ = 90 • it just starts from this value and is continuously decreasing up to the critical point keeping the corresponding tilting constant through the evolution in the aplanar phase. The correspondence between the constant value of the azimuthal angle acquired at the critical point and the triaxiality degree is visualized in Fig.5. IV. EMERGENCE OF CHIRAL BANDS As the energy function has always a single minimum only in the ϕ variable, one chooses to expand it around the corresponding minimum points for fixed values of r: H(r, ϕ) ≈ H(r, ϕ 0 (r)) + 1 2 ∂ 2 H ∂ϕ 2 ϕ0(r)φ 2 ,(4.1) whereφ = ϕ − ϕ 0 (r) with ϕ 0 (r) being the value which minimizes the energy function for a fixed r. ϕ 0 (r) is therefore defined as the solution of the following equation Lacking an analytical expression for the general solution of the above equation, one will further pursue only the special case of γ = 90 • , for which A 1 = A 2 and ϕ 0 = 45 • = const. The general case for γ = 90 • implies a numerical part and will be presented elsewhere. (2I − 1)r(2I − r)(A 2 − A 1 ) cos ϕ 0 sin ϕ 0 = 2I r(2I − r) (A 2 j ′ cos ϕ 0 − A 1 j sin ϕ 0 ) . (4.2) γ 80°L R φ In order to have a better view of the chiral dynamics, a new chiral variable is introduced, namelyr = r − I. This quantity is just the classical third component of the total angular momentum (2.20), and varies between −I and I. The pair of variablesr andφ are also canonical conjugate, i.e. {r,φ} = 1. Symmetrizing the products of r andφ functions one proceeds to the quantization of the approximate classical energy function (4.1) by making the substitutionsr = x,φ = i d dx ,(4.3) rather than quantizing the classical trajectories by means of a WKB-like approximation as was performed in Ref. [45]. This differential representation of the conjugate canonical coordinates is equivalent to working in the momentum space represented by the generalized momentum variabler. After the quantization procedure, one arrives at a quantum Hamiltonian expressed as the differential operator H c = − 1 2B(x) d 2 dx 2 + B ′ (x) 2 [B(x)] 2 d dx + H(x, ϕ 0 ) + B ′′ (x) 4 [B(x)] 2 − [B ′ (x)] 2 2 [B(x)] 3 ,(4.4) where B(x) = ∂ 2 H(x, ϕ) ∂ϕ 2 −1 ϕ0 . (4.5) Suppose now that the wave function corresponding to above quantum Hamiltonian is F (x) and is normalized to unity. Then making the change of function f (x) = [B(x)] −1/4 F (x), one can write the final Hamiltonian for f (x) in the following form H c = − 1 2 1 B(x) d dx 1 B(x) d dx + V (x),(4.6) where B(x) plays the role of an one dimensional mass which depends on x, while the corresponding potential is given as: V (x) = H(x, ϕ 0 ) + B ′′ (x) 8 [B(x)] 2 − 9 [B ′ (x)] 2 32 [B(x)] 3 . (4.7) The identification of B(x) as the mass of the system is confirmed also by the normalization condition for the function f (x) which reads as f (x)f * (x) B(x)dx = 1. (4.8) It can be easily checked that both mass function and the potential are invariant under the parity transformation x → −x. All these ingredients are reminiscent of the one dimensional collective Hamiltonian obtained in [36]. The difference here is that both mass term and potential are products of the original quantum Hamiltonian, and are determined solely on the basis of the rotational geometry, in comparison to the approach of Ref. [36] where the kinetic and potential terms are obtained in separate ways. Moreover, due to the canonical conjugate character of the two semiclassical coordinates r and ϕ, there is an additional relation between the polar and azimuthal angles. Thus, although the chiral Hamiltonian is one-dimensional, the quantum fluctuations of both directional angles are included. From the graphical representation of the chiral potential and mass term shown in Fig.6, one can see that while the emergence of the double minimum profile for the chiral potential is similar to that of Ref. [36], the dynamical evolution of the mass is quite different. Indeed, the mass term determined in Ref. [36] starts from being shallow at low rotational frequencies acquiring a more localized minimum as the frequency is increased. In the present case, the evolution is opposite as can be seen from Fig.6(b). This distinction comes from the fact that in the present case, the mass is defined in the momentum space. Otherwise the picture is consistent with the formalism of Ref. [36]. The states of the two chiral partner bands are then defined by the first two eigensolutions of the differential operator (4.4). Although the associated mass term and the chiral potential have analytical expressions, the corresponding Schrödinger equation cannot be exactly solved. Consequently, the energies are determined through a diagonalization in a suitable basis. In order to avoid large dimension diagonalizations, it is customary to use different basis states for even and odd parity solutions when symmetric potentials are involved. Choosing particle in the box eigenstates as basis functions, one assigns for even parity the basis states g 1 n (x) = 1 √ I cos (2n − 1)πx 2I , n = 1, 2, ..., (4.9) while for the odd parity the following basis states are used: g −1 n (x) = 1 √ I sin 2nπx 2I , n = 1, 2, ... and where shown to be very performant as basis states in symmetrical multiple minima problems [46,47]. The eigenvalues of Eq.(4.6) are then obtained by diagonalization in the above defined basis space which is truncated such that to accommodate a satisfactory convergence of the results. The same procedure will give the coefficients a n of the basis expansion where N denotes the dimension of the truncated space. The splitting between energies of the two chiral solutions is shown in Fig.7 relative to the barrier hight and the depth of the minimum for a series of integer values of total angular momentum. The splitting persists along many angular momentum states, vanishing only when the two energy states become considerably lower than the barrier peak, that is around I = 15. The results are consistent with the well known behaviour of the spectra for double well potentials [48]. V. TOTAL WAVE-FUNCTIONS AND ELECTROMAGNETIC TRANSITIONS Expressing the original complex variable in terms of the chiral one as In order to couple the rotational motion described by the above state with the information regarding the chiral vibration, one will weight the coherent state in ϕ = ϕ 0 = 45 • with the density probability for the oscillating chiral variable z = I − x I + x e iϕ ,(5.ρ I p (x) = F I p (x) 2 , p = −1, 1. (5.3) The evolution of this quantity with total angular momentum can be tracked in Fig.8, where one plotted the interpolated density probability as function of x and I. The wave functions with restored chiral symmetry can be then expressed as: I−K 2 (I − x) I+K 2 dx, while N Ip is a redefined normalization constant. Using these wave functions, one can now proceed to the calculation of the quadrupole transition probabilities using the following transition operator: M(E2, µ) = 5 16π Q ′ 0 D 2 µ0 + Q ′ 2 √ 2 D 2 µ2 + D 2 µ−2 . (5.6) Q ′ 0 and Q ′ 2 are intrinsic quadrupole moments for a reference frame where the MOI on the third principal axis is maximal. They can be related to the commonly used components Q 0 = Q cos γ and Q 2 = Q sin γ/ √ 2 defined in a system of reference with the maximal MOI along the first axis, by Q ′ 0 = − 1 2 Q 0 + 3 2 Q 2 = −Q cos γ + π 3 , (5.7) Q ′ 2 = − 1 2 3 2 Q 0 + Q 2 = −Q sin γ + π 3 √ 2 , (5.8) where Q = 3 √ 5π R 2 0 Zβ with β being the axial deformation, Z is the charge number, while R 0 is the nuclear radius. The reduced transition probability is determined with B(E2, Ip → I ′ p ′ ) = | Ip||M(E2)||I ′ p ′ | 2 . (5.9) The expression for the involved reduced matrix element of the quadrupole transition operator in the considered particular case of γ = 90 • and ϕ 0 = 45 • can be readily deduced: Ip||M(E2)||I ′ p ′ = Q 8 15 πÎ ′ I e i π 4 (I ′ −I) (5.10) × I K=−I S IKp S I ′ Kp ′ C I ′ 2 I K 0 K . The simple form is obtained by dismissing the nondiagonal quadrupole components which cancel each other when the summation is performed on positive and negative projections. Another observable related to chiral partner bands, is the magnetic dipole transition probability [15,20,[49][50][51][52]. It is however predominantly given by the single-particle degrees of freedom which are neglected in the present study. VI. COMPARISON WITH EXPERIMENT The formalism is applied to the chiral bands of 134 Pr, which are among the most extended in what concerns the number of observed different spin states. For the calculation of the energy levels corresponding to the two partner bands the following formula is used: E Ip = E 0 + E c Ip ,(6.11) where E 0 is an energy reference, while E c Ip is the eigenvalue of the chiral quantum Hamiltonian, obtained from 9. Comparison of yrast and non-yrast energy levels between theoretical results and experimental data [51] for 134 Pr. ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ • • • • • • • • • • • • • ■ © Exp. • n -Exp. ! " # $ % & ' ( ) -01234 5 6 7 V I 8 9 @ A B C D E F P G Q R 0 S T U W X Y I Energy [w`a ] FIG. the diagonalization procedure. The dimension of the diagonalization basis is truncated at 50 states, assuring thus a convergence of the energies up to spin I = 21. As the asymmetry of the triaxial core γ is considered fixed at 90 • , the only free parameters remain the reference energy E 0 and the inertial constant J 0 . Fitting the experimental data against the two parameters one obtains the following values: E 0 = 2.746 MeV, J 0 = 33.196 MeV −1 , which correspond to an rms of 59.28 keV. The exceptionally good reproduction of data can be better seen in Fig.9, where all aspects of the data evolution, such as the general rotational behaviour of the two bands, energy splitting between bands, as well as the angular momentum of critical point for the transition between chiral vibration and static chirality, are well reproduced. For the calculation of E2 transition probabilities, the value Q = 3.5 e b is considered as in Refs. [18,20,52]. The theoretical results are compared to experimentally available data on 134 Pr in Fig.10. The agreement with experiment is satisfactory, with a better reproduction of the data for intra-band transitions. Especially well reproduced is the descending trend of intra-band transitions for I = 15 − 17. The evolution of theoretical results with angular momentum is similar for both intraand inter-band transitions. In both cases, the transitions from yrast states are, with few exceptions, greater than those from the non-yrast states up to I = 16. The same is true for the measured values. For I ≥ 17, the two transition probabilities become equal due to the stabilization of the static chirality. The difference between B(E2) from yrast and those from non-yrast states is almost constant up to I = 13. Starting form this angular momentum value, all transition rates undergo a kind of second order phase transition to lower values [53]. The difference becomes first larger and then smaller for the intra-band transitions, while the inter-band ones become continuously closer intersecting each other between I = 15 and The transitional region I = 14 − 16 coincides with the angular momentum interval where the density probability of the vibrational states is most extended. Indeed, although from the semiclassical analysis, the critical angular momentum where chiral minima appear in the classical energy function is I = 11 (Table I), from quantum point of view the two chiral solutions become distinguishable only around I = 16 where the quantum tunneling subsides. Fig.10 shows that transition probabilities involving the state I = 14 act as critical points for the change from high to low B(E2) values. Coming back to the density probability distribution depicted in Fig.8, one can see that the density probability for I = 14 in ground state covers both chiral minima with undistinguishable peaks, while for the excited state, the height of the two vibrational peaks is minimal. In the first case there exists a coexistence between the two chiral solutions. As a matter of fact the broadening of the probability density distribution attributed to coexistence phenomena have immediate repercussions on the electromagnetic properties [54][55][56]. □ □ □ □ □ □ □ □ ○ ○ ○ ○ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ • • • • • • • • • • • (a) □ Y Exp. ○ NY Exp. ■ Y b c d • NY e f g h i p q r s t u v x y 0.00 d e f g h i j k l m o I B(E2,Ip→I-2p) [e 2 b 2 ] □ □ □ □ □ ○ ○ ○ ○ ○ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ • • • • • • • • • • • (b) □ Y → NY Exp. ○ NY → Y Exp. ■ Y → NY p q r • NY → Y The good agreement with experimental energy levels, and electromagnetic transitions at least for a small interval of angular momentum states, indicates that chiral geometry is a viable hypothesis in what concerns the interpretation of the doublet bands observed in increasingly more nuclei. There are however alternative interpretations of the fingerprints usually attributed to nuclear chirality [2]. For example the interacting boson-fermionfermion model analysis made on partner bands of 134 Pr point to the domination of shape fluctuations over the chiral geometry [50,51]. Among the alternative mechanisms of the doublet bands generation in 134 Pr, one must mention also the shape coexisting scenario [49] where the two bands are considered to have different quadrupole moments. Therefore the chiral symmetry breaking cannot be considered the unique or the sole mechanism responsible for the experimentally observed doublet bands. VII. CONCLUSIONS Through a time dependent variational principle one associated a classical energy function to a system of three mutually perpendicular spins corresponding to a triaxial core and two single particle configurations of valence nucleons. A coherent state for the angular momentum operators is used as a variational state, whose stereographic parametrization, gives the dependence of the classical energy function on azimuthal angle ϕ and a canonical conjugate coordinate r related to the polar angle θ. Maintaining rigid the trihedral configuration of the three spins, it is found that the classical energy function goes from a single minimum to a double minima surface in the space of canonical variables (ϕ, r) as the total angular momentum is increased. The analytical expression for these critical points identifies the solution with a single minimum as planar, while the double minimum solution is associated to an aplanar case. The two degenerated minima in the later case describe distinct chiral configurations of the three spin vectors involved in the dynamics of the total system. The single minimum and double minima conditions define two distinct rotational phases which are delimited by a separatrix represented by a critical angular momentum value. The dependence of the critical spin on traiaxiality γ for different single-particle configurations revealed that its minimum lies at maximum triaxiality γ = 90 • . By studying the evolution with total angular momentum of the spherical angles associated to energy minima, a distinct dynamical behaviour was observed for γ = 90 • . Speculating the symmetry of the classical energy function for this particular case, one quantized the energy function by replacing some redefined canonical conjugate coordinates with their corresponding differential operators after performing a harmonic approximation against one of the original coordinates. The resulting differential operator is written in terms of a new variable which is just the total angular momentum projection on the quantization axis. It was shown that the differential equation can be brought to a Schrödinger form containing a kinetic operator with a variable-dependent mass term and an effective symmetrical potential which can have a single or double degenerated minima, depending on the total angular momentum. The energy states of the chiral partner bands for a given angular momentum are obtained through diagonalization of the quantum Hamiltonian in a trigonometric basis with symmetric and antisymmetric basis states. The solutions are then used to calculate B(E2) transition probabilities with a redefined total wave-function having an incorporated coupling between the rotational motion and chiral vibration. The model was applied to the description of the chiral bands of 134 Pr. The agreement with experiment is very good in what concerns the energy levels considering that the triaxiality is a priori fixed to γ = 90 • . Although the single-particle degrees of freedom are ignored because one considered rigid alignments of the single-particle spins, the agreement between theoretical calculations for the transition probabilities and experimental data is quite satisfactory. Especially good closeness to data is obtained for the transitional interval of angular momenta defining the change from chiral vibration to static chirality. Although the considered system is drastically restrained, it provides a good reference picture for how the chiral symmetry breaking occurs and how it affects the system's rotation. The rotational aspect is mainly given by the classical analysis which sorts the relevant degrees of freedom further used to quantize the fluctuations around or between stable rotational configurations. Therefore, the proposed semiclassical approach is able to describe consistently the complex dynamics of a nucleus undergoing a transition from chiral vibration to static chirality. FIG. 1 . 1Schematic representation of the chiral geometry. FIG. 2 . 2Classical energy surfaces as a function of the generalized coordinate ϕ and momentum r for γ = 80 • and γ = 90 • and selected values of the total angular momentum. The single or double minima are indicated with crosses, while the difference between two consecutive contours is 10 arbitrary units. The increase goes from dark to light. functions (3.3), (3.4) and (3.6) have under unity values. FIG. 3 . 3The evolution as a function of triaxiality γ of the separatrix represented by the critical angular momentum Ic for few quasiparticle configurations expected to break the chiral symmetry. FIG. 4 .FIG. 5 . 45Evolution of the spherical angles as a function of angular momentum for different degrees of triaxiality: (a) γ = 80 • and (b) γ = 90 • . The correspondence between the triaxiality measure γ and the final value of the azimuthal angle ϕa, which remains invariant with total angular momentum. F p (FIG. 6 . 6p n (x), p = −1, Chiral potential (a) and mass (b) as function of the chiral variable x for few values of total angular momentum. FIG. 7 . 7The lowest two eigenvalues of the chiral potential for I = 8 − 16 are visualized relative to the potential profile. The lowest energy state corresponds to the symmetric wave function (p = 1). S IKp e iϕ0(I+K) |IM K , (5.4) FIG. 8. Density probability distribution as function of total angular momentum and the chiral variable x = K for ground (a) and first excited (b) states corresponding to p = 1 and respectively p = −1. Consecutive contours denote a variation of probability of 0.01 arbitrary units. The increase goes from dark to light. where S IKp = 1 (2I) I (2I)! (I − K)!(I + K)! (5.5) × I −I ρ I p (x)(I + x) FIG. 10 . 10Theoretical values of B(E2) are compared to the experimental data measured for 134 Pr for intra-band (a) and inter-band (b) transitions involving yrast (Y) and non-yrast (NY) states. I = 16. TABLE I . IMinimal classical value of the critical angular momentum where the classical energy function starts to allow stable chiral solutions, for few one particle one hole quasiparticle configurations known to generate chiral doublet bands.Mass region Configuration j j ′ I min From ACKNOWLEDGMENTSThe author is grateful to Dr. Q. B. Chen for inspiring discussions. This work was supported by a grant of Ministry of Research and Innovation, CNCS -UE-FISCDI, project number PN-III-P1-1.1-TE-2016-0268, within PNCDI III. . S Frauendorf, J Meng, Nucl. Phys. A. 617131S. Frauendorf and J. Meng, Nucl. Phys. A 617, 131 (1997). . J Meng, S Q Zhang, J. Phys. G: Nucl. Part. Phys. 3764025J. Meng and S. Q. Zhang, J. Phys. G: Nucl. Part. Phys. 37, 064025 (2010). . K Starosta, T Koike, C J Chiara, D B Fossan, D R Lafosse, A A Hecht, C W Beausang, M A Caprio, J R Cooper, R Krücken, J R Novak, N V Zamfir, K E Zyromski, D J Hartley, D L Balabanski, J.-Y Zhang, S Frauendorf, V I Dimitrov, Phys. Rev. Lett. 86971K. Starosta, T. Koike, C. J. Chiara, D. B. Fossan, D. R. LaFosse, A. A. Hecht, C. W. Beausang, M. A. Caprio, J. R. Cooper, R. Krücken, J. R. Novak, N. V. Zamfir, K. E. Zyromski, D. J. Hartley, D. L. Balabanski, J.-Y. Zhang, S. Frauendorf, and V. I. Dimitrov, Phys. Rev. Lett. 86, 971 (2001). . R A Bark, E O Lieder, R M Lieder, E A Lawrie, J J Lawrie, S P Bvumbi, N Y Kheswa, S S Ntshangase, T E Madiba, P L Masiteng, S M Mullins, S Murray, P Papka, O Shirinda, Q B Chen, S Q Zhang, Z H Zhang, P W Zhao, C Xu, J Meng, D G Roux, Z. P. R. A. Bark, E. O. Lieder, R. M. Lieder, E. A. Lawrie, J. J. Lawrie, S. P. Bvumbi, N. Y. Kheswa, S. S. Ntshangase, T. E. Madiba, P. L. Masiteng, S. M. Mullins, S. Murray, P. Papka, O. Shirinda, Q. B. Chen, S. Q. Zhang, Z. H. Zhang, P. W. Zhao, C. Xu, J. Meng, D. G. Roux, Z. P. . J Li, B Peng, S Y Qi, Z G Wang, Xiao, Int. J. Mod. Phys. E. 231461001Li, J. Peng, B. Qi, S. Y. Wang and Z. G. Xiao, Int. J. Mod. Phys. E 23, 1461001 (2014). . K Starosta, T Koike, Phys. Scr. 9293002K. Starosta and T. Koike, Phys. Scr. 92, 093002 (2017). . A Bohr, B R Mottelson, Nuclear Structure. 2A. Bohr and B. R. Mottelson, Nuclear Structure, Vol. 2 (Benjamin, Reading, Massachusetts, 1975). . S Frauendorf, Nucl. Phys. A. 557259S. Frauendorf, Nucl. Phys. A 557, 259c (1993). . S Brant, D Vretenar, A Ventura, Phys. Rev. C. 6917304S. Brant, D. Vretenar, and A. Ventura, Phys. Rev. C 69, 017304 (2004). . H G Ganev, A I Georgieva, S Brant, A Ventura, Phys. Rev. C. 7944322H. G. Ganev, A. I. Georgieva, S. Brant, and A. Ventura, Phys. Rev. C 79, 044322 (2009). . A A Raduta, C M Raduta, A Faessler, J. Phys. G: Nucl. Part. Phys. 4135105A. A. Raduta, C. M. Raduta, and A. Faessler, J. Phys. G: Nucl. Part. Phys. 41, 035105 (2014). . A A Raduta, Al H Raduta, C M Petrache, J. Phys. G: Nucl. Part. Phys. 4395107A. A. Raduta, Al. H. Raduta, and C. M. Petrache, J. Phys. G: Nucl. Part. Phys. 43, 095107 (2016). . K Higashiyama, N Yoshinaga, K Tanabe, Phys. Rev. C. 7224315K. Higashiyama, N. Yoshinaga, and K. Tanabe, Phys. Rev. C 72, 024315 (2005). . G H Bhat, J A Sheikh, R Palit, Phys. Lett. B. 707250G. H. Bhat, J. A. Sheikh, and R. Palit, Phys. Lett. B 707, 250 (2012). . J Peng, J Meng, S Q Zhang, Phys. Rev. C. 6844324J. Peng, J. Meng, and S. Q. Zhang, Phys. Rev. C 68, 044324 (2003). . T Koike, K Starosta, I Hamamoto, Phys. Rev. Lett. 93172502T. Koike, K. Starosta, and I. Hamamoto, Phys. Rev. Lett. 93, 172502 (2004). . S Q Zhang, B Qi, S Y Wang, J Meng, Phys. Rev. C. 7544307S. Q. Zhang, B. Qi, S. Y. Wang, and J. Meng, Phys. Rev. C 75, 044307 (2007). . K Higashiyama, N Yoshinaga, Eur. Phys. J. A. 33355K. Higashiyama and N. Yoshinaga, Eur. Phys. J. A 33, 355 (2007). . S Y Wang, S Q Zhang, B Qi, J Meng, Phys. Rev. C. 7524309S. Y. Wang, S. Q. Zhang, B. Qi and J. Meng, Phys. Rev. C 75, 024309 (2007). . S Y Wang, S Q Zhang, B Qi, J Peng, J M Yao, J Meng, Phys. Rev. C. 7734314S. Y. Wang, S. Q. Zhang, B. Qi, J. Peng, J. M. Yao, and J. Meng, Phys. Rev. C 77, 034314 (2008). . Q B Chen, J M Yao, S Q Zhang, B Qi, Phys. Rev. C. 8267302Q. B. Chen, J. M. Yao, S. Q. Zhang, and B. Qi, Phys. Rev. C 82, 067302 (2010). . E A Lawrie, O Shirinda, Phys. Lett. B. 68966E. A. Lawrie and O. Shirinda, Phys. Lett. B 689, 66 (2010). . S G Rohozinski, L Prochniak, K Starosta, C Droste, Eur. Phys. J. A. 4790S. G. Rohozinski, L. Prochniak, K. Starosta, and C. Droste, Eur. Phys. J. A 47, 90 (2011). . H Zhang, Q B Chen, Chin. Phys. C. 4024101H. Zhang and Q. B. Chen, Chin. Phys. C 40, 024101 (2016). . V I Dimitrov, S Frauendorf, F Dönau, Phys. Rev. Lett. 845732V. I. Dimitrov, S. Frauendorf and F. Dönau, Phys. Rev. Lett. 84, 5732 (2000). . V I Dimitrov, F Dönau, S Frauendorf, Phys. Rev. C. 6224315V. I. Dimitrov, F. Dönau and S. Frauendorf, Phys. Rev. C 62, 024315 (2000). . A A Hecht, C W Beausang, K E Zyromski, D L Balabanski, C J Barton, M A Caprio, R F Casten, J R Cooper, D J Hartley, R Krucken, D Meyer, H Newman, J R Novak, E S Paul, N Pietralla, A Wolf, N V Zamfir, J.-Y Zhang, F Dönau, Phys. Rev. C. 6351302A. A. Hecht, C. W. Beausang, K. E. Zyromski, D. L. Balabanski, C. J. Barton, M. A. Caprio, R. F. Casten, J. R. Cooper, D. J. Hartley, R. Krucken, D. Meyer, H. Newman, J. R. Novak, E. S. Paul, N. Pietralla, A. Wolf, N. V. Zamfir, J.-Y. Zhang, and F. Dönau, Phys. Rev. C 63, 051302(R) (2001). . S Zhu, U Garg, B K Nayak, S S Ghugre, N S Pattabiraman, D B Fossan, T Koike, K Starosta, C Vaman, R V F Janssens, R S Chakrawarthy, M Whitehead, A O Macchiavelli, S Frauendorf, Phys. Rev. Lett. 91132501S. Zhu, U. Garg, B. K. Nayak, S. S. Ghugre, N. S. Pat- tabiraman, D. B. Fossan, T. Koike, K. Starosta, C. Va- man, R. V. F. Janssens, R. S. Chakrawarthy, M. White- head, A. O. Macchiavelli, and S. Frauendorf, Phys. Rev. Lett. 91, 132501 (2003). . P Olbratowski, J Dobaczewski, J Dudek, W Plóciennik, Phys. Rev. Lett. 9352501P. Olbratowski, J. Dobaczewski, J. Dudek, and W. Plóciennik, Phys. Rev. Lett. 93, 052501 (2004). . P Olbratowski, J Dobaczewski, J Dudek, Phys. Rev. C. 7354308P. Olbratowski, J. Dobaczewski, and J. Dudek, Phys. Rev. C 73, 054308 (2006). . P W Zhao, Phys. Lett. B. 7731P. W. Zhao, Phys. Lett. B 773, 1 (2017). . B Qi, S Q Zhang, J Meng, S Y Wang, S Frauendorf, Phys. Lett. B. 675175B. Qi, S. Q. Zhang, J. Meng, S. Y. Wang, and S. Frauen- dorf, Phys. Lett. B 675, 175 (2009). . B Qi, S Q Zhang, S Y Wang, J Meng, T Koike, Phys. Rev. C. 8334303B. Qi, S. Q. Zhang, S. Y. Wang, J. Meng, and T. Koike, Phys. Rev. C 83, 034303 (2011). . O Shirinda, E A Lawrie, Eur. Phys. J. A. 52344O. Shirinda and E. A. Lawrie, Eur. Phys. J. A 52, 344 (2016). . S Mukhopadhyay, D Almehed, U Garg, S Frauendorf, T Li, P V M Rao, X Wang, S S Ghugre, M P Carpenter, S Gros, Phys. Rev. Lett. 99172501S. Mukhopadhyay, D. Almehed, U. Garg, S. Frauendorf, T. Li, P. V. M. Rao, X. Wang, S. S. Ghugre, M. P. Carpenter, S. Gros, et al., Phys. Rev. Lett. 99, 172501 (2007). . D Almehed, F Dönau, S Frauendorf, Phys. Rev. C. 8354308D. Almehed, F. Dönau, and S. Frauendorf, Phys. Rev. C 83, 054308 (2011). . Q B Chen, S Q Zhang, P W Zhao, R V Jolos, J Meng, Phys. Rev. C. 8724314Q. B. Chen, S. Q. Zhang, P. W. Zhao, R. V. Jolos, and J. Meng, Phys. Rev. C 87, 024314 (2013). . Q B Chen, S Q Zhang, P W Zhao, R V Jolos, J Meng, Phys. Rev. C. 9444301Q. B. Chen, S. Q. Zhang, P. W. Zhao, R. V. Jolos, and J. Meng, Phys. Rev. C 94, 044301 (2016). . A A Raduta, R Poenaru, L Gr, Ixaru, Phys. Rev. C. 9654320A. A. Raduta, R. Poenaru, and L. Gr. Ixaru, Phys. Rev. C 96, 054320 (2017). . R Budaca, Phys. Rev. C. 9724302R. Budaca, Phys. Rev. C 97, 024302 (2018). . W.-X Shi, Q B Chen, Chinese Phys. C. 3954105W.-X. Shi and Q. B. Chen, Chinese Phys. C 39 054105 (2015). . J M Radcliffe, J. Phys. A: Gen. Phys. 4313J. M. Radcliffe, J. Phys. A: Gen. Phys. 4, 313 (1971). . S Takeno, S Homma, Prog. Theor. Phys. 641193S. Takeno and S. Homma, Prog. Theor. Phys. 64 1193 (1980). . A A Raduta, R Budaca, C M Raduta, Phys. Rev. C. 7664309A. A. Raduta, R. Budaca and C. M. Raduta, Phys. Rev. C 76, 064309 (2007). . S Iida, M Yamamura, Prog. Theor. Phys. 70783S. Iida and M. Yamamura, Prog. Theor. Phys. 70 783 (1983). . S S Malik, Nucl. Phys. A. 940279S. S. Malik, Nucl. Phys. A 940, 279 (2015). . H Taseli, Int. J. Quantum Chem. 46319H. Taseli, Int. J. Quantum Chem. 46, 319 (1993). . A Chandra, K Bhattacharyya, Pramana J. Phys. 43117A. Chandra and K. Bhattacharyya, Pramana J. Phys. 43, 117 (1994). . K S Pitzer, J. Chem. Phys. 7251K. S. Pitzer, J. Chem. Phys. 7, 251 (1939). . C M Petrache, G B Hagemann, I Hamamoto, K Starosta, Phys. Rev. Lett. 96112502C. M. Petrache, G. B. Hagemann, I. Hamamoto, and K. Starosta, Phys. Rev. Lett. 96, 112502 (2006). . D Tonev, G De Angelis, P Petkov, A Dewald, S Brant, S Frauendorf, D L Balabanski, P Pejovic, D Bazzacco, P Bednarczyk, F Camera, A Fitzler, A Gadea, S Lenzi, S Lunardi, N Marginean, O Möller, D R Napoli, A Paleni, C M Petrache, G Prete, K O Zell, Y H Zhang, Jing-Ye Zhang, Q Zhong, D Curien, Phys. Rev. Lett. 9652501D. Tonev, G. de Angelis, P. Petkov, A. Dewald, S. Brant, S. Frauendorf, D. L. Balabanski, P. Pejovic, D. Bazza- cco, P. Bednarczyk, F. Camera, A. Fitzler, A. Gadea, S. Lenzi, S. Lunardi, N. Marginean, O. Möller, D. R. Napoli, A. Paleni, C. M. Petrache, G. Prete, K. O. Zell, Y. H. Zhang, Jing-ye Zhang, Q. Zhong, and D. Curien, Phys. Rev. Lett. 96, 052501 (2006). . D Tonev, G De Angelis, S Brant, S Frauendorf, P Petkov, A Dewald, F Dönau, D L Balabanski, Q Zhong, P Pejovic, D Bazzacco, P Bednarczyk, F Camera, D Curien, F Della Vedova, A Fitzler, A Gadea, G Lo, S Bianco, S Lenzi, N Lunardi, O Marginean, D R Möller, R Napoli, E Orlandi, A Sahin, J Saltarelli, K O Valiente Dobon, J.-Y Zell, Y H Zhang, Zhang, Phys. Rev. 7644313D. Tonev, G. de Angelis, S. Brant, S. Frauendorf, P. Petkov, A. Dewald, F. Dönau, D. L. Balabanski, Q. Zhong, P. Pejovic, D. Bazzacco, P. Bednarczyk, F. Cam- era, D. Curien, F. Della Vedova, A. Fitzler, A. Gadea, G. Lo Bianco, S. Lenzi, S. Lunardi, N. Marginean, O. Möller, D. R. Napoli, R. Orlandi, E. Sahin, A. Saltarelli, J. Valiente Dobon, K. O. Zell, J.-Y. Zhang, and Y. H. Zhang, Phys. Rev. 76, 044313 (2007). . B Qi, S Q Zhang, S Y Wang, J M Yao, J Meng, Phys. Rev. C. 7941302B. Qi, S. Q. Zhang, S. Y.Wang, J. M. Yao, and J. Meng, Phys. Rev. C 79, 041302(R) (2009). . L Landau, D. Ter Haar193216Pergaman, OxfordCollected Papers of L. D. LandauL. Landau, Collected Papers of L. D. Landau, pp. 193216 (ed. D. Ter Haar, Pergaman, Oxford, 1965). . P Buganu, R Budaca, Phys. Rev. C. 9114306P. Buganu and R. Budaca, Phys. Rev. C 91, 014306 (2015). . R Budaca, P Buganu, A I Budaca, Phys. Lett. B. 77626R. Budaca, P. Buganu, and A. I. Budaca, Phys. Lett. B 776, 26 (2018). . R Budaca, P Buganu, A I Budaca, Bulg. J. Phys. 44319R. Budaca, P. Buganu, and A. I. Budaca, Bulg. J. Phys. 44, 319 (2017).
{'fraction_non_alphanumeric': 0.07232890618391, 'fraction_numerical': 0.039928940021994756, 'mean_word_length': 3.6494591937069814, 'pattern_counts': {'":': 0, '<': 6, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 47, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'A triaxial particle-rotor Hamiltonian for three mutually perpendicular angular momentum vectors corresponding to two high-j quasiparticles and the rotation of a triaxial collective core, is treated within a time-dependent variational principle. The resulting classical energy function is used to investigate the rotational dynamics of the system. It is found that the classical energy function exhibits two minima starting from a critical angular momentum value which depends on the singleparticle configuration and the asymmetry measure γ. The emergence of the two minima is attributed to the breaking of the chiral symmetry. Quantizing the energy function for a given angular momentum, one obtains a Schrödinger equation with a coordinate dependent mass term for a symmetrical potential which changes from a single to a double well shape as the angular momentum pass the critical value. The energies of the chiral partner bands for a given angular momentum are then given by the lowest two eigenvalues. The procedure is exemplified for maximal triaxiality and two h 11/2 quasiparticles, with the results used for the description of the chiral doublet bands in 134 Pr.', 'arxivid': '1806.08846', 'author': ['R Budaca \nHoria Hulubei" National Institute for Physics and Nuclear Engineering\nStr. Reactorului 30, POB-MG6RO-077125Bucharest-MǎgureleRomania\n\nAcademy of Romanian Scientists\n54 Splaiul IndependenţeiRO-050094BucharestRomania\n'], 'authoraffiliation': ['Horia Hulubei" National Institute for Physics and Nuclear Engineering\nStr. Reactorului 30, POB-MG6RO-077125Bucharest-MǎgureleRomania', 'Academy of Romanian Scientists\n54 Splaiul IndependenţeiRO-050094BucharestRomania'], 'corpusid': 119105862, 'doi': '10.1103/physrevc.98.014303', 'github_urls': [], 'n_tokens_mistral': 15881, 'n_tokens_neox': 13613, 'n_words': 8402, 'pdfsha': '9e21e22695077518ffa8dce2a2c3b93a81cd5692', 'pdfurls': ['https://arxiv.org/pdf/1806.08846v1.pdf'], 'title': ['Semiclassical description of chiral geometry in triaxial nuclei', 'Semiclassical description of chiral geometry in triaxial nuclei'], 'venue': []}
arxiv
On the effect of blockage objects in dense MIMO SWIPT networks January 14, 2018 Ayse Ipek Akin Institute of Information and Communication Technologies Electronics and Applied Mathematics (ICTEAM) Université catholique de Louvain Louvain la NeuveBelgium Ivan Stupia Institute of Information and Communication Technologies Electronics and Applied Mathematics (ICTEAM) Université catholique de Louvain Louvain la NeuveBelgium Fellow, IEEELuc Vandendorpe Institute of Information and Communication Technologies Electronics and Applied Mathematics (ICTEAM) Université catholique de Louvain Louvain la NeuveBelgium On the effect of blockage objects in dense MIMO SWIPT networks January 14, 20181 Simultaneous information and power transfer (SWIPT) is characterised by the ambiguous role of multi-user interference. In short, the beneficial effect of multi-user interference on RF energy harvesting is obtained at the price of a reduced link capacity, thus originating nontrivial trade-offs between the achievable information rate and the harvestable energy. Arguably, in indoor environments, this tradeoff might be affected by the propagation loss due to blockage objects like walls. Hence, a couple of fundamental questions arise. How much must the network elements be densified to counteract the blockage attenuation? Is blockage always detrimental on the achievable rate-energy trade-off? In this paper, we analyse the performance of an indoor multiple-input multiple-output (MIMO) SWIPT-enabled network in the attempt to shed a light of those questions. The effects of the obstacles are examined with the help of a stochastic approach in which energy transmitters (also referred to as power heads) are located by using a Poisson Point Process and walls are generated through a Manhattan Poisson Line Process. The stochastic behaviour of the signal attenuation and the multi-user interference is studied to obtain the Joint Complementary Cumulative Distribution Function (J-CCDF) of information rate and harvested power. Theoretical results are validated through Monte Carlo simulations. Eventually, the rateenergy trade-off is presented as a function of the frequency of walls to emphasise the cross-dependences between the deployment of the network elements and the topology of the venue. A. I. Akin, I. Stupia and L. Vandendorpe are with the Index Terms Simultaneous wireless information and power transfer (SWIPT), stochastic geometry, indoor environment, Manhattan Poisson Line Process, blockage modeling I. INTRODUCTION A. Background and motivations Simultaneous information and power transfer (SWIPT) is an emerging concept to increase the battery life of low power wireless devices. The main idea behind SWIPT is to use the same signal to transfer information and power simultaneously. The achievable trade-off between harvested energy and information rate in a single SWIPT link was originally presented by Varshney in [1]. Few years later, the authors of [2] showed the benefits of multi-input multi-output (MIMO) techniques on the performance of the SWIPT link with practical receiver architectures. At the network level, particular interest in designing SWIPT systems is the ambivalent role of the multi-user interference (MUI). Traditionally, this is considered as an undesired factor for wireless information transfer (WIT) due to the negative effect on the coverage probability and the information rate [3], [4]. On the contrary, for wireless power transfer (WPT), the interference can be used as an additional source of energy to be harvested [5], [6]. It is doubtless that interference may have a major impact in determining the achievable rate-energy trade-off in SWIPT-enabled networks. MUI is even more crucial when the constraints on the minimum received power needed by the current RF harvesting technologies (tens of microwatts) are considered. These constraints, together with the limitations on RF emissions for guaranteeing the public safety, suggest that the deployment of SWIPT power heads (PHs) must be denser than that of traditional access points in wireless networks. Another straightforward consequence of the network elements' densification is that the performance of a SWIPT network is also highly dependent on the spatial locations of the network elements, thus making the Wyner model and the regular hexagonal or square grid models (see [7], [8]) inadequate to describe SWIPT systems whose PHs are likely to be randomly located. Recently, stochastic geometry has emerged as a random and spatial approach for modelling such kinds of dense networks, [9]. A first study that analyses the performance of SWIPT-enabled systems with a stochastic approach is reported in [10]. In this paper, the authors have proposed a tractable mathematical approach for the system-level analysis and optimization of SWIPT-enabled outdoor cellular networks. In [11], a mathematical framework is presented for MIMO SWIPT-enabled outdoor cellular networks. Though the methodology that is presented in both papers is extremely valuable in providing a tool for analysing the performance of stochastic SWIPT-enabled networks, their studies consider an outdoor cellular network that is too sparse to satisfy the constraints on the minimum received power that would enable RF energy harvesting. Moreover, whereas the distance is the determinant factor for outdoor wireless networks, propagation in indoor environments is also affected by the blockage due to the walls. Therefore, distance-dependent functions are not sufficient to model the propagation-losses for SWIPT indoor systems. In this perspective, a new stochastic geometry analysis for in-building systems, modelling the walls' distribution as a Manhattan Poisson Line Process (MPLP), [12], was presented in [13]. However, the assumption behind this work is that the blockage-based penetration loss is the dominant factor and free-space propagation loss can be neglected. This approximation conflicts with the typical operating conditions of SWIPT systems, in which the minimum received energy must be in the order of few microwatts to enable RF harvesting, so that a small difference on the distance can have a huge impact on the achievable rate-energy trade-off. More realistically, in [14], the authors jointly considered distance-dependent path loss, wall blockage and small-scale fading in information-only wireless networks by examining the average number of blockages for several wall generation methods. In this study, they considered different fixed transmitter configurations that provide the best scenario with respect to the interference. To the best of our knowledge, for SWIPT-enabled indoor networks, there has not been any study showing the effect of the obstacles combined with distance-dependent path-loss. B. Novelty and contributions All these concerns motivate us to propose an accurate stochastic geometry analysis for dense in-building MIMO power splitted SWIPT networks. MPLP is considered for the distribution of the walls and PHs are located by using Poisson Point Process (PPP). As a channel model, distance dependent path-loss, blockage-based path loss and fast fading are considered. In order to study the trade-off between information rate and harvested power, an analytical expression of the Joint Complementary Cumulative Distribution Function (J-CCDF) is derived by jointly considering the effects of obstacles, distance and multiple antennas. Eventually, our mathematical framework is validated through Monte Carlo simulations. The main contributions of this paper can therefore be summarised as follows: • In contrast to prior works relevant to SWIPT, [10], [11], we consider the effect of randomly located blockage objects like walls. This is of particular importance for any scenario accounting for the realistic levels of power enabling RF harvesting. • Differently from [13], we consider the joint effect of the distance dependent propagation loss and the blockages. Moreover, the locations of both PHs and walls are modelled through a stochastic process. This allows a macro-level investigation of the interactions between the network and the venue in which it is deployed. • An analytical expression of the cumulative distribution function of the multi-user interference as a function of the frequency of walls is provided. A similar result is also obtained for the distribution of the minimum propagation loss. • The proposed numerical results provide important insights on the interplay between the network and the in-building environment from the perspective of the energy-rate trade-off. We also clarify the role of multiple antennas and SWIPT receiver architecture when an ultra-dense deployment of the network elements is considered. The rest of this paper is organized as follows. Section II describes the system and path loss DRAFT January 14, 2018 model. We focus on performance analyses in Section III. In Section IV, numerical results are presented and finally the paper is concluded in Section V. II. SYSTEM MODEL In this section, we first introduce the major elements constituting a MIMO SWIPT network. Then, some blanket assumptions on the spatial distribution of both the network elements and the blockage objects are made clear. For illustrative purposes, an instance of indoor SWIPT network over a finite area is shown in Fig.1, where the lines represent walls and the markers symbolize PHs. It is worth recalling that this paper targets a stochastic characterisation of the SWIPT performance for a generic low power device (LPD) equipped with a power splitting receiver. Without loss of generality, in the remainder of the paper it is assumed that the LPD is conveniently located at the origin of the x− and y− axes. A. SWIPT Enabled Indoor Model We consider a MIMO SWIPT network deployed in a two-dimensional finite indoor area. The SWIPT waveforms are generated through a set a randomly deployed PHs equipped with n t antennas. The LPD embedding a SWIPT receiver with n r antennas is located at the centre of the investigated region. It is assumed that the data delivered to the LPD comes from the PH providing the minimum average signal attenuation. In order to increase the performance of the information transfer process, the received signals captured from the different LPD antennas are processed using maximum ratio combining (MRC), while maximum ratio transmission (MRT) is implemented at the transmitter side. The SWIPT receiver operates according to the power B. Signal Propagation Similarly to what has been done in [14], we assume that the signals are subject to distance dependent path loss, wall blockages and small scale fading. The path loss and the wall blockages are jointly modelled through the following log-distance dependent law: l N (r) = κr β K N (1) where r is the distance between the PH and the device, β is the path-loss exponent, K ∈ [0, 1) is the so called penetration loss, and N is a random variable representing the number of walls between a generic PH and the device. The path loss constant is defined as κ = ( 4π v ) 2 , where v = c 0 /f c is the transmission wavelength, f c and c 0 being the carrier frequency (Hz) and the DRAFT speed of light (m/sec), respectively. The small scale fading is modelled through random variables following a Rayleigh distribution to account for the multi-path propagation effect. C. Spatial distribution of PHs For analysing an ultra-dense MIMO SWIPT network in an indoor environment, we capitalise on a stochastic geometry approach. To achieve this, the possible sets of PHs deployed in a given area are modelled as instances of an homogeneous PPP Ψ of density λ P H . Thanks to this stochastic approach, we will obtain a statistical characterisation of the performance metrics for a typical LPD. As already mentioned, we assume that the information transfer towards the typical LPD is guaranteed by the PH ensuring the smallest average signal attenuation, also referred to as serving PH. From the information transfer perspective the other PHs are considered as interferers. The set of interfering PHs is denoted by Ψ (\0) . D. Random wall placement Differently from previous studies (e.g. [11]), this paper focuses on an indoor environment. It follows that our signal propagation model must comprise wall blockages. Again, we adopt a stochastic approach in which the position of wall is modelled as Manhattan Poisson Line Processes (MPLP). Generally speaking, MPLP is used to generate random lines in an Euclidean space R n . For the 2-D case, this is achieved by defining two homogeneous PPPs Ψ x and Ψ y , of identical frequency λ w , over the x-axis and the y-axis, respectively. Each set of points is obtained as a single realisation of Ψ x and Ψ y and it represents the midpoints of infinite length walls. Hence, the walls grow parallel to the x-axis and the y-axis at every point of these processes and each wall divides the plane into infinite rectangular boxes. Each of these boxes is considered as a room of our indoor environment. The room that contains the origin, called typical room, is identified by the couple (0, 0), while the other rooms are labelled according to their position with respect to the typical room, e.g. the signal associated with a PH located in the room (i, j) Fig. 2: SWIPT indoor network shall cross |i| walls on the x-axis and |j| walls on the y-axis to reach the typical LPD. In order to justify our assumptions, we mention here the work presented in [14], wherein MPLP modelling is identified as the most promising wall generation method to approximate a realistic indoor scenario while guaranteeing a mathematically tractability of the wall blockages analysis. III. PERFORMANCE ANALYSIS In this section we provide a methodology to study the effects of the blockage objects on the SWIPT performance. Then, we provide an expression for the J-CCDF of information rate and harvested power as a function of the network parameters. An illustration of a SWIPT network in an indoor scenario is shown in Fig. 2, where the serving PH (the red one in Fig. 2) is transmitting data and power towards the LPD with power gain g (0) (including MRT, MRC) and the signals, coming from all the PHs, experience Rayleigh fading. A. Wall Blockage Analysis The main analytical contribution of this paper is the study of the effect of blockage objects on the SWIPT performance. To achieve this, we first decompose the original PPP, Ψ, with density λ P H into the sum of equivalent inhomogeneous PPPs, Ψ N , with densities given by λ N (r, θ) = λ P H P N (r, θ)(2) where P N (r, θ) = (λ w r| cos(θ)| + λ w r| sin(θ)|) N N! exp {− (λ w r| cos(θ)| + λ w r| sin(θ)|)}(3) is the probability that a PH located at the point defined by the polar coordinates (r, θ) experiences the blockage effect of N obstacles . Since the choice of the serving PH is associated with the geometry of both the PHs' spatial distribution and the placement of the walls, the preliminary step enabling the analysis of the performance for the system depicted in Fig. 2 is the characterisation of the stochastic behaviour of the minimum path-loss, L (0) , and the multi-user interference, I M U . This will be the objective of the following subsections. 1) Minimum Path Loss: As it has been already mentioned, we assume that the serving PH is the one associated with the minimum path loss, i.e. L (0) = min N min n∈Ψ N l N (r (n) )(4) where n is the index of the PHs belonging to the inhomogeneous PPPs associated with N obstacles, Ψ N . Following the procedure proposed in [9], we capitalise on the well-known displacement theorem to end up with the following proposition. Proposition 1: Let the minimum path loss being the one defined in (4) and define a function χ η (λ w ) as χ η (λ w ) = λ η w 2 η 2 √ π Γ( η+1 2 ) Γ( η+2 2 ) − √ 2 2 F 1 1 2 , η+1 2 , η+3 2 , 1 2 η + 1(5) where Γ(· , · ) is the upper-incomplete Gamma function and 2 F 1 (· ) is the Gaussian hypergeometric function [15]. Then the cumulative distribution function (CDF) of L (0) is given by F L (0) (α) = Pr{L (0) ≤ α} = 1 − Nmax N =1 exp {−Λ N ([0, α))}(6) where N max is the maximum number of obstacles that can be encountered in a circular region of ray R D and Λ N ([0, α)) =          4λ P H N ! ∞ η=N (−1) η−N χη(λw) (η−N )!(η+2) αK N κ η+2 β if α < R β D κ K N 4λ P H N ! ∞ η=N (−1) η−N χη(λw) (η−N )!(η+2) R η+2 D if α ≥ R β D κ K N(7) is the intensity of the process L N = l N (r (n) ), n ∈ Ψ N . Proof: See Appendix A. From (7), it is apparent that the signal attenuation is a process whose intensity can be expressed as the weighted sum of power functions of α. Interestingly, the effects of the blockage due to the walls is summarised through the weights χ η (λ w ) that can be computed offline and tabulated for given values of η and λ w as shown in Table I. Since χ η (λ w ) is a decreasing function of η, from (6) and (7), we can infer that the greater the number of obstacles N, the lower the impact of the blockage objects on the CDF of L (0) . On the contrary, χ η (λ w ) is an increasing function of λ w and, not surprisingly, the impact of the blockage objects on F L (0) (α) increases with λ w . 2) Multi-User Interference: The normalized (with respect to 1W of transmit power) multi-user interference can be expressed as I M U = Nmax N =0 n∈Ψ N h (n) l N (r (n) ) ½ l N (r (n) ) > L (0)(8) where h (n) is an exponentially distributed random variable with unit variance representing the gain of the nth interfering link, and r (n) denotes the distance from a generic PH to the LPD. Proposition 2: Assume that the minimum path loss is given and equal to L (0) , consider the functions χ η (λ w ), η ∈ R + as in (5), and define the function ∆ η,N ω; L (0) = L (0) K N κ η+2 β 1 − 2 F 1 1, − η + 2 β , 1 − (η + 2) β , jω L (0) − R η+2 D 1 − 2 F 1 1, − η + 2 β , 1 − (η + 2) β , jωK N R β D κ .(9) Then, capitalizing on the Gil-Pelaez inversion theorem, [16], the CDF of I M U is given by F I M U (z; L (0) ) = Pr{I M U ≤ z|L (0) } = 1/2 − ∞ 0 1 πω Im e −jωz Nmax N =1 Φ N ω; L (0) dω (10) where Φ N ω; L (0) =          exp 4λ P H N ! ∞ η=N (−1) η−N χη(λw) (η−N )!(η+2) ∆ η,N ω; L (0) if L (0) < R β D κ K N 1 if L (0) ≥ R β D κ K N(11) is the characteristic function of the multi-user interference produced by the PHs whose signals are subject to the attenuation of N obstacles to reach the LPD. Proof: See Appendix B. B. SWIPT Performance Analysis The goal of this section is to provide an analytical expression for studying the stochastic behaviour of the SWIPT performance as a function of the PHs' density λ P H and the frequency of blockage objects λ w . Because of the multiobjective nature of the system, two different performance metrics are considered, namely the average throughput, expressed in bits/sec and denoted with R, and the average harvested power, measured in Watt and referred to as Q. To achieve this goal, the instantaneous metrics are first defined as R = B c log 2 1 + P g (0) /L (0) P I M U + σ 2 n + σ 2 c /(1 − ρ) Q = ρξP g (0) L (0) + I M U(12) where B c is the signal bandwidth, P is the average transmit power, ρ is the power splitting ratio and ξ is the energy harvesting efficiency factor. The variance of the thermal noise is denoted by σ 2 n , while σ 2 c indicates the variance of the noise due to the conversion of the received signal from radio frequency to baseband. In addition to the minimum path loss L (0) and the multi-user interference I M U , whose statistical properties have been studied in the previous subsections, the other source of randomness is the power gain g (0) , which encompasses the small scale fading experienced by the serving PH's signal and the effect of both the MRT and the MRC processing at the transmitter and the receiver side, respectively. The PDF of g (0) is given in [17] as, where m = min(n t , n r ) and n = max(n t , n r ). Here K m,n = m k=1 ((m − k)!(n − k)!) −1 is a normalising factor and a s,t are some coefficients that can be easily obtained by using [17, Algorithm 1]. In SWIPT enabled networks the performance of the system can be described in terms of achievable trade-offs between the information rate and the harvested power. This trade-off will be analysed taking advantage of the J-CCDF of R and Q, as originally proposed in [10]. The J-CCDF is defined as F c (R * , Q * ) = Pr{R ≥ R * , Q ≥ Q * }(14) where Q * ≥ 0 is the sensitivity of the energy harvester and R * ≥ 0 is the minimum achievable rate. A convenient reformulation of the J-CCDF has been proposed in [11], and it amounts to DRAFT January 14,2018 computing the probability that the multi-user interference belongs to an interval for which a minimum SINR γ can be achieved conditioned to a minimum amount of received power q * , i.e. F c (R * , Q * ) = E L (0) +∞ (T * /P )L (0) F I M U xγ/L (0) − σ 2 * /P   L (0) f g (0) (x)dx − E L (0) +∞ (T * /P )L (0) F I M U −x/L (0) + q * /P   L (0) f g (0) (x)dx(15) where σ 2 * = σ 2 n + σ 2 c (1 − ρ) −1 , q * = Q * /(ρξ), T * = (q * + σ 2 * )/(γ + 1) and γ = 1/ 2 R * /Bc − 1 . Proposition 3: Let define, Φ ω; L (0) = ∞ N =0 Φ N ω; L (0) , Λ([0, α)) = ∞ N =0 Λ N ([0, α)),(16) and Λ([0, α)) as the derivative of Λ([0, α)) with respect to α, whose expression is provided in (27) in the Appendix B . Given the statistical characterisation of L (0) , I M U , and g (0) expressed as in equations (6), (29), and (13), respectively, the J-CCDF F c (R * , Q * ) can be computed as [11], F c (R * , Q * ) = K m,n m s=1 (n+m−2s)s t=n−m a s,t J (1) s,t − J (2) s,t(17) where the functions J s,t are defined as, J (1) s,t = ∞ 0 ∞ 0 1 πw Im exp −jw q * P s − jw y −(1+t) Γ 1 + t, T * P (sy − jw) Φ(w; y) Λ([0, α)) exp {−Λ([0, α))} dwdy,(18) and J (2) s,t = ∞ 0 ∞ 0 1 πw Im exp jw σ 2 * P s + jwγ y −(1+t) Γ 1 + t, T * P (sy + jwγ) Φ(w; y) Λ([0, α)) exp {−Λ([0, α))} dwdy.(19) Proof: The proof trivially follows from [11, Proposition 1]. Using Proposition 3 the J-CCDF can be numerically computed for illustrating the average trade-off between the information rate and the harvested power without the need of time consuming Monte Carlo simulations. IV. NUMERICAL RESULTS In this section, we show the performance of an in-building SWIPT network by means of numerical results. The analytical findings presented in Section III will be validated through Monte Carlo simulations. A. Setup We considered a circular area of radius R D = 60m in which a set of PHs is transmitting with an average power of 1W, i.e. P = 30dBm. The SWIPT signals have a bandwidth of B c = 200kHz centred around f c = 2.1 GHz. The thermal noise has a variance σ 2 n = −174 + 10 log 10 (B c ) + F n , where F n = 10dB is the noise figure. The variance of the noise due to the RF to DC conversion is set to σ 2 c = −70dBm. The PHs are assumed to be deployed with a density λ P H = 1/(πd 2 P H ), where d P H is half of the average distance between PHs. The efficiency of the RF energy harvesting process is ξ = 0.8 and the path-loss exponent is β = 2.5. Unless otherwise stated, the power splitting factor is ρ = 0.5, the number of receive antennas is n r = 2 and the number of transmit antennas is n t = 4. For the chosen frequency, according to [18, (7) and (11) has to be cut at some point, the almost perfect match between theoretical results and simulations is apparent. Clearly, the harvested power always benefits from the increment of the PHs' density λ P H , while the maximum achievable information rate decreases because of the larger level of multi-user interference. We can therefore identify a first compromise to be achieved when the network harvested power when ρ is equal to 0.5 and 0.9, respectively. We can conclude that the power splitting ratio must be as large as possible and its fine tuning is of limited interest in the design of ultra-dense SWIPT networks. It is worth remarking that this conclusion is radically different from previous works (see [11]) in which the authors consider values of harvested power below −60dBm. In that case, multi-user interference has a limited role while the optimisation of the power splitting ratio is paramount. On the contrary, in this paper we show that the technological limitations of the harvesting process make the topology of the venue and its effect on the interference the most important parameters in defining the achievable rate-energy trade-off. λ w = 0.05. It is apparent that the additional attenuation of walls is beneficial for the achievable trade-off and rate. For example, an information rate of 300 Kbps can be achieved with −20dBm of harvested power in the presence of blockages, while this value is reduced to 240 kbps when those obstacles are removed. In Fig. 8 First, we note that the process collecting all the propagation losses L = l(r (n) ), n ∈ Ψ can be characterised as a transformation of the points of Ψ. Hence, invoking the displacement theorem, [9, Theorem 1.3.9], and recalling that Ψ can be expressed as the superposition of the PPPs Ψ N , the processes L N = l N (r (n) ), n ∈ Ψ N , N ∈ {0, N max }, are PPPs whose intensity is given by 3) MIMO Effect Λ N ([0, α)) = Pr κ(r (n) ) β K N ∈ [0, α), n ∈ Ψ N =λ P H 2π 0 ∞ 0 H α − κr β K N P N (r, θ) rdrdθ.(20) Now, by substituting (3) in (20) and integrating it over r, we get Λ N ([0, α)) = λ P H N! 2π 0 (λ w | cos(θ)| + λ w | sin(θ)|) −2 × Γ(2 + N) − Γ 2 + N, αK N κ 1/β (λ w | cos(θ)| + λ w | sin(θ)|) H α − R β D κ K N + Γ(2 + N) − Γ (2 + N, R D (λ w | cos(θ)| + λ w | sin(θ)|)) H α − R β D κ K N dθ.(21) In order to have an analytical expression for the integration over the variable θ, we can express the the upper-incomplete Gamma functions of θ through their Taylor expansions, i.e., Γ 2 + N, αK N κ 1/β (λ w | cos(θ)| + λ w | sin(θ)|) = Γ(2 + N) − ∞ i=0 (−1) i αK N κ (N +2+i)/β i!(i + N + 2) (λ w | cos(θ)| + λ w | sin(θ)|) (N +2+i) ,(22) and, substituting (22) and (23) into (21), we get Λ N ([0, α)) = 4λ P H N! ∞ i=0 (−1) i λ N +i w i!(N + i + 2) 2 N+i 2 √ π Γ( N +i+1 2 ) Γ( N +i+2 2 ) − √ 2 2 F 1 1 2 , N +i+1 2 , N +i+3 2 , 1 2 N + i + 1 × R N +2+i D H α − R β D κ K N + αK N κ N+i+2 βH α − R β D κ K N .(24) Eventually, it worth recalling that the Heaviside function is defined as H (x) =        1 if x ≥ 0 0 if x < 0 and thatH (x) = 1 − H (x). Hence, by introducing the slack variable η = N + i in (24) and defining the function χ η (λ w ) as in (5) where Λ N ([0, α)) is the first derivative of Λ N ([0, α)) with respect to α, given by Λ N ([0, α)) =          4λ P H N ! K N κβ ∞ η=N (−1) η−N χη(λw) (η−N )! αK N κ η+2 β −1 if α < R β D κ K N 0 if α ≥ R β D κ K N(27) with χ η (λ w ) defined as in (5). Hence, similarly to what has been done in [10], we can use the notable result and compute the expectation with respect to h (n) by taking advantage of the identity in [15,Eq. 7.521]. After some manipulations, the expression of the characteristic function is given by Φ N ω; L (0) = exp 4λ P H N! ∞ η=N (−1) η−N χ η (λ w ) (η − N)!(η + 2) × L (0) K N κ η+2 β 1 − 2 F 1 1, − η + 2 β , 1 − (η + 2) β , jω L (0) − R η+2 D 1 − 2 F 1 1, − η + 2 β , 1 − (η + 2) β , jωK N R β D κ H L (0) − R β D κ K N(28) Then, invoking the Gil-Pelaez inversion theorem F I M U (z; L (0) ) = 1/2 − ∞ 0 1 πω Im e −jωz Φ ω; L (0) dω,(29) and recalling (9) and (25) the imaginary unit. E(·) is the expectation operator. ½{·} is the indicator function. Im{·} denotes the imaginary part. Γ(· , · ) is the upper-incomplete Gamma function [15, Eq. 8.350.2]. pF q (a 1 , ..., a p ; b 1 , ..., b q ; · )is the generalized hypergeometric function[15, Eq. 9.14.1]. H(·) denotes the Heaviside function andH(·) = 1 − H(·). Fig. 1 : 1One realization of the MPL and PP processes. splitting (PS) scheme: the received signal is split in two streams of different power levels by using a power splitting ratio ρ, where 0 ≤ ρ ≤ 1. While one signal stream is sent to the rectenna circuit for energy harvesting, the other stream is used for information decoding. ,t ζ t exp(−sζ) Fig. 3 : 3Monte Carlo simulations and theoretical results of rate-energy trade-off for different λ P H when λ w = 0.05, n t = 4, n r = 2. B. Results 1) Validation of the analytical findings: Fig. 3 shows the trade-off between the information rate and the harvested power parametrised with respect to λ P H when F c (R * , Q * ) = 0.75. The results are presented for different values of d P H = 3, 5, 7 meters i.e. λ P H = 1 28 , 1 79 , 1 154 , while the frequency of the walls is kept fixed to λ w = 0.05. The analytical results (solid lines) are compared to the ones obtained through Monte Carlo simulations (squares). Despite of the approximation due to the fact that the series in the equations Fig. 4 :Fig. 5 :Fig. 6 : 456Monte Carlo simulations and theoretical results of rate-energy trade-off for different λ w when d P H = 5m, n t = 4, n r = 2.elements are densified. For example, if a minimum harvested energy of −23dBm is required, the optimum PHs' density is the one corresponding to d P H = 3 meters (blue line), while if a harvested energy of −28dBm is sufficient, it would be better to have a less dense network, d P H = 5 meters, to benefit of a higher information rate (red line).In order to investigate the effect of the walls on the SWIPT performance, Monte Carlo simulations (markers) and theoretical results (solid lines) are provided inFig. 4when F c (R * , Q * ) = 0.75 and d P H = 5 meters . The blue curve is used as a benchmark and represents the case in which all the blockage objects have been removed. The red and black curves are associated with two different values of λ w , and specifically with λ w = 0.03 and λ w = 0.05, respectively. It is apparent that an increment of the frequency of blockage objects is beneficial for the maximum achievable information rate because of the reduced multi-user interference. On the other side, we notice a reduction of the harvestable power level when λ w increases. We can conclude, that the topology DRAFT Theoretical results of rate-energy trade-off for different ρ values when F c (R * , Q * ) = 0.75, λ w = 0.03, n t = 4 and n r = 2.of the venue impacts the achievable rate-energy trade-off in view of the ambivalent role of the multi-user interference in SWIPT networks.2) SWIPT Receiver Analysis:We now analyse the role of the SWIPT receiver architecture on the achievable performance. The rate-energy trade-offs for different values of the power splitting ratio when F c (R * , Q * ) = 0.75 are presented inFig. 5. The curves are provided for 3 different values of λ P H when λ w = 0.03. Obviously, the receiver harvests more power with higher power splitting ratio for all cases. More interestingly, we obtain the same value of maximum achievable information rate for all values of ρ. This is due to the fact that, with the level of densification required to receive a total power belonging to the microwatt region, the system is essentially interference limited and the SINR does not depend on ρ for realistic level of noise. For instance, considering the case d P H = 3m (solid lines), with an information rate of 300 Kbps, −27dBm of harvested power can be achieved when ρ = 0.1, while we can obtain −20dBm and −17dBm of Theoretical results for rate-energy trade-off as a function of n t when n r = 2, λ w = 0.03 and d P H = 5m. Fig. 7 :Fig. 8 : 78Analysis: The previous subsections showed how the ambivalent role of the interference impacts the rate-energy trade-off. On the other side, both harvested energy and information rate can be improved by increasing the power gain from the serving PH to the LPD. This can be achieved by increasing the antenna array gain. In order to illustrate the benefits DRAFT Theoretical results for the envelope of rate-energy trade-off for different λ P H and λ w when F c (R * , Q * ) = 0.75, n t = 4, n r = 2.of MIMO on the SWIPT performance in such a dense interference-limited scenario,Fig. 6provides the rate-energy trade-off for different numbers of transmit antennas when F c (R * , Q * ) is 0.75 and n r = 2. The proposed results show how increasing the number of antennas is of great help in dealing with the MUI while improving the level of harvestable power. Here, for illustrative purposes, we present numerical results only for λ w = 0.03, but the same trend has been observed for all the tested frequencies of walls.4) Joint effect of λ P H and λ w on the achievable rate-energy trade-off:Eventually, we focus on how the the level of network densification must be wisely chosen in accordance with the topological characteristic of the indoor environment. To this purpose,Fig. 7shows the envelopes of the rate-energy trade-offs relevant to different values of λ P H and for a J-CCDF equal to 0.75. In particular, the red curve represents a situation in which all blockage objects have been removed, while the blue curve is associated with the presence of walls with frequency equal to Theoretical results of rate-energy trade-off for different λ P H and λ w when F c (R * , Q * ) = 0.75, n t = 4 and n r = 2. and Γ (2 + N, R D (λ w | cos(θ)| + λ w | sin(θ)w | cos(θ)| + λ w | sin(θ)|) (N +2+i) , FirstΦ , let note that the multi-user interference I M U can be expressed as the sum of the interferences produced by the PHs belonging to Ψ N , N ∈ {0, N max }, where the processes Ψ N are assumed to be independent. Then, given the minimum propagation loss L (0) , the characteristic function of I M U is given byΦ ω; L (0) = E exp jωI M U (L (0) N ω; L (0) .(25)where Φ N ω; L (0) is the characteristic function of the interference generating from the PHs associated with Ψ N . In order to compute Φ N ω; L (0) , we can invoke the Probability GeneratingFunctional (PGFL) theorem for PPPs (see [9, Proposition 1.2.2]), thus getting Φ N ω; L (0) = exp E h (n) ∞ L (0) exp jωh (n) /α − 1 Λ N ([0, α))dα (26) {jωA/x} − 1) x v−1 dx = (1/v)N v (1 − 1 F 1 (−v, 1 − v, jωA/N )) was supported by F.R.S.-FNRS under the EOS program (EOS project 30452698) and by INNOVIRIS under the COPINE-IOT project. TABLE I : IValues of χ η (λ w ) Table 3 ] 3, the penetration loss K can be reasonably set to −10dB per crossed wall.DRAFT we plotted the energy-rate trade-off for different values of λ w with d P H = 3m and d P H = 7m. In the denser case, the information rate always benefits from the reduced MUI due to the presence of walls. On the contrary, when the network elements are less dense, the blockage experienced by the serving PH is not compensated by the MUI reduction.Hence, neither the harvested power nor the information rate benefits from the interference. Hence, we can argue that the best achievable trade-off is a multidimensional factor in which the crossdependences between the network infrastructure and the topology of the venue have a major impact.V. CONCLUDING REMARKSIn this paper, we considered a dense in-building MIMO SWIPT network by proposing an accurate stochastic geometry analysis. In this scenario, we addressed the issue of the MUI in relation with both the network infrastructure and the topology of the venue. We discovered that there is a nontrivial rate-energy trade-off between the PHs' density and the frequency of the obstacles. Blockage objects can be beneficial in some cases depending on which trade-off is considered. Their effect in reducing the received power is compensated by the decrement of the MUI. Moreover, we investigated the power splitting ratio and showed that it has no effect on the maximum achievable information rate while more power can be harvested with its higher values. Finally, it is demonstrated that multi antenna processing is a valuable strategy to counteract the effect of the densification. Our theoretical findings are validated through Monte Carlo simulations.APPENDIX A PROOF OF PROPOSITION 1 January 14, 2018 DRAFT January 14, 2018 Transporting information and energy simultaneously. L R Varshney, Information Theory. IEEEL. R. Varshney, "Transporting information and energy simultaneously," in Information Theory, 2008. ISIT 2008. IEEE International Symposium on. IEEE, 2008, pp. 1612-1616. Mimo broadcasting for simultaneous wireless information and power transfer. R Zhang, C K Ho, IEEE Transactions on Wireless Communications. 125R. Zhang and C. K. Ho, "Mimo broadcasting for simultaneous wireless information and power transfer," IEEE Transactions on Wireless Communications, vol. 12, no. 5, pp. 1989-2001, 2013. A tractable approach to coverage and rate in cellular networks. J G Andrews, F Baccelli, R K Ganti, IEEE Transactions on Communications. 5911J. G. Andrews, F. Baccelli, and R. K. Ganti, "A tractable approach to coverage and rate in cellular networks," IEEE Transactions on Communications, vol. 59, no. 11, pp. 3122-3134, 2011. Average rate of downlink heterogeneous cellular networks over generalized fading channels: A stochastic geometry approach. M Di Renzo, A Guidotti, G E Corazza, IEEE Transactions on Communications. 617M. Di Renzo, A. Guidotti, and G. E. Corazza, "Average rate of downlink heterogeneous cellular networks over generalized fading channels: A stochastic geometry approach," IEEE Transactions on Communications, vol. 61, no. 7, pp. 3050-3071, 2013. Wireless information transfer with opportunistic energy harvesting. L Liu, R Zhang, K.-C Chua, IEEE Transactions on Wireless Communications. 121L. Liu, R. Zhang, and K.-C. Chua, "Wireless information transfer with opportunistic energy harvesting," IEEE Transactions on Wireless Communications, vol. 12, no. 1, pp. 288-300, 2013. Rethinking the role of interference in wireless networks. G Zheng, I Krikidis, C Masouros, S Timotheou, D.-A Toumpakaris, Z Ding, IEEE Communications Magazine. 5211G. Zheng, I. Krikidis, C. Masouros, S. Timotheou, D.-A. Toumpakaris, and Z. Ding, "Rethinking the role of interference in wireless networks," IEEE Communications Magazine, vol. 52, no. 11, pp. 152-158, 2014. DRAFT January 14, 2018 Shannon-theoretic approach to a gaussian cellular multiple-access channel. A D Wyner, IEEE Transactions on Information Theory. 406A. D. Wyner, "Shannon-theoretic approach to a gaussian cellular multiple-access channel," IEEE Transactions on Information Theory, vol. 40, no. 6, pp. 1713-1727, 1994. Advanced mobile phone service: The cellular concept. V H Mac Donald, Bell Labs Technical Journal. 581V. H. Mac Donald, "Advanced mobile phone service: The cellular concept," Bell Labs Technical Journal, vol. 58, no. 1, pp. 15-41, 1979. Stochastic geometry and wireless networks: Volume i theory. F Baccelli, B Błaszczyszyn, Foundations and Trends in Networking. 33-4F. Baccelli, B. Błaszczyszyn et al., "Stochastic geometry and wireless networks: Volume i theory," Foundations and Trends in Networking, vol. 3, no. 3-4, pp. 249-449, 2009. On stochastic geometry analysis and optimization of wireless-powered cellular networks. W Lu, M Di Renzo, T Duong, Global Communications Conference (GLOBECOM). W. Lu, M. Di Renzo, and T. Duong, "On stochastic geometry analysis and optimization of wireless-powered cellular networks," in Global Communications Conference (GLOBECOM), 2015 IEEE. IEEE, 2015, pp. 1-7. Mimo cellular networks with simultaneous wireless information and power transfer. T L Thanh, M Di Renzo, J P Coon, Signal Processing Advances in Wireless Communications (SPAWC). IEEEIEEE 17th International Workshop onT. L. Thanh, M. Di Renzo, and J. P. Coon, "Mimo cellular networks with simultaneous wireless information and power transfer," in Signal Processing Advances in Wireless Communications (SPAWC), 2016 IEEE 17th International Workshop on. IEEE, 2016, pp. 1-5. An introduction to the theory of point processes: volume II: general theory and structure. D J Daley, D Vere-Jones, Springer Science & Business MediaD. J. Daley and D. Vere-Jones, An introduction to the theory of point processes: volume II: general theory and structure. Springer Science & Business Media, 2007. An indoor correlated shadowing model. X Zhang, F Baccelli, R W Heath, Global Communications Conference (GLOBECOM). X. Zhang, F. Baccelli, and R. W. Heath, "An indoor correlated shadowing model," in Global Communications Conference (GLOBECOM), 2015 IEEE. IEEE, 2015, pp. 1-7. Analyzing wireless indoor communications by blockage models. M K Müller, M Taranetz, M Rupp, IEEE Access. 5M. K. Müller, M. Taranetz, and M. Rupp, "Analyzing wireless indoor communications by blockage models," IEEE Access, vol. 5, pp. 2172-2186, 2017. I S Gradshteyn, I M Ryzhik, Table of integrals, series, and products. Academic pressI. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products. Academic press, 2014. Note on the inversion theorem. J Gil-Pelaez, Biometrika. 383-4J. Gil-Pelaez, "Note on the inversion theorem," Biometrika, vol. 38, no. 3-4, pp. 481-482, 1951. Closed-form expressions for the outage and ergodic shannon capacity of mimo mrc systems. A Maaref, S Aïssa, IEEE Transactions on Communications. 537A. Maaref and S. Aïssa, "Closed-form expressions for the outage and ergodic shannon capacity of mimo mrc systems," IEEE Transactions on Communications, vol. 53, no. 7, pp. 1092-1095, 2005. Propagation data and prediction methods for the planning of indoor radiocommunication systems and radio local area networks in the frequency range 900 mhz to 100 ghz. P Series, ICU. P. Series, "Propagation data and prediction methods for the planning of indoor radiocommunication systems and radio local area networks in the frequency range 900 mhz to 100 ghz," ICU: Genève, Switzerland, 2012.
{'fraction_non_alphanumeric': 0.0617634325845399, 'fraction_numerical': 0.02253595651546491, 'mean_word_length': 3.977136700981566, 'pattern_counts': {'":': 0, '<': 4, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 26, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'Simultaneous information and power transfer (SWIPT) is characterised by the ambiguous role of multi-user interference. In short, the beneficial effect of multi-user interference on RF energy harvesting is obtained at the price of a reduced link capacity, thus originating nontrivial trade-offs between the achievable information rate and the harvestable energy. Arguably, in indoor environments, this tradeoff might be affected by the propagation loss due to blockage objects like walls. Hence, a couple of fundamental questions arise. How much must the network elements be densified to counteract the blockage attenuation? Is blockage always detrimental on the achievable rate-energy trade-off? In this paper, we analyse the performance of an indoor multiple-input multiple-output (MIMO) SWIPT-enabled network in the attempt to shed a light of those questions. The effects of the obstacles are examined with the help of a stochastic approach in which energy transmitters (also referred to as power heads) are located by using a Poisson Point Process and walls are generated through a Manhattan Poisson Line Process. The stochastic behaviour of the signal attenuation and the multi-user interference is studied to obtain the Joint Complementary Cumulative Distribution Function (J-CCDF) of information rate and harvested power. Theoretical results are validated through Monte Carlo simulations. Eventually, the rateenergy trade-off is presented as a function of the frequency of walls to emphasise the cross-dependences between the deployment of the network elements and the topology of the venue. A. I. Akin, I. Stupia and L. Vandendorpe are with the', 'arxivid': '1801.04558', 'author': ['Ayse Ipek Akin \nInstitute of Information and Communication Technologies\nElectronics and Applied Mathematics (ICTEAM)\nUniversité catholique de Louvain\nLouvain la NeuveBelgium\n', 'Ivan Stupia \nInstitute of Information and Communication Technologies\nElectronics and Applied Mathematics (ICTEAM)\nUniversité catholique de Louvain\nLouvain la NeuveBelgium\n', 'Fellow, IEEELuc Vandendorpe \nInstitute of Information and Communication Technologies\nElectronics and Applied Mathematics (ICTEAM)\nUniversité catholique de Louvain\nLouvain la NeuveBelgium\n'], 'authoraffiliation': ['Institute of Information and Communication Technologies\nElectronics and Applied Mathematics (ICTEAM)\nUniversité catholique de Louvain\nLouvain la NeuveBelgium', 'Institute of Information and Communication Technologies\nElectronics and Applied Mathematics (ICTEAM)\nUniversité catholique de Louvain\nLouvain la NeuveBelgium', 'Institute of Information and Communication Technologies\nElectronics and Applied Mathematics (ICTEAM)\nUniversité catholique de Louvain\nLouvain la NeuveBelgium'], 'corpusid': 1405830, 'doi': '10.1109/tcomm.2018.2876308', 'github_urls': [], 'n_tokens_mistral': 12644, 'n_tokens_neox': 11102, 'n_words': 7230, 'pdfsha': 'dec007b148304ee1c378d7f3350d58e2a5706a80', 'pdfurls': ['https://arxiv.org/pdf/1801.04558v1.pdf'], 'title': ['On the effect of blockage objects in dense MIMO SWIPT networks', 'On the effect of blockage objects in dense MIMO SWIPT networks'], 'venue': []}
arxiv
A NOTE ON PROPERLY DISCONTINUOUS ACTIONS 28 Jan 2023 M Kapovich A NOTE ON PROPERLY DISCONTINUOUS ACTIONS 28 Jan 2023 We compare various notions of proper discontinuity for group actions. We also discuss fundamental domains and criteria for cocompactness.To the memory of Sasha Anan'inIntroductionThis note is meant to clarify the relation between different commonly used definitions of proper discontinuity without the local compactness assumption for the underlying topological space. Much of the discussion applies to actions of nondiscrete locally compact Hausdorff topological groups, but, since my primary interest is geometric group theory, I will mostly work with discrete groups. All group actions are assumed to be continuous, in other words, for discrete groups, these are homomorphisms from abstract groups to groups of homeomorphisms of topological spaces. This combination of continuous and properly discontinuous, sadly, leads to the ugly terminology "a continuous properly discontinuous action." A better terminology might be that of a properly discrete action, since it refers to proper actions of discrete groups.Throughout this note, I will be working only with topological spaces which are 1st countable, since spaces most common in metric geometry, geometric topology, algebraic topology and geometric group theory satisfy this property. One advantage of this assumption is that if (x n ) is a sequence converging to a point x ∈ X, then the subset {x} ∪ {x n : n ∈ N} is compact, which is not true if we work with nets instead of sequences. However, I will try to avoid the local compactness assumption whenever possible, since many spaces appearing in metric geometry and geometric group theory (e.g. asymptotic cones) and algebraic topology (e.g. CW complexes) are not locally compact. (Recall that topological space X is locally compact if every point has a basis of topology consisting of relatively compact subsets.) In the last two sections of the note I also discuss cocompact group actions and fundamental sets/domains of properly discontinuous group actions.Acknowledgement. I am grateful to Boris Okun for pointing out several typos and the reference to[12]. I am also grateful to the referee of the paper for useful suggestions and corrections.Group actionsA topological group is a group G equipped with a topology such that the multiplication and inversion maps G × G → G, (g, h) → gh, G → G, g → g −1 are both continuous. A discrete group is a group with discrete topology. Every discrete group is clearly a topological group. A left continuous action of a topological group G on a topological space X is a continuous map λ : G × X → X satisfying 1. λ(1 G , x) = x for all x ∈ X. 2. λ(gh, x) = λ(g, λ(h, x)), for all x ∈ X, g, h ∈ G. From this, it follows that the map ρ : G → Homeo(X) ρ(g)(x) = λ(g, x), is a group homomorphism, where the group operation φψ on Homeo(X) is the composition φ • ψ. If G is discrete, then every homomorphism G → Homeo(X) defines a left continuous action of G on X. The shorthand for ρ(g)(x) is gx or g · x. Similarly, for a subset A ⊂ X, GA or G · A, denotes the orbit of A under the G-action: GA = g∈G gA. The quotient space X/G (also frequently denoted G\X), of X by the G-action, is the set of Gorbits of points in X, equipped with the quotient topology: The elements of X/G are equivalence classes in X, where x ∼ y when Gx = Gy (equivalently, y ∈ Gx). The stabilizer of a point x ∈ X under the G-action is the subgroup G x < G given by {g ∈ G : gx = x}. An action of G on X is called free if G x = {1} for all x ∈ X. Assuming that X is Hausdorff, G x is closed in G for every x ∈ X. Example 1. An example of a left action of G is the action of G on itself via left multiplication: λ(g, h) = gh. In this case, the common notation for ρ(g) is L g . This action is free. Proper maps Properness of certain maps is the most common form of defining proper discontinuity; sadly, there are two competing notions of properness in the literature. A continuous map f : X → Y of topological spaces is proper in the sense of Bourbaki, or simply Bourbaki-proper (cf. [3, Ch. I, §10, Theorem 1]) if f is a closed map (images of closed subsets are closed) and point-preimages f −1 (y), y ∈ Y , are compact. A continuous map f : X → Y is proper (and this is the most common definition) if for every compact subset K ⊂ X, f −1 (K) is compact. It is noted in [3, Ch. I, §10; Prop. 7] that if X is Hausdorff and Y is locally compact, then f is Bourbaki-proper if and only if f is proper. The advantage of the notion of Bourbaki-properness is that it applies in the case of Zariski topology, where spaces tend to be compact 1 (every subset of a finite-dimensional affine space is Zariski-compact) and, hence, the standard notion of properness is useless. Since our goal is to trade local compactness for 1st countability, I will prove a lemma which appears as a Corollary in [12]: 1 quasicompact in the Bourbaki terminology Lemma 2. If f : X → Y is proper, and X, Y are Hausdorff and 1st countable, then f is Bourbakiproper. Proof. We only have to verify that f is closed. Suppose that A ⊂ X is a closed subset. Since Y is 1st countable, it suffices to show that for each sequence (x n ) in A such that (f (x n )) converges to y ∈ Y , there is a subsequence (x n k ) which converges to some x ∈ A such that f (x) = y. The subset C = {y} ∪ {f (x n ) : n ∈ N} ⊂ Y is compact. Hence, by properness of f , K = f −1 (C) is also compact. Since X is Hausdorff, and K is compact, follows that (x n ) subconverges to a point x ∈ K. By continuity of f , f (x) = y. Since A is closed, x ∈ A. Remark 3. This lemma still holds if one were to replace the assumption that X is 1st countable by surjectivity of f , see [12]. The converse (each Bourbaki-proper map is proper) is proven in [3, Ch. I, §10; Prop. 6] without any restrictions on X, Y . Hence: Corollary 4. For maps between 1st countable Hausdorff spaces, Bourbaki-properness is equivalent to properness. Proper discontinuity Suppose that X is a 1st countable Hausdorff topological space, G a discrete group and G×X → X a (continuous) action. I use the notation g n → ∞ in G to indicate that g n converges to ∞ in the 1-point compactification G ∪ {∞} of G, i.e. for every finite subset F ⊂ G, card({n : g n ∈ F }) < ∞. Given a group action G × X → X and two subsets A, B ⊂ X, the transporter subset (A|B) G is defined as (A|B) G := {g ∈ G : gA ∩ B = ∅}. Properness of group actions is (typically) stated using certain transporter sets. Definition 5. Two points x, y ∈ X are said to be G-dynamically related if there is a sequence g n → ∞ in G and a sequence x n → x in X such that g n x n → y. A point x ∈ X is said to be a wandering point of the G-action if there is a neighborhood U of x such that (U |U ) G is finite. Lemma 6. Suppose that the action G × X → X is wandering at a point x ∈ X. Then the Gaction has a G-slice at x, i.e. a neighborhood W x ⊂ U which is G x -stable and for all g / ∈ G x , gW x ∩ W x = ∅. Proof. For each g ∈ (U |U ) G − G x we pick a neighborhood V g ⊂ U of x such that gV g ∩ V g = ∅. Then the intersection V := g∈(U|U)G−Gx V g satisfies the property that (V |V ) G = G x . Lastly, take W x := g∈Gx V. The next lemma is clear: Lemma 7. Assuming that X is Hausdorff and 1st countable, the action G × X → X is wandering at x if and only if x is not dynamically related to itself. Given a group action α : G × X → X, we have the natural map α := α × id X : G × X → X × X where id X : (g, x) → x. Definition 8. An action α of a discrete group G on a topological space X is Bourbaki-proper if the mapα is Bourbaki-proper. Lemma 9. If the action α : G × X → X of a discrete group G on a Hausdorff topological space X is Bourbaki-proper, then the quotient space X/G is Hausdorff. Proof. The quotient map X → X/G is an open map by the definition of the quotient topology on X/G. Since α is Bourbaki-proper, the image of the mapα is closed in X × X. This image is the equivalence relation on X × X which use used to form the quotient X/G. Now, Hausdorffness of X/G follows from [3, Proposition 8 in I.8.3]. Definition 10. An action α of a discrete group G on a topological space X is proper if the mapα is proper. Note that the equivalence of (1) and (5) in the following theorem is proven in [3, Ch. III, §4.4, Proposition 7] without any assumptions on X. Theorem 11. Assuming that X is Hausdorff and 1st countable, the following are equivalent: (1) The action α : G × X → X is Bourbaki-proper. (2) For every compact subset K ⊂ X, card((K|K) G ) < ∞. (3) The action α : G × X → X is proper, i.e. the mapα is proper. (4) For every compact subset K ⊂ X, there exists an open neighborhood U of K such that card((U |U ) G ) < ∞. (5) For any pair of points x, y ∈ X there is a pair of neighborhoods U x , V x (of x, y respectively) such that card((U x |V y ) G )) < ∞. (6) There are no G-dynamically related points in X. (7) Assuming, that G is countable and X is completely metrizable 2 : The G-stabilizer of every x ∈ X is finite and for any two points x ∈ X, y ∈ X−Gx, there exists a pair of neighborhoods U x , V y (of x, resp. y) such that ∀g ∈ G, gU x ∩ V y = ∅. (8) Assuming that X is a metric space and the action G × X → X is equicontinuous 3 : There is no x ∈ X and a sequence h n → ∞ in G such that h n x → x. (9) Assuming that X is a metric space and the action G × X → X is equicontinuous: Every x ∈ X is a wandering point of the G-action. (10) Assuming that X is a CW complex and the action G × X → X is cellular: Every point of X is wandering. (11) Assuming that X is a CW complex the action G × X → X is cellular: Every cell in X has finite G-stabilizer. Proof. The action α is Bourbaki-proper if and only if the mapα is proper (see Corollary 4) which is equivalent to the statement that for each compact K ⊂ X, the subset (K|K) G × K is compact. Hence, (1) ⇐⇒ (2). Assume that (3) holds, i.e. α is proper, equivalently, the mapα is proper. This means that for each compact K ⊂ X,α −1 (K × K) = {(g, x) ∈ G × K : x ∈ K, gx ∈ K} is compact. This subset is closed in G × X and projects onto (K|K) G in the first factor and to the subset (⋆) g∈(K|K)G g −1 (K) in the second factor. Hence, properness of the action α implies finiteness of (K|K) G , i.e. (2). Conversely, if (K|K) G is finite, compactness of g −1 (K) for every g ∈ G implies compactness of the union (⋆). Thus, (2) ⇐⇒ (3). In order to show that (2)⇒(6), suppose that x, y are G-dynamically related points: There exists a sequence g n → ∞ in G and a sequence x n → x such that g n (x n ) → y. The subset K = {x, y} ∪ {x n , g n (x n ) : n ∈ N} is compact. However, y n ∈ g n (K) ∩ K for every n. A contradiction. (6)⇒(5): Suppose that the neighborhoods U x , V y do not exist. Let {U n } n∈N , {V n } n∈N be countable bases at x, y respectively. Then for every n there exists g n ∈ G, such that g n (U n ) ∩ V n = ∅ for infinitely many g n 's in G. After extraction, g n → ∞ in G. This yields points x n ∈ U n , y n = g n (x n ) ∈ V n . Hence, x n → x, y n → y. Thus, x is G-dynamically related to y. A contradiction. (5)⇒(4). Consider a compact K ⊂ X. Then for each x ∈ K, y ∈ K there exist neighborhoods U x , V y such that (U x |V y ) G is finite. The product sets U x × V y , x, y ∈ K constitute an open cover of K 2 . By compactness of K 2 , there exist x 1 , ..., x n , y 1 , ..., y m ∈ K such that K ⊂ U x1 ∪ ... ∪ U xn K ⊂ V y1 ∪ ... ∪ V ym and for each pair (x i , y j ), card({g ∈ G : gU xi ∩ V yj = ∅}) < ∞. Setting W := n i=1 U xi , V := m j=1 V yj , we see that card((W |V ) G ) < ∞. Taking U := V ∩ W yields the required subset U . The implication (4)⇒(2) is immediate. This concludes the proof of equivalence of the properties (1)-(6). (5)⇒(7): Finiteness of G-stabilizers of points in X is clear. Let x, y be points in distinct G- orbits. Let U ′ x , V ′ y be neighborhoods of x, y such that (U ′ x |V ′ y ) G = {g 1 , ..., g n }. For each i, since X is Hausdorff, there are disjoint neighborhoods V i of y and W i of g i (x i ). Now set V y := n i=1 V i , U x := n i=1 g −1 i (W i ). Then gU x ∩ V y = ∅ for every g ∈ G. (7)⇒(6): It is clear that (7) implies that there are no dynamically related points with distinct G-orbits. In particular, every G-orbit in X is closed. Assume now that X is completely metrizable and G is countable. Suppose that a point x ∈ X is G-dynamically related to itself. Since the stabilizer G x is finite, the point x is an accumulation point of Gx; moreover, Gx is closed in X. Hence, Gx is a closed perfect subset of X. Since X admits a complete metric, so does its closed subset Gx. Thus, for each g ∈ G, the complement U g := Gx − {gx} is open and dense in Gx. By the Baire Category Theorem, the countable intersection g∈G U g is dense in Gx. However, this intersection is empty. A contradiction. It is clear that (6)⇒(8) (without any extra assumptions). (8)⇒ (6). Suppose that X is a metric space and the G-action is equicontinuous. Equicontinuity implies that for each z ∈ X, a sequence z n → z and g n ∈ G, g n z n → gz. Suppose that there exist a pair of G-dynamically related points x, y ∈ X: ∃x n → x, g n ∈ G, g n x n → y. By the equicontinuity of the action, g n x → y. Since g n → ∞, there exist subsequences g ni → ∞ and g mi → ∞ such that the products h i := g −1 ni g mi are all distinct. Then, by the equicontinuity, h i x → x. A contradiction. The implications (5)⇒(9)⇒(8) and (5)⇒(10)⇒(11) are clear. Lastly, let us prove the implication (11)⇒(2). We first observe that every CW complex is Hausdorff and 1st countable. Furthermore, every compact K ⊂ X intersects only finitely many open cells e λ in X. (Otherwise, picking one point from each nonempty intersection K ∩ e λ we obtain an infinite closed discrete subset of K.) Thus, there exists a finite subset E := {e λ : λ ∈ Λ} of open cells in X such that for every g ∈ (K|K) G , gE ∩ E = ∅. Now, finiteness of (K|K) G follows from finiteness of cell-stabilizers in G. Unfortunately, the property that every point of X is a wandering point is frequently taken as the definition of proper discontinuity for G-actions, see e.g. [9,11]. Items (8) and (10) in the above theorem provide a (weak) justification for this abuse of terminology. I feel that the better name for such actions is wandering actions. Example 12. Consider the action of G = Z on the punctured affine plane X = R 2 − {(0, 0)}, where the generator of Z acts via (x, y) → (2x, 1 2 y). Then for any p ∈ X, the G-orbit Gp has no accumulation points in X. However, any two points p = (x, 0), q = (0, y) ∈ X are dynamically related. Thus, the action of G is not proper. This example shows that the quotient space of a wandering action need not be Hausdorff. Lemma 13. Suppose that G × X → X is a wandering action. Then each G-orbit is closed and discrete in X. In particular, the quotient space X/G is T1. Proof. Suppose that Gx accumulates at a point y. Then Gx ∩ W y is nonempty, where W y is a G-slice at y. It follows that all points of Gx ∩ W y lie in the same W y -orbit, which implies that Gx ∩ W y = {y}. There are several reasons to consider proper actions of discrete (and, more generally, locally compact) groups; one reason is that such each proper action of a discrete group yields an orbicovering map in the case of smooth group actions on manifolds: M → M/G is an orbi-covering provided that the action of G on M is smooth (or, at least, locally smoothable). Another reason is that for a proper action on a Hausdorff space, G × X → X, the quotient X/G is again Hausdorff, see Lemma 9. Question 14. Suppose that G is a discrete group, G × X → X is a free continuous action on an n-dimensional topological manifold X such that the quotient space X/G is a (Hausdorff ) ndimensional topological manifold. Does it follow that the G-action on X is proper? The answer to this question is negative if one merely assumes that X is a locally compact Hausdorff topological space and X/G is Hausdorff, see [7] (the action given there was even cocompact). Below is a different example. We begin by constructing a non-proper free continuous R-action on a manifold, such that the quotient space is not just Hausdorff but is a manifold with boundary. Example 15. This is a variation on Example 12. We start with the space Z = {(x, y) : x, y ∈ [0, ∞), (x, y) = (0, 0)}. Take the quotient space X of Z by the equivalence relation (x, 0) ∼ (0, 1 x ). The space X is homeomorphic to the open Moebius band. The group G = R acts on Z continuously by (t, (x, y)) → (2 t x, 2 −t y). The above equivalence relation on X is preserved by the G-action and, hence, the G-action descends to a continuous G-action on X. It is easy to see that this action is free but not proper: The equivalence class of (1, 0) is dynamically related to itself. Lastly, the quotient X/G is Hausdorff, homeomorphic to [0, 1) (the equivalence class of (1, 0) maps to 0 ∈ [0, 1)). Lastly, we use Example 15 to construct a non-proper free Z-action with Hausdorff quotient. We continue with the notation of the previous example. Let W denote the projection of Y to X. We take Γ = Z < G = R. This subgroup preserves Y and, hence, W . The quotient W/Γ is homeomorphic to Y ∩ {(0, y) : y ∈ R}, hence, is Hausdorff. At the same time, the Γ-action on W is non-proper. Cocompactness There are two common notions of cocompactness for group actions: (1) G × X → X is cocompact if there exists a compact K ⊂ X such that G · K = X. (2) G × X → X is cocompact if X/G is compact. It is clear that (1)⇒(2), as the image of a compact under the continuous (quotient) map p : X → X/G is compact. Lemma 17. If X is locally compact then (2)⇒(1). Proof. For each x ∈ X let U x denote a relatively compact neighborhood of x in X. Then V x := p(U x ) = p(G · U x ), is compact since G · U x is open in X. Thus, we obtain an open cover {V x : x ∈ X} of X/G. Since X/G is compact, this open cover contains a finite subcover V x1 , ..., V xn . It follows that p( n i=1 U xi ) = X/G. The set K = n i=1 U xi is compact and p(K) = X/G. Hence, G · K = X. Lemma 18. Suppose that X is normal and Hausdorff, G × X → X is a proper action of a discrete group, such that X/G is locally compact. Then X is locally compact. Proof. Pick x ∈ X. Let W x be a slice for the G-action at x; then W x /G x → X/G is a topological embedding. Thus, our assumptions imply that W x /G x is compact for every x ∈ X. Let (x α ) be a net in W x . Since W x /G x is compact, the net (x α )/G contains a convergent subnet. Thus, after passing to a subnet, there exists g ∈ G x such that (gx α ) converges to some x ∈ W x . Hence, (x α ) subconverges to g −1 (x). Thus, W x is relatively compact. Since X is assumed to be normal, x admits a basis of relatively compact neighborhoods. Corollary 19. For normal Hausdorff spaces X the two notions of cocompactness agree for proper discrete group actions on X. On the other hand, if the drop the properness condition, the two notions are not equivalent even for Z-actions with Hausdorff quotients, see the example by R. de la Vega in [16]. Fundamental sets Definition 20. A closed subset F ⊂ X is a fundamental set for the action of G on X if G · F = X and there exists an open neighborhood U = U F of F such that for every compact K ⊂ X, the transporter set (U |K) G is finite (the local finiteness condition). Fundamental sets appear naturally in the reduction theory of arithmetic groups (Siegel sets), see [13] and [2]. There are several existence theorems for fundamental sets. The next proposition, proven in [10, Lemma 2], guarantees the existence of fundamental sets under the paracompactness assumption on X/G. Proposition 21. Each proper action G × X → X of a discrete group G on a locally compact Hausdorff space X with paracompact quotient X/G admits a fundamental set. One frequently encounters a sharper version of fundamental sets, called fundamental domains. A domain in a topological space X is an open connected subset U ⊂ X which equals the interior of its closure. Definition 22. Suppose that G × X → X is a proper action of a discrete group. A subset F in X is called a fundamental domain for an action G × X → X if the following hold: (1) F is a domain in X. (2) G · F = X. (3) gF ∩ F = ∅ if and only if g = 1. (4) For every compact subset K ⊂ X, the transporter set (F |K) G is finite, i.e. the family {gF } g∈G of subsets in X is locally finite. Suppose that (X, d) is a proper geodesic metric space, i.e. a space where every closed metric ball is compact and every two points are connected by a geodesic segment. Suppose, furthermore, that G × X → X is a proper isometric action of a discrete group, x ∈ X is a point which is fixed only by the identity element. Remark 23. If G is countable and fixed point sets in X of nontrivial elements of G are nowhere dense, then Baire's Theorem implies existence of such x. One defines the Dirichlet domain of the action as D = D x = {y ∈ X : d(y, x) < d(y, gx) ∀g ∈ G \ G x }. Note that gD x = D gx . Proposition 24. Each Dirichlet domain D is a fundamental domain for the G-action. Proof. 1. The closure D is contained in D =D x = {y ∈ X : d(y, x) ≤ d(y, gx) ∀g ∈ G \ G x }. As before, gD x =D gx . I claim thatD is the closure of D and D is the interior ofD; this will prove that D is a domain. Clearly, D is contained in the interior ofD andD is closed. Hence, it suffices to prove that each point ofD is the limit of a sequence in D. Consider a point z ∈D \ D and let c : [0, T ] → X be a geodesic connecting x to z. Then for each t ∈ [0, T ) and g ∈ G \ {1}, d(x, c(t)) < d(x, c(t)) + d(c(t), z) = d(x, z) ≤ d(z, gx), i.e. c(t) ∈ D. Thus, indeed, z lies in the closure of D, as claimed. This argument also proves that D is connected. 2. Let us prove that gD = X. For each y ∈ X the function g → d(z, gx) is a proper function on G, hence, it attains its minimum at some g ∈ G. Then, clearly, y ∈D gx , hence, y ∈ gD x . Thus, gD = X. 3. Suppose that g ∈ G \ {1} is such that gD = D gx ∩ D = ∅. Then each point y of intersection is closer to x than to gx (since y ∈ D x ) and also y is closer to gx than to g −1 gx = x (since y ∈ D gx ). This is clearly impossible. 4. Lastly, we verify local finiteness. Consider a compact K ⊂ X. Then K ⊂ B = B(x, R) for some R. For every g ∈ G such that gB ∩ B = ∅, d(x, gx) ≤ 2R. Since (X, d) is a proper metric space and the action of G on X is proper, the set of such elements of G is finite. We will now prove existence of fundamental domains for proper discrete group actions on a certain class of topological spaces, cf. [14]. Theorem 25. Suppose that X is a 2nd countable, connected and locally connected locally compact Hausdorff topological space. Suppose that G × X → X is a proper action of a discrete countable group such that the fixed-point set of each nontrivial element of G is nowhere dense in X. Then this action admits a fundamental domain. Proof. Our goal is to construct a G-invariant geodesic metric metrizing X. Then the result will follow from the proposition. Lemma 26. The quotient space Y = X/G is locally compact, connected, locally connected and metrizable. Proof. Local compactness and connectedness of Y follows from that of X. The 2nd countability of X implies the 2nd countability of Y . By Lemma 9, Y is Hausdorff. Since Y is locally compact and Hausdorff, its one-point compactification is compact and Hausdorff, hence, regular. It follows that Y itself is regular. In view of the 2nd countability of Y , Urysohn's metrization theorem implies that Y is metrizable. Remark 27. Note that each locally compact metrizable space is also locally path-connected. It is proven in [15] that each locally compact, connected, locally connected metrizable space, such as Y , admits a complete geodesic metric d Y which we fix from now on. Consider the projection p : X → Y . According to [4,Theorem 6.2] (see also [1,Lemma 2]), the map p satisfies the pathlifting property: Given any path c : [0, 1] → Y , a point x ∈ X satisfying p(x) = c(0), there exists a pathc : [0, 1] → X such that p •c = c. (This result is, of course, much easier if the G-action is free, i.e. p : X → Y is a covering map.) We let L X denote the set of paths in X which are lifts of rectifiable paths c : [0, 1] → Y . Clearly, the postcomposition ofc ∈ L X with an element of G is again in L X . Our next goal is to equip X with a G-invariant length structure using the family of paths L X . Such a structure is a function on L X with values in [0, ∞), satisfying certain axioms that can be found in [5,Section 2.1]. Verification of most of these axioms is straightforward, I will check only some (items 1, 2, 3 and 4 below). 1. Ifc ∈ L X is a lift of a path c in Y , then we declare ℓ(c) to be equal to the length of c. 2. Ifc i , i = 1, 2, are paths in L X (which are lifts of the paths c 1 , c 2 respectively) whose concatenation b =c 1 ⋆c 2 is defined, then b is a lift of the concatenation c 1 ⋆ c 2 . Clearly, ℓ(b) = ℓ(c 1 ) + ℓ(c 2 ). 3. Let U be a neighborhood of some x ∈ X. We need to prove that (28) inf γ {ℓ(γ)} > 0, where the infimum is taken over all γ =c ∈ L X connecting x to points of X \ U . It suffices to prove this claim in the case when U is G x -invariant, satisfies (29) U ∩ gU = ∅ ⇐⇒ g ∈ G x , and γ connects x to points of ∂U . Then V = p(U ) is a neighborhood of y = p(x) in Y and the paths c = p • γ connect y to points in ∂V . But the lengths of the paths c are clearly bounded away from zero and are equal to the lengths of their liftsc. Thus, we obtain the required bound (28). 4. Let us verify that any two points in X are connected by a path in L X . Since X is connected, it suffices to verify the claim locally. Let U is G x -invariant neighborhood of x satisfying (29), such that V = p(U ) is an open metric ball in Y centered at y = p(x). Take u ∈ U , v := p(u) ∈ V . Let c : [0, T ] → V be a geodesic connecting v to y. Then there exists a liftc : [0, T ] → U of c with c(0) = u. Since x ∈ U is the only point projecting to y, we getc(T ) = x. By taking concatenations of pairs of such radial paths in U , we conclude that any two points in U are connected by a path c ∈ L X . Given a length structure on X, one defines a path-metric (metrizing the topology of X) by d X (x 1 , x 2 ) = inf γ {ℓ(γ)} where the infimum is taken over all γ ∈ L X connecting x 1 to x 2 . By the construction, the projection p : (X, d X ) → (Y, d Y ) is 1-Lipschitz. Lemma 30. The metric d X is complete. Proof. Let (x n ) be a Cauchy sequence in (X, d X ). By the construction of the metric d X , there exists a finite length pathc : [0, 1) → (X, d X ) and a sequence t n ∈ [0, 1) such thatc(t n ) = x n ,c(0) = x = x 1 . Since the map p is 1-Lipschitz, the path c = p •c : [0, 1) → (Y, d Y ) also has finite length. Since the metric d Y was complete to begin with, the path c extends to a pathc : [0, 1] → Y ; set y ′ :=c(1). Assume for a moment that G acts freely on X. Then we have the uniqueness of lifts of paths from Y to X. Thus, the unique liftc ofc starting at the point x satisfies the property that its restriction to [0, 1) equalsc. It follows that the sequence (x n ) converges toc(1). Below we generalize this argument to the case of non-free actions. Let U be a neighborhood of y ′ =c(1) which is the projection to Y of a relatively compact slice neighborhoodŨ of some x ′ ∈ p −1 (y ′ ). Without loss of generality (by removing finitely many initial terms of the sequence (x n )) we can assume that the image of the path c lies entirely in U . Applying the path-lifting property to the path c with the prescribed terminal point x ′ , we obtain a lift of the pathc that terminates at x ′ . This lift has to be entirely contained inŨ and its initial point has to be of the form g(x) for some g ∈ G. Applying g −1 to this lift, we obtain another lift ofc, denoted c, which starts at x and terminates at g −1 (x ′ ). Consider the restriction ofc to [0, 1). This restriction is also a lift to the path c| [0,1) and the image of the latter lies entirely in U . Hence, the image ofc| [0,1) lies entirely in the relatively compact subset g −1 (Ũ ) ⊂ X. Thus, the Cauchy sequence (x n ) lies in a relatively compact subset of X, and it follows that this sequence converges in X. Since (X, d X ) is locally compact and complete, by Theorem 2.5.28 (and Remark 2.5.29) in [5], (X, d X ) is a geodesic metric space. Lastly, we note that, by the construction, the length structure on X and, hence, the metric d X , is G-invariant. This concludes the proof of the theorem. Question 31. Local compactness and local connectivity were critical for the proof of the theorem. Does the theorem hold without these assumptions? For each fundamental set F of a G-action on a topological space X we define its quotient space F/G as the quotient space of the equivalence relation x ∼ y ⇐⇒ ({x}|{y}) G = ∅. The following proposition explains why fundamental sets are useful: They allow one to describe quotient spaces of proper actions by discrete groups using less information than is contained in the description of that action. Proposition 32. Suppose that F is a fundamental set for proper action by discrete group G on a 1st countable and Hausdorff space X. Then the natural projection map p : F/G → X/G is a homeomorphism. Proof. The map p is continuous by the definition of the quotient topology. It is also obviously a bijection. It remains to show that p is a closed map. Since F is closed, it suffices to show that the projection q : F → X/G is a closed map. Suppose that (x n ) is a sequence in F such that q(x n ) converges to some y ∈ X/G, y is represented by a point x ∈ F . Then there is a sequence g n ∈ G such that g n (x n ) converges to x. Since {g n (x n ) : n ∈ N} ∪ {x} is compact which, without loss of generality is contained in U F , the local finiteness assumption implies that the sequence (g n ) is finite. Hence, after extraction, g n = g for all n. The fact that F is closed then implies that x ∈ F . It follows that x is an accumulation point of (x n ). Thus, q : F → F/G is a closed map. Example 16 . 16Let Y ⊂ Z denote the following subset of Z (with the subspace topology): Y = {(2 m , 0) : m ∈ Z} ∪ {(0, 2 n ) : n ∈ Z} ∪ {(2 m , 2 n ) : (m, n) ∈ Z 2 }. It suffices to assume that X is hereditarily Baire: Every closed subset of X is Baire. 3 E.g. an isometric action. The fundamental group of the orbit space of a discontinuous group. M Armstrong, Proc. Cambridge Philos. Soc. 64M. Armstrong, The fundamental group of the orbit space of a discontinuous group, Proc. Cambridge Philos. Soc. 64 (1968) 299-301. Compactifications of Symmetric and Locally Symmetric Spaces. A Borel, L Ji, Birkhauser Verlag, Series "Mathematics: Theory and Applications. A. Borel, L. Ji, "Compactifications of Symmetric and Locally Symmetric Spaces." Birkhauser Verlag, Series "Mathematics: Theory and Applications", 2005. . N Bourbaki, Elements of Mathematics. General Topology." Parts I-IV. N. Bourbaki, "Elements of Mathematics. General Topology." Parts I-IV, Hermann, Paris, 1966. Introduction to Compact Transformation Groups. G Bredon, Academic PressG. Bredon, "Introduction to Compact Transformation Groups," Academic Press, 1972. A Course in Metric Geometry. D Burago, Y Burago, S Ivanov, Graduate Studies in Mathematics. 33American Mathematical SocietyD. Burago, Y. Burago and S. Ivanov, "A Course in Metric Geometry." Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. H Chapdelaine, A question about group action on topological space. H. Chapdelaine, A question about group action on topological space, https://mathoverflow.net/q/50128. An answer to "Finite stabilizers + compact orbit space ⇒ proper action?. Y De Cornulier, Y. de Cornulier, An answer to "Finite stabilizers + compact orbit space ⇒ proper action?", https://mathoverflow.net/q/202244 M David, Properly discontinuous action. M. David, Properly discontinuous action, https://mathoverflow.net/q/55726. Algebraic Topology. A Hatcher, Cambridge University PressA. Hatcher, "Algebraic Topology." Cambridge University Press, 2001. Lectures on Groups of Transformations. J L Koszul, Tata Institute of Fundamental Research. J. L. Koszul, "Lectures on Groups of Transformations." Tata Institute of Fundamental Research, Bombay, 1965. . J Munkres, Topology, Prentice Hall, IncUpper Saddle River, NJSecond editionJ. Munkres, "Topology." Second edition, Prentice Hall, Inc., Upper Saddle River, NJ, 2000. When proper maps are closed. R Palais, Proc. of AMS. 24R. Palais, When proper maps are closed, Proc. of AMS, 24 (1970), 835-836. C L Siegel, Discontinuous groups. C. L. Siegel, Discontinuous groups, Ann. of Math. (2) 44 (1943), 674-689. . I Smilga, I. Smilga,
{'fraction_non_alphanumeric': 0.07472366858500046, 'fraction_numerical': 0.013611034986754362, 'mean_word_length': 3.369611495476317, 'pattern_counts': {'":': 0, '<': 10, '<?xml version=': 0, '>': 1, 'https://': 3, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'We compare various notions of proper discontinuity for group actions. We also discuss fundamental domains and criteria for cocompactness.To the memory of Sasha Anan\'inIntroductionThis note is meant to clarify the relation between different commonly used definitions of proper discontinuity without the local compactness assumption for the underlying topological space. Much of the discussion applies to actions of nondiscrete locally compact Hausdorff topological groups, but, since my primary interest is geometric group theory, I will mostly work with discrete groups. All group actions are assumed to be continuous, in other words, for discrete groups, these are homomorphisms from abstract groups to groups of homeomorphisms of topological spaces. This combination of continuous and properly discontinuous, sadly, leads to the ugly terminology "a continuous properly discontinuous action." A better terminology might be that of a properly discrete action, since it refers to proper actions of discrete groups.Throughout this note, I will be working only with topological spaces which are 1st countable, since spaces most common in metric geometry, geometric topology, algebraic topology and geometric group theory satisfy this property. One advantage of this assumption is that if (x n ) is a sequence converging to a point x ∈ X, then the subset {x} ∪ {x n : n ∈ N} is compact, which is not true if we work with nets instead of sequences. However, I will try to avoid the local compactness assumption whenever possible, since many spaces appearing in metric geometry and geometric group theory (e.g. asymptotic cones) and algebraic topology (e.g. CW complexes) are not locally compact. (Recall that topological space X is locally compact if every point has a basis of topology consisting of relatively compact subsets.) In the last two sections of the note I also discuss cocompact group actions and fundamental sets/domains of properly discontinuous group actions.Acknowledgement. I am grateful to Boris Okun for pointing out several typos and the reference to[12]. I am also grateful to the referee of the paper for useful suggestions and corrections.Group actionsA topological group is a group G equipped with a topology such that the multiplication and inversion maps G × G → G, (g, h) → gh, G → G, g → g −1 are both continuous. A discrete group is a group with discrete topology. Every discrete group is clearly a topological group.', 'arxivid': '2301.05325', 'author': ['M Kapovich ', 'M Kapovich '], 'authoraffiliation': [], 'corpusid': 11392456, 'doi': '10.1007/s40863-023-00353-z', 'github_urls': [], 'n_tokens_mistral': 10658, 'n_tokens_neox': 9816, 'n_words': 6420, 'pdfsha': 'c886bac29677a151c8c5cb5e8feb34900063f94c', 'pdfurls': ['https://export.arxiv.org/pdf/2301.05325v2.pdf'], 'title': ['A NOTE ON PROPERLY DISCONTINUOUS ACTIONS', 'A NOTE ON PROPERLY DISCONTINUOUS ACTIONS', 'A NOTE ON PROPERLY DISCONTINUOUS ACTIONS', 'A NOTE ON PROPERLY DISCONTINUOUS ACTIONS'], 'venue': []}
arxiv
A Compactness Result for H−holomorphic Curves in Symplectizations February 17, 2022 Alexandru Doicu [email protected] Institut für Mathematik Mathematisches Institut Universität Augsburg Universität Heidelberg Urs Fuchs [email protected] Institut für Mathematik Mathematisches Institut Universität Augsburg Universität Heidelberg A Compactness Result for H−holomorphic Curves in Symplectizations February 17, 2022 H−holomorphic curves are solutions of a specific modification of the pseudoholomorphic curve equation in symplectizations involving a harmonic 1−form as perturbation term. In this paper we compactify the moduli space of H−holomorphic curves with a priori bounds on the harmonic 1−forms.A Holomorphic disks with fixed boundary 46 B Half cylinders with small energy 49 C Special coordinates 51 Introduction Let M be a closed, connected, 3−dimensional manifold and α a 1−form on M such that (M, α) is a contact manifold. Further, let X α be the Reeb vector field with respect to the contact form α on M, defined by ι X α α ≡ 1 and ι X α dα ≡ 0. Denote by ξ = ker(α) the contact structure and π α : T M → ξ the canonical projection along the Reeb vector field X α . Denote by φ α ρ the flow of X α , and note that φ α ρ preserves the contact structure and the Reeb vector field X α . Consider a dα−compatible almost complex structure J ξ : ξ → ξ, and let J be the extension of J ξ to a R−invariant almost complex structure on R × M by mapping 1 ∈ T R to X α and X α to −1 ∈ T R. M is equipped with the metric g = α ⊗ α + dα(·, J ξ ·) while we equip the R × M with the metric g = dr ⊗ dr + g (1.1) where r is the coordinate on R. Throughout this paper we assume that all periodic orbits are non-degenerate. This means that for every periodic orbit x of period T , the linear map dφ α T (x(0)) : ξ x(0) → ξ x(T ) does not contain 1 in its spectrum. Let (S, j) be a closed Riemann surface and P ⊂ S a finite subset whose elements are called "punctures". A proper and non-constant map u = (a, f) : sums the contribution of the α−energy (first term) and the dα−energy (second term) of u on S\P. The set A consists of all smooth maps ϕ : R → [0, 1] with ϕ (r) 0 for all r ∈ R. Note that if the perturbation 1−form γ vanishes, the energy of u is equal to the Hofer energy defined in [1]. This modification of the pseudoholomorphic curve equation, which was first suggested by Hofer [3], was used by Abbas et al. [4] to prove the generalized Weinstein conjecture in dimension three. However, due to a lack of a compactness result of the moduli space of H−holomorphic curves, the generalized Weinstein conjecture was proved only in the planar case, i.e. when the leaves of the holomorphic open book decomposition [14] have zero genus. In this paper we describe a compactification of the moduli space of finite energy H−holomorphic curves by imposing some additional conditions. These are outlined below. The L 2 −norm of the harmonic perturbation 1−form γ is defined by , respectively. The significance of these two quantities will become apparent later on. S\P → R × M is called H−holomorphic if π α df • j = J ξ (u) • π α df f * α • j = da + γ E(u; S\P) < ∞γ 2 L 2 (S) = S γ • j ∧ γ. The compactness result will be established for finite energy H−holomorphic curves with harmonic perturbation 1−forms having uniformly bounded L 2 −norms and uniformly bounded conformal periods and co-periods. Specifically, we will consider a sequence of H−holomorphic curves u n = (a n , f n ) : (S n \P n , j n ) → R × M with harmonic perturbations γ n , satisfying the following conditions: A1 (S n , j n ) are compact Riemann surfaces of the same genus and P n ⊂ S n is a finite set of punctures whose cardinality is independent of n. A2 The energy of u n , as well as the L 2 −norm of γ n are uniformly bounded by the constants E 0 > 0 and C 0 > 0, respectively. In [12] Bergmann introduced a model for the compactification of the moduli space of H−holomorphic curves satisfying the conditions A1 and A2. However, he neglects the phenomena occuring when conformal periods and conformal coperiods of the sequence of the harmonic perturbation 1−forms γ n are unbounded. In this case, the convergence behaviour can be very complicated. For instance, near a node, the H−holomorphic curve can follow along a Reeb trajectory which is dense in M. To avoid these complications we make additional assumptions on the conformal period and co-period. The task is to establish a notion of convergence of such curves as well as the description of the limit object similar to [1] and [8]. Essentially, we will prove the convergence of a sequence of H−holomorphic maps to a stratified broken H−holomorphic building. The concept of a stratified broken H−holomorphic building of a certain level is similar to that given in [1]. Each level consists of a nodal H−holomorphic curve having the same asymptotic properties at the positive and negative punctures. However, as compared to [1], there are two differences: 1. The nodes on each level are not just points; in our setting they are replaced by a finite length Reeb trajectory. 2. The breaking orbits between the levels have a twist in the sense made precise in Definition 14 from Section 2.3. Remark 1. For a sequence of punctured Riemann surfaces (S n , j n , P n ), the Deligne-Mumford convergence result implies that there exists a punctured nodal Riemann surface (S, j, P, D) and a sequence of diffeomorphisms ϕ n : S D,r → S n , such that ϕ * n j n converges outside certain circles in C ∞ loc to j. Here, S D,r is the surface obtained by blowing up the points from D and identifying them via the decoration r (see Section 2). Denote by Γ nod i , for i = 1, ..., |D|/2, the equivalence classes of the boundary circles of S D in S D,r . Let Γ nod n,i = (ϕ n ) * Γ nod i for all n ∈ N and i = 1, ..., |D|/2. The main result of our analysis is the following Theorem 2. Let (S n , j n , u n , P n , γ n ) be a sequence of H−holomorphic curves in R×M satisfying assumptions A1 and A2. Then there exists a subsequence that converges to a H−holomorphic curve (S, j, u, P, D, γ) in the sense of Definition 16 from Section 2.3.3. Moreover, if there exists a constant C > 0 such that for all n ∈ N and all 1 i |D|/2 we have |τ [Γ nod n,i ],γ n |, |σ [Γ nod n,i ],γ n | < C then (S, j, u, P, D, γ) is a stratified H−holomorphic building of height N and after going over to a subsequence the H−holomorphic curves (S n , j n , u n , P n , γ n ) converges to (S, j, u, P, D, γ) in the sense of Definition 18 from Section 2.3.3. For applications we would like to get rid of the a priori bounds on the L 2 −norm of γ n . Such bounds are automatically satisfied if all the leafs of the foliation, given by ker(f * α • j), are compact. Outline of the paper The paper is organized as follows. In Chapter 2 we review the basic concepts related to the compactness of H−holomorphic curves. More precisely, in Section 2.1 we recall the Deligne-Mumford convergence theorem for stable Riemann surfaces by following the analysis given in [1] and [6]. In Section 2.2 we provide the precise definition of H−holomorphic curves. By Proposition 9, we recall a result established by Hofer et al. [10] stating that the behaviour of H−holomorphic curves in a neighbourhood of the punctures is similar to that of usual pseudoholomorphic curves. This result will enable us to split the set of punctures into positive and negative punctures as in [1]. In Section 2.3 we introduce the notion of convergence and describe the limit object. In particular, the description of the limit object is given in Section 2.3.1, Definition 14, while the notion of convergence is defined in Section 2.3.3, Definition 16 and Definition 18. The limit object known as a stratified H−holomorphic building of a certain hight is similar to the broken pseudoholomorphic building introduced in [1], [6] and [8]; the difference is that we allow two points, lying in the same level, to be connected by a finite length trajectory of the Reeb vector field. Essentially, the H−holomorphic curves converge in C ∞ loc away from the punctures and certain loops that degenerate to nodes, while the projections of the H−holomorphic curves to M converge in C 0 . The proof of the main compactness results on the thick part with certain points removed, as well as on the thin part and in a neighbourhood of the removed points, are carried out in Sections 3.1 and 3.2 of Chapter 3, respectively. For the thick part, we use the Deligne-Mumford convergence and the thick-thin decomposition to show that the domains converge in the Deligne-Mumford sense to a punctured nodal Riemann surface. By using bubbling-off analysis and the results of Appendix C (to generate a sequence of holomorphic coordinates that behaves well under Deligne-Mumford limit process) we prove, after introducing additional punctures, that the H−holomorphic curves have uniformly bounded gradients in the complement of the special circles and certain marked points. By using the elliptic regularity theorem for pseudoholomorphic curves and Arzela-Ascoli theorem we show that the H−holomorphic curves togeher with the harmonic perturbations converge in C ∞ loc on the thick part with certain points removed to a H−holomorphic curve with harmonic perturbation. This result is similar to the bubbling-off analysis used in [1]. However, in contrast to Lemma 10.7 of [1], we do not change the hyperbolic structure each time after adding the additional marked point generated by the bubbling-off analysis. The thin part is decomposed into cusps corresponding to neighbourhoods of punctures and hyperbolic cylinders corresponding to nodes in the limit. As the perturbation harmonic 1−forms are exact in a neighbourhood of the punctures or the points that were removed in the first part, by means of a change of the R−coordinate, the H−holomorphic curves are turned into usual pseudoholomorphic curves on which the classical theory [1], [8] is applicable. The case of hyperbolic cylinders is more interesting because the difference from the classical SFT compactness result is evident. Due to a lack of the monotonicity lemma, we cannot expect the H−holomorphic curves to have uniformly bounded gradients, and so, to apply the classical SFT convergence theory. To deal with this problem we decompose the hyperbolic cylinder into a finite uniform number of smaller cylinders; some of them having conformal modulus tending to infinity but dα−energies strictly smaller than h, and the rest of them having bounded modulus but dα−energies possibly larger than h. Here, h > 0 is defined by h := min{|P 1 − P 2 | | P 1 , P 2 ∈ P α , P 1 = P 2 , P 1 , P 2 E 0 } (1. 6) where P α is the action spectrum of α as defined in [9] and E 0 > 0 is the uniform bound on the energy. We refer to these cylinders as cylinders of types ∞ and b 1 , respectively, and note that they appear alternately. Convergence results are derived for each cylinder type, and then glued together to obtain a convergence result on the whole hyperbolic cylinder. As cylinders of type ∞ have small dα−energies, we assume by the classical bubbling-off analysis, that the H−holomorphic curves have uniformly bounded gradients. To turn these maps into pseudoholomorphic curves, we perform a transformation by pushing them along the Reeb flow up to some specific time characterized by a uniformly bounded conformal period. These transformed curves are now pseudoholomorphic with respect to a domain-dependent almost complex structure on M, which due to the uniform boundedness of the conformal period varies in a compact set. For this part of our analysis, we use the results established in [15]. In the case of cylinders of type b 1 we proceed as follows. Relying on a bubbling-off argument, as we did in the case of the thick part, we assume that the gradient blows up only in a finite uniform number of points and remains uniformly bounded on a compact complement of them. In this compact region we use Arzelá-Ascoli theorem to show that the H−holomorphic curves together with the harmonic perturbations converge in C ∞ to some H−holomorphic curve. What is then left is the convergence in a neighbourhood of the finitely many punctures, where the gradient blows up. Here, a neighbourhood of a puncture is a disk on which the harmonic perturbation can be made exact and can be encoded in the R−coordinate of the H−holomorphic curve. By this procedure we transform the H−holomorphic curve into a usual pseudoholomorphic curve defined on a disk D. By the C ∞ −convergence on any compact complement of the punctures, established before, we assume that the transformed curves converge on an arbitrary neighbourhood of ∂D. Then we use the results of [8], especially Gromov compactness with free boundary, to obtain a convergence results for cylinders of type b 1 . This part of our analysis uses extensively the results established in Appendix A and some of [15]. In Chapter 4 we discuss the condition imposed on the conformal period and co-period, namely their uniformly boundedness. The conformal period and co-period can be seen as a link between the conformal data and the topology on the Riemann surface, as well as the harmonic perturbation 1−form. Without these conditions, the transformation performed in [15] cannot be established. The reason is that the domain dependent almost complex structure, which was constructed in order to change the H−holomorphic curve into a usual pseudoholomorphic curve, does not vary in a compact space, and so, the results established in [9] cannot be applied. By means of a counterexample stated in Proposition 40 we show that the condition on the uniform bound of the conformal period is not always satisfied. It should be pointed out that Bergmann [12] claimed to have established a compactification of the space of H−holomorphic curves by performing the same transformation as we did in [15], i.e. by pushing the M−component of the H−holomorphic curve by the Reeb flow up to some specific time determined by the conformal period, and then by assuming that the conformal period can be universally bounded by a quantity which depends only on the periods of the harmonic perturbation 1−form (note that if the L 2 −norm of a sequence of harmonic 1−forms is uniformly bounded then their periods are also uniformly bounded). In this context, Proposition 40 contradicts his argument. Definitions and main results In this chapter we present the basic concepts related to the compactness of H−holomorphic curves. In particular, we provide the Deligne-Mumford compactness in order to describe the convergence of a sequence of Riemann surfaces, introduce the concept of a stratified H−holomorphic buildings of height N, which serves as limit object, and discuss the convergence of such maps. Deligne-Mumford convergence In this section we review the Deligne-Mumford convergence following the analysis given in [1] and [6]. Consider the surface (S, j, M D), where (S, j) is a closed Riemann surface, and M and D are finite disjoint subsets of S. Assume that the cardinality of D is even. The points from M are called marked points, while the points from D are called nodal points. The points from D are organized in pairs, D = {d 1 , d 1 , d 2 , d 2 , ..., d k , d k }. A nodal surface (S, j, M D) is said to be stable if the stability condition 2g + µ 3 is satisfied for each component of the surface S, where µ = |M ∪ D|. With a nodal surface (S, j, M D) we can associate the following singular surface with double points,Ŝ D = S/{d i ∼ d i | i = 1, ..., k}. The identified points d i ∼ d i are called nodes. The nodal surface (S, j, M D) is said to be connected if the singular surfaceŜ D is connected. For each marked point p ∈ M D of a stable nodal Riemann surface (S, j, M D), we define the surface S p with boundary as the oriented blow-up of S at the point p. Thus S p is the circle compactification of S\{p}; it is a compact surface bounded by the circle Γ p = (T p S\{0})/R + . The canonical projection π : S p → S sends the circle Γ p to the point p and the maps S p \Γ p diffeomorphically to S\{p}. Similarly, given a finite set M = {p 1 , ..., p k } ⊂ M D of punctures, we consider a blow-up surface S M with k boundary components Γ 1 , ..., Γ k . It comes with the projection π : S M → S, which collapses the boundary circles Γ 1 , ..., Γ k to points p 1 , ..., p k and the maps S M \ k i=1 Γ i diffeomorphically toṠ = S\M . If the nodal surface (S, j, M D) is connected its arithmetic genus g is defined as g = 1 2 |D| − b 0 + b 0 i=1 g i + 1, where |D| = 2k is the cardinality of D, b 0 is the number of connected components of the surface S, and b 0 i=1 g i is the sum of the genera of the connected components of S. The signature of a nodal curve (S, j, M D) is the pair (g, µ), where g is the arithmetic genus and µ = |M|. A stable nodal Riemann surface (S, j, M D) is called decorated if for each node there is an orientation reversing orthogonal map r i : Γ i = (T d i S\{0})/R + → Γ i = (T d i S\{0})/R + . (2.1) For the orthogonal orientation reversing map r i , we must have that r i (e 2πiϑ p) = e −2πiϑ r(p) for all p ∈ Γ i . In the following we argue as in [1]. Consider the oriented blow-up S D at the points of D as described above. The circles Γ i and Γ i defined by (2.1) are boundary circles for the points d i , d i ∈ D. The canonical projection π : S D → S, collapsing the circles Γ i and Γ i to the points d i and d i , respectively, induces a conformal structure on S D \ k i=1 Γ i Γ i . The smooth structure of S D \ k i=1 Γ i Γ i extends to S D , while the extended conformal structure degenerates along the boundary circles Γ i and Γ i . Let (S, j, M D, r) be a decorated surface, where r = (r 1 , ..., r k ). By means of the mappings r i , i = 1, ..., k, Γ i and Γ i can be glued together to yield a closed surface S D,r . The genus of the surface S D,r is equal to the arithmetic genus of (S, j, M D). There exists a canonical projection p : S D,r →Ŝ D which projects the circle Γ i = {Γ i , Γ i } to the node d i = {d i , d i }. The projection p induces on the surface S D,r a conformal structure in the complement of the special circles Γ i ; the conformal structure is still denoted by j. The continous extension of j to S D,r degenerates along the special circles Γ i . According to the uniformization theorem, for a stable surface (S, j, M D) there exists a unique complete hyperbolic metric of constant curvature −1 of finite volume, in the given conformal class j onṠ = S\(M D). This metric is denoted by h j,M D . Each point in M D corresponds to a cusp of the hyperbolic metric h j,M D . Assume that for a given stable Riemann surface (S, j, M D), the punctured surfaceṠ = S\(M D) is endowed with the uniformizing hyperbolic metric h j,M D . Fix δ > 0, and denote by Thick δ (S, h j,M D ) = x ∈Ṡ | ρ(x) δ and Thin δ (S, h j,M D ) = x ∈Ṡ | ρ(x) < δ , the δ−thick and δ−thin parts, respectively, where ρ(x) is the injectivity radius of the metric h j,M D at the point x ∈Ṡ. A fundamental result of hyperbolic geometry states that there exists a universal constant δ 0 = sinh −1 (1) such that for any δ < δ 0 , each component C of Thin δ (S, h j,M D ) is conformally equivalent either to a finite cylinder [−R, R] × S 1 if the component C is not adjacent to a puncture, or to the punctured disk D\{0} ∼ = [0, ∞) × S 1 if it is adjacent to a puncture (see, for example, [5] and [6]). Each compact component C of the thin part contains a unique closed geodesic of length 2ρ(C) denoted by Γ C , where ρ(C) = inf x∈C ρ(x). When considering the δ−thick−thin decompositions we always assume that δ is chosen smaller than δ 0 . The uniformization metric h j,M D can be lifted to a metric h j,M D onṠ D,r := S D,r \M. The lifted metric degenerates along each circle Γ i in the sense that the length of Γ i is 0, and the distance of Γ i to any other point inṠ D,r is infinite. However, we can still speak about geodesics onṠ D,r which are orthogonal to Γ i , i.e., two geodesics rays, whose asymptotic directions at the cusps d i and d i are related via the map r i , and which correspond to a compact geodesic interval in S D,r intersecting orthogonally the circle Γ i . It is convenient to regard Thin δ (S, h j,M D ) and Thick δ (S, h j,M D ) as subsets ofṠ D,r . This interpretation provides a compactification of the non-compact components of Thin δ (S, h j,M D ) not adjacent to points from M. Any compact component C of Thin δ (S, h j,M D ) ⊂Ṡ D,r is a compact annulus; it contains either a closed geodesic Γ C , or one of the special circles, still denoted by Γ C , which projects to a node (as described above). Consider a sequence of decorated stable nodal marked Riemann surfaces (S n , j n , M n D n , r n ) indexed by n ∈ N. Definition 3. The sequence (S n , j n , M n D n , r n ) is said to converge to a decorated stable nodal surface (S, j, M D, r) if for sufficiently large n, there exists a sequence of diffeomorphisms ϕ n : S D,r → S D n ,r n n with ϕ n (M) = M n such that the following are satisfied. 1. For any n 1, the images ϕ n (Γ i ) of the special circles Γ i ⊂ S D,r for i = 1, ..., k, are special circles or closed geodesics of the metrics h j n ,M n D n onṠ D n ,r n . All special circles on S D n ,r n are among these images. 2. h n → h in C ∞ loc (Ṡ D,r \ k i=1 Γ i ), where h n := ϕ * n h j n ,M n D n and h := h j,M D . 3. Given a component C of Thin δ (S, h j,M D ) ⊂Ṡ D,r containing a special circle Γ i , and given a point c i ∈ Γ i , let δ n i be the geodesic arc corresponding to the induceed metric h n = ϕ * n h j n ,M n D n for any n 1, intersecting Γ i orthogonally at the point c i , and having the ends in the δ−thick part of the metric h n . Then, in the limit n → ∞, (C ∩ δ n i ) converge in C 0 to a continous geodesic for a metric h passing through the point c i . Remark 4. In view of the uniformization theorem, one can see that the second property of Definition 3 is equivalent to the condition ϕ * n j n → j in C ∞ loc (Ṡ D,r \ k i=1 Γ i ) which in turn, by the removable singularity theorem, is equivalent to the convergence in C ∞ loc (S D,r \ k i=1 Γ i ). We are now in the position to state the Deligne-Mumford convergence. Theorem 5. (Deligne-Mumford) Any sequence of nodal stable Riemann surfaces (S n , j n , M n D n , r n ) of signature (g, µ) has a subsequence which converges in the sense of Definition 3 to a decorated nodal stable Riemann surface (S, j, M D, r) of signature (g, µ). Corollary 6. Any sequence of stable Riemann surfaces (S n , j n , M n ) of signature (g, µ) has a subsequence which converges to a decorated nodal stable Riemann surface (S, j, M D, r) of signature (g, µ). Asymptotic behaviour of H−holomorphic curves To describe the behaviour of a H−holomorphic curve near the puncture from P we need some auxiliary tools. One of these is the lemma about the removal of singularity. Consider a H−holomorphic curve (S, j, P, u, γ), and assume that the set of punctures P ⊂ S is not empty. For p ∈ P, consider a neighbourhood U(p) = U ⊂ S, which is biholomorphic to the standard open unit disk D ⊂ C, such that, under this biholomorphism, the point p is mapped to 0. First we mention a removable singularity result for a harmonic 1−form γ defined on the punctured unit disk D\{0}. Lemma 7. If γ is a harmonic 1−form defined on the punctured disk D\{0}, and having a bounded L 2 −norm with respect to the standard complex structure i on D, i.e. γ 2 L 2 (D\{0}) < ∞ then γ can be extended accross the puncture. Proof. With z = s + it = (s, t) being the coordinates on D, we express γ as γ = f(s, t)ds + g(s, t)dt, where f, g : D\{0} → R are harmonic functions. As γ is harmonic with respect to the standard complex structure i, F := f + ig : D\{0} → C is a mermorphic function with a bounded L 2 −norm, i.e., D\{0} |F(s, t)| 2 dsdt = D\{0} |f(s, t)| 2 + |g(s, t)| 2 dsdt < ∞. Consider the Laurent series of F, F(z) = ∞ n=−∞ F n z n , where F n ∈ C. Since the Laurent series converges in C 0 loc to F and e 2πinθ is an orthonormal system in L 2 (S 1 ), we infer that for every fixed 0 < ρ < 1, 1 0 |F(ρe 2πiθ )| 2 dθ = ∞ n=−∞ |F n | 2 ρ 2n . Consequently, due to Fubini's theorem, D\{0} |F(z)| 2 dsdt = 2π (0,1]×S 1 ρ|F(ρe 2πiθ )| 2 dθdρ = 2π 1 0 ∞ n=−∞ |F n | 2 ρ 2n+1 dρ. As the terms in the sum are all non-negative, it follows that D\{0} |F(z)| 2 dsdt 2π|F n | 2 1 0 ρ 2n+1 dρ for all n ∈ Z. However, for n < 0 and because of 1 0 ρ 2n+1 dρ = ∞, this yields a contradiction to the finiteness of the L 2 −norm of F. Hence F −n = 0 for all n 1, and so, F can be extended to a holomorphic function on D. Therefore γ can be extended accross the puncture. A removable singularity result for H−holomorphic curves is the following Lemma 8. Let (D, i, {0}, u, γ) be a H−holomorphic curve defined on D\{0} such that the image of u lies in a compact subset of R × M. Then u extends continously to a H−holomorphic map on the whole disk D. Proof. Since D is contractible and dγ = d(γ • i) = 0, the harmonic perturbation γ can be written as γ = dΓ , where Γ : D → R is a harmonic function. Hence u = (a, f) := (a + Γ , f) is a pseudoholomorphic curve (unperturbed), which still has the property that its image lies in a (maybe larger) compact subset of R × M. Application of the usual removable singularity theorem (see Lemma 5.5 of [1]) finishes the proof of the lemma. In a neighbourhood of a puncture, the map a is either bounded or unbounded. In the first case, Lemma 8 can be used to extend the H−holomorphic curve across the puncture. In the second case, in which a : D\{0} → R is unbounded we have the following | of X α , where T = 0 such that lim s→∞ f(s, t) = x(T t) and lim s→∞ a(s, t) s = T in C ∞ (S 1 ) where (s, t) denote the coordinates on [0, ∞) × S 1 . Proof. As we restrict the curve to the disk, the harmonic perturbation γ is exact, i.e. there exists a harmonic function Γ defined on the open unit disk such that γ = dΓ . The new curve u = (a, f) = (a + Γ , f) is pseudoholomorphic. Let ψ : R + × S 1 → D\{0}, (s, t) → e −2π(s+it) be a biholomorphism which maps D\{0} to the cylinder R + × S 1 . We consider the pseudoholomorphic curve u as beeing defined on the cylinder R + × S 1 with finite energy and having an unbounded image in R × M. Since we assumed that X α is of Morse type, we obtain by Proposition 5.6 of [1], that there exists T = 0 and a periodic orbit x of X α of period |T | such that lim s→∞ f(s, t) = x(T t) and lim s→∞ a(s, t) s = T in C ∞ (S 1 ). By the boundedness of the harmonic function Γ , we have lim s→∞ a(s, t) s = T in C ∞ (S 1 ), and the proof is finished. The puncture p ∈ P is called positive or negative depending on the sign of the coordinate function a when approaching the puncture. Keep in mind that the holomorphic coordinates near the puncture affects only the choice of the origin on the orbit x of X α ; the parametrization of the asymptotic orbits induceed by the holomorphic polar coordinates remains otherwise the same. Hence, the orientation induced on x by the holomorphic coordinates coincides with the orientation defined by the vector field X α if and only if the puncture is positive. Let S P be the oriented blow-up of S at the punctures P = {p 1 , ..., p k } as defined in the previous section or in Section 4.3 of [1]. S P is a compact surface with boundary circles Γ 1 , ..., Γ k . Noting that each of these circles is endowed with a canonical S 1 −action and letting ϕ i : S 1 → Γ i be (up to a choice of the base point) the canonical parametrization of the boundary circle Γ i , for i = 1, ..., k, we reformulate Proposition 9 as follows. Proposition 10. Let (S, j, P, u, γ) be a H−holomorphic map without removable singularities. The map f : S → M extends to a continous map f : S P → M such that f(ϕ i (e 2πit )) = x i (T t), (2.2) where x i : S 1 = R/Z → M is a periodic orbit of the Reeb vector field X α of period T , parametrized by the vector field X α . The sign of T coincides with the sign of the puncture p i ∈ P. In a second step we define a stratified H−holomorphic building of height N. Let (S 1 , j 1 ), ..., (S N , j N ) be closed (possibly disconected) Riemann surfaces, and for any i ∈ {1, ..., N}, let P i ⊂ S i and P i ⊂ S i be the sets of negative and positive punctures on level i, respectively. There is a one-to-one correspondence between the elements P i−1 and P i given by a bijective map ϕ i : Notion of convergence For each {d i , d i } ∈ D, τ i , σ i ∈ R the points u(d i ) and u(d i ) are connected by the map [−1/2, 1/2] → R × M, s → (−2σ i s + b, φ α −2τ i s (w f )) for some b ∈ R and w f ∈ M such that u(d i ) = (σ i + b, φ α τ i (w f )) and u(d i ) = (−σ i + b, φ α −τ i (w f )).P i−1 → P i . The pair {p i−1,j , p ij }, where p ij = ϕ i (p i−1,j ) , is called the breaking point between the levels S i−1 and S i . Let P = N i=1 P i P i be the set of punctures, P i = P i P i the set of punctures at level i, D i = {d i1 , d i1 , ..., d ik i , d ik i } the set of nodes at level i, and D = N i=1 D i the set of all nodes. If S P i i is the blow-up of S i at the punctures P i = P i P i , then accounting of the splitting of the punctures P i , we denote the boundary components of S P i i by Γ i and Γ i ; they correspond to the negative and positive punctures P i and P i , respectively. There is a one-to-one correspondence between the elements of Γ i−1 and Γ i given by an orientation reversing diffeomorphism Φ i : Γ i−1 → Γ i . The pair {Γ i−1,j , Γ ij }, where Γ ij = Φ i (Γ i−1,j ) , is called a breaking orbit for all i = 2, ..., N. This gives an identification of the boundary components Γ i−1 from S P i−1 i−1 and the boundary components Γ i from S P i i . Further on, let S P,Φ := S P 1 1 ∪ Φ 2 S P 2 2 ∪ Φ 3 ... ∪ Φ N S P N N := N i=1 S P i i / ∼ where ∼ is defined by identifying the circles Γ i−1,j and Γ ij via the diffeomorphism Φ i for all i = 2, ..., N and j = 1, ..., |P i |. Obviously, S P,Φ is a compact surface with |P 1 | + |P N | boundary components. The equivalence class of Γ i−1,j in S P,Φ , denoted by Γ ij for all i = 2, ..., N and j = 1, ..., |P i |, is called a special circle; the collection of all special circles is denoted by Γ . A tuple (S, j, P, D) with the properties described above will be called a broken building of height N. We are now well prepared to introduce a stratified H−holomorphic building of height N. (S i , j i , u i , P i P i , D i , γ i , {τ ij i | j i = 1, ..., |D i |/2}, {σ ij i | j i = 1, ..., |D i |/2}) is a stratified H−holomorphic building of height 1, where u i = u| S i \P i , and j i is the complex structure on S i . 2. For all breaking points {p i−1,j , p ij } and τ ij ∈ τ, there exist T ij > 0 such that the H−holomorphic building of height 1, u i−1 :Ṡ i−1 → R × M is asymptotic at p i−1,j to a trivial cylinder over the Reeb orbit x ij of period T ij > 0, and u i :Ṡ i → R × M is asymptotic at p ij to the trivial cylinder over the Reeb orbit x ij (· + τ ij ) of period −T ij < 0. Remark 15. The energy of a stratified H−holomorphic buidling of height N is defined by E(u) = max 1 i N E α (u i ) + N i=1 E dα (u i ). Collar Blow-up In this section we introduce a version of blow-up similar to that from [1]. Let (S, j, P, D) be a broken building of height N, and consider the setting used in the previous section. In addition, let S P i ∪D i i be the blow-up of S i at the punctures P i and nodes D i . To each pair of nodes {d ij , d ij }, the corresponding boundary of S P i ∪D i i is denoted by {Γ ij , Γ ij }, and for each such pair of boundary circles, let r ij : Γ ij → Γ ij be orientation reversing diffeomorphisms. The diffeomorphisms r ij are used to glue the boundary circles Γ ij and Γ ij together. Consider the surfaceŜ := S P∪D,Φ∪r which is obtained from S by blowing-up the punctures P and the nodes D, and by using the orientation reversing diffomorphisms Φ and r ij .Ŝ is a compact surface with boundary components given by the sets Γ 1 and Γ N . The equivalence class of Γ ij inŜ is denoted by Γ nod ij and is called nodal special circles; the set of all nodal special circles is denoted by Γ nod . The collar blow-up S is a modification of the usual blow-upŜ defined in [1]. Essentially, we insert the cylinders [−1/2, 1/2] × S 1 between the special circles Γ i−1,j and Γ ij , and between Γ ij and Γ ij . To obtain a         Γ ij Γ i−1,j − 1 2 × S 1 1 2 × S 1 Φ ij Φ ijΦ ij : Γ i−1,j → {−1/2} × S 1 and Φ ij : {1/2} × S 1 → Γ i,j . surface with boundary components Γ 1 and Γ N that has the same topology asŜ we modify the orientation reversing the diffeomorphismsm Φ ij and r ij as follows: B1 The orientation reversing diffeomorphisms Φ ij correspond to two orientation reversing diffeomorphisms Φ ij : Γ i−1,j → {−1/2} × S 1 and Φ ij : {1/2} × S 1 → Γ ij for all i = 2, ..., N and j = 1, ..., |P i |. B2 Instead of glueing Γ i−1,j and Γ ij via the orientation reversing diffeomorphisms Φ ij , we glue Γ i−1,j , the cylinder [−1/2, 1/2] × S 1 , and Γ ij via the orientation reversing diffeomorphisms Φ ij and Φ ij (see Figure 2.1). B3 For the nodal special circles Γ ij and Γ ij , we proceed analogously, and denote by r ij : Γ ij → {−1/2} × S 1 and r ij : {1/2} × S 1 → Γ ij the orientation reversing diffeomorphisms that glue Γ ij , the cylinder [−1/2, 1/2] × S 1 and Γ ij together. Let S be the surface obtained by applying the above construction to all special and nodal special circles. The equivalence class of the cylinder [−1/2, 1/2] × S 1 in S corresponding to the special circle Γ ij is denoted by A ij , and is called special cylinder. The equivalence class of the cylinder [−1/2, 1/2] × S 1 in S corresponding to the nodal special circle Γ nod ij is denoted by A nod ij , and is called nodal special cylinder. The boundary circles of A ij are still denoted by Γ i−1,j and Γ ij , while the boundary circles of A nod ij are also still denoted by Γ ij and Γ ij . Finally, the collections of all special and nodal special cylinders are denoted by A and A nod , respectively. Take notice that there exists a natural projection between the collar blow-up S and the blow-up surfaceŜ, which is defined similarly to [1], i.e. it maps S\(A A nod ) diffeomorphically toŜ\(Γ Γ nod ) and the annulli A and A nod are mapped to Γ and Γ nod . This induces a conformal structure on S\(A A nod ). LetS be the closed surface obtained from S by identifying the boundary components Γ 1 and Γ N to points. Having now a stratified H−holomorphic building (S, j, u, P, D, γ, τ, σ) of height N, we define the continous extension f of f on the surface S and the continous extension a of a on S\A. The extension f may be defined on the clinders A ij and A nod ij , while the extension a is defined only on A nod ij . Set f(s, t) = φ α −2sτ ij (w f ), for all (s, t) ∈ A nod ij = [−1/2, 1/2] × S 1 , f(s, t) = φ α −(s+ 1 2 )τij (x ij (T ij t)), for all (s, t) ∈ A ij = [−1/2, 1/2] × S 1 and a(s, t) = 2σ ij s + b, for all (s, t) ∈ A nod ij = [−1/2, 1/2] × S 1 for some b ∈ R and w f ∈ M. Here x ij is the Reeb orbit of period T ij > 0. Convergence In this section we define the notion of convergence using the notation from the previous two sections. Definition 16. A sequence of H−holomorphic curves (S n , j n , u n , P n = P n P n , γ n ) converges in the C ∞ loc sense to a H−holomorphic curve (S, j, u, P, D, γ), if the tuple (S, j, P, D) is a broken building of height N and there exists a sequence of diffeomorphisms ϕ n :S → S n , whereS is the modified collar blow-up defined in Section 2.3.2, such that ϕ −1 n (P n ) = P 1 and ϕ −1 n (P n ) = P N and such that the following conditions are satisfied: 1. The sequence of complex structures (ϕ n ) * j n converges in C ∞ loc onS\(A A nod ) to j. 2. The special circles of (S n , j n , P n ) are mapped by ϕ −1 n bijectively onto {0} × S 1 of A ij or A nod ij . For every special cylinder A ij there exists an annulus A ij ∼ = [−1, 1] × S 1 such that A ij ⊂ A ij and (A ij , (ϕ n ) * j n ) and (A ij , (ϕ n ) * j n ) are conformally equivalent to ([−R n , R n ] × S 1 , i) and ([−R n + h n , R n − h n ] × S 1 , i), respectively, where R n , h n , R n /h n → ∞ as n → ∞, i is the standard complex structure and the diffeomorphisms are of the form (s, t) → (κ(s), t). 3. The H−holomorphic curves u n • ϕ n :Ṡ :=S\(P 1 P N ) → R × M together with the harmonic perturbation (ϕ n ) * γ n which are defined onS converge in C ∞ loc onṠ\ A A nod to the H−holomorphic curve u with harmonic perturbation γ. Note thatṠ\(A A nod ) may be conformally identified with S\(P D). Next we describe the C 0 −convergence. Let (S n , j n , u n , P n , γ n ) be a sequence of H−holomorphic curves. For any special circle Γ ij , let τ n ij ∈ R and σ n ij ∈ R be the conformal period of ϕ * n γ n on Γ ij with respect to the complex structure ϕ * n j n , and the conformal co-period of ϕ * n γ n on Γ ij with respect to the complex structure ϕ * n j n , respectively. For any nodal special circle Γ nod ij consider the numbersτ n ij ∈ R andσ n ij ∈ R, whereτ n ij is the conformal period of ϕ * n γ n on Γ nod ij with respect to the complex structure ϕ * n j n , andσ n ij is the conformal co-period of ϕ * n γ n on Γ nod ij with respect to the complex structure ϕ * n j n , respectively. Remark 17. For a sequence (S n , j n , u n , P n , γ n ) of H−holomorphic curves that converges to a H−holomorphic curve (S, j, u, P, D, γ) in the sense of Definition 16, the quantities τ n ij , σ n ij ,τ n ij andσ n ij can be unbounded (see, e.g, Section 4). If τ n ij , σ n ij ,τ n ij andσ n ij are bounded, then after going over to a further subsequence, and assuming that there exist the real numbers τ ij , σ ij ,τ ij ,σ ij ∈ R such that τ n ij → τ ij , (2.3) σ n ij → σ ij , (2.4) τ n ij →τ ij , (2.5) σ n ij →σ ij (2.6) as n → ∞, we are able to derive a C 0 − convergence result. The convergence of a sequence of H−holomorphic curves to a stratified H−holomorphic building of height N should be understood in the following sense: Definition 18. A sequence of H−holomorphic curves (S n , j n , P n , u n , γ n ) converges in the C 0 sense to a statified H−holomorphic building (S, j, u, P, D, γ, τ, σ) of height N if the following conditions are satisfied. 1. The parameters τ n ij , σ n ij ,τ n ij andσ n ij converge as in (2.3)-(2.6). 2. The sequence (S n , j n , P n , u n , γ n ) converges to the underlying H−holomorphic curve (S, j, u, P, D, γ) in the sense of Definition 16 with respect to a sequence of diffeomorphisms ϕ n :S → S n . 3. (S, j, u, P, D, γ, τ, σ) is a stratified H−holomorphic building of height N corresponding to the constants τ ij , τ ij andσ ij , as in Definition 14. 4. The maps u n • ϕ n converges in C 0 loc on S\A to the blow-up map u defined onṠ\A. 5. The maps f n • ϕ n converges in C 0 on S to the blow-up map f defined on S. Proof of the Compactness Theorem Let (S n , j n , u n , P n , γ n ) be a sequence of H−holomorphic curves satisfying Assumptions A1 and A2. After introducing an additional finite set of points M n disjoint from the set of punctures P n we assume that the domains (S n , j n , P n M n ) of the sequence of H−holomorphic curves are stable. This condition enables us to use the Deligne-Mumford convergence (see Section 2.1) which makes it possible to formulate a convergence result for the domains (S n , j n , P n M n ). Note that M n can be choosen in such a way that their cardinality is independent of the index n. As an additional structure, let h j n be the hyperbolic metric onṠ n := S n \(P n M n ). By the Deligne-Mumford convergence result (Corollary 6) there exists a stable nodal decorated surface (S, j, P M, D, r) and a sequence of diffeomorphisms ϕ n : S D,r → S n , where S D,r is the closed surface obtained by blowing up the nodes and glueing pairs of nodal points according to the decoration r as described in Section 2.1, such that the following holds: Let h be the hyperbolic metric on S\(P M D). The diffeomorphisms ϕ n map marked points into marked points and punctures into punctures, i.e. ϕ n (M) = M n and ϕ n (P) = P n . Via ϕ n we pull-back the complex structures j n and the hyperbolic metrics h j n , i.e. we define j (n) := ϕ * n j n on S D,r and h n := ϕ * n h j n onṠ D,r := S D,r \(M P). By the Deligne-Mumford convergence, h n → h in C ∞ loc (Ṡ D,r \ j Γ j ) as n → ∞, where Γ j are the special circles in S D,r (see Section 2.1 for the definition of special circles). This yields j (n) → j in C ∞ loc (S D,r \ j Γ j ) as n → ∞. Consider now the mapsũ n = (ã n ,f n ) := u n • ϕ n : S D,r \P → R × M andγ n := ϕ * n γ n ∈ H 1 j (n) (S D,r ) . Thenũ n is a H−holomorphic curve with harmonic perturbationγ n ; it satisfies the equation π α df n • j (n) = J(f n ) • π α df n (f * n α) • j (n) = dã n +γ n on S D,r \P and has uniformly bounded energies, i.e. for E 0 > 0 and all n ∈ N we have E(ũ n ; S D,r \P) E 0 . The L 2 −norm ofγ n over S D,r is equal to the L 2 −norm of γ n over S n and it is apparent that the L 2 −norm ofγ n is uniformly bounded by the constant C 0 > 0. Hence A1 and A2 are satisfied forũ n . In the following, we first establish a convergence result on the thick part, i.e. on S D,r away from special circles, punctures and certain additional marked points, and then treat the components from the thin part. With the convergence on the thick components, the first statement of Theorem 2 is proved. The Thick Part For the sequenceũ n : S D,r \P → R × M as defined above, we prove the C ∞ loc −convergence in the complement of the special circles and of a finite collection of points inṠ D,r := S D,r \(P M). SetṠ D,r :=Ṡ D,r \ j Γ j . To simplify the notation we continue to denote the mapsũ n by u n andγ n by γ n . The main result of this section is the following Theorem 19. There exists a subsequence of u n , still denoted by u n , a finite subset Z ⊂Ṡ D,r , and a H−holomorphic curve u :Ṡ D,r \Z → R × M with harmonic perturbation γ defined onṠ D,r with respect to the complex structure j such that u n → u and γ n → γ in C ∞ loc (Ṡ D,r \Z). Before proving Theorem 19 we establish some preliminary results. Assume that there exists a point z 1 ∈ K ⊂Ṡ D,r , where K is compact, and a sequence z n ∈ K such that z n → z 1 and du n (z n ) → ∞ as n → ∞. The next lemma describing the convergence of conformal structures on Riemann surfaces is similar to Lemma 10.7 of [1]. Lemma 20. There exist the open neighbourhoods U n (z 1 ) = U n and U(z 1 ) = U of z 1 , and the diffeomorphisms ψ n : D → U n , ψ : D → U such that 1. ψ n are i − j (n) −biholomorphisms and ψ is a i − j−biholomorphism; 2. ψ n → ψ in C ∞ loc (D) as n → ∞ with respect to the Euclidean metric on D and the hyperbolic metric h on their images; 3. ψ n (0) = z 1 for every n and ψ(0) = z 1 ; 4. z n ∈ U n for every sufficiently large n; 5. z (n) := ψ −1 n (z n ) → 0 as n → ∞. Proof. Lemma 48 applied to the compact Riemann surface with boundary K and the interior point z 1 , yields the diffeomorphisms ψ n : D → U n and ψ : D → U for which the first three assertions hold true. The fourth and fifth assertions are obvious since z n converge to z 1 . Remark 21. The coordinate maps ψ n and ψ have uniformly bounded gradients with respect to the Euclidian metric on D and the hyperbolic metric h on their images. This follows from the second assertion of Lemma 20. Let h > 0 be defined by (1.6). The next lemma essentially states that the dα−energy concentrates around the point z 1 and is at least h/2 > 0. The proof relies on bubbling-off analysis and proceeds as in Section 5.6 of [1]. Lemma 22. For every open neighbourhood U(z 1 ) = U ⊂Ṡ D,r we have 0 < h lim n→∞ E dα (u n ; U) E 0 . In particular, for each open neighbourhood U of z 1 there exists an integer N 1 ∈ N such that for all n N 1 we have E dα (u n ; U) h 2 . Proof. Consider the mapsû n := u n • ψ n : D → R × M, where ψ n are the biholomorphisms given by Lemma 20. They satisfy the H−holomorphic equations π α df n • i = J(f n ) • π α df n (f * n α) • i = dâ n +γ n on D, whereγ n := ψ * n γ n is a harmonic 1−form on D with respect to i. The energy ofû n on D is uniformly bounded as E(û n ; D) E 0 , while the L 2 −norm of the i−harmonic 1−formγ n is uniformly bounded on D by the constant C 0 . Furthermore, for z (n) := ψ −1 n (z n ), dû n (z (n) ) → ∞ as n → ∞. This can be seen as follows. If v n ∈ T z (n) D with v n eucl. = 1 is such that du n (z n ) dψ n (z (n) )v n dψ n (z (n) )v n h n g = du n (z n ) , then, dû n (z (n) )v n g = du n (z n ) dψ n (z (n) )v n dψ n (z (n) )v n h n g dψ n (z (n) )v n h n = du n (z n ) dψ n (z (n) )v n h n du n (z n ) 1 2 dψ n (z (n) ) du n (z n ) 1 4 dψ(0) → ∞ as n → ∞. The first inequality follows from the i − j (n) −holomorphicity of ψ n . Applying Hofer's topological lemma (Lemma 2.39 of [6]), we obtain another sequence {z (n) } n∈N ⊂ D withz (n) → 0, R n := dû n (z (n) ) → ∞, n 0, n R n → ∞ as n → ∞ and dû n (z) 2R n for all z ∈ D n (z (n) ). Doing rescaling we define the maps v n (z) = (b n (z), g n (z)) := â n z (n) + z R n −â n (z (n) ),f n z (n) + z R n for all z ∈ D n R n (0). The maps v n = (b n , g n ) : D n R n (0) → R × M satisfy dv n (0) = 1 and dv n (z) 2 for all z ∈ D n R n (0), and we have E α (v n ; D n R n (0)) = E α (û n ; D n (z (n) )) E α (û n ; D) and E dα (v n ; D n R n (0)) = E dα (û n ; D n (z (n) )) E dα (û n ; D) giving E(v n ; D n R n (0)) E 0 . Moreover, v n solves the H−holomorphic equations π α dg n • i = J • π α dg n , (g * n α) • i = db n + γ n , where γ n :=γ n /R n . Because v n has a bounded gradient, there exists a smooth map v : C → R × M with a bounded energy (by E 0 ) such that v n → v in C ∞ loc (C) as n → ∞. Nevertheless, becauseγ n is bounded in L 2 −norm, γ n → 0 as n → 0. Thus v = (b, g) : C → R × M is a pseudoholomorphic plane, i.e. it solves the pseudoholomorphic curve equation π α dg • i = J • π α dg, (g * α) • i = db. We prove now that the α− and dα−energies of v are bounded. Let R > 0 be arbitrary and for some τ 0 ∈ A consider D R (0) τ 0 (b)db • i ∧ db = lim n→∞ D R (0) τ 0 (b n )db n • i ∧ db n = lim n→∞ D R/Rn (z (n) ) τ 0 (â n −â n (z (n) ))dâ n • i ∧ dâ n = lim n→∞ D R/Rn (z (n) ) τ n (â n )dâ n • i ∧ dâ n lim n→∞ sup τ∈A D R/Rn (z (n) ) τ (â n )dâ n • i ∧ dâ n = lim n→∞ E α (û n ; D R/R n (z (n) )), where τ n = τ 0 (· −â n (z (n) )) is a sequence of functions that belong to A. Taking the supremum of the left-hand side over τ ∈ A, we get E α (v; D R (0)) lim n→∞ E α (û n ; D R/R n (z (n) )), while picking some arbitrary > 0, we obtain E α (v; D R (0)) lim n→∞ E α (û n ; D R/R n (z (n) )) lim n→∞ E α (û n ; D (0)). For the dα−energy, we proceed analogously: for R > 0 we have E dα (v; D R (0)) = lim n→∞ D R (0) g * n dα = lim n→∞ D R/Rn (z (n) )f * n dα, while picking some arbitrary > 0, we find E dα (v; D R (0)) = lim n→∞ D R/Rn (z (n) )f * n dα lim n→∞ D (0)f * n dα lim n→∞ E dα (û n ; D (0)). Because the α− and dα−energies are non-negative, E(v; D R (0)) = E α (v; D R (0)) + E dα (v; D R (0)) lim n→∞ E α (û n ; D (0)) + lim n→∞ E dα (û n ; D (0)) = lim n→∞ E(û n ; D (0)) E 0 , and since R > 0 was arbitrary, we obtain E dα (v; C) E 0 . As v is a usual pseudoholomorphic curve, it follows that E(v; C) = E H (v; C), where E H is the Hofer energy defined in [1]; thus E H (v; C) E 0 . Moreover, as v is non-constant we have by Remark 2.38 of [6], that for any > 0, 0 < h E dα (v; C) lim n→∞ E dα (û n ; D (0)) lim n→∞ E dα (u n ; ψ n (D (0))). Choosing > 0 such that ψ n (D (0)) ⊂ U for all n, we end up with 0 < h lim n→∞ E dα (u n ; U) E 0 , and the proof is finished. The next proposition is proved by contradiction by means of Lemma 22. Proposition 23. There exists a subsequence of u n , still denoted by u n , and a finite subset Z ⊂Ṡ D,r such that for every compact subset K ⊂Ṡ D,r \Z, there exists a constant C K > 0 such that du n (z) := sup v∈T z S D,r , v hn =1 du n (z)v g C K for all z ∈ K. Proof. For the sequence u n and any finite subset Z ⊂Ṡ D,r , we define Z {u n },Z := z ∈Ṡ D,r \Z | there exists a subsequence u n k of u n and a sequence z k ∈Ṡ D,r \Z such that z k → z and du n k (z k ) → ∞ as k → ∞ . If Z {u n },∅ is empty then the assertion is fullfilled for the sequence u n and the finite set Z = ∅. Otherwise, we choose z 1 ∈ Z {u n },∅ . In this case, there exists a sequence z 1 n ∈Ṡ D,r and a subsequence u 1 n of u n such that z 1 n → z 1 and du 1 n (z 1 n ) → ∞. Consider now the set Z {u 1 n },{z 1 } . If Z {u 1 n },{z 1 } is empty then the assertion is fullfilled for the subsequence u 1 n and the finite set Z = {z 1 }. Otherwise, we choose an element z 2 ∈ Z {u 2 n },{z 1 } . In this case, by definition, there exists a sequence z 2 n ∈Ṡ D,r \{z 1 } and a subsequence u 2 n of u 1 n such that z 2 n → z 2 and du 2 n (z 2 n ) → ∞. Let us show that the set of points Z = {z 1 , z 2 , ...} constructed in this way is finite, or more precisely, that |Z| 2E 0 / h. Assume |Z| > 2E 0 / h and pick an integer k > 2E 0 / h and pairwise different points z 1 , ..., z k ∈ Z. Let U 1 , ..., U k ⊂Ṡ D,r be some open pairwise disjoint neighbourhoods of z 1 , ..., z k . Applying Lemma 22 inductively, we deduce that there exists a positive integer N such that for every n N, E dα (u n ; U i ) h/2 for all i = 1, ..., k. Since the U i are disjoint, we obtain k h 2 k i=1 E dα (u n ; U i ) E dα (u n ;Ṡ D,r ) E 0 . Thus k 2E 0 / h which is a contradiction to our assumption. By means of Proposition 23 we can prove the convergence of the H−holomorphic maps in a punctured thick part of the Riemann surface. Proof. (of Theorem 19) For some sufficiently small k ∈ N we consider the subsets Ω k := Thick 1/k (Ṡ D,r , h)\ N i=1 D h 1/k (z i ), where Z = {z 1 , ..., z N } is the subset in Proposition 23 and D h 1/k (z i ) is the open disk around z i of radius 1/k with respect to the metric h. In order to keep the notation simple, the subsequence obtained by applying Proposition 23 is still denoted by u n . Obviously, Ω k build an exhaustion by compact sets ofṠ D,r \Z. These sets are compact surfaces with boundary. By Proposition 23, the maps u n have uniformly bounded gradients on Ω 1 . Thus after a suitable translation of the maps u n in the R−coordinate, there exists a subsequence u 1 n of u n that converges in ∞ b 1 ∞ ∞ b 1 Figure 3.1: The component of the thin part, which is biholomorphic to a cylinder, is divided in cylinders of types b 1 and ∞ in an alternating order. C ∞ (Ω 1 ) to a map u : Ω 1 → R × M. Iteratively, at step k + 1 there exists a subsequence u k+1 n of u k n that converges in C ∞ (Ω k+1 ) to a map u : Ω k+1 → R × M which is an extension from Ω k to Ω k+1 . This procedure allows us to define a map u :Ṡ D,r \Z → R × M. After passing to some diagonal subsequene u n n , the maps u n n converge in C ∞ loc (Ṡ D,r \Z) to the map u :Ṡ D,r \Z → R × M. Since the L 2 −norms of γ n are uniformly bounded on S D,r , they converge in C ∞ loc (Ṡ D,r ) to some harmonic 1−form γ with a bounded L 2 −norm onṠ D,r . Hence the map u is a H−holomorphic curve onṠ D,r \Z with harmonic perturbation γ. Convergence on the thin part and around the points from Z In this section we investigate the convergence of the H−holomorphic curves u n on the components of the thin part and in the neighbourhood of the points from Z that were constructed in Theorem 19. For a sufficient small δ > 0, the set Thin δ (Ṡ D,r , h n ) can be decomposed in two types of connected components: (I) the non-compact components that are called cusps, which are neighbourhoods of punctures with respect to the hyperbolic metric, and (II) the compact components called hyperbolic cylinders. Each of these cylinders can be biholomorphically identified with the standard cylinder [−R, R] × S 1 for a suitable R > 0. In the Deligne-Mumford limiting process R may tend to ∞ and nodes appear. For more details we refer to Chapter 1 of [6]. This section is organized as follows. First, we analyze the convergence of u n on components that can be indentified with hyperbolic cylinders, and describe the limit object. Second, we treat the convergence of u n on components that can be identified with cusps, and as before, describe the limit object. The convergence results estabished here can be used to describe the convergence of u n in a neighbourhood of the points from Z. Third, we use the description of the convergence of the H−holomorpic curves u n on the thick part (established in Section 3.1), the thin part, and in the neighbourhood of the points from Z (established in this section) to define a new surface by glueing the two parts together. On this surface we decribe the convergence of u n completely. Cylinders We analyze the convergence of u n on compact components of the thin part which are biholomorphic to hyperbolic cylinders. When restricted to these cylinders, the curves u n can have a dα−energy larger than the constant h > 0 defined in (1.6). Since we do not have a version of the monotonicity lemma in the H−holomorphic case, the classical results on the asymptotic of holomorphic cylinders from [1] and [9] are not directly applicable. To deal with this problem we shift the maps by the Reeb flow to make them pseudoholomorphic. Actually we proceed as follows. We decompose the hyperbolic cylinder into a finite uniform number of smaller cylinders; some of them having conformal modulus tending to infinity but a dα−energy strictly smaller than h, and the rest of them having bounded modulus but a dα−energy possibly larger than h. We refer to these cylinders as cylinders of types ∞ and b 1 , respectively. We consider an alternating appearance of these cylinders, as it can be seen in Figure 3.1. The convergence and the description of the limit object are first treated for cylinders of type ∞, and then for cylinders of type b 1 . As cylinders of type ∞ have a small dα−energy, we can assume, by the classical bubbling-off analysis, that the maps u n have uniformly bounded gradients. To make the curves u n pseudoholomorphic, we perform a transformation by pushing them along the Reeb flow up to some specific time. This procedure is made precise in [15]. As the gradients of these transformed curves still remain uniformly bounded, we can adapt the results of [9] to formulate a convergence result for the transformed curves (see [15]). Undoing the transformation we obtain a convergence result for the H−holomorphic curves. In the case of cylinders of type b 1 we proceed as follows. Relying on a bubbling-off argument, as we did in the case of the thick part (see Section 3.1), we assume that the gradients blow up only in a finite uniform number of points and remain uniformly bounded in a compact complement of them. In this compact region, the Arzelá-Ascoli theorem shows that the curves u n together with the harmonic perturbations γ n converge in C ∞ to some H−holomorphic curve. What is then left is the convergence in a neighbourhood of the finitely many punctures where the gradients blow up. Here, a neighbourhood of a puncture is a disk on which the harmonic perturbation can be made exact and can be encoded in the R−coordinate of the curve u n . By this procedure we transform the H−holomorphic curve into a usual pseudoholomorphic curve defined on a disk D. By the C ∞ −convergence of u n on any compact complement of the punctures, we assume that the transformed curves converge on an arbitrary neighbourhood of ∂D. This approach, which is described in detail in Section 3.2.3, uses a convergence result established in Appendix A. As for cylinders of type ∞, we undo the transformation and derive a convergence result for the H−homolorphic curves on cylinders of type b 1 . Finally, glueing all cylinders together, we are led to a convergence result for the entire component which is biholomorphic to a hyperbolic cylinder from the thin part. Let C n be a component of Thin δ (Ṡ D,r , h n ) which is conformal equivalent to the cylinder [−σ δ n , σ δ n ] × S 1 via a map ρ n : [−σ δ n , σ δ n ] × S 1 → C n . Observe that from the definition of Deligne-Mumford convergence, σ δ n → ∞ as n → ∞. In the following, we drop the fixed, sufficiently small constant δ > 0, and assume that the curves u n are defined on [−σ n , σ n ] × S 1 . Let u n = (a n , f n ) : [−σ n , σ n ] × S 1 → R × M be a sequence of H−holomorphic curves with harmonic perturbations γ n , i.e., π α df n • i = J(f n ) • π α df n , (f * n α) • i = da n + γ n on [−σ n , σ n ] × S 1 , and let us assume that the energy of u n , as well as the L 2 −norm of γ n on the cylinders are uniformly bounded, i.e. for the constants E 0 , C 0 > 0 we have E(u n ; [−σ n , σ n ]×S 1 ) E 0 and γ n 2 L 2 ([−σ n ,σ n ]×S 1 ) C 0 for all n ∈ N. Before describing the decomposition of [−σ n , σ n ] × S 1 into cylinders of types ∞ and b 1 we give a proposition which states that the C 1 −norm of the harmonic perturbation γ n is uniformly bounded. This result will play an essential role in Section 3.2.3. We set γ n = f n ds + g n dt, where f n and g n are harmonic functions defined on [−σ n , σ n ] × S 1 with coordinates (s, t) such that f n + ig n is holomorphic. By the uniform L 2 −bound of γ n , we have γ n 2 L 2 ([−σ n ,σ n ]×S 1 ) = [−σ n ,σ n ]×S 1 f 2 n + g 2 n dsdt C 0 for all n ∈ N. As a result, the L 2 −norm of the holomorphic function f n + ig n is uniformly bounded. Denote this function by G n = f n + ig n . Proposition 24. For any δ > 0 there exists a constant C δ > 0 such that G n C 1 ([−σ n +δ,σ n −δ]×S 1 ) C δ for all n ∈ N. Proof. First, we prove that the sequence G n is uniformly bounded in C 0 −norm. As G n : [−σ n , σ n ] × S 1 → C is holomorphic, f n = (G n ) and g n = (G n ) are harmonic functions defined on [−σ n , σ n ] × S 1 . For a sufficiently small δ > 0 we establish C 0 −bounds for f n on the subcylinders [−σ n + (δ/2), σ n − (δ/2)] × S 1 . By the mean value theorem for harmonic function, we have f n (p) = 16 πδ 2 D δ 4 (p) f n (s, t)dsdt for all p ∈ [−σ n + (δ/2), σ n − (δ/2)] × S 1 , where D δ/4 (p) ⊂ [−σ n , σ n ] × S 1 . Then Hölder's inequality yields |f n (p)| 4 √ πδ   D δ 4 (p) |f n (s, t)| 2 dsdt   1 2 4 √ πδ C 0 for all n ∈ N. As a results, we obtain f n C 0 ([−σn+ δ 2 ,σ n − δ 2 ]×S 1 ) 4 √ πδ C 0 , and note that the same result holds for g n . By means of bubbling-off analysis we prove now that the gradient of G n is uniformly bounded. Assume sup p∈[−σ n +δ,σ n −δ]×S 1 |∇G n (p)| → ∞ as n → ∞. Let p n ∈ [−σ n + δ, σ n − δ] × S 1 be such that |∇G n (p n )| = sup p∈[−σ n +δ,σ n −δ]×S 1 |∇G n (p)| ; then R n := |∇G n (p n )| → ∞ as n → ∞. Set n := R − 1 2 n 0 as n → ∞, and observe that n R n → ∞ as n → ∞. Choose n 0 ∈ N 0 sufficiently large such that D 10 n (p n ) ⊂ [−σ n , σ n ] × S 1 for all n n 0 . By Hofer's topologial lemma there exist n ∈ (0, n ] and p n ∈ [−σ n , σ n ] × S 1 satisfying: 1. n R n n R n ; 2. p n ∈ D 2 n (p n ) ⊂ D 10 n (p n ); 3. |∇G n (p)| 2R n , for all p ∈ D n (p n ) ⊂ D 10 n (p n ), where R n := |du n (p n )|. Via rescaling consider the mapsG n : D n R n (0) → C, defined bỹ G n (w) := G n p n + w R n for w ∈ D n R n (0). Observe that p n + (w/R n ) ∈ D n (p n ) for w ∈ D n R n (0), and that forG n we have: 1. |∇G n (0)| = 1; 2. |∇G n (w)| 2 for w ∈ D n R n (0); 3.G n is holomorphic on D n R n (0); −σ n = h 0 n h 1 n h 2 n h 3 n h 4 n h 5 n σ n = h 6 n Figure 3.2: Decomposition of [−σ n , σ n ] × S 1 into smaller cylinders [h (m) n , h (m+1) n ] × S 1 having dα−energy h/4 or less. 4.G n is uniformly bounded on [−σ n + δ, σ n − δ] × S 1 (by Assertion 1). By the usual regularity theory for pseudoholomorphic maps and Arzelá-Ascoli theorem,G n converge in C ∞ loc (C) to a bounded holomorphic mapG : C → C with |∇G(0)| = 1. By Liouville theorem this map can be only the constant map, and so, we arrive at a contradiction with |∇G(0)| = 1. Thus for δ > 0 we can replace the cylinder [−σ n +δ, σ n −δ]×S 1 by [−σ n , σ n ]×S 1 if we consider Thin δ (Ṡ D,r , h n ) for a smaller δ > 0. We come now to the decomposition of [−σ n , σ n ] × S 1 into cylinders of types ∞ and b 1 . Consider the parameter-dependent function with parameter h ∈ [−σ n , σ n ] defined by F n,h : [h, σ n ] → R, s → [h,s]×S 1 f * n dα.E dα (u n ; [−σ n , σ n ] × S 1 ) = (N n − 1) h 4 + E dα (u n ; [h (N n −1) n , h (N n ) n ] × S 1 ), which implies the following bound on N n : 0 N n 4E 0 h + 1. After going over to a subsequence, we can further assume that N n is also independent of n; for this reason, we set N n = N. Thus the cylinders [−σ n , σ n ] × S 1 have been decomposed into N smaller subcylinders [h (0) n , h(1)n ] × S 1 , ..., [h (N−1) n , h (N) n ] × S 1 on which we have E dα (u n ; [h (m−1) n , h (m) n ] × S 1 ) = h/4 for m ∈ {1, ..., N − 1} and E dα (u n ; [h (N−1) n , h (N) n ] × S 1 ) h/4. A sequence of cylinders [a n , b n ] × S 1 , where a n , b n ∈ R and a n < b n is called of type b 1 if b n − a n is bounded from above, and of type ∞ if b n − a n → ∞ as n → ∞. This is illustrated in Figure 3 du n (z) C 0 = sup v eucl =1 du n (z)v < C h for all z ∈ [h (m−1) n + h, h (m) n − h] × S 1 and n ∈ N. Proof. The proof makes use of bubbling-off analysis. Assume that there exists h > 0 such that h (m) n −h (m−1) n −2h > 0 and sup z∈[h (m−1) n +h,h (m) n −h]×S 1 du n (z) C 0 = ∞. (3.1) Then there exists a sequence z n ∈ (h (m−1) n + h, h (m) n − h) × S 1 with the property R n := du n (z n ) C 0 → ∞ as n → ∞. Let n = R1. n R n n R n ; 2. z n ∈ D 2 n (z n ) ⊂ D 10 n (z n ); 3. du n (z) C 0 2R n , for all z ∈ D n (z n ) ⊂ D 10 n (z n ), where R n := du n (z n ) C 0 . Applying rescaling consider the map v n : D n R n (0) → R × M, defined by v n (w) = (b n (w), g n (w)) := u n z n + w R n − a n (z n ) for w ∈ D n R n (0). Note that z n + (w/R n ) ∈ D n (z n ) for w ∈ D n R n (0), and that for v n we have 1. dv n (0) C 0 = 1; 2. dv n (w) C 0 2 for w ∈ D n R n (0); 3. E dα (v n ; D n R n (0)) h/4 (straightforward calculation shows that the α−energy is also uniformly bounded); 4. v n solves π α dg n • i = J • π α dg n , (g * n α) • i = db n + γ n R n on D n R n (0). As the gradients of v n are uniformly bounded, v n converge in C ∞ loc (C) to a finite energy plane v = (b, g) : C → R×M characterized by: 1. dv(0) C 0 = 1; 2. dv(w) C 0 2 for w ∈ C; 3. E dα (v; C) h/4; 4. v is a finite energy holomorphic plane. Assertion 3 follows from the fact that for an arbitray R > 0 we have E dα (v, D R (0)) = lim n→∞ E dα (v n ; D R (0)) lim n→∞ E dα (v n ; D n R n (0)) h 4 , while Assertion 4 follows from the fact that γ n has a uniformly bounded L 2 −norm. Note that by employing the above argument, a bound for the α−energy can be also obtained. Now, as v is non-constant, Theorem 31 of [2] gives E dα (v; C) h, which is a contradiction to Assertion 3. Thus Assumption (3.1) does not hold, and the gradient of u n on cylinders of type ∞ is uniformly bounded. Now we change the above decomposition so that the lengths of the cylinders of type b 1 are also bounded by below and describe the alternating appearance of cylinders of types ∞ and b 1 . This process is necessary, because on the cylinders of type b 1 whose length tends to zero we cannot analyze the convergence behaviour of the maps u n and cannot describe their limit object. We proceed as follows. Step n , h (m+1) n ] × S 1 of type ∞ is decomposed into three smaller cylinders: two cylinders [h (m) n , h (m) n + h] × S 1 , [h (m+1) n − h, h (m+1) n ] × S 1 of type b 1 and one cylinder [h (m) n + h, h (m+1) n − h] × S 1 of type ∞. The length of these two cylinders of type b 1 is h > 0. To any other cylinder of type ∞ we apply the same procedure with a fixed constant h > 0. Step 2. We combine all cylinders of type b 1 , which are next to each other, to form a bigger cylinder of type b 1 . This can be seen in Figure 3.3. By this procedure, we guarantee that in a constellation consisting of three cylinders that lie next to each other, the type of the middle cylinder is different to the types of the left and right cylinders. Thus we got rid of the cylinders of type b 1 with length tending to zero, and make sure that the cylinders of types ∞ and b 1 appear alternately. We aditionally assume that the first and last cylinders in the decomposition are of type ∞, since otherwise, we can glue the cylinder of type b 1 to the thick part of the surface and consider Thin δ (Ṡ D,r , h n ) for a smaller δ > 0. By this procedure, we decompose [−σ n , σ n ] × S 1 into cylinders of types ∞ and b 1 , while the first and last cylinders in the decomposition are of type ∞. Step 3. ForẼ 0 = 2(E 0 + C h ) and in view of the non-degeneracy of the contact manifold (M, α), let the constant h 0 be given by h 0 := min{|T 1 − T 2 | | T 1 , T 2 ∈ P α , T 1 = T 2 , T 1 , T 2 Ẽ 0 }. (3.2) Observe that because ofẼ 0 E 0 , h 0 h. If [h (m−1) n , h(m) n ]×S 1 is a cylinder of type ∞ for some m ∈ {1, ..., N}, we define the constant h 0 as above and apply Step 1 and Step 2 to decompose this cylinder into cylinders of types ∞ and b 1 , while the first and last cylinders in the decomposition are of type ∞. The cylinders of type ∞ have now a dα−energy smaller than h 0 /4. We apply this procedure to all cylinders of type ∞. In summary, [−σ n , σ n ] × S 1 is decomposed into cylinders of type ∞ with a dα−energy smaller than h 0 /4 and cylinders of type b 1 , with the first and last cylinders being of type ∞. By the above procedure, the cylinder [−σ n , σ n ] × S 1 is decomposed into an alternating constellation of cylinders of types ∞ and b 1 . On cylinders of type ∞, the dα−energy is smaller than h 0 /4, while on cylinders of type b 1 , the dα−energy can be larger than h 0 /4. By Lemma 25, the gradients of the H−holomorphic curves on the cylinders of type ∞ are uniformly bounded by the constant C h > 0 with respect to the Euclidean metric on the domain, and to the metric described in (1.1) on the target space R × M. Finally, the cylinders of types ∞ and b 1 overlap. We are now well prepared to analyse the convergence of the H−holomorphic curves on cylinders of types ∞ and b 1 . After obtaining separate convergence results, we glue the limit objects of these cylinders on the overlaps, and obtain a limit object on the whole cylinder [−σ n , σ n ] × S 1 . Sections 3.2.2 and 3.2.3 deal with the convergence and the description of the limit object on cylinders of types ∞ and b 1 , while in Section 3.2.4 we carry out the glueing of these two convergence results.     h (m) n h (m+1) n h (m+2) n h (m) n h (m+2) n b 1 b 1 b 1 sum up to Cylinders of type ∞ We describe the convergence and the limit object of the sequence of H−holomorphic curves u n , defined on cylinders of type ∞. Let m ∈ {1, ..., N} be such that [h C5 The map u n together with the 1−form γ n solve the H−holomorphic curve equation π α df n • i = J(f n ) • π α df n , (f * n α) • i = da n + γ n . C6 The harmonic 1−form γ n has a uniformly bounded L 2 −norm, i.e. for the constant C 0 > 0 we have γ n 2 L 2 ([−R (m) n ,R (m) n ]×S 1 ) C 0 for all n. C7 The map u n has a uniformly bounded gradient due to Lemma 25 and Step 4 of Section 3.2.1, i.e. for the constant C h > 0 we have du n (z) C 0 = sup v eucl =1 du n (z)v < C h for all z ∈ [−R (m) n , R(m) n ] × S 1 and all n ∈ N. C8 If P n := P γ n ({0} × S 1 ) is the period of γ n over the closed curve {0} × S 1 , as defind in (1.4), we assume that the sequence R n P n is bounded by the constant C > 0. Moreover, after going over to some subsequence, we assume that R n P n converges to some real number τ. C9 If S n := S γ n ({0} × S 1 ) is the co-period of γ n over the curve {0} × S 1 as defined in (1.5), we assume that S n R n → σ as n → ∞. n ] × S 1 . In the non-contractible case, Γ nod i also lies in the isotopy class of (ρ n • ψ n )({0} × S 1 ), and by the assumptions of Theorem 2, conditions C1-C9 are satisfied. To simplify notation we drop the index m. By Theorem 11 from [15] we consider two cases. In Case 1, there exists a subsequence of u n with vanishing center action, and we use Theorem 2 from [15] to describe the convergence of the H−holomorphic curves with harmonic perturbations γ n . In Case 2, each subsequence of u n has a center action larger than h 0 , and we use Theorem 4 from [15] to describe the convergence. Remark 27. For every sequence h n ∈ R + with h n < R n and h n , R n /h n → ∞ as n → ∞, consider a sequence of diffeomorphisms θ n : [−R n , R n ] → [−1, 1] having the following properties: 2. On [−R n + h n , R n − h n ] we define the diffeomorphism θ n to be linear by requiring Note that the diffeomorphism θ n give rise to a diffeomorphism between the cylinders [−R n , R n ]×S 1 and [−1, 1]×S 1 , according to [−R n , R n ] × S 1 → [−1, 1] × S 1 , (s, t) → (θ n (s), t). By abuse of notation these diffeomorphisms will be still denoted by θ n . Denote by u ± n (s, t) := u n (s ± R n , t) the left and right shifts of the maps u n , and by γ ± n := γ n (s ± R n , t) the left and right shifts of the harmonic perturbation, which are defined on Theorem 28. Let u n be a sequence of H−holomorphic cylinders with harmonic perturbations γ n that satisfy C1-C9 and possessing a subsequence having vanishing center action. Then there exists a subsequence of u n , still denoted by u n , the H−holomorphic cylinders u ± defined on (−∞, 0]×S 1 and [0, ∞)×S 1 , respectively, and a point w = (w a , w f ) ∈ R × M such that for every sequence h n ∈ R + and every sequence of diffeomorphisms θ n : [−R n , R n ] → [−1, 1] constructed as in Remark 27 the following C ∞ loc − and C 0 −convergence results hold (after a suitable shift of u n in the R−coordinate) C ∞ loc −convergence: θ n : Op([−R n + h n , R n − h n ]) → Op − 1 2 , 1 2 , s → s 2(R n − h n ) , 1. For any sequence s n ∈ [−R n + h n , R n − h n ] there exists τ {s n } ∈ [−τ, τ] such that after passing to a subsequence, the shifted maps u n (s + s n , t) + S n s n , defined on [−R n + h n − s n , R n − h n − s n ] × S 1 , converge in C ∞ loc to (w a , φ α −τ {sn } (w f )). The shifted harmonic perturbation 1−forms γ n (s + s n , t) posses a subsequence converging in C ∞ loc to 0. 2. The left shifts u − n (s, t) − R n S n := u n (s − R n , t) − R n S n , defined on [0, h n ) × S 1 , posses a subsequence that converges in C ∞ loc to a pseudoholomorphic half cylinder u − = (a − , f − ), defined on [0, +∞) × S 1 . The curve u − is asymptotic to (w a , φ α τ (w f )). The left shifted harmonic perturbation 1−forms γ − n converge in C ∞ loc to an exact harmonic 1−form dΓ − , defined on [0, +∞) × S 1 . Their asymptotics are 0. 3. The right shifts u + n (s, t) + R n S n := u n (s + R n , t) + R n S n , defined on (−h n , 0] × S 1 , posses a subsequence that converges in C ∞ loc to a pseudoholomorphic half cylinder u + = (a + , f + ), defined on (−∞, 0] × S 1 . The curve u + is asymptotic to (w a , φ α −τ (w f )). The right shifted harmonic perturbation 1−forms γ + n converge in C ∞ loc to an exact harmonic 1−form dΓ + , defined on (−∞, 0] × S 1 . Their asymptitics are 0. C 0 −convergence: 1. The maps v n : [−1/2, 1/2] × S 1 → R × M defined by v n (s, t) = u n (θ −1 n (s), t), converge in C 0 to (−2σs + w a , φ α −2τs (w f )). The maps v − n − R n S n : [−1, −1/2] × S 1 → R × M defined by v − n (s, t) = u n ((θ − n ) −1 (s), t), converge in C 0 to a map v − : [−1, −1/2] × S 1 → R × M such that v − (s, t) = u − ((θ − ) −1 (s), t) and v − (−1/2, t) = (w a , φ α τ (w f )). The maps v + n + R n S n : [1/2, 1] × S 1 → R × M defined by v + n (s, t) = u n ((θ + n ) −1 (s), t), converge in C 0 to a map v + : [1/2, 1] × S 1 → R × M such that v + (s, t) = u + ((θ + ) −1 (s), t) and v + (1/2, t) = (w a , φ α −τ (w f )). An immediate Corollary is Corollary 29. Unter the same hypothesis of Theorem 28 the following C ∞ loc −convergence results hold. 1. The maps v − n − R n S n converges in C ∞ loc to v − , where v − is asymptotic to (w a , φ α τ (w f )) as s → −1/2. The harmonic 1−forms [(θ − n ) −1 ] * γ − n with respect to the complex structrue [(θ − n ) −1 ] * i converges in C ∞ loc to a harmonic 1−form [(θ − ) −1 ] * dΓ − with respect to the complex structure [(θ − ) −1 ] * i which is asymptotic to some constant as s → −1/2. The maps v + n + R n S n converges in C ∞ loc to v + , where v + is asymptotic to (w a , φ α −τ (w f )) as s → 1/2. The harmonic 1−forms [(θ + n ) −1 ] * γ − n with respect to the complex structrue [(θ + n ) −1 ] * i converges in C ∞ loc to a harmonic 1−form [(θ + ) −1 ] * dΓ + with respect to the complex structure [(θ + ) −1 ] * i which is asymptotic to some constant as s → 1/2. Next we formulate the convergence in the case when there is no subsequence of u n with a vanishing center action. This result follows from Theorem 4 in [15]. Theorem 30. Let u n be a sequence of H−holomorphic cylinders with harmonic perturbations γ n satisfy C1-A9 and possesing no subsequence with vanishing center action. Then there exist a subsequence of u n , still denoted by u n , the H−holomorphic half cylinders u ± defined on (−∞, 0]×S 1 and [0, ∞)×S 1 , respectively, a periodic orbit x of period T ∈ R\{0}, and the sequences r ± n ∈ R with |r + n − r n n | → ∞ as n → ∞ such that for every sequence h n ∈ R + and every sequence of diffeomorphisms θ n : [−R n , R n ] → [−1, 1] as in Remark 27, the following convergence results hold (after a suitable shift of u n in the R−coordinate). C ∞ loc −convergence: 1. For any sequence s n ∈ [−R n + h n , R n − h n ] there exists τ {s n } ∈ [−τ, τ] such that after passing to a subsequence, the shifted maps u n (s + s n , t) − s n T − S n s n , defined on [−R n + h n − s n , R n − h n − s n ] × S 1 , converge in C ∞ loc to (T s + a 0 , φ α −τ {sn } (x(T t)) = x(T t + τ {s n } )). The shifted harmonic perturbation 1−forms γ n (s + s n , t) posses a subsequence converging in C ∞ loc to 0. The left shifts u − n (s, t) − R n S n , defined on [0, h n ) × S 1 , posses a subsequence that converges in C ∞ loc to a H−holomorphic half cylinder u − = (a − , f − ), defined on [0, +∞) × S 1 . The curve u − is asymptotic to (T s + a 0 , φ α τ (x(T t)) = x(T t + τ)) . The left shifted harmonic perturbation 1−forms γ − n converge in C ∞ loc to an exact harmonic 1−form dΓ − , defined on [0, +∞) × S 1 . Their asymptotics are 0. 3. The right shifts u + n (s, t) + R n S n , defined on (−h n , 0] × S 1 posses a subsequence that converges in C ∞ loc to a H−holomorphic half cylinder u + = (a + , f + ), defined on (−∞, 0] × S 1 . The curve u + is asymptotic to (T s + a 0 , φ α −τ (x(T t)) = x(T t − τ)). The right shifted harmonic perturbation 1−forms γ + n converge in C ∞ loc to an exact harmonic 1−form dΓ + , defined on (−∞, 0] × S 1 . Their asymptotics are 0. C 0 −convergence: 1. The maps f n • θ −1 n : [−1/2, 1/2] × S 1 → M converge in C 0 to φ α −2τs (x(T t)) = x(T t − 2τs). The maps f − n • (θ − n ) −1 : [−1, −1/2] × S 1 → M converge in C 0 to a map f − • (θ − ) −1 : [−1, −1/2] × S 1 → M such that f − ((θ − ) −1 (−1/2), t) = φ α τ (x(T t)) = x(T t + τ). 3. The maps f + n • (θ + n ) −1 : [1/2, 1] × S 1 → M converge in C 0 to a map f + • (θ + ) −1 : [1/2, 1] × S 1 → M such that f + ((θ + ) −1 (1/2), t) = φ α −τ (x(T t)) = x(T t − τ). 4. There exist C > 0, ρ > 0 and N ∈ N such that for any R > 0, a n • θ −1 n (s, t) ∈ [r − n + R − C, r + n − R + C] for all n N and all (s, t) ∈ [−ρ, ρ] × S 1 . An immediate corallary is Corollary 31. Unter the same hypothesis of Theorem 30 and the notations from Theorem 27 we have the following C ∞ loc −convergence results. 1. The maps v − n − R n S n converges in C ∞ loc to v − where f − ((θ − ) −1 (−1/2), t) = x(T t + τ). The harmonic 1−forms [(θ − n ) −1 ] * γ − n with respect to the complex structrue [(θ − n ) −1 ] * i converges in C ∞ loc to a harmonic 1−form [(θ − ) −1 ] * dΓ − with respect to the complex structure [(θ − ) −1 ] * i which is asymptotic to some constant as s → −1/2. 2. The maps v + n +R n S n converges in C ∞ loc to v + where f + ((θ + ) −1 (1/2), t) = x(T t−τ). The harmonic 1−forms [(θ + n ) −1 ] * γ − n with respect to the complex structrue [(θ + n ) −1 ] * i converges in C ∞ loc to a harmonic 1−form [(θ + ) −1 ] * dΓ + with respect to the complex structure [(θ + ) −1 ] * i which is asymptotic to some constant as s → 1/2. Since θ − : [0, ∞) × S 1 → [−1, −1/2) × S 1 is a biholomorphism with respect to the standard complex structrue i on the domain and the pull-back structureĩ := [(θ − ) −1 ] * i, we can identify [−1, −1/2) × S 1 with the punctured disk equipped with the standard complex structure, that extends over the puncture. We use now Theorems 28 and 30 to describe the limit object. In Case 1, the "limit surface" in the symplectization consists of two disks which are connected by a straight line at the origin. The limit map u = (a, f) : [−1, 1] × S 1 → R × M with the limit perturbation 1−form γ can be described as follows (see Figure 3.5). D1 On [−1, −1/2)×S 1 , u is a H−holomorphic curve with harmonic perturbation γ such that at the puncture it is asymptotic to (σ + w a , φ α τ (w f )), while the harmonic perturbation is asymptotic to a constant. D2 On (1/2, 1] × S 1 , u is a H−holomorphic curve with harmonic perturbation γ such that at the puncture it is asymptotic to (−σ + w a , φ α −τ (w f )), while the harmonic perturbation is asymptotic to a constant. D3 On the middle part [−1/2, 1/2] × S 1 , u is given by u(s, t) = (−2σs + w a , φ α −2τs (w f )). On this part the 1−form γ is not defined. h (m−1) n − 3h h (m−1) n h (m) n h (m) n + 3h D r (z 1 ) D r (z 2 ) D r (z 3 ) D r (z 4 ) D r (z 5 )z i ) contained in (h (m−1) n − h, h (m) n + h) × S 1 . Cylinders of type b 1 We analyze the convergence on cylinders of type b 1 by using the results of Appendix A. Let m ∈ {1, ..., N} be such that the cylinders [h Under these assumptions, the H−holomorphic curves converge in C ∞ on the complement of the union of these disks (centered at the punctures) to a H−holomorphic curve. What is left to prove is the convergence in each D r ; for this we use the results of Appendix A. In the final step, we glue the convergence results on the disks to the rest of the cylinder, and obtain the desired description on the entire cylinder of type b 1 . → H (m) as n → ∞. Consider the translated H−holomorphic curves u n − a n (0, 0) = (a n , f n ) − a n (0, 0) : [0, H (m) n ] × S 1 → R × M with harmonic perturbations γ n . In order to keep the notation simple, let the curve u n − a n (0, 0) be still denoted by u n . The analysis is performed in the following setting: Under the biholomorphic map [h (m−1) n − 3h, h (m) n + 3h] × S 1 → [0, H (m) n ] × S 1 ,(s, t) → (s − h (m−1) n + 3h, t), where H (m) n := h (m) n − hdu n (z) = sup v eucl. =1 du n (z)v g < C K for all z ∈ K and n ∈ N. Proof. The proof relies on the same arguments of bubbling-off analysis, which have been employed in Theorem 19 from Section 3.1 for the thick part. Pick some r > 0 such that r < h/2, and let D r (Z (m) ) consists of |Z (m) | pairwise disjoint closed disks of radius r > 0, centered at the punctures of Z (m) . Obviously, D r (Z (m) ) ⊂ (2h, H The latter result is needed to describe the convergence of the harmonic perturbations γ n on the disks D r (Z (m) ). As in the previous section, we set γ n = f n ds + g n dt, where f n and g n are harmonic functions defined on [0, H (m) n ] × S 1 such that f n + ig n are holomorphic. By the uniform L 2 −bound of γ n it follows that γ n 2 L 2 ([0,H (m) n ]×S 1 ) = [0,H (m) n ]×S 1 f 2 n + g 2 n dsdt C 0 for all n ∈ N, and so, that the L 2 −norms of the holomorphic functions f n + ig n are uniformly bounded. Letting G n = f n + ig n we state the following Proposition 33. There exists a subsequence of G n , also denoted by G n , that converges in C ∞ to some holomorphic map G defined on [0, H (m) ] × S 1 . Moreover, the harmonic perturbations γ n converge in C ∞ to a harmonic map γ. Proof. By Proposition 24, G n has a uniformly bounded C 1 −norm, while by the standard regularity results from the theory of pseudoholomorphic curves (see, for example, Section 2.2.3 of [6]), the C k derivatives of G n are also uniformly bounded. Hence, in view of Arzelá-Ascoli theorem, we can extract a subsequence that converges to some holomorphic function G. Let us analyze the convergence of the H−holomorphic curves in a neighbourhood of the punctures of Z (m) , which are given by Lemma 32. For r > 0 as above and z ∈ Z (m) , consider the closed disks D r (z) and the H−holomorphic curves u n = (a n , f n ) : D r (z) → R × M with harmonic perturbations γ n that converge in C ∞ to some harmonic 1−form γ. According to the biholomorphism D → D r (z), p → rp + z, where D is the standard closed unit disk, regard the H−holomorphic curves u n together with the harmonic perturbations as beeing defined on D instead of D r (z). The following setting is pertinent to our analysis: F1 The maps u n = (a n , f n ) : D → R × M are H−holomorphic curves with harmonic perturbations γ n with respect to the standard complex structure i on D and the almost complex structure J on ξ. F2 The maps u n = (a n , f n ) and γ n have uniformly bounded energies and L 2 −norms. F3 For any constant 1 > τ > 0, u n | A 1,τ = (a n , f n )| A 1,τ converge in C ∞ to a H−holomorphic map with harmonic perturbation γ, where A 1,τ = {z ∈ D | τ |z| 1}. As the domain of definition D is simply connected, we infer that γ n is exact, i.e. it can be written as γ n = dΓ n , whereΓ n : D → R is a harmonic function. By Condition F2,Γ n has a uniformly bounded gradient ∇Γ n in the L 2 −norm, and it is apparent that the existence ofΓ n is unique up to addition by a constant. Let us make some remarks on the choice ofΓ n and discuss some of its properties. By using the mean value theorem for harmonic functions as in Proposition 24 we conclude (after eventually, shrinking D) that the gradient ∇Γ n are uniformly bounded in C 0 . Denote by z = s + it the coordinates on D, and let K n = 1 π DΓ n (s, t)dsdt be the mean value ofΓ n , so that by the mean value theorem for harmonic functions, K n =Γ n (0). Finally, define the map Γ n (z) :=Γ n (z) −Γ n (0) which obviously satisfies γ n = dΓ n . Remark 34. From Poincaré inequality it follows that Γ n L 2 (D) c ∇Γ n L 2 (D) for some constant c > 0 and so, that Γ n is uniformly bounded in L 2 −norm. Again, by using the mean value theorem for harmonic functions, we deduce (after maybe shrinking D) that Γ n has a uniformly bounded C 0 −norm, and consequently, that Γ n has a uniformly bounded C 1 −norm. Because γ n = dΓ n is a harmonic 1−form, ∂ s Γ n + i∂ t Γ n is a holomorphic function. In this context, by Proposition 24, Γ n converge in C ∞ to a harmonic function Γ : D → R. In the following we transform the H−holomorphic curves defined on the disk in a usual pseudoholomorphic curve by encoding the harmonic perturbation γ n = dΓ n in the R−coordinate of the H−holomorphic curve u n . Specifically, we define the maps u n = (a n , f n ) = (a n + Γ n , f n ) which are obviously pseudoholomorphic. The transformation is usable if we ensure that the energy bounds are still satisfied. For an ordinary pseudoholomorphic curve, the sum of the α− and dα−energies, that are both positive, yield the Hofer energy E H (u n ; D). A uniform bound on the Hofer energy, which ensures a uniform bound on the α− and dα−energies of u n , is E H (u n ; D) = sup ϕ∈A D u * n d(ϕα) = sup ϕ∈A ∂D ϕ(a n )f * n α ∂D |f * n α| C h . Here, the last inequality follows from Condition F3, according to which, u n converge in C ∞ in a fixed neighbourhood of ∂D. Note that the constant C h is guaranteed by Lemma 32. In a next step we use the results of Appendix A to establish the convergence of the maps u n and to describe their limit object. Then we undo the transformation in the R−coordinate (more precisely, the encoding of γ n in the R−coordinate of the curve u n ) and give a convergence result together with a description of the limit object for u n . Before proceeding we state the setting corresponding to the pseudoholomorphic curves u n . G1 The maps u n = (a n , f n ) : D → R × M solve the pseudoholomorphic curve equation π α df n • i = J(f n ) • π α df n , f * n α • i = da n on D. G2 The maps u n have uniformly bounded energies. G3 For any τ > 0, u n | A 1,τ = (a n , f n )| A 1,τ converge in C ∞ to a pseudoholomorphic map. We consider two cases. In the first case, the R−components of u n are uniformly bounded, while in the second case they are not. Actually, the first case does not occur. We will prove this result in the next lemma by using standard bubbling-off analysis. Let z n ∈ D be the sequence choosen from the bubbling-off argument of Lemma 32, i.e. for which we have that du n (z n ) C 0 = sup z∈D du n (z) C 0 → ∞ (3.3) as n → ∞. Lemma 35. The R−coordinates of the maps u n are unbounded on D. Proof. We prove by contradiction using bubbling-off analysis. Assume that the R−coordinates of the maps u n are uniformly bounded. Employing the same arguments as in the proof of Lemma 25 for the sequence R n := du n (z n ) C 0 , we find that the maps v n : D n R n (0) → R × M converge in C ∞ loc (C) to a non-constant finite energy holomorphic plane v. Note that the boundedness of E dα (v; C) follows from the fact that for an arbitray R > 0 we have E dα (v, D R (0)) = lim n→∞ E dα (v n ; D R (0)) lim n→∞ E dα (v n ; D n R n (0)) C h , yielding E dα (v; C) C h . As we have assumed that the R−coordinates of u n are uniformly bounded it follows that the R−coordiantes of v n , and so, of v, are also uniformly bounded. By singularity removal, v can be extended to a pseudoholomorphic sphere. Thus the dα−energy vanishes and by the maximum principle, the function a is constant. For this reason, v must be constant and we are lead to a contradiction. We consider now the second case in which the R−coordinates of the maps u n are unbounded, and make extensively use of the results of Appendix A. By the maximum principle, the function a n tends to −∞, while by Proposition 43, the maps u n = (a n , f n ) : (D, i) → R × M converge to a broken holomorphic curve u = (a, f) : (Z, j) → R × M. Here, Z is obtained as follows. Let Z be a surface diffeomorphic to D, and let ∆ = ∆ n ∆ p ⊂ Z be a collection of finitely many disjoint loops away from ∂Z. Further on, let Z\∆ p = N+1 ν=0 Z (ν) for some N ∈ N as described in Appendix A. For a loop δ ∈ ∆ p , there exists ν ∈ {0, ..., N} such that δ is adjacent to Z (ν) and Z (ν+1) . Fix an embedded annuli A δ,ν ∼ = [−1, 1] × S 1 ⊂ Z\∆ n such that {0} × S 1 = δ, {−1} × S 1 ⊂ Z (ν) , and {1} × S 1 ⊂ Z (ν+1) . In this context, there exist a sequence of diffeomorphism ϕ n : D → Z and a sequence of negative real numbers min(a n ) = r (0) n < r (1) n < ... < r (N+1) n = −K−2, where K ∈ R is the constant determined in Appendix A and r (ν+1) n − r (ν) n → ∞ as n → ∞ such that the following hold: H1 i n := (ϕ n ) * i → j in C ∞ loc on Z\∆. H2 The sequence u n • ϕ −1 n | Z (ν) : Z (ν) → R × M converges in C ∞ loc on Z (ν) \∆ n to a punctured nodal pseudoholomorphic curve u (ν) : (Z (ν) , j) → R × M, and in C 0 loc on Z (ν) . H3 The sequence f n •ϕ −1 n : Z → M converges in C 0 to a map f : Z → M, whose restriction to ∆ p parametrizes the Reeb orbits and to ∆ n parametrizes points. H4 For any S > 0 , there exist ρ > 0 andÑ ∈ N such that a n • ϕ −1 n (s, t) ∈ [r (ν) n + S, r (ν+1) n − S] for all n Ñ and all (s, t) ∈ A δ,ν with |s| ρ. To establish a convergence result for the H−holomorphic curve u n we undo the tranformation. The maps u n are given by u n = u n − Γ n , where Γ n : D → R is the harmonic function defined in Remark 34. Observe that by Remark 34, the Γ n converge in C ∞ (D) to some harmonic function and are uniformly bounded in C 0 (D). Via the above diffeomorphisms ϕ n : D → Z, consider the functions G n := Γ n • ϕ −1 n : Z → R. Since Γ n are harmonic functions with respect to i, G n are harmonic functions on Z with respect to i n . Moreover, their gradients and absolute values are bounded in L 2 − and C 0 −norms, respectively, i.e. Z dG n • i n ∧ dG n C 0 (3.4) and G n C 0 (Z) C 1 (3.5) for some constant C 1 > 0 and for all n ∈ N, respectively. Lemma 36. For any compact subset K ⊂ (Z\∆) there exists a subsequence of G n , also denoted by G n , such that G n → G in C ∞ (K) as n → ∞, where G is a harmonic function defined on a neighbourhood of K. Proof. Let K ⊂ (Z\∆) be a compact subset. By Lemma 48 there exists a finite covering of K by the charts ψ For some r ∈ (0, 1), the following hold: 1. ψ (l) n are i − i n −biholomorphisms and ψ (l) is an i − j−biholomorphism; 2. ψ (l) n → ψ (l) in C ∞ loc (D) as n → ∞; 3. K ⊂ N l=1 ψ (l) n (D r (0)) for all n ∈ N, and K ⊂ N l=1 ψ (l) (D r (0)). n converges in C 0 (D 3r/2 (0)) to a harmonic function G (l) defined on D 3r/2 (0). By the mean value theorem for harmonic functions, there exists a constant c > 0 such that ∇G (l) n C 0 (D 4r/3 (0)) c for all n ∈ N. Hence G (l) n is uniformly bounded in C 1 (D 4r/3 (0)). Because dG (l) n defines a harmonic 1−form, ∂ s G Consider the function G (l) n + i∂ t G (l) n is a uniformly bounded holomorphic function defined on D 4r/3 (0), where s, t are the coordinates on D 4r/3 (0). By means of the Cauchy integral formula, all derivatives of ∂ s G (l) n + i∂ t G (l) n are uniformly bounded on D 5r/4 (0). From this and the fact that G (l) n converges uniformly to G (l) we deduce that there exists a further subsequence, also denoted by G (l) n , that converges in C ∞ (D 6r/5 (0)) to a harmonic function G (l) : D 6r/5 (0) → R. For n sufficiently large, ψ (l) (D r (0)) ⊂ ψ (l) (D 6r/5 (0)) and ψ (l) (D r (0)) ⊂ ψ (l) n (D 6r/5 (0)). Hence the harmonic function G n = G (l) D r (0))) to a harmonic functionG (l) := G (l) • (ψ (l) ) −1 : ψ (l) (D r (0)) → R. Obviously, if l, l ∈ {1, ..., N} are such that ψ (l) (D r (0)) ∩ ψ (l ) (D r (0)) = ∅, the uniqueness of the limit yieldsG (l) | ψ (l) (D r (0))∩ψ (l ) (D r (0)) = G (l ) | ψ (l) (D r (0))∩ψ (l ) (D r (0)) . Hence allG (l) glue together to a harmonic function defined in a neighbourhood of K. n • (ψ (l) n ) −1 : ψ (l) (D r (0)) → R converges in C ∞ (ψ (l) ( By Lemma 36 it is apparent that after going over to a diagonal subsequence, G n converges in C ∞ loc (Z\∆) to a harmonic function G : Z\∆ → R with respect to j. This shows that the H−holomorphic curve u n • ϕ −1 n | Z (ν) : Z (ν) → R × M with harmonic perturbation dG n converges in C ∞ loc on Z (ν) \∆ n to a H−holomorphic curve u (ν) : (Z (ν) , j) → R × M with harmonic perturbation dG, where u (ν) = u (ν) − G for all ν. What is left is the description of the convergence of the H−holomorphic curves u n • ϕ −1 n with harmonic perturbation dG n in a neighbourhood of the loops from ∆ n , i.e. accross the nodes from ∆ n . Observe that, from (3.5), G n is uniformly bounded on Z by the constant C 1 and the L 2 −norm of dG n is uniformly bounded by the constant C 0 . A neighbourhood C n of a loop in ∆ n can be biholomorphically parametrized as [−r n , r n ] × S 1 by the biholomorphism ψ n : [−r n , r n ] × S 1 → C n , where r n → ∞ as n → ∞. From the C 0 bound of G n on Z, the maps u n • ϕ −1 n are uniformly bounded in C 0 on C n (maybe after some shift in the R−coordinaten). Thus we consider the H−holomorphic cylinder u n • ϕ −1 n • ψ n with harmonic perturbation ψ * n dG n defined on [−r n , r n ] × S 1 . Note that the energy of u n • ϕ −1 n • ψ n is uniformly bounded by the constant E 0 . As in Section 3.2 we divide the cylinder [−r n , r n ] × S 1 into cylinders of type ∞ with an energy less than h 0 /2 and cylinders of type b 1 . We apply the result of Section 3.2.2 to cylinders of type ∞. Keep in mind that according to Remark 26), conditions C1-C9 are satisfied. For cylinders of type b 1 , the maps u n • ϕ −1 n • ψ n , after a specific shift in the R−coordinate, are contained in a compact subset of R × M. By the usual bubbling-off analysis and the maximum principle, these maps together with the harmonic perturbation converge in C ∞ on cylinders of type b 1 . We "glue" the convergence result for the ∞-type subcylinders of [−r n , r n ] × S 1 introduced in Section 3.2.2 together with the C ∞ −convergence result for the cylinders of type b 1 . This process is similar to that described in Section 3.2.4. However, we are faced with a more simple situation because we can choose the surface C (m) as the cylinder [−1 − 4h, 1 + 4h] × S 1 , where h is the constant defined in Section 3.2.1. By the method described in Section 3.2.4, the diffeomorphism ϕ n , the surface Z, and the set of nodes ∆ n having properties H1-H4 are replaced by some modified versions, which are still denoted by ϕ n , Z and ∆ n . More precisely, the loops from ∆ n are the center loops that correspond to subcylinders of type ∞ in the decomposition of the components from the thick part which are conformal equivalent to [−r n , r n ] × S 1 , where r n → ∞ as n → ∞. As in the convergence description (see Section 2.3.3), we choose the nodal special cylinders A nod around the elements from ∆ n . Remark 37. Around a puncture from Z (ν) , the H−holomorphic curve u n • ϕ n is asymptotic to a trivial cylinder over a Reeb orbit (see Section 2.2). This result is a consequence of the uniform C 0 −bound of the harmonic functions G n . We are now in the position to formulate the convergence result for the H−holomorphic curves u n with harmonic perturbation γ n defined on the disk D. There exist the diffeomorphisms ϕ n : D → Z such that the following hold: I1 i n → j in C ∞ loc on Z\∆ p A nod . I2 For every special cylinder A ij of Z there exists an annulus A ij ∼ = [−1, 1] × S 1 such that A ij ⊂ A ij and (A ij , i n ) and (A ij , i n ) are conformally equivalent to ([−R n , R n ] × S 1 , i) and ([−R n + h n , R n − h n ] × S 1 , i), respectively, where R n − h n , h n → ∞ as n → ∞, i is the standard complex structure and the diffeomorphisms are of the form (s, t) → (κ(s), t). I3 The sequence of H−holomorphic curves (D, i, u n , γ n ) with boundary converges to a stratified H−holomorphic building (Z, j, u, P, D, γ) in the sense of Definition 18 from Section 2.3.3. Moreover, the curves converge in C ∞ in a neighbourhood of the boundary ∂D. This convergence result can be applied to disks such as neighbourhoods of all points of Z (m) . To deal with the entire cylinder of type b 1 , we glue the obtained convergence result on disks centered at points of Z (m) to the complement of disk neighbourhoods of Z (m) . During the convergence description of the H−holomorphic curves u n restricted to disk neighbourhoods of the points of Z (m) , the diffeomorphism ϕ n , describing the convergence, have the property that in a neighbourhood of ∂D they are independent of n (see Appendix A). Coming back to the puncture z ∈ Z (m) we focus on the neighbourhood D r (z). Considering the translation and streching diffeomorphism D → D r (z), p → z + rp, we see that ϕ n : D r (z)\D rτ (z) → Z is independent of n; hereafter, we drop the index n and denote it by ϕ : D r (z)\D rτ (z) → Z. This map is used to glue Z and ([0, H (m) ] × S 1 )\D rτ (z) along the collar D r (z)\D rτ (z). Consider the surface C (m) = (([0, H (m) ] × S 1 )\D rτ (z)) Z ∼, where x ∼ y if and only if x ∈ D r (z)\D rτ (z), y ∈ ϕ(D r (z)\D rτ (z)) and ϕ(x) = y. This gives rise to the diffeomorphism ψ (m) n : [0, H (m) n ] × S 1 → C (m) , defined by ψ (m) n (x) = x, x ∈ C (m) \D r (z) ϕ n (x), x ∈ D r (z). We are now able to describe the convergence on cylinders of type b 1 . Let ∆ n , ∆ p and A nod be the collection of loops from C (m) obtained by the above convergence process for each point of Z (m) . Take notice that the complex structure j (m) on C (m) is given by j (m) (p) := i, p ∈ C (m) \D r (Z (m) ) j, p ∈ Z and that it is well-defined since ϕ is a biholomorphism. There exists a sequence of diffeomorphisms ψ n ] × S 1 → C (m) such that the following hold: J1 (ψ (m) n ) * i → j (m) in C ∞ loc on C (m) \∆ p A nod . J2 For every special cylinder A ij of C (m) there exists an annulus A ij ∼ = [−1, 1] × S 1 such that A ij ⊂ A ij and (A ij , i n ) and (A ij , i n ) are conformally equivalent to ([−R n , R n ] × S 1 , i) and ([−R n + h n , R n − h n ] × S 1 , i), respectively, where R n , R n − h n → ∞ as n → ∞, i is the standard complex structure and the diffeomorphisms are of the form (s, t) → (κ(s), t). n ] × S 1 , i, u n , γ n ) with boundary converges to a stratified broken H−holomorphic building (C (m) , j, u, P, D, γ) with boundary in the sense of Definition 18 from Section 2.3.3. By construction, the curves converge in C ∞ in a neighbourhood of the boundary ∂D. Glueing cylinders of type ∞ with cylinders of type b 1 By a modified version of the diffeomorphisms θ n we identify the cylinders of type ∞ with the cylinder [−1 − 4h, 1 + 4h] × S 1 where h > 0 is the constant from Lemma 25, so that after the glueing process, we end up with a bigger cylinder of finite length and a sequence of diffeomorphisms. Let us make this precedure more precise. Let [h (m−1) n , h (m) n ] × S 1 and [h (m) n − 4h, h (m+1) n + 4h] × Sθ n | ([h (m−1) n ,h (m−1) n +3h]×S 1 ) ([h (m) n −3h,h (m) n ]×S 1 ) = id, θ − | [−4h,−h]×S 1 = id, θ + | [h,4h]×S 1 = id, θ − n → θ − in C ∞ ([−4h, −h] × S 1 ), and θ + n → θ + in C ∞ ([h, 4h] × S 1 ) . We consider now the cylinders of type b 1 and note that the diffomorphisms ψ n : [h (m) n − 4h, h (m+1) n + 4h] × S 1 → C (m) have the property that ψ n | ([h (m) n −4h,h (m) n −h]×S 1 ) ([h (m+1) n −3h,h (m+1) n ]×S 1 ) = id. In this regard we consider the surface ([−1 − 4h, 1 + 4h] × S 1 ) C (m) / ∼ where x ∼ y if and only if x ∈ [h, 4h] × S 1 and y ∈ [h (m) n − 4h, h (m) n − h] × S 1 such that θ n (y) = x. By this procedure we glue all cylinders of types ∞ and b 1 , and obtain a bigger cylinder C n together with a sequence of diffeomorphisms Φ n : [−σ n , σ n ] × S 1 → C n , where [−σ n , σ n ] × S 1 is the parametrization of the δ−thin part, i.e. of Thin δ (Ṡ D,r , h n ). Let ϕ n : C n → [−σ n , σ n ]×S 1 be the conformal parametrization of the cylindrical component of Thin (Ṡ D,r , h n ). Since both ends of [−σ n , σ n ] × S 1 contain cylinders of type ∞, we infer by the above construction, that Φ n is identity near the boundary. Specifically, with the constant h > 0 we have Φ n | ([−σ n ,−σ n +3h]×S 1 ) ([σ n −3h,σ n ]×S 1 ) = id. Then we consider the surface Ṡ D,r \ϕ −1 n ([−σ n + 3h, σ n − 3h] × S 1 ) C n / ∼ where x ∼ y if and only if x ∈Ṡ D,r \ϕ −1 n ([−σ n + 3h, σ n − 3h] × S 1 ) and y ∈ C n such that Φ n • ϕ n (x) = y. In this way we handle all components of Thin δ (Ṡ D,r , h n ) that are conformal equivalent to hyperbolic cylinders. Punctures and elements of Z We analyze the convergence of u n on components of the thin part which are biholomorphic to cusps, as well as, in a neighbourhood of the points from Z. Recall that cusps correspond to neighbourhoods of punctures. Let p ∈ S D,r be a puncture or an element from Z. By Lemma 20 of Appendix C, there exist the open neighbourhoods U n and U of p, and the biholomorphisms ψ n : D → U n and ψ : D → U such that ψ n converge in C ∞ to ψ. We consider the sequence of H−holomorphic curves u n with harmonic perturbations γ n restricted to U n . By the convergence of u n on the thick part, for every open neighbourhoods U and V of p, such that V U, the H−holomorphic curves u n together with the harmonic perturbations γ n converge in C ∞ on U\V to some H−holomorphic curve u with harmonic perturbation γ. Via the biholomorphisms ψ n and ψ, we consider the H−holomorphic curves u n and the harmonic perturbations γ n as beeing defined on D\{0}. Actually, we consider the following setup: For the sequence of H−holomorphic curves u n = (a n , f n ) : D\{0} → R × M with the harmonic perturbations γ n defined on the whole disk D, the following are satisfied: K1 The energy of u n is uniformly bounded, i.e. with the constant E 0 > 0 we have E(u n ; D\{0}) E 0 for all n ∈ N. K2 The L 2 −norms of γ n are uniformly bounded, i.e. with the constant C 0 > 0 we have γ n 2 L 2 (D\{0}) C 0 for all n ∈ N. K3 For every open neighbourhoods U and V of p such that V U, the H−holomorphic curves u n with harmonic perturbations γ n converge in C ∞ on U\V to a H−holomorphic curve u with harmonic perturbation γ. We consider two cases. In the first case there exists a subsequence of u n for which the singularity at 0 is removable, i.e. the R−coordinate a n is bounded in a neighbourhood of 0, but not necessarily uniformly bounded. In particular, this case is typically for neighbourhoods of points from Z. Hence the sequence of H−holomorphic curves u n can be defined accross the puncture 0 and we end up with a sequence of H−holomorphic disks with fixed boundary. To describe the compactness we use the results of Section 3.2.3. In the second case, there exists no subsequence of the u n that has a bounded R−coordinate a n near 0. Since D is simply connected, there exists a harmonic functionΓ n : D → R such that γ n = dΓ n . By the second condition from above, the gradients ∇Γ n are uniformly bounded in L 2 −norm by the constant C 0 > 0. Denote by K n = 1 π DΓ n (x, y)dxdy the mean value ofΓ n on the disk D. Furthermore, define Γ n :=Γ n − K n ; Γ n is a harmonic function on the disk with vanishing average and satisfying γ n = dΓ n , while the gradients ∇Γ n have uniformly bounded L 2 −norms. By Poincaré inequality, the L 2 −norm of Γ n is uniformly bounded, i.e. with the constant C 0 > 0 we have Γ n L 2 (D) C 0 for all n ∈ N. Pick τ ∈ (0, 1) and denote by D τ the disk around 0 of radius τ. L3 The H−holomorphic curves u n converge in C ∞ loc to a H−holomorphic curve u with harmonic perturbation γ. L4 The harmonic perturbations γ n satisfy γ n = dΓ n , where Γ n : [0, ∞) × S 1 → R is a harmonic function with a uniformly bounded gradient ∇Γ n in L 2 −norm. Furthermore, Γ n is uniformly bounded in C 0 ([0, ∞) × S 1 ). By using the decomposition discussed in Section 3.2.1 we split the half cylinder into smaller cylinders with dα−energies smaller than h 0 /2. As described in Section 3.2.1 we end up with a sequence of finitely many cylinder of types ∞ and b 1 , and a half cylinder with a small dα−energy. The appearance of the cylinders of types b 1 and ∞ is alternating; the decomposition starts with a cylinder of type ∞ and ends with a cylinder of type b 1 followed by the half cylinder (see Figure 3.7). For the cylinders of types ∞ and b 1 we formulate the convergence results as in Sections 3.2.3 and 3.2.2. Since the harmonic 1−forms γ n are defined over the puncture p, the period of the harmonic perturbation γ n over each cylinder (either of type ∞ or type b 1 ) is 0. Hence, the converge properties of the cylinders of type ∞ are the same as in the classical theory of Hofer (see [9]), and we are left with the half cylinder having a dα−energy smaller than h 0 /2. We have the following setup: M1 u n = (a n , f n ) : [0, ∞) × S 1 → R × M is a H−holomorphic curve with harmonic perturbation γ n . M2 The energy of u n and the L 2 −norm of γ n are uniformly bounded by the constants E 0 and C 0 , respectively, while the dα−energy of u n is smaller than h 0 /2. M3 The harmonic perturbations γ n satisfy γ n = dΓ n , where Γ n : [0, ∞)×S 1 → R is a harmonic function with a uniformly bounded gradient ∇Γ n in L 2 −norm. Furthermore, Γ n is uniformly bounded in C 0 ([0, ∞) × S 1 ). M4 The gradients of u n are uniformly bounded, i.e. there exists a constantC > 0 such that du n (z) = sup v eucl. =1 du n (z)(v) g C (3.6) for all z ∈ [0, ∞) × S 1 and all n ∈ N. By bubbling-off analysis and in view of the uniformly small dα−energy, Assumption (3.6) is also valid. Moreover, by the mean value thorem for harmonic functions and the uniformly boundedness of the L 2 −norms of ∇Γ n , the harmonic perturbation γ n is uniformly bounded in C 0 on [0, ∞)×S 1 with respect to the standard Euclidean metric. We turn the H−holomorphic curve u n with harmonic perturbation γ n into a usual pseudoholomorphic curve u n by setting u n = (a n , f n ) = (a n + Γ n , f n ) as in Section 3.2.3. In the following we show that the α− and dα− energies of u n are uniformly bounded. As f n = f n we have E dα (u n ; [0, ∞) × S 1 ) = E dα (u n ; [0, ∞) × S 1 ) h 0 2 and therefore the dα−energy is uniformly small. By definition and accounting on the uniform bound on the gradients (3.6) and the uniform C 0 −bound of the harmonic 1−forms γ n , we obtain E α (u n ; [0, ∞) × S 1 ) − sup ϕ∈A [0,∞)×S 1 d(ϕ(a n )da n • i) + E dα (u n ) = − sup ϕ∈A lim r→∞ {r}×S 1 ϕ(a n )da n • i − {0}×S 1 ϕ(a n )da n • i + E dα (u n ) 2C + h 0 2 . Thus the α−energy is uniformly bounded. From the definition ofẼ 0 (see Section 3.2.1) we have E(u n ; [0, ∞)×S 1 ) Ẽ 0 for all n ∈ N. In this regard, we consider the following setup: N1 u n = (a n , f n ) : [0, ∞) × S 1 → R × M is a pseudoholomorphic curve. N2 The energy of u n is uniformly bounded, while the dα−energy of u n is uniformly smaller than h 0 /2. Using the diffeomorphism θ defined above together with the notation (B.2), and employing Theorem 47 of Appendix B we have the following Theorem 38. There exists a subsequence of u n , still denoted by u n such that the following is satisfied. 1. u n is asysmptotic to the same Reeb orbit, i.e. there exists a Reeb orbit x of period T = 0 with |T | Ẽ 0 and a sequence c n ∈ S 1 such that lim s→∞ f n (s, t) = x(T (t + c n )) and lim s→∞ a n (s, t) s = T for all n ∈ N. 2. u n converge in C ∞ loc to a pseudoholomorphic half cylinder u : [0, ∞) × S 1 → R × M having a bounded energy and a dα−energy smaller than h 0 /2. Moreover, there exists c * ∈ S 1 such that lim s→∞ f(s, t) = x(T (t + c * )) and lim s→∞ a(s, t) s = T . The maps g n = f n • θ −1 : [0, 1] × S 1 → M, where g n (1, t) = x(T (t + c n )) converge in C 0 to a map g : [0, 1] × S 1 → M, that satisfy g(1, t) = x(T (t + c * )) , where x is a Reeb orbit of period T = 0 from part 1. With this result we are in the position to formulate the convergence of the sequence of H−holomorphic half cylinders u n with harmonic perturbations γ n . Theorem 39. There exists a subsequence u n still denoted by u n such that the following is satisfied. 1. u n is asysmptotic to the same Reeb orbit, i.e. there exists a Reeb orbit x of period T = 0 with |T | Ẽ 0 and a sequence c n ∈ S 1 such that lim s→∞ f n (s, t) = x(T (t + c n )) and lim s→∞ a n (s, t) s = T for all n ∈ N. 2. u n converge in C ∞ loc to a H−holomorphic half cylinder u : [0, ∞)×S 1 → R×M with harmonic perturbation γ having a bounded energy and a dα−energy smaller than h 0 /2. Moreover, there exists c * ∈ S 1 such that lim s→∞ f(s, t) = x(T (t + c * )) and lim s→∞ a(s, t) s = T . 3. The maps g n = f n • θ −1 : [0, 1] × S 1 → M, where g n (1, t) = x(T (t + c n )) converge in C 0 to a map g : [0, 1] × S 1 → M, and satisfy g(1, t) = x(T (t + c * )), where x is a Reeb orbit of period T = 0 from part 1. Proof. Since the Γ n are uniformly bounded in C 0 −norm, the first assertion is obvious. Employing the same arguments as in [15], i.e. the mean value theorem for harmonic functions and Cauchy integral formula, we deduce that Γ n have uniformly bounded derivatives, and so, converge in C ∞ loc on [0, ∞) × S 1 to a harmonic function (s+it) we assume that the harmonic functions Γ n and Γ are defined on the disk D. Then, since the Γ n are uniformly bounded in C 0 and have gradients with uniformly bounded L 2 −norms, it follows that Γ n → Γ in C ∞ (D ρ (0)) for some 0 < ρ < 1. This shows that Γ is uniformly bounded on D and hence, via the conformal map [0, ∞) × S 1 → D\{0} it is uniformly bounded on [0, ∞) × S 1 . Thus, the second assertion is proved, and by means of f n = f n , the third assertion is evident. Γ : [0, ∞) × S 1 → R with a gradient bounded in L 2 −norm. Let us show that Γ : [0, ∞) × S 1 → R is bounded in C 0 . Via the conformal diffomorphism [0, ∞) × S 1 → D\{0}, (s, t) → e −2π By cutting a smal piece of finite length from the infinite half cylinder, we can make the cylinder preceding the infinite half cylinder to be of type b 1 . Assuming that the infinite half cylinder is of type ∞, we glue all cylinders of types ∞ and b 1 together (as described in the previous section). Via the map [0, 1) × S 1 → D\{0}, (s, t) → (1 − s)e 2πit , Discussion on conformal period In this section we analyze Condition C8 and C9 of Section 3.2.2 dealing with the boundedness of the sequence R n P n , and which can be regarded as a connection between the conformal data of the Riemann surface and the harmonic 1−forms γ n . Without this additional condition the convergence result from [15] can not be estabished. The reason is that the almost complex structure constructed on the contact manifold M might not vary in a compact interval. We show that this condition is not automatically satisfied by giving a counterexample. It should be pointed out that this example contradicts Lemma A.2 of [12]. Essentially, we will construct a sequence of harmonic 1−forms γ n on a sequence of stable Riemann surfaces, that degenerate along a single circle, have uniformly bounded L 2 −norms but unbounded P n / n , where P n denotes the period of γ n along the degenerating circle and n its length with respect to the hyperbolic metric. Observe that the quantity 1/ n is similar to R n . Let (S n , j n , M n ) be a sequence of stable Riemann surfaces of genus g, where M n ⊂ S n are finite sets of marked points with the same cardinality. Choose a basis c 1 , ..., c 2g ∈ H 1 (S n ; Z) which is independent of n. This choice is possible because all S n have genus g and are closed (they are topologically the same). By the Deligne-Mumford convergence, (S n , j n , M n ) → (S, j, M, D, r), where (S, j, M, D, r) is a decorated nodal Riemann surface. Again, according to the definition of the Deligne-Mumford convergence, there exist the diffeomorphisms ϕ n : S D,r → S n , such that j n → j on S D,r \ l j=1 Γ j or equivalently, h n → h onṠ D,r \ l j=1 Γ j where Γ j are special circles, and h n and j n are the pull-back of the complex structure and the hyperbolic metric from S n andṠ n via the diffeomorphism ϕ n . Assume that l = 1, i.e. that there exists only one degenerating geodesic in the Deligne-Mumford convergence. Denote this geodesic by Γ . Furthermore, assume that Γ = c 1 (Γ lies in the class of c 1 ). The main result of this section is the following Proposition 40. There exists a sequence of harmonic 1−forms γ n ∈ H 1 j n (S n ) with uniformly bounded L 2 −norms, periods, and co-periods, but unbounded conformal periods. Proof. Choose a sequence of harmonic 1−forms γ n ∈ H 1 j n (S D,r ) with vanishing periods except on Γ (on all of c i with i = 1 except on c 1 = Γ ). By normalization, assume that γ n L 2 (S D,r ) = 1. The uniform bounds on the L 2 −norms imply that the periods P n of γ n over Γ converge to 0. Thus γ n converge in C ∞ loc to γ on S D,r \Γ which can be seen as a harmonic 1−form on S with vanishing periods. By Hodge theory, we have γ = 0. For n sufficiently large, the L 2 −norms of γ n concentrate in the collar neighbourhood around Γ . Indeed, from 1 = γ n 2 L 2 (S D,r ) = γ n 2 L 2 (C n ) + γ n 2 L 2 (S D,r \C n ) , where C n is the cylindrical component of the δ−thin part for some sufficiently small but fixed δ > 0, it follows that S D,r \C n is contained in a compact subset of S D,r \Γ , and so, that γ n 2 L 2 (S D,r \C n ) converge to 0, and for n sufficiently large we have γ n L 2 (C n ) 1 and γ n L 2 (C n ) → 1 as n → ∞. If F n is the unique holomorphic 1−form with Re(F n ) = γ n , F n 2 L 2 (S D,r ) = i 2 S D,r F n ∧ F n . The collar C n is conformaly equivalent to [−R n , R n ] × S 1 , where R n ∼ 1/ n and n is the length of Γ with respect to h n . On C n we write γ n = f n ds + g n dt, where f n and g n are harmonic functions on the cylinder [−R n , R n ] × S 1 (s is the coordinate in [−R n , R n ] and t is the coordinate on S 1 ), express the holomorphic 1−form F n as F n = (f n − ig n )dz = (f n − ig n )(ds + idt), and note that F n L 2 (C n ) = γ n L 2 (C n ) . Consider the quantity | F n L 2 (C n ) − |b 0 | dz L 2 (C n ) |, where b 0 = −S n − iP n andS n is the co-period defined bỹ S n = Γ γ n • j n = − {0}×S 1 f n (0, t)dt. Recalling that P n = Γ γ n = {0}×S 1 g(0, t)dt, we obtain F n L 2 (C n ) − |b 0 | dz L 2 (C n ) = (f n +S n ) − i(g n − P n ) dz L 2 (C n ) (f n +S n )dz L 2 (C n ) + (g n − P n )dz L 2 (C n ) . Further calculation gives dz L 2 (C n ) = √ 2R n , (f n +S n )dz L 2 (C n ) = f n +S n L 2 ([−R n ,R n ]×S 1 ) and similarly (g n − P n )dz L 2 (C n ) = g n − P n L 2 ([−R n ,R n ]×S 1 ) . Application of Lemma 41 yields f n +S n 2 L 2 ([−R n ,R n ]×S 1 ) = R n −R n f n (s) +S n 2 L 2 (S 1 ) ds 36 R n −R n ρ 2 (s)ds max f n (±R n ) +S n 2 L 2 (S 1 ) and g n − P n 2 L 2 ([−R n ,R n ]×S 1 ) 36 R n −R n ρ 2 (s)ds max g n (±R n ) − P n 2 L 2 (S 1 ) . Using R n −R n ρ 2 (s)ds = 4(1 − e −4R n ) 4, we obtain f n +S n 2 L 2 ([−R n ,R n ]×S 1 ) 144 max f n (±R n ) +S n 2 L 2 (S 1 ) , g n − P n 2 L 2 ([−R n ,R n ]×S 1 ) 144 max g n (±R n ) − P n 2 L 2 (S 1 ) . Because the harmonic 1−forms γ n converge to 0 in C ∞ loc (S D,r \Γ ), f n (±R n ) , g n (±R n ),S n , and P n converge to zero. Hence F n L 2 (C n ) − √ 2|b 0 | R n → 0 as n → ∞. As F n L 2 (C n ) is almost 1, there exists the constants C 0 , C 1 > 0 such that C 0 1 √ 2R n |b 0 | C 1 1 √ 2R n giving C 0 1 √ 2R n P 2 n +S 2 n C 1 1 √ 2R n , or equivalently, C 0 R n 2 (P n R n ) 2 + (S n R n ) 2 C 1 R n 2 . These inequalities show that either P n R n orS n R n tend to ∞, although P n andS n stay uniformly bounded (γ n have uniformly bounded L 2 −norms). If P n R n remains uniformly bounded then we replace γ n by γ n • j n . Lemma 41. For any harmonic functions f and g on the cylinder [−R, R] × S 1 such that η = fds + gdt is a harmonic 1−form on [−R, R] × S 1 we have f(s) +S L 2 (S 1 ) 6ρ(s) max f(±R) +S L 2 (S 1 ) g(s) − P L 2 (S 1 ) 6ρ(s) max g(±R) − P L 2 (S 1 ) for all s ∈ [−R, R]. HereS and P are the co-period and the period of η, respectively, and ρ(s) 2 = 8e −2R cosh(2s). Proof. Any harmonic 1−form η defined on the cylinder [−R, R]×S 1 can be written as η = (−Sds+Pdt)+f(s, t)ds+ g(s, t)dt wheref andg are harmonic functions on [−R, R] × S 1 with vanishing average. Note that the average of f corresponds to the co-periodS and the average of g corresponds to −P. To show this, write η in the form η = f(s, t)ds + g(s, t)dt and compute the averages of f and g as 1 2R [−R,R]×S 1 f(s, t)ds ∧ dt = 1 2R [−R,R]×S 1 η ∧ dt = {0}×S 1 η • j = −S and 1 2R [−R,R]×S 1 g(s, t)ds ∧ dt = 1 2R [−R,R]×S 1 ds ∧ η = − 1 2R R −R {s}×S 1 η ds = P, respectively. Hence the 1−form η − (−Sds + Pdt) =f(s, t)ds +g(s, t)dt has vanishing average twist and vanishing periods. The Fourier series off andg in the t variable arẽ f(s, t) = a 0 (s) 2 + ∞ k=1 a k (s) cos(kt) + b k (s) sin(kt), g(s, t) = α 0 (s) 2 + ∞ k=1 α k (s) cos(kt) + β k (s) sin(kt). Sincef andg are harmonic, the Fourier expansion coefficients solve a k = k 2 a k , b k = k 2 b k , α k = k 2 α k and β k = k 2 β k for k ∈ N 0 . The solutions to these ordinaty differential equations are of the form a 0 (s) = c 0 + sd 0 , a k (s) = c k cosh(ks) + d k sinh(ks), b k (s) = e k cosh(ks) + f k sinh(ks), α 0 (s) = δ 0 + 0 s, α k (s) = δ k cosh(ks) + k sinh(ks), β k (s) = η k cosh(ks) + θ k sinh(ks). Since dη = d(η • j) = 0 we obtain ∂ tf = ∂ sg and ∂ sf = −∂ tg , giving a 0 (s) = c 0 and α 0 (s) = δ 0 . Asfds +gdt has vanishing co-period and vanishing period, we find a 0 (s) = α 0 (s) = 0, and the following relations relating the coefficients a k , b k , α k , and β k for k ∈ N: δ k = f k , k = e k , η k = −d k and θ k = −c k . Consequently, a k , b k , α k , and β k can be written as Let us expressf andg asf (s, t) = ∞ k=1 a k (s) cos(kt) + b k (s) sin(kt) = k∈Z\{0} F k (s)e 2πikt , g(s, t) = ∞ k=1 α k (s) cos(kt) + β k (s) sin(kt) = k∈Z\{0} Γ k (s)e 2πikt , where F k = 1 2 (a k − ib k ), F −k = 1 2 (a k + ib k ), Γ k = 1 2 (α k − iβ k ) , and Γ −k = 1 2 (α k + iβ k ) for k 1. From cosh(ks) cosh(kR) 3e −R cosh(s) 3ρ(s) and | sinh(ks)| | sinh(Rs)| 3e −R cosh(s) 3ρ(s), where ρ(s) 2 = 8e −2R cosh(2s), it follows that cosh(ks) 3ρ(s) cosh(kR) and | sinh(ks)| 3ρ(s) sinh(Rs). For s ∈ [0, R] × S 1 we then have f (s) 2 L 2 (S 1 ) = k∈Z\{0} |F k (s)| 2 = 1 2 ∞ k=1 (c k cosh(ks) + d k sinh(ks)) 2 + 1 2 L 2 (S 1 ) . The same inequality holds for negative s, and a similar estimate can be derived for the harmonic functiong. Thus f (s) 2 L 2 (S 1 ) 9ρ(s) 2 f (R) 2 L 2 (S 1 ) + f (−R) 2 L 2 (S 1 ) , g(s) 2 L 2 (S 1 ) 9ρ(s) 2 g(R) 2 L 2 (S 1 ) + g(−R) 2 L 2 (S 1 ) , and fromf(s, t) := f(s, t) +S andg(s, t) := g(s, t) − P, we end up with f(s) +S 2 L 2 (S 1 ) 9ρ(s) 2 f(R) +S 2 L 2 (S 1 ) + f(−R) +S 2 L 2 (S 1 ) 18ρ(s) 2 max f(R) +S 2 L 2 (S 1 ) , f(−R) +S 2 L 2 (S 1 ) , g(s) − P 2 L 2 (S 1 ) 9ρ(s) 2 g(R) − P 2 L 2 (S 1 ) + g(−R) − P 2 L 2 (S 1 ) 18ρ(s) 2 max g(R) − P 2 L 2 (S 1 ) , g(−R) − P 2 L 2 (S 1 ) . Remark 42. In [12], a notion of convergence for H−holomorphic curves is derived by using a result (Lemma A.2) which states that the conformal co-period of a harmonic 1−form on a Riemann surface can be universally controlled by its periods. Proposition 40 gives a counterexample to this statement. A Holomorphic disks with fixed boundary This appendix is devoted to the description of the convergence of pseudoholomorphic disks with fixed boundaries in symplectization, as well as, of their limit object. The results are used for proving the convergence of a cylinder of "finite length", i.e. of type b 1 as discussed in Section 3.2.3. Let u n = (a n , f n ) : D → R × M be a sequence of pseudoholomorphic curves in the symplectization R × M of the contact manifold (M, α), and being defined on the open unit disk D with respect to the standard complex structure i and the cylindrical almost complex structure J on R × M. For any τ > 0 we assume that there exists a subsequence of u n , also denoted by u n , such that u n → u (A.1) as n → ∞ in C ∞ (D\D τ (0)). Furthermore, we assume that the Hofer energy E H (u n ; D) of u n is uniformly bounded. In the following we analyze the convergence of u n . The functions a n can be supposed to be not uniformly bounded. If this is not the case, we may deduce using standard bubbling-off analysis that the gradients of u n are uniformly bounded on all of D, which in turn, implies                                                       bottom boundary      Figure A.1: Decomposition of a −1 n ((−∞, −K − 2]) that u n converge in C ∞ (D) to a pseudoholomorphic disk with finite Hofer energy. To describe the convergence and the limit object we use the arguments from [8] and [11]. However, we drop the details and explain only the strategy and mention the convergence result. As we have assumed that the R−coordinates of u n are unbounded, the maximum principle for subharmonic functions gives a n → −∞. By (A.1) we have the C ∞ −convergence of u n on an arbitrary neighbourhood of ∂D, and by a specific choice of this neighbourhood, we assume that the R−components of u n , when restricted to this neighbourhood, do not leave a fixed interval [−K, K] for some K ∈ R with K > 0. Thus from level −K − 2 we start with the decomposition of a −1 n ((−∞, −K − 2]) into cylindrical, essential and one "bottom" boundary components. This decomposition which is identical to the decomposition done in [8] and [11] is illustrated in Figure A.1. From [8] and [11] we know that there are at most N 0 ∈ N cylindrical components. In addition to the above decomposition, we add one more boundary components, namely the "upper" boundary component. This surface has two types of boundaries. The first one is the boundary ∂D which lies in a specific neighbourhood such that its image under u n belongs to [−K, K] × M. The second one is the boundary which connects certain cylindrical components. For the cylindrical, essential and bottom boundary components we use the results established in [8] and [11] to describe the convergence and the limit object. For the "upper" boundary component we use Theorem 3.2 of [8], also known as "Gromov compactness with free boundary". Here the choice of the neighbourhood of ∂D, on which the R−components of u n lie in [−K, K], plays an essential role. The existence of a special parametrization of a neighbourhood of ∂D will enable us to apply "Gromov compactness with free boundary" in the analysis of the convergence property of the upper boundary component. Essentially, the application of "Gromov compactness with free boundary", requires that the properties (A4) and (A5) under Definition 3.1 of [8] are satisfied. The following considerations ensure these conditions: Choose L 0 1 as in Remark 3.3 after Theorem 3.2 of [8]. More precisely, L 0 depends only on the genus g of the surface, the number of boundary components m, the number of marked points q, the uniform bound C on the area of the considered pseudoholomorphic curves, the constant 0 from Remark II.4.3 of [5], and the constant C ML from Lemma 3.17 of [8] (the classical monotonicity lemma). For this L 0 we write L 0 (g, m, q, C, 0 , C ML ). Further on, choose L 0 as L 0 := max {L 0 (0, 1, 2, C, 0 , C ML ), L 0 (0, 2, 1, C, 0 , C ML ), L 0 (0, 3, 0, C, 0 , C ML ), ..., L 0 (0, 2N 0 , 0, C, 0 , C ML )} . Note that when determining the constant L 0 in the first two cases, we introduce one and two artificial punctures, i.e. q = 2 or q = 1, in order to make our surface stable. Set τ 0 = e −10πL 0 and choose τ < τ 0 . In view of (A.1), assume that there exists a constant K > 0 such that u n (D\D τ (0)) ⊂ [−K, K] × M for all n ∈ N. Hence the boundary is fixed in the symplectization. The boundary region can be conformaly parametrized as follows. Consider the map β ∂D,0 : [0, 5L 0 ] × S 1 → D\D τ 0 (0), (s, t) → e −2π(s+it) . This map is obviously a conformal parametrization of the boundary region. Let now L = − ln(τ)/10π. Obviously, L L 0 and the map β ∂D : [0, 5L] × S 1 → D\D τ (0), (s, t) → e −2π(s+it) is a conformal parametrization of a neighbourhood of the boundary circle ∂D. Fix this boundary. This conformal parametrization is obviously independent of n and will be used in conjunction with "Gromov compactness with free boundary". Finally, glue the upper boundary component to the rest of the surface, and obtain the resulting limit surface together with the convergence description. To formulate the convergence result we introduce some notations. Let Z be an oriented surface diffeomorphis to the standard unit disk D and ∆ = ∆ n ∆ p ⊂ Z a collection of finitely many disjoint simple loops divided into two disjoint sets. Denote by Z ∆ n the surface obtained by collapsing the curves in ∆ n to points. Write Z * := Z ∆ n \∆ p =: Z (0) N ν=1 Z (ν) Z (N+1) as a disjoint union of components Z (ν) . Here Z (0) is the bottom boundary component which is the disjoint union of finetly many disks, while Z (N+1) is the upper boundary component whose boundary is of two types. One type is the boundary of the disk D and the other boundary components are certain loops from ∆ p . Let j be a conformal structure on Z\∆ such that (Z\∆, j) is a punctured Riemann surface together with an identification of distinct pairs of punctures given by the elements of ∆. This shows that Z * has the structure of a nodal punctured Riemann surface with a remaining identification of punctures given by the loops {δ i } i∈I = ∆ p , for some index set I. A broken pseudoholomorphic curve (with N + 2 levels) is a map F = (F (0) , F (1) , ..., F (N+1) ) : (Z * , j) → X, where X = N+1 ν=0 (R × M) such that F (ν) : (Z (ν) , j) → R × M is a punctured pseudoholomorphic curve with the additional property that F extends to a continous map F : Z → X. Here X is obtained as follows. The negative end of the compactification of R×M of the ν−th copy is glued to the positive end of the compactification of R×M of the copy ν + 1. This procedure is done for ν = 0, ..., N. For a loop δ ∈ ∆ p , there exists ν ∈ {0, ..., N} such that δ is adjacent to Z (ν) and Z (ν+1) . Fix an embedded annulli A δ,ν ∼ = [−1, 1] × S 1 ⊂ Z\∆ n such that {0} × S 1 = δ, {−1} × S 1 ⊂ Z (ν) and {1} × S 1 ⊂ Z (ν+1) . In this context, we state a convergence result which has been established in [8] and [11]. Proposition 43. The sequence of pseudoholomorphic disks u n = (a n , f n ) : (D, i) → R×M satisfying (A.1) and having a uniformly bounded Hofer energy has a subsequence that converges to a broken pseudoholomorphic curve u = (a, f) : (Z, j) → R × M with N + 2 levels in the following sense: There exists a sequence of diffomorphisms ϕ n : D → Z and a sequence of negative real numbers min(a n ) = r n → ∞ as n → ∞ such that the following hold: 1. Z with the circles ∆ collapsed to points is a nodal Riemann surface (in the sense of the above discussion, but with boundary). i n := (ϕ n ) * i → j in C ∞ loc on Z\∆. For every i ∈ I, the annulus (A i , (ϕ n ) * i) is conformally equivalent to a standard annulus [−R n , R n ] × S 1 by a diffeomorphism of the form (s, t) → (κ(s), t) with R n → ∞ as n → ∞. 2. The sequence u n • ϕ −1 n | Z (ν) : Z (ν) → R × M converges in C ∞ loc on Z (ν) \∆ n to a punctured nodal pseudoholomorphic curve u (ν) : (Z (ν) , j) → R × M, and in C 0 loc on Z (ν) . 3. The sequence f n •ϕ −1 n : Z → M converges in C 0 to a map f : Z → M, whose restriction to ∆ p parametrizes the Reeb orbits and to ∆ n parametrizes points. 4. For any S > 0 , there exist ρ > 0 and K ∈ N such that a n • ϕ −1 n (s, t) ∈ [r (ν) n + S, r (ν+1) n − S] for all n K and all (s, t) ∈ A δ,ν with |s| ρ. 5. The diffeomorphisms ϕ n • β ∂D : [0, 5L] × S 1 → Z are independent of n. B Half cylinders with small energy This appendix is devoted to the description of the convergence of a sequence of pseudoholomorphic half cylinders u n = (a n , f n ) : [0, ∞) × S 1 → R × M with uniformly bounded α− and dα−energies. More precisely, we assume that there exists a constant E 0 > 0 such that E(u n ; [0, ∞) × S 1 ) E 0 and E dα (u n ; [0, ∞) × S 1 ) h 2 , (B.1) where h > 0 is defined as in Section 3.2.1. Since the dα−energy is smaller than h/2 it follows, from the usual bubbling-off analysis, that the gradients of u n are uniformly bounded with respect to the standard Euclidian metric on the cylinder [0, ∞) × S 1 and the induced cylindrical metric on R × M. To analyze the convergence of such a sequence we use the results of Appendix A and [15]. As before we split the analysis of the convergence in two parts, namely the C ∞ loc − and the C 0 −convergence. Before stating the convergence results we need some auxiliary results similar to those from [15]. We begin with a remark on the asymptotic of a pseudoholomorphic half cylinder. Remark 44. Let u = (a, f) : [0, ∞) × S 1 → R × M be a pseudoholomorphic half cylinder with E(u; [0, ∞) × S 1 ) E 0 and E dα (u; [0, ∞) × S 1 ) h/2. To describe the behaviour of u as s → ∞, we first assume that u has a bounded image in R × M. Consider the conformal transformation h : [0, ∞) × S 1 → D\{0}, (s, t) → e −2π(s+it) . Then u • h −1 = (a • h −1 , f • h −1 ) is a pseudoholomorphic punctured disk satisfying the same assumption as u does. By the removal of singularity, u • h −1 can be defined on the whole disk D. In this case we use the results from Appendix A to describe the convergence. If u has an unbounded image in R × M, then due to Proposition 5.6 from [1], there exists T = 0 and a periodic orbit x of X α such that x is of period |T | and lim s→∞ f(s, t) = x(T t) and lim s→∞ a(s, t) s = T in C ∞ (S 1 ). To analyze the convergence of the sequence of pseudoholomorphic half cylinders u n = (a n , f n ) : [0, ∞)×S 1 → R×M we destinguish two cases. In the first case each element of a subsequence of u n , still denoted by u n , has a bounded image in the symplectization R × M. By Remark 44 we consider the sequence of pseudoholomorphic disks u n • h −1 : D → R × M having uniformly bounded energies and small dα−energies. After applying bubbling-off analysis and accounting on the uniform energy bounds as well as on the small dα−energies, we obtain a subsequence having uniform gradient bounds with respect to the Euclidian metric on the domains and the induced metric on R × M. After a specific shift in the R−coordinate, u n • h −1 converge in C ∞ to a pseudoholomorphic disk u : D → R × M. In the second case each element of a subsequence of u n , still denoted by u n , has an unbounded image in R × M. In the following we assume that after a specific shift in the R−coordinate, a n (0, 0) = 0. Before describing the convergence of u n , we prove an asymptotic result for punctures which is similar to that given in [1]. Proposition 45. After going over to a subsequence the pseudoholomorphic half cylinders u n are asysmptotic to the same Reeb orbit, i.e. there exists a Reeb orbit x and T = 0 with |T | C and a sequence c n ∈ S 1 such that lim s→∞ f n (s, t) = x(T (t+c n )) and lim s→∞ a n (s, t) s = T . Moreover, u n → u in C ∞ loc , where u is a pseudoholomorphic half cylinder u : [0, ∞) × S 1 → R × M which is asymptotic to the same Reeb orbit x(T (t+c * )) of period T as above. Here, c * ∈ S 1 and c n → c * as n → ∞. Proof. Let the sequence u n be asymptotic to some Reeb orbit. More precisely, for all n ∈ N there exist T n = 0 and a periodic orbit x n of period |T n | such that lim s→∞ f n (s, t) = x n (T n t) and lim s→∞ a n (s, t) s = T n in C ∞ (S 1 ). For simplicity, choose a subsequence of T n , also denoted by T n , which is always positive (positive puncture). Since we are in the non-degenerate case and T n E 0 , assume, after going to some subsequence, that T n = T > 0 and x n (T t) = x(T (t+c n )), where c n ∈ S 1 for all n. Thus after going over to some subsequence we may assume that c n → c * ∈ S 1 . From the uniform boundedness of the gradients of u n , the elliptic regularity, and Arzelá-Ascoli theorem, we have u n → u : [0, ∞) × S 1 → R × M in C ∞ loc . Here u is a pseuhoholomorphic half cylinder with bounded energy and a small dα−energy which is asymptotic to some periodic orbit with period T or a point; both being denoted by x . Choose the sequences N n , N n n→∞ −→ ∞ and N n < N n such that after going over to a subsequence we have lim n→∞ f n (N n , t) = x(T t) and lim n→∞ f n (N n , t) = x(T (t+c * )) in C ∞ (S 1 ), and consider the maps v n = u n | [N n ,N n ]×S 1 . which have by construction dα−energy tending to 0. Performing the same analysis as in [1] we conclude that x = x and T = T . To describe the C 0 −convergence of u n we use the results established in [15]. In view of Proposition 45, choose a sequence R n > 0 such that R n → ∞ and a n (R n , t) − T R n → 0 as n → ∞. Consider the shifted maps u n (s, t) := u n (s + R n , t) − T R n for (s, t) ∈ [−R n , R n ] × S 1 . These are pseudoholomorphic cylinders with uniformly bounded α− and dα−energies and a dα−energy smaller than h/2. Recall that these pseudoholomorphic cylinders are a special case of the H−holomorphic cylinders described in [15]. We distinguish two cases corresponding to subsequences with vanishing and non-vanshing center actions. In latter case, the cater action is greater than h > 0. By Proposition 45, the first case does not appear and we are left with the case in which A(u n ) h. By Corollary 25 of [15], for every > 0 there exists h > 0 such that for all n ∈ N and R n > h, dist g 0 (f n (s, t), x(T t + c n )) < and |a n (s, t) − T s − a 0 | < for all (s, t) ∈ [−R n + h, R n − h] × S 1 . On the other hand, we have the following result: For every > 0 there exists h > 0 such that for all n ∈ N and R n > h, dist g 0 (f n (s, t), x(T t + c n )) < and |a n (s, t) − T s − a 0 | < for all (s, t) ∈ [h, 2R n − h] × S 1 . As R n can be choosen arbitrary large the following equivalent statement readily follows: Corollary 46. For every > 0 there exist h > 0 and N ∈ N such that for all n N , dist g 0 (f n (s, t), x(T t + c n )) < and |a n (s, t) − T s − a 0 | < for all (s, t) ∈ [h, ∞) × S 1 . which by Proposition 45 converge in C ∞ loc to a map g := f • θ −1 : [0, 1) × S 1 → M. By Corollary 46, the maps g n and g can be continously extended to [0, 1] × S 1 by g n (1, t) = g(1, t) = x(T t+c n ) for all n ∈ N and all t ∈ S 1 . Hence due to Corollary 46, g n converge in C 0 to g. As a consequence, we formulate the following compactness property of the sequence of pseudoholomorphic half cylinders u n : [0, ∞) × S 1 → R × M with uniformly bounded energies and dα−energies less than h/2: F n and F are j (n) − i−holomorphic and i − i−holomorphic, respectively, they solve the equations dF n + i • dF n • j (n) = 0 and dF + i • dF • i = 0, respectively, which in turn, are equivalent to db n = −df n • j (n) and db = −df • i, respectively. By the harmonicity of f n and f, and the application of Poincare lemma on D 1 (0), we find the solutions b n and b which are unique up to addition with some constant. They can be make unique by requiring that b n (0) = 0 and b(0) = 0. In particular, we find F(x, y) = x + iy. Then we get db n → db in C ∞ loc (D 1 (0)) as n → ∞, and from b n (0) = 0 and b(0) = 0, we actually get b n → b in C ∞ loc (D 1 (0)) as n → ∞. Hence F n → F = id in C ∞ loc (D 1 (0)) as n → ∞. For n large, F n is bijective onto its image (maybe after shrinking the domain). This follows from the proof of the inverse function theorem. WithF n = F n − f n (0), the maps ψ n and ψ are defined by ψ n = c •F n : D 1 (0) → U n and ψ = c • F : D 1 (0) → U for sufficiently large n, respectively. harmonic 1−form γ ∈ H 1 j (S), i.e. dγ = d(γ • j) = 0. The energy E(u; S\P) = sup ϕ∈A S\P ϕ (a)da • j ∧ da + S\P f * dα (1.3) For an isotopy class [c] which is represented by a smooth loop c the period and co-period of γ over [c] Let R [c] the conformal modulus of [c], as defined in [8]. The conformal period and co-period of γ over [c] are τ [c],γ = R [c] P γ ([c]) and σ [c],γ = R [c] S γ ([c]) Proposition 9 . 9Let (D, i, {0}, u, γ) be a H−holomorphic curve defined on D\{0} such that the image of u is unbounded in R × M. Then u is asymptotic to a trivial cylinder over a periodic orbit of X α , i.e. after identifying D\{0} with the half open cylinder [0, ∞) × S 1 there exists a periodic orbit x of period |T H−holomorphic buildings In this section we introduce the notion of a stratified H−holomorphic building. These are the objects which are needed for the compactification of the moduli space of H−holomorphic curves. First we define a H−holomorphic building of height 1, and then we introduce the general notion of a H−holomorphic building of height greater than 1. Let (S, j) be a Riemann surface, and P ⊂ S and P ⊂ S two disjoint unordered finite subsets called the sets of negative and positive punctures, respectively. Let P = {p 1 , ..., p l }, P = {p 1 , ..., p f } and P = P P. The set of nodes, defined by D = {d 1 , d 1 , ..., d k , d k } ⊂ S, is a finite subset of S, where the significance of the pair {d i , d i } will be clarified later on. Denote by S P the blow-up of the surfaceṠ = S\P at the punctures P. The surface S P has |P| boundary components, which due to the splitting of P, are denoted by Γ = {Γ 1 , ..., Γ l } and Γ = {Γ 1 , ..., Γ f }. Definition 11. (S, j, u, P, D, γ, τ, σ), where τ = {τ i } i=1,...,|D|/2 , σ = {σ i } i=1,...,|D|/2 and τ i , σ i ∈ R for all i = 1, ..., |D|/2 is called a stratified H−holomorphic building of height 1 if the following conditions are satisfied. 1. (S, j, u, P, γ) is a H−holomorphic curve as defined in (1.2). Remark 12 . 12The M−component f :Ṡ → M of a stratified H−holomorphic building u = (a, f) :Ṡ → R × M of height 1 can be continously extended to S P . For the extension f : S P → M, it is apparent that f| Γ , where Γ = Γ Γ , defines parametrizations of Reeb orbits. Remark 13. The energy of a stratified H−holomorphic building of height 1 is the sum of the α− and dα−energies of the H−holomorphic curve, as defined in (1.3). Definition 14 . 14A tuple (S, j, u, P, D, γ, τ, σ), where τ = {τ ij i | i = 1, ..., N and j i = 1, ..., |D i |/2} ∪ {τ ij i | i = 1, ..., N − 1 and j i = 1, ..., |Γ i |}, σ = {σ ij i | i = 1, ..., N and j i = 1, ..., |D i |/2} and (S, j, P, D) is a broken building of height N, is called a stratified H−holomorphic building of height N if the following are satisfied: 1. For any i = 1, ..., N, Figure 2 . 1 : 21The glueing of Γ i−1,j , the cylinder [−1/2, 1/2] × S 1 and Γ ij via the orientation reversing diffeomorphisms ]] As f * n dα is non-negative, F n,h is positive and monotone. For the constant h defined in (1.6), we set h dα (u n ;[−σ n , σ n ] × S 1 ) < E 0 , the sequence {h (m)n } m∈N 0 has to end after N n steps, where h × S 1 , the dα−energy of u n can be smaller than h/4. Obviously, we have −σ n = h n giving E dα (u n ; × S 1 ) = h/4 for m = 1, ..., N n − 1 and E dα (u n ; × S 1 ) h/4. Hence the dα−energy can be written as . 2 . 2Lemma 25. Let [h (m−1) n , h (m) n ] × S 1 be a cylinder of type ∞ and let h > 0 be chosen small enough such that h (m) n − h (m−1) n − 2h = (h (m) n − h) − (h (m−1) n + h) > 0 for all n ∈ N. Then there exists a constant C h > 0 such that n → ∞, and observe that n R n → ∞ as n → ∞. Choose n 0 ∈ N sufficiently large such that D 10 n (z n ) ⊂ [h × S 1 for all n n 0 . By Hofer's topological lemma, there exist n ∈ (0, n ] and z n ∈ [h Step 4 . 4We enlarge the cylinders of type b 1 without changing their type. Let h > 0 be as in Lemma 25 and pick m ∈ {1, ..., N} such that [h (m−1) n , h (m) n ] × S 1 is of type b 1 . For n sufficiently large, we replace the cylinder [h (m−1) n , h (m) n ] × S 1 by the bigger cylinder [h (m−1) n − 3h, h (m) n + 3h] × S 1 , and apply this procedure Figure 3 . 3 :Figure 3 . 4 :]+] 3334Two cylinders of type b 1 are combined to form a bigger cylinder of type b 1 . Decomposition of [−σ n , σ n ] × S 1 into cylinders of types ∞ and b 1 in an alternating order. to all cylinders of type b 1 . As a result, neighbouring cylinders will overlap. Essentially, this means that if [h × S 1 is a cylinder of type ∞, which lies to the left of a cylinder [h 3h] × S 1 of type b 1 , then their intersection is [h × S 1 . This can be seen in Figure 3.4. × S 1 is a cylinder of type ∞ as described in Section 3.2.1, i.e. h →n ∞ as n → ∞. Consider the diffeomorphism ψ n : the H−holomorphic maps u n = (a n , f n ) : ] × S 1 → R × M with harmonic perturbation γ n . For deriving a C ∞ loc −convergence result we consider the following setting:C1 R (m) n → ∞ as n → ∞. × S 1 with respect to the standard complex structure i, i.e. dγ n = dγ n • i = 0.C3 The dα−energy of u n is uniformly small, i.e. E dα (u n ; × S 1 ) h 0 /2 for all n, where h 0 is the constant defined in(3.2).C4The energy of u n is uniformly bounded, i.e. for the constant E 0 > 0 we have E(u n ; × S 1 ) E 0 for all n ∈ N. Remark 26 . 26The special circles Γ nod i in Remark 1 are of two types: contractible and non-contractible. In the contractible case, Γ nod i lies in the isotopy class of (ρ n • ψ n )({0} × S 1 ) and the conformal periods and co-periods of the harmonic 1−forms γ n vanish. Hence, conditions C1-C9 are satisfied on the sequence of degenerating cylinders [ 1 . 1The left and right shifts θ ± n (s) := θ n (s ± R n ) defined on [0, h n ] → [−1, −1/2] and [−h n , 0] → [1/2, 1], respectively, converge in C ∞ loc to the diffeomorphisms θ − : [0, ∞) → [−1, −1/2) and θ + : (−∞, 0] → (1/2, 1], respectively. [ 0 , 0h n ] × S 1 and [−h n , 0] × S 1 , respectively. In both cases we use the diffeomorphisms θ n to pull the structures back to the cylinder [−1, 1]×S 1 . Let i n := dθ n •i•dθ −1 n be the induced complex structure on [−1, 1]×S 1 . Then u n •θ −1 n : [−1, 1]×S 1 → R×M is a sequence of H−holomorphic curves with harmonic perturbations (θ −1 n ) * γ n with respect to the complex structure i n on [−1, 1] × S 1 and the cylindrical almost complex structure J on the target space R × M. From the result θ −1 n (s) = (θ − n ) −1 (s) − R n , and the fact that θ − n and θ + n converge in C ∞ loc to θ − on [−1, −1/2) and θ + on (1/2, 1], respectively, it follows that the complex structures i n converge in C ∞ loc to a complex structureĩ on [−1, −1/2) × S 1 and (1/2, 1] × S 1 . First, we formulate the convergence in the case when there exists a subsequence of u n , still denoted by u n , with a vanishing center action. In Case 2, the limit surface is the disjoint union of the cylinders [−1, −1/2)×S 1 and (1/2, 1]×S 1 . The H−holomorphic curve u = (a, f) : ([−1, −1/2) (1/2, 1]) × S 1 → R × M with harmonic perturbation γ can be described as follows. D1' u is asymptotic on [−1, −1/2) × S 1 and (1/2, 1] × S 1 to a trivial cylinder over the Reeb orbit x(T t + τ) or x(T t − τ), respectively, while the harmonic perturbation is asymptotic to a constant. D2' On the middle part [−1/2, 1/2] × S 1 , the M−component f is given by f(s, t) = x(T t − 2τs). Figure 3 . 5 : 35The limit surface consists of two cones connected by a straight line. Figure 3 . 6 : 36The gradient might blow up on the discs D r ( ++n+ 3h] × S 1 are of type b 1 . By the construction described in the previous section and Lemma 25, the H−holomorphic curves have uniform gradient bounds on the two boundary cylinders[h 3h] × S 1 .The convergence analysis is organized as follows. As in Section 3.1 we apply bubbling-off analysis on the cylinder [h ] × S 1 to show that on any compact set in the complement of a finite number of points Z (m) 3h] × S 1 , the gradient of u n is uniformly bounded. The points on which the gradient might blow up are located in (h ) × S 1 . Each resulting puncture from Z (m) lies in a disk D r of radius r smaller than h/2. For a smaller radius r, we assume that all disks D r are pairwise disjoint and that their union lies in (h ) × S 1 (seeFigure 3.6). + 6h, assume that the H−holomorphic curves u n together with the harmonic perturbations γ n are defined on [0, H (m) n ] × S 1 . By going over to a subsequence, we have H(m) n E1 The maps u n = (a n , f n ) are H−holomorphic curves with harmonic perturbation γ n on [0, H (m) n ] × S 1 with respect to the standard complex structure i on the domain and the almost complex structure J on ξ. E2E3 The maps u n have uniformly bounded energies, while the harmonic perturbations γ n have uniformly bounded L 2 −norms, i.e., with the constants E 0 , C 0 > 0 we have E(u n ; The maps u n have uniformly bounded gradients on [0, 3h] × S 1 and [H × S 1 with respect to the Euclidean metric on the domain and the cylindrical metric on the target space R × M, i.e.du n (z) = sup v eucl. =1 du n (z)v g < C h for all z ∈ ([0, 3h] ∪ [H ) × S 1 and n ∈ N.The next lemma states the existence of a finite set Z (m) of punctures on which the gradient of u n blows up.Lemma 32. There exists a finite set of points Z (m) ⊂ [3h, H (m) n − 3h] × S 1 such that for any compact subset K ⊂ ([0, H (m) n ] × S 1 )\Z (m) there exists a constant C K > 0 such that − 2h) × S 1 . Then by Lemma 32, u n has a uniformly bounded gradient on ([0, H (m) n ] × S 1 )\D r (Z (m) ). As ([0, H (m) n ] × S 1 )\D r (Z (m) ) is connected, we assume, after going over to some subsequence, that u n | ([0,H (m) n ]×S 1 )\D r (Z (m) ) converge in C ∞ to some smooth map u| ([0,H (m) ]×S 1 )\D r (Z (m) ) = (a, f)| ([0,H (m) ]×S 1 )\D r (Z (m) ) . Before treating the convergence of the H−holomorphic curves in a neighbourhood of the punctures of Z (m) , we establish the convergence of the harmonic perturbations γ n on [0, H (m) n ] × S 1 , so that at the end u n | ([0,H (m) n ]×S 1 )\D r (Z (m) ) converge in C ∞ to a H−holomorphic curve u| ([0,H (m) ]×S 1 )\D r (Z (m) ) , and the harmonic perturbations γ n have uniformly bounded C k −norms on the disks D r (Z (m) ) for all k ∈ N 0 . n and ψ (l) : D → U (l) , where l ∈ {1, ..., N} and N ∈ N. nC 1 . 1: D → R for some l ∈ {1, ..., N}. Because ψ Relying on the compactness result for harmonic functions we assume that G (l) J3 The H−holomorphic curves ([0, H (m) −− 1 be cylinders of types ∞ and b 1 , respectively. First we consider the cylinders [h × S 1 of type ∞. With the constant h > 0 defined in Section 3.4h] × S 1 be a subcylinder. By the uniform gradient bounds of u n on cylinders of type ∞, we conclude that the H−holomorphic curves u n together with the harmonic perturbations γ n converge in C ∞ on [h × S 1 to a H−holomorphic curve u with harmonic perturbation γ. 4h] × S 1 we perform the same analysis as in Theorems 28 and 30. After going over to a subsequence we obtain a sequence of diffeomorphisms θ n : [h 4h]×S 1 . Next we extend the diffeomorphisms θ n , θ − and θ + to [h × S 1 , [−4h, ∞) × S 1 and (−∞, 4h] × S 1 , respectively, such that Figure 3 . 7 : 37From the mean value inequality for harmonic functions, Γ n is uniformly bounded in C 0 (D τ ). Via the biholomorphism [0, ∞) × S 1 → D\{0}, (s, t) → e −2π(s+it) we consider the H−holomorphic maps u n together with the harmonic perturbations γ n as beeing defined on the half open cylinder [0, ∞) × S 1 . Specifically we consider the following setup: For the sequence u n = (a n , f n ) : [0, ∞)×S 1 → R×M of H−holomorphic half cylinders with harmonic perturbations γ n the following are satisfied: L1 The energy of u n and the L 2 −norm of the harmonic pertuabtions γ n are uniformly bounded, i.e. with the constants E 0 , C 0 > 0 we have E(u n ; [0, ∞) × S 1 ) E 0 and γ n 2 L 2 (D\{0}) C 0 for all n ∈ N. Decomposition of a punctured neighbourhood into cylinders of type ∞, b 1 and a half open cylinder. we identify the cylinder [0, 1) × S 1 , which is diffeomorphic with the infinite half open cylinder, with a punctured disk D\{0}. In this way the upper half open cylinder [0, 1) × S 1 can be identified with a neighbourhood of a puncture. a k (s) = c k cosh(ks) + d k sinh(ks), b k (s) = e k cosh(ks) + f k sinh(ks), α k (s) = f k cosh(ks) + e k sinh(ks), β k (s) = −d k cosh(ks) − c k sinh(ks). K (k) = +1, c k and d k have the same parity −1, otherwise and G(k) = +1, e k and f k have the same parity −1, otherwise. Consider the diffeomorphism θ : [0, ∞) × S 1 → R × M and the mapsg n := f n • θ −1 : [0, 1) × S 1 → M, (B.2) ] × S 1 of type ∞, on which we apply Lemma 25. When doing this we choose a sufficiently small constant h > 0, so that the gradients are uniformly bounded only on [h − h] × S 1 by the constant C h > 0, which in turn, is again a cylinder of type ∞. By this procedure, a cylinder [h1. We consider a cylinder [h (m) n , h (m+1) n (m) n + h, h (m+1) n (m) where Op([−R n + h n , R n − h n ]) and Op([−1/2, 1/2]) are sufficiently small neighbourhoods of the intervals [−R n + h n , R n − h n ] and [−1/2, 1/2], respectively. Acknowledgement. We thank Peter Albers and Kai Cieliebak who provided insight and expertise that greatly assisted the research. U.F. is supported by the SNF fellowship 155099, a fellowship at Institut Mittag-Leffler and the GIF Grant 1281.Theorem 47. Let u n be a sequence of pseudoholomorphic curves having uniformly bounded energy by E 0 and satisfying condition (B.1). Then there exists a subsequence of u n , still denoted by u n , such that the following is satisfied.1. u n is asysmptotic to the same Reeb orbit, i.e. there exists a Reeb orbit x and T = 0 with |T | C and a sequence c n ∈ S 1 such that lim s→∞ f n (s, t) = x(T t+c n ) and lim s→∞ a n (s, t) s = T .for all n ∈ N.2. u n converge in C ∞ loc to a pseudoholomorphic half cylinder u : [0, ∞) × S 1 → R × M having uniformly bounded energy by the constant E 0 and satisfying condition (B.1).The maps gwhere x is a Reeb orbit of period T = 0.C Special coordinatesLet S be a compact surface with boundary, and let j n and j be complex structures on S for all n ∈ N. Additionally, let h n and h be the hyperbolic structures on S with respect to j n and j, respectively. Assume that j n → j and h n → h in C ∞ (S). In this appendix we construct a sequence of biholomorphic coordinates around some point in S with respect to the complex structure j n that converges in a certain sense to the biholomorphic coordinates with respect to j. This result is used in Section 3 for proving the convergence on the thick part.Lemma 48. For each z ∈ int(S) there exist the open neighbourhoods U n (z) = U n and U(z) = U of z and the diffeomorphisms2. ψ n → ψ in C ∞ loc (D 1 (0)) as n → ∞ with respect to the Euclidian metric on D 1 (0) and h on S;3. ψ n (0) = z for every n and ψ(0) = z.Proof. Around z ∈ int(S), choose the i − j−holomorphic coordinates c : D 2 (0) → U such that U ⊂ int(S) and c(0) = z, and consider the complex structures j (n) := c * j n . Since j n → j asn be the operator defined by d C n f = df • j (n) and let d C be the operator defined by d C f = df • i. Denote by p x : R 2 → R, (x, y) → x the projection onto the first coordinate. Consider the problem of finding a smooth fumction f :for all n andAs the second problem translates intowhere ∆ is the standard Laplace operator in R, the unique solution is f(x, y) = x for all (x, y) ∈ D 1 (0). To see the uniqueness observe that the difference of f with any other solution of (C.3) solves ∆u = 0 with u| ∂D 1 (0) = 0. Thus from the maximum principle for harmonic functions we deduce that u ≡ 0, and so, that (C.3) has the unique solution f. In coordinates representation, j (n) can be written asand take notice that j (n) → i in C ∞ on D 1 (0) as n → ∞. The solutions of (C.1) are equivalent to the solutions ofwhere t n = −dd C n p x . Hence dd C n is an elliptic and coercive operator, and thus by Proposition 5.10 from[13], the problem (C.4) has a uniquely weak solutionf n ∈ W 1,2 (D 1 (0)) for all n. From regularity theorem, the solutions f n are smooth for all n. Thus f n :=f n + p x is the smooth unique solution of (C.1). Let us show that f n → f in C ∞ loc (D 1 (0)) as n → ∞. For u n := f n − f we have dd C n u n = g n on D 1 (0), u n = 0 on ∂D 1 (0).Here, g n ∈ C ∞ (D 1 (0)) is defined by g n := dd C n f, and because of j (n) → i in C ∞ (D 1 (0)) as n → ∞, g n converges to 0 in C ∞ loc (D 1 (0)) as n → ∞. For every m ∈ N 0 we consider the bounded operator dd C n : W 2+m,2consists of maps from W 2+m,2 (D 1 (0), R) that vanish at the boundary. By Proposition 5.10 together with Propositons 5.18 and 5.19 of[13]we deduce that the operator dd C n is bounded invertible; hence u n = (dd C n ) −1 g n . Since dd C n → ∆ in operator norm, (dd C n ) −1 is a uniformly bounded family, and so, u n W m+2,2 → 0 as n → ∞. Further on, as m ∈ N 0 was arbitrary, the Sobolev embedding theorem yields u n → 0 in C ∞ loc (D 1 (0)) as n → ∞. Thus we have constructed a unique sequence of solutions {f n : D 1 (0) → R} n∈N of (C.1), and a unique solution f : D 1 (0) → R, (x, y) → x of (C.2) satisfying f n → f in C ∞ loc (D 1 (0)) as n → ∞. According to Lemma 6.8.1 of[7], there exists a j (n) −i−holomorphic function F n : D 1 (0) → C and a i−i−holomorhic function F : D 1 (0) → C such that f n = (F n ) and f = (F). Let us investigate the unique extensions of the functions F n and F. For doing this we set F n = f n + ib and F = f + ib, where b n , b : D 1 (0) → R are harmonic functions. As F Bourgeois, Y Eliashberg, H Hofer, K Wysocki, E Zehnder, Compactness results in Symplectic Field Theory. 7F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki, E. Zehnder, Compactness results in Symplectic Field Theory, Geom. Topol. 7 (2003) 799-888 Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three. H Hofer, Invent. Math. 1143H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math. 114, no. 3 (1993) 515-563 Holomorphic curves and real three-dimensional dynamics. H Hofer, Geom. Funct. Anal. Special. Part IIH. Hofer, Holomorphic curves and real three-dimensional dynamics, Geom. Funct. Anal. Special Volume 2000, Part II (2000) 674-704 The Weinstein conjecture for planar contact structures in dimension three. C Abbas, K Cieliebak, H Hofer, Comment. Math. Helv. 80C. Abbas, K. Cieliebak, H. Hofer, The Weinstein conjecture for planar contact structures in dimension three, Comment. Math. Helv. 80 (2005) 771-793 Gromov's compactness theorem for pseudo-holomorphic curves. C Hummel, Progress in Mathematics. 151Birkhäuser VerlagC. Hummel, Gromov's compactness theorem for pseudo-holomorphic curves, Progress in Mathematics, 151 Birkhäuser Verlag, Basel (1997) C Abbas, An Introduction to Compactness Results in Symplectic Field Theory. Springer VerlagC. Abbas, An Introduction to Compactness Results in Symplectic Field Theory, Springer Verlag (2014) An Introduction to Riemann Surfaces. T Napier, M Ramachandran, Birkhäuser VerlagT. Napier, M. Ramachandran, An Introduction to Riemann Surfaces, Birkhäuser Verlag (2011) Compactness for punctured holomorphic curves. K Cieliebak, K Mohnke, J. Symp. Geom. 34K. Cieliebak, K. Mohnke, Compactness for punctured holomorphic curves, J. Symp. Geom. 3 (4) (2005) 589-654. Finite energy cylinders of small area. H Hofer, K Wysocki, E Zehnder, Dyn. Syst. Ergod. Theory. 225H. Hofer, K. Wysocki, E. Zehnder, Finite energy cylinders of small area, Dyn. Syst. Ergod. Theory 22 (5) (2002), 1451-1486. H Hofer, K Wysocki, E Zehnder, Properties of pseudoholomorphic curves in symplectisations I: Asymptotics. 13H. Hofer, K. Wysocki, E. Zehnder, Properties of pseudoholomorphic curves in symplectisations I: Asymp- totics, Ann. Inst. Henri Poincare Anal. Non Lineaire 13 (3) (1996), 337-379. Pseudoholomorphic curves in symplectic and contact geometry and their application in dynamics. U Fuchs, 3636017PhD dissertation, Purdue University. Ann Arbor: ProQuest/UMIU. Fuchs, Pseudoholomorphic curves in symplectic and contact geometry and their application in dynam- ics, PhD dissertation, Purdue University. Ann Arbor: ProQuest/UMI, 2014. (Publication No. AAT 3636017.) J V Bergmann, arXiv:0904.1603Compactness Results for H-Holomorphic Maps. J. v. Bergmann, Compactness Results for H-Holomorphic Maps, arXiv:0904.1603. Lecture notes. B Schmidt, Partielle Differentialgleichungen, University of AugsburgB. Schmidt, Partielle Differentialgleichungen, Lecture notes, University of Augsburg 2012. (https://www.math.uni-augsburg.de/prof/ana/arbeitsgruppe/schmidt/skripten/pdg1.pdf) Holomorphic open book decompositions. C Abbas, Duke Math J. 1581C. Abbas, Holomorphic open book decompositions, Duke Math J. 158 (1) (2011), 29-82 A Doicu, U Fuchs, arXiv:1802.05573Asymptotic Behaviour for H−holomorphic Cylinders of Small Area. A. Doicu, U. Fuchs, Asymptotic Behaviour for H−holomorphic Cylinders of Small Area, arXiv:1802.05573.
{'fraction_non_alphanumeric': 0.08795670698985562, 'fraction_numerical': 0.022325382919433364, 'mean_word_length': 3.068176438096741, 'pattern_counts': {'":': 0, '<': 34, '<?xml version=': 0, '>': 96, 'https://': 1, 'lorem ipsum': 0, 'www.': 1, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 182, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'H−holomorphic curves are solutions of a specific modification of the pseudoholomorphic curve equation in symplectizations involving a harmonic 1−form as perturbation term. In this paper we compactify the moduli space of H−holomorphic curves with a priori bounds on the harmonic 1−forms.A Holomorphic disks with fixed boundary 46 B Half cylinders with small energy 49 C Special coordinates 51', 'arxivid': '1802.08991', 'author': ['Alexandru Doicu [email protected] \nInstitut für Mathematik\nMathematisches Institut\nUniversität Augsburg\nUniversität Heidelberg\n\n', 'Urs Fuchs [email protected] \nInstitut für Mathematik\nMathematisches Institut\nUniversität Augsburg\nUniversität Heidelberg\n\n'], 'authoraffiliation': ['Institut für Mathematik\nMathematisches Institut\nUniversität Augsburg\nUniversität Heidelberg\n', 'Institut für Mathematik\nMathematisches Institut\nUniversität Augsburg\nUniversität Heidelberg\n'], 'corpusid': 119723541, 'doi': '10.4310/jsg.2021.v19.n1.a2', 'github_urls': [], 'n_tokens_mistral': 55481, 'n_tokens_neox': 50114, 'n_words': 32810, 'pdfsha': '4bd549eda81e2b61e4f0946c1576198afebbbf98', 'pdfurls': ['https://export.arxiv.org/pdf/1802.08991v1.pdf'], 'title': ['A Compactness Result for H−holomorphic Curves in Symplectizations', 'A Compactness Result for H−holomorphic Curves in Symplectizations'], 'venue': []}
arxiv
Single crystal growth and characterizations of Cu 0.03 TaS 2 superconductors X D Zhu Institute of Solid State Physics Key Laboratory of Materials Physics Chinese Academy of Sciences 230031HefeiPeople's Republic of China Y P Sun Institute of Solid State Physics Key Laboratory of Materials Physics Chinese Academy of Sciences 230031HefeiPeople's Republic of China High Magnetic Field Laboratory Chinese Academy of Sciences 230031HefeiPeople's Republic of China X B Zhu Institute of Solid State Physics Key Laboratory of Materials Physics Chinese Academy of Sciences 230031HefeiPeople's Republic of China X Luo Institute of Solid State Physics Key Laboratory of Materials Physics Chinese Academy of Sciences 230031HefeiPeople's Republic of China B S Wang Institute of Solid State Physics Key Laboratory of Materials Physics Chinese Academy of Sciences 230031HefeiPeople's Republic of China G Li Institute of Solid State Physics Key Laboratory of Materials Physics Chinese Academy of Sciences 230031HefeiPeople's Republic of China Z R Yang Institute of Solid State Physics Key Laboratory of Materials Physics Chinese Academy of Sciences 230031HefeiPeople's Republic of China W H Song Institute of Solid State Physics Key Laboratory of Materials Physics Chinese Academy of Sciences 230031HefeiPeople's Republic of China J M Dai Institute of Solid State Physics Key Laboratory of Materials Physics Chinese Academy of Sciences 230031HefeiPeople's Republic of China Single crystal growth and characterizations of Cu 0.03 TaS 2 superconductors Superconducting materials. * Corresponding author. 1A1 Charge density waveA2 Single crystal growthA2 Chemical vapor transportB2 Single crystal of Cu 0.03 TaS 2 with low copper intercalated content was successfully grown via chemical vapor transport method. The structural characterization results show that the copper intercalated 2H-Cu 0.03 TaS 2 single crystal has the same structure of the CdI 2 -type structure as the parent 2H-TaS 2 crystal. Electrical resistivity and magnetization measurements reveal that 2H-Cu 0.03 TaS 2 becomes a superconductor below 4.2 K. Besides, electrical resistivity and Hall effects results show that a charge density wave transition occurs at T CDW = 50 K. Introduction Diverse physical properties such as Charge-Density-Wave (CDW) and superconductivity in layered transition metal dichalcogenides (TMDC) have been abroad studied [1,2]. TMDC have a general formula TX 2 , where T is usually a transition metal atom from group of IVB, VB, VIB of the periodic table of elements and X is one of sulfur, selenium, or tellurium. The layered TMDC can be regarded as stacking two-dimensional X-T-X sandwiches. The bonding within each sandwich is covalent, while the bonding between them is weak van der Waals type. The crystal structures of the layered TMDC are usually described as 1T, 2H, 3R, 4H a , 4H b , 6R phases [1,2]. The integer denotes the number of X-T-X layers per unit cell perpendicular to the layers, while T, H, and R denote trigonal, hexagonal, rhombohedral symmetries, respectively. Introducing foreign atoms or molecules between the weak coupling sandwiches or layers is the process of intercalation. Intercalation not only increases the layer separation, but also provides a powerful way to tune the electronic structures of the host materials. Intercalated TMDC, graphite, and layered structural transition metal nitrides have received considerable attention due to their special structures and transport properties especially for superconductivities [2][3][4]. New interests have been aroused since the recent discoveries of superconductivities in YbC 6 , CaC 6 and Cu x TiSe 2 [5][6][7]. 2H-TaS 2 is one of the layered TMDC [8]. Metallic 2H-TaS 2 undergoes a CDW transition at 78K, and becomes a superconductor below 0.8 K [8]. Many inorganic atoms and organic molecules have been intercalated into TaS 2 [3,[9][10][11][12][13][14]. Intriguingly, superconductivity has been found in Py 1/2 TaS 2 [9], A 0.33 TaS 2 (A = Li, Na, K, Rb, Cs) [14], and 2H-Fe x TaS 2 (x = 0.05) et al. [15]. Recently, the competition driven by intercalation between superconductivity and CDW in Na x TaS 2 has been discovered [16]. Copper intercalations in TaS 2 have been achieved by chemical vapor transport (CVT) method to produce Cu 1/2 TaS 2 [17], by direct reaction from copper and TaS 2 to produce single phase Cu x TaS 2 (0.33 < x < 0.75) [18], and by electro-chemical method [19]. However, superconductivity has not been observed in any high copper intercalation Cu x TaS 2 [17][18][19]. Further, there is also no any report about lower copper intercalation in Cu x TaS 2 . Thus, it is interesting to explore whether low copper intercalation in Cu x TaS 2 can induce superconductivity or not. The low copper intercalation experiments were carried out in order to search for possible superconductivity 2 in Cu x TaS 2 (x < 0.33). In this paper, we report the single crystal growth of Cu 0.03 TaS 2 with low copper intercalated content via CVT method. Electrical resistivity and magnetization measurements reveal that 2H-Cu 0.03 TaS 2 becomes a superconductor below 4.2 K. Besides, resistivity and Hall effects results show that a charge density wave transition occurs at T CDW = 50 K. Experimental procedure Single crystal Cu 0.03 TaS 2 was grown via CVT method. Firstly, polycrystalline TaS 2 powder was synthesized by solid state reaction. Stoichiometric amounts of Ta and S powders were mixed and sealed in an evacuated quartz tube. The tube was heated at 900°C for 4 days. Secondly, Cu, TaS 2 powders in mol ratio of 0.5:1, and 100 mg iodine were mixed and sealed in an evacuated quartz tube with a 12 mm diameter and a 19 cm length. The tube was inserted in a two-zone horizontal tube furnace with a source-zone temperature of 1000°C and a growth-zone temperature of 900°C for 10 days. Both zones were decreased to 440°C in several days, and then cooled to room temperature. Dark blue, mirror-like plates in a typical size of 3 × 3 × 0.2 mm 3 were obtained, and the photograph is shown in Fig. 1. F. R. Gamble et al. has reported that 2H-TaS 2 single crystals grown via CVT method were black and somewhat wrinkle [9]. The grown Cu 0.03 TaS 2 plates were cleaned by ultrasonic in supersaturated aqueous solutions of KI, de-ionized water, and alcohol, respectively. The composition of Cu 0.03 TaS 2 single crystal was characterized by energy dispersive x-ray spectroscopy (EDS) and inductively coupled plasma atomic emission spectrometry (ICP-AES, Atomscan Advantage). The crystal structure and phase purity were examined by powder and single-crystal X-ray diffraction pattern (XRD) using a Philips Results and discussions Chemical analysis The EDS pattern for the grown single crystal is shown in Fig. 2. The estimated mol ratio of Ta : S is about 1 : 2. Though the copper peak is evident in the EDS spectrum, the accurate copper content can not be obtained because the copper content is too small. Thus, the copper content is determined by ICP-AES. The determined mol ratio of the Cu : Ta is 0.03 : 1. Structural analysis The XRD patterns for Cu 0.03 TaS Several single crystals were crushed into powders, which were used in the powder XRD experiment. The powder XRD patterns of crushed Cu 0.03 TaS 2 and 2H-TaS 2 crystals are shown in Fig. 3b. All the peaks can be well indexed to the 2H structure [20]. The present Cu 0.03 TaS 2 is single phase with no detectable secondary phases. The lattice constants were obtained from powder XRD patterns. The lattice constant of Cu 0.03 TaS 2 is almost equal to that of 2H-TaS 2 (a = 3.31Å), while the lattice constant c of Cu 0.03 TaS 2 is 12.13(7) Å, which is slightly larger than that of 2H-TaS 2 (c = 12.08(0) Å). Co-existence of superconductivity and charge density wave The temperature dependence of electrical resistivity in ab plane (ρ ab -T) for 2H-Cu 0.03 TaS 2 is plotted in Fig. 4. Obviously, there is a kink around T = 50 K, which can be attributed to CDW transition. The inset shows the enlarged view of the ρ ab -T curve near the superconducting region. The resistivity sharply drops to zero around T = 4.2 K, which indicates the superconductivity. The onset superconducting transition temperature is 4.2 K, and the transition width (10%-90%) is 0.2 K. 4 The temperature dependence of dc magnetic susceptibility for Cu 0.03 TaS 2 is plotted in Fig. 5. The magnetic field of 2 Oe was applied parallel to the ab plane of a Cu 0.03 TaS 2 plate. A sharp drop of magnetization at 4.2 K was observed for both zero-field-cooling (ZFC) and field-cooling (FC) measurements，which further confirms the existence of superconductivity. The ZFC curve shows an almost perfect shielding effect, while the magnetic flux exclusion fraction estimated from the FC curve is as low as ~4% at 2 K, using -4πχ v to estimate the volume fraction. Similar phenomena have been reported in other intercalated compounds such as Py 1/2 TaS 2 [21], YbC 6 and CaC 6 [5,6], LiPd 3 BB 2 [22]. Prober et al. suggested that the small FC magnetization was due to the complicated flux trapping effect in these intercalated compounds [23]. Thus, it is suggested that the bulk superconductivity is found in Cu 0.03 TaS 2 . The temperature dependence of the Hall (R H ) coefficient for Cu 0.03 TaS 2 is shown in the Fig. 6. The R H versus T is determined from the Hall resistivity (ρ xy ) results using R H = ρ xy /H. The inset in Conclusion In summary, Cu 0.03 TaS 2 single crystals with low copper intercalated content was successfully grown via CVT. Structure analysis shows that the low copper intercalated Cu 0.03 TaS 2 and the parent compound 2H-TaS 2 share the same structure. Electrical resistivity and magnetization measurements reveal that 2H-Cu 0.03 TaS 2 becomes a superconductor below 4.2 K. Besides, electrical resistivity and X ' pert PRO x-ray diffractometer with Cu Kα radiation at room temperature. The electrical resistivity was performed by the standard four-probe method using a Quantum Design Physical Property Measurement System (PPMS) (1.8 K ≤ T ≤ 400 K, 0 T ≤ H ≤ 9 T). The Hall effects experiments were also performed using PPMS. Magnetization measurements as a function of temperature were performed in a Quantum Design Superconducting Quantum Interference Device (SQUID) system (1.8 K ≤ T ≤ 400 K, 0 T ≤ H ≤ 5 T). 2 and 2H-TaS 2 single crystals are shown in Fig. 3a. It can be observed that the orientations of the crystal surfaces are both (00l) planes. The magnification plots of (006) peaks are shown in the inset of Fig. 3a. Obviously, the peak positions of Cu 0.03 TaS 2 shift to lower 2θ value. According to the Bragg equation nλ = 2dsinθ, the copper intercalation leads to expansion of the lattice constant c. Fig. 6 6shows the magnetic field dependence of the ρ xy measured at different temperatures. The R H increases from ~ 2.8×10 -4 cm 3 /C above 60 K to ~5.5×10 -4 cm 3 /C below 30K. In contrast, the sign of the R H of matrix 2H-TaS 2 even change from positive to negative near the CDW transition, which was reported by A. H. Thompson et al[21]. The change of the R H of Cu 0.03 TaS 2 in the region of 30 K< T < 60 K confirms the CDW transition revealed by the resistivity results. Apparently, copper intercalation suppresses the T CDW from T CDW ~ 78 K (2H-TaS 2 ) to T CDW ~ 50 K. reduction of carrier density is consistent with the partial gapping of the Fermi surface due to the occurrence of CDW transition. Figure Captions： Figure Captions： Fig. 1 . 1Photograph of Cu 0.03 TaS 2 single crystal. Fig. 2 . 2The EDS pattern for Cu 0.03 TaS 2 . Fig. 3 . 3The single crystal XRD patterns for Cu 0.03 TaS 2 and 2H-TaS 2 ; Inset: the magnification plot of single crystal XRD patterns; (b) The powder XRD patterns for Cu 0.03 TaS 2 and 2H-TaS 2 . Fig. 4 . 4Temperature dependence of the in-plane resistivity (ρ ab ) for Cu 0.03 TaS 2 . The inset shows the ρ ab -T curve near the superconducting region. The arrows show the charge density wave transition temperature (T CDW ) and the onset transition temperature (T Conset ). Fig. 5 . 5The temperature dependence of ZFC and FC susceptibility χ g measured in an applied field of 2 Oe parallel to ab plane for Cu 0.03 TaS 2 : FC (filled circles); ZFC (open circles). Fig. 6 . 6Temperature dependence of the Hall coefficients (R H ) for Cu 0.03 TaS 2 . The inset shows the measured Hall resistivity (ρ xy ) versus H measured at different temperatures. 8 Hall effects results show that a charge density wave transition occurs at T CDW = 50 K. . J A Wilson, A D Yoffe, Adv. Phys. 18193J. A. Wilson and A. D. Yoffe, Adv. Phys. 18 (1969) 193. . R H Friend, A D Yoffe, Adv. Phys. 361R. H. Friend and A. D. Yoffe, Adv. Phys. 36 (1987) 1. . M S Dresselhaus, G Dresselhaus, Adv. Phys. 561M. S. Dresselhaus and G. Dresselhaus, Adv. Phys. 56 (2002) 1. . S Yamanaka, Annu. Rev. Mater. Sci. 3053S. Yamanaka, Annu. Rev. Mater. Sci. 30 (2000) 53. . T E Weller, T E Weller, M Ellerby, S S Saxena, R P Smith, N T Skipper, Nature Physics. 130T. E. Weller, T. E. Weller, M. Ellerby, S. S. Saxena, R. P. Smith, and N. T. Skipper, Nature Physics 1 (2005) 30. . N Mery, C Hérold, M Ďastuto, V Garcia, Ch Bellin, J F Marêché, P Lagrange, G Loupias, Phys. Rev. Lett. 9587003N.E mery, C. Hérold, M. ďAstuto, V. Garcia, Ch. Bellin, J. F. Marêché, P. Lagrange, and G. Loupias, Phys. Rev. Lett. 95 (2005) 087003. . E Morosan, H W Zandbergen, B S Dennis, J W G Bos, Y Onose, T Klimczuk, A P Amirez, N P Ong, R J Cava, Nature Physics. 2544E. Morosan, H. W. Zandbergen, B. S. Dennis, J. W. G. Bos, Y. Onose, T. Klimczuk, A. P. R amirez, N. P. Ong and R. J. Cava, Nature Physics 2 (2006) 544. . J M E Harper, T H Geballe, F J Di Salvo, Phys. Rev. B. 152943J. M. E. Harper, T. H. Geballe and F. J. Di Salvo, Phys. Rev. B 15 (1977), 2943. . F R Gamble, F J Di Salvo, R A Klemm, T H Geballe, Science. 168568F. R. Gamble, F. J. Di Salvo, R. A. Klemm and T. H. Geballe, Science 168 (1970) 568. . D W Murphy, F J Di Salvo, G W Hull, J V , Waszczak，Inorganic Chemistry. 1517D. W. Murphy, F. J. Di Salvo, G. W. Hull, and J. V. Waszczak，Inorganic Chemistry 15 (1976) 17. . R M Fleming, R V Coleman, Phys. Rev. Lett. 341502R. M. Fleming and R. V. Coleman, Phys. Rev. Lett. 34 (1975) 1502. . A Schlicht, A Lerf, W Biberacher, Synthetic Metals. 1021483A. Schlicht, A. Lerf and W. Biberacher, Synthetic Metals 102 (1999) 1483. . S F Meyer, R E Howard, G R Stewart, J V Acrivos, T H Geballe, J. Chem. Phys. 62S. F. Meyer, R. E. Howard, G. R. Stewart, J. V. Acrivos, and T. H. Geballe, J. Chem. Phys. 62 . A Lerf, F Sernetz, W Biberacher, R Schöllhorn, Mat. Res. Bull. 14797A. Lerf, F. Sernetz, W. Biberacher, R. Schöllhorn, Mat. Res. Bull. 14 (1979) 797. . R M Fleming, R V Coleman, Phys. Rev. Lett. 341502R. M. Fleming and R. V. Coleman, Phys. Rev. Lett. 34 (1975) 1502. . L Fang, Y Wang, P Y Zou, L Tang, Z Xu, H Chen, C Dong, L Shan, H H Wen, Phys. Rev. B. 7214534L. Fang, Y. Wang, P. Y. Zou, L. Tang, Z. Xu, H. Chen, C. Dong, L. Shan, and H. H. Wen, Phys. Rev. B 72 (2005) 014534. . R De Ridder, G Van Tendeloo, J Van Landuyt, D Van Dyck, S Amelinckx, Phys. Stat. Sol. (a). 37691R. de Ridder, G. van Tendeloo, J. van Landuyt, D. van. Dyck, and S. Amelinckx, Phys. Stat. Sol. (a) 37 (1976) 691. . T Uchida, S Sato, M Wakihara, M Tanigychi, Nippon Kagaku Kaishi. 101306T. Uchida, S. Sato, M. Wakihara and M. Tanigychi, Nippon Kagaku Kaishi 10 (1991) 1306. . C Ramos, A Lerf, T Butz, Hyperfine Interactions. 1209C. Ramos, A. Lerf and T. Butz, Hyperfine Interactions 61 (1990) 1209. . A Meetsm, A Meetsma, G A Wiegers, R J Haange, J L De Boer, Acta. Cryst. C. 461598A. Meetsm, A. Meetsma, G. A. Wiegers, R. J. Haange and J. L. de Boer, Acta. Cryst. C 46 (1990) 1598. . A H Thompson, F R Gamble, R F Koehler, Jr , Phys. Rev. B. 52811A. H. Thompson, F. R. Gamble and R. F. Koehler, Jr. Phys. Rev. B 5 (1972) 2811. . K Togano, P Badica, Y Nakamori, S Orimo, H Takeya, K Hirata, Phys. Rev. Lett. 93247004K. Togano, P. Badica, Y. Nakamori, S. Orimo, H. Takeya, and K. Hirata, Phys. Rev. Lett. 93 (2004) 247004. . D E Prober, M R Beasley, R E Schwall, Phys. Rev. B. 157D. E. Prober and M. R. Beasley and R. E. Schwall, Phys. Rev. B 15 (1977) 5245. 7
{'fraction_non_alphanumeric': 0.06299938537185003, 'fraction_numerical': 0.05009219422249539, 'mean_word_length': 3.670206659012629, 'pattern_counts': {'":': 0, '<': 5, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'Single crystal of Cu 0.03 TaS 2 with low copper intercalated content was successfully grown via chemical vapor transport method. The structural characterization results show that the copper intercalated 2H-Cu 0.03 TaS 2 single crystal has the same structure of the CdI 2 -type structure as the parent 2H-TaS 2 crystal. Electrical resistivity and magnetization measurements reveal that 2H-Cu 0.03 TaS 2 becomes a superconductor below 4.2 K. Besides, electrical resistivity and Hall effects results show that a charge density wave transition occurs at T CDW = 50 K.', 'arxivid': '0808.2357', 'author': ["X D Zhu \nInstitute of Solid State Physics\nKey Laboratory of Materials Physics\nChinese Academy of Sciences\n230031HefeiPeople's Republic of China\n", "Y P Sun \nInstitute of Solid State Physics\nKey Laboratory of Materials Physics\nChinese Academy of Sciences\n230031HefeiPeople's Republic of China\n\nHigh Magnetic Field Laboratory\nChinese Academy of Sciences\n230031HefeiPeople's Republic of China\n", "X B Zhu \nInstitute of Solid State Physics\nKey Laboratory of Materials Physics\nChinese Academy of Sciences\n230031HefeiPeople's Republic of China\n", "X Luo \nInstitute of Solid State Physics\nKey Laboratory of Materials Physics\nChinese Academy of Sciences\n230031HefeiPeople's Republic of China\n", "B S Wang \nInstitute of Solid State Physics\nKey Laboratory of Materials Physics\nChinese Academy of Sciences\n230031HefeiPeople's Republic of China\n", "G Li \nInstitute of Solid State Physics\nKey Laboratory of Materials Physics\nChinese Academy of Sciences\n230031HefeiPeople's Republic of China\n", "Z R Yang \nInstitute of Solid State Physics\nKey Laboratory of Materials Physics\nChinese Academy of Sciences\n230031HefeiPeople's Republic of China\n", "W H Song \nInstitute of Solid State Physics\nKey Laboratory of Materials Physics\nChinese Academy of Sciences\n230031HefeiPeople's Republic of China\n", "J M Dai \nInstitute of Solid State Physics\nKey Laboratory of Materials Physics\nChinese Academy of Sciences\n230031HefeiPeople's Republic of China\n"], 'authoraffiliation': ["Institute of Solid State Physics\nKey Laboratory of Materials Physics\nChinese Academy of Sciences\n230031HefeiPeople's Republic of China", "Institute of Solid State Physics\nKey Laboratory of Materials Physics\nChinese Academy of Sciences\n230031HefeiPeople's Republic of China", "High Magnetic Field Laboratory\nChinese Academy of Sciences\n230031HefeiPeople's Republic of China", "Institute of Solid State Physics\nKey Laboratory of Materials Physics\nChinese Academy of Sciences\n230031HefeiPeople's Republic of China", "Institute of Solid State Physics\nKey Laboratory of Materials Physics\nChinese Academy of Sciences\n230031HefeiPeople's Republic of China", "Institute of Solid State Physics\nKey Laboratory of Materials Physics\nChinese Academy of Sciences\n230031HefeiPeople's Republic of China", "Institute of Solid State Physics\nKey Laboratory of Materials Physics\nChinese Academy of Sciences\n230031HefeiPeople's Republic of China", "Institute of Solid State Physics\nKey Laboratory of Materials Physics\nChinese Academy of Sciences\n230031HefeiPeople's Republic of China", "Institute of Solid State Physics\nKey Laboratory of Materials Physics\nChinese Academy of Sciences\n230031HefeiPeople's Republic of China", "Institute of Solid State Physics\nKey Laboratory of Materials Physics\nChinese Academy of Sciences\n230031HefeiPeople's Republic of China"], 'corpusid': 94840576, 'doi': '10.1016/j.jcrysgro.2008.10.023', 'github_urls': [], 'n_tokens_mistral': 5743, 'n_tokens_neox': 4781, 'n_words': 2793, 'pdfsha': 'ccc47f2f1d07d6bb91c5867210cabb2b6f5407a7', 'pdfurls': ['https://arxiv.org/pdf/0808.2357v3.pdf'], 'title': ['Single crystal growth and characterizations of Cu 0.03 TaS 2 superconductors', 'Single crystal growth and characterizations of Cu 0.03 TaS 2 superconductors'], 'venue': []}
arxiv
A sufficient condition for a set of calibrated surfaces to be area-minimizing under diffeomorphisms 11 Aug 2005 November 22, 2021 Doan The Hieu Department of Mathematics College of Education Hue University Le Loi HueViet Nam A sufficient condition for a set of calibrated surfaces to be area-minimizing under diffeomorphisms 11 Aug 2005 November 22, 2021 We extend Choe's idea in [Ch] to nonpolyhedral calibrated surfaces and give some examples of polyhedral sets over right prisms and nonpolyhedral calibrated surfaces. * This work was completed at SNU, with the support of Korean Foundation for Advanced Studies. Introduction In [Ch], Choe proved "Every stationary polyhedral set is area-minimizing under diffeomophisms leaving the boundary fixed". In his proof, a system of differential forms and orientations (of faces) was chosen at each singular edge. In fact, the differential forms are calibrations that calibrate the faces at each singular edge and have the vanishing sum. We observe that, the suitable orientations of faces at each singular edge determine the same orientation on it whenever it lies on the boundary of faces. By the above observation, we extend Choe's idea by proving a sufficient condition for certain sets of calibrated surfaces (including polyhedral sets) to be area-minimizing under diffeomophisms leaving the boundary fixed. This sufficient condition, when applies to polyhedral sets, is also necessary. We give some more examples of polyhedral sets over right prisms and first examples of nonpolyhedral calibrated surfaces (2-dimensional ones with singular sets of dimension 1 in R 4 ). The author would like to thank the gracious host scholar, Professor Hong-Jong Kim, for his thoughtfulness and assistance. Many thanks to Professor Jaigyoung Choe for his help, suggestions and the invitation talking this result at his seminar. This paper was written while the author was visiting RIM-GARC, Department of Mathematics of the Seoul National University, Korea. We would like to thank that institution for its hospitality and generous support. The theorem We refer the readers to [Ch] for the definition of polyhedral sets. Let {C i } i∈I be a set of calibrated surfaces of dimension m in R n (m < n) and {w i } i∈I be the set of correspondent calibrations. That means for each i ∈ I, w i calibrates C i with a suitable orientation. Note that if ω i calibrates C i , then −ω i calibrates C i with opposite orientation. Depending on a chosen orientation on C i we have the corespondent calibration to be ω i or −ω i . Let Σ ⊂ R n be a set satisfies the following conditions: (i) Σ ⊂ ∪ i∈I C i , (ii) the set E = Σ ∩ (C i ∩ C j ) is of dimension m − 1 for every i, j ∈ I, i = j. We call each F i = Σ ∩ C i a face, each E a singular edge, the union of all singular edges E the singular set S, the closure of ∂F i ∼ S the boundary edge of Σ in F i , the union ∪ i∈I (∂F i ∼ S) the boundary ∂Σ of Σ. Σ is said to be area-minimizing under diffeomorphisms leaving the boundary fixed if V ol(Σ) ≤ V ol(ϕ(Σ)), for any diffeomorphism ϕ of R n leaving the boundary of Σ fixed. Suppose {E j } j∈J is the set of all singular edges and {F i } i∈I is the set of all faces of Σ. Denote I Ej = {i : F i ⊃ E j } ⊂ I, J Fi = {j : E j ⊂ F i } ⊂ J. Theorem 2.1 Let Σ be a set defined as above. Suppose that every singular edge E j lies on the boundary ∂F i , ∀i ∈ I Ej and for each E j we can choose suitable orientations on F i , ∀i ∈ I Ej , such that: (i) the orientations on F i , ∀i ∈ I Ej determine the same orientation on E j , (ii) the corespondent calibrations have vanishing sum. Then Σ is area-minimizing under diffeomorphisms leaving ∂Σ fixed. Proof. The reasonings of the proof are very similar as that of the main theorem in [Ch] with some little changes. Let ϕ be a diffeomorphism leaving ∂Σ fixed and ϕ t be the homotopy from the identity to ϕ. Suppose G j is the m-dimensional smooth surface swept out by ϕ t (E j ) and D i is (m + 1)-dimensional surface swept out by ϕ t (F i ). We have ∂D i = F i ∪ ϕ(F i ) ∪ j∈JF i G j , and hence ∂Di w i = Fi w i + ϕ(Fi) w i + j∈JF i Gj w i . Since w i is a calibration that calibrates F i , we get the following inequality: V ol(F i ) ≤ V ol(ϕ(F i )) − j∈JF i Gj w i , and finally V ol(Σ) ≤ V ol(ϕ(Σ)) − j∈J i∈IE j Gj w i . By virtue of the assumtions of the theorem, we can assume the orientations on F i , ∀i ∈ I Ej , determine the same orientation on G j and since i∈IE j w i = 0, the last term equals zero. The theorem is proved. Corollary 2.2 Let Σ be a polyhedral set. Then Σ is area-minimizing under diffeomorphisms leaving ∂Σ fixed if and only if Σ satisfies the assumptions in the Theorem 2.1. Proof. The sufficiency follows from the above theorem and the necessity follows from the proof of the main theorem in [Ch]. Let C 2 ≡ R 4 be complex plane with the standard complex structure J 1 , J 1 e 1 = e 3 ; J 1 e 2 = e 4 . Let R 2 , R 3 , . . . , R n be the rotations of angles α, 2α, . . . , (n − 1)α about the plane {x 3 = x 4 = 0}, respectively, where α satisfies the condition nα = 2π, n ∈ N. And let J 2 , J 3 , . . . , J n be (n − 1) complex structures on R 4 induced by R 2 , R 3 , . . . , R n ; J i (e 1 ) = R i (e 3 ), J i (e 2 ) = R i (e 4 ); i = 2, 3, . . . , n. Denote w 1 , w 2 , . . . , w n the Kähler forms correspondent to J 1 , J 2 , . . . , J n . We can easily to see that: Consider the complex curves: n i=1 w i = 0.C = {(z, w) ∈ C 2 : z = w 2 } = {(x 1 , x 2 , x 3 , x 4 ) : x 2 = x 2 1 − x 2 3 ; x 4 = 2x 1 x 3 }. Let D be the intersection of C and { 4 i=1 x 2 i = 1; x 1 ≥ 0; x 3 ≥ 0}. Note that D contains two planar curves {x 2 = x 2 1 ; x 3 = x 4 = 0; x 2 1 + x 2 2 ≤ 1; x 1 ≥ 0}, and {x 2 = −x 2 3 ; x 1 = x 4 = 0; x 2 2 + x 2 3 ≤ 1; x 3 ≥ 0}. By using the rotations of angles kα, k = 1, 2. . . . , n − 1 about the plane {x 3 = x 4 = 0} we get the images D i (i = 2, 3, . . . , n) of D. Obviously, w 1 calibrates D and w i calibrates D i , i = 2, 3, . . . n. The set Σ = D D i contains one singular edge and is area-minimizing under diffeomophisms leaving the boundary fixed by virtue of Theorem 2.1. Similarly, by using the rotations of angles kβ, k = 1, 2. . . . , m − 1; mβ = 2π about the plane {x 1 = x 4 = 0}, we get the images D ′ j (j = 2, 3, . . . , m) of D and the images Σ j (j = 2, 3, . . . , m) of Σ. The set D D i D ′ j contains two singular edges. The set Σ ′ = Σ Σ j contains many singular edges. By the same reasoning as above, they are also area-minimizing under diffeomophisms leaving the boundary fixed. Ken Brakke's homepage, http://www.susqu.edu/facstaff/b/brakke/, we can see eight nice polyhedral cones, that are made of flat sheets meeting along triple lines with an equal angle 120 0 . All of them are area-minimizing under diffeomophisms leaving the boundary fixed by virtue of Theorem 2.1. Figures 1 provide more three polyhedral sets over right prisms. 2. Below are examples of nonpolyhedral calibrated surfaces that is areaminimizing under diffeomophisms leaving the boundary fixed. Figure 1 : 1Polyhedral sets over right prisms. Every stationary polyhedral set in R n is area minimizing under diffeomorphisms. J Choe, Pacific J. Math. 175J. Choe, Every stationary polyhedral set in R n is area minimizing under diffeomorphisms, Pacific J. Math. 175 (1996), 439-446. Calibrated geometries. R Harvey, H B Lawson, Acta Math. 104R. Harvey and H. B. Lawson, Calibrated geometries, Acta Math., 104 (1982), 47-157.
{'fraction_non_alphanumeric': 0.06852248394004283, 'fraction_numerical': 0.024759100642398287, 'mean_word_length': 3.19125070106562, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 1, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 22, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': "We extend Choe's idea in [Ch] to nonpolyhedral calibrated surfaces and give some examples of polyhedral sets over right prisms and nonpolyhedral calibrated surfaces. * This work was completed at SNU, with the support of Korean Foundation for Advanced Studies.", 'arxivid': 'math/0508193', 'author': ['Doan The ', 'Hieu ', '\nDepartment of Mathematics College of Education\nHue University\n\n', '\nLe Loi\nHueViet Nam\n'], 'authoraffiliation': ['Department of Mathematics College of Education\nHue University\n', 'Le Loi\nHueViet Nam'], 'corpusid': 15936183, 'doi': '10.1007/s00013-005-1317-y', 'github_urls': [], 'n_tokens_mistral': 2553, 'n_tokens_neox': 2230, 'n_words': 1439, 'pdfsha': '37942840f012f1bfdbc68a4fb9f5e76c92a4cd8f', 'pdfurls': ['https://export.arxiv.org/pdf/math/0508193v1.pdf'], 'title': ['A sufficient condition for a set of calibrated surfaces to be area-minimizing under diffeomorphisms', 'A sufficient condition for a set of calibrated surfaces to be area-minimizing under diffeomorphisms'], 'venue': []}
arxiv
Topics in the Haystack: Extracting and Evaluating Topics beyond Coherence Anton Thielmann [email protected] Chair of Data Science and Applied Statistics TU Clausthal Germany Quentin Seifert Chair of Spatial Data Science and Statistical Learning University of Göttingen Germany Arik Reuter Chair of Data Science and Applied Statistics TU Clausthal Germany Elisabeth Bergherr Chair of Spatial Data Science and Statistical Learning University of Göttingen Germany Benjamin Säfken Chair of Data Science and Applied Statistics TU Clausthal Germany Topics in the Haystack: Extracting and Evaluating Topics beyond Coherence Extracting and identifying latent topics in large text corpora has gained increasing importance in Natural Language Processing (NLP). Most models, whether probabilistic models similar to Latent Dirichlet Allocation (LDA) or neural topic models, follow the same underlying approach of topic interpretability and topic extraction. We propose a method that incorporates a deeper understanding of both sentence and document themes, and goes beyond simply analyzing word frequencies in the data. This allows our model to detect latent topics that may include uncommon words or neologisms, as well as words not present in the documents themselves. Additionally, we propose several new evaluation metrics based on intruder words and similarity measures in the semantic space. We present correlation coefficients with human identification of intruder words and achieve near-human level results at the word-intrusion task. We demonstrate the competitive performance of our method with a large benchmark study, and achieve superior results compared to state-of-the-art topic modeling and document clustering models. Introduction Identifying latent topics in large text corpora is a central task in Natural Language Processing (NLP). With the ever-growing availability of textual data in virtually all languages and about every possible topic, automated topic extraction is gaining increasing importance. Hence, the approaches are manifold. For almost all models, a topic is intuitively defined by a set of words with each word having a probability of occurrence for the given topic. Different topics can share words, and a document can be linked to more than one topic. Generative probabilistic models, such as probabilistic latent semantic analysis (PLSA) [23] and Latent Dirichlet Allocation (LDA) [12], are still widely used and inspired multiple adaptations as e.g. [1,11,14,38, 41] all drawing heavily from word-co-occurrences. Due to its popularity and general good performance on benchmark datasets, the interpretation of a topic from LDA is seldomly challenged. Neural topic models, like e.g. [18,52], further improve upon the existing methods by integrating word-embeddings or variational autoencoders [45] into the modeling approach, but still heavily rely on the ideas from [12]. New methods that challenge the typical idea of topic modeling also integrate wordand document-embeddings [4,20,43]. However, improvement over the current state of the art is usually measured in terms of performance as determined by evaluation arXiv:2303.17324v1 [cs.CL] 30 Mar 2023 metrics on standard benchmark datasets. While older models were still evaluated using likelihood-based perplexity metrics [28,29,41], empirical results showed a negative correlation between perplexity based metrics and human evaluation of a topic model [13]. Additionally, Chang et al. [13] first introduced the idea of intruder words. According to this idea, a topic is considered coherent or simply put, good, if a randomly chosen word, not belonging to that topic, can clearly be identified by humans. As human evaluation of models is cost and time intensive, researchers used new evaluation methods that correlated with human evaluation [30,37]. Hoyle et al. [24] even found no contemporary model at all that used human feedback as a form of model evaluation. Newer models were hence evaluated using coherence scores [4,18,20,43,45]. However, Hoyle et al. [24] found severe flaws in coherence scores. First, they find that coherence scores exaggerate differences between models and second, they validate the findings from Bhatia et al. [6] and find much lower Pearson correlations between automated coherence scores and human evaluation as compared to [30]. We identify two shortcomings in the current state-of-the-art in topic modelling. The first is the significant gap in validated automatic evaluation methods for topic models. The second stems from the continued reliance on evaluation methods based on word co-occurrences and outdated definitions of topics from older models. Current methods rely on limited corpora from which the topic representations are created. However, integrating larger corpora into the modeling process can enhance topic quality by including contextually relevant words that were missing from the original corpus. Contributions The contributions of this paper are hence twofold and can be summarized as follows: -We propose the Context Based Topic Model (CBTM) that, with only a few adaptations, integrates linguistic ideas into its modeling. Soft-clustering on the document level is integrated, such that P(document | topic) is modeled. -We introduce new topic modeling performance metrics. The validation of the proposed metrics is validated by demonstrating impressive correlations with human judgement. -We conduct a benchmark study comparing the presented approach to state-of-theart topic modeling and document clustering methods and outperform common benchmark models on both, coherence scores and the presented new metrics for topic evaluation. The remainder of the paper is structured as follows: First, a short introduction into the used linguistic ideas and the definition of topics is presented. Second, the method of extracting latent topics from documents, incorporating the aforementioned definitions, is presented. Third, new evaluation metrics are introduced and validated by presenting correlations with human annotators. Fourth, the proposed model is applied to two common data sets and compared with state-of-the-art topic models. Finally, a discussion of the limitations as well as a conclusion is given in sections 6 and 7. The word best representing a sentence (or document) does not necessarily needs to be included in that text. The figure represents a New York Times headline from the financial crisis in 2009: "Lehman had to die so Global Finance could live". All words present in that text and additional words are mapped into a high dimensional feature space. The dimensions are reduced to visually demonstrate, that words not occurring in that sentence, e.g. banking crisis are better suited to summarize that sentence than words present in the sentence, e.g. global. 2 On the Nature of Topics While there have been numerous approaches to extracting latent topics from large text corpora, little effort has been made in adapting those models to more refined definitions of a topic. We propose a topic model that follows ideas from linguistic definitions of topics [16,17]. We present two ideas from linguistic theory in order to construct more humanly interpretable topics: i) A word that most accurately expresses the topic of a document may not necessarily occur in that document. ii) Only using nouns and noun phrases is more appropriate for representing understandable topics. i) closely follows Guijarro [21]: "a topic is, above all, a textual category that is determined by the context and not by purely formal or structural aspects." Therefore, the topic of a document or even a sentence may go beyond the mere occurrence of all the words in that document. That is, a word that most accurately expresses the topic of a document may not necessarily occur in that document. We leverage a simple example from a New York Times headline to demonstrate that: "Lehman had to die so Global Finance could live" That sentence pertains to the financial crisis and the collapse of the Lehman Brothers bank, but neither phrase is explicitly mentioned. A bag-of-words model that only considers words present in the document corpus would not be able to accurately capture the document's topic. Contextually relevant words, even if not present in the document, can provide better representations. Figure 1 shows the described example. Comparing the cosine distance in a reduced embedding space between the complete embedded sentence (TEXT) and each embedded word demonstrates how words and phrases not occurring in that text can be a meaningful summary of that text. "Banking crisis" is a more meaningful representation of the sentence than e.g. "global" and lies closer to the text in the semantic space. Common topic models, such as [9,12,18,45], as well as document clustering methods, such as [4,20,43], face a limitation in that they only consider words that appear in the reference corpus when generating topic representations. This limitation can lead to incorrect topic interpretations, as shown in the example above. Through expanding the reference corpus and leveraging pre-trained embedding models, we make sure that "the indispensability of frame knowledge for understanding texts" [5] is accounted for. ii) closely follows Beghto [5], after whom one of the features of generalized titles is the absence of verbal forms. Following the idea that a title is the highest macroproposition of a textual unit [5], we apply this idea to the construction of topics and hence propose to only consider nouns and noun phrases for the proposed method of topic extraction. Methodology Let V = {w 1 , . . . , w n } be the vocabulary of words and D = {d 1 , . . . , d M } be a corpus, i.e. a collection of documents. Each document is a sequence of words d i = [w i1 , . . . , w ini ] where w ij ∈ V and n i denotes the length of document d i . Further, let D = {δ 1 , . . . , δ M } be the set of documents represented in the embedding space, such that δ i is the vector representation of d i and let W = {ω 1 , . . . , ω n } be the vocabulary's representation in the same embedding space. Hence, each word w i in the embedding space represented as ω i ∈ R L has the same dimensionality L as a document vector δ i ∈ R L . There are different representations of topics, but mostly a topic t k from a set of topics T = {t 1 , . . . , t K } is represented as a discrete probability distribution over the vocabulary [12], such that t k is often expressed as (φ k,1 , . . . , φ k,n ) T and n i=1 φ k,i = 1 for every k 3 . Based upon the idea expressed in section 2, we form clusters from the documents embeddings, D and subsequently extract topics, t k , that represent these clusters best. Hence, after transforming the raw documents into document vectors, they are clustered. Due to the curse of dimensionality [2] we reduce the dimensions before clustering using UMAP [36], closely following [4] and [20]. However, we allow each document to belong to more than one cluster resulting in document topic matrices θ and word topic matrices β, similar to LDA [12]. The documents are clustered with a Gaussian mixture model [40], as it not only allows for soft-clustering, but also has the advantage of optimizing hyperparameters via, for instance, the Akaike information criterion or the Bayesian information criterion. As a results, CBTM, in contrast to [4,20,43] offers not only word-topic distributions but also document-topic distributions. Topic Extraction To find the words that best represent the corpus' topics, we first extract the centroids of the k clusters, µ k ∈ R L , in the original embedding space. Second, we filter the given vocabulary for nouns and enhance this vocabulary by any specified external vocabulary of nouns, resulting in a new dictionaryV = {w 1 , . . . , w n , w n+1 , . . . , w n+z }. The word vectors ω i closest to µ k in the embedding space, are the words that represent cluster k's centroid best [4], where it could happen, that a word represents a topic ideally where w / ∈ V but always w ∈V . To compute the words best representing a topic, we compute the cosine similarity between every word inV and all cluster centroids in the embedding space. For a single word w, its embedding ω and a single cluster with centroid µ, we hence compute: sim(ω, µ) = ω · µ ω µ ,(1) where ω · µ = L i=1 ω i µ i and ω µ = L i=1 (ω i ) 2 L i=1 (µ i ) 2 . L denotes the vectors dimension in the feature space which is identical for ω and µ. To avoid having words in a topic that are semantically overly similar, as e.g. economics and economy, each topic can be cleaned. The cosine similarity between the top Z words contained in a topic can be computed and all words that exceed a certain threshold, e.g. 0.85 4 , are removed in descending order of the similarity with the clusters centroid. An additional advantage of the corpus expansion is the possibility to model documents in one language, but create topics in a different language, when using a multi-language embedding model. Evaluation Given the described approach, we are effectively losing any idea of co-occurence based coherence for model evaluation. The words best describing a cluster of documents or topic do not necessarily have to occur together often in documents. In fact, a word capturing the topic of a single document optimally, does not necessarily have to be contained in that same document. Additionally, by enhancing the corpus, it might be possible that neologisms are the words best representing a topic. Imagine, e.g. a set of documents being equally about software and hardware issues. The neologism softwarehardware would be an understandable and reasonable word describing that topic, but would perform poorly in any word-co-occurence based evaluation measure. Fig. 2. The expressivity of a model is captured by averaging over the topics centroids cosine similarity to the null space, defined as the centroid of all embedded stopwords. For visualization the vector dimensions are heavily reduced, but the overall expressivity is still visualized. Due to the dimensionality reduction, the axes are just labelled "X" and "Y" respectively. The visualized topics are created from the 20 Newsgroups data set with the CBTM method and a single topic, "would", created with a LDA model. The topic's top word is annotated at the topic's position in the reduced embedding space. Evaluation Metrics For evaluation, we hence propose new, non word-co-occurence based measures and use existing measures leveraging word embeddings [47]. We validate the intruder based metrics by computing correlations with human annotations. Topic Expressivity (EXPRS) First, we propose a novel measure inherently representing the meaningfulness of a topic. For that, we leverage stopwords, which widely recognized fulfill a grammatical purpose, but transport nothing about the meaning of a document [42,54]. Hence, we compute the vector embeddings of all stopwords and calculate a centroid embedding. Subsequently, we compute the cosine similarity between a topic centroid and the stopword centroid (see Figure 2). The weighted topic vector centroid, γ k , is computed by taking the top Z words and normalizing their weights, such that Z i=1 φ k,i = 1. The complete vector is hence computed as γ k = 1 Z Z i=1 φ k, i ω i and the overall metric, which we call the models expressivity, where we sum over all K topics is defined as: EXP RS(γ, ψ) = 1 K K i=1 sim(γ i , ψ)(2) with ψ being the centroid vector representation of all stopwords. Note, that γ i = µ i , as µ i is the centroid of the document cluster and γ i is the centroid of topic t i . Fig. 3. The intruder word detection in the embedding space. A topic, covering "religion", and an intruder word, "medicine" are plotted with heavily reduced dimensions, using a PCA. The intruder word clearly separates from the otherwise coherent topic, even in a two-dimensional space. Due to the dimension reduction, the axis are just labelled with "X" and "Y" respectively. The topic is again created with the CBTM method on the 20 Newsgroups data set. Embedding Coherence (COH) A measure, generally introduced by Aletras and Stevenson [3] and reformulated by Fang et al. [19] resembling classical coherence scores, is constructed by computing the similarity between the top Z words in a topic. While Aletras and Stevenson [3] compute the word vectors using word co-occurrences we follow Fang et al. [19] and use the created word-embeddings. In contrast to classical coherence, we compute the similarity between every top-Z word in the topic and do not implement a sliding-window approach. Hence, for Z words, we sum over Z(Z−1) 2 cosine similarities: COH(t k ) = Z−1 i=1 Z j=i+1 sim(ω i , ω j ),(3) where the overall average coherence of a model is hence computed as: 2 K(Z − 1)Z K k=1 COH(t k ). Word embedding-based Weighted Sum Similarity (WESS) A metric representing the diversity or the similarity between the topics of a topic model was introduced by [47] as the Word embedding-based Weighted Sum Similarity and is slightly adjusted for comparing models with a different number of topics as: W ESS(T ) = (K − 1)K 2 K−1 i=1 K j=i+1 sim(γ i , γ j ),(4) where γ i represents the weighted topic centroid for topic i. While this metric certainly captures the similarity between topics, it does also reflect the diversity of the model. Hence, if W ESS(T ) is close to 1, the model would have created topics that are extremely similar to one another. Additionally, we propose three different new metrics, leveraging the idea of intruder words [13] and similarly integrating an idea of topic diversity. First, a metric that is based upon unweighted topic centroids. Intruder Shift (ISH) Given the top Z words from a topic, we calculate the topics unweighted centroid, denoted asγ i . Subsequently, we randomly select a word from that topic and replace it with a randomly selected word, from a randomly selected different topic. The centroid of the resulting words is again computed, denoted asγ i . Given a coherent topic and generally diverse topics, one would expect a larger shift in the topics centroids. Therefore we calculate the intruder shift of every topic and average over the number of topics: ISH(T ) = 1 K K i=1 sim(γ i ,γ i )(5) Hence, one would expect a coherent and diverse topic model to have a lower ISH score than an incoherent and non-diverse topic model. Intruder Accuracy (INT) The second intruder-word based metric follows the classical approach of identifying an intruder word more closely. Given Z top words of a topic, we again randomly select an intruder word from a randomly drawn topic. Subsequently, we calculate the cosine similarity for every possible pair of words within the set of the top Z words. Then we calculate the cosine similarity of each top word and the intruderω. Finally, our metric reports the fraction of top words to which the intruder has the least similar word embedding. IN T (t k ) = 1 Z Z i=1 1(∀j : sim(ωi,ω) < sim(ωi, ωj))(6) Hence we return the number of words from the set where the farthest word from them in the embedding space is the intruder word, divided by the number of words, Z, taken into account (See Figure 3 for a visualization). Average Intruder Similarity (ISIM) As a last metric, we propose the average cosine similarity between every word in a topic and an intruder word: ISIM (t k ) = 1 Z Z i=1 sim(ω i ,ω)(7) To account for any induced randomness in the metrics ISH, IN T and ISIM due to the random choice of a particular intruder from a particular topic, we propose to calculate those metrics multiple times with differently chosen random intruder words and subsequently average the results. Hence, the robustness against the specific selection of intruder words is increased. Validation of Metrics To validate the intruder word based evaluation metrics we take the publicly available data from Chang et al. [13]. Similar to Lau et al. [30] we compute the metrics over all topics and all models provided in [13] for the 20 Newsgroups dataset. However, for clear interpretability, we reduce all words that include hyphens, due to the representations from [13]. Hence, we compute the metrics for 7,004 topics in total. We compute the accuracy of the metrics in terms of the true intruder and the humanly detected intruder for all metrics as well as the Pearson-r. While the important measures are here the correlation with the human annotations, reporting the correlations with the true intruder word ensures that the metrics are not inherently biased towards machine selection. For the accuracy, we consider a pre-selected or human-selected intruder to be correctly identified, if the score for this word is the lowest or highest, respectively, among all displayed top words. The results are shown in Table 1. For all results it must be noted that the human answers have some ambiguity in them. As reported by Lau et al. [30], the Pearson-r between the human answers was 0.77. Hence, the results for IN T with a maximum correlation of 0.728 is highly credible and outperforms the reported correlations [30] for coherence evaluation metrics. Interestingly, ISIM performs best, when considering the accuracy for the true intruder word, but significantly worse when considering the human selected word. We find that, independent of the chosen model, the newly introduced metrics strongly outperform the results reported by Lau et al. [30] at the topic-level with reported Pearson correlations of around r = 0.6. Results To evaluate the proposed model, we compare the model results with different benchmark models. We also demonstrate the validity of our two hypotheses on corpus expansion and noun phrases stated in Section 2. As comparison models, we use BERTopic [20] and Top2Vec [4] as closely related models and representatives of clustering based topic models, LDA [12] as a model not leveraging pre-trained embeddings, CTM [9] as a generative probabilistic model leveraging pre-trained embeddings, a simple K-Means model -closely following the architecture from [20], but replacing HDBSCAN with a K-Means clustering approach, ETM [18] leveraging word2vec [31] and NeuralLDA and ProdLDA [45]. All models are Table 1. Metric Evaluation: Accuracy and Pearson correlation with the reported true (Intruder) and humanly selected (Human) intruder word from Chang et al. [13] for all models and all topics on the 20 Newsgroups dataset. As embedding models we consider the Paraphrase-MiniLM-L6-v2 model [39], the All-MiniLM-L12-v2 model [53], the All-mpnet-base-v2 model [44], the Multiqa-mpnet-base-dot-v1 model [44] and the All-distilroberta-v1 model [34] as well as a word2vec model pre-trained on the GoogleNews corpus and a Glove model pre-trained on a Wikipedia corpus. The three best results for the human correlation and accuracy are marked in bold. One can see that the metric evaluation for different embedding models produces impressive results, given the correlation between participants of 0.77. The paraphrase-MiniLM-L6-v2 performs best, considering IN T and ISIM , closely followed by the Glove model. fit using the OCTIS framework [46]. Where applicable the same pre-trained embedding model as for CBTM, all-MiniLM-L6-v2 [39] is used. Note, that we perform extensive hyperparameter tuning for all models except for CBTM. A detailed description of the benchmark models hyperparameters and the hyperparameter tuning can be found in the Appendix. As a corpus expanding the reference corpus in CBTM for topic extraction we use the Brown corpus taken from nltk [10], which we also use for filtering the vocabulary for noun-phrases. We compute the proposed metrics from Section 4 except for the ISH metric due to its inferior performance on the intruder word detection task (Table 1). Additionally, we compute normalized pointwise mutual information (NPMI) scores [30] with the input corpus as the reference corpus and Topic Diversity (WESS) and Wordembedding Pairwise Coherence scores (COHPW) using the OCTIS framework [46]. All word-embedding based metrics are computed with the paraphrase-MiniLM-L6-v2 model [39] due to the results from Table 1, except for WESS and COHPW where we use OCTIS' default pre-trained word2vec [31] model 5 . Table 2. Comparison of noun-based topic extraction vs. non-noun-based model extraction for the CBTM model. The reported metrics are averaged over the results for three datasets, the 20 Newsgroups dataset, the BBC News dataset and the M10 dataset. All datasets are taken from OCTIS. All models are fitted using the all-MiniLM-L6-v2 model [39]. Given the results from Table 1, paraphrase-MiniLM-L6-v2 is used for the embedding based evaluation metrics. We report the baseline metrics for a model not using an expanded corpus and using all word types and report the differences to that baseline. We find that especially expanding the reference corpus leads to better topics, represented by nearly all metrics. As expected, the NPMI coherence scores are considerably worse, when expanding the reference corpus. That is due to the fact, that we used the original corpus the models where fit on as the NPMI coherence reference corpus. Additionally, we find that only considering nouns for topic words, can increase the evaluation metrics, especially when we clean the topics. To confirm our two hypotheses from Section 2 that expanding the reference corpus and only considering nouns for topic extraction can increase the topic quality, we perform several analyses. We compare the presented method with and without reference corpus expansion and with and without noun phrase filtering. The averaged results over 3 datasets can be seen in Table 2. Hypothesis I: Corpus Expansion Our results confirm our hypothesis that expanding the reference corpus leads to creating better topics depicted by nearly all metrics. Unsurprisingly, we find that NPMI coherence scores, only using the reference corpus for computing the coherence are decreased when expanding the reference corpus during topic extraction. Additionally, we find that using a smaller pre-trained model for computing the metrics, as the leveraged word2vec [31] model for COHPW and WESS also shows a decrease in performance when expanding the reference corpus. That is presumably due to the smaller vocabulary size used in these models. Hypothesis II: Noun Phrases We find that the noun-based models perform worse than the models that consider all types of words and for the different embedding models used to construct the evaluation metrics. However, we find that when cleaning the topics the topic quality increases when using only nouns as compared to using all word types. Additionally we find that expanding the reference corpus and only considering nouns achieves better performance than no expansion and using all word types. Benchmarks For comparing CBTM with other models we use two standard benchmark datasets, 20 Newsgroups and Reuters [33] as shown in Table 3. We fix the number of topics to the true number of topics of 20 and 90, respectively (see Appendix for additional benchmarks on two further datasets). CBTM outperforms all models, concerning INT, COH and COHPW for both datasets for all configurations. Additionally, CBTM performs well on topic diversity for the 20 Newsgroups dataset and EXPRS for both datasets. Interestingly, it also performs very well concerning classical NPMI coherence scores for the 20 Newsgroups dataset when not expanding the reference corpus. As expected, the models closely related to CBTM perform also well on both datasets. However, while Top2Vec, BERTopic and the used K-Means model are closely related to the proposed CBTM, CBTM achieves much better results concerning all metrics. Interestingly, CTM performs very well on smaller datasets (see supplemental material for additional benchmarks). Additionally, our results do not confirm that models that use a hard clustering approach perform considerably worse for a multi-label dataset (Reuters) as compared to models that integrate soft-clustering (see e.g. CTM/ETM vs Top2Vec/BERTopic results). Conclusion We develop a novel model for topic extraction beyond the mere occurrence of words in the reference corpus. We are able to show that expanding the reference corpus improves model performance. Additionally, we can confirm, that restricting the word types for Table 3. Benchmark results on the 20 Newsgroups and Reuters datasets. All models are fit using the all-MiniLM-L6-v2 pre-trained embedding model [39] where applicable. paraphrase-MiniLM-L6-v2 is used for the evaluation metrics ISIM, INT, TOP DIV and EXPRS. For the metrics available in OCTIS we use the default embeddings which are pre-trained word2vec embeddings on the Google News corpus. Extensive hyperparameter tuning is performed for the comparison models (see Appendix). All models, except BERTopic and Top2Vec, are fit with a pre-specified number of 20 or 90 topics respectively. BERTopic and Top2Vec detect the optimal number of topics automatically, hence we fit the model as intended by the authors. However, we additionally fit a K-Means model using the class based tf-idf topic extraction method from BERTopic with 20 and 90 topics respectively and hierarchically reduce the number of topics in Top2Vec. topic extraction by only considering nouns can also lead to improved topic quality, under certain conditions. CBTM outperforms commonly used state-of-the-art topic models on multiple benchmark datasets, even in cases where the comparison models underwent extensive hyperparameter tuning while no hyperparameter tuning was performed for CBTM (see supplemental material for details on the hyperparameter tuning). Given that almost all newly introduced topic models are evaluated automatically [24], automatic evaluation metrics are of outmost importance. Hoyle et al. [24] even postulated that automatic topic model evaluation is broken, as the current used metrics have overall low correlations with human judgement of topic quality. We present multiple novel evaluation metrics closely following state of the art human evaluation of topic model quality and achieve great correlations with human evaluation. We greatly improve upon the correlation with human evaluation compared to the currently most often used metric, NPMI, achieving correlations of around r = 0.73 compared to NPMI correlations of r = 0.63. The proposed approach of using word embeddings and cosine similarity achieves impressive results given the overall lower agreement between human responses (Pearson-r=0.77). Additionally, we introduce a novel evaluation metric, based upon the centroid cluster of stopwords in the embedding space. Given the approach of enhancing the reference corpus, the described model might be especially useful when evaluating short texts or identifying sparsely represented topics in a corpus [48,49]. Through the inherent sparsity of the data, the words best describing a topic might not be included in the reference corpus and an enhancement could thus greatly improve the creation of topics. Limitations Automated evaluation of topic model quality is inherently difficult. That difficulty is considerably increased by the fact there is no gold standard or even a ground truth for the quality of a topic. Chang et al. [13] introduced the reasonable approach of evaluating the coherence of a set of words with intruder-words. However, one cannot expect 100% agreement between people when it comes to judging whether a word is an intruder word in a topic. The proposed evaluation metrics achieve impressive results with human annotations, they cannot, however, reflect human ambiguity or extreme subtlety in perceived topic quality. Additionally, as all evaluation metrics based upon human evaluation and hence experimental results achieved with human participants, the metrics might reflect a selection bias (WEIRD) [22]. Further embedding models could be evaluated and tested and larger human evaluation studies could be conducted. Recent findings about the dominance of certain dimensions in transformer embeddings [51] suggest an inherent bias in transformer embeddings that could negatively affect similarity measures in the semantic space. Our results do not suggest that such a bias negatively influences the modeling results, however, this study does not look into the dimensionality effects which could be the topic of further research. Moreover, the creation of transformer models solely for the purpose of topic extraction that emphasize, for example, the beginnings of phrases due to their increased importance to the underlying topics of a subsubsection [25, 26] could greatly improve upon the existing methods. A Supplemental Methodology To make reading easier, we provide a full notation list. All used variables and their notation can be found here. All modeling steps from the proposed method are presented here in extensive form. First, the target corpus should be embedded. This can be done, either using contextualized transformer embeddings, as e.g. Bianchi et al. [8] showed that contextualized embeddings can improve topic quality. However, approaches as used by Sia et al. [43] where every word is embedded singularly and the documents are represented as centroid vectors of all occurring words are also possible. Second, the dimensions of the embedded documents, δ i , are reduced due to the curse of dimensionality. Afterwards, the reduced embeddings, δ i , are clustered e.g. using GMM such that soft clustering is possible. The centroids for each document cluster, µ k , are computed. Next, the corpus is filtered for nouns and all nouns present in the corpus supplemented by all nouns present in an expansion corpus are embedded. Note, that here the same embedding procedure must be chosen as for the documents ( see e.g. [4,20]). Then, the similarity between all candidate words and all document cluster centroids is computed. Based on the candidate embeddings and the similarity to the document clusters µ k , the topic centroids γ k are computed and similar to LDA, we get a document topic matrix, θ, and a word topic matrix, β. Last, a cleaning step can be performed to remove overly similar words from the topics. B Human Topic Evaluation As automated evaluation of topic model quality is inherently difficult, creating great questionnaires and adequately operationalizing what researchers are interested in is adamantly important. Lund et al. [35] introduced a topic-word matching task, weighting and selecting answers from participants that have a high confidence and performed well on test questions. Choosing that approach reduces ambiguity in answers, but also induces a bias towards highly confident participants and neglects the subtle differences in perceived quality from humans. [37], chose a straight-forward approach of letting humans rate the created topics quality. Choosing a 3-point scale for model evaluation, however, can induce unreliability of responses [27]. [6,7] introduce a document-level topic model evaluation leveraging the intruder-topic task, also introduced in Chang et al. [13]. However, for direct annotation they also resort to a 3-point ordinal scale. Clark et al. [15] even question human judgement all together; however, the used questionnaire design not only does not provide a midpoint but additionally can strongly induce a bias in preference due to a highly biasing follow up question [15] (See e.g. [32]). B.1 Additional Benchmark Results In addition to the 20 Newsgroups and Reuters dataset, we fit all models on the M10 and BBC News datasets. Both datasets are taken from OCTIS [46]. CBTM again outperforms most other models on nearly all metrics. Interestingly, CTM achieves good results for the BBC News dataset, which is comparably small with <2.000 documents. For the M10 dataset, which is comprised of scientific papers and hence a more difficult dataset, we find that topic expansion strongly improves the model performance. C Experimental Setup For all tested models, we use the same pre-trained embedding model all-MiniLM-L6-v2 [39], where applicable. NPMI Coherence scores are calculated as presented by [30]. For the best possible comparison, we use the same dimensionality reduction for CBTM as is used in Doc2Vec [4] and BERTopic [20]. Hence, we use Umap [36] and reduce the dimensions to 5, explicitly using the same hyperparameters as done in the mentioned models. The same is done for the simple K-Means model. C.1 Hyperparameter Tuning For CBTM we do not implement any form of hyperparameter tuning. Hence, the Gaussian Mixture Model is fit using scikit-learns default parameters. Hence the convergence threshhold for the Expectation Maximization (EM) Algorithm is 0.0001, each component has its own general covariance matrix and 1e-6 is added to the covariance diagonals for regularization purposes. The maximum number of iterations in the EM algorithm is set to 100 and K-Means is used to initialize the weights. Hence, the results achieved by CBTM could be further optimized, by e.g. optimizing GMM with respect to the Bayesian-or Akaike Information Criterion. Additionally, the pre-trained embedding could be fine-tuned, which is true for all models leveraging pre-trained embeddings and could additionally improve the models performance [8,50]. For LDA, ProdLDA, NeuraLLDA, ETM and CTM, we optimize over various hyperparameters with Bayesian optimization as provided by the OCTIS package [46]. We use Table 5. Benchmark results on the M10 dataset. All models are fit using the all-MiniLM-L6-v2 pre-trained embedding model [39] where applicable. paraphrase-MiniLM-L6-v2 is used for the evaluation metrics ISIM, INT, TOP DIV and EXPRS. For the metrics available in OCTIS we use the default embeddings which are pre-trained word2vec embeddings on the Google News corpus. Extensive Hyperparameter tuning is performed for the comparison models (See Appendix). All models, except BERTopic and Top2Vec, are fit with a pre-specified number of 10 topics. BERTopic and Top2Vec detect the optimal number of topics automatically, hence we fit the model as intended by the authors. However, we additionally fit a KMeans model using the class based tf-idf topic extraction method from BERTopic with 10 topics and hierarchically reduce the number of topics in Top2Vec. model perplexity, measured based on the evidence lower bound of a validation sample of documents, as the objective function in order to not rely on metrics, such as NPMI coherence or WESS, that measure either cohesion or separation of topics. LDA is optimized over the parameters of the two symmetric Dirichlet priors on the topic-specific word distribution and the document-specific topic distribution. For ProdLDA, NeuralLDA and CTM, the learning rate parameter, as well as the number of layers and the number of neurons per layer in the inference network are considered. Finally, for ETM, we tune the learning rate, the number of hidden units in the encoder and the embedding size. Since BERTopic and Top2Vec are highly insensitive to different hyperparameter settings of the underlying HDBSCAN algorithm and also do not provide a way to measure the (marginal) likelihood of data, we choose the default hyperparameters for those models. While finding the optimal hyperparameters for these models might improve their performances compared to the models where we implemented hyperparameter tuning, the same is true for CBTM. Table 6. Benchmark results on the BBC News dataset. All models are fit using the all-MiniLM-L6-v2 pre-trained embedding model [39] where applicable. paraphrase-MiniLM-L6-v2 is used for the evaluation metrics ISIM, INT, TOP DIV and EXPRS. For the metrics available in OCTIS we use the default embeddings which are pre-trained word2vec embeddings on the Google News corpus. Extensive Hyperparameter tuning is performed for the comparison models (See Appendix). All models, except BERTopic and Top2Vec, are fit with a pre-specified number of 10 topics. BERTopic and Top2Vec detect the optimal number of topics automatically, hence we fit the model as intended by the authors. However, we additionally fit a KMeans model using the class based tf-idf topic extraction method from BERTopic with 5 topics and hierarchically reduce the number of topics in Top2Vec. interpretation, truth, assert, argue, claim, consideration, logic, insist, complain, belief 20 secure, encryption, security, encrypt, privacy, protect, protection, scheme, enforcement, access Table 7. The CBTM model fit on the 20 Newsgroups dataset. The reference corpus is expanded with the brown corpus taken from the nltk package [10]. Coherence Measures Fig. 1 . 1Fig. 1. The word best representing a sentence (or document) does not necessarily needs to be included in that text. The figure represents a New York Times headline from the financial crisis in 2009: "Lehman had to die so Global Finance could live". All words present in that text and additional words are mapped into a high dimensional feature space. The dimensions are reduced to visually demonstrate, that words not occurring in that sentence, e.g. banking crisis are better suited to summarize that sentence than words present in the sentence, e.g. global. (↑) COHPW (↑) COH (↑) TOP DIV (↑) WESS (↓) EXPRS (↓) ISIM (↓) INT (↑) (↑) COHPW (↑) COH (↑) TOP DIV (↑) WESS (↓) EXPRS (↓) ISIM (↓) INT (↑) 21 . 21Guijarro, A.J.M.: Towards a definition and hierarchization of topic. Talk and Text: Studies on Spoken and Written Discourse, ed. by A. Rothwell, A. Guijarro & J. Albentosa pp. 97-116 (2000) 22. Henrich, J., Heine, S.J., Norenzayan, A.: Most people are not weird. Nature 466(7302), 29-29 (2010) 23. Hofmann, T.: Unsupervised learning by probabilistic latent semantic analysis. Machine learning 42(1), 177-196 (2001) 24. Hoyle, A., Goel, P., Hian-Cheong, A., Peskov, D., Boyd-Graber, J., Resnik, P.: Is automated topic model evaluation broken? the incoherence of coherence. Advances in Neural Information Processing Systems 34 (2021) 25. Kieras, D.E.: Initial mention as a signal to thematic content in technical passages. Memory & Cognition 8(4), 345-353 (1980) 26. Kieras, D.E.: Topicalization effects in cued recall of technical prose. Memory & Cognition 9(6), 541-549 (1981) 27. Krosnick, J.A.: Questionnaire design. In: The Palgrave handbook of survey research, pp. 439-455. Springer (2018) 28. Lafferty, J., Blei, D.: Correlated topic models. Advances in neural information processing systems 18 (2005) 29. Larochelle, H., Lauly, S.: A neural autoregressive topic model. Advances in Neural Information Processing Systems 25 (2012) 30. Lau, J.H., Newman, D., Baldwin, T.: Machine reading tea leaves: Automatically evaluating topic coherence and topic model quality. In: Proceedings of the 14th Conference of the European Chapter of the Association for Computational Linguistics. pp. 530-539 (2014) 31. Le, Q., Mikolov, T.: Distributed representations of sentences and documents. In: International conference on machine learning. pp. 1188-1196. PMLR (2014) 32. Lehman, D.R., Krosnick, J.A., West, R.L., Li, F.: The focus of judgment effect: A question wording effect due to hypothesis confirnation bias. Personality and Social Psychology Bulletin 18(6), 690-699 (1992) 33. Lewis, D.D.: Reuters-21578 text categorization collection data set (1997) 34. Liu, Y., Ott, M., Goyal, N., Du, J., Joshi, M., Chen, D., Levy, O., Lewis, M., Zettlemoyer, L., Stoyanov, V.: Roberta: A robustly optimized bert pretraining approach. arXiv preprint arXiv:1907.11692 (2019) 35. Lund, J., Armstrong, P., Fearn, W., Cowley, S., Byun, C., Boyd-Graber, J., Seppi, K.: Automatic evaluation of local topic quality. arXiv preprint arXiv:1905.13126 (2019) 36. McInnes, L., Healy, J., Melville, J.: Umap: Uniform manifold approximation and projection for dimension reduction. arXiv preprint arXiv:1802.03426 (2018) 37. Newman, D., Lau, J.H., Grieser, K., Baldwin, T.: Automatic evaluation of topic coherence. In: Human language technologies: The 2010 annual conference of the North American chapter of the association for computational linguistics. pp. 100-108 (2010) 38. Ramage, D., Hall, D., Nallapati, R., Manning, C.D.: Labeled lda: A supervised topic model for credit attribution in multi-labeled corpora. In: Proceedings of the 2009 conference on empirical methods in natural language processing. pp. 248-256 (2009) 39. Reimers, N., Gurevych, I.: Sentence-bert: Sentence embeddings using siamese bert-networks. In: Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing. Association for Computational Linguistics (11 2019), https://arxiv.org/abs/ 1908.10084 40. Reynolds, D.A.: Gaussian mixture models. Encyclopedia of biometrics 741(659-663) (2009) 41. Rosen-Zvi, M., Griffiths, T., Steyvers, M., Smyth, P.: The author-topic model for authors and documents. arXiv preprint arXiv:1207.4169 (2012) 42. Salton, G.: Automatic text processing: The transformation, analysis, and retrieval of. Reading: Addison-Wesley 169 (1989) (↑) COHPW (↑) COH (↑) TOP DIV (↑) WESS (↓) EXPRS (↓) ISIM (↓) INT (↑) (↑) COHPW (↑) COH (↑) TOP DIV (↑) WESS (↓) EXPRS (↓) ISIM (↓) INT (↑) Table 4 . 4Variable listV Vocabulary D Corpus M Number of documents in the corpus di Document i wi Word i in V ωi Word i represented in the embedding space δi Document i represented in the embedding spacê δi di represented in the reduced embedding space t k Topic k T Set of topics φ k,i Probability of word i in topic k γ k Topic centroid vector of topic k µ k Mean of document cluster k θ Document cluster/topic matrix β Word cluster/topic matrix ψ Null Space/centroid of all stopwords HDBSCAN results with > 10 topics * Only Nouns + Expanded topic corpusKmeans -0.108 0.063 0.254 0.940 0.354 0.458 0.149 0.320 BERTopic † -0.318 0.056 0.231 0.628 0.424 0.514 0.165 0.219 Top2Vec † -0.345 0.083 0.315 0.060 0.547 0.478 0.220 0.326 TOP2Vec -0.270 0.100 0.335 0.780 0.496 0.454 0.198 0.484 LDA -0.176 0.035 0.244 0.830 0.330 0.440 0.208 0.177 ProdLDA -0.251 0.074 0.222 0.970 0.425 0.508 0.170 0.220 NeuralLDA -0.571 0.030 0.186 0.373 0.582 0.581 0.185 0.118 ETM -0.204 0.044 0.255 0.330 0.591 0.500 0.268 0.151 CTM -0.322 0.060 0.239 0.950 0.247 0.353 0.172 0.271 CBTM + -0.8411 0.338 0.512 0.855 0.322 0.383 0.179 0.827 CBTM * -0.5762 0.441 0.419 0.770 0.394 0.420 0.193 0.719 CBTM * + -0.8033 0.358 0.451 0.825 0.339 0.395 0.166 0.818 † HDBSCAN results with > 5 topics * Only Nouns + Expanded topic corpus Topic Words 1 game, league, player, play, baseball, sport, pitch, hockey, team, batf 2 application, program, software, workstation, code, window, file, programming, print, tool 3 bullet, firearm, weapon, attack, shoot, kill, action, armed, protect, protection 4 homosexual, homosexuality, sexual, insist, reject, accept, morality, contrary, disagree, oppose 5 machine, chip, circuit, electronic, hardware, equipment, device, computer, workstation, processor 6 vehicle, auto, engine, rear, tire, driver, truck, motor, wheel, bike 7 israeli, conflict, oppose, attack, peace, struggle, arab, turkish, armenian, kill 8 action, consideration, complain, oppose, bother, rule, issue, policy, insist, accept 9 complain, respond, response, consideration, suggestion, idea, bother, challenge, influence, accept 10 orbit, satellite, solar, planet, shuttle, mission, earth, rocket, moon, plane 11 mailing, mail, send, email, contact, message, telephone, address, customer, request 12 printer, print, font, format, digital, make, manufacture, manufacturer, machine, workstation 13 sell, sale, purchase, offer, brand, customer, supply, vendor, deal, price 14 send, inform, publish, message, newsgroup, reader, mailing, post, topic, mail 15 lose, result, score, loss, beat, challenge, division, note, gain, fall 16 belief, faith, doctrine, accept, truth, religion, notion, religious, trust, interpretation 17 hardware, computer, device, drive, machine, monitor, electronic, chip, shareware, modem 18 patient, complain, care, affect, effect, issue, treat, suffer, response, treatment 19Kmeans -0.868 0.088 0.333 1.000 0.297 0.490 0.139 0.667 BERTopic † -0.307 0.053 0.232 0.623 0.423 0.513 0.166 0.218 Top2Vec † -0.339 0.082 0.314 0.059 0.542 0.477 0.218 0.329 TOP2Vec -0.324 0.097 0.334 0.920 0.419 0.435 0.173 0.528 LDA -0.150 0.029 0.208 0.840 0.447 0.480 0.202 0.098 ProdLDA -0.290 0.050 0.212 0.960 0.484 0.541 0.171 0.199 NeuralLDA -0.460 0.077 0.190 1.000 0.574 0.558 0.170 0.136 ETM -0.184 0.043 0.249 0.600 0.510 0.489 0.252 0.182 CTM -0.299 0.050 0.232 1.000 0.236 0.369 0.148 0.241 CBTM + -0.851 0.351 0.456 0.810 0.368 0.444 0.186 0.701 CBTM * 0.055 0.440 0.402 0.765 0.433 0.518 0.202 0.602 CBTM * + -0.772 0.373 0.403 0.795 0.397 0.474 0.181 0.656 † See table 4in the Appendix for a complete variable and notation list The cosine similarity between the words "economy" and "economies", using the paraphrase-MiniLM-L6-v2 embedder [39] is for instance 0.9. The word2vec model is trained on the GoogleNews corpus. The number of top words, Z, taken into account for the metrics EXP RS, COH, W ESS, IN T and ISIM is 10. For IN T and ISIM , we randomly select an intruder word from a randomly selected topic 50 times and report the averages. flda: matrix factorization through latent dirichlet allocation. D Agarwal, B C Chen, Proceedings of the third ACM international conference on Web search and data mining. the third ACM international conference on Web search and data miningAgarwal, D., Chen, B.C.: flda: matrix factorization through latent dirichlet allocation. In: Proceedings of the third ACM international conference on Web search and data mining. pp. 91-100 (2010) On the surprising behavior of distance metrics in high dimensional space. C C Aggarwal, A Hinneburg, D A Keim, International conference on database theory. SpringerAggarwal, C.C., Hinneburg, A., Keim, D.A.: On the surprising behavior of distance metrics in high dimensional space. In: International conference on database theory. pp. 420-434. Springer (2001) Evaluating topic coherence using distributional semantics. N Aletras, M Stevenson, Proceedings of the 10th International Conference on Computational Semantics (IWCS 2013)-Long Papers. the 10th International Conference on Computational Semantics (IWCS 2013)-Long PapersAletras, N., Stevenson, M.: Evaluating topic coherence using distributional semantics. In: Proceedings of the 10th International Conference on Computational Semantics (IWCS 2013)- Long Papers. pp. 13-22 (2013) D Angelov, arXiv:2008.09470Top2vec: Distributed representations of topics. arXiv preprintAngelov, D.: Top2vec: Distributed representations of topics. arXiv preprint arXiv:2008.09470 (2020) Bibliographic classification theory and text linguistics: aboutness analysis, intertextuality and the cognitive act of classifying documents. C Beghtol, Journal of documentation. Beghtol, C.: Bibliographic classification theory and text linguistics: aboutness analysis, inter- textuality and the cognitive act of classifying documents. Journal of documentation (1986) An automatic approach for document-level topic model evaluation. S Bhatia, J H Lau, T Baldwin, arXiv:1706.05140arXiv preprintBhatia, S., Lau, J.H., Baldwin, T.: An automatic approach for document-level topic model evaluation. arXiv preprint arXiv:1706.05140 (2017) Topic intrusion for automatic topic model evaluation. S Bhatia, J H Lau, T Baldwin, Association for Computational LinguisticsBhatia, S., Lau, J.H., Baldwin, T.: Topic intrusion for automatic topic model evaluation. Association for Computational Linguistics (2020) F Bianchi, S Terragni, D Hovy, arXiv:2004.03974Pre-training is a hot topic: Contextualized document embeddings improve topic coherence. arXiv preprintBianchi, F., Terragni, S., Hovy, D.: Pre-training is a hot topic: Contextualized document embeddings improve topic coherence. arXiv preprint arXiv:2004.03974 (2020) Cross-lingual contextualized topic models with zero-shot learning. F Bianchi, S Terragni, D Hovy, D Nozza, E Fersini, Proceedings of the 16th Conference of the European Chapter of the Association for Computational Linguistics: Main Volume. the 16th Conference of the European Chapter of the Association for Computational Linguistics: Main VolumeAssociation for Computational LinguisticsBianchi, F., Terragni, S., Hovy, D., Nozza, D., Fersini, E.: Cross-lingual contextualized topic models with zero-shot learning. In: Proceedings of the 16th Conference of the European Chapter of the Association for Computational Linguistics: Main Volume. pp. 1676-1683. Association for Computational Linguistics, Online (Apr 2021), https://www.aclweb. org/anthology/2021.eacl-main.143 Natural language processing with Python: analyzing text with the natural language toolkit. S Bird, E Klein, E Loper, O'Reilly MediaBird, S., Klein, E., Loper, E.: Natural language processing with Python: analyzing text with the natural language toolkit. " O'Reilly Media, Inc." (2009) The nested chinese restaurant process and bayesian nonparametric inference of topic hierarchies. D M Blei, T L Griffiths, M I Jordan, Journal of the ACM (JACM). 572Blei, D.M., Griffiths, T.L., Jordan, M.I.: The nested chinese restaurant process and bayesian nonparametric inference of topic hierarchies. Journal of the ACM (JACM) 57(2), 1-30 (2010) Latent dirichlet allocation. D M Blei, A Y Ng, M I Jordan, Journal of machine Learning research. 3Blei, D.M., Ng, A.Y., Jordan, M.I.: Latent dirichlet allocation. Journal of machine Learning research 3(Jan), 993-1022 (2003) Reading tea leaves: How humans interpret topic models. J Chang, S Gerrish, C Wang, J Boyd-Graber, D Blei, Advances in neural information processing systems. 22Chang, J., Gerrish, S., Wang, C., Boyd-Graber, J., Blei, D.: Reading tea leaves: How humans interpret topic models. Advances in neural information processing systems 22 (2009) Latent dirichlet mixture model. J T Chien, C H Lee, Z H Tan, Neurocomputing. 278Chien, J.T., Lee, C.H., Tan, Z.H.: Latent dirichlet mixture model. Neurocomputing 278, 12-22 (2018) All that's' human'is not gold: Evaluating human evaluation of generated text. E Clark, T August, S Serrano, N Haduong, S Gururangan, N A Smith, arXiv:2107.00061arXiv preprintClark, E., August, T., Serrano, S., Haduong, N., Gururangan, S., Smith, N.A.: All that's' human'is not gold: Evaluating human evaluation of generated text. arXiv preprint arXiv:2107.00061 (2021) A systematic definition of sentence topic. A Davison, Center for the Study of ReadingTechnical ReportDavison, A.: A systematic definition of sentence topic. Center for the Study of Reading Technical Report; no. 264 (1982) Syntactic markedness and the definition of sentence topic. A Davison, Language. 604Davison, A.: Syntactic markedness and the definition of sentence topic. Language 60(4), 797-846 (1984) Topic modeling in embedding spaces. A B Dieng, F J Ruiz, D M Blei, Transactions of the Association for Computational Linguistics. 8Dieng, A.B., Ruiz, F.J., Blei, D.M.: Topic modeling in embedding spaces. Transactions of the Association for Computational Linguistics 8, 439-453 (2020) Using word embedding to evaluate the coherence of topics from twitter data. A Fang, C Macdonald, I Ounis, P Habel, Proceedings of the 39th International ACM SIGIR conference on Research and Development in Information Retrieval. the 39th International ACM SIGIR conference on Research and Development in Information RetrievalFang, A., Macdonald, C., Ounis, I., Habel, P.: Using word embedding to evaluate the coherence of topics from twitter data. In: Proceedings of the 39th International ACM SIGIR conference on Research and Development in Information Retrieval. pp. 1057-1060 (2016) Bertopic: Neural topic modeling with a class-based tf-idf procedure. M Grootendorst, arXiv:2203.05794arXiv preprintGrootendorst, M.: Bertopic: Neural topic modeling with a class-based tf-idf procedure. arXiv preprint arXiv:2203.05794 (2022) S Sia, A Dalmia, S J Mielke, arXiv:2004.14914Tired of topic models? clusters of pretrained word embeddings make for fast and good topics too! arXiv preprint. Sia, S., Dalmia, A., Mielke, S.J.: Tired of topic models? clusters of pretrained word embed- dings make for fast and good topics too! arXiv preprint arXiv:2004.14914 (2020) Mpnet: Masked and permuted pre-training for language understanding. K Song, X Tan, T Qin, J Lu, T Y Liu, Advances in Neural Information Processing Systems. 33Song, K., Tan, X., Qin, T., Lu, J., Liu, T.Y.: Mpnet: Masked and permuted pre-training for language understanding. Advances in Neural Information Processing Systems 33, 16857- 16867 (2020) Autoencoding variational inference for topic models. A Srivastava, C Sutton, arXiv:1703.01488arXiv preprintSrivastava, A., Sutton, C.: Autoencoding variational inference for topic models. arXiv preprint arXiv:1703.01488 (2017) S Terragni, E Fersini, B G Galuzzi, P Tropeano, A Candelieri, Octis: Comparing and optimizing topic models is simple! In: Proceedings of the 16th Conference of the European Chapter of the Association for Computational Linguistics: System Demonstrations. Terragni, S., Fersini, E., Galuzzi, B.G., Tropeano, P., Candelieri, A.: Octis: Comparing and optimizing topic models is simple! In: Proceedings of the 16th Conference of the European Chapter of the Association for Computational Linguistics: System Demonstrations. pp. 263- 270 (2021) Word embedding-based topic similarity measures. S Terragni, E Fersini, E Messina, International Conference on Applications of Natural Language to Information Systems. SpringerTerragni, S., Fersini, E., Messina, E.: Word embedding-based topic similarity measures. In: International Conference on Applications of Natural Language to Information Systems. pp. 33-45. Springer (2021) One-class support vector machine and lda topic model integration-evidence for ai patents. A Thielmann, C Weisser, A Krenz, Soft computing: Biomedical and related applications. SpringerThielmann, A., Weisser, C., Krenz, A.: One-class support vector machine and lda topic model integration-evidence for ai patents. In: Soft computing: Biomedical and related applications, pp. 263-272. Springer (2021) Unsupervised document classification integrating web scraping, one-class svm and lda topic modelling. A Thielmann, C Weisser, A Krenz, B Säfken, Journal of Applied Statistics. Thielmann, A., Weisser, C., Krenz, A., Säfken, B.: Unsupervised document classification integrating web scraping, one-class svm and lda topic modelling. Journal of Applied Statistics pp. 1-18 (2021) Human in the loop: How to effectively create coherent topics by manually labeling only a few documents per class. A Thielmann, C Weisser, B Säfken, arXiv:2212.09422arXiv preprintThielmann, A., Weisser, C., Säfken, B.: Human in the loop: How to effectively create coherent topics by manually labeling only a few documents per class. arXiv preprint arXiv:2212.09422 (2022) All bark and no bite: Rogue dimensions in transformer language models obscure representational quality. W Timkey, M Van Schijndel, arXiv:2109.04404arXiv preprintTimkey, W., van Schijndel, M.: All bark and no bite: Rogue dimensions in transformer language models obscure representational quality. arXiv preprint arXiv:2109.04404 (2021) Atm: Adversarial-neural topic model. R Wang, D Zhou, Y He, Information Processing & Management. 566102098Wang, R., Zhou, D., He, Y.: Atm: Adversarial-neural topic model. Information Processing & Management 56(6), 102098 (2019) Minilm: Deep self-attention distillation for task-agnostic compression of pre-trained transformers. W Wang, F Wei, L Dong, H Bao, N Yang, M Zhou, Advances in Neural Information Processing Systems. 33Wang, W., Wei, F., Dong, L., Bao, H., Yang, N., Zhou, M.: Minilm: Deep self-attention distillation for task-agnostic compression of pre-trained transformers. Advances in Neural Information Processing Systems 33, 5776-5788 (2020) The automatic identification of stop words. W J Wilbur, K Sirotkin, Journal of information science. 181Wilbur, W.J., Sirotkin, K.: The automatic identification of stop words. Journal of information science 18(1), 45-55 (1992)
{'fraction_non_alphanumeric': 0.051639547369290495, 'fraction_numerical': 0.034789791029982656, 'mean_word_length': 4.534973027338393, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 2, 'https://': 2, 'lorem ipsum': 0, 'www.': 1, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 11, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'Extracting and identifying latent topics in large text corpora has gained increasing importance in Natural Language Processing (NLP). Most models, whether probabilistic models similar to Latent Dirichlet Allocation (LDA) or neural topic models, follow the same underlying approach of topic interpretability and topic extraction. We propose a method that incorporates a deeper understanding of both sentence and document themes, and goes beyond simply analyzing word frequencies in the data. This allows our model to detect latent topics that may include uncommon words or neologisms, as well as words not present in the documents themselves. Additionally, we propose several new evaluation metrics based on intruder words and similarity measures in the semantic space. We present correlation coefficients with human identification of intruder words and achieve near-human level results at the word-intrusion task. We demonstrate the competitive performance of our method with a large benchmark study, and achieve superior results compared to state-of-the-art topic modeling and document clustering models.', 'arxivid': '2303.17324', 'author': ['Anton Thielmann [email protected] \nChair of Data Science and Applied Statistics\nTU Clausthal\nGermany\n', 'Quentin Seifert \nChair of Spatial Data Science and Statistical Learning\nUniversity of Göttingen\nGermany\n', 'Arik Reuter \nChair of Data Science and Applied Statistics\nTU Clausthal\nGermany\n', 'Elisabeth Bergherr \nChair of Spatial Data Science and Statistical Learning\nUniversity of Göttingen\nGermany\n', 'Benjamin Säfken \nChair of Data Science and Applied Statistics\nTU Clausthal\nGermany\n'], 'authoraffiliation': ['Chair of Data Science and Applied Statistics\nTU Clausthal\nGermany', 'Chair of Spatial Data Science and Statistical Learning\nUniversity of Göttingen\nGermany', 'Chair of Data Science and Applied Statistics\nTU Clausthal\nGermany', 'Chair of Spatial Data Science and Statistical Learning\nUniversity of Göttingen\nGermany', 'Chair of Data Science and Applied Statistics\nTU Clausthal\nGermany'], 'corpusid': 257833712, 'doi': '10.48550/arxiv.2303.17324', 'github_urls': [], 'n_tokens_mistral': 17370, 'n_tokens_neox': 14992, 'n_words': 9137, 'pdfsha': 'e4eda298d376d27488fee88f6ca52ca945c19848', 'pdfurls': ['https://export.arxiv.org/pdf/2303.17324v1.pdf'], 'title': ['Topics in the Haystack: Extracting and Evaluating Topics beyond Coherence', 'Topics in the Haystack: Extracting and Evaluating Topics beyond Coherence'], 'venue': []}
arxiv
Deep Structural Causal Models for Tractable Counterfactual Inference Nick Pawlowski Imperial College London Imperial College London Imperial College London Daniel C Castro Imperial College London Imperial College London Imperial College London Ben Glocker [email protected] Imperial College London Imperial College London Imperial College London Deep Structural Causal Models for Tractable Counterfactual Inference We formulate a general framework for building structural causal models (SCMs) with deep learning components. The proposed approach employs normalising flows and variational inference to enable tractable inference of exogenous noise variables-a crucial step for counterfactual inference that is missing from existing deep causal learning methods. Our framework is validated on a synthetic dataset built on MNIST as well as on a real-world medical dataset of brain MRI scans. Our experimental results indicate that we can successfully train deep SCMs that are capable of all three levels of Pearl's ladder of causation: association, intervention, and counterfactuals, giving rise to a powerful new approach for answering causal questions in imaging applications and beyond. The code for all our experiments is available at https://github.com/biomedia-mira/deepscm. Introduction Many questions in everyday life as well as in scientific inquiry are causal in nature: "How would the climate have changed if we'd had less emissions in the '80s?", "How fast could I run if I hadn't been smoking?", or "Will my headache be gone if I take that pill?". None of those questions can be answered with statistical tools alone, but require methods from causality to analyse interactions with our environment (interventions) and hypothetical alternate worlds (counterfactuals), going beyond joint, marginal, and conditional probabilities [1]. Even though these are natural lines of reasoning, their mathematical formalisation under a unified theory is relatively recent [2]. In some statistics-based research fields, such as econometrics or epidemiology, the use of causal inference methods has been established for some time [3,4]. However, causal approaches have been introduced into deep learning (DL) only very recently [5]. For example, research has studied the use of causality for disentanglement [6,7], causal discovery [8,9], and for deriving causality-inspired explanations [10,11] or data augmentations [12]. Causal DL models could be capable of learning relationships from complex high-dimensional data and of providing answers to interventional and counterfactual questions, although current work on deep counterfactuals is limited by modelling only direct cause-effect relationships [11] or instrumental-variable scenarios [13], or by not providing a full recipe for tractable counterfactual inference [14]. The integration of causality into DL research promises to enable novel scientific advances as well as to tackle known shortcomings of DL methods: DL is known to be susceptible to learning spurious correlations and amplifying biases [e.g. 15], and to be exceptionally vulnerable to changes in the input distribution [16]. By explicitly modelling causal relationships and acknowledging the difference between causation and correlation, causality becomes a natural field of study for improving the transparency, fairness, and robustness of DL-based systems [17,18]. Further, the tractable inference of deep counterfactuals enables novel research avenues that aim to study causal reasoning on a per-instance rather than population level, which could lead to advances in personalised medicine as well as in decision-support systems, more generally. In this context, our work studies the use of DL-based causal mechanisms and establishes effective ways of performing counterfactual inference. Our main contributions are: 1) a unified framework for structural causal models using modular deep mechanisms; 2) an efficient approach to estimating counterfactuals by inferring exogenous noise via variational inference or normalising flows; 3) case studies exemplifying how to apply deep structural causal models and perform counterfactual inference. The paper is organised as follows: we first review structural causal models and discuss how to leverage deep mechanisms and enable tractable counterfactual inference. Second, we compare our work to recent progress in deep causal learning in light of Pearl's ladder of causation [19]. Finally, we apply deep structural causal models to a synthetic experiment as well as to modelling brain MRI scans, demonstrating the practical utility of our framework in answering counterfactual questions. Deep Structural Causal Models We consider the problem of modelling a collection of K random variables x = (x 1 , . . . , x K ). By considering causal relationships between them, we aim to build a model that not only is capable of generating convincing novel samples, but also satisfies all three rungs of the causation ladder [19]. The first level, association, describes reasoning about passively observed data. This level deals with correlations in the data and questions of the type "What are the odds that I observe...?", which relates purely to marginal, joint, and conditional probabilities. Intervention concerns interactions with the environment. It requires knowledge beyond just observations, as it relies on structural assumptions about the underlying data-generating process. Characteristic questions ask about the effects of certain actions: "What happens if I do...?". Lastly, counterfactuals deal with retrospective hypothetical scenarios. Counterfactual queries leverage functional models of the generative processes to imagine alternative outcomes for individual data points, answering "What if I had done A instead of B?". Arguably, such questions are at the heart of scientific reasoning (and beyond), yet are less well-studied in the field of machine learning. The three levels of causation can be operationalised by employing structural causal models (SCMs) 2 , recapitulated in the next section. Background on structural causal models A structural causal model G := (S, P ( )) consists of a collection S = (f 1 , . . . , f K ) of structural assignments x k := f k ( k ; pa k ) (called mechanisms), where pa k is the set of parents of x k (its direct causes), and a joint distribution P ( ) = K k=1 P ( k ) over mutually independent exogenous noise variables (i.e. unaccounted sources of variation). As assignments are assumed acyclic, relationships can be represented by a directed acyclic graph (DAG) with edges pointing from causes to effects, called the causal graph induced by G. Every SCM G entails a unique joint observational distribution P G (x), satisfying the causal Markov assumption: each variable is independent of its non-effects given its direct causes. It therefore factorises as P G (x) = K k=1 P G (x k |pa k ), where each conditional distribution P G (x k |pa k ) is determined by the corresponding mechanism and noise distribution [1]. Crucially, unlike conventional Bayesian networks, the conditional factors above are imbued with a causal interpretation. This enables G to be used to predict the effects of interventions, defined as substituting one or multiple of its structural assignments, written as 'do( · · · )'. In particular, a constant reassignment of the form do(x k := a) is called an atomic intervention, which disconnects x k from all its parents and represents a direct manipulation disregarding its natural causes. While the observational distribution relates to statistical associations and interventions can predict causal effects, SCMs further enable reasoning about counterfactuals. These are hypothetical retrospective interventions, given an observed outcome: 'What would x i have been if x j were different, given that we observed x?'. This type of question effectively offers explanations of the data, since we can analyse the changes resulting from manipulating each variable. Counterfactual queries can be mathematically formulated as a three-step procedure [2, Ch. 7]: 1. Abduction: Predict the 'state of the world' (the exogenous noise, ) that is compatible with the observations, x, i.e. infer P G ( |x). k x k pa k f k (a) Invertible explicit likelihood z k u k x k pa k e k h k g k (b) Amortised explicit likelihood k x k pa k f k e k (c) Amortised implicit likelihood Figure 1: Classes of deep causal mechanisms considered in this work. Bi-directional arrows indicate invertible transformations, optionally conditioned on other inputs (edges ending in black circles). Black and white arrowheads refer resp. to the generative and abductive directions, while dotted arrows depict an amortised variational approximation. Here, f k is the forward model, e k is an encoder that amortises abduction in non-invertible mechanisms, g k is a 'high-level' non-invertible branch (e.g. a probabilistic decoder), and h k is a 'low-level' invertible mapping (e.g. reparametrisation). 2. Action: Perform an intervention (e.g. do(x k := x k )) corresponding to the desired manipulation, resulting in a modified SCM G = G x;do( x k ) = ( S, P G ( |x)) [1, Sec. 6.4]. Prediction: Compute the quantity of interest based on the distribution entailed by the counterfactual SCM, P G (x). With these operations in mind, the next section explores a few options for building flexible, expressive, and counterfactual-capable functional mechanisms for highly structured data. Deep mechanisms In statistical literature (e.g. epidemiology, econometrics, sociology), SCMs are typically employed with simple linear mechanisms (or generalised linear models, involving an output non-linearity). Analysts attach great importance to the regression weights, as under certain conditions these may be readily interpreted as estimates of the causal effects between variables. While this approach generally works well for scalar variables and can be useful for decision-making, it is not flexible enough to model higher-dimensional data such as images. Solutions to this limitation have been proposed by introducing deep-learning techniques into causal inference [8,14]. We call an SCM that uses deep-learning components to model the structural assignments a deep structural causal model (DSCM). In DSCMs, the inference of counterfactual queries becomes more complex due to the potentially intractable abduction step (inferring the posterior noise distribution, as defined above). To overcome this, we propose to use recent advances in normalising flows and variational inference to model mechanisms for composable DSCMs that enable tractable counterfactual inference. While here we focus on continuous data, DSCMs also fully support discrete variables without the need for relaxations (see Appendix C). We consider three types of mechanisms that differ mainly in their invertibility, illustrated in Fig. 1. Invertible, explicit: Normalising flows model complex probability distributions using transformations from simpler base distributions with same dimensionality [20]. For an observed variable x, diffeomorphic transformation f , and base variable ∼ P ( ) such that x = f ( ), the output density p(x) can be computed as p(x) = p( )|det ∇f ( )| −1 , evaluated at = f −1 (x) [21,22]. For judicious choices of f , the Jacobian ∇f may take special forms with efficiently computable determinant, providing a flexible and tractable probabilistic model whose parameters can be trained via exact maximum likelihood. Furthermore, flows can be made as expressive as needed by composing sequences of simple transformations. For more information on flow-based models, refer to the comprehensive survey by Papamakarios et al. [22]. Note that this class of models also subsumes the typical location-scale and inverse cumulative distribution function transformations used in the reparametrisation trick [23,24], as well as the Gumbel trick for discrete variable relaxations [25,26]. Although normalising flows were originally proposed for unconditional distributions, they have been extended to conditional densities [27], including in high dimensions [28,29], by parametrising the transformation as x = f ( ; pa X ), assumed invertible in the first argument. In particular, conditional flows can be adopted in DSCMs to represent invertible, explicit-likelihood mechanisms (Fig. 1a): x i := f i ( i ; pa i ), p(x i |pa i ) = p( i ) · |det ∇ i f i ( i ; pa i )| −1 i=f −1 i (xi;pa i ) . (1) Amortised, explicit: Such invertible architectures typically come with heavy computational requirements when modelling high-dimensional observations, because all intermediate operations act in the space of the data. Instead, it is possible to use arbitrary functional forms for the structural assignments, at the cost of losing invertibility and tractable likelihoods p(x k |pa k ). Here, we propose to separate the assignment f k into a 'low-level', invertible component h k and a 'high-level', non-invertible part g k -with a corresponding noise decomposition k = (u k , z k )-such that x k := f k ( k ; pa k ) = h k (u k ; g k (z k ; pa k ), pa k ), P ( k ) = P (u k )P (z k ) .(2) In such a decomposition, the invertible transformation h k can be made shallower, while the upstream non-invertible g k maps from a lower-dimensional space and is expected to capture more of the high-level structure of the data. Indeed, a common implementation of this type of model for images would involve a probabilistic decoder, where g k may be a convolutional neural network, predicting the parameters of a simple location-scale transformation performed by h k [24]. As the conditional likelihood p(x k |pa k ) in this class of models is no longer tractable because z k cannot be marginalised out, it may alternatively be trained with amortised variational inference. Specifically, we can introduce a variational distribution Q(z k |x k , pa k ) to formulate a lower bound on the true marginal conditional log-likelihood, which will be maximised instead: log p(x k |pa k ) ≥ E Q(z k |x k ,pa k ) [log p(x k |z k , pa k )] − D KL [Q(z k |x k , pa k ) P (z k )] .(3) The argument of the expectation in this lower bound can be calculated similarly to Eq. (1): p(x k |z k , pa k ) = p(u k ) · |det ∇ u k h k (u k ; g k (z k , pa k ), pa k )| −1 u k =h −1 k (x k ;g k (z k ,pa k ),pa k ) . (4) The approximate posterior distribution Q(z k |x k , pa k ) can for example be realised by an encoder function, e k (x k ; pa k ), that outputs the parameters of a simple distribution over z k (Fig. 1b), as in the auto-encoding variational Bayes (AEVB) framework [24]. Amortised, implicit: While the models above rely on (approximate) maximum-likelihood as training objective, it is admissible to train a non-invertible mechanism as a conditional implicitlikelihood model (Fig. 1c), optimising an adversarial objective [30][31][32]. Specifically, a deterministic encoder e j would strive to fool a discriminator function attempting to tell apart tuples of encoded real data (x j , e j (x j ; pa j ), pa j ) and generated samples (f j ( j ; pa j ), j , pa j ). Deep counterfactual inference Now equipped with effective deep models for representing mechanisms in DSCMs, we discuss the inference procedure allowing us to compute answers to counterfactual questions. Abduction: As presented in Section 2.1, the first step in computing counterfactuals is abduction, i.e. to predict the exogenous noise, , based on the available evidence, x. Because each noise variable is assumed to affect only the respective observed variable, ( k ) K k=1 are conditionally independent given x, therefore this posterior distribution factorises as P G ( |x) = K k=1 P G ( k |x k , pa k ). In other words, it suffices to infer the noise independently for each mechanism, given the observed values of the variable and of its parents 3 . For invertible mechanisms, the noise variable can be obtained deterministically and exactly by just inverting the mechanism: i = f −1 i (x i ; pa i ). Similarly, implicit-likelihood mechanisms can be approximately inverted by using the trained encoder function: j ≈ e j (x j ; pa j ). Some care must be taken in the case of amortised, explicit-likelihood mechanisms, as the 'high-level' noise z k and 'low-level' noise u k are not independent given x k . Recalling that this mechanism is trained along with a conditional probabilistic encoder, Q(z k |e k (x k ; pa k )), the noise posterior can be approximated as follows, where δ w ( · ) denotes the Dirac delta distribution centred at w: P G ( k |x k , pa k ) = P G (z k |x k , pa k ) P G (u k |z k , x k , pa k ) ≈ Q(z k |e k (x k ; pa k )) δ h −1 k (x k ;g k (z k ;pa k ),pa k ) (u k ) .(5) Action: The causal graph is then modified according to the desired hypothetical intervention(s), as in the general case (Section 2.1). For each intervened variable x k , its structural assignment is replaced either by a constant, x k := x k -making it independent of its former parents (direct causes, pa k ) and of its exogenous noise ( k )-or by a surrogate mechanism x k := f k ( k ; pa k ), forming a set of counterfactual assignments, S. This then defines a counterfactual SCM G = ( S, P G ( |x)). Prediction: Finally, we can sample from G. Noise variables that were deterministically inverted (either exactly or approximately) can simply be plugged back into the respective forward mechanism to determine the new output value. Notice that this step is redundant for observed variables that are not descendants of the ones being intervened upon, as they will be unaffected by the changes. As mentioned above, the posterior distribution over (z k , u k ) for an amortised, explicit-likelihood mechanism does not factorise (Eq. (5)), and the resulting distribution over the counterfactual x k cannot be characterised explicitly. However, sampling from it is straightforward, such that we can approximate the counterfactual distribution via Monte Carlo as follows, for each sample s: z (s) k ∼ Q(z k |e k (x k ; pa k )) u (s) k = h −1 k (x k ; g k (z (s) k ; pa k ), pa k ) x (s) k = h k (u (s) k ; g k (z (s) k ; pa k ), pa k ) .(6) Consider an uncorrelated Gaussian decoder for images as a concrete example, predicting vectors of means and variances for each pixel of x k : g k (z k ; pa k ) = (µ(z k ; pa k ), σ 2 (z k ; pa k )). Exploiting the reparametrisation trick, counterfactuals that preserve x k 's mechanism can be computed simply as u (s) k = (x k − µ(z (s) k ; pa k )) σ(z (s) k ; pa k ), x (s) k = µ(z (s) k ; pa k ) + σ(z (s) k ; pa k ) u (s) k , where and denote element-wise division and multiplication, respectively. In particular, in the constant-variance setting adopted for our experiments, counterfactuals further simplify to x (s) k = x k + [µ(z (s) k ; pa k ) − µ(z (s) k ; pa k )] . This showcases how true image counterfactuals are able to retain pixel-level details. Typical conditional generative models would output only µ(z k ; pa k ) (which is often blurry in vanilla variational auto-encoders [33]), or would in addition have to sample P (u k ) (resulting in noisy images). Related Work Deep generative modelling has seen a wide range of contributions since the popularisation of variational auto-encoders (VAEs) [24], generative adversarial networks (GANs) [34], and normalising flows [21]. These models have since been employed to capture conditional distributions [27,29,32,35], and VAEs and GANs were also extended to model structured data by incorporating probabilistic graphical models [36][37][38]. In addition, deep generative models have been heavily used for (unsupervised) representation learning with an emphasis on disentanglement [39][40][41][42]. However, even when these methods faithfully capture the distribution of observed data, they are capable of fulfilling only the association rung of the ladder of causation. Interventions build on the associative capabilities of probabilistic models to enable queries related to changes in causal mechanisms. By integrating a causal graph into the connectivity of a deep model, it is possible to perform interventions with GANs [14] and causal generative NNs [8]. VAEs can also express causal links using specific covariance matrices between latent variables, which however restrict the dependences to be linear [6]. Despite reaching the second rung of the causal ladder, these methods lack tractable abduction capabilities and therefore cannot generate counterfactuals. Some machine-learning tasks such as explainability, image-to-image translation, or style transfer are closely related to counterfactual queries of the sort 'How would x (have to) change if we (wished to) modify y?'. Here, y could be the style of a picture for style transfer [43], the image domain (e.g. drawing to photo) for image-to-image translation [44], the age of a person in natural images [45] or medical scans [46], or a predicted output for explainability [11]. However, these approaches do not explicitly model associations, interventions, nor causal structure. Potentially closest to our work is a method for counterfactual explainability of visual models, which extends CausalGANs [14] to predict reparametrised distributions over image attributes following an assumed causal graph [10]. However, this approach performs no abduction step, instead resampling the noise of attributes downstream from the intervention(s), and does not include a generative model of imaging data. To the best of our knowledge, the proposed DSCM framework is the first flexible approach enabling end-to-end training and tractable inference on all three levels of the ladder of causation for high-dimensional data. T t I i z X x u X t i x (a) Independent T t I i z X x u X t i x (b) Conditional T t I i z X x u X t i x (c) Full Case Study 1: Morpho-MNIST We consider the problem of modelling the causal model of a synthetic dataset based on MNIST digits [47], where stroke thickness causes the brightness of the digit: thicker digits are brighter whereas thinner digits are dimmer. This simple dataset allows for examining the three levels of causation in a controlled and measurable environment. We use morphological transformations on MNIST [48] to generate a dataset with known causal structure and access to the 'true' process of generating counterfactuals. The SCM for this synthetic dataset is as follows: t := f * T ( * T ) = 0.5 + * T , * T ∼ Γ(10, 5) . i := f * I ( * I ; t) = 191 · σ(0.5 · * I + 2 · t − 5) + 64 , * I ∼ N (0, 1) . x := f * X ( * X ; i, t) = SetIntensity(SetThickness( * X ; t); i) , * X ∼ MNIST ,(7) where SetIntensity( · ; i) and SetThickness( · ; t) refer to the operations that act on an image of a digit and set its intensity to i and thickness to t (see Appendix A.1 for details), x is the resulting image, * is the exogenous noise for each variable and σ( · ) is the logistic sigmoid. We use this setup to study the capabilities of our framework in comparison to models with less causal structure. We adapt the true causal graph from Eq. (7) and model thickness and intensity using (conditional) normalising flows and employ a conditional VAE for modelling the image. In particular, we adopt the causal graphs shown in Fig. 2 and test a fully independent model (Fig. 2a), a conditional decoder model (Fig. 2b), as well as our full causal model (Fig. 2c). All our experiments were implemented within PyTorch [49] using the Pyro probabilistic programming framework [50], and implementation details can be found in Appendices A.2 and B.2. We quantitatively compare the associative capabilities of all models by evaluating their evidence lower bound (Eq. (3)), log-likelihoods and reconstruction errors as shown in Table 1. We find that performance improves consistently with the model's capabilities: enabling conditional image generation improves p(x|t, i), and adding a causal dependency between t and i improves p(i|t). Further, we examine samples of the conditional and unconditional distributions in Appendix A.3.1. The interventional distributions can be directly compared to the true generative process. Figure 3 shows that the densities predicted by our full model after intervening on t closely resemble the p(t, i) p(t, i | do(t + 1)) p(t, i | do(t 0.5)) True data Lastly, we examine the full model's ability to generate counterfactuals. The other two models were omitted as they are incapable of accomplishing interventions, a prerequisite for counterfactual inference. Examples of previously unseen images and generated counterfactuals are shown in Fig. 4. We see that our model is capable of generating convincing counterfactuals that preserve the digit identity while changing thickness and intensity consistently with the underlying causal model. Case Study 2: Brain Imaging Our real-world application touches upon fundamental scientific questions in the context of medical imaging: how would a person's anatomy change if particular traits were different? We illustrate with a (simplified) example that our DSCM framework may provide the means to answer such counterfactual queries, which may enable entirely new research into better understanding the physical manifestation of lifestyle, demographics, and disease. Here, we model the appearance of brain MRI scans given the person's age and biological sex, as well as brain and ventricle volumes 4 , using population data from the UK Biobank [51]. Ventricle and total brain volumes are two quantities that are closely related to brain age [52] and can be observed relatively easily. We adopt the causal graph shown in Fig. 5a and otherwise follow the same training procedure as for the MNIST experiments. The learned DSCM is capable of all three levels of the causal hierarchy. We present the analysis of lower levels in Appendix B.3.1 and focus here on counterfactuals, shown in Fig. 5b (more examples in Appendix B.3.2). The difference maps show plausible counterfactual changes: increasing age causes slightly larger ventricles while decreasing the overall brain volume (first column). In contrast, directly changing brain volume has an opposite effect on the ventricles compared to changing age (second column). Intervening on ventricle volume has a much more localised effect (third column), while intervening on the categorical variable of biological sex has smaller yet more diffuse effects. Note how the anatomical 'identity' (such as the cortical folding) is well preserved after each intervention. Conclusion We introduce a novel general framework for fitting SCMs with deep mechanisms. Our deep SCM (DSCM) framework fulfils all three rungs of Pearl's ladder of causation-in particular, it is the first to enable efficient abduction of exogenous noise, permitting principled counterfactual inference. We demonstrate the potential of DSCMs with two case studies: a synthetic task of modelling Morpho-MNIST digits with a known causal structure and a real-world example with brain MRI. The ability to correctly generate plausible counterfactuals could greatly benefit a wide variety of possible applications, e.g.: explainability, where differences between observed and counterfactual data can suggest causal explanations of outcomes; data augmentation, as counterfactuals can extrapolate beyond the range of observed data (e.g. novel combinations of attributes); and domain adaptation, since including the source of the data as an indicator variable in the causal model could enable generating counterfactual examples in a relevant target domain. The proposed method does not come without limitations to be investigated in future work. Like the related approaches, the current setup requires all variables to be observed when computing a counterfactual, which may limit its applicability in certain scenarios. This could be alleviated by imputing the missing data via MCMC or learning auxiliary distributions. Further work should study more closely the dynamic behaviour of deep mechanisms in SCMs. While not observed in our experiments, neural networks may not learn to cleanly separate the roles of its inputs on the output as expected-which could require custom counterfactual regularisation similar to losses used in image-to-image translation [46] and explainability work [11]. The use of such flexible models also raises questions about the identifiability of the 'true' mechanism, as counterfactuals may not be uniquely defined. Lastly, it would be interesting to examine whether this framework can be applied to causal discovery, attempting to uncover plausible causal structures from data. Broader Impact Causal inference can be applied to a wide range of applications, promising to provide a deeper understanding of the observed data and prevent the fitting of spurious correlations. Our research presents a methodological contribution to the causal literature proposing a framework that combines causal models and deep learning to facilitate modelling high-dimensional data. Because of the general applicability of deep learning and causal inference, our framework could have a broad impact of enabling fairer machine learning models explicitly modelling causal mechanisms, reducing spurious correlations and tackling statistical and societal biases. The resulting models offer better interpretability due to counterfactual explanations and could yield novel understanding through causal discovery. However, causal modelling relies on strong assumptions and cannot always unambiguously determine the true causal structure of observational data. It therefore is necessary to carefully consider and communicate the assumptions being made by the analyst. In this light, our methodology is susceptible to being used to wrongly claim the discovery of causal structures due to careless application or intentional misuse. Particularly, the use of 'black-box' components as causal mechanisms may exacerbate concerns about identifiability, already present even for simple linear models. Whereas deep causal models can be useful for deriving insights from data, we must be cautious about their use in consequential decision-making, such as in informing policies or in the context of healthcare. [55] Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014. [ A Synthetic Morpho-MNIST Experiment A.1 Data Generation We use the original MNIST dataset [47] together with the morphometric measurements introduced with Morpho-MNIST [48] to add functionality to measure intensity as well as set the intensity and thickness to a given value. We implement MeasureIntensity by following the processing steps proposed by Castro et al. [48], and measure the intensity i of an image as the median intensity of pixels within the extracted binary mask. Once the intensity is measured, the entire image is rescaled to match the target intensity, with values clamped between 0 and 255 (images are assumed to be in unsigned 8-bit format). Originally, Morpho-MNIST only proposed relative thinning and thickening operations. We expand those operations to absolute values by calculating the amount of dilation or erosion based on the ratio between target thickness and measured thickness. Finally, we follow Eq. A.2 Experimental Setup We use (conditional) normalising flows for all variables apart from the images, which we model using (conditional) deep encoder-decoder architectures. The flows consist of components that constrain the support of the output distribution (where applicable) and components relevant for fitting the distribution. We use unit Gaussians as base distributions for all exogenous noise distributions P ( ) and, if available, we use the implementations in PyTorch [49] or Pyro [50] for all transformations. Otherwise, we adapt the available implementations, referring to [53] for details. We indicate with θ the modules with learnable parameters. We model the mechanisms of the thickness t and intensity i as t := f T ( T ) = (exp • AffineNormalisation • Spline θ )( T ) , (A.1) i := f I ( I ; t) = AffineNormalisation • sigmoid • ConditionalAffine θ (t) ( I ) . (A.2) In the independent model, where i is not conditioned on t, we use instead i := f I ( I ) = (AffineNormalisation • sigmoid • Spline θ • Affine θ )( I ) . (A.3) We found that including normalisation layers help learning dynamics 5 and therefore include flows to perform commonly used normalisation transformations. For doubly bounded variable y we learn the flows in unconstrained space and then constrain them by a sigmoid transform and rescale to the original range using fixed affine transformations with bias min(Y ) and scale [max(Y ) − min(Y )]. We constrain singly bounded values by applying an exponential transform to the unbounded values and using an affine normalisation equivalent to a whitening operation in unbounded log-space. We denote those fixed normalisation transforms as AffineNormalisation and use a hat to refer to the unconstrained, normalised values (e.g. pa k ). The Spline θ transformation refers to first-order neural spline flows [53], Affine θ is an element-wise affine transformation, and sigmoid refers to the logistic function. ConditionalAffine θ (·) is a regular affine transform whose transformation parameters are predicted by a context neural network taking · as input. In the case of f I ( I ; t), the context network is represented by a simple linear transform. Further, we model x using a low-level flow: h X (u X ; pa X ) = [Preprocessing • ConditionalAffine θ ( pa X )](u X ) , (A.4) where the ConditionalAffine transform practically reparametrises the noise distribution into another Gaussian distribution and Preprocessing describes a fixed preprocessing transformation. We follow the same preprocessing as used with RealNVP [54]. The context network for the conditional affine transformation is the high-level mechanism g X (z X ; pa X ) and is implemented as a decoder network that outputs the bias for of the affine transformation, while the log-variance is fixed to log σ 2 = −5. We implement the decoder network as a CNN: g X (z X ; pa X ) = (Conv θ (1; 1; 1; 0) • ConvTranspose θ (1; 4; 2; 1) • ReLU • BN θ • ConvTranspose θ (64; 4; 2; 1) • Reshape(64, 7, 7) • ReLU • BN θ • Linear θ (1024) • ReLU • BN θ • Linear θ (1024))([z X , pa X ]) , (A.5) where the operators describe neural network layers as follows: BN is batch normalisation; ReLU the ReLU activation function; Conv(c; k; s; p) and ConvTranspose(c; k; s; p) are a convolution or transposed convolution using a kernel with size k, a stride of s, a padding of p and outputting c channels; Linear(h) is a linear layer with h output neurons; and Reshape(·) reshapes its inputs into the given shape ·. Lastly, [z X , pa X ] denotes the concatenation of z X and pa X , and z X ∈ R 16 . Equivalently, we implement the the encoder function as a simple CNN that outputs mean and log-variance of a independent Gaussian: e X (x; pa X ) = [Linear θ (16), Linear θ (16)] • [LeakyReLU(0.1), pa X ] • BN θ • Linear θ (100) • Reshape(128 · 7 · 7) • LeakyReLU(0.1) • BN θ • Conv θ (128; 4; 2, 1) • LeakyReLU(0.1) • BN θ • Conv θ (64; 4; 2, 1) (x) , (A.6) where LeakyReLU( ) is the leaky ReLU activation function with a leakiness of . We use Adam [55] for optimisation with batch size of 256 and a learning rate of 10 −4 for the encoder-decoder and 0.005 for the covariate flows. We set the number of particles (MC samples) for estimating the ELBO to 4. We use 32 MC samples for estimating reconstruction and counterfactuals. We train all models for 1000 epochs and report the results of the model with the best validation loss. A.3 Additional Results Here we further illustrate the associative, interventional, and counterfactual capabilities of the trained independent, conditional, and full models. (Continued on the next page.) A.3.1 Association E Q(z X |e X (x;pa X )) [g X (z X ; pa X )], where e X and g X are the image encoder and decoder networks. All models seem capable of producing faithful reconstructions. Since t causes i, notice how p(t|i) (left) is markedly different from p(t|do(i)) (middle), which collapses to p(t). On the other hand, p(i|do(t)) and p(i|t) (right) are identical. A.3.3 Counterfactual Original do(t = 1.0) do(t = 3.0) do(t = 5.0) do(i = 64) do(i = 160) do(i = 255) t = 1.8 i = 119 Original do(t = 1.0) do(t = 3.0) do(t = 5.0) do(i = 64) do(i = 160) do(i = 255) t = 3.6 i = 242 Original We observe that all counterfactuals preserve the digits' identity and style. Our model even generates sensible counterfactual images (with some artefacts) in very low-density regions, e.g. '0' with do(i = 64) (thick but dim), and very far from the original, e.g. '2' with do(t = 5.0). do(t = 1.0) do(t = 3.0) do(t = 5.0) do(i = 64) do(i = 160) do(i = 255) t = 2.6 i = 142 B Brain Modelling B.1 Data Generation The original three-dimensional (3D) T1-weighted brain MRI scans have been pre-processed by the data providers of the UK Biobank Imaging study using the FSL neuroimaging toolkit [56]. The pre-processing involves skull removal, bias field correction, and automatic segmentation of brain structures. In addition, we have rigidly registered all scans to the standard MNI atlas space using an in-house image registration tool, which enabled us to extract anatomically corresponding mid-axial 2D slices that were used for the experiments presented in this paper. The 2D slices were normalised in intensity by mapping the minimum and maximum values inside the brain mask to the range [0, 255]. Background pixels outside the brain were set to zero. Age and biological sex for each subject were retrieved from the UK Biobank database along with the pre-computed brain and ventricle volumes. These volumes are derived from the 3D segmentation maps obtained with FSL, and although these are image-derived measurements, they may serve as reasonable proxies of the true measurements within our (simplified yet plausible) causal model of the physical manifestation of the brain anatomy. B.2 Experimental Setup The setup for the brain imaging experiment closely follows the MNIST example as described in Appendix A.2. We randomly split the available 13, 750 brain images into train, validation and test sets with the respective ratios 70%, 15% and 15%. During training, we randomly crop the brain slices from their original size of 233 px × 197 px to 192 px × 192 px and use center crops during validation and testing. The cropped images are downsampled by a factor of 3 to a size of 64 px × 64 px. We use the same low-level mechanism for the image x as with MNIST images but change the encoder and decoder functions to a deeper architecture with 5 scales consisting of 3 blocks of (LeakyReLU(0.1) • BN θ • Conv θ ) each as well as a linear layer that converts to and from the latent space with 100 dimensions. We directly learn the binary probability of the sex s and use the following invertible transforms to model the age a, brain volume b, and ventricle volume v as a := f A ( A ) = exp • AffineNormalisation • Spline θ ( A ) , (B.1) b := f B ( B ; s, a) = exp • AffineNormalisation • ConditionalAffine θ ([s, a]) ( B ) , (B.2) v := f V ( V ; a, b) = exp • AffineNormalisation • ConditionalAffine θ ([b, a]) ( V ) , (B.3) where the context networks are implemented as a fully-connected network with 8 and 16 hidden units, and a LeakyReLU(0.1) nonlinearity. B.3 Additional Results Likewise, we present more detailed analyses of the model trained on UK Biobank brain images and covariates, in terms of modelling the observational distribution and computing various counterfactual queries. (Continued on the next page.) Original Original Recon. Recon. Full model (a) Age vs. brain volume: p(a, b|s). Here we see differences in head size across biological sexes (reflected in brain volume), as well as a downward trend in brain volume as age progresses. (b) Age vs. ventricle volume: p(a, v |b ∈ · ). As expected from the literature [52], we observe a consistent increase in ventricle volume with age, in addition to a proportionality relationship with the overall brain volume. C Discrete counterfactuals As mentioned in the main text, the DSCM framework supports not only low-and high-dimensional continuous data, but also discrete variables. In particular, discrete mechanisms with a Gumbel-max parametrisation have been shown to lead to counterfactuals satisfying desirable properties [57]. For example, they are invariant to category permutations and are stable, such that increasing the odds only of the observed outcome cannot produce a different counterfactual outcome. More computational details and properties of the Gumbel distribution are found in Maddison and Tarlow [58]. Consider a discrete random variable over K categories, y, with a conditional likelihood described by logits λ, assumed to be a function g Y of its parents, pa Y : P (y = k |pa Y ) = e λ k K l=1 e λ l , λ = g Y (pa Y ) . (C.1) Under the Gumbel-max parametrisation, the mechanism generating y can be described as y := f Y ( Y ; pa Y ) = arg max 1≤l≤K ( l Y + λ l ), l Y ∼ Gumbel(0, 1) . (C.2) Samples from the Gumbel(0, 1) distribution can be generated by computing − log(− log U ), where U ∼ Unif(0, 1). The Gumbel distribution has certain special properties [58] that enable tractable abduction. Given that we observed y = k, samples can be generated from the exact posterior P ( Y |y = k, pa Y ): k Y = G k + log l e λ l − λ k , G k ∼ Gumbel(0, 1), l Y = − log(e −G l −λ l + e − k Y −λ k ) − λ l , G l ∼ Gumbel(0, 1), ∀l = k . (C.3) Finally, given an upstream counterfactual intervention such that λ = g Y ( pa Y ), the counterfactual outcome for y can be determined simply as y = f Y ( Y ; pa Y ) = arg max 1≤l≤K ( l Y + λ l ) . (C.4) Note that this entire derivation applies to a truly discrete variable, without the need for continuous relaxations as commonly used in deep generative models [25,26], as the likelihood is given in closed form and no gradients of expectations are necessary. Figure 2 : 2Computational graphs of the structural causal models for the Morpho-MNIST example. The image is denoted by x, stroke thickness by t, and image intensity by i. The corresponding causal diagrams are displayed in the top-right corners. Figure 3 :Figure 4 : 34Distributions of thickness and intensity in the true data (left), and learned by the full (centre) and conditional (right) models. Contours depict the observational (red, shaded) and interventional joint densities for do(t := f T ( T ) + 1) (blue, solid) and do(t := f T ( T ) − 0.5) (green, dashed). Original A do(t = 5) do(i = 64) do(t = 3, i = 180) Original B do(t = 1.5) do(t =1.5, i = 224) do(t = 3, Counterfactuals generated by the full model. (left) Counterfactual 'trajectories' of two original samples, A and B, as their thickness and intensity are modified, overlaid on the learned joint density p(t, i). (right) Original and counterfactual images corresponding to samples A and B. true behaviour. The conditional and independent models operate equivalently and are incapable of modelling the relationship between t and i, capturing only their marginal distributions. Figure 5 : 5(b = 800 ml) do(v = 110 ml) do(s = male) s = female; a = 49 y; b = 1153 ml; v = 26.62 ml (b) Original image, counterfactuals, and difference maps Brain imaging example. Variables are image (x), age (a), sex (s), and brain (b) and ventricle (v) volumes. The counterfactuals show different interventions on the same original brain. ( 7 ) 216 Figure A. 1 : 72161to modify each image within the MNIST dataset and randomly split the original training set into a training and validation set. We show random samples from the resulting test set in Fig. A.1. t = 2.3; i = 145 t = 3.2; i = 229 t = 2.9; i = 191 t = 2.6; i = 134 t = 2.6; i = 125 t = 2.7; i = 170 t = 4.0; i = 243 t = 2.3; i = 122 t = 2.1; i = 103 t = 3.6; i = 226 t = 2.2; i = 106 t = 3.3; i = 223 t = 2.9; i = 189 t = 3.6; i = 242 t = 3.5; i = 224 t = 3.1; i = Random exemplars from the synthetically generated Morpho-MNIST test dataset Figure A. 2 :Figure A. 3 :Figure A. 4 : 234Random samples generated by the independent, conditional and full model. Note how all models appear to have the same unconditional generation capacity. Conditional samples generated by the independent, conditional, and full model. The high-level noise, z X , is shared for all samples from each model, ensuring the same 'style' of the generated digit. The independent model generates images independent of the thickness and intensity values, resulting in identical samples. For the conditional and full models, thickness and intensity change consistently along each column and row, respectively. Reconstructions. These are computed as Monte Carlo averages approximating Figure A.5: Comparison of the target covariates and the corresponding values measured from the generated images. The leftmost column refers to the accuracy of the SetThickness and SetIntensity transforms used in generating the synthetic dataset, and the remaining three columns describe the fidelity of samples generated by each of the learned models. While images sampled from the independent model are trivially inconsistent with the sampled covariates, the conditional and full models show comparable conditioning performance. i|do(t)) = p(i|t)Figure A.6: Difference between conditioning and intervening, based on the trained full model. The joint density p(t, i) is shown as contours in the background, for reference, and the 'violin' shapes represent the density of one variable when conditioning or intervening on three different values of the other variable. Figure A. 7 : 7Original samples and counterfactuals from the full model. The first column shows the original image and true values of the non-imaging data. The even rows show the difference maps between the original image and the corresponding counterfactual image. Figure B. 1 : 1Random examplars from the test set of the adopted UK Biobank dataset Figure B. 2 : 2Random samples from the model trained on the UK Biobank dataset b = 800 ml v = 10 ml v = 100 ml v = 1000 ml b = 1200 ml b = 1600 ml v = 10 ml v = 100 ml v = 1000 ml v = 10 ml v = 100 ml v = 1000 ml Figure B.3: Conditional samples from the model trained on the UK Biobank dataset. Images in each 3×3 block share the same the high-level noise vector, z X . Each row consistently changes the brain size, whereas each column changes the ventricle volume. Figure B. 4 : 4Original samples and reconstructions from the model trained on the UK Figure B. 5 : 5Densities for the true data (KDE) and for the learned model. The overall trends and interactions present in the true data distribution seem faithfully captured by the model. do(s = male) do(a = 40 y) do(a = 80 y) do(b = 800 ml) do(b = 1600 ml) do(v = 11 ml) do(v = 110 ml) s = female a = 49 y b = 1153 ml v = 26.62 ml Original do(s = female) do(a = 40 y) do(a = 80 y) do(b = 800 ml) do(b = 1600 ml) do(v = 11 ml) do(v = 110 ml) s = male a = 68 y b = 1078 ml v = 19.89 ml Original do(s = male) do(a = 40 y) do(a = 80 y) do(b = 800 ml) do(b = 1600 ml) do(v = 11 ml) do(v = 110 ml) s = female a = 50 y b = 1095 ml v = 46.84 ml Original do(s = male) do(a = 40 y) do(a = 80 y) do(b = 800 ml) do(b = 1600 ml) do(v = 11 ml) do(v = 110 ml) s = female a = 60 y b = 1035 ml v = 24.29 ml Original do(s = female) do(a = 40 y) do(a = 80 y) do(b = 800 ml) do(b = 1600 ml) do(v = 11 ml) do(v = 110 ml) s = male a = 70 y b = 1062 ml v = 34.87 ml Figure B. 6 : 6Original samples and counterfactuals from the model trained on the UK Biobank dataset. The first column shows the original image and true values of the non-imaging data. The even rows show the difference maps between the original image and the corresponding counterfactual image. Table 1 : 1Comparison of the associative abilities of the models shown inFig. 2. The image is denoted by x, thickness by t, and intensity by i. Quantities with ≥ are lower bounds. MAE refers to the mean absolute error between pixels of the original image and of its reconstruction.Model log p(x, t, i) ≥ log p(x|t, i) ≥ log p(t) log p(i|t) MAE(x, x ) Independent −5925.26 −5919.14 −0.93 −5.19 4.50 Conditional −5526.50 −5520.37 −0.93 −5.19 4.26 Full −5692.94 −5687.71 −0.93 −4.30 4.43 2 4 6 thickness (t) 100 150 200 250 intensity (i) SCMs are also known as (nonlinear) structural equation models or functional causal models. Note that here we assume full observability, i.e. no variables are missing when predicting counterfactuals. We discuss challenges of handling partial evidence in Section 6. Ventricles are fluid-filled cavities identified as the dark areas in the centre of the brain. We observed that not normalising the inputs can lead to the deep models prioritising learning the dependence on the variable with largest magnitude. This phenomenon should be investigated further. AcknowledgementsWe thank Thanos Vlontzos for helpful comments on a draft of this paper. This Elements of Causal Inference: Foundations and Learning Algorithms. Jonas Peters, Dominik Janzing, Bernhard Schölkopf, MIT PressCambridge, MAJonas Peters, Dominik Janzing, and Bernhard Schölkopf. Elements of Causal Inference: Foundations and Learning Algorithms. MIT Press, Cambridge, MA, 2017. Causality: Models, Reasoning, and Inference. Judea Pearl, Cambridge University Press2nd editionJudea Pearl. Causality: Models, Reasoning, and Inference. Cambridge University Press, 2nd edition, 2009. Causality and econometrics. H Wold, Econometrica. 222H. Wold. Causality and econometrics. Econometrica, 22(2):162-177, 1954. URL http: //www.jstor.org/stable/1907540. Causal diagrams for epidemiologic research. Sander Greenland, Judea Pearl, James M Robins, Epidemiology. 101Sander Greenland, Judea Pearl, and James M. Robins. Causal diagrams for epidemiologic re- search. Epidemiology, 10(1):37-48, 1999. URL http://www.jstor.org/stable/3702180. Causality for machine learning. Bernhard Schölkopf, arXiv:1911.10500arXiv preprintBernhard Schölkopf. Causality for machine learning. arXiv preprint arXiv:1911.10500, 2019. CausalVAE: Structured causal disentanglement in variational autoencoder. Mengyue Yang, Furui Liu, Zhitang Chen, Xinwei Shen, Jianye Hao, Jun Wang, arXiv:2004.08697arXiv preprintMengyue Yang, Furui Liu, Zhitang Chen, Xinwei Shen, Jianye Hao, and Jun Wang. CausalVAE: Structured causal disentanglement in variational autoencoder. arXiv preprint arXiv:2004.08697, 2020. Learning independent causal mechanisms. Giambattista Parascandolo, Niki Kilbertus, Mateo Rojas-Carulla, Bernhard Schölkopf, PMLRProceedings of the 35th International Conference on Machine Learning. the 35th International Conference on Machine LearningPMLR80Giambattista Parascandolo, Niki Kilbertus, Mateo Rojas-Carulla, and Bernhard Schölkopf. Learning independent causal mechanisms. In Proceedings of the 35th International Conference on Machine Learning, volume 80 of PMLR, pages 4036-4044. PMLR, 2018. Learning functional causal models with generative neural networks. Olivier Goudet, Diviyan Kalainathan, Philippe Caillou, Isabelle Guyon, David Lopez-Paz, Michèle Sebag, 10.1007/978-3-319-98131-4_3Explainable and Interpretable Models in Computer Vision and Machine Learning. Hugo Jair Escalante, Sergio Escalera, Isabelle Guyon, Xavier Baró, Yagmur Güçlütürk, Umut Güçlü, and Marcel van GervenChamSpringer International PublishingOlivier Goudet, Diviyan Kalainathan, Philippe Caillou, Isabelle Guyon, David Lopez-Paz, and Michèle Sebag. Learning functional causal models with generative neural networks. In Hugo Jair Escalante, Sergio Escalera, Isabelle Guyon, Xavier Baró, Yagmur Güçlütürk, Umut Güçlü, and Marcel van Gerven, editors, Explainable and Interpretable Models in Computer Vision and Machine Learning, pages 39-80. Springer International Publishing, Cham, 2018. doi: 10.1007/978-3-319-98131-4_3. A meta-transfer objective for learning to disentangle causal mechanisms. Yoshua Bengio, Tristan Deleu, Nasim Rahaman, Nan Rosemary Ke, Sebastien Lachapelle, Olexa Bilaniuk, Anirudh Goyal, Christopher Pal, International Conference on Learning Representations. Yoshua Bengio, Tristan Deleu, Nasim Rahaman, Nan Rosemary Ke, Sebastien Lachapelle, Olexa Bilaniuk, Anirudh Goyal, and Christopher Pal. A meta-transfer objective for learning to disentangle causal mechanisms. In International Conference on Learning Representations, 2020. URL https://openreview.net/forum?id=ryxWIgBFPS. Explaining visual models by causal attribution. Parafita Álvaro, Jordi Martínez, Vitrià Marca, 2019 IEEE/CVF International Conference on Computer Vision Workshop (ICCVW). IEEEÁlvaro Parafita Martínez and Jordi Vitrià Marca. Explaining visual models by causal attribution. In 2019 IEEE/CVF International Conference on Computer Vision Workshop (ICCVW), pages 4167-4175. IEEE, 2019. Explanation by progressive exaggeration. Sumedha Singla, Brian Pollack, Junxiang Chen, Kayhan Batmanghelich, International Conference on Learning Representations. Sumedha Singla, Brian Pollack, Junxiang Chen, and Kayhan Batmanghelich. Explanation by progressive exaggeration. In International Conference on Learning Representations, 2020. URL https://openreview.net/forum?id=H1xFWgrFPS. Learning the difference that makes a difference with counterfactually-augmented data. Divyansh Kaushik, Eduard Hovy, Zachary C Lipton, International Conference on Learning Representations. Divyansh Kaushik, Eduard Hovy, and Zachary C. Lipton. Learning the difference that makes a difference with counterfactually-augmented data. In International Conference on Learning Representations, 2020. URL https://openreview.net/forum?id=Sklgs0NFvr. Deep IV: A flexible approach for counterfactual prediction. Jason Hartford, Greg Lewis, Kevin Leyton-Brown, Matt Taddy, PMLRProceedings of the 34th International Conference on Machine Learning. the 34th International Conference on Machine Learning70Jason Hartford, Greg Lewis, Kevin Leyton-Brown, and Matt Taddy. Deep IV: A flexible approach for counterfactual prediction. In Proceedings of the 34th International Conference on Machine Learning, volume 70 of PMLR, pages 1414-1423. PMLR, 06-11 Aug 2017. Causal-GAN: Learning causal implicit generative models with adversarial training. Murat Kocaoglu, Christopher Snyder, Alexandros G Dimakis, Sriram Vishwanath, International Conference on Learning Representations. Murat Kocaoglu, Christopher Snyder, Alexandros G. Dimakis, and Sriram Vishwanath. Causal- GAN: Learning causal implicit generative models with adversarial training. In International Conference on Learning Representations, 2018. URL https://openreview.net/forum? id=BJE-4xW0W. Men also like shopping: Reducing gender bias amplification using corpus-level constraints. Jieyu Zhao, Tianlu Wang, Mark Yatskar, Vicente Ordonez, Kai-Wei Chang, Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing. the 2017 Conference on Empirical Methods in Natural Language ProcessingJieyu Zhao, Tianlu Wang, Mark Yatskar, Vicente Ordonez, and Kai-Wei Chang. Men also like shopping: Reducing gender bias amplification using corpus-level constraints. In Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing, pages 2979-2989, 2017. Dumitru Erhan, Ian Goodfellow, and Rob Fergus. Intriguing properties of neural networks. Christian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, International Conference on Learning Representations. Christian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan, Ian Good- fellow, and Rob Fergus. Intriguing properties of neural networks. In International Conference on Learning Representations, 2014. URL http://arxiv.org/abs/1312.6199. Counterfactual fairness. Matt J Kusner, Joshua Loftus, Chris Russell, Ricardo Silva, Advances in Neural Information Processing Systems. 30Matt J. Kusner, Joshua Loftus, Chris Russell, and Ricardo Silva. Counterfactual fairness. In Advances in Neural Information Processing Systems 30 (NIPS 2017), pages 4066-4076, 2017. Preventing failures due to dataset shift: Learning predictive models that transport. Adarsh Subbaswamy, Peter Schulam, Suchi Saria, PMLRProceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics (AISTATS 2019). the Twenty-Second International Conference on Artificial Intelligence and Statistics (AISTATS 2019)PMLR89Adarsh Subbaswamy, Peter Schulam, and Suchi Saria. Preventing failures due to dataset shift: Learning predictive models that transport. In Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics (AISTATS 2019), volume 89 of PMLR, pages 3118-3127. PMLR, 2019. The seven tools of causal inference, with reflections on machine learning. Judea Pearl, 10.1145/3241036Communications of the ACM. 623Judea Pearl. The seven tools of causal inference, with reflections on machine learning. Commu- nications of the ACM, 62(3):54-60, feb 2019. doi: 10.1145/3241036. A family of nonparametric density estimation algorithms. Esteban G Tabak, Cristina V Turner, 10.1002/cpa.21423Communications on Pure and Applied Mathematics. 662Esteban G. Tabak and Cristina V. Turner. A family of nonparametric density estimation algorithms. Communications on Pure and Applied Mathematics, 66(2):145-164, feb 2013. doi: 10.1002/cpa.21423. Variational inference with normalizing flows. Danilo Rezende, Shakir Mohamed, PMLRProceedings of the 32nd International Conference on Machine Learning. the 32nd International Conference on Machine LearningPMLR37Danilo Rezende and Shakir Mohamed. Variational inference with normalizing flows. In Proceedings of the 32nd International Conference on Machine Learning, volume 37 of PMLR, pages 1530-1538. PMLR, 2015. Normalizing flows for probabilistic modeling and inference. George Papamakarios, Eric Nalisnick, Danilo Jimenez Rezende, Shakir Mohamed, Balaji Lakshminarayanan, arXiv:1912.02762arXiv preprintGeorge Papamakarios, Eric Nalisnick, Danilo Jimenez Rezende, Shakir Mohamed, and Balaji Lakshminarayanan. Normalizing flows for probabilistic modeling and inference. arXiv preprint arXiv:1912.02762, 2019. Stochastic backpropagation and approximate inference in deep generative models. Danilo Jimenez Rezende, Shakir Mohamed, Daan Wierstra, PMLRProceedings of the 31st International Conference on Machine Learning. the 31st International Conference on Machine LearningPMLR32Danilo Jimenez Rezende, Shakir Mohamed, and Daan Wierstra. Stochastic backpropagation and approximate inference in deep generative models. In Proceedings of the 31st International Conference on Machine Learning, volume 32 of PMLR, pages 1278-1286. PMLR, 2014. Auto-encoding variational Bayes. P Diederik, Max Kingma, Welling, International Conference on Learning Representations. Diederik P. Kingma and Max Welling. Auto-encoding variational Bayes. In International Conference on Learning Representations, 2014. URL http://arxiv.org/abs/1312.6114. Categorical reparameterization with Gumbel-softmax. Eric Jang, Shixiang Gu, Ben Poole, International Conference on Learning Representations. Eric Jang, Shixiang Gu, and Ben Poole. Categorical reparameterization with Gumbel-softmax. In International Conference on Learning Representations, 2017. URL https://openreview. net/forum?id=rkE3y85ee. The Concrete distribution: A continuous relaxation of discrete random variables. Chris J Maddison, Andriy Mnih, Yee Whye Teh, International Conference on Learning Representations. Chris J. Maddison, Andriy Mnih, and Yee Whye Teh. The Concrete distribution: A con- tinuous relaxation of discrete random variables. In International Conference on Learning Representations, 2017. URL https://openreview.net/forum?id=S1jE5L5gl. Conditional density estimation with Bayesian normalising flows. L Brian, Richard E Trippe, Turner, NIPS 2017 Workshop on Bayesian Deep Learning. Brian L. Trippe and Richard E. Turner. Conditional density estimation with Bayesian normalising flows. In NIPS 2017 Workshop on Bayesian Deep Learning, 2017. URL http://arxiv.org/abs/1802.04908. Structured output learning with conditional generative flows. You Lu, Bert Huang, Proceedings of the Thirty-Fourth AAAI Conference on Artificial Intelligence. the Thirty-Fourth AAAI Conference on Artificial Intelligence2020You Lu and Bert Huang. Structured output learning with conditional generative flows. In Proceedings of the Thirty-Fourth AAAI Conference on Artificial Intelligence (AAAI 2020). . AAAI. To appearAAAI, 2020. URL http://arxiv.org/abs/1905.13288. To appear. Christina Winkler, Daniel Worrall, Emiel Hoogeboom, Max Welling, arXiv:1912.00042Learning likelihoods with conditional normalizing flows. arXiv preprintChristina Winkler, Daniel Worrall, Emiel Hoogeboom, and Max Welling. Learning likelihoods with conditional normalizing flows. arXiv preprint arXiv:1912.00042, 2019. Adversarial feature learning. Jeff Donahue, Philipp Krähenbühl, Trevor Darrell, International Conference on Learning Representations. Jeff Donahue, Philipp Krähenbühl, and Trevor Darrell. Adversarial feature learning. In International Conference on Learning Representations, 2017. URL https://openreview. net/forum?id=BJtNZAFgg. Adversarially learned inference. Ishmael Vincent Dumoulin, Ben Belghazi, Olivier Poole, Alex Mastropietro, Martin Lamb, Aaron Arjovsky, Courville, International Conference on Learning Representations. Vincent Dumoulin, Ishmael Belghazi, Ben Poole, Olivier Mastropietro, Alex Lamb, Martin Arjovsky, and Aaron Courville. Adversarially learned inference. In International Conference on Learning Representations, 2017. URL https://openreview.net/forum?id=B1ElR4cgg. Mehdi Mirza, Simon Osindero, arXiv:1411.1784Conditional generative adversarial nets. arXiv preprintMehdi Mirza and Simon Osindero. Conditional generative adversarial nets. arXiv preprint arXiv:1411.1784, 2014. Autoencoding beyond pixels using a learned similarity metric. Anders Boesen Lindbo Larsen, Søren Kaae Sønderby, Hugo Larochelle, Ole Winther, PMLRProceedings of The 33rd International Conference on Machine Learning. The 33rd International Conference on Machine LearningPMLR48Anders Boesen Lindbo Larsen, Søren Kaae Sønderby, Hugo Larochelle, and Ole Winther. Autoencoding beyond pixels using a learned similarity metric. In Proceedings of The 33rd International Conference on Machine Learning, volume 48 of PMLR, pages 1558-1566. PMLR, 2016. Generative adversarial nets. Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, Yoshua Bengio, Advances in Neural Information Processing Systems 27 (NIPS 2014). Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in Neural Information Processing Systems 27 (NIPS 2014), pages 2672-2680, 2014. Learning structured output representation using deep conditional generative models. Kihyuk Sohn, Honglak Lee, Xinchen Yan, Advances in Neural Information Processing Systems 28 (NIPS 2015). Kihyuk Sohn, Honglak Lee, and Xinchen Yan. Learning structured output representation using deep conditional generative models. In Advances in Neural Information Processing Systems 28 (NIPS 2015), pages 3483-3491, 2015. Composing graphical models with neural networks for structured representations and fast inference. Matthew J Johnson, David K Duvenaud, Alex Wiltschko, Ryan P Adams, Sandeep R Datta, Advances in Neural Information Processing Systems 29 (NIPS 2016). Matthew J. Johnson, David K. Duvenaud, Alex Wiltschko, Ryan P. Adams, and Sandeep R. Datta. Composing graphical models with neural networks for structured representations and fast inference. In Advances in Neural Information Processing Systems 29 (NIPS 2016), pages 2946-2954, 2016. Variational message passing with structured inference networks. Wu Lin, Mohammad Emtiyaz Khan, Nicolas Hubacher, International Conference on Learning Representations. Wu Lin, Mohammad Emtiyaz Khan, and Nicolas Hubacher. Variational message passing with structured inference networks. In International Conference on Learning Representations, 2018. URL https://openreview.net/forum?id=HyH9lbZAW. Graphical generative adversarial networks. Chongxuan Li, Max Welling, Jun Zhu, Bo Zhang, Advances in Neural Information Processing Systems. 31Chongxuan Li, Max Welling, Jun Zhu, and Bo Zhang. Graphical generative adversarial networks. In Advances in Neural Information Processing Systems 31 (NeurIPS 2018), pages 6069-6080, 2018. Deep convolutional inverse graphics network. D Tejas, William F Kulkarni, Pushmeet Whitney, Josh Kohli, Tenenbaum, Advances in Neural Information Processing Systems 28 (NIPS 2015). Tejas D. Kulkarni, William F. Whitney, Pushmeet Kohli, and Josh Tenenbaum. Deep convo- lutional inverse graphics network. In Advances in Neural Information Processing Systems 28 (NIPS 2015), pages 2539-2547, 2015. Info-GAN: Interpretable representation learning by information maximizing generative adversarial nets. Xi Chen, Yan Duan, Rein Houthooft, John Schulman, Advances in Neural Information Processing Systems 29 (NIPS 2016). Ilya Sutskever, and Pieter AbbeelXi Chen, Yan Duan, Rein Houthooft, John Schulman, Ilya Sutskever, and Pieter Abbeel. Info- GAN: Interpretable representation learning by information maximizing generative adversarial nets. In Advances in Neural Information Processing Systems 29 (NIPS 2016), pages 2172-2180, 2016. Shakir Mohamed, and Alexander Lerchner. β-VAE: Learning basic visual concepts with a constrained variational framework. Irina Higgins, Loic Matthey, Arka Pal, Christopher Burgess, Xavier Glorot, Matthew Botvinick, International Conference on Learning Representations. Irina Higgins, Loic Matthey, Arka Pal, Christopher Burgess, Xavier Glorot, Matthew Botvinick, Shakir Mohamed, and Alexander Lerchner. β-VAE: Learning basic visual concepts with a constrained variational framework. In International Conference on Learning Representations, 2017. URL https://openreview.net/forum?id=Sy2fzU9gl. Attribute-based regularization of VAE latent spaces. Ashis Pati, Alexander Lerch, arXiv:2004.05485arXiv preprintAshis Pati and Alexander Lerch. Attribute-based regularization of VAE latent spaces. arXiv preprint arXiv:2004.05485, 2020. Image style transfer using convolutional neural networks. Leon A Gatys, Alexander S Ecker, Matthias Bethge, Proceedings of the 2016 IEEE Conference on Computer Vision and Pattern Recognition. the 2016 IEEE Conference on Computer Vision and Pattern RecognitionLeon A. Gatys, Alexander S. Ecker, and Matthias Bethge. Image style transfer using convolu- tional neural networks. In Proceedings of the 2016 IEEE Conference on Computer Vision and Pattern Recognition, pages 2414-2423, 2016. Image-to-image translation with conditional adversarial networks. Phillip Isola, Jun-Yan Zhu, Tinghui Zhou, Alexei A Efros, Proceedings of the 2017 IEEE Conference on Computer Vision and Pattern Recognition. the 2017 IEEE Conference on Computer Vision and Pattern RecognitionPhillip Isola, Jun-Yan Zhu, Tinghui Zhou, and Alexei A. Efros. Image-to-image translation with conditional adversarial networks. In Proceedings of the 2017 IEEE Conference on Computer Vision and Pattern Recognition, pages 1125-1134, 2017. Face aging with conditional generative adversarial networks. G Antipov, M Baccouche, J Dugelay, 2017 IEEE International Conference on Image Processing (ICIP). G. Antipov, M. Baccouche, and J. Dugelay. Face aging with conditional generative adversarial networks. In 2017 IEEE International Conference on Image Processing (ICIP), pages 2089- 2093, 2017. Learning to synthesise the ageing brain without longitudinal data. Tian Xia, Agisilaos Chartsias, Chengjia Wang, Sotirios A Tsaftaris, arXiv:1912.02620arXiv preprintTian Xia, Agisilaos Chartsias, Chengjia Wang, and Sotirios A. Tsaftaris. Learning to synthesise the ageing brain without longitudinal data. arXiv preprint arXiv:1912.02620, 2019. Gradient-based learning applied to document recognition. Yann Lecun, Léon Bottou, Yoshua Bengio, Patrick Haffner, 10.1109/5.726791Proceedings of the IEEE. 8611Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278-2324, 1998. doi: 10.1109/5.726791. Morpho-MNIST: Quantitative assessment and diagnostics for representation learning. C Daniel, Jeremy Castro, Bernhard Tan, Ender Kainz, Ben Konukoglu, Glocker, Journal of Machine Learning Research. 20178Daniel C. Castro, Jeremy Tan, Bernhard Kainz, Ender Konukoglu, and Ben Glocker. Morpho- MNIST: Quantitative assessment and diagnostics for representation learning. Journal of Machine Learning Research, 20(178), 2019. PyTorch: An imperative style, high-performance deep learning library. Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Kopf, Edward Yang, Zachary Devito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, Soumith Chintala, Advances in Neural Information Processing Systems. 32Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Kopf, Edward Yang, Zachary DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. PyTorch: An imperative style, high-performance deep learning library. In Advances in Neural Information Processing Systems 32 (NeurIPS 2019), pages 8024-8035, 2019. Pyro: Deep universal probabilistic programming. Eli Bingham, Jonathan P Chen, Martin Jankowiak, Fritz Obermeyer, Neeraj Pradhan, Theofanis Karaletsos, Rohit Singh, Paul Szerlip, Paul Horsfall, Noah D Goodman, Journal of Machine Learning Research. 2028Eli Bingham, Jonathan P. Chen, Martin Jankowiak, Fritz Obermeyer, Neeraj Pradhan, Theofanis Karaletsos, Rohit Singh, Paul Szerlip, Paul Horsfall, and Noah D. Goodman. Pyro: Deep universal probabilistic programming. Journal of Machine Learning Research, 20(28), 2019. UK Biobank: an open access resource for identifying the causes of a wide range of complex diseases of middle and old age. Cathie Sudlow, John Gallacher, Naomi Allen, Valerie Beral, Paul Burton, John Danesh, Paul Downey, Paul Elliott, Jane Green, Martin Landray, PLoS Medicine. 123Cathie Sudlow, John Gallacher, Naomi Allen, Valerie Beral, Paul Burton, John Danesh, Paul Downey, Paul Elliott, Jane Green, Martin Landray, et al. UK Biobank: an open access resource for identifying the causes of a wide range of complex diseases of middle and old age. PLoS Medicine, 12(3), 2015. Ageing and the brain. Ruth Peters, Postgraduate Medical Journal. 82964Ruth Peters. Ageing and the brain. Postgraduate Medical Journal, 82(964):84-88, 2006. Neural spline flows. Conor Durkan, Artur Bekasov, Iain Murray, George Papamakarios, Advances in Neural Information Processing Systems. 32Conor Durkan, Artur Bekasov, Iain Murray, and George Papamakarios. Neural spline flows. In Advances in Neural Information Processing Systems 32 (NeurIPS 2019), pages 7511-7522, 2019. Density estimation using real nvp. Laurent Dinh, Jascha Sohl-Dickstein, Samy Bengio, Laurent Dinh, Jascha Sohl-Dickstein, and Samy Bengio. Density estimation using real nvp, 2016. URL https://arxiv.org/abs/1605.08803.
{'fraction_non_alphanumeric': 0.05436814643265716, 'fraction_numerical': 0.026814364240073663, 'mean_word_length': 4.757811872439554, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 12, 'lorem ipsum': 0, 'www.': 2, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 18, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': "We formulate a general framework for building structural causal models (SCMs) with deep learning components. The proposed approach employs normalising flows and variational inference to enable tractable inference of exogenous noise variables-a crucial step for counterfactual inference that is missing from existing deep causal learning methods. Our framework is validated on a synthetic dataset built on MNIST as well as on a real-world medical dataset of brain MRI scans. Our experimental results indicate that we can successfully train deep SCMs that are capable of all three levels of Pearl's ladder of causation: association, intervention, and counterfactuals, giving rise to a powerful new approach for answering causal questions in imaging applications and beyond. The code for all our experiments is available at https://github.com/biomedia-mira/deepscm.", 'arxivid': '2006.06485', 'author': ['Nick Pawlowski \nImperial College London\nImperial College London\nImperial College London\n\n', 'Daniel C Castro \nImperial College London\nImperial College London\nImperial College London\n\n', 'Ben Glocker [email protected] \nImperial College London\nImperial College London\nImperial College London\n\n'], 'authoraffiliation': ['Imperial College London\nImperial College London\nImperial College London\n', 'Imperial College London\nImperial College London\nImperial College London\n', 'Imperial College London\nImperial College London\nImperial College London\n'], 'corpusid': 219573236, 'doi': None, 'github_urls': ['https://github.com/biomedia-mira/deepscm.'], 'n_tokens_mistral': 20647, 'n_tokens_neox': 17778, 'n_words': 10571, 'pdfsha': 'd4155cc64d16e3a8f516e4e51e6b4ee86d8d34d4', 'pdfurls': ['https://arxiv.org/pdf/2006.06485v1.pdf'], 'title': ['Deep Structural Causal Models for Tractable Counterfactual Inference', 'Deep Structural Causal Models for Tractable Counterfactual Inference'], 'venue': []}
arxiv
Probing the Top-Higgs Yukawa CP Structure in dileptonic tth with M 2 -Assisted Reconstruction 4 Oct 2022 Dorival Gonçalves [email protected] PITT-PACC Department of Physics and Astronomy University of Pittsburgh USA Jeong Han Kim [email protected] Department of Physics and Astronomy University of Kansas 66045LawrenceKSUSA Kyoungchul Kong [email protected] Department of Physics and Astronomy University of Kansas 66045LawrenceKSUSA Probing the Top-Higgs Yukawa CP Structure in dileptonic tth with M 2 -Assisted Reconstruction 4 Oct 2022Beyond Standard ModelPhenomenological ModelsHiggs PhysicsTop physicsLHC Constraining the Higgs boson properties is a cornerstone of the LHC program. We study the potential to directly probe the Higgs-top CP-structure via the tth channel at the LHC with the Higgs boson decaying to a bottom pair and top-quarks in the dileptonic mode. We show that a combination of laboratory and tt rest frame observables display large CP-sensitivity, exploring the spin correlations in the top decays. To efficiently reconstruct our final state, we present a method based on simple mass minimization and prove its robustness to shower, hadronization and detector effects. In addition, the mass reconstruction works as an extra relevant handle for background suppression. Based on our results, we demonstrate that the Higgs-top CP-phase (α) can be probed up to cos α < 0.7 at the high luminosity LHC. Introduction After the discovery of the Higgs boson at the Large Hadron Collider (LHC) [1,2], the determination of its properties has become a prominent path in the search for physics beyond the Standard Model (SM) [3][4][5]. So far, measurements based on the Higgs signal strengths conform to the SM predictions [6,7]. However, the tensor structure of the Higgs couplings to other matter fields remains relatively unconstrained. A particularly interesting option is that the Higgs interactions present new sources of CP-violation, which could be a key element in explaining the matter-antimatter unbalance in the Universe [8,9]. CP-violation in the Higgs sector has been searched for at the LHC mostly via Higgs couplings with W ± and Z gauge bosons throughout the Higgs decays h → W + W − and ZZ [10][11][12][13][14][15][16][17][18][19]. However, these possible CP-violating interactions are one-loop suppressed, arising only via operators of dimension-6 or higher [20,21]. On the other hand, CP-odd Higgs fermion interactions could manifest already at the tree level, being naturally more sensitive to new physics [22? ? ? ? ? ? ? -33]. Of special interest is the Higgs coupling to top quarks, as y SM t ∼ 1. Relevant constraints to the CP-structure of the top-Higgs coupling can be indirectly probed via loop-induced interactions in electric dipole moment (EDM) experiments and gluon fusion hjj production at the LHC [13,[34][35][36]. While electron and neutron EDM can set very stringent bounds on CP-mixed top Yukawa, it critically assumes the Yukawa coupling with the first generation fermions the same as in the SM, and that new CP-violating interactions are limited to the third generation. A minor modification on the strength and CP-structure of the Higgs interactions to first generation can considerably degrade these constraints [34]. Similarly, possible new physics loop-induced contributions can spoil the measurement through gluon fusion hjj production [37][38][39][40][41][42]. Therefore, the direct measurement of this coupling is required to disentangle potential additional new physics effects. Analogously to the direct (model independent) measurement of the top Yukawa strength, the direct measurement for its CP-phase also has the pp → tth channel as its most natural path. Going beyond the signal strength analysis for this channel becomes even further motivated given i) the recent CMS result, showing observation for the tth signal with 5.2σ observed (4.2σ expected) [43,44]; and ii) the High-Lumi LHC (HL-LHC) projections, indicating that this channel will be measured with a very high precision, δy t < 10% [45]. Hence, that is the approach which we follow in the present study, exploring the spin correlations in the top pair decays. The different Higgs-top CP-structure affects the top-spin correlation, that can propagate to the top quark decay products. The most natural channel to perform such a study is the dileptonic top decay, as the spin analyzing power for charged leptons is maximal. Spin correlations can be enhanced looking at the tt rest frame, however the large experimental uncertainties at hadron collider due to top reconstruction and frame change make this measurement challenging. We will present a method for the top reconstruction that will address these issues, allowing the construction of relevant CP observables at tt rest frame. The aim of this paper is twofold. First, we will study direct Higgs-top CP measurement via the tth production, exploiting full kinematic reconstruction in the dilepton channel. For this purpose we adopt a kinematic reconstruction method presented in Ref. [46]. Second, since this reconstruction method was studied only at the parton-level, we would like to investigate its performance further beyond the parton-level, including more realistic effects such as parton-shower, hadronization and detector resolution. Although this reconstruction method was initially presented for the top quark pair production tt, we will show that it can be easily adopted to the tth channel. This paper is structured as follows. In section 2, we will present our setup and the kinematic observables to access the CP-phase. In section 3, we will discuss the method for kinematic reconstruction of the dileptonic tops. In section 4.1, we show that the angular correlations can be obtained via this method, presenting the results at the parton level, while in section 4.2, we perform a full signal and background analysis, including parton-shower, hadronization and detector effects, and discuss the prospects of the CP measurement in the tth channel with dileptonic top-quarks and h → bb decays. Setup and angular observables We start with the following Lagrangian containing the top Yukawa coupling L ⊇ − m t v K t (cos α + iγ 5 sin α) t h ,(2.1) where v = 246 GeV is the SM Higgs vacuum expectation value, K is a real number and α represents the Higgs-top CP-phase. Hence, the SM Higgs-top interaction is represented by the pure CP-even coupling (K, α) = (1, 0), while (K, α = π/2) parametrizes a pure CP-odd Higgs boson. Various observables have been explored in the literature to access the Higgs-top CP-phase in tth events, e.g., total cross-section, transverse Higgs momentum, invariant tt mass, and spin correlations in the top quark decay products [22][23][24][25][26][27][28][29][30][31]. The latter is specially interesting as it can accurately probe the Higgs-top interaction, exploring the spin polarization of the tt pair via a shape analysis. While at hadron colliders the top quarks are unpolarized, the top and anti-top pair are highly correlated. This fact can be experimentally revealed by spin correlations between the top decay products [47]. The top-quark spin polarization is transferred to the top decays, t → W + b with W + → + ν or d + u, where the spin analyzing power is maximal for the charged lepton + and the down quark d. Exploring this, Ref. [23] demonstrates that the difference in azimuthal angle between the leptons ∆φ lab (from top decays) in the laboratory frame can directly reveal the CP-structure of the Higgs-top interaction with the sensitivity of the measurement substantially enhanced in the boosted Higgs regime, as shown in the left panel of Fig. 1. This study shows that the Higgs-top coupling strength and the CP structure can be directly probed with achievable luminosity at the HL-LHC, using boosted Higgs substructure in the dileptonic channel. In the present paper, we would like to include observables in the center-of-mass frame of tt system, exploiting novel kinematic reconstruction methods. Among several distributions studied in the tt differential cross-section measurements, we find that the production angle θ * in the Collins-Soper reference frame brings an interesting correlation, as shown in Fig. 1 (middle). This θ * is a collision angle of the top with respect to a beam axis in the tt centerof-mass frame and therefore the two top quarks have equal and opposite momenta, with each making the same angle θ * with the beam direction [48]. See Ref. [29] for a recent application of a similar observable which probes the spin and parity of a new light resonance. All these variables, including ∆φ lab and θ * , are sensitive only to the square terms cos 2 α and sin 2 α (CP-even observables), providing only an indirect measure of CP-violation, missing the interference term between CP-even and odd couplings, cos α sin α, that can capture a relative coupling sign. To define CP-odd observables, we have to further explore the spin polarization of the tt pair. Remarkably, tensor product relations of the top-pair and the final state particles, that follow from totally antisymmetric expressions (p a , p b , p c , p d ) ≡ µνρσ p µ a p ν b p ρ c p σ d (with 0123 = 1) , are examples of such observables. In the present work, we will focus on a relevant tensor product that has information on the top and anti-top and the charged leptons from top-quark decays, maximizing the spin analyzing power: (p t , p t , p + , p − ). In general, this expression leads to several terms, making it difficult to define an observable that extracts all its information. However, this relation opportunely simplifies at the tt center of mass (CM) frame, resulting in a single triple product (p t , p t , p + , p − )| tt CM ∝ p t · (p + × p − ) ,(2.2) provided that we can fully reconstruct the tt CM frame. We further explore this relation to define our CP-odd observable ∆φ tt = sgn [ p t · ( p + × p − )] arccos p t × p + | p t × p + | · p t × p − | p t × p − | , (2.3) that is defined in the [−π, π] range. In Fig. 1 (right), we display the ∆φ tt distributions at the parton-truth level for different CP hypotheses α. The CP-mixed cases α = π/4 from −π/4 display different distribution shapes, confirming that ∆φ tt is a truly CP-odd observable. One may quantify these differences via an asymmetry, comparing the number of events with positive and negative ∆φ tt [25]: A ≡ N (∆φ tt > 0) − N (∆φ tt < 0) N (∆φ tt > 0) + N (∆φ tt < 0) ,(2.4) where A ∈ [−1, 1]. While the asymmetry A results in deviations from the SM hypothesis of at maximum O(4%) (for α ≈ ± π 4 ), ∆φ tt presents parameter space regions that can reach up to O(10%) of difference in ratio 1 σ(α) · dσ(α) d∆φ tt / 1 σ(0) · dσ(0) d∆φ tt , as shown in the subfigure of the right plot. The latter leads to a potentially stronger distinguishing power that can be explored via a shape analysis. Due to difficulty in event reconstruction to go to the tt rest frame, the ∆φ tt observable has not been investigated in a realistic analysis so far. In this study, we shall attempt to reconstruct the θ * and ∆φ tt variable at hadron-level including detector resolution. We will then examine how these two observables (∆φ tt and θ * ) would improve the existing analysis with the laboratory angle (∆φ lab ). We will make a brief comment on the sign of CP angle as well. t W + v t W - v b l + b l - (b) (l ) (bl ) h Brief review of kinematic reconstruction In this section, we briefly review the reconstruction method that we adopt. Our algorithm is entirely based on mass minimization. Thus, it is more flexible for new physics analyses and robust for our spin-correlation study 1 . The event topology considered in this paper is shown in Fig. 2, together with three possible subsystems. The blue dotted, the green dot-dashed, and the black solid boxes indicate the subsystems (b), ( ), and (b ), respectively. We consider that the Higgs (denoted as h) is fully reconstructed, in which case the only source of the missing transverse momentum is two neutrinos from the top decays. In the presence of two missing particles at the end of a cascade decay, M T 2 provides a good estimate of mass information in the involved decay [49,51,53,58]. Following notations and conventions of Ref. [46], we define M T 2 as follows: M T 2 (m) ≡ min q 1T , q 2T {max [M T P 1 ( q 1T ,m), M T P 2 ( q 2T ,m)]} , (3.1) q 1T + q 2T = / P T , where M T P i (i = 1, 2) is the transverse mass of the decaying particle in the i-th side andm is a test mass, which we set to zero in our study. q iT is the unknown transverse momentum of the i-th missing particle, which is a neutrino in this case. Individual values ( q 1T and q 2T ) are unknown and only their sum ( q 1T + q 2T ) is constrained by the total missing transverse momentum, / P T . Another mass-constraining variable is the M N [46,53,59], which is the (3+1)-dimensional version of Eq. (3.1): M 2 (m) ≡ min q 1 , q 2 {max [M P 1 ( q 1 ,m), M P 2 ( q 2 ,m)]} , (3.2) q 1T + q 2T = / P T , where the actual parent masses (M P i ) are considered instead of their transverse masses (M T P i ). Note that the minimization is now performed over the 3-component momentum vectors q 1 and q 2 [53]. In fact, at this point the two definitions (3.1) and (3.2) are equivalent, in the sense that the resulting two variables, M T 2 and M 2 , will have the same numerical value [53,54,60]. However, M 2 begins to differ from M T 2 when applying additional kinematic constraints beyond the missing transverse momentum condition q 1T + q 2T = / P T . Then, the M 2 variable can be further refined and one can obtain non-trivial variants as shown below [54]: M 2XX ≡ min q 1 , q 2 {max [M P 1 ( q 1 ,m), M P 2 ( q 2 ,m)]} , (3.3) q 1T + q 2T = / P T M 2CX ≡ min q 1 , q 2 {max [M P 1 ( q 1 ,m), M P 2 ( q 2 ,m)]} , (3.4) q 1T + q 2T = / P T M P 1 = M P 2 M 2XC ≡ min q 1 , q 2 {max [M P 1 ( q 1 ,m), M P 2 ( q 2 ,m)]} , (3.5) q 1T + q 2T = / P T M 2 R 1 = M 2 R 2 M 2CC ≡ min q 1 , q 2 {max [M P 1 ( q 1 ,m), M P 2 ( q 2 ,m)]} . (3.6) q 1T + q 2T = / P T M P 1 = M P 2 M 2 R 1 = M 2 R 2 Here M P i (M R i ) is the mass of the parent (relative) particle in the i-th decay chain and a subscript "C" indicates that an equal mass constraint is applied for the two parents (when "C" is in the first position) or for the relatives (when "C" is in the second position). A subscript "X" simply means that no such constraint is applied. Note that M 2XX in Eq. M T 2 = M 2XX = M 2CX ≤ M 2XC ≤ M 2CC . (3.7) More specifically, in the tt-like production (tt + X where X is fully reconstructed), we could use the experimentally measured W -boson mass, m W , and introduce the following variable: M (b ) 2CW ≡ min q 1 , q 2 {max [M t 1 ( q 1 ,m), M t 2 ( q 2 ,m)]} . (3.8) q 1T + q 2T = / P T M t 1 = M t 2 M W 1 = M W 2 = m W Similarly, taking the mass m t of the top quark in the minimization, we can define a new variable in the ( ) subsystem: M ( ) 2Ct ≡ min q 1 , q 2 {max [M W 1 ( q 1 ,m), M W 2 ( q 2 ,m)]} . (3.9) q 1T + q 2T = / P T M W 1 = M W 2 M t 1 = M t 2 = m t Although these mass-constraining variables are proposed for mass measurement originally, one could use them for other purposes such as measurement of spins and couplings [61]. In our study, we use these variables to fully reconstruct the final state of our interest, with the unknown momenta obtained via minimization procedure. These momenta may or may not be true particle momenta but they provide important non-trivial correlations with other visible particles in the final state, which helps reconstruction. Based on Ref. [46], we define the following parameter space: (x, y, z) ≡ m max b − max i {m (i) b }, m t − M (b ) 2CW , m W − M ( ) 2Ct ,(3.10) where m 2Ct , leading to positive x, y, and z. On the other hand, the wrong pairing could give either sign. Finally, by requiring that the partition which gives more "plus" sign as the "correct" one, we can resolve two-fold ambiguity. Then we treat the corresponding momenta of two missing particles (which are obtained via the minimization procedure) as "real" momenta of two missing neutrinos. If both partitions give the same numbers of positive and negative signs (called "unresolved case"), we discard those events. From Ref. [46], the efficiency of this method is known to be about 88%, including unresolved events with a coin flip, 50% probability of picking the right combination. Since we ignore those events to obtain a highpurity sample, the corresponding efficiency becomes 83%. We also note that we assign the (1/σ) dσ/dmb parton level (resolved) α = 1 2 π α = 1 4 π α = 0 α = − 1 4 π α = − 1 2 π 0(1/σ) dσ/dM ( ) 2Ct parton level (resolved) α = 1 2 π α = 1 4 π α = 0 α = − 1 4 π α = − 1(1/σ) dσ/dM (b ) 2CW parton level (resolved) α = 1 2 π α = 1 4 π α = 0 α = − 1 4 π α = − 1 2 π Figure 3: Distributions of m b (left), M ( ) 2Ct (middle), and M (b ) 2CW (right) for different CP phases. negative sign for a partitioning, if a viable solution is not found during minimization. This is because the wrong pairing would fail more often than the correct paring. With the obtained neutrino momenta, now we can reconstruct momenta of W s and top quarks for the CP measurement of the top-Yukawa coupling. Top-Higgs Yukawa coupling with M 2 -assisted reconstruction We show our parton-level results in section 4.1, and detector-level (including parton-shower, hadronization, and detector resolution for signal and backgrounds) in section 4.2. For our parton-level study, we assume that the Higgs is fully reconstructed. We separate these semirealistic effects to better examine the capability and feasibility of reconstruction methods in the dileptonic tth production. Throughout our study, we use OPTIMASS [55] to obtain momenta of two invisible neutrinos, following the reconstruction method described in the previous section. Parton-level analysis Parton level events are generated at leading order by MadGraph5 aMC@NLO [62] in chain with FeynRules package [63] without any generation level cuts. We use the default NNPDF2.3QED parton distribution function [64] with dynamical renormalization and factorization scales set to m 2 T (transverse mass of the visible system) at the 14 TeV LHC. In this section, we focus on comparison between Monte-Carlo truth and parton-level results without worrying about effects of hadronization and parton-shower, which will be the topic in the next section. Performing the procedure described in the previous section, we obtain the momenta of two neutrinos and also resolve two fold ambiguity in the dilepton final state, which allows full reconstruction of the final state. the purity of the samples is known to be 96% [46]. Note that, throughout this paper, all plots are generated with the "resolved" events, after discarding "unresolved" ones. We find that the efficiency of our method is = 81.38%, which is consistent with 83% as in Ref. [46]. The resolved events contain both correct and wrong combinations and the fraction of the correct combination out of the resolved events is defined as purity. To examine performance of momentum reconstruction, we show in Fig. 4 correlations between ∆p x ≡ p x,true − p x and ∆p z ≡ p z,true − p z , and between the difference in magnitude | p | − | p true | and the direction mismatch ∆R( p, p true ) for M (b ) 2CW for the SM case (α = 0). Other CP angles show similar results. Here p true is the true momentum of a neutrino and p is the momentum from the minimization using OPTIMASS. In the upper panel, the scatter plots are generated without any cuts, while a mass cut (165 GeV < M As shown in Ref. [23], the difference in azimuthal angles of two isolated leptons in the laboratory frame ∆φ lab provides a good discrimination of different CP angles at the boosted regime. We reproduce this result as already shown in the left panel of Fig. 1. Once the cuts of p T (h) > 200 GeV and m > 75 GeV are applied, the distributions acquire high distinguishing power, as shown in the figure. Thanks to the fact that it depends only on the leptons, and it is reconstructed at the laboratory frame, this observable displays small uncertainties. Having reconstructed full four-momenta of each top, we form θ * shown in Fig. 5, which is the production angle in the Collins-Soper reference frame [48]. This distribution exhibits very little sensitivity to the adopted reconstruction procedure and retains the corresponding shape at Mone-Carlo truth (see the middle plot in Fig. 1 for comparison). This is partially due to a much simpler structure of θ * as compared to the shape of other distributions such as ∆φ tt . In Fig. 6, we present ∆φ tt in the center-of-mass frame of the tt system (see Eq. 2.3) for various values of α. While Fig. 1 assumes prior knowledge (parton-truth) of correct final state particles pairs, Fig. 6 is obtained via the M 2 reconstruction. This distribution gets degraded as shown in the left panel of Fig. 6, once we include all the resolved events (admixture of both correct and wrong combinations). However, one can make an improvement with a mass cut on M 2CW < 175 GeV, and restore their original shapes, as shown in the right panel of Fig. 6. In the case of CP mixed eigenstate (e.g. α = ±π/4), the ∆φ tt distributions are asymmetric with respect to ∆φ tt = 0. On the other hand, θ * distributions are symmetric. Numerical values of ∆φ tt asymmetry are summarized in Table 1. A (α = 0, ±π/2) = 0 is expected but we obtain nonzero values due to statistical uncertainties. We observe that the wrong combinatorics can be further suppressed with the M closer to the idealistic parton-truth asymmetries. Detector level analysis and LHC sensitivity After proving that our top mass reconstruction method dovetails nicely with CP-sensitive observables at the tt rest frame, we perform a full Monte Carlo study, including the Higgs boson decay to a pair of b-quarks. We require four bottom tagged jets and two opposite sign leptons in our signal. The major backgrounds for this signature in order of relevance are ttbb and ttZ. Both signal and background events are showered and hadronized by PYTHIA 6 [68]. Jets are clustered with the FastJet [69] implementation of the anti-k T algorithm [70] with a fixed cone size of R = 0.4 (1.2) for a slim (fat) jet. We include simple detector effects based on the ATLAS detector performances [71], and smear momenta and energies of reconstructed jets and leptons according to their energy values. See Appendix A for more details. In the phase space where the Higgs is kinematically boosted, its decay products are collimated in the same direction. In this regime, the Higgs can be better reconstructed using a single fat jet evading its possible intervention to the tt-system. Therefore, our previous method of resolving a combinatorial problem can be repeatedly applicable in the boosted Higgs configuration. The boosted Higgs jet with a two-pronged substructure is a rare feature that the SM backgrounds retain. Thus, it delivers a further handle to disentangle the backgrounds from our signal events. The first demonstration of the use of a jet substructure technique in the dileptonic tth(bb) channel can be found in Ref. [23], where it effectively kills both ttbb and ttZ backgrounds. Here we follow similar steps, employing the TemplateTagger v.1.0 [72] implementation of the Template Overlap Method (TOM) [73,74] as a boosted Higgs tagger, due to its robustness against pile-up contaminations. We first require at least one R = 1.2 fat jet with p J T > 200 GeV, and |η J | < 2.5. (4.3) For a fat jet to be tagged as a Higgs, we demand a two-pronged Higgs template overlap score Ov h 2 > 0.5. (4.4) We require exactly one Higgs-tagged fat jet that passes the cuts in Eqs. (4.3-4.4) and has 2b-tagged slim jets inside 2 : Additionally, we require at least two slim jets that are isolated from the Higgs-tagged fat jet p j T > 30 GeV, and |η j | < 2.5, (4.6) in which we require exactly two b-tagged slim jets. We demand exactly two isolated leptons passing the cuts in Eq. (4.2) and N h = 1.(1/σ) dσ/dM ( ) 2Ct α = 1 2 π α = 1 4 π α = 0 α = − 1 4 π α = − 1 2 π tt + jets(1/σ) dσ/dM (b ) 2CW α = 1 2 π α = 1 4 π α = 0 α = − 1 4 π α = − 1 2 π tt + jetsp T /p Σ T > 0.7,(4.7) where p Σ T is the sum of transverse momenta of final state particles (including a lepton) within ∆R = 0.3 isolation cone. In Fig. 7 (upper-left), we show the reconstructed invariant mass distributions for Higgstagged fat jet, laid out with the dominant ttbb background. The distributions are insensitive to different CP structures, but provide more separation from the background. Hence, we 2CW becomes broader due to parton shower, hadronization and detector resolution effects, compared to parton-level results in Fig. 3, but the basic shape remains the same. We resolve the combinatorial ambiguity of the two b-lepton pairs based on the prescription in Eq. (3.10). The efficiency of the method for our signal is 82% (comparable to the efficiency at parton level), yet at the same time ttbb and ttZ backgrounds are cut down to 64% and 70%, respectively. Hence, the top mass reconstruction method works as an extra relevant handle in the background suppression, eliminating wrong combinations from b-jets that are not from the top decays. (1/σ) dσ/dφ lab α = 1 2 π α = 1 4 π α = 0 α = − 1 4 π α = − 1 2 π Figure 9: Distributions of ∆φ lab , after resolving the combinatorial problem and 4b-tagging, without (left) and with (right) an additional m > 75 GeV selection. Momentum reconstructions of two neutrinos are displayed in Fig. 8. The level of accuracy in reconstructing neutrino momenta also degrades to some extent, where the uncertainty in p z direction is greater than the transverse components. Additional mass cut 155 GeV < M (b ) 2CW < 180 GeV reduces the reconstruction efficiency to = 32%, but would increase the purity of the sample and improve the momentum resolution. We observe that the reconstruction method is robust to parton-shower, hadronization, and detector resolution effects, presenting similar efficiencies to the parton level analysis. Our reconstruction is better than (or comparable to) existing results. For example, Ref. [77] performs a conventional kinematic mass reconstruction with the missing transverse momentum and attempts resolving the two-fold sign ambiguity using a likelihood based on transverse momenta of the involved particles. This method leads to 62% efficiency with 50% purity for signal, and 51% efficiency for backgrounds. Since our method is purely based on mass minimization, it is less sensitive to new physics modifications and is a suitable element for a robust spin-correlation analysis. We note that one can further improve the efficiency of our method by utilizing those discarded "unresolved" events and deploying a hybrid method [46] together with M 2 reconstruction. We acknowledge that there is a certain degree of uncertainty in the precision compared to parton-level results in Fig. 4, where the peaks are broadened. We attribute this change to contaminations in the total missing transverse momentum where additional neutrinos from h → bb system, via the semi-leptonic decays of the b-hadrons, can disrupt the relations in Eqs. (3.8)-(3.9), in combination with detector effects. Nevertheless, overall net shapes stay the same showing its resilience over the procedures. Distributions of ∆φ lab , ∆φ tt , and θ * are presented in Figs. 9 and 10. The θ * and ∆φ lab distributions remain very similar to those at parton level ( Fig. 1 and Fig. 5), while ∆φ tt distribution gets more distorted (see Fig. 6). Table 2 summarizes the impact of a series of cuts for the signal (α = 0) and background cross sections. In the last column, we show the significances (σ), which are calculated for a , with L(x|n) = x n n! e −x , (4.9) where S and B are the expected number of signal and background events, respectively [78]. We find that our results are roughly in agreement with those from Ref. [23]. Although we obtain high significance as shown in the first row of Table 2, we would impose more stringent cuts for high-purity sample of tth production. We obtain σ = 8.1 with the resolved combinatorics. For an additional mass cut, we could retrieve even higher purity but we would suffer from statistics. In the following analysis, we do not impose this mass cut but instead require the dilepton invariant mass cut, m > 75 GeV. The asymmetry results at detectorlevel are summarized in Table 3 In Fig. 11 (left panel) we display the 95% C.L. bound to distinguish the CP-α Higgs-top interaction from the SM via tth production. Our limits are based on a binned log-likelihood analysis invoking the CL s method for (∆φ lab , θ * ) (blue dashed), and (∆φ lab , θ * , ∆φ tt ) (blue full) [79]. The bounds are obtained, including backgrounds, parton-shower, hadronization and semi-realistic detector effects. To illustrate the robustness of the top reconstruction method when going from the parton to the detector level, we also show the bounds using the parton-level distributions (∆φ lab , θ * ) with the rates rescaled to the full detector analysis (black full). The red-solid curve, labelled as "(∆φ lab , ∆Φ jj )", was extracted for comparison from Ref. [23], which runs a different analysis. To focus only on measurement of the CPphase, we fix the number of signal tth events to the SM prediction α = 0, comparing only the shapes between the null and pseudo-hypotheses. We note that the top reconstruction in the dileptonic channel, where the top spin analyzing power is maximal, results in relevant sensitivity improvements for the direct Higgs-top CP-phase measurement. While the lab-observables (∆φ lab , ∆φ jj ) result in the limit cos α < 0.5 at 95% CL for the high-lumi LHC with 3 ab −1 , the addition of our observables defined at the top pair rest frame in two scenarios (∆φ lab , θ * ) and (∆φ lab , ∆φ tt , θ * ), result in relevant improvements of cos α < 0.65 and cos α < 0.7, respectively. Figure 11: Left: Luminosity required to distinguish an arbitrary CP-α state from the SM Higgs via tth production. Our limits are based on a binned log-likelihood analysis for (∆φ lab , ∆φ jj ) (red full), (∆φ lab , θ * ) (blue dashed), and (∆φ lab , θ * , ∆φ tt ) (blue full), accounting for the full detector level analysis. To illustrate the robustness of the top reconstruction method when going from the parton to the detector level, we also show the bounds using the parton-level distributions (∆φ lab , θ * ) with the rates rescaled to the full detector analysis (black full). Right: CL s as a function of the luminosity to distinguish CP(π/4) from CP(−π/4) state, based on ∆φ tt distribution. CP-phase A A (cut) α = 1 2 π 0.001 −0.004 α = 1 4 π 0.015 0.024 α = 0 0.007 −0.001 α = − 1 4 π −0.021 −0.020 α = − 1 2 π 0.001 −0.003 As we are able to probe ∆φ tt , that is sensitive to the sign of α, we can go beyond and inquire if the LHC will be able to capture also the CP-phase sign. In Fig. 11 (right panel), we show the luminosity needed to disentangle the CP(α = π 4 ) from the CP(α = − π 4 ) state based on ∆φ tt distribution. We chose ± π 4 for an illustration, since they give the largest difference. The observation of the sign for the maximal CP violation case requires at least 8 ab −1 of data at the 14 TeV LHC even at 1σ-level. Summary Characterizing the Higgs boson is a critical component of the LHC program. In this paper, we have studied the direct Higgs-top CP-phase determination via the tth channel with Higgs decaying to bottom quarks and the top-quarks in the dileptonic mode. Although this tt decay mode leads to maximal spin analyzing power, it always accompanies two neutrinos in the final state, making the analysis and reconstruction challenging. We show that kinematic reconstruction can be obtained via the M 2 algorithm. This method is entirely based on mass minimization, being more flexible for new physics studies and robust for our spin-correlation analysis. We expanded the previous M 2 -assisted reconstruction studies, investigating effects of parton-shower, hadronization and detector resolution. We found that the algorithm performance in resolving two fold ambiguity still remains superior despite the slightly worse momentum reconstruction when compared to the parton level. We prove however that an additional mass selection on M (b ) 2CW can efficiently improve the reconstruction efficiencies. We then studied the Higgs-top CP-phase discrimination via a realistic Monte Carlo analysis. We show that the CP sensitivity of the azimuthal angle between two leptons in the laboratory frame ∆φ lab can be relevantly enhanced when combined with tt rest of frame observables: top quark production angle θ * and ∆φ tt , where the latter is a truly CP-odd observable, sensitive to the sign of the CP-phase. Including the relevant backgrounds, we have performed a binned log-likelihood analysis and computed the luminosity required to distinguish the SM Higgs from an arbitrary CP-phase at 95% confidence level. Based on our results, the Higgs-top CP-phase can be probed up to cos α < 0.7 at the high luminosity LHC. A Parameterization of detector resolution effects The jet energy resolution is parametrized by a noise (N ), a stochastic (S), and a constant (C) terms σ E = N E 2 + S √ E 2 + C 2 , (A.1) where in our analysis we use N = 5.3, S = 0.74 and C = 0.05 respectively [71]. The electron energy resolution is based on the parameterization σ E = 0.3 E + 0.1 √ E + 0.01 . (A.2) The muon energy resolution is derived by the Inner Detector (ID) and Muon Spectrometer (MS) resolution functions σ = σ ID σ MS σ 2 ID + σ 2 MS , (A.3) where σ ID = E a 2 1 + (a 2 E) 2 (A.4) σ MS = E b 0 E 2 + b 2 1 + (b 2 E) 2 . (A.5) We choose a 1 = 0.023035, a 2 = 0.000347, b 0 = 0.12, b 1 = 0.03278 and b 2 = 0.00014 in our study [71]. Figure 1 : 1Left: ∆φ lab distribution between the two leptons from the tt decay in the laboratory frame after p T (h) > 200 GeV and m > 75 GeV selections. Middle: Distribution of a collision angle (θ * ) of the top with respect to a beam axis in the tt rest frame. Right: ∆φ tt distribution between the two leptons in the tt rest frame for tth production. We display the SM α = 0 and beyond the SM scenarios with α = ±π/4, ±π/2. The ratios of the different hypotheses to the SM are shown in the sub-figure (top right). The results are at the parton-truth level, fully reconstructing the particles' momenta at the 14 TeV LHC. Figure 2 : 2The event topology considered in this paper, together with the three possible subsystems. The blue dotted, the green dot-dashed, and the black solid boxes indicate the subsystems (b), ( ), and (b ), respectively. same as the original definition of M 2 in Eq. (3.2) and the subscript (XX) is added explicitly to indicate that no extra constraints are imposed during the minimization. In any given subsystem ((b), ( ) or (b )), these variables (3.1-3.6) are related as follows[54] the invariant mass of b and in i-th pairing (i = 1, 2), and m max b = m 2 t − m 2 W (in the m b → 0 limit). Since there are two possible ways of paring b and in the dilepton channel of the tt-like events, we repeat the same calculation for each partitioning. Then the correct combination would respect the anticipated end points of m b , M Figure 4 : 4No cuts are employed for parton-level studies, unless we mention explicitly. Distributions of m b , bounded above by the mass of the W boson and top quark, M ( ) 2Ct ≤ m W and M (b ) 2CW ≤ m t . A small fraction of events which leak beyond the expected mass bounds is due to finite width effects of the top quark and W boson. Also there is small contamination coming from wrong pair of b-quark and , although Correlation between ∆p z and ∆p x , and | p | − | p true | and ∆R( p, p true ) for M (b ) 2CW (top) without and (bottom) with a mass cut 165 GeV < M (b ) 2CW < 175 GeV. The corresponding efficiencies are 81.38% with 96.4% of purity and 24.86% with 97.9% purity, respectively. Figure 5 : 5175 GeV) is applied in the bottom panel, leading to the = 24.86% efficiency with 97.9% of purity. A relaxed cut, 160 GeV< M (b ) 2CW < 175 GeV, gives a slightly higher efficiency = 35.82 % with with 97.7% of purity. At this point, purity of the resolved sample is already high but the Distributions of θ * for various values of α before (left) and after ( Figure 6 : 6∆φ tt distributions for various choices of CP-phase α at parton-level. The top pair restframe reconstruction is obtained via the mass minimization procedure using OPTIMASS. No cuts are applied on the left plot, while the distributions in the right panel exploits a mass cut on M (b ) 2CW to improve the purity of the resolved events. CP-phase A (parton-truth) A (resolved) A (resolved, cut) Both signal and SM backgrounds are simulated by the MadGraph5 aMC@NLO with leading order accuracy in QCD at √ s = 14 TeV. Higher order effects are included by normalizing the tth rate to the next-to-leading order (NLO) QCD+EW cross-section 614 fb [? ], and the ttbb and ttZ to their NLO QCD predictions 2.64 pb [66] and 1.06 pb [67], respectively. At generation level, we demand all partons to pass the following cuts: p T > 20 GeV, and |η| < 5 , (4.1) while leptons are required to have p T > 20 GeV, and |η | < 2.5 . (4.2) Figure 7 : 7Higgs tagged fat jet reconstructed mass m h (upper-left) and m b (upper-right) distributions after the boosted selection for different CP phases. We also show fully reconstructed M lower-right) distributions. All plots are generated after resolving the combinatorial problem and 4b-tagging. Figure 8 : 8Correlations between (left panels) ∆p z and ∆p x and (right panels) | p | − | p true | and ∆R( p, p true ) for M (b ) 2CW with respect to α = 0 case. All plots are generated after resolving the combinatorial problem, 4b-tagging and 105 GeV < m h < 145 GeV, without (top panels) and with (bottom panels) an additional mass cut 155 GeV < M reconstructed invariant mass distributions m b (upper-right), lower-right) are also shown in Fig. 7. The distribution of reconstructed M (b ) Figure 10 : 10∆φ tt (left panels) and θ * (right panels) distributions, after resolving the combinatorial problem, 4b-tagging and 105 GeV < m h < 145 GeV, without (top panels) and with (bottom panels) an additional mass cut 155 GeV < M(b ) 2CW < 180 GeV. luminosity of 3 ab −1 , using the expression σ ≡ −2 ln L(B|S +B) L(S +B|S +B) Table 1 : 1Asymmetry variable A for different CP phases calculated for three different samples at parton-level. Here "resolved" samples include basic cuts only, while "resolved, cut" samples include the mass cut 165 < m t1,2 = M(b ) 2CW < 175 GeV. . They can be compared against those at parton-level in Tables 1.cuts tth (α = 0) ttbb ttZ S/B σ N h = 1, 4b-tags, p T > 20 GeV, |η | < 2.5 0.075 0.25 0.012 0.23 6.64 p j T > 30 GeV, |η j | < 2.5, N j ≥ 2, N = 2 105 GeV < m h < 145 GeV 0.056 0.12 0.0067 0.35 7.00 Resolving combinatorics 0.046 0.077 0.0047 0.45 7.07 m > 75 GeV 0.038 0.058 0.0038 0.49 6.68 Table 2 : 2Cumulative cut-flow table showing the SM background and signal (α = 0) cross sections in fb. Significances (σ) are calculated for a luminosity of 3 ab −1 . Table 3 : 3Asymmetry variables A after resolving the combinatorial problem, 4b-tagging and 105 GeV < m h < 145 GeV, without and with an additional mass cut 155 GeV < M(b ) 2CW < 180 GeV. See Refs.[49][50][51][52] for MT 2 and its various extensions and Refs.[53][54][55][56][57] for four dimensional variables. We refer to Refs.[53,58] for reviews on various kinematic variables. In our b-tagging algorithm, R = 0.4 jets are classified into three categories: If a b-hadron (c-hadron) is found inside a slim jet, it is classified as a b-jet (c-jet). The remaining unmatched jets are called light-jets. Each jet candidate is multiplied by an appropriate tag-rate[75]. We apply a flat b-tag rate of b→b = 0.7 and a mis-tag rate that a c-jet (light-jet) is misidentified as a b-jet of c→b = 0.2 ( j→b = 0.01). For a R = 1.2 fat jet to be b-tagged, we require that a b-tagged slim jet is found inside a fat jet. To take into account the case where more than one b-jet lands inside a fat jet, we reweight a b-tagging efficiency based on a scheme described in Ref.[76]. AcknowledgmentsWe are grateful to HTCaaS group of the Korea Institute of Science and Technology Information (KISTI) for providing the necessary computing resources. KK thanks the PITT-PACC for hospitality and support during the initial stage of this work. DG was funded by U.S. National Science Foundation under the grant PHY-1519175. This work is supported in part by U.S. National Science Foundation (PHY-1519175) and U.S. Department of Energy (DE-SC0017965, DE-SC0017988). Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC. G Aad, ATLAS collaboration10.1016/j.physletb.2012.08.0201207.7214Phys.Lett. 716ATLAS collaboration, G. Aad et al., Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC, Phys.Lett. B716 (2012) 1-29, [1207.7214]. Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC. S Chatrchyan, CMS collaboration10.1016/j.physletb.2012.08.0211207.7235Phys.Lett. 716CMS collaboration, S. Chatrchyan et al., Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC, Phys.Lett. B716 (2012) 30-61, [1207.7235]. Broken symmetries, massless particles and gauge fields. P W Higgs, 10.1016/0031-9163(64)91136-9Phys. Lett. 12P. W. Higgs, Broken symmetries, massless particles and gauge fields, Phys. Lett. 12 (1964) 132-133. Broken Symmetries and the Masses of Gauge Bosons. P W Higgs, 10.1103/PhysRevLett.13.508Phys. Rev. Lett. 13P. W. Higgs, Broken Symmetries and the Masses of Gauge Bosons, Phys. Rev. Lett. 13 (1964) 508-509. Broken Symmetry and the Mass of Gauge Vector Mesons. F Englert, R Brout, 10.1103/PhysRevLett.13.321Phys. Rev. Lett. 13F. Englert and R. Brout, Broken Symmetry and the Mass of Gauge Vector Mesons, Phys. Rev. Lett. 13 (1964) 321-323. Measurements of the Higgs boson production and decay rates and constraints on its couplings from a combined ATLAS and CMS analysis of the LHC pp collision data at √ s = 7 and 8 TeV. G Aad, CMS collaboration10.1007/JHEP08(2016)0451606.02266ATLAS. 0845JHEPATLAS, CMS collaboration, G. Aad et al., Measurements of the Higgs boson production and decay rates and constraints on its couplings from a combined ATLAS and CMS analysis of the LHC pp collision data at √ s = 7 and 8 TeV, JHEP 08 (2016) 045, [1606.02266]. The Higgs Legacy of the LHC Run I. T Corbett, O J P Eboli, D Goncalves, J Gonzalez-Fraile, T Plehn, M Rauch, 10.1007/JHEP08(2015)1561505.05516JHEP. 08156T. Corbett, O. J. P. Eboli, D. Goncalves, J. Gonzalez-Fraile, T. Plehn and M. Rauch, The Higgs Legacy of the LHC Run I, JHEP 08 (2015) 156, [1505.05516]. Violation of CP Invariance, c Asymmetry, and Baryon Asymmetry of the Universe. A D Sakharov, 10.1070/PU1991v034n05ABEH002497Pisma Zh. Eksp. Teor. Fiz. 5A. D. Sakharov, Violation of CP Invariance, c Asymmetry, and Baryon Asymmetry of the Universe, Pisma Zh. Eksp. Teor. Fiz. 5 (1967) 32-35. Electroweak Baryogenesis in Non-minimal Composite Higgs Models. J R Espinosa, B Gripaios, T Konstandin, F Riva, 10.1088/1475-7516/2012/01/0121110.2876JCAP. 120112J. R. Espinosa, B. Gripaios, T. Konstandin and F. Riva, Electroweak Baryogenesis in Non-minimal Composite Higgs Models, JCAP 1201 (2012) 012, [1110.2876]. Determining the structure of Higgs couplings at the LHC. T Plehn, D L Rainwater, D Zeppenfeld, 10.1103/PhysRevLett.88.051801hep-ph/0105325Phys. Rev. Lett. 8851801T. Plehn, D. L. Rainwater and D. Zeppenfeld, Determining the structure of Higgs couplings at the LHC, Phys. Rev. Lett. 88 (2002) 051801, [hep-ph/0105325]. Jet angular correlation in vector-boson fusion processes at hadron colliders. K Hagiwara, Q Li, K Mawatari, 10.1088/1126-6708/2009/07/101JHEP. 071010905.4314K. Hagiwara, Q. Li and K. Mawatari, Jet angular correlation in vector-boson fusion processes at hadron colliders, JHEP 07 (2009) 101, [0905.4314]. On the spin and parity of a single-produced resonance at the LHC. S Bolognesi, Y Gao, A V Gritsan, K Melnikov, M Schulze, N V Tran, 10.1103/PhysRevD.86.0950311208.4018Phys. Rev. 8695031S. Bolognesi, Y. Gao, A. V. Gritsan, K. Melnikov, M. Schulze, N. V. Tran et al., On the spin and parity of a single-produced resonance at the LHC, Phys. Rev. D86 (2012) 095031, [1208.4018]. Higgs Quantum Numbers in Weak Boson Fusion. C Englert, D Goncalves-Netto, K Mawatari, T Plehn, 10.1007/JHEP01(2013)1481212.0843JHEP. 01148C. Englert, D. Goncalves-Netto, K. Mawatari and T. Plehn, Higgs Quantum Numbers in Weak Boson Fusion, JHEP 01 (2013) 148, [1212.0843]. Higgs CP Properties From Early LHC Data. A Freitas, P Schwaller, 10.1103/PhysRevD.87.0550141211.1980Phys. Rev. 8755014A. Freitas and P. Schwaller, Higgs CP Properties From Early LHC Data, Phys. Rev. D87 (2013) 055014, [1211.1980]. Does the 'Higgs' have Spin Zero?. J Ellis, D S Hwang, 10.1007/JHEP09(2012)0711202.6660JHEP. 0971J. Ellis and D. S. Hwang, Does the 'Higgs' have Spin Zero?, JHEP 09 (2012) 071, [1202.6660]. Distinguishing 'Higgs' spin hypotheses using γγ and W W * decays. J Ellis, R Fok, D S Hwang, V Sanz, T You, 10.1140/epjc/s10052-013-2488-51210.5229Eur. Phys. J. 732488J. Ellis, R. Fok, D. S. Hwang, V. Sanz and T. You, Distinguishing 'Higgs' spin hypotheses using γγ and W W * decays, Eur. Phys. J. C73 (2013) 2488, [1210.5229]. The shape of spins. C Englert, D Goncalves, G Nail, M Spannowsky, 10.1103/PhysRevD.88.0130161304.0033Phys. Rev. 8813016C. Englert, D. Goncalves, G. Nail and M. Spannowsky, The shape of spins, Phys. Rev. D88 (2013) 013016, [1304.0033]. Constraints on the spin-parity and anomalous HVV couplings of the Higgs boson in proton collisions at 7 and 8 TeV. V Khachatryan, CMS collaboration10.1103/PhysRevD.92.0120041411.3441Phys. Rev. 9212004CMS collaboration, V. Khachatryan et al., Constraints on the spin-parity and anomalous HVV couplings of the Higgs boson in proton collisions at 7 and 8 TeV, Phys. Rev. D92 (2015) 012004, [1411.3441]. Better Higgs-CP Tests Through Information Geometry. J Brehmer, F Kling, T Plehn, T M P Tait, 1712.02350J. Brehmer, F. Kling, T. Plehn and T. M. P. Tait, Better Higgs-CP Tests Through Information Geometry, 1712.02350. Effective Lagrangian Analysis of New Interactions and Flavor Conservation. W Buchmuller, D Wyler, 10.1016/0550-3213(86)90262-2Nucl. Phys. 268W. Buchmuller and D. Wyler, Effective Lagrangian Analysis of New Interactions and Flavor Conservation, Nucl. Phys. B268 (1986) 621-653. Dimension-Six Terms in the Standard Model Lagrangian. B Grzadkowski, M Iskrzynski, M Misiak, J Rosiek, 10.1007/JHEP10(2010)0851008.4884JHEP. 1085B. Grzadkowski, M. Iskrzynski, M. Misiak and J. Rosiek, Dimension-Six Terms in the Standard Model Lagrangian, JHEP 10 (2010) 085, [1008.4884]. Disentangling Higgs-Top Couplings in Associated Production. J Ellis, D S Hwang, K Sakurai, M Takeuchi, 10.1007/JHEP04(2014)0041312.5736JHEP. 044J. Ellis, D. S. Hwang, K. Sakurai and M. Takeuchi, Disentangling Higgs-Top Couplings in Associated Production, JHEP 04 (2014) 004, [1312.5736]. Boosting the Direct CP Measurement of the Higgs-Top Coupling. M R Buckley, D Goncalves, 10.1103/PhysRevLett.116.0918011507.07926Phys. Rev. Lett. 11691801M. R. Buckley and D. Goncalves, Boosting the Direct CP Measurement of the Higgs-Top Coupling, Phys. Rev. Lett. 116 (2016) 091801, [1507.07926]. Lab-frame observables for probing the top-Higgs interaction. F Boudjema, R M Godbole, D Guadagnoli, K A Mohan, 10.1103/PhysRevD.92.0150191501.03157Phys. Rev. 9215019F. Boudjema, R. M. Godbole, D. Guadagnoli and K. A. Mohan, Lab-frame observables for probing the top-Higgs interaction, Phys. Rev. D92 (2015) 015019, [1501.03157]. Pseudoscalar top-Higgs coupling: exploration of CP-odd observables to resolve the sign ambiguity. N Mileo, K Kiers, A Szynkman, D Crane, E Gegner, 10.1007/JHEP07(2016)0561603.03632JHEP. 0756N. Mileo, K. Kiers, A. Szynkman, D. Crane and E. Gegner, Pseudoscalar top-Higgs coupling: exploration of CP-odd observables to resolve the sign ambiguity, JHEP 07 (2016) 056, [1603.03632]. Constraining anomalous Higgs boson couplings to the heavy flavor fermions using matrix element techniques. A V Gritsan, R Röntsch, M Schulze, M Xiao, 10.1103/PhysRevD.94.0550231606.03107Phys. Rev. 9455023A. V. Gritsan, R. Röntsch, M. Schulze and M. Xiao, Constraining anomalous Higgs boson couplings to the heavy flavor fermions using matrix element techniques, Phys. Rev. D94 (2016) 055023, [1606.03107]. Determining the CP parity of Higgs bosons at the LHC in their tau decay channels. S Berge, W Bernreuther, J Ziethe, 10.1103/PhysRevLett.100.171605Phys. Rev. Lett. 1001716050801.2297S. Berge, W. Bernreuther and J. Ziethe, Determining the CP parity of Higgs bosons at the LHC in their tau decay channels, Phys. Rev. Lett. 100 (2008) 171605, [0801.2297]. Measuring CP violation in h → τ + τ − at colliders. R Harnik, A Martin, T Okui, R Primulando, F Yu, 10.1103/PhysRevD.88.0760091308.1094Phys. Rev. 8876009R. Harnik, A. Martin, T. Okui, R. Primulando and F. Yu, Measuring CP violation in h → τ + τ − at colliders, Phys. Rev. D88 (2013) 076009, [1308.1094]. Determining the quantum numbers of simplified models in ttX production at the LHC. M J Dolan, M Spannowsky, Q Wang, Z.-H Yu, 10.1103/PhysRevD.94.0150251606.00019Phys. Rev. 9415025M. J. Dolan, M. Spannowsky, Q. Wang and Z.-H. Yu, Determining the quantum numbers of simplified models in ttX production at the LHC, Phys. Rev. D94 (2016) 015025, [1606.00019]. Angular distributions in ttH(H?bb) reconstructed events at the LHC. S P Amor, Santos, 10.1103/PhysRevD.92.0340211503.07787Phys. Rev. 9234021S. P. Amor dos Santos et al., Angular distributions in ttH(H?bb) reconstructed events at the LHC, Phys. Rev. D92 (2015) 034021, [1503.07787]. Pseudoscalar searches with dileptonic tops and jet substructure. D Goncalves, D Lopez-Val, 10.1103/PhysRevD.94.0950051607.08614Phys. Rev. 9495005D. Goncalves and D. Lopez-Val, Pseudoscalar searches with dileptonic tops and jet substructure, Phys. Rev. D94 (2016) 095005, [1607.08614]. Measuring the CP property of Higgs coupling to tau leptons in the VBF channel at the LHC. T Han, S Mukhopadhyay, B Mukhopadhyaya, Y Wu, 10.1007/JHEP05(2017)1281612.00413JHEP. 05128T. Han, S. Mukhopadhyay, B. Mukhopadhyaya and Y. Wu, Measuring the CP property of Higgs coupling to tau leptons in the VBF channel at the LHC, JHEP 05 (2017) 128, [1612.00413]. Probing CP violation in h → τ − τ + at the LHC. K Hagiwara, K Ma, S Mori, 10.1103/PhysRevLett.118.1718021609.00943Phys. Rev. Lett. 118171802K. Hagiwara, K. Ma and S. Mori, Probing CP violation in h → τ − τ + at the LHC, Phys. Rev. Lett. 118 (2017) 171802, [1609.00943]. Constraints on CP-violating Higgs couplings to the third generation. J Brod, U Haisch, J Zupan, 10.1007/JHEP11(2013)1801310.1385JHEP. 11180J. Brod, U. Haisch and J. Zupan, Constraints on CP-violating Higgs couplings to the third generation, JHEP 11 (2013) 180, [1310.1385]. Monte Carlo studies of the jet activity in Higgs + 2 jet events. V Duca, G Klamke, D Zeppenfeld, M L Mangano, M Moretti, F Piccinini, 10.1088/1126-6708/2006/10/016hep-ph/0608158JHEP. 1016V. Del Duca, G. Klamke, D. Zeppenfeld, M. L. Mangano, M. Moretti, F. Piccinini et al., Monte Carlo studies of the jet activity in Higgs + 2 jet events, JHEP 10 (2006) 016, [hep-ph/0608158]. Constraining CP -violating Higgs Sectors at the LHC using gluon fusion. M J Dolan, P Harris, M Jankowiak, M Spannowsky, 10.1103/PhysRevD.90.0730081406.3322Phys. Rev. 9073008M. J. Dolan, P. Harris, M. Jankowiak and M. Spannowsky, Constraining CP -violating Higgs Sectors at the LHC using gluon fusion, Phys. Rev. D90 (2014) 073008, [1406.3322]. Probing top-partners in Higgs+jets. A Banfi, A Martin, V Sanz, 10.1007/JHEP08(2014)0531308.4771JHEP. 0853A. Banfi, A. Martin and V. Sanz, Probing top-partners in Higgs+jets, JHEP 08 (2014) 053, [1308.4771]. Probing Higgs couplings with high p T Higgs production. A Azatov, A Paul, 10.1007/JHEP01(2014)0141309.5273JHEP. 0114A. Azatov and A. Paul, Probing Higgs couplings with high p T Higgs production, JHEP 01 (2014) 014, [1309.5273]. Very boosted Higgs in gluon fusion. C Grojean, E Salvioni, M Schlaffer, A Weiler, 10.1007/JHEP05(2014)0221312.3317JHEP. 0522C. Grojean, E. Salvioni, M. Schlaffer and A. Weiler, Very boosted Higgs in gluon fusion, JHEP 05 (2014) 022, [1312.3317]. Boosted Higgs Shapes. M Schlaffer, M Spannowsky, M Takeuchi, A Weiler, C Wymant, 10.1140/epjc/s10052-014-3120-z1405.4295Eur. Phys. J. 743120M. Schlaffer, M. Spannowsky, M. Takeuchi, A. Weiler and C. Wymant, Boosted Higgs Shapes, Eur. Phys. J. C74 (2014) 3120, [1405.4295]. Resolving the Higgs-Gluon Coupling with Jets. M Buschmann, C Englert, D Goncalves, T Plehn, M Spannowsky, 10.1103/PhysRevD.90.0130101405.7651Phys. Rev. 9013010M. Buschmann, C. Englert, D. Goncalves, T. Plehn and M. Spannowsky, Resolving the Higgs-Gluon Coupling with Jets, Phys. Rev. D90 (2014) 013010, [1405.7651]. Mass Effects in the Higgs-Gluon Coupling: Boosted vs Off-Shell Production. M Buschmann, D Goncalves, S Kuttimalai, M Schonherr, F Krauss, T Plehn, 10.1007/JHEP02(2015)038JHEP. 02381410.5806M. Buschmann, D. Goncalves, S. Kuttimalai, M. Schonherr, F. Krauss and T. Plehn, Mass Effects in the Higgs-Gluon Coupling: Boosted vs Off-Shell Production, JHEP 02 (2015) 038, [1410.5806]. Evidence for the associated production of the Higgs boson and a top quark pair with the ATLAS detector. ATLAS-CONF-2017-077GenevaCERNTech. Rep.ATLAS collaboration, Evidence for the associated production of the Higgs boson and a top quark pair with the ATLAS detector, Tech. Rep. ATLAS-CONF-2017-077, CERN, Geneva, Nov, 2017. Observation of ttH production. A M Sirunyan, CMS collaboration1804.02610CMS collaboration, A. M. Sirunyan et al., Observation of ttH production, 1804.02610. Projected Performance of an Upgraded CMS Detector at the LHC and HL-LHC: Contribution to the Snowmass Process. Proceedings, 2013 Community Summer Study on the Future of U.S. Particle Physics: Snowmass on the Mississippi (CSS2013). 2013 Community Summer Study on the Future of U.S. Particle Physics: Snowmass on the Mississippi (CSS2013)Minneapolis, MN, USACMS collaboration, Projected Performance of an Upgraded CMS Detector at the LHC and HL-LHC: Contribution to the Snowmass Process, in Proceedings, 2013 Community Summer Study on the Future of U.S. Particle Physics: Snowmass on the Mississippi (CSS2013): Minneapolis, MN, USA, July 29-August 6, 2013, 2013. 1307.7135. Resolving Combinatorial Ambiguities in Dilepton tt Event Topologies with Constrained M 2 Variables. D Debnath, D Kim, J H Kim, K Kong, K T Matchev, 10.1103/PhysRevD.96.0760051706.04995Phys. Rev. 9676005D. Debnath, D. Kim, J. H. Kim, K. Kong and K. T. Matchev, Resolving Combinatorial Ambiguities in Dilepton tt Event Topologies with Constrained M 2 Variables, Phys. Rev. D96 (2017) 076005, [1706.04995]. Angular correlations in top quark pair production and decay at hadron colliders. G Mahlon, S J Parke, 10.1103/PhysRevD.53.4886hep-ph/9512264Phys. Rev. 53G. Mahlon and S. J. Parke, Angular correlations in top quark pair production and decay at hadron colliders, Phys. Rev. D53 (1996) 4886-4896, [hep-ph/9512264]. ATLAS-CONF-2016-100Measurements of tt differential cross-sections in the all-hadronic channel with the ATLAS detector using highly boosted top quarks in pp collisions at √ s = 13 TeV. ATLAS collaboration, Measurements of tt differential cross-sections in the all-hadronic channel with the ATLAS detector using highly boosted top quarks in pp collisions at √ s = 13 TeV, ATLAS-CONF-2016-100 (2016) . Measuring masses of semiinvisibly decaying particles pair produced at hadron colliders. C G Lester, D J Summers, 10.1016/S0370-2693(99)00945-4hep-ph/9906349Phys. Lett. 463C. G. Lester and D. J. Summers, Measuring masses of semiinvisibly decaying particles pair produced at hadron colliders, Phys. Lett. B463 (1999) 99-103, [hep-ph/9906349]. Superpartner Mass Measurement Technique using 1D Orthogonal Decompositions of the Cambridge Transverse Mass Variable M T 2. P Konar, K Kong, K T Matchev, M Park, 10.1103/PhysRevLett.105.051802Phys. Rev. Lett. 105518020910.3679P. Konar, K. Kong, K. T. Matchev and M. Park, Superpartner Mass Measurement Technique using 1D Orthogonal Decompositions of the Cambridge Transverse Mass Variable M T 2 , Phys. Rev. Lett. 105 (2010) 051802, [0910.3679]. Using Subsystem MT2 for Complete Mass Determinations in Decay Chains with Missing Energy at Hadron Colliders. M Burns, K Kong, K T Matchev, M Park, 10.1088/1126-6708/2009/03/143JHEP. 031430810.5576M. Burns, K. Kong, K. T. Matchev and M. Park, Using Subsystem MT2 for Complete Mass Determinations in Decay Chains with Missing Energy at Hadron Colliders, JHEP 03 (2009) 143, [0810.5576]. P Konar, K Kong, K T Matchev, M Park, 10.1007/JHEP04(2010)086Dark Matter Particle Spectroscopy at the LHC: Generalizing M(T2) to Asymmetric Event Topologies. 860911.4126P. Konar, K. Kong, K. T. Matchev and M. Park, Dark Matter Particle Spectroscopy at the LHC: Generalizing M(T2) to Asymmetric Event Topologies, JHEP 04 (2010) 086, [0911.4126]. Guide to transverse projections and mass-constraining variables. A J Barr, T J Khoo, P Konar, K Kong, C G Lester, K T Matchev, 10.1103/PhysRevD.84.0950311105.2977Phys. Rev. 8495031A. J. Barr, T. J. Khoo, P. Konar, K. Kong, C. G. Lester, K. T. Matchev et al., Guide to transverse projections and mass-constraining variables, Phys. Rev. D84 (2011) 095031, [1105.2977]. On-shell constrained M 2 variables with applications to mass measurements and topology disambiguation. W S Cho, J S Gainer, D Kim, K T Matchev, F Moortgat, L Pape, 10.1007/JHEP08(2014)0701401.1449JHEP. 0870W. S. Cho, J. S. Gainer, D. Kim, K. T. Matchev, F. Moortgat, L. Pape et al., On-shell constrained M 2 variables with applications to mass measurements and topology disambiguation, JHEP 08 (2014) 070, [1401.1449]. OPTIMASS: A Package for the Minimization of Kinematic Mass Functions with Constraints. W S Cho, J S Gainer, D Kim, S H Lim, K T Matchev, F Moortgat, 10.1007/JHEP01(2016)0261508.00589JHEP. 0126W. S. Cho, J. S. Gainer, D. Kim, S. H. Lim, K. T. Matchev, F. Moortgat et al., OPTIMASS: A Package for the Minimization of Kinematic Mass Functions with Constraints, JHEP 01 (2016) 026, [1508.00589]. √ŝ min : A Global inclusive variable for determining the mass scale of new physics in events with missing energy at hadron colliders. P Konar, K Kong, K T Matchev, 10.1088/1126-6708/2009/03/085JHEP. 03850812.1042P. Konar, K. Kong and K. T. Matchev, √ŝ min : A Global inclusive variable for determining the mass scale of new physics in events with missing energy at hadron colliders, JHEP 03 (2009) 085, [0812.1042]. RECO level √ s min and subsystem √ s min : Improved global inclusive variables for measuring the new physics mass scale in / E T events at hadron colliders. P Konar, K Kong, K T Matchev, M Park, 10.1007/JHEP06(2011)0411006.0653JHEP. 0641P. Konar, K. Kong, K. T. Matchev and M. Park, RECO level √ s min and subsystem √ s min : Improved global inclusive variables for measuring the new physics mass scale in / E T events at hadron colliders, JHEP 06 (2011) 041, [1006.0653]. A Review of the Mass Measurement Techniques proposed for the Large Hadron Collider. A J Barr, C G Lester, 10.1088/0954-3899/37/12/1230011004.2732J. Phys. 37123001A. J. Barr and C. G. Lester, A Review of the Mass Measurement Techniques proposed for the Large Hadron Collider, J. Phys. G37 (2010) 123001, [1004.2732]. Testing Invisible Momentum Ansatze in Missing Energy Events at the LHC. D Kim, K T Matchev, F Moortgat, L Pape, 10.1007/JHEP08(2017)1021703.06887JHEP. 08102D. Kim, K. T. Matchev, F. Moortgat and L. Pape, Testing Invisible Momentum Ansatze in Missing Energy Events at the LHC, JHEP 08 (2017) 102, [1703.06887]. Mass determination of new states at hadron colliders. G G Ross, M Serna, 10.1016/j.physletb.2008.06.003Phys. Lett. 6650712.0943G. G. Ross and M. Serna, Mass determination of new states at hadron colliders, Phys. Lett. B665 (2008) 212-218, [0712.0943]. Revisiting Combinatorial Ambiguities at Hadron Colliders with M T 2. P Baringer, K Kong, M Mccaskey, D Noonan, 10.1007/JHEP10(2011)1011109.1563JHEP. 10101P. Baringer, K. Kong, M. McCaskey and D. Noonan, Revisiting Combinatorial Ambiguities at Hadron Colliders with M T 2 , JHEP 10 (2011) 101, [1109.1563]. The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations. J Alwall, R Frederix, S Frixione, V Hirschi, F Maltoni, O Mattelaer, 10.1007/JHEP07(2014)0791405.0301JHEP. 0779J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer et al., The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations, JHEP 07 (2014) 079, [1405.0301]. FeynRules 2.0 -A complete toolbox for tree-level phenomenology. A Alloul, N D Christensen, C Degrande, C Duhr, B Fuks, 10.1016/j.cpc.2014.04.012Comput. Phys. Commun. 1851310.1921A. Alloul, N. D. Christensen, C. Degrande, C. Duhr and B. Fuks, FeynRules 2.0 -A complete toolbox for tree-level phenomenology, Comput. Phys. Commun. 185 (2014) 2250-2300, [1310.1921]. Parton distributions with QED corrections. R D Ball, NNPDF collaborationV Bertone, NNPDF collaborationS Carrazza, NNPDF collaborationL Debbio, NNPDF collaborationS Forte, NNPDF collaborationA Guffanti, NNPDF collaboration10.1016/j.nuclphysb.2013.10.010Nucl. Phys. 8771308.0598NNPDF collaboration, R. D. Ball, V. Bertone, S. Carrazza, L. Del Debbio, S. Forte, A. Guffanti et al., Parton distributions with QED corrections, Nucl. Phys. B877 (2013) 290-320, [1308.0598]. Associated top quark Higgs boson production at the LHC. S Dawson, L H Orr, L Reina, D Wackeroth, 10.1103/PhysRevD.67.071503hep-ph/0211438Phys. Rev. 6771503S. Dawson, L. H. Orr, L. Reina and D. Wackeroth, Associated top quark Higgs boson production at the LHC, Phys. Rev. D67 (2003) 071503, [hep-ph/0211438]. NLO QCD corrections to pp -¿ t anti-t b anti-b + X at the LHC. A Bredenstein, A Denner, S Dittmaier, S Pozzorini, 10.1103/PhysRevLett.103.0120020905.0110Phys. Rev. Lett. 10312002A. Bredenstein, A. Denner, S. Dittmaier and S. Pozzorini, NLO QCD corrections to pp -¿ t anti-t b anti-b + X at the LHC, Phys. Rev. Lett. 103 (2009) 012002, [0905.0110]. Associated production of a top-quark pair with vector bosons at NLO in QCD: impact on ttH searches at the LHC. F Maltoni, D Pagani, I Tsinikos, 10.1007/JHEP02(2016)1131507.05640JHEP. 02113F. Maltoni, D. Pagani and I. Tsinikos, Associated production of a top-quark pair with vector bosons at NLO in QCD: impact on ttH searches at the LHC, JHEP 02 (2016) 113, [1507.05640]. PYTHIA 6.4 Physics and Manual. T Sjostrand, S Mrenna, P Z Skands, 10.1088/1126-6708/2006/05/026hep-ph/0603175JHEP. 0526T. Sjostrand, S. Mrenna and P. Z. Skands, PYTHIA 6.4 Physics and Manual, JHEP 05 (2006) 026, [hep-ph/0603175]. FastJet User Manual. M Cacciari, G P Salam, G Soyez, 10.1140/epjc/s10052-012-1896-21111.6097Eur. Phys. J. C72. 1896M. Cacciari, G. P. Salam and G. Soyez, FastJet User Manual, Eur. Phys. J. C72 (2012) 1896, [1111.6097]. The Anti-k(t) jet clustering algorithm. M Cacciari, G P Salam, G Soyez, 10.1088/1126-6708/2008/04/063JHEP. 04630802.1189M. Cacciari, G. P. Salam and G. Soyez, The Anti-k(t) jet clustering algorithm, JHEP 04 (2008) 063, [0802.1189]. Performance assumptions for an upgraded ATLAS detector at a High-Luminosity LHC. ATL-PHYS-PUB-2013-004GenevaCERNTech. Rep.Performance assumptions for an upgraded ATLAS detector at a High-Luminosity LHC, Tech. Rep. ATL-PHYS-PUB-2013-004, CERN, Geneva, Mar, 2013. TemplateTagger v1.0.0: A Template Matching Tool for Jet Substructure. M Backović, J Juknevich, 10.1016/j.cpc.2013.12.018Comput. Phys. Commun. 1851212.2978M. Backović and J. Juknevich, TemplateTagger v1.0.0: A Template Matching Tool for Jet Substructure, Comput. Phys. Commun. 185 (2014) 1322-1338, [1212.2978]. Sterman and I. Sung, Template Overlap Method for Massive Jets. L G Almeida, S J Lee, G Perez, G , 10.1103/PhysRevD.82.054034Phys. Rev. 82540341006.2035L. G. Almeida, S. J. Lee, G. Perez, G. Sterman and I. Sung, Template Overlap Method for Massive Jets, Phys. Rev. D82 (2010) 054034, [1006.2035]. Measuring boosted tops in semi-leptonic tt events for the standard model and beyond. M Backovic, O Gabizon, J Juknevich, G Perez, Y Soreq, 10.1007/JHEP04(2014)1761311.2962JHEP. 04176M. Backovic, O. Gabizon, J. Juknevich, G. Perez and Y. Soreq, Measuring boosted tops in semi-leptonic tt events for the standard model and beyond, JHEP 04 (2014) 176, [1311.2962]. Expected performance for an upgraded ATLAS detector at High-Luminosity LHC. ATL-PHYS-PUB-2016-026GenevaCERNTech. Rep.ATLAS Collaboration collaboration, Expected performance for an upgraded ATLAS detector at High-Luminosity LHC, Tech. Rep. ATL-PHYS-PUB-2016-026, CERN, Geneva, Oct, 2016. Search Strategies for TeV Scale Fermionic Top Partners with Charge 2/3. M Backovic, T Flacke, J H Kim, S J Lee, 10.1007/JHEP04(2016)0141507.06568JHEP. 0414M. Backovic, T. Flacke, J. H. Kim and S. J. Lee, Search Strategies for TeV Scale Fermionic Top Partners with Charge 2/3, JHEP 04 (2016) 014, [1507.06568]. Probing the CP nature of the Higgs coupling in tth events at the LHC. S. Amor Dos Santos, 10.1103/PhysRevD.96.0130041704.03565Phys. Rev. 9613004S. Amor Dos Santos et al., Probing the CP nature of the Higgs coupling in tth events at the LHC, Phys. Rev. D96 (2017) 013004, [1704.03565]. Asymptotic formulae for likelihood-based tests of new physics. G Cowan, K Cranmer, E Gross, O Vitells, 10.1140/epjc/s10052-011-1554-0,10.1140/epjc/s10052-013-2501-z1007.1727Eur. Phys. J. 711554G. Cowan, K. Cranmer, E. Gross and O. Vitells, Asymptotic formulae for likelihood-based tests of new physics, Eur. Phys. J. C71 (2011) 1554, [1007.1727]. Presentation of search results: The CL(s) technique. A L Read, 10.1088/0954-3899/28/10/313J. Phys. 28A. L. Read, Presentation of search results: The CL(s) technique, J. Phys. G28 (2002) 2693-2704.
{'fraction_non_alphanumeric': 0.07141245996300807, 'fraction_numerical': 0.07112674997368461, 'mean_word_length': 3.9780672206003445, 'pattern_counts': {'":': 0, '<': 37, '<?xml version=': 0, '>': 17, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 3, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 45, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'Constraining the Higgs boson properties is a cornerstone of the LHC program. We study the potential to directly probe the Higgs-top CP-structure via the tth channel at the LHC with the Higgs boson decaying to a bottom pair and top-quarks in the dileptonic mode. We show that a combination of laboratory and tt rest frame observables display large CP-sensitivity, exploring the spin correlations in the top decays. To efficiently reconstruct our final state, we present a method based on simple mass minimization and prove its robustness to shower, hadronization and detector effects. In addition, the mass reconstruction works as an extra relevant handle for background suppression. Based on our results, we demonstrate that the Higgs-top CP-phase (α) can be probed up to cos α < 0.7 at the high luminosity LHC.', 'arxivid': '1804.05874', 'author': ['Dorival Gonçalves [email protected] \nPITT-PACC\nDepartment of Physics and Astronomy\nUniversity of Pittsburgh\nUSA\n', 'Jeong Han Kim [email protected] \nDepartment of Physics and Astronomy\nUniversity of Kansas\n66045LawrenceKSUSA\n', 'Kyoungchul Kong [email protected] \nDepartment of Physics and Astronomy\nUniversity of Kansas\n66045LawrenceKSUSA\n'], 'authoraffiliation': ['PITT-PACC\nDepartment of Physics and Astronomy\nUniversity of Pittsburgh\nUSA', 'Department of Physics and Astronomy\nUniversity of Kansas\n66045LawrenceKSUSA', 'Department of Physics and Astronomy\nUniversity of Kansas\n66045LawrenceKSUSA'], 'corpusid': 119202894, 'doi': '10.1007/jhep06(2018)079', 'github_urls': [], 'n_tokens_mistral': 24156, 'n_tokens_neox': 19354, 'n_words': 10677, 'pdfsha': '6d646c07b671c1e842686305f358797aa3d07b6b', 'pdfurls': ['https://export.arxiv.org/pdf/1804.05874v2.pdf'], 'title': ['Probing the Top-Higgs Yukawa CP Structure in dileptonic tth with M 2 -Assisted Reconstruction', 'Probing the Top-Higgs Yukawa CP Structure in dileptonic tth with M 2 -Assisted Reconstruction'], 'venue': []}
arxiv
NLO automated tools for QCD and beyond 26 Feb 2012 Nikolas Kauer [email protected] Department of Physics Royal Holloway University of London TW20 0EXEghamUK NLO automated tools for QCD and beyond LC11 Proceedings Frascati Physics Series 26 Feb 2012 Theoretical predictions for scattering processes with multi-particle final states at next-to-leading order (NLO) in perturbative QCD are essential to fully exploit the physics potential of present and future high-energy colliders. The status of NLO QCD calculations and tools is reviewed.4The important topics of next-to-next-to-leading order (NNLO) calculations and combining parton-level fixed-order calculations and parton-shower event generators are beyond the scope of this review.5The Born amplitude is assumed to be at tree level. 6 An alternative to the widely used subtraction formalism [4] is the phase space slicing method[5]. 7 The POWHEG BOX [22] library project[23,24]was inspired by these collections. Introduction The study of hard scattering processes at the Large Hadron Collider (LHC) [1] and a future TeV-scale linear collider is our primary means to probe and extend the Standard Model of particle physics. It is driven by the comparison of experimental measurements with theoretical predictions, which depends on our ability to compute collider cross sections in perturbative QCD with adequate accuracy [2,3]. This can only be achieved by going beyond leading order (LO) in QCD. When using conventional measures, LO scale uncertainties are typically large compared to experimental uncertainties. Moreover, for theoretical reasons a reliable estimation of the scale uncertainty is not feasible at LO. Consequently, an assessment of different scale choices, which is particularly important for many-particle/jet processes, is not possible. Furthermore, the convergence of the perturbative series cannot be assessed at LO. When going beyond LO by including NLO corrections, the situation improves significantly. 2 At NLO, scale uncertainties can be assessed more reliably, and the residual uncertainties are often comparable to experimental uncertainties. 3 NLO calculations thus deliver accurate predictions not only for the overall normalisation, but also for kinematic distributions including peripheral phase space regions. This is in part due to the fact that new subprocesses often become active at NLO, which modify the normalisation and kinematic distributions. Our ability to determine the uncertainty of parton distribution functions (PDF) and to model the structure of jets is also greatly enhanced at NLO. In Section 2, the state-of-the-art methods, implementations and tools for parton-level NLO calculations are briefly reviewed. In Section 3, the status of collider physics applications is described. The review ends with a summary. 4 Methods, implementations and tools The structure and implied modularity of NLO calculations is illustrated in Eqs. (1)- (3): σ NLO = σ Born + σ corr (1) σ Born = dφ n 1 2ŝ |A LO | 2 (2) σ corr = dφ n α s 2ŝ j dφ j D j + A LO A * NLO,V + A * LO A NLO,V + dφ n+1 α s 2ŝ |M NLO,R | 2 − j D j(3) The new components of the NLO correction σ corr are: 5 the virtual corrections (involving oneloop amplitudes), the real corrections (involving tree amplitudes) and the infrared subtraction terms. 6 The resulting procedure for NLO calculations is given in Table 1. The Binoth Les Houches Accord, a standard interface for combining the tree-level and loop-level contributions, has been defined in Ref. [6] and is implemented in many automated tools (see below). Until circa 2005, the limiting factor of NLO calculations was the computation of the virtual corrections, which typically applied Passarino-Veltman (PV) [7] or PV-inspired [8] tensor integral reduction methods to evaluate the form factors of a Feynman-diagram-based amplitude representation. Several one-loop integral libraries are available as public codes: LoopTools [9,10], QCDLoop [11], Golem95 [12], OneLOop [13] and PJFry [14]. The PV approach is general, but practical limitations arise due to the factorial growth of the number of Feynman graphs with N = n + 2, the strong growth of the number of reduction terms with N and due to numerical instabilities for exceptional kinematic configurations, which are caused by vanishing Gram determinants. It has nevertheless been used successfully to create collections of NLO calculations based on analytic formulae and semi-automated methods, such as MCFM [15,16], MC@NLO [17] and VBFNLO [18,19,20,21]. 7 Since 2004, tremendous improvements have been achieved for the calculation of multi-leg one-loop amplitudes due to the exploitation of on-shell 1. Real correction: generate and evaluate 2 → n + 1 tree-level amplitudes 2. Subtract soft and collinear singularities due to single unresolved real radiation to obtain finite result 3. Integrate over (n + 1)-particle phase space 4. Virtual correction: generate and evaluate UV-renormalised 2 → n one-loop amplitude after extraction of soft and collinear singularities to obtain finite result 5. Confirm cancellation of soft/collinear singularities (absorb initial state collinear singularities into PDF) 6. Integrate over n-particle phase space 7. Combine 2 → n + 1 and 2 → n contributions 8. Convolve with NLO PDF 9. Repeat for all contributing subprocesses Table 1: Steps to calculate the NLO QCD corrections for a 2 → n process. n excludes electroweak decays. recursion relations and generalized-unitarity-cut constructibility as well as the possibility to even reconstruct the full rational terms [25,26]. On-shell reduction related tools are CutTools [27], Rocket [28] and Samurai [29]. Further innovative, complementary methods are also being developed [30]. A comprehensive review of methods for multi-leg one-loop calculations can be found in Ref. [31]. Three widely-used algorithms for the generation of process-independent infrared subtraction terms are Catani-Seymour dipole subtraction [32], Frixione-Kunszt-Signer (FKS) subtraction [33] and antenna subtraction [34]. 8 Several implementations for these standard schemes are available: Sherpa-Dipoles [36], MadDipole [37], HELAC-Dipoles [38], MadFKS [39], TeVJet [40] and AutoDipole [41]. The following programs aim to provide a comprehensive, automated solution for NLO calculations: aMC@NLO [27,39,42], BlackHat/Sherpa [26,36,43], HELAC-NLO [13,27,38,44], GoSam [45], FeynArts/FormCalc/LoopTools [10,46] and MadGolem [47]. Collider physics applications Discussions at the Les Houches 2005 Physics at TeV Colliders Workshop resulted in a list of processes for which the knowledge of NLO corrections was considered of particular importance for the LHC physics programme [48]. This experimenter's NLO "wish list" has guided theoretical efforts and was subsequently revised and updated in 2007 [49] as well as 2009 [50]. The most recent version is displayed in Table 2. Due to the groundbreaking advances outlined in Section 2, since 2009 the frontier for collider physics applications of NLO techniques has also advanced considerably. The following 2 → 4 processes -most are on the wish list -have now been calculated at NLO QCD: 9 pp → W γγ+jet [21], pp → W +3 jets [62,63,66,67], pp → Z, γ * +3 jets [68], pp → ttbb [59,60,61,69], pp → ttjj [64,70], pp → bbbb [71], pp → W + W − bb [72], pp → W ± W ± jj [24,73], pp → W + W − jj [74] and most recently pp → 4 jets [75]. Leptonic decays of weak bosons can be included trivially. At the same level of complexity, complete off-shell effects for pp → tt with dileptonic decay, i.e. pp → e + ν e bµ −ν µb , have been calculated at NLO QCD in Ref. [76], which allowed to explicitly confirm the O(α s Γ/M) effect predicted by Ref. [77]. Advancing the frontier for linear collider physics, the process e + e − → 5 jets has recently been calculated at NLO [78], which allowed to extract a competitive value of α s (M Z ) from 5-jet LEP data. Going beyond 4-particle final states in general requires the computation of 7-point one-loop amplitudes or higher. This is the current complexity frontier. At this level, NLO cross sections in leading-colour approximation have been calculated for V + 4 jets by the BlackHat/Sherpa collaboration (pp → W + 4 jets [79] and pp → Z + 4 jets [80]) and for e + e − → n jets up to n = 7 [81]. 10 The n = 7 case required the computation of a one-loop 8-point function. Summary NLO QCD predictions for multi-particle processes are essential to fully exploit the physics potential of the LHC and a future linear collider. In recent years, tremendous progress has been made in developing the calculational methods and tools that are required to compute NLO corrections for hard scattering processes with 6, 7 or more external particles. At this level a (semi-)manual approach is no longer feasible, and the transition from collections of codes for specific processes to automated code generation for any process up to a maximum complexity has now been achieved. Several such automated tools are available or will become public in the near future. The modularity of NLO calculations allows to interface many tool components on the basis of the Binoth Les Houches Accord. 9 pp is given as initial state, but pp is also implied. 10 Recently, the full-colour virtual contribution to pp → W + 4 jets has been calculated [82]. Process (V ∈ {Z, W, γ})Comments Calculations completed since Les Houches 20051. pp → V V +jet W W +jet completed by Dittmaier/Kallweit/Uwer [51, 52]; Campbell/Ellis/Zanderighi [53]. ZZ+jet completed by Binoth/Gleisberg/Karg/Kauer/Sanguinetti [54] 2. pp → Higgs+2jets NLO QCD to the gg channel completed by Campbell/Ellis/Zanderighi [16]; NLO QCD+EW to the VBF channel completed by Ciccolini/Denner/Dittmaier [55, 56] 3. pp → V V V ZZZ completed by Lazopoulos/Melnikov/Petriello [57] and W W Z by Hankele/Zeppenfeld [19] (see also Binoth/Ossola/Papadopoulos/Pittau [58]) 4. pp → tt bb relevant for ttH computed by Bredenstein/Denner/Dittmaier/Pozzorini [59, 60] and Bevilacqua/Czakon/Papadopoulos/Pittau/Worek [61] 5. pp → V +3jets calculated by the Blackhat/Sherpa [62] and Rocket [63] collaborations Calculations remaining from Les Houches 2005 6. pp → tt+2jets relevant for ttH computed by Bevilacqua/Czakon/Papadopoulos/Worek [64] 7. pp → V V bb, relevant for VBF → H → V V , ttH 8. pp → V V +2jets relevant for VBF → H → V V VBF contributions calculated by (Bozzi/)Jäger/Oleari/Zeppenfeld [20] NLO calculations added to list in 2007 9. pp → bbbb qq channel calculated by Golem collaboration [65] NLO calculations added to list in 2009 10. pp → V +4jets top pair production, various new physics signatures 11. pp → W bbj top, new physics signatures 12. pp → tttt various new physics signatures Calculations beyond NLO added in 2007 13. gg → W * W * O(α 2 α 3 s ) backgrounds to Higgs 14. NNLO pp → tt normalisation of a benchmark process 15. NNLO to VBF and Z/γ+jet Higgs couplings and SM benchmark Calculations including electroweak effects 16. NNLO QCD+NLO EW for W/Z precision calculation of a SM benchmark Table 2 : 21103.4613 [hep-ph]]; G. Bozzi, F. Campanario, M. Rauch and D. Zeppenfeld, Phys. Rev. D 84 (2011) 074028 [arXiv:1107.3149 [hepph]]; F. Campanario, C. Englert, M. Spannowsky and D. Zeppenfeld, Europhys. Lett. 88 (2009) 11001 [arXiv:0908.1638 [hep-ph]]; F. Campanario, C. Englert and M. Spannowsky, Phys. Rev. D 83 (2011) 074009 [arXiv:1010.1291 [hep-ph]]; F. Campanario, C. Englert, S. Kallweit, M. Spannowsky and D. Zeppenfeld, JHEP 1007 (2010) 076 [arXiv:1006.0390 [hep-ph]]; F. Campanario, C. Englert and M. Spannowsky, Phys. Rev. D 82 (2010) 054015 [arXiv:1006.3090 [hep-ph]]; V. Del Duca, W. Kilgore, C. Oleari, C. Schmidt and D. Zeppenfeld, Phys. Rev. Lett. 87 (2001) 122001 [hep-ph/0105129]; V. Del Duca, W. Kilgore, C. Oleari, C. Schmidt and D. Zeppenfeld, Nucl. Phys. B 616 (2001) 367 [hep-ph/0108030]; V. Del Duca, G. Klamke, D. Zeppenfeld, M. L. Mangano, M. Moretti, F. Piccinini, R. Pittau and A. D. Polosa, JHEP 0610 (2006) 016 [hep-ph/0608158]; G. Klamke and D. Zeppenfeld, JHEP 0704 (2007) 052 [hep-ph/0703202]; F. Campanario, M. Kubocz and D. Zeppenfeld, Phys. Rev. D 84 (2011) 095025 [arXiv:1011.3819 [hep-ph]]; K. Arnold, M. Bahr, G. Bozzi, F. Campanario, C. Englert, T. Figy, N. Greiner and C. Hackstein et al., Comput. Phys. Commun. 180 (2009) 1661 [arXiv:0811.4559 [hep-ph]]; K. Arnold, J. Bellm, G. Bozzi, M. Brieg, F. Campanario, C. Englert, B. Feigl and J. Frank et al., arXiv:1107.4038 [hepph]. [19] V. Hankele and D. Zeppenfeld, Phys. Lett. B 661 (2008) 103 [arXiv:0712.3544 [hep-ph]].[20] B. Jager, C.Oleari and D. Zeppenfeld, JHEP 0607 (2006) 015 [hep-ph/0603177]; B. Jager, C. Oleari and D. Zeppenfeld, Phys. Rev. D 73 (2006) 113006 [hep-ph/0604200]; G. Bozzi, B. Jager, C. Oleari and D. Zeppenfeld, Phys. Rev. D 75 (2007) 073004 [hep-ph/0701105]. [21] F. Campanario, C. Englert, M. Rauch and D. Zeppenfeld, Phys. Lett. B 704 (2011) 515 [arXiv:1106.4009 [hep-ph]]; F. Campanario, JHEP 1110 (2011) 070 [arXiv:1105.0920 [hepph]]. [22] P. Nason, JHEP 0411 (2004) 040 [hep-ph/0409146]; S. Frixione, P. Nason and C. Oleari, JHEP 0711 (2007) 070 [arXiv:0709.2092 [hep-ph]]; S. Alioli, P. Nason, C. Oleari and E. Re, JHEP 1006 (2010) 043 [arXiv:1002.2581 [hep-ph]].The experimenter's wish list for LHC processes in early 2010 (from [50]). Presented at Linear Collider 2011: Understanding QCD at Linear Colliders in searching for old and new physics, 12-16 September 2011, ECT*, Trento, Italy 2 For processes with vastly differing scales, the resummation of large logarithms of ratios of scales may also be necessary.3 Notable exceptions are the hadroproduction of Higgs and W bb with σ NLO /σ LO ≈ 2. Research on alternative subtraction schemes is also being carried out[35]. AcknowledgmentsI would like to thank the organisers for the invitation to speak at Linear Collider 2011 and commend G. Pancheri and her team for hosting this well-organised and thoroughly enjoyable meeting. The hospitality of the European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*) as well as partial support from ECT* and INFN are gratefully acknowledged. This work was carried out as part of the research programme of the Royal Holloway and Sussex Particle Physics Theory Consortium and the NExT Institute. Financial support under the SEPnet Initiative from the Higher Education Funding Council for England and the Science and Technology Facilities Council (STFC) is gratefully acknowledged. This work was supported by STFC grant ST/J000485/1. . J M Butterworth, G Dissertori, G P Salam, arXiv:1202.0583hep-exJ. M. Butterworth, G. Dissertori and G. P. Salam, arXiv:1202.0583 [hep-ex]. . G P Salam, arXiv:1103.1318PoS. 2010556hep-phG. P. Salam, PoS ICHEP 2010 (2010) 556 [arXiv:1103.1318 [hep-ph]]. . G Zanderighi, arXiv:1201.3905hep-phG. Zanderighi, arXiv:1201.3905 [hep-ph]. . R K Ellis, D A Ross, A E Terrano, Nucl. Phys. B. 178421R. K. Ellis, D. A. Ross and A. E. Terrano, Nucl. Phys. B 178 (1981) 421. . K Fabricius, G Kramer, G Schierholz, I Schmitt, Z. Phys. C. 11315K. Fabricius, G. Kramer, G. Schierholz and I. Schmitt, Z. Phys. C 11 (1981) 315; . G Kramer, B Lampe, Fortsch. Phys. 37161G. Kramer and B. Lampe, Fortsch. Phys. 37 (1989) 161; . B W Harris, J F Owens, hep-ph/0102128Phys. Rev. D. 6594032B. W. Harris and J. F. Owens, Phys. Rev. D 65 (2002) 094032 [hep-ph/0102128]. . T Binoth, arXiv:1001.1307Comput. Phys. Commun. 1811612hep-phT. Binoth et al., Comput. Phys. Commun. 181 (2010) 1612 [arXiv:1001.1307 [hep-ph]]. . G Hooft, M J G Veltman, Nucl. Phys. B. 153365G. 't Hooft and M. J. G. Veltman, Nucl. Phys. B 153 (1979) 365; . G Passarino, M J G Veltman, Nucl. Phys. B. 160151G. Passarino and M. J. G. Veltman, Nucl. Phys. B 160 (1979) 151. . A Denner, S Dittmaier, hep-ph/0212259Nucl. Phys. B. 658175A. Denner and S. Dittmaier, Nucl. Phys. B 658 (2003) 175 [hep-ph/0212259]; . T Binoth, J P Guillet, G Heinrich, E Pilon, C Schubert, hep-ph/0504267JHEP. 051015T. Binoth, J. P. Guillet, G. Heinrich, E. Pilon and C. Schubert, JHEP 0510 (2005) 015 [hep-ph/0504267]; . A Denner, S Dittmaier, hep-ph/0509141Nucl. Phys. B. 73462A. Denner and S. Dittmaier, Nucl. Phys. B 734 (2006) 62 [hep-ph/0509141]. . G J Van Oldenborgh, J A M Vermaseren, Z. Phys. C. 46425G. J. van Oldenborgh and J. A. M. Vermaseren, Z. Phys. C 46 (1990) 425. . T Hahn, M Perez-Victoria, hep-ph/9807565Comput. Phys. Commun. 118153T. Hahn and M. Perez-Victoria, Comput. Phys. Commun. 118 (1999) 153 [hep-ph/9807565]. . R K Ellis, G Zanderighi, arXiv:0712.1851JHEP. 08022hep-phR. K. Ellis and G. Zanderighi, JHEP 0802 (2008) 002 [arXiv:0712.1851 [hep-ph]]. . T Binoth, J. -P Guillet, G Heinrich, E Pilon, T Reiter, arXiv:0810.0992Comput. Phys. Commun. 1802317hep-phT. Binoth, J. -P. Guillet, G. Heinrich, E. Pilon and T. Reiter, Comput. Phys. Commun. 180 (2009) 2317 [arXiv:0810.0992 [hep-ph]]; . G Cullen, J P Guillet, G Heinrich, T Kleinschmidt, E Pilon, T Reiter, M Rodgers, arXiv:1101.5595Comput. Phys. Commun. 1822276hep-phG. Cullen, J. P. Guillet, G. Heinrich, T. Klein- schmidt, E. Pilon, T. Reiter and M. Rodgers, Comput. Phys. Commun. 182 (2011) 2276 [arXiv:1101.5595 [hep-ph]]. . A Van Hameren, arXiv:1007.4716Comput. Phys. Commun. 1822427hep-phA. van Hameren, Comput. Phys. Commun. 182 (2011) 2427 [arXiv:1007.4716 [hep-ph]]. . J Fleischer, T Riemann, arXiv:1009.4436Phys. Rev. D. 8373004hep-phJ. Fleischer and T. Riemann, Phys. Rev. D 83 (2011) 073004 [arXiv:1009.4436 [hep-ph]]. . R K Ellis, S Veseli, hep-ph/9810489Phys. Rev. D. 6011501R. K. Ellis and S. Veseli, Phys. Rev. D 60 (1999) 011501 [hep-ph/9810489]; . J M Campbell, R K Ellis, hep-ph/9905386Phys. Rev. D. 60113006J. M. Campbell and R. K. Ellis, Phys. Rev. D 60 (1999) 113006 [hep-ph/9905386]; . J M Campbell, R K Ellis, hep-ph/0006304Phys. Rev. D. 62114012J. M. Campbell and R. K. Ellis, Phys. Rev. D 62 (2000) 114012 [hep-ph/0006304]; . J M Campbell, R K Ellis, hep-ph/0202176Phys. Rev. D. 65113007J. M. Campbell and R. K. El- lis, Phys. Rev. D 65 (2002) 113007 [hep-ph/0202176]; . J M Campbell, R K Ellis, F Maltoni, S Willenbrock, hep-ph/0204093Phys. Rev. D. 6795002J. M. Campbell, R. K. Ellis, F. Mal- toni and S. Willenbrock, Phys. Rev. D 67 (2003) 095002 [hep-ph/0204093]; . J M Campbell, R K Ellis, D L Rainwater, hep-ph/0308195Phys. Rev. D. 6894021J. M. Camp- bell, R. K. Ellis and D. L. Rainwater, Phys. Rev. D 68 (2003) 094021 [hep-ph/0308195]; . J M Campbell, R K Ellis, F Maltoni, S Willenbrock, hep-ph/0312024Phys. Rev. D. 6974021J. M. Campbell, R. K. Ellis, F. Maltoni and S. Willenbrock, Phys. Rev. D 69 (2004) 074021 [hep-ph/0312024]; . J M Campbell, R K Ellis, F Tramontano, hep-ph/0408158Phys. Rev. D. 7094012J. M. Campbell, R. K. Ellis and F. Tramontano, Phys. Rev. D 70 (2004) 094012 [hep-ph/0408158]; . J M Campbell, F Tramontano, hep-ph/0506289Nucl. Phys. B. 726109J. M. Campbell and F. Tramontano, Nucl. Phys. B 726 (2005) 109 [hep-ph/0506289]; . J M Campbell, R K Ellis, F Maltoni, S Willenbrock, hep-ph/0510362Phys. Rev. D. 7354007Erratum-ibid. D 77 (2008) 019903J. M. Campbell, R. K. Ellis, F. Maltoni and S. Willenbrock, Phys. Rev. D 73 (2006) 054007 [Erratum-ibid. D 77 (2008) 019903] [hep-ph/0510362]; . J M Campbell, R Frederix, F Maltoni, F Tramontano, arXiv:0903.0005Phys. Rev. Lett. 102182003hep-phJ. M. Campbell, R. Frederix, F. Maltoni and F. Tramontano, Phys. Rev. Lett. 102 (2009) 182003 [arXiv:0903.0005 [hep-ph]]; . J M Campbell, R K Ellis, C Williams, arXiv:1001.4495Phys. Rev. D. 8174023hep-phJ. M. Campbell, R. K. Ellis and C. Williams, Phys. Rev. D 81 (2010) 074023 [arXiv:1001.4495 [hep-ph]]; . S Badger, J M Campbell, R K Ellis, arXiv:1011.6647JHEP. 110327hep-phS. Badger, J. M. Campbell and R. K. Ellis, JHEP 1103 (2011) 027 [arXiv:1011.6647 [hep-ph]]; . J M Campbell, R K Ellis, C Williams, arXiv:1105.0020JHEP. 110718hep-phJ. M. Campbell, R. K. El- lis and C. Williams, JHEP 1107 (2011) 018 [arXiv:1105.0020 [hep-ph]]; . J M Campbell, R K Ellis, C Williams, arXiv:1107.5569JHEP. 11105hep-phJ. M. Campbell, R. K. Ellis and C. Williams, JHEP 1110 (2011) 005 [arXiv:1107.5569 [hep-ph]]. . J M Campbell, R K Ellis, G Zanderighi, hep-ph/0608194JHEP. 061028J. M. Campbell, R. K. Ellis and G. Zanderighi, JHEP 0610 (2006) 028 [hep-ph/0608194]. . S Frixione, B R Webber, hep-ph/0204244JHEP. 020629S. Frixione and B. R. Webber, JHEP 0206 (2002) 029 [hep-ph/0204244]; . S Frixione, P Nason, B R Webber, hep-ph/0305252JHEP. 03087S. Frixione, P. Nason and B. R. Webber, JHEP 0308 (2003) 007 [hep-ph/0305252]; . S Frixione, E Laenen, P Motylinski, B R Webber, hep-ph/0512250JHEP. 060392S. Frixione, E. Lae- nen, P. Motylinski and B. R. Webber, JHEP 0603 (2006) 092 [hep-ph/0512250]; . S Frixione, E Laenen, P Motylinski, B R Webber, C D White, arXiv:0805.3067JHEP. 080729hep-phS. Frix- ione, E. Laenen, P. Motylinski, B. R. Webber and C. D. White, JHEP 0807 (2008) 029 [arXiv:0805.3067 [hep-ph]]; . S Frixione, F Stoeckli, P Torrielli, B R Webber, arXiv:1010.0568JHEP. 110153hep-phS. Frixione, F. Stoeckli, P. Torrielli and B. R. Webber, JHEP 1101 (2011) 053 [arXiv:1010.0568 [hep-ph]]; . B Fuks, M Klasen, F Ledroit, Q Li, J Morel, arXiv:0711.0749Nucl. Phys. B. 797322hep-phB. Fuks, M. Klasen, F. Ledroit, Q. Li and J. Morel, Nucl. Phys. B 797 (2008) 322 [arXiv:0711.0749 [hep-ph]]. . T Figy, C Oleari, D Zeppenfeld, hep-ph/0306109Phys. Rev. D. 6873005T. Figy, C. Oleari and D. Zeppenfeld, Phys. Rev. D 68 (2003) 073005 [hep-ph/0306109]; . T Figy, S Palmer, G Weiglein, arXiv:1012.4789hep-phT. Figy, S. Palmer and G. Weiglein, arXiv:1012.4789 [hep-ph]; . T Figy, V Hankele, D Zeppenfeld, arXiv:0710.5621JHEP. 080276hep-phT. Figy, V. Hankele and D. Zeppenfeld, JHEP 0802 (2008) 076 [arXiv:0710.5621 [hep-ph]]; . C Oleari, D Zeppenfeld, hep-ph/0310156Phys. Rev. D. 6993004C. Oleari and D. Zep- penfeld, Phys. Rev. D 69 (2004) 093004 [hep-ph/0310156]; . C Englert, B Jager, M Worek, C. Englert, B. Jager, M. Worek . S Alioli, P Nason, C Oleari, E Re, arXiv:0805.4802JHEP. 080760hepphS. Alioli, P. Nason, C. Oleari and E. Re, JHEP 0807 (2008) 060 [arXiv:0805.4802 [hep- ph]]; . S Alioli, P Nason, C Oleari, E Re, arXiv:1009.5594JHEP. 110195hep-phS. Alioli, P. Nason, C. Oleari and E. Re, JHEP 1101 (2011) 095 [arXiv:1009.5594 [hep-ph]]; . S Alioli, P Nason, C Oleari, E Re, arXiv:0907.4076JHEP. 0909111Erratumibid. 1002 (2010) 011. hep-phS. Alioli, P. Nason, C. Oleari and E. Re, JHEP 0909 (2009) 111 [Erratum- ibid. 1002 (2010) 011] [arXiv:0907.4076 [hep-ph]]; . E Re, arXiv:1009.2450Eur. Phys. J. C. 711547hep-phE. Re, Eur. Phys. J. C 71 (2011) 1547 [arXiv:1009.2450 [hep-ph]]; . S Alioli, P Nason, C Oleari, E Re, arXiv:0812.0578JHEP. 09042hep-phS. Alioli, P. Nason, C. Oleari and E. Re, JHEP 0904 (2009) 002 [arXiv:0812.0578 [hep-ph]]; . P Nason, C Oleari, arXiv:0911.5299JHEP. 100237hep-phP. Nason and C. Oleari, JHEP 1002 (2010) 037 [arXiv:0911.5299 [hep-ph]]; . S Alioli, K Hamilton, P Nason, C Oleari, E Re, arXiv:1012.3380JHEP. 110481hep-phS. Alioli, K. Hamilton, P. Nason, C. Oleari and E. Re, JHEP 1104 (2011) 081 [arXiv:1012.3380 [hep-ph]]; . S Frixione, P Nason, G Ridolfi, arXiv:0707.3088JHEP. 0709126hep-phS. Frixione, P. Nason and G. Ridolfi, JHEP 0709 (2007) 126 [arXiv:0707.3088 [hep-ph]]; . T Melia, P Nason, R Rontsch, G Zanderighi, arXiv:1102.4846Eur. Phys. J. C. 711670hep-phT. Melia, P. Nason, R. Rontsch and G. Zan- derighi, Eur. Phys. J. C 71 (2011) 1670 [arXiv:1102.4846 [hep-ph]]; . T Melia, P Nason, R Rontsch, G Zanderighi, arXiv:1107.5051JHEP. 111178hep-phT. Melia, P. Nason, R. Rontsch and G. Zanderighi, JHEP 1111 (2011) 078 [arXiv:1107.5051 [hep-ph]]; . C Oleari, L Reina, arXiv:1105.4488JHEP. 110861Erratum-ibid. 1111 (2011) 040. hep-phC. Oleari and L. Reina, JHEP 1108 (2011) 061 [Erratum-ibid. 1111 (2011) 040] [arXiv:1105.4488 [hep-ph]]; . E Bagnaschi, G Degrassi, P Slavich, A Vicini, arXiv:1111.2854hep-phE. Bagnaschi, G. Degrassi, P. Slavich and A. Vicini, arXiv:1111.2854 [hep-ph]; . C Bernaciak, D Wackeroth, arXiv:1201.4804hep-phC. Bernaciak and D. Wackeroth, arXiv:1201.4804 [hep-ph]; . L Barze, &apos; , G Montagna, P Nason, O Nicrosini, F Piccinini, arXiv:1202.0465hep-phL. Barze', G. Montagna, P. Na- son, O. Nicrosini and F. Piccinini, arXiv:1202.0465 [hep-ph]. . B Jager, G Zanderighi, arXiv:1108.0864JHEP. 111155hep-phB. Jager and G. Zanderighi, JHEP 1111 (2011) 055 [arXiv:1108.0864 [hep-ph]]. . Z Bern, L J Dixon, D C Dunbar, D A Kosower, hep-ph/9403226Nucl. Phys. B. 425217Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, Nucl. Phys. B 425 (1994) 217 [hep-ph/9403226]; . Z Bern, L J Dixon, D C Dunbar, D A Kosower, hep-ph/9409265Nucl. Phys. B. 43559Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, Nucl. Phys. B 435 (1995) 59 [hep-ph/9409265]; . Z Bern, L J Dixon, D A Kosower, arXiv:hep-ph/9708239Nucl. Phys. B. 5133Z. Bern, L. J. Dixon and D. A. Kosower, Nucl. Phys. B 513 (1998) 3 [arXiv:hep-ph/9708239]; . R Britto, F Cachazo, B Feng, arXiv:hep-th/0412103Nucl. Phys. B. 725275R. Britto, F. Cachazo and B. Feng, Nucl. Phys. B 725 (2005) 275 [arXiv:hep-th/0412103]; . G Ossola, C G Papadopoulos, R Pittau, arXiv:hep-ph/0609007Nucl. Phys. B. 763147G. Ossola, C. G. Papadopoulos and R. Pit- tau, Nucl. Phys. B 763 (2007) 147 [arXiv:hep-ph/0609007]; . R K Ellis, W T Giele, Z Kunszt, arXiv:0708.2398JHEP. 08033hep-phR. K. Ellis, W. T. Giele and Z. Kunszt, JHEP 0803 (2008) 003 [arXiv:0708.2398 [hep-ph]]; . W T Giele, Z Kunszt, K Melnikov, arXiv:0801.2237JHEP. 080449hep-phW. T. Giele, Z. Kunszt and K. Melnikov, JHEP 0804 (2008) 049 [arXiv:0801.2237 [hep-ph]]; . R K Ellis, W T Giele, Z Kunszt, K Melnikov, arXiv:0806.3467Nucl. Phys. B. 822270hep-phR. K. Ellis, W. T. Giele, Z. Kunszt and K. Melnikov, Nucl. Phys. B 822 (2009) 270 [arXiv:0806.3467 [hep-ph]]. . C F Berger, Z Bern, L J Dixon, F Cordero, D Forde, H Ita, D A Kosower, D Maitre, arXiv:0803.4180Phys. Rev. D. 7836003hep-phC. F. Berger, Z. Bern, L. J. Dixon, F. Febres Cordero, D. Forde, H. Ita, D. A. Kosower and D. Maitre, Phys. Rev. D 78 (2008) 036003 [arXiv:0803.4180 [hep-ph]]. . G Ossola, C G Papadopoulos, R Pittau, arXiv:0711.3596JHEP. 080342hep-phG. Ossola, C. G. Papadopoulos and R. Pittau, JHEP 0803 (2008) 042 [arXiv:0711.3596 [hep-ph]]. . W T Giele, G Zanderighi, arXiv:0805.2152JHEP. 080638hep-phW. T. Giele and G. Zanderighi, JHEP 0806 (2008) 038 [arXiv:0805.2152 [hep-ph]]; . R K Ellis, W T Giele, Z Kunszt, K Melnikov, G Zanderighi, arXiv:0810.2762JHEP. 090112hep-phR. K. Ellis, W. T. Giele, Z. Kunszt, K. Melnikov and G. Zanderighi, JHEP 0901 (2009) 012 [arXiv:0810.2762 [hep-ph]]. . P Mastrolia, G Ossola, T Reiter, F Tramontano, arXiv:1006.0710JHEP. 100880hep-phP. Mastrolia, G. Ossola, T. Reiter and F. Tramontano, JHEP 1008 (2010) 080 [arXiv:1006.0710 [hep-ph]]. . A Van Hameren, arXiv:0905.1005JHEP. 090788hep-phA. van Hameren, JHEP 0907 (2009) 088 [arXiv:0905.1005 [hep-ph]]; . S Becker, C Reuschle, S Weinzierl, arXiv:1010.4187JHEP. 101213hep-phS. Becker, C. Reuschle and S. Weinzierl, JHEP 1012 (2010) 013 [arXiv:1010.4187 [hep-ph]]; . F Cascioli, P Maierhofer, S Pozzorini, arXiv:1111.5206hep-phF. Cascioli, P. Maier- hofer and S. Pozzorini, arXiv:1111.5206 [hep-ph]. . R K Ellis, Z Kunszt, K Melnikov, G Zanderighi, arXiv:1105.4319hep-phR. K. Ellis, Z. Kunszt, K. Melnikov and G. Zanderighi, arXiv:1105.4319 [hep-ph]. . S Catani, M H Seymour, hep-ph/9605323Nucl. Phys. B. 485291Erratum-ibid. B 510 (1998) 503S. Catani and M. H. Seymour, Nucl. Phys. B 485 (1997) 291 [Erratum-ibid. B 510 (1998) 503] [hep-ph/9605323]; . S Catani, S Dittmaier, M H Seymour, Z Trocsanyi, hep-ph/0201036Nucl. Phys. B. 627189S. Catani, S. Dittmaier, M. H. Seymour and Z. Trocsanyi, Nucl. Phys. B 627 (2002) 189 [hep-ph/0201036]. . S Frixione, Z Kunszt, A Signer, hep-ph/9512328Nucl. Phys. B. 467399S. Frixione, Z. Kunszt and A. Signer, Nucl. Phys. B 467 (1996) 399 [hep-ph/9512328]; . S Frixione, hep-ph/9706545Nucl. Phys. B. 507295S. Frixione, Nucl. Phys. B 507 (1997) 295 [hep-ph/9706545]. . D A Kosower, hep-ph/9710213Phys. Rev. D. 575410D. A. Kosower, Phys. Rev. D 57 (1998) 5410 [hep-ph/9710213]; . A Gehrmann-De Ridder, T Gehrmann, E W N Glover, hep-ph/0505111JHEP. 050956A. Gehrmann-De Ridder, T. Gehrmann and E. W. N. Glover, JHEP 0509 (2005) 056 [hep-ph/0505111]; . A Daleo, T Gehrmann, D Maitre, hep-ph/0612257JHEP. 070416A. Daleo, T. Gehrmann and D. Maitre, JHEP 0704 (2007) 016 [hep-ph/0612257]. . C H Chung, M Kramer, T Robens, arXiv:1012.4948JHEP. 1106144hepphC. H. Chung, M. Kramer and T. Robens, JHEP 1106 (2011) 144 [arXiv:1012.4948 [hep- ph]]. . T Gleisberg, F Krauss, arXiv:0709.2881Eur. Phys. J. C. 53501hep-phT. Gleisberg and F. Krauss, Eur. Phys. J. C 53 (2008) 501 [arXiv:0709.2881 [hep-ph]]. . R Frederix, T Gehrmann, N Greiner, arXiv:0808.2128JHEP. 0809122hepphR. Frederix, T. Gehrmann and N. Greiner, JHEP 0809 (2008) 122 [arXiv:0808.2128 [hep- ph]]; . R Frederix, T Gehrmann, N Greiner, arXiv:1004.2905JHEP. 100686hep-phR. Frederix, T. Gehrmann and N. Greiner, JHEP 1006 (2010) 086 [arXiv:1004.2905 [hep-ph]]. . M Czakon, C G Papadopoulos, M Worek, arXiv:0905.0883JHEP. 090885hep-phM. Czakon, C. G. Papadopoulos and M. Worek, JHEP 0908 (2009) 085 [arXiv:0905.0883 [hep-ph]]. . R Frederix, S Frixione, F Maltoni, T Stelzer, arXiv:0908.4272JHEP. 09103hep-phR. Frederix, S. Frixione, F. Maltoni and T. Stelzer, JHEP 0910 (2009) 003 [arXiv:0908.4272 [hep-ph]]. . M H Seymour, C Tevlin, arXiv:0803.2231hep-phM. H. Seymour and C. Tevlin, arXiv:0803.2231 [hep-ph]. . K Hasegawa, S Moch, P Uwer, arXiv:0911.4371Comput. Phys. Commun. 1811802hep-phK. Hasegawa, S. Moch and P. Uwer, Comput. Phys. Commun. 181 (2010) 1802 [arXiv:0911.4371 [hep-ph]]. . V Hirschi, R Frederix, S Frixione, M V Garzelli, F Maltoni, R Pittau, arXiv:1103.0621JHEP. 110544hep-phV. Hirschi, R. Frederix, S. Frixione, M. V. Garzelli, F. Maltoni and R. Pittau, JHEP 1105 (2011) 044 [arXiv:1103.0621 [hep-ph]]; . R Frederix, S Frixione, V Hirschi, F Maltoni, R Pittau, P Torrielli, arXiv:1104.5613Phys. Lett. B. 701427hep-phR. Frederix, S. Frixione, V. Hirschi, F. Mal- toni, R. Pittau and P. Torrielli, Phys. Lett. B 701 (2011) 427 [arXiv:1104.5613 [hep-ph]]; . R Frederix, S Frixione, V Hirschi, F Maltoni, R Pittau, P Torrielli, arXiv:1106.6019JHEP. 110961hep-phR. Frederix, S. Frixione, V. Hirschi, F. Maltoni, R. Pittau and P. Torrielli, JHEP 1109 (2011) 061 [arXiv:1106.6019 [hep-ph]]; . R Frederix, S Frixione, V Hirschi, F Maltoni, R Pittau, P Torrielli, arXiv:1110.4738hep-phR. Frederix, S. Frixione, V. Hirschi, F. Maltoni, R. Pittau and P. Torrielli, arXiv:1110.4738 [hep-ph]; . R Frederix, S Frixione, V Hirschi, F Maltoni, R Pittau, P Torrielli, arXiv:1110.5502JHEP. 120248hep-phR. Frederix, S. Frixione, V. Hirschi, F. Maltoni, R. Pittau and P. Torrielli, JHEP 1202 (2012) 048 [arXiv:1110.5502 [hep-ph]]; . J Alwall, M Herquet, F Maltoni, O Mattelaer, T Stelzer, arXiv:1106.0522JHEP. 1106128hep-phJ. Alwall, M. Herquet, F. Maltoni, O. Mattelaer and T. Stelzer, JHEP 1106 (2011) 128 [arXiv:1106.0522 [hep-ph]]. . T Gleisberg, S Hoeche, F Krauss, M Schonherr, S Schumann, F Siegert, J Winter, arXiv:0811.4622JHEP. 09027hep-phT. Gleisberg, S. Hoeche, F. Krauss, M. Schonherr, S. Schumann, F. Siegert and J. Winter, JHEP 0902 (2009) 007 [arXiv:0811.4622 [hep-ph]]. . A Van Hameren, C G Papadopoulos, R Pittau, arXiv:0903.4665JHEP. 0909106hep-phA. van Hameren, C. G. Papadopoulos and R. Pittau, JHEP 0909 (2009) 106 [arXiv:0903.4665 [hep-ph]]; . G Bevilacqua, M Czakon, M V Garzelli, A Van Hameren, A Kardos, C G Papadopoulos, R Pittau, M Worek, arXiv:1110.1499hep-phG. Bevilacqua, M. Czakon, M. V. Garzelli, A. van Hameren, A. Kardos, C. G. Papadopoulos, R. Pittau and M. Worek, arXiv:1110.1499 [hep-ph]; . A Cafarella, C G Papadopoulos, M Worek, arXiv:0710.2427Comput. Phys. Commun. 1801941hep-phA. Ca- farella, C. G. Papadopoulos and M. Worek, Comput. Phys. Commun. 180 (2009) 1941 [arXiv:0710.2427 [hep-ph]]. . G Cullen, N Greiner, G Heinrich, G Luisoni, P Mastrolia, G Ossola, T Reiter, F Tramontano, arXiv:1111.2034hep-phG. Cullen, N. Greiner, G. Heinrich, G. Luisoni, P. Mastrolia, G. Ossola, T. Reiter and F. Tramontano, arXiv:1111.2034 [hep-ph]. . T Hahn, hep-ph/0012260Comput. Phys. Commun. 140418T. Hahn, Comput. Phys. Commun. 140 (2001) 418 [hep-ph/0012260]. . T Binoth, D Netto, D Lopez-Val, K Mawatari, T Plehn, I Wigmore, arXiv:1108.1250Phys. Rev. D. 8475005hep-phT. Binoth, D. Goncalves Netto, D. Lopez-Val, K. Mawatari, T. Plehn and I. Wigmore, Phys. Rev. D 84 (2011) 075005 [arXiv:1108.1250 [hep-ph]]. . C Buttar, S Dittmaier, V Drollinger, S Frixione, A Nikitenko, S Willenbrock, S Abdullin, E Accomando, hep-ph/0604120C. Buttar, S. Dittmaier, V. Drollinger, S. Frixione, A. Nikitenko, S. Willenbrock, S. Ab- dullin and E. Accomando et al., hep-ph/0604120. . Z Bern, NLO Multileg Working Group CollaborationarXiv:0803.0494hep-phZ. Bern et al. [NLO Multileg Working Group Collaboration], arXiv:0803.0494 [hep-ph]. . J R Andersen, SM and NLO Multileg Working Group CollaborationarXiv:1003.1241hep-phJ. R. Andersen et al. [SM and NLO Multileg Working Group Collaboration], arXiv:1003.1241 [hep-ph]. . S Dittmaier, S Kallweit, P Uwer, arXiv:0710.1577Phys. Rev. Lett. 10062003hep-phS. Dittmaier, S. Kallweit and P. Uwer, Phys. Rev. Lett. 100 (2008) 062003 [arXiv:0710.1577 [hep-ph]]. . S Dittmaier, S Kallweit, P Uwer, arXiv:0908.4124Nucl. Phys. B. 82618hep-phS. Dittmaier, S. Kallweit and P. Uwer, Nucl. Phys. B 826 (2010) 18 [arXiv:0908.4124 [hep-ph]]. . J M Campbell, R K Ellis, G Zanderighi, arXiv:0710.1832JHEP. 071256hep-phJ. M. Campbell, R. K. Ellis and G. Zanderighi, JHEP 0712 (2007) 056 [arXiv:0710.1832 [hep-ph]]. . T Binoth, T Gleisberg, S Karg, N Kauer, G Sanguinetti, arXiv:0911.3181Phys. Lett. B. 683154hep-phT. Binoth, T. Gleisberg, S. Karg, N. Kauer and G. Sanguinetti, Phys. Lett. B 683 (2010) 154 [arXiv:0911.3181 [hep-ph]]. . M Ciccolini, A Denner, S Dittmaier, arXiv:0707.0381Phys. Rev. Lett. 99161803hep-phM. Ciccolini, A. Denner and S. Dittmaier, Phys. Rev. Lett. 99 (2007) 161803 [arXiv:0707.0381 [hep-ph]]. . M Ciccolini, A Denner, S Dittmaier, arXiv:0710.4749Phys. Rev. D. 7713002hep-phM. Ciccolini, A. Denner and S. Dittmaier, Phys. Rev. D 77 (2008) 013002 [arXiv:0710.4749 [hep-ph]]. . A Lazopoulos, K Melnikov, F Petriello, arXiv:hep-ph/0703273Phys. Rev. D. 7614001A. Lazopoulos, K. Melnikov and F. Petriello, Phys. Rev. D 76 (2007) 014001 [arXiv:hep-ph/0703273]. . T Binoth, G Ossola, C G Papadopoulos, R Pittau, arXiv:0804.0350JHEP. 080682hep-phT. Binoth, G. Ossola, C. G. Papadopoulos and R. Pittau, JHEP 0806 (2008) 082 [arXiv:0804.0350 [hep-ph]]. . A Bredenstein, A Denner, S Dittmaier, S Pozzorini, arXiv:0905.0110Phys. Rev. Lett. 10312002hep-phA. Bredenstein, A. Denner, S. Dittmaier and S. Pozzorini, Phys. Rev. Lett. 103 (2009) 012002 [arXiv:0905.0110 [hep-ph]]. . A Bredenstein, A Denner, S Dittmaier, S Pozzorini, arXiv:1001.4006JHEP. 100321hep-phA. Bredenstein, A. Denner, S. Dittmaier and S. Pozzorini, JHEP 1003 (2010) 021 [arXiv:1001.4006 [hep-ph]]. . G Bevilacqua, M Czakon, C G Papadopoulos, R Pittau, M Worek, arXiv:0907.4723JHEP. 0909109hep-phG. Bevilacqua, M. Czakon, C. G. Papadopoulos, R. Pittau and M. Worek, JHEP 0909 (2009) 109 [arXiv:0907.4723 [hep-ph]]. . C F Berger, Z Bern, L J Dixon, F Cordero, D Forde, T Gleisberg, H Ita, D A Kosower, arXiv:0907.1984Phys. Rev. D. 8074036hep-phC. F. Berger, Z. Bern, L. J. Dixon, F. Febres Cordero, D. Forde, T. Gleisberg, H. Ita and D. A. Kosower et al., Phys. Rev. D 80 (2009) 074036 [arXiv:0907.1984 [hep-ph]]. . R K Ellis, K Melnikov, G Zanderighi, arXiv:0901.4101JHEP. 090477hep-phR. K. Ellis, K. Melnikov and G. Zanderighi, JHEP 0904 (2009) 077 [arXiv:0901.4101 [hep-ph]]. . G Bevilacqua, M Czakon, C G Papadopoulos, M Worek, arXiv:1002.4009Phys. Rev. Lett. 104162002hep-phG. Bevilacqua, M. Czakon, C. G. Papadopoulos and M. Worek, Phys. Rev. Lett. 104 (2010) 162002 [arXiv:1002.4009 [hep-ph]]. . T Binoth, arXiv:1001.4905PoS. 200926hep-phT. Binoth et al., PoS RADCOR2009 (2010) 026 [arXiv:1001.4905 [hep-ph]]. . C F Berger, Z Bern, L J Dixon, F Cordero, D Forde, T Gleisberg, H Ita, D A Kosower, arXiv:0902.2760Phys. Rev. Lett. 102222001hep-phC. F. Berger, Z. Bern, L. J. Dixon, F. Febres Cordero, D. Forde, T. Gleisberg, H. Ita and D. A. Kosower et al., Phys. Rev. Lett. 102 (2009) 222001 [arXiv:0902.2760 [hep-ph]]. . R K Ellis, K Melnikov, G Zanderighi, arXiv:0906.1445Phys. Rev. D. 8094002hep-phR. K. Ellis, K. Melnikov and G. Zanderighi, Phys. Rev. D 80 (2009) 094002 [arXiv:0906.1445 [hep-ph]]. . C F Berger, Z Bern, L J Dixon, F Cordero, D Forde, T Gleisberg, H Ita, D A Kosower, arXiv:1004.1659Phys. Rev. D. 8274002hep-phC. F. Berger, Z. Bern, L. J. Dixon, F. Febres Cordero, D. Forde, T. Gleisberg, H. Ita and D. A. Kosower et al., Phys. Rev. D 82 (2010) 074002 [arXiv:1004.1659 [hep-ph]]. . A Bredenstein, A Denner, S Dittmaier, S Pozzorini, arXiv:0807.1248JHEP. 0808108hep-phA. Bredenstein, A. Denner, S. Dittmaier and S. Pozzorini, JHEP 0808 (2008) 108 [arXiv:0807.1248 [hep-ph]]. . G Bevilacqua, M Czakon, C G Papadopoulos, M Worek, arXiv:1108.2851Phys. Rev. D. 84114017hep-phG. Bevilacqua, M. Czakon, C. G. Papadopoulos and M. Worek, Phys. Rev. D 84 (2011) 114017 [arXiv:1108.2851 [hep-ph]]. . T Binoth, N Greiner, A Guffanti, J Reuter, J P Guillet, T Reiter, arXiv:0910.4379Phys. Lett. B. 685293hep-phT. Binoth, N. Greiner, A. Guffanti, J. Reuter, J. P. Guillet and T. Reiter, Phys. Lett. B 685 (2010) 293 [arXiv:0910.4379 [hep-ph]]; . N Greiner, A Guffanti, T Reiter, J Reuter, arXiv:1105.3624Phys. Rev. Lett. 107102002hep-phN. Greiner, A. Guffanti, T. Reiter and J. Reuter, Phys. Rev. Lett. 107 (2011) 102002 [arXiv:1105.3624 [hep-ph]]. . A Denner, S Dittmaier, S Kallweit, S Pozzorini, arXiv:1012.3975Phys. Rev. Lett. 10652001hep-phA. Denner, S. Dittmaier, S. Kallweit and S. Pozzorini, Phys. Rev. Lett. 106 (2011) 052001 [arXiv:1012.3975 [hep-ph]]. . T Melia, K Melnikov, R Rontsch, G Zanderighi, arXiv:1007.5313JHEP. 101253hep-phT. Melia, K. Melnikov, R. Rontsch and G. Zanderighi, JHEP 1012 (2010) 053 [arXiv:1007.5313 [hep-ph]]. . T Melia, K Melnikov, R Rontsch, G Zanderighi, arXiv:1104.2327Phys. Rev. D. 83114043hep-phT. Melia, K. Melnikov, R. Rontsch and G. Zanderighi, Phys. Rev. D 83 (2011) 114043 [arXiv:1104.2327 [hep-ph]]. . Z Bern, G Diana, L J Dixon, F Cordero, S Hoeche, D A Kosower, H Ita, D Maitre, arXiv:1112.3940hep-phZ. Bern, G. Diana, L. J. Dixon, F. Febres Cordero, S. Hoeche, D. A. Kosower, H. Ita and D. Maitre et al., arXiv:1112.3940 [hep-ph]. . G Bevilacqua, M Czakon, A Van Hameren, C G Papadopoulos, M Worek, arXiv:1012.4230JHEP. 110283hep-phG. Bevilacqua, M. Czakon, A. van Hameren, C. G. Papadopoulos and M. Worek, JHEP 1102 (2011) 083 [arXiv:1012.4230 [hep-ph]]. . V S Fadin, V A Khoze, A D Martin, Phys. Rev. D. 492247V. S. Fadin, V. A. Khoze and A. D. Martin, Phys. Rev. D 49 (1994) 2247. . R Frederix, S Frixione, K Melnikov, G Zanderighi, arXiv:1008.5313JHEP. 101150hep-phR. Frederix, S. Frixione, K. Melnikov and G. Zanderighi, JHEP 1011 (2010) 050 [arXiv:1008.5313 [hep-ph]]. . C F Berger, Z Bern, L J Dixon, F Cordero, D Forde, T Gleisberg, H Ita, D A Kosower, arXiv:1009.2338Phys. Rev. Lett. 10692001hep-phC. F. Berger, Z. Bern, L. J. Dixon, F. Febres Cordero, D. Forde, T. Gleisberg, H. Ita and D. A. Kosower et al., Phys. Rev. Lett. 106 (2011) 092001 [arXiv:1009.2338 [hep-ph]]. . H Ita, Z Bern, L J Dixon, F F Cordero, D A Kosower, D Maitre, arXiv:1108.2229hep-phH. Ita, Z. Bern, L. J. Dixon, F. F. Cordero, D. A. Kosower and D. Maitre, arXiv:1108.2229 [hep-ph]. . S Becker, D Goetz, C Reuschle, C Schwan, S Weinzierl, arXiv:1111.1733Phys. Rev. Lett. 10832005hep-phS. Becker, D. Goetz, C. Reuschle, C. Schwan and S. Weinzierl, Phys. Rev. Lett. 108 (2012) 032005 [arXiv:1111.1733 [hep-ph]]. . H Ita, K Ozeren, arXiv:1111.4193hep-phH. Ita and K. Ozeren, arXiv:1111.4193 [hep-ph].
{'fraction_non_alphanumeric': 0.11894878975795159, 'fraction_numerical': 0.12822564512902582, 'mean_word_length': 3.6557625145518045, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'Theoretical predictions for scattering processes with multi-particle final states at next-to-leading order (NLO) in perturbative QCD are essential to fully exploit the physics potential of present and future high-energy colliders. The status of NLO QCD calculations and tools is reviewed.4The important topics of next-to-next-to-leading order (NNLO) calculations and combining parton-level fixed-order calculations and parton-shower event generators are beyond the scope of this review.5The Born amplitude is assumed to be at tree level. 6 An alternative to the widely used subtraction formalism [4] is the phase space slicing method[5]. 7 The POWHEG BOX [22] library project[23,24]was inspired by these collections.', 'arxivid': '1202.4608', 'author': ['Nikolas Kauer [email protected] \nDepartment of Physics\nRoyal Holloway\nUniversity of London\nTW20 0EXEghamUK\n'], 'authoraffiliation': ['Department of Physics\nRoyal Holloway\nUniversity of London\nTW20 0EXEghamUK'], 'corpusid': 118363720, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 20109, 'n_tokens_neox': 15216, 'n_words': 5935, 'pdfsha': '157434a6b67426bf8385122ad29eefc321ca9517', 'pdfurls': None, 'title': ['NLO automated tools for QCD and beyond', 'NLO automated tools for QCD and beyond'], 'venue': ['LC11 Proceedings Frascati Physics Series']}
arxiv
An overview of new measurements of flow, chirality, and vorticity from STAR experiment June 5. 2023 Chunjian Zhang [email protected] (for the STAR Collaboration) Chemistry Department Stony Brook University 11794Stony BrookNew YorkUSA An overview of new measurements of flow, chirality, and vorticity from STAR experiment International Journal of Modern Physics E June 5. 2023Heavy-ion collisionscollective flowchiralityvorticity PACS numbers: 2675Nq, 2575Ld, 2575Ag, 2575G In relativistic heavy-ion collisions, the properties of quark-gluon plasma (QGP) and complex dynamics of multi-scale processes in Quantum Chromodynamics (QCD) are studied by analyzing the final state produced particles in a variety of different ways. In these proceedings, we present an overview of new detailed measurements of flow, chirality, and vorticity by the STAR experiment at RHIC. Furthermore, STAR's future opportunities for the precision measurements on small systems, fixed-target (FXT) mode, and Beam Energy Scan (BES-II) program are discussed. Introduction The initial conditions and dynamics of a hot and dense phase of QCD matter, the strongly interacting QGP, 1, 2 is naturally created in the nuclear collisions at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC). After the collision, the subsequent fluid motion and the expansion of QGP flow hydrodynamically, [3][4][5] later on the QGP turns into a lower-temperature hadronic phase. Thus, such nuclear collisions offer an ideal environment to explore fundamental physics. During the QGP fireball expansion, spatial anisotropies in the initial state, lead to final state momentum anisotropies. The large azimuthal modulations in the final distributions of the produced particles, known as collective flow phenomena, are typically characterized by Fourier coefficients. 6 It is also of fundamental importance to explore and understand the topological and electromagnetic properties of QGP. In the early stage of the nuclear collisions, a strong electromagnetic field exists and could induce an electric current along the direction of the strong magnetic field B for chirality imbalanced domains with a nonzero topological charge inside the hot chiral-symmetric QGP, which is known as Chiral Magnetic Effect (CME). [7][8][9] The search for the CME, in such a unique micro-universe environment created by relativistic nuclear collision experiments, has been pursued for more than a decade. The non-central heavy-ion collisions have large orbital angular momentum that could result in strong fluid shear and nonvanishing local fluid vorticity. 10,11 In such vorticity of the fluid cell, the spin-orbit coupling effect could lead to preferential orientation of particle spins along the direction of local fluid vorticity. 12,13,17 The first measurement of final state Λ hyperon polarization by STAR 14 sheds light on such vortical structure and its transport properties. Measurements of Ξ and Ω hyperons polarizations, 15 Λ-hyperon polarization at lower BES energies and FXT collisions, 16 and the theoretical model calculations 12,13,[17][18][19][20] are crucial for understanding the vorticity and polarization phenomena. In these proceedings, we present recent measurements of the flow, chirality, and vorticity measurements by the STAR experiment at RHIC, and discuss future opportunities. Flow and Fluctuations Anisotropic Flow in Small Systems The origin of a sizeable azimuthal anisotropy in small systems is still unknown, although the anisotropic flow for different harmonics and different particle species have been extensively measured via two-and multi-particle correlations at RHIC 21-23 and the LHC. [24][25][26] Some of the unsolved questions in understanding the behavior of small system collisions are 1) what determines the initial geometry ? 2) what is the connection between initial state and final state correlations? 3) what are the roles of nucleonic and sub-nucleonic fluctuations? Figure 1 shows v 2,3 (p T ) from template-fit method 29 and the comparisons with Sonic, 30 Supersonic, 31 and IP-Glasma+MUSIC+UrQMD 32 calculations. The measurements from STAR Collaboration show a hierarchy: v p+Au 2 < v d+Au 2 ∼ v 3 He+Au 2 and v p+Au 3 ∼ v d+Au 3 ∼ v 3 He+Au 3 . The Sonic calculations with initial geometry eccentricity from nucleon Glauber model predict the v 2 well but underpredict v 3 in p/d/ 3 He+Au collisions. After including the pre-equilibrium flow, the Supersonic calculations match the v n better. The calculations from the IP-Glasma+MUSIC+UrQMD model with sub-nucleonic fluctuations over-predict the v 2 , while it reproduces the v 3 in the three collision systems. are compared to published PHENIX measurements. 27 v 3 (p T ) shows a reasonable agreement in 3 He+Au collisions between STAR and PHENIX. However, within the statistical and systematic uncertainties, there is a factor 3 -4 discrepancy in p/d+Au collisions between STAR and PHENIX measurements. STAR results imply, the fluctuation-driven v 3 (p T ) is system-independent. The final result has been reported in Ref. 28 Future measurements including proper nonflow treatment, enhanced detector acceptance, and various other collisions, such as O+O, could provide additional constraints and insights on the origin of QGP fluidity in small system collisions. Flow Correlations with Mean Transverse Momentum The correlation between flow harmonics (v n ) and the mean transverse momentum ([p T ]), estimated by using a Pearson correlation coefficient ρ(v 2 n , [p T ]), is proposed to reveal interesting information both on the correlations in the initial state between the geometric size and the eccentricities. 34 In relativistic heavy-ion collisions, the shape and size of the QGP may depend on the fluctuations and the shape of the colliding nuclei where the spatial distribution of nucleons is often described by a Woods-Saxon density profile: ρ(r, θ) = ρ0 1+e [r−R0(1+β2Y2,0(θ)+β3Y3,0(θ))/a0] , where ρ 0 denotes the nucleon density at the center of the nucleus, R 0 = 1.2A 1/3 is the nuclear radius, and a 0 is the surface diffuseness parameter (known as skin depth). Y n,0 (θ) (n=2,3) are spherical harmonics. Therefore, ρ(v 2 n , [p T ]) is of particular interest is to distinguish the information of the initial geometry effect induced by the nuclear deformation. [35][36][37][38] In the hydrodynamic calculations, elliptic flow (v 2 ) emerges as a response to the initial eccentricity with v 2 = k 2 ϵ 2 . This leads to an enhanced fluctuations of the observed v 2 39, 40, 42 in collisions of deformed nuclei. As shown in the talk, 41 the expected anticorrelation between v 2 and normalized average transverse momentum ⟨p T ⟩ / ⟨⟨p T ⟩⟩ − 1 is observed in collisions of prolate 238 U nuclei. On the other hand, in collisions of oblate 197 Au, v 2 is observed to be essentially flat with only a slight increase of v 2 with ⟨p T ⟩ / ⟨⟨p T ⟩⟩ − 1 due to the increasing impact parameter. Note that TRENTo with initial state fluctuations can capture the trend for Au+Au collisions. The sign-change of Pearson correlation coefficient ρ(v 2 2 , [p T ]) in U+U collisions is observed towards central collisions, whereas the result from Au+Au collisions is positive throughout. ρ(v 2 3 , [p T ]), which is expected to be fluctuation driven, is almost identical between Au+Au and U+U collision systems across the whole centrality range. The comparison of ρ(v 2 2 , [p T ]) with state-of-the-art hydrodynamic calculations shows hierarchical trends, and suggests the most striking signature of nuclear deformation β 2 of 238 U to be around 0.3, observed for the first time in high-energy nuclear experiments so far. Moreover, to further constrain the initial conditions and transport properities in hydrodynamic evolution, the experiments at the LHC 44-46 and phenomenological studies 35,43,47,48 have also reported the studies of the ρ(v 2 n , [p T ]). In future, such calculations also could be conducted in the Ru+Ru and Zr+Zr collisions, since the final state effects are totally canceled. 49 Transverse Momentum Fluctuations In relativistic heavy-ion collisions, the event-by-event mean transverse momentum fluctuations are sensitive to overlap area and energy density fluctuations in the initial state. 53 Therefore, the shape of the nucleus could also be imaged and the size fluctuations could be used to isolate the β 2 dependence, especially in central and ultra-central collisions. The analytical estimation of shape and size are strongly correlated with nuclear deformation as illustrated in Ref. 54 So far, experimental measurements are limited to the mean transverse momentum and and its variance. [55][56][57][58][59] To save the computational overhead from loop-calculations, a framework for calculating the higher-order dynamical p T cumulants up to fourth-order using the standard and subevent methods is established and detailed in Ref. 60 As shown in the talk, 41 the normalized variance (δp T ) 2 / ⟨p T ⟩ approxi- mately follows a power-law dependence as a function of multiplicity owing to dynamical correlations on top of correlations arising from independent superposition picture. 61 The additional fluctuation induced by the nuclear deformation β 2 of 238 U collisions is observed as an enhancement in normalized vari- ance (δp T ) 2 / ⟨p T ⟩, normalized skewness (δp T ) 3 ⟨δp T ) 2 3/2 , and inten- sive skewness (δp T ) 3 * ⟨p T ⟩ / (δp T ) 2 2 . Interestingly, the normalized kurtosis (δp T ) 4 c / (δp T ) 2 2 in U+U collisions shows a clear and significant sign-change behavior in ultracentral regions. Remarkably, p T fluctuations from mean to kurtosis could be used as a complementary tool to probe nuclear structure in 238 U, 96 Ru, and 96 Zr with heavy-ion colliders in the future. 54 Longitudinal Flow Decorrelations Initial state fluctuations and final state dynamics of QGP are important properties in heavy-ion collisions. The distributions of particle production sources and the associated eccentricity, fluctuate along the pseudorapidity (η), which causes a non boost-invariant flow, known as flow decorrelations. [62][63][64][65] The flow decorrelations are usually quantified by the factorization ratio r n (η) = ⟨qn(−η)q * n (η ref )+qn(η)q * n (−η ref )⟩ ⟨qn(η)q * n (η ref )+qn(−η)q * n (−η ref )⟩ where the flow vector q n ≡ ω i e inϕi / ω i and ω i is the efficiency correction. [66][67][68] To improve the understanding of the longitudinal structure, a broad range of energy dependence of longitudinal flow decorrelations from the LHC to RHIC is crucial. In STAR, such an analysis was performed using the charged particles with 0. GeV Au+Au collisions, respectively. To investigate the energy dependence of flow decorrelations, a comparison between 27 and 200 GeV has been shown in Fig. 3 with a beam rapidity normalization. The r 2 shows slight energy dependence while r 3 shows clear energy dependence and a stronger decorrelation at 27 GeV after beam-rapidity normalization. In future, collision energy scan using high statistics BES-II data and system-size scan would help to better understand the logitudinal dynamics in heavy-ion collisions. Collectivity measurements at FXT Program To study the possible first-order phase transition and a QCD critical point, the BES-I and BES-II data-taking 69, 70 with adequate luminosity were achieved. Moreover, RHIC also pursued the FXT heavy-ion program 71, 72 at high baryon density region, by inserting a gold target into the beam pipe and circulating one beam, to broaden the reach of BES-II data-taking and allow the STAR to access energies below √ s N N =7.7 GeV. 74 show that partonic interaction is no longer dominant and baryonic scatterings take over at 3 GeV. Figure 5 reports the slope of dv 1 /dy| y=0 for baryons (left) and mesons (right) in 4.5 GeV Au+Au FXT mode. Currently, the results of proton and Λ directed flow are in agreement within the uncertainties. The proton v 1 agrees with the E895 4.3 GeV energy data within errors. Interestingly, the observed difference between π + and π − might be due to the isospin effect at lower energy or Coulomb dynamics. 73 Chiral Magnetic Effect Relativistic heavy-ion collisions can create the strongest electromagnetic field of eB ∼ 10 18 G in the universe. 75 An imbalance between the numbers of left-and right-handed (anti)quarks occurs due to the locally violated parity (P) and chargeparity (CP ) symmetries in such a strong B field. 76,77 A charge separation along the direction of the magnetic field, a novel CME phenomenon, has been extensively studied using STAR data. 78-80 Figure 6 left panel shows the cartoon for isobar collisions at RHIC: the stronger magnetic field in Ru+Ru collisions will result in a greater separation of charged particles in Zr+Zr collisions assuming the same background effects in these isobars. The most widely used observable in the CME search is the γ correlator, γ = ⟨cos (ϕ α + ϕ β − 2Ψ RP )⟩, where ϕ α and ϕ β are the azimuthal angles of particles of interest (POIs) and Ψ RP is the reaction plane. The magnitude (left axis) and significance (right axis) of the projected difference in γ correlator in isobar runs change accordingly as shown in Fig. 6 right panel when a different background level is assumed. It has been proposed to be able to determine the CME signal with 5 σ significance if 1.2 billion events for each collision system at 200 GeV are taken. The details of the observables for CME search can be found in Ref. 81 no CME signature that satisfies the predefined criteria has been observed in this blind isobar analysis by STAR. However, the future unblinded analysis with more comprehensive baselines, background estimations from the difference of nuclear structure, and further endeavors based on BES-II data are still ongoing. 15 The centrality dependence ofP Λ in Au+Au 3 GeV FXT mode comparing to the model calculations from Ref. 16 is shown in the right panel. Vorticity and Polarization Experimental measurements of the hyperon polarization and the theoretical calculations from hydrodynamics and transport models reveal that the QGP is a vortical fluid. [10][11][12][13][14][15][16][17][18][19][20] However, many questions are raised including the sign problems in differential measurements of local polarizations, uniform rapidity dependence but energy dependence in global polarization. Whether a significant difference between Λ andΛ global polarization exists, the underlying differences in various theoretical calculations and spin/thermal equilibration timescale are also interesting works. Therefore, the precise measurements based on the BES-II data and FXT mode are necessary. Figure 7 left panel shows the energy dependence of the hyperon global polarization measurements with the newly added Ξ and Ω Au+Au 200 GeV results. 15 The difference of two methods for Ξ polarization extractions is within 1 σ with given uncertainties. However, the averaged vaule of two Ξ polarization extractions (⟨P Ξ ⟩ (%) = 0.47 ± 0.10 (stat) ± 0.23 (syst)) is larger than those for Λ values (⟨P Λ ⟩ (%) = 0.24 ± 0.03 ± 0.03). The global polarization value of Ω was also measured to be ⟨P Ω ⟩ (%) = 1.11 ± 0.87( stat ) ± 1.97( syst ) for 20%−80% centrality. The larger hyperon polarization for more peripheral collisions indicates the increased vorticity of the system and is observed in data and are compared to the calculations from 3FD and AMPT. 16 Thanks to the efficient RHIC operation, we would take the bonus d+Au runs with 100 million minimumbias and 100 million central events as shown in Table. 1. In addition to critical point search, these data will enable STAR to explore, with unprecedented precision, numerous important physics. Briefly, some potential works on flow, chirality, and vorticity sides are as follows: Future Opportunities • The high statistics isobar Ru+Ru and Zr+Zr collisions could be used to perform a new and compelling experimental evidence of the nuclear structure including nuclear deformation and neutron skin thickness in relativistic nuclear collisions. A more profound understanding of the 96 Ru and 96 Zr nuclei allows us to gain the nuclear structure and its effect on the CME search, i.e. isobar baseline and background estimations. Theoretical model calculations [83][84][85][86][87][88][89][90][91][92][93][94][95] are necessary to understand above physics. Unlike Radioactive Ion Beam Line in Lanzhou (RIBLL), Facility for Rare Isotope Beams (FRIB), and Nuclotron-based Ion Collider fAcility (NICA) at low and medium energies, isobar collisions open up new opportunity to study nuclear structure at a very short time scale (∼ 10 −23 s) through heavy-ion collisions. • One fundamental property of light atomic nuclei in unusual nuclear structure regimes is the α cluster structure. 96,97 It is a good opportunity to directly provide experimental evidence using relativistic nuclear collisions for the first time. 98 Intuitively, the configuration of α nucleonic cluster could be deposited in the initial state, therefore such effect could be traced via final state harmonic flow. [99][100][101][102][103] In conjunction with the measurements of nuclear deformation and neutron skin thickness, the basic understanding of the nucleon topological structure could be achieved by investigating the α cluster in 16 O nuclei at STAR. • The interpretation of a fluid-like state in small collisions has been challenged due to the small collision size and short thermalization/evolution. 52 In understanding the early-time conditions of small systems, O+O runs would allow for a direct comparison with a similarly proposed higher-energy O+O run at the LHC. Whether the small system collectivity arises from the initial momentum correlation or from the final state interaction could be distinguished. 22, 28 • There is a disagreement of triangular flow v 3 between STAR and PHENIX 27 in the small system p/d/ 3 He+Au collisions. The origin of the difference has hitherto been not fully understood. More d+Au events with iTPC and EPD detectors could help to decipher this puzzle. • To study the effect of initial state momentum correlations in small collision systems, the correlator ρ v 2 2 , [p T ] has been proposed to be a key experimental measurement. 50,51 The d+Au and O+O collision data in 2021 run provide a potential chance to prove the presence and importance of the initial state momentum anisotropies predicted by an effective theory of QCD at RHIC energies. • 2 billion events at Au+Au 3 GeV FXT mode providing enhanced statistics enable the measurements of proton high-order moments/cumulants. Furthermore, at lower energies, the large baryon chemical potential allows to precisely measure ϕ meson flow, hypernuclei lifetime, and binding energy. 82 • The large data taking in BES-II program and FXT mode is intriguing to study the polarization and vorticity of the QGP. The energy and pseudorapidity dependence of the global polarization at lower energies below 7.7 GeV would be better understood. 82 • A good precision from the RHIC BES-II datasets with EPD detector providing a modern versatility for the CME search could be achieved at lower energies, where the electromagnetic field may still be larger and the flow/nonflow related background may be smaller. Summary Recently, the STAR experiment has reported important measurements: anisotropic flow in small systems, the nuclear structure probe based on flow correlations with mean transverse momentum (ρ(v 2 n , [p T ])) and mean transverse momentum fluctuations, the energy dependence of longitudinal decorrelations, collectivity measurements in FXT mode, CME search and vorticity/polarization measurements. Besides, based on the Run-21 efficient data-takings, future opportunities for precise measurements are also elaborated. Fig. 1 . 1Comparison of v 2,3 (p T ) values in the central p/d/ 3 He+Au collisions at √ s N N = 200 GeV with the calculations from Sonic, 30 Supersonic, 31 and IP-Glasma+MUSIC+UrQMD 32 calculations. Fig. 2 . 2Comparison of v 3 (p T ) measurements obtained by STAR and PHENIX in the central p/d/ 3 He+Au collisions at √ s N N = 200 GeV. The solid lines in the top panels represent a fit to the STAR data. The bottom panels show the ratio of the respective data to this fit. Figure 2 2presents the measurements of v 3 (p T ) from peripheral subtraction method 33 in the central p/d/ 3 He+Au collisions at √ s N N = 200 GeV at STAR, and 4 < p T < 4 GeV/c from the Time Projection Chamber (TPC, |η| < 1), and the reference flow vector is calculated from the Event Plane Detector (EPD, 2.1 < |η ref | < 5.1) and Forward Meson Sepctrometer (FMS, 2.5 < |η ref | < 4) for √ s N N = 27 and 200 Fig. 3 . 3The r 2 (η) (left panel) and r 3 (η) (right panel) as a function of η/y beam in 0-10% in Au+Au collisions at 27 and 200 GeV. A linear fit is in dashed line. Fig. 4 . 4v 2 measured by several experiments and STAR 4.5 GeV Au+Au FXT points for protons and pions are near the transition region. 73 v 2 scaled by the number of constituent quarks (v 2 /nq) as a function of the scaled transverse kinetic energy ((m T − m 0 ) /nq) for pions, kaons and protons from 3 GeV Au+Au FXT. 74 Left panel in Fig. 4 shows the measurements of the beam energy dependence of elliptic flow v 2 for all charged particles integrated over p T . The current results from 4.5 GeV Au+Au FXT are consistent with the trends established by the previously published data for various experiments. From squeeze-out to in-plane elliptic expansion, the v 2 changes sign around 3 -4 GeV collision energies. Such phenomenon has been observed in 3 GeV Au+Au FXT results where π, K and p are shown by the filled triangles, open triangles, and filled stars in the middle and right panels of Fig. 4 . 4The breakdown of number of constituent quark (NCQ) scaling indicates the disappearance of partonic collectivity in such low energy collisions. The detailed model comparisons in Ref. Fig. 5 . 5The directed flow slope of dv 1 /dy| y=0 at midrapidity for baryons (left panel) and mesons (right panel) are measured at 4.5 GeV Au+Au FXT comparing the STAR BES energies and AGS E895 experimental results. 74 The directed flow reflects the early time expansion, Equation of State (EOS), and the nature of phase transition. Fig. 6 . 6The isobar collisions at RHIC: the stronger magnetic field of Ru+Ru collisions resulting in greater separation of charged particles is expected than Zr+Zr collisions. Magnitude and significance of the relative difference in the projected γ correlator between Ru+Ru and Zr+Zr at 200 GeV.79 Fig. 7 . 7The energy dependence of the hyperon global polarization measurements with the newly added Ξ and Ω in 200 GeV Au+Au 200 results in the left panel from Ref. STAR has finished the scientific data taking for Run-21: 1) The highest priority is to complete the second phase of the BES-II program. 2) The second-highest priority is four short FXT runs with the detector upgrade of the iTPC and eTOF. 3) The third-highest priority is to collect data of O+O runs at √ s N N = 200 GeV, Au+Au runs at √ s N N = 17.1 GeV and 2 billion events at √ s N N = 3 GeV in FXT mode.82 Table 1 . 1STAR Run-21 efficient runs and data-taking. 82 Single-Beam Energy √ s NN Run Time Species Events Priority (GeV /nucleon ) (Rad/s) (Rad/s) 3.85 7.7 11-20 weeks Au+Au 100M 1 3.85 3(FXT) 3 days Au+Au 300M 2 44.5 9.2(FXT) 0.5 days Au+Au 50M 2 70 11.5(FXT) 0.5 days Au+Au 50M 2 100 13.7(FXT) 0.5 days Au+Au 50M 2 100 200 1 week O+O 400M + 200M(central) 3 8.35 17.1 2.5 weeks Au+Au 250M 3 3.85 3(FXT) 3 weeks Au+Au 2B 3 100 200 1 week d+Au 100M MB + 100M(central) 4 AcknowledgmentsAuthor acknowledges the STAR Collaboration, FCV PWG for the tremendous contributions, critical comments, and helpful suggestions. C. Zhang is supported by DOE Award No. DEFG0287ER40331. . R Paul, R Ulrike, Cambridge Monographs on Mathematical Physics. 5Cambridge University PressR. Paul and R. Ulrike, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 5 (2019). . S Schlichting, D Teaney, Annu. Rev. Nucl. Part. Sci. 69447S. Schlichting and D. Teaney, Annu. Rev. Nucl. Part. Sci. 69, 447 (2019). . U Heinz, R Snellings, Annu. Rev. Nucl. Part. Sci. 63123U. Heinz and R. Snellings, Annu. Rev. Nucl. Part. Sci. 63, 123 (2013). . F G Gardim, G Giacalone, M Luzum, J Y Ollitrault, Nature Phys. 16615F.G. Gardim, G. Giacalone, M. Luzum and J.Y. Ollitrault, Nature Phys. 16, 615 (2020). . H C Song, S A Bass, U Heinz, T Hirano, C Shen, Phys. Rev. Lett. 109139904H.C. Song, S.A. Bass, U. Heinz, T. Hirano and C. Shen, Phys. Rev. Lett. 109, 139904 (2012). . S Voloshin, Y Zhang, Z. Phys. C. 70665S. Voloshin and Y. Zhang, Z. Phys. C 70, 665 (1996). . D Kharzeev, Phys. Lett. B. 633260D. Kharzeev, Phys. Lett. B 633, 260 (2006). . D Kharzeev, A Krasnitz, R Venugopalan, Phys. Lett. B. 545298D. Kharzeev, A. Krasnitz and R. Venugopalan, Phys. Lett. B 545, 298 (2002). . S A Voloshin, Phys. Rev. C. 7057901S.A. Voloshin, Phys. Rev. C 70, 057901 (2004). . Z T Liang, X N Wang, Phys. Rev. Lett. 94102301Z.T. Liang and X.N. Wang, Phys. Rev. Lett. 94, 102301 (2005). . F Becattini, M A Lisa, Annu. Rev. Nucl. Part. Sci. 70395F. Becattini and M.A. Lisa, Annu. Rev. Nucl. Part. Sci. 70, 395 (2020). . F Becattini, Iu, Karpenko, Phys. Rev. Lett. 12012302F. Becattini and Iu. Karpenko, Phys. Rev. Lett. 120, 012302 (2018). . B C Fu, S Liu, L G Pang, H C Song, Y Yin, Phys. Rev. Lett. 127142301B.C. Fu, S. Liu, L.G. Pang, H.C. Song and Y. Yin, Phys. Rev. Lett. 127, 142301 (2021). . L Adamczyk, STAR CollaborationNature. 54862L. Adamczyk et al. (STAR Collaboration), Nature 548, 62 (2017). . J Adam, STAR CollaborationPhys. Rev. Lett. 126162301J. Adam et al. (STAR Collaboration), Phys. Rev. Lett. 126, 162301 (2021). . M S Abdallah, STAR CollaborationPhys. Rev. C. 10461901M.S. Abdallah et al. (STAR Collaboration), Phys. Rev. C. 104, L061901 (2021). . L G Pang, H Petersen, Q Wang, X N Wang, Phys. Rev. Lett. 117192301L.G. Pang, H. Petersen, Q. Wang and X.N. Wang, Phys. Rev. Lett. 117, 192301 (2016). . Y C Liu, X G Huang, Nucl. Sci. Tech. 3156Y.C. Liu and X.G. Huang, Nucl. Sci. Tech. 31, 56 (2020). . Y F Sun, C M Ko, Phys. Rev. C. 9624906Y.F. Sun and C.M. Ko, Phys. Rev. C 96, 024906 (2017). . X G Deng, X G Huang, Y G Ma, S Zhang, Phys. Rev. C. 10164908X.G. Deng, X.G. Huang, Y.G. Ma and S. Zhang, Phys. Rev. C 101, 064908 (2020). . L Adamczyk, STAR CollaborationPhys. Lett. B. 747265L. Adamczyk et al. (STAR Collaboration), Phys. Lett. B 747, 265 (2015). . A Adare, PHENIX CollaborationPhys. Rev. Lett. 111212301A. Adare et al. (PHENIX Collaboration), Phys. Rev. Lett. 111, 212301 (2013). . A Adare, PHENIX CollaborationPhys. Rev. Lett. 115142301A. Adare et al. (PHENIX Collaboration), Phys. Rev. Lett. 115, 142301 (2015). . S Chatrchyan, CMS CollaborationPhys. Lett. B. 718795S. Chatrchyan et al. (CMS Collaboration), Phys. Lett. B 718, 795 (2013). . S Chatrchyan, ALICE CollaborationPhys. Lett. B. 71929S. Chatrchyan et al. (ALICE Collaboration), Phys. Lett. B 719, 29 (2013). . G Aad, ATLAS CollaborationPhys. Rev. Lett. 110182302G. Aad et al. (ATLAS Collaboration), Phys. Rev. Lett. 110, 182302 (2013). . C Aidala, PHENIX CollaborationNature. Phys. 15214C. Aidala et al. (PHENIX Collaboration), Nature. Phys. 15, 214 (2019). . G Aad, ATLAS CollaborationPhys. Rev. Lett. 116172301G. Aad et al. (ATLAS Collaboration), Phys. Rev. Lett. 116, 172301 (2016). . J L Nagle, A Adare, S Beckman, T Koblesky, J O Koop, D Mcglinchey, P Romatschke, J Carlson, J E Lynn, M Mccumber, Phys. Rev. Lett. 113112301J.L. Nagle, A. Adare, S. Beckman, T. Koblesky, J.O. Koop, D. McGlinchey, P. Ro- matschke, J. Carlson, J.E. Lynn and M. McCumber, Phys. Rev. Lett. 113, 112301 (2014). . P Romatschke, arXiv:1502.04745P. Romatschke, arXiv: 1502.04745. . B Schenke, C Shen, P Tribedy, Phys. Lett. B. 803135322B. Schenke, C. Shen and P. Tribedy, Phys. Lett. B 803, 135322 (2020). . Roy A Lacey, STAR CollaborationNucl. Phys. A. 105122041Roy A. Lacey (for the STAR Collaboration), Nucl. Phys. A 105, 122041 (2016). . P Bozek, Phys. Rev. C. 9344908P. Bozek, Phys. Rev. C 93, 044908 (2016). . G Giacalone, G G Gardim, J N Hostler, J Ollitrault, Phys. Rev. C. 10324909G. Giacalone, G.G. Gardim, J.N. Hostler and J.Y Ollitrault, Phys. Rev. C 103, 024909 (2021). . J Y Jia, S L Huang, C J Zhang, Phys. Rev. C. 10514906J.Y. Jia, S.L. Huang and C.J. Zhang, Phys. Rev. C 105, 014906 (2022). . G Giacalone, B Schenke, C Shen, Phys. Rev. Lett. 12842301G. Giacalone, B. Schenke and C. Shen, Phys. Rev. Lett. 128, 042301 (2022). . B Bally, M Bender, G Giacalone, V Soma, Phys. Rev. Lett. 12882301B. Bally, M. Bender, G. Giacalone and V. Soma, Phys. Rev. Lett. 128, 082301 (2022). . G Giacalone, Phys. Rev. Lett. 124202301G. Giacalone, Phys. Rev. Lett. 124, 202301 (2020). . G Giacalone, J Y Jia, C J Zhang, Phys. Rev. Lett. 127242301G. Giacalone, J.Y. Jia and C.J. Zhang, Phys. Rev. Lett. 127, 242301 (2021). . C J Zhang, STAR CollaborationC.J. Zhang,(STAR Collaboration), https://indico.cern.ch/event/1025480/contributions /4457027/attachments/2300377/3912763/ICNFP2021 ChunjianZhang STAR v6.pdf . Q Y Shou, Y G Ma, P Sorensen, A H Tang, F Videbaek, H Wang, Phys. Lett. B. 749215Q.Y. Shou, Y.G. Ma, P. Sorensen, A.H. Tang, F. Videbaek and H. Wang, Phys. Lett. B 749, 215 (2015). . B Schenke, C Shen, D Teaney, Phys. Rev. C. 10234905B. Schenke, C. Shen and D. Teaney, Phys. Rev. C 102, 034905 (2020). . G Aad, ATLAS CollaborationEur. Phys. J. C. 79985G. Aad et al. (ATLAS Collaboration), Eur. Phys. J. C 79, 985 (2019). . S Acharya, ALICE CollaborationPhys. Lett. B. 834137393S. Acharya et al. (ALICE Collaboration), Phys. Lett. B 834 137393 (2022). B Somadutta, ATLAS CollaborationPos EPS-HEP2021. 398305B. Somadutta (for the ATLAS Collaboration), Pos EPS-HEP2021 398, 0305 (2022). . C J Zhang, A Behera, S Bhatta, J Y Jia, Phys. Lett. B. 822136702C.J. Zhang, A. Behera, S. Bhatta and J.Y. Jia, Phys. Lett. B 822, 136702 (2020). . N Magdy An, R A Lacey, Phys. Lett. B. 821136625N. Magdy an R.A. Lacey, Phys. Lett. B 821, 136625 (2021). . C J Zhang, S Bhatta, J Y Jia, Phys. Rev. C. 10631901C.J. Zhang, S. Bhatta and J.Y. Jia, Phys. Rev. C 106, L031901 (2022). . G Giacalone, B Schenke, C Shen, Phys. Rev. Lett. 125192301G. Giacalone, B. Schenke and C. Shen, Phys. Rev. Lett. 125, 192301 (2020). . B Schenke, Rep. Prog. Phys. 8482301B. Schenke, Rep. Prog. Phys. 84, 082301 (2021). . S L Huang, Z Y Chen, W Li, J Y Jia, Phys. Rev. C. 10121901S.L. Huang, Z.Y. Chen, W. Li, and J.Y. Jia, Phys. Rev. C 101, 021901 (2020). . G Giacalone, F G Gardim, J N Hostler, J Y Ollitrault, Phys. Rev. C. 10324910G. Giacalone, F.G. Gardim, J.N. Hostler, J.Y. Ollitrault, Phys. Rev. C 103, 024910 (2021). . J Y Jia, Phys. Rev. C. 10544905J.Y. Jia, Phys. Rev. C 105, 044905 (2022). . J Adam, STAR CollaborationPhys. Rev. C. 7244902J. Adam et al. (STAR Collaboration), Phys. Rev. C 72, 044902 (2005). . B B Abelev, ALICE CollaborationPhys. Lett. B. 727371B. B. Abelev et al. (ALICE Collaboration), Phys. Lett. B 727, 371 (2013). . B B Abelev, ALICE CollaborationEur. Phys. J. C. 743077B. B. Abelev et al. (ALICE Collaboration), Eur. Phys. J. C 74, 3077 (2014). . L Adamczyk, STAR CollaborationPhys. Rev. C. 9644904L. Adamczyk et al. (STAR Collaboration), Phys. Rev. C 96, 044904 (2017). . J Adam, STAR CollaborationPhys. Rev. C. 9944918J. Adam et al. (STAR Collaboration), Phys. Rev. C 99, 044918 (2019). . S Bhatta, C J Zhang, J Y Jia, Phys. Rev. C. 10524904S. Bhatta, C.J. Zhang and J.Y. Jia, Phys. Rev. C 105, 024904 (2022). . X N Wang, M Gyulassy, Phys. Rev. D. 443501X.N. Wang and M. Gyulassy, Phys. Rev. D 44, 3501 (1991). . P Bozek, W Broniowski, J Moreira, Phys. Rev. C. 8334911P. Bozek, W. Broniowski and J. Moreira, Phys. Rev. C 83, 034911 (2011). . V Khachatryan, CMS CollaborationPhys. Rev. C. 9234911V. Khachatryan et al. (CMS Collaboration), Phys. Rev. C 92, 034911 (2015). . G Aad, ATLAS CollaborationPhys. Rev. Lett. 126122301G. Aad et al. (ATLAS Collaboration), Phys. Rev. Lett. 126, 122301 (2021). . L G Pang, H Petersen, G Y Qin, V Roy, X N Wang, Eur. Phys. J. A. 5297L.G. Pang, H. Petersen, G.Y. Qin, V. Roy and X.N. Wang, Eur. Phys. J. A 52, 97 (2016). . M W Nie, STAR CollaborationNucl. Phys. A. 1005121783M.W. Nie (for the STAR Collaboration), Nucl. Phys. A 1005, 121783 (2020). . J Jia, P Huo, Phys. Rev. C. 9034915J. Jia and P.Huo, Phys. Rev. C 90 034915 (2014). . J Jia, P Huo, G Ma, M Nie, J. Phys. G. 4475106J. Jia, P. Huo, G. Ma and M. Nie, J. Phys. G 44, 075106 (2017). . M M Aggarwal, STAR CollaborationarXiv:1007.2613M.M. Aggarwal et al. (STAR Collaboration), arXiv: 1007.2613. . J Adam, STAR CollaborationPhys. Rev. Lett. 12692301J. Adam et al. (STAR Collaboration), Phys. Rev. Lett. 126 092301 (2021). . K Meehan, STAR CollaborationNucl. Phys. A. 967808K. Meehan (for the STAR Collaboration), Nucl. Phys. A 967, 808 (2017). . J Adam, STAR CollaborationPhys. Rev. C. 10334908J. Adam et al. (STAR Collaboration), Phys. Rev. C 103, 034908 (2021). . M S Abdallah, STAR CollaborationPhys. Lett. B. 827137003M.S. Abdallah et al. (STAR Collaboration), Phys. Lett. B 827, 137003 (2022). . D E Kharzeev, Prog. in Part. and Nucl. 75133D.E. Kharzeev, Prog. in Part. and Nucl. 75, 133 (2014). . D E Kharzeev, L D Mclerran, H J Warringa, Nucl. Phys. A. 803227D.E. Kharzeev, L.D. Mclerran, H.J. Warringa, Nucl. Phys. A 803, 227 (2008). . D E Kharzeev, J F Liao, S A Voloshin, G Wang, Prog. in Part. and Nucl. 881D.E. Kharzeev, J.F. Liao, S.A. Voloshin and G. Wang, Prog. in Part. and Nucl. 88, 1 (2016). . B I Abelev, STAR CollaborationPhys. Rev. Lett. 103251601B.I. Abelev et al. (STAR Collaboration), Phys. Rev. Lett. 103, 251601 (2009). RHIC Beam Use Request For Runs. 17STAR Collaboration, RHIC Beam Use Request For Runs 17 and 18, (2016). . S Choudhury, Chinese Phys. C. 4614101S. Choudhury et al., Chinese Phys. C 46, 014101 (2022). . M S Abdallah, STAR CollaborationPhys. Rev. C. 10514901M.S. Abdallah et al. (STAR Collaboration), Phys. Rev. C 105, 014901 (2022). The STAR Beam Use Request for Run-21, Run-22 and data taking in 2023-25. STAR Collaboration, The STAR Beam Use Request for Run-21, Run-22 and data taking in 2023-25, (2020). . C J Zhang, J Y Jia, Phys. Rev. Lett. 12822301C.J. Zhang and J.Y. Jia, Phys. Rev. Lett. 128, 022301 (2022). . G Giacalone, J Y Jia, V Soma, Phys. Rev. C. 10441903G. Giacalone, J.Y. Jia and V. Soma, Phys. Rev. C 104, L041903 (2021). . J Y Jia, Phys. Rev. C. 10514905J.Y. Jia, Phys. Rev. C 105, 014905 (2021). . J Y Jia, C J Zhang, arXiv:2111.15559J.Y. Jia and C.J. Zhang, arXiv: 2111.15559. . H L Li, H J Xu, Y Zhou, X B Wang, J Zhao, L W Chen, F Wang, Phys. Rev. Lett. 125222301H.L. Li, H.J. Xu, Y. Zhou, X.B. Wang, J. Zhao, L.W. Chen and F.Q Wang, Phys. Rev. Lett. 125, 222301 (2020). . H J Xu, H L Li, X B Wang, C W Shen, F Q Wang, Phys. Lett. B. 8191136453H.J. Xu, H.L. Li, X.B. Wang, C.W. Shen, F.Q. Wang, Phys. Lett. B 819, 1136453 (2021). . H J Xu, X B Wang, H L Li, J Zhao, Z W Lin, C W Shen, F Q Wang, Phys. Rev. Lett. 12122301H.J. Xu, X.B. Wang, H.L. Li, J. Zhao, Z.W. Lin, C.W. Shen, F.Q. Wang, Phys. Rev. Lett. 121, 022301 (2018). . H J Xu, W B Zhao, H L Li, Y Zhou, L Chen, F Q Wang, arXiv:2111.14812H.J. Xu, W.B. Zhao, H.L. Li, Y. Zhou, L.W Chen, F.Q. Wang, arXiv: 2111.14812. . G Nijs, W Schee, arXiv:2112.13771G. Nijs and W. Schee, arXiv: 2112.13771. . F Li, Y G Ma, S Zhang, G L Ma, Q Y Shou, Phys. Rev. C. 10614906F. Li, Y.G. Ma, S. Zhang, G.L. Ma and Q.Y. Shou, Phys. Rev. C 106 014906 (2022). . E Shuryak, arXiv:2201.11064E. Shuryak, arXiv: 2201.11064. Flow and Spin polarization at RHIC isobar run (RBRC workshop 2022). H C Song, H.C. Song, Flow and Spin polarization at RHIC isobar run (RBRC workshop 2022). . L M Liu, C J Zhang, J Zhou, J Xu, J Y Jia, G X Peng, Phys. Lett. B. 834137441L.M. Liu, C.J. Zhang, J. Zhou, J. Xu, J.Y. Jia and G.X. Peng, Phys. Lett. B 834 137441 (2022). . J P Ebran, E Khan, T Niksic, D Vretenar, Nature. 487341J.P. Ebran, E. Khan, T. Niksic and D. Vretenar, Nature 487, 341 (2012). . W B He, Y G Ma, X G Cao, X Z Cai, G Q Zhang, Phys. Rev. Lett. 11332506W.B. He, Y.G. Ma, X.G. Cao, X.Z. Cai and G.Q. Zhang, Phys. Rev. Lett. 113, 032506 (2014). . W Broniowski, E R Arriola, Phys. Rev. Lett. 112112501W. Broniowski and E.R. Arriola, Phys. Rev. Lett. 112, 0112501 (2014). . P Bozek, W Broniowski, E R Arrola, M Rybczynski, Phys. Rev. C. 9064902P. Bozek, W. Broniowski, E.R. Arrola and M. Rybczynski, Phys. Rev. C 90, 064902 (2014). . S Zhang, Y G Ma, J H Chen, W B He, C Zhong, Phys. Rev. C. 9564904S. Zhang, Y.G. Ma, J.H. Chen, W.B. He and C. Zhong, Phys. Rev. C 95, 064904 (2017). . M Rybczynski, M Piotrowska, W Broniowski, Phys. Rev. C. 9734912M. Rybczynski, M. Piotrowska and W. Broniowski, Phys. Rev. C 97, 034912 (2018). . Y A Li, S Zhang, Y G Ma, Phys. Rev. C. 10254907Y.A. Li, S. Zhang and Y.G. Ma, Phys. Rev. C 102, 054907 (2020). . N Summerfield, B N Lu, C Plumberg, D Lee, J N Hostler, A Timmins, Phys. Rev. C. 10441901N. Summerfield, B.N. Lu, C. Plumberg, D. Lee, J.N. Hostler and A. Timmins, Phys. Rev. C 104, L041901 (2021).
{'fraction_non_alphanumeric': 0.07939189189189189, 'fraction_numerical': 0.07089232089232089, 'mean_word_length': 3.716959656001012, 'pattern_counts': {'":': 0, '<': 8, '<?xml version=': 0, '>': 0, 'https://': 1, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': "In relativistic heavy-ion collisions, the properties of quark-gluon plasma (QGP) and complex dynamics of multi-scale processes in Quantum Chromodynamics (QCD) are studied by analyzing the final state produced particles in a variety of different ways. In these proceedings, we present an overview of new detailed measurements of flow, chirality, and vorticity by the STAR experiment at RHIC. Furthermore, STAR's future opportunities for the precision measurements on small systems, fixed-target (FXT) mode, and Beam Energy Scan (BES-II) program are discussed.", 'arxivid': '2203.13106', 'author': ['Chunjian Zhang [email protected] \n(for the STAR Collaboration) Chemistry Department\nStony Brook University\n11794Stony BrookNew YorkUSA\n'], 'authoraffiliation': ['(for the STAR Collaboration) Chemistry Department\nStony Brook University\n11794Stony BrookNew YorkUSA'], 'corpusid': 247627881, 'doi': '10.1142/s021830132341001x', 'github_urls': [], 'n_tokens_mistral': 14295, 'n_tokens_neox': 11428, 'n_words': 6041, 'pdfsha': '7be36f8071bdb058cc03c69c3a326da3b59b0d12', 'pdfurls': ['https://export.arxiv.org/pdf/2203.13106v2.pdf'], 'title': ['An overview of new measurements of flow, chirality, and vorticity from STAR experiment', 'An overview of new measurements of flow, chirality, and vorticity from STAR experiment'], 'venue': ['International Journal of Modern Physics E']}
arxiv
Spontaneous emission from a two-level atom in anisotropic one-band photonic crystals: a fractional calculus approach 3 Jun 2009 Jing-Nuo Wu Department of Photonics and Institute of Electro-Optical Engineering National Chiao Tung University HsinchuTaiwan, R. O. C Chih-Hsien Huang Department of Photonics and Institute of Electro-Optical Engineering National Chiao Tung University HsinchuTaiwan, R. O. C Szu-Cheng Cheng Department of Physics Chinese Culture University TaipeiTaiwan, R. O. C. Wen-Feng Hsieh Department of Photonics and Institute of Electro-Optical Engineering National Chiao Tung University HsinchuTaiwan, R. O. C Department of Electro-Optical Engineering National Cheng Kung University TainanTaiwan, R. O. C Spontaneous emission from a two-level atom in anisotropic one-band photonic crystals: a fractional calculus approach 3 Jun 2009(Received June 3, 2009)numbers: 0540-a4250-p3280-t * Electronic address: sccheng@faculty Spontaneous emission (SE) from a two-level atom in a photonic crystal (PC) with anisotropic oneband model is investigated using the fractional calculus. Analytically solving the kinetic equation in terms of the fractional exponential function, the dynamical discrepancy of SE between the anisotropic and isotropic systems is discussed on the basis of different photon density of states (DOS) and the existence of incoherent diffusion field that becomes even more clearly as the atomic transition frequency lies close to the band edge. With the same atom-field coupling strength and detuning in the forbidden gap, the photon-atom bound states in the isotropic system turn into the unbound ones in the anisotropic system that is consistent with the experimental observation in P hys. Rev. Lett. 96, 243902 (2006). Dynamics along different wavevectors with various curvatures of dispersion is also addressed with the changes of the photon DOS and the appearance of the diffusion fields. I. INTRODUCTION Photonic crystals (PCs), a new class of optical materials with periodic dielectric structure, provide a way to control spontaneous emission (SE) through redistributing the photon density of states (DOS) near photonic band gap (PBG). This control offers the key technology of manipulating light, such as in light emitting devices [1], quantum information processing [2], or solar cells [3]. The special photon density of states near band edge changes the optical behavior of an atom in a photonic crystal including the appearance of photon-atom bound states [4,5,6,7,8], spectral splitting [9], enhanced quantum interference effects [10], coherent control of SE [11], non-Markovian effects [4,5,6,12], etc. Among these studies, the photon reservoir of a photonic crystal could be well described by one band edge frequency ω c at a certain edge wavevector and a dispersion relation which leads to the special photon DOS. The dispersion relation near the band edge is assumed to be isotropic in early studies [4,5,6,7,9,12]. In a real three-dimensional photonic crystal with an allowed point-group symmetry, the photonic band structure is highly anisotropic, namely, the equal frequency surface near the band edge is no longer spherical. A vector form of photon dispersion relation is required to describe a more realistic picture of the band edge behavior. In this vector form of dispersion relation, the photon DOS is proportional to √ ω − ω c while that of an isotropic dispersion relation is proportional to 1/ √ ω − ω c , where ω stands for the eigenmode frequency and ω c for the band edge frequency [13]. This discrepancy of DOS will cause dynamical difference of spontaneous emission between the anisotropic and isotropic systems. The most prominent difference is the existence of diffusion field [14,15,16,17]. It was predicted theoretically by Yang et al. [14,15,16,17] that the bare atomic transition frequency lying in the region near band edge will be shifted into the forbidden gap by the interaction with the radiation modes where an atom-photon bound state is generated. That is, the emitter with frequency lying in the forbidden gap will not yield SE. Recently, Barth et al. observed experimentally [18] that the anisotropic properties of a PC could be detected by employing single emitters such as quantum dots (QDs). When an emitter is placed inside a PC with the anisotropic band structure, the additional anisotropy will imprint on the spontaneous emission of the system if the emission frequency lies in the forbidden gap. In this observation, the CdSe/ZnS quantum dots embedded inside artificial colloidal opals with direction-dependent band structure gave fluorescence image (SE) with an extra anisotropy but did not emit light if embedded inside a PC with weak anisotropy of band structure for the emission frequency of the QDs lying in the forbidden gap. There exists inconsistency between the theoretical prediction and experimental observation because SE appears in the anisotropic PC system even for the emitter's frequency lying in the forbidden gap. These inconsistency arouses our attention for the physical properties of the anisotropic PC system. We found that the presence of SE from the QDs in the anisotropic PC results from the dynamical difference of SE from the isotropic (or weak anisotropic) PC. In this paper, we study the dynamics of SE from a two-level atom embedded in a PC with anisotropic one-band model (Fig. 1). Fractional calculus is applied to solve the non-Markovian dynamics of the anisotropic optical system with threshold-like DOS. We found that the dynamical discrepancy of SE between the anisotropic and isotropic systems is the consequence of the different DOS in these two systems and the existence of the diffusion field in the anisotropic system. With the same atom-field coupling strength and detuning in the forbidden gap, the photon-atom bound states near the band edge of the isotropic reservoir turn into the decaying states in anisotropic system. This alteration leads to the presence of SE in the anisotropic PC system which is consistent with the experimental observation by Barth et al. [18]. The new topic of how the curvature of the anisotropic dispersion relation affects the dynamical behavior of the system has also been investigated. The dynamical difference between the systems with various curvatures of dispersion is observed and discussed on the basis of the curvature-dependent DOS and coupling strength. The paper is organized as follows. In Sec. II, the basic theory of the anisotropic system is given. In Sec. III, we solve the kinetic equation of the anisotropic system through fractional calculus and express the analytical solution in terms of the fractional exponential function. The dynamical behavior of the anisotropic system is compared with that of the isotropic system. And the influence of curvature of the anisotropic dispersion relation on the dynamical behavior is also discussed here. Finally, we summarize our results in Sec. IV. II. BASIC THEORY OF A TWO-LEVEL ATOM IN AN ANISOTROPIC ONE- BAND PHOTONIC CRYSTAL When the system of a two-level atom coupled to the field reservoir in a photonic crystal with anisotropic one-band model is considered, the Hamiltonian for this atom-field interact-ing system can be written as H = ω 21 σ 22 + k ω k a + k a k + i k g k (a + k σ 12 − σ 21 a k ),(1) where σ ij = |i j| (i,j =1,2) are the atomic operators for a two-level atom with excited state |2 , ground state |1 , and resonant transition frequency ω 21 ; a k and a + k are the annihilation and creation operators of the radiation field; ω k is the radiation frequency of mode k in the reservoir, and the atom-field coupling constant g k = ω 21 d 21 [ 2ǫ 0 ω k V ] 1 2ê k ·û d is assumed to be independent of atomic position with the fixed atomic dipole moment d 21 = d 21ûd . Here V is the sample volume,ê k is the polarization unit vector of the reservoir mode k, and ǫ 0 is the Coulomb constant. In the single photon sector, the wave function of the system has the form |ψ(t) = B(t)e −iω 21 t |2, {0} + k C k (t)e −iω k t |1, {1 k }(2) with initial condition B(0) = 1 and C k (0) = 0. Here B(t) labels the probability amplitude for the atom in its excited state |2 with an electromagnetic vacuum state and C k (t) for the atom in its ground state |1 with a single photon in mode k with frequency ω k . We got the equations of motion for the amplitudes by projecting the time-dependent Schrödinger equation on the one-photon sector of the Hilbert space as d dt B(t) = − k g k C k (t)e −iΩ k t (3) d dt C k (t) = g k B(t)e iΩ k t(4) with detuning frequency Ω k = ω k − ω 21 . By substituting the time integration of equation (4) into equation (3), we have the time evolving equation of the excited-state probability amplitude d dt B(t) = − t 0 G(t − τ )B(τ )dτ (5) with the memory kernel G(t − τ ) = k g 2 k e −iΩ k (t−τ ) . This memory kernel is related to the dispersion relation of photon reservoir. For the anisotropic photonic band gap (PBG) reservoir, the dispersion relation has a vector form and could be expressed by the effectivemass approximation as [19] ω k ≈ ω c + A k − k c 2(6) where A ∼ = f ω c /k 2 c signifies different curvatures in different directions with scaling factor f whose value depends on the nature of the dispersion relation near the band edge ω c . The related memory kernel of the anisotropic system could be further expressed as G(t − τ ) = ω 2 21 d 2 21 12π 2 ǫ 0 ω c ρ(ω)e −i(ω−ω 21 )(t−τ ) dω(7) with the curvature-dependent density of states [4,5] ρ(ω) = ω − ω c A 3 Θ(ω − ω c ),(8) where Θ(ω − ω c ) is the Heaviside step function. This density of states has threshold-like behavior near band edge ω c which results in the different dynamical behavior of this atomfield interacting system from that of free space. By applying the complex Fresnel integral [20], we could integrate this memory kernel as ∞ 0 x p−1 e −ax dx = Γ(p)/a pG(t − τ ) = β 1/2 /f 3/2 √ π(t − τ ) 3/2 e −i[3π/4−∆c(t−τ )](9) with the detuning frequency ∆ c = ω 21 − ω c of the atomic resonance frequency ω 21 from the band edge ω c , coupling constant β 1/2 = (ω 2 21 d 2 21 ω 1/2 c )/(24πǫ 0 c 3 ) , and t > τ in the long time limit [21]. Substituting this memory kernel into the time evolving equation (5) and making the transformation B(t) = e i∆ct D(t), we have the kinetic equation of this anisotropic system as d dt D(t) + i∆ c D(t) = β 1/2 e iπ/4 √ πf 3/2 t 0 D(τ ) (t − τ ) 3/2 dτ.(10) III. DYNAMICS OF SPONTANEOUS EMISSION In this section, we will use the mathematical method of fractional calculus to solve the kinetic equation of the anisotropic system and discuss the dynamics of SE on the basis of the obtained analytical solution. When the right-hand-side term of the kinetic equation (10) That is, t 0 D(τ ) (t − τ ) 3/2 dτ = Γ(−1/2) d 1/2 dt 1/2 D(t)(11) with Gamma function Γ(x). The kinetic equation thus has a fractional differential form as d dt D(t) + i∆ c D(t) + 2β 1/2 e iπ/4 f 3/2 d 1/2 dt 1/2 D(t) = 0.(12) In order to solve this fractional kinetic equation, we manipulated the fractional operators including the integral operator (d −1 /dt −1 ) and the fractional differentiation operator d 1/2 /dt 1/2 . Mathematically, the adopted manipulation is not unique provided that one could justify the function arrived at is the solution of the original fractional differential equation. The first step of our manipulation yielded D(t) − D(0) + i∆ c d −1 dt −1 D(t) + 2β 1/2 e iπ/4 f 3/2 d −1/2 dt −1/2 D(t) = 0.(13) The second step gave d 1/2 dt 1/2 D(t) + i∆ c d −1/2 dt −1/2 D(t) + 2β 1/2 e iπ/4 f 3/2 D(t) = t −1/2 √ π .(14) Here the initial condition D(0) = B(0) = 1 has been applied. The probability amplitude D(t) could be solved by performing the Laplace transform of the fractional derivative in- cluding [24] L d 1/2 dt 1/2 D(t) = s 1/2D (s) − d −1/2 dt −1/2 D(0) t=0 ,(15)L d −1/2 dt −1/2 D(t) = s 1/2D (s),(16) and L[t −1/2 ] = Γ(1/2) √ s ,(17) whereD(s) is the Laplace transform of D(t). These procedures gave the Laplace transform of D(t) asD (s) = 1 s + i∆ c + 2β 1/2 e iπ/4 s 1/2 /f 3/2 . In order to express this equation as a sum of partial fractions, we need to find the roots of the indicial equation Y 2 + 2β 1/2 e iπ/4 Y /f 3/2 + i∆ c = 0,(s) = 1 ( √ s − Y 1 ) − 1 ( √ s − Y 2 ) 1 (Y 1 − Y 2 )(19) with Y 1 = e iπ/4 − β 1/2 f 3/2 + β f 3 − ∆ c(20) and Y 2 = e iπ/4 − β 1/2 f 3/2 − β f 3 − ∆ c .(21) For the degenerate case, we have β f 3 = ∆ c . The indicial equation becomes Y 2 + 2 β 1/2 e iπ/4 f 3/2 Y + βe iπ/2 f 3 = 0 (22) or Y + β 1/2 e iπ/4 f 3/2 2 = 0.(23) The partial fractions ofD(s) can thus be written as D(s) = 1 √ s + β 1/2 e iπ/4 f 3/2 2 .(24) The dynamical solution of the probability amplitude D(t) could be obtained by applying the inverse Laplace transform on the partial-fractional forms ofD(s) for the two cases of different roots and degenerate root. The applied formula of the inverse Laplace transform include L −1 1 ( √ s − a) = E t − 1 2 , a 2 + aE t 0, a 2(25) and L −1 1 ( √ s − a) 2 = 2atE t − 1 2 , a 2 + 1 + 2a 2 t E t 0, a 2 + aE t 1 2 , a 2 (26) with E t (α, a) = t α ∞ n=0 (at) n Γ(α+n+1) = d −α dt −D(t) = 1 2e iπ/4 β/f 3 − ∆ c × Y 2 1 E t (1/2, Y 2 1 ) − Y 2 2 E t (1/2, Y 2 2 ) + Y 1 e Y 2 1 t − Y 2 e Y 2 2 t .(27) For β/f 3 = ∆ c , D(t) = −2 β 3/2 e i3π/4 f 9/2 tE t ( 1 2 , iβ/f 3 ) − β 1/2 e iπ/4 f 3/2 E t ( 1 2 , iβ/f 3 ) +(1 + 2itβ/f 3 )e iβt/f 3 − 2 β 1/2 e iπ/4 f 3/2 t 1/2 / √ π.(28) Here we have applied the relation of the fractional exponential function for special values E t (−1/2, a) = aE t (1/2, a) + t −1/2 / √ π and E t (0, a) = e at . This analytical solution, which has so far not been obtained, determines the dynamical behavior of the atomic excitation B(t) and the amplitude of the radiation field which could be obtained via B(t) in a standard way [16,25]. Different kinds of the indicial roots Y 1 and Y 2 gave different dynamical behavior of the system which depend on the lying regions of the atomic transition frequency ( see Eq. (20) and (21)). As the atomic frequency lies in the region ω 21 < ω c + β/f 3 (∆ c < β/f 3 ), the square of these roots is pure imaginary and the dynamical solution has some non-decaying terms. Near band edge (ω 21 ∼ = ω c , Y 2 1 ∼ = Y 2 2 ), these non-decaying terms interfere each other severely which leads to the decaying solution. Instead of analyzing the integration contours for the probability amplitudes in previous solving procedures, we plot the dynamical behavior of this anisotropic system directly from the analytical solution. Based on the excited-state probability amplitude P (t) = |B(t)| 2 = |D(t)| 2 , the dynamical behavior of the anisotropic system is shown in Fig. 2. It could be observed from Fig. 2 that typical characteristic of non-Markovian dynamics including nonexponential decay and atom-photon bound states exists in the system which results from the special (threshold-like) density of states (see Eq. (8)). When the atomic transition frequency lies in the bandgap (∆ c < 0), the system exhibits photon-atom bound states and decaying states in the allowed band (∆ c > 0). The dynamical behavior of the anisotropic system is almost the same as that of the isotropic system [8] except for the smaller probability of bound states and faster decaying behavior of decaying states. The dynamical difference of spontaneous emission in the two systems results from the different DOS in the two systems and the existence of diffusion field in the anisotropic system. As the atomic transition frequency moves from the bandgap to the allowed band, the density of states "seen" by the emitted photon is singularly large near band edge in the isotropic system and small in the anisotropic case. The singularity of density of states in isotropic system leads to the appearance of coherent propagating field while not large enough density of states results in the coexistence of incoherent diffusion field and coherent propagating field in anisotropic system. The energy transfer from the localized field to the diffusion field for the bound states of the anisotropic system leads to the smaller probability in excited level and coexisting energy of diffusion field and propagating field for decaying states results in faster decaying of the excited population. This dynamical difference in the isotropic and anisotropic systems is more obvious as the atomic transition frequency lying close to the edge of the PBG reservoir which is shown in Fig. 3. As the atomic transition frequency lying close to the band edge, states in the isotropic system exhibit bound (∆ c < 0) or slow decaying (∆ c > 0) behavior. In the anisotropic system, however, almost all of these states display fast decaying behavior. That were embedded inside a PC with weak anisotropy of band structure. That is, spontaneous emission appears only in the anisotropic photonic crystal as the emission frequency of the embedded quantum dots lies in the forbidden gap, but they did not emit light with bound states in the weak anisotropic PC system. The correctness of our results is validated by the presence of spontaneous emission in the anisotropic photonic crystal which differs from the prediction of the previous studies [14,15,16]. Dynamical difference of the isotropic and anisotropic PC systems leads to the appearance of fluorescence image in the anisotropic system. Close to the band edge, DOS in the anisotropic case is nearly zero so that the incoherent diffusion field emerges to release some of the radiation energy. This diffusion field increases as the atomic frequency shifts from the bandgap to the band edge and decreases as the frequency shifts to the allowed band. The transferred radiation energy from localized field and propagating field reaches maximum at the band edge. The phenomenon of the dynamical behavior of the two systems arriving at the great difference close to the band edge illustrates the existence of the diffusion field in the anisotropic system. In Fig. 4, we show how the curvature of the dispersion relation affects the dynamical behavior of the anisotropic system which has so far not been explored. The solid lines are plotted for the system with the larger curvature (f = 1) of dispersion relation and dotted lines for the system with the smaller curvature (f = 0.8). For the bound states (∆ c < 0), the excited-state probability P (t) of the system with the smaller curvature has the smaller value than that of the system with the larger curvature. On the other hand, the excited-state probability of the system with smaller curvature decays faster than that of the larger-curvature system for the decaying states (∆ c > 0). Without changing the units of energy (coupling constant β) and time (1/β), the dispersion relation of the smaller curvature has the larger DOS and coupling strength. As the atomic transition frequency moves from the bandgap to the allowed band, this larger DOS leads to the earlier appearance of the diffusion field that corresponds to the earlier energy transfer from the localized field to the diffusion field for the bound states and earlier coexisting energy of the diffusion field and the propagating field for the decaying states. This earlier energy transfer and coexistence resulting in the smaller value of probability in the bound states and fast decaying in the decaying states cause different optical behavior of the anisotropic system along different wavevectors. IV. CONCLUSION We have used fractional calculus to solve the non-Markovian dynamics of the optical system consisting of a two-level atom coupled to a PBG reservoir with anisotropic one-band model. It is the first time in the anisotropic system that the analytical solution of the kinetic equation is obtained in terms of fractional exponential functions. The dynamical spontaneous emission has almost the same behavior as that of the isotropic system except for the smaller probability of the photon-atom bound states (∆ c < 0) and the faster decaying rates for the decaying states (∆ c > 0). The dynamical difference originates from the different DOS in the two systems and the existence of diffusion field in the anisotropic system. Not large enough DOS near band edge in the anisotropic system leads to the appearance of the incoherent diffusion field and energy transfer from the localized field to the diffusion field. The dynamical difference between the anisotropic and isotropic systems manifests itself more clearly as the atomic transition frequency lies close to the edge of the PBG reservoir. With the same atom-field coupling strength and detuning frequency in the forbidden gap, the bound states in the isotropic system turn into the unbound states in the anisotropic system. This change leads to the presence of SE in the anisotropic system that agrees with the experimental observation in Ref. 18 where spontaneous emission happens only in the strong anisotropic PCs but not in the weak anisotropic system for the emission frequency lying in the forbidden gap. The presence of spontaneous emission in the anisotropic photonic crystal validates the correctness of our results while illustrates the inconsistency with the prediction of the previous studies [14,15,16]. The existence of the diffusion field in the anisotropic system is elucidated by the dynamical behavior of the two systems arriving at the great difference close to the band edge. We also investigated the new topic of how the curvature of the anisotropic dispersion relation affects the dynamical behavior of the anisotropic system. Without changing the units of energy and time, the dispersion relation of the smaller curvature has larger DOS and coupling strength which leads to the earlier appearance of the diffusion field and energy transfer. This earlier energy transfer resulting in the smaller and faster-decaying probability causes the different optical behavior along different wavevectors in the anisotropic PC system. where we have converted the variable s 1/2 into Y . There are two kinds of roots for this indicial equation: one with different roots Y 1 = Y 2 and the other with degenerate root Y 1 = Y 2 . For the case of different roots,D(s) could be expressed asD α e at being the two-parameter fractional exponential function of variable t, order α, and constant a [24]. The analytical solution for the fractional kinetic equation (Eq. (12)) of the anisotropic photonic crystal system is thus obtained. For β/f 3 = ∆ c , These non-decaying terms oscillate individually with time and form atom-photon bound states as the atomic frequency lies deeply in the forbidden gap (ω 21 << ω c ). On the other hand, when the atomic frequency is in the higher-energy region ω 21 > ω c + β/f 3 (region of allowed band), the square of these roots is complex and the solution decays with time quickly. These two regions of atomic transition frequency given by the roots of the indicial equation are the discussing basis of the previous studies[14,15,16,17]. It was predicted in the previous studies that the bare atomic transition frequency lying in the region ω 21 < ω c + β shifted into the forbidden gap by the interaction with the radiation modes where a photon-atom bound state is generated. That is, there will not exist spontaneous emission in the anisotropic PC system if the atomic transition frequency lies in the region near band edge (ω 21 ∼ = ω c ). This result is inconsistent with the experimental observation in P hys. Rev.Lett. 96, 243902 (2006) where SE appeared with an extra angular anisotropy in the anisotropic PC system as the emission frequency of the embedded QDs lies in the forbidden gap. is, the bound states close to the band edge of the isotropic system change to the decaying states which lead to the appearance of SE in the anisotropic system. This result is consistent with the experimental observation in P hys. Rev. Lett. 96, 243902 (2006). In order to investigate the local optical properties of PCs, Barth et al. doped the PCs of artificial colloidal opals with CdSe/ZnS core-shell quantum dots whose emission frequency lies inside the forbidden gap and the linewidth was narrower than the width of the band gap. They demonstrated that the characteristic patterns of fluorescence image from different quantum dots carried information on the modification of the optical mode density which arose from the direction-dependent photonic stop band. The anisotropic band structure of the artificial colloidal opals, which corresponds to the direction-dependent photonic stop band, brought an extra angular anisotropy of fluorescence image that was detected by defocused wide-field image of the single CdSe/ZnS quantum dots. These quantum dots did not emit light if they FIG. 1 : 1(a) A two-level atom with excited state |2 and ground state |1 . The transition frequency ω 21 is nearly resonant with the frequency range of the PBG reservoir. (b) Directional dependent dispersion relation near band edge with edge frequency ω c . (c) DOS of the anisotropic one-band effective mass model. [22] K. B. Oldham and J. Spanies, Fractional Calculus (New York: Academic, 1974). [23] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (London: Gordon and Breach, 1993). [24] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (New York: Wiley, 1993). [25] M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge Univ. Press, Cambridge, 1997). FIG. 4: Dynamics of SE of the anisotropic system with two kinds of curvatures in the dispersion relation. Solid lines for the system with smaller-curvature dispersion relation and dashed lines for the larger-curvature system. is considered, we could express it as a Riemann-Liouville fractional differentiation operator [22, 23, 24] with order ν = 1/2. AcknowledgmentsWe would like to gratefully acknowledge partially financial support from the National . L Chen, A V Nurmikko, Appl. Phys. Lett. 853663L. Chen and A. V. Nurmikko, Appl. Phys. Lett. 85, 3663 (2004). . N Vats, T Rudolph, J. Mod. Opt. 481495N. Vats and T. Rudolph, J. Mod. Opt. 48, 1495 (2001). . S Nishimura, N Abrams, B A Lewis, L I Halaoui, T E Mallouk, K D Benkstein, J Van De Lagemaat, A J Frank, J. Am. Chem. Soc. 1256306S. Nishimura, N. Abrams, B. A. Lewis, L. I. Halaoui, T. E. Mallouk, K. D. Benkstein, J. van de Lagemaat, and A. J. Frank, J. Am. Chem. Soc. 125, 6306 (2003). . S John, J Wang, Phys. Rev. Lett. 642418S. John and J. Wang, Phys. Rev. Lett. 64, 2418 (1990). . S John, J Wang, Phys. Rev. B. 4312772S. John and J. Wang, Phys. Rev. B 43, 12772 (1991). . S John, T Quang, Phys. Rev. Lett. 781888S. John and T. Quang, Phys. Rev. Lett. 78, 1888 (1997). . S Bay, P Lambropoulos, K Molmer, Phys. Rev. Lett. 792654S. Bay, P. Lambropoulos, and K. Molmer, Phys. Rev. Lett. 79, 2654 (1997). . S.-C Cheng, J.-N Wu, M.-R Tsai, W.-F Hsieh, J. Phys.: Condens. Matter. 2115503S.-C. Cheng, J.-N. Wu, M.-R. Tsai, and W.-F. Hsieh, J. Phys.: Condens. Matter 21, 015503 (2009). . S John, T Quang, Phys. Rev. A. 501764S. John and T. Quang, Phys. Rev. A 50, 1764 (1994). . S Y Zhu, H Chen, H Huang, Phys. Rev. Lett. 79205S. Y. Zhu, H. Chen, and H. Huang, Phys. Rev. Lett. 79, 205 (1997). . T Quang, Phys. Rev. Lett. 795238T. Quang et al., Phys. Rev. Lett. 79, 5238 (1997). . A G Kofman, G Kurizki, B Sherman, J. Mod. Opt. 41353A. G. Kofman, G. Kurizki, and B. Sherman, J. Mod. Opt. 41, 353 (1994). . S John, T Quang, Phys. Rev. Lett. 743419S. John and T. Quang, Phys. Rev. Lett. 74, 3419 (1995). . S.-Y Zhu, Y Yang, H Chen, H Zheng, M S Zubairy, Phys. Rev. Lett. 842136S.-Y. Zhu, Y. Yang, H. Chen, H. Zheng, and M. S. Zubairy, Phys. Rev. Lett. 84, 2136 (2000). . Y Yang, M Fleischhauer, S.-Y Zhu, Phys. Rev. A. 6822103Y. Yang, M. Fleischhauer, and S.-Y. Zhu, Phys. Rev. A 68, 022103 (2003). . Y Yang, M Fleischhauer, S.-Y Zhu, Phys. Rev. A. 6843805Y. Yang, M. Fleischhauer, and S.-Y. Zhu, Phys. Rev. A 68, 043805 (2003). . Y Yang, G Li, H Chen, S.-Y Zhu, Opt. Commun. 265559Y. Yang, G. Li, H. Chen, and S.-Y. Zhu, Opt. Commun. 265, 559 (2006). . M Barth, R Schuster, A Gruber, F Cichos, Phys. Rev. Lett. 96243902M. Barth, R. Schuster, A. Gruber, and F. Cichos, Phys. Rev. Lett. 96, 243902 (2006). . S John, Phys. Rev. Lett. 582486S. John, Phys. Rev. Lett. 58, 2486 (1987). I S Gradshteyn, I M Ryzhik, Tables of Integrals, Series and Products. Academic PressI. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, 1980). . N Vats, S John, Phys. Rev. A. 584168N. Vats and S. John, Phys. Rev. A 58, 4168 (1998). Dynamics of SE of the anisotropic PC system as a function of βt for various values of the. FIG. 2: Dynamics of SE of the anisotropic PC system as a function of βt for various values of the Dynamics of SE of the anisotropic (Aniso., solid lines) and isotropic (Iso., dashed lines) systems with detuning frequency close to the band edge of PBG reservoir (∆ c = ω 21 − ω c → 0). 3FIG. 3: Dynamics of SE of the anisotropic (Aniso., solid lines) and isotropic (Iso., dashed lines) systems with detuning frequency close to the band edge of PBG reservoir (∆ c = ω 21 − ω c → 0).
{'fraction_non_alphanumeric': 0.05801464048710406, 'fraction_numerical': 0.032872682492987616, 'mean_word_length': 3.9225458831453106, 'pattern_counts': {'":': 0, '<': 9, '<?xml version=': 0, '>': 6, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 14, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'Spontaneous emission (SE) from a two-level atom in a photonic crystal (PC) with anisotropic oneband model is investigated using the fractional calculus. Analytically solving the kinetic equation in terms of the fractional exponential function, the dynamical discrepancy of SE between the anisotropic and isotropic systems is discussed on the basis of different photon density of states (DOS) and the existence of incoherent diffusion field that becomes even more clearly as the atomic transition frequency lies close to the band edge. With the same atom-field coupling strength and detuning in the forbidden gap, the photon-atom bound states in the isotropic system turn into the unbound ones in the anisotropic system that is consistent with the experimental observation in P hys. Rev. Lett. 96, 243902 (2006). Dynamics along different wavevectors with various curvatures of dispersion is also addressed with the changes of the photon DOS and the appearance of the diffusion fields.', 'arxivid': '0906.0743', 'author': ['Jing-Nuo Wu \nDepartment of Photonics and Institute of Electro-Optical Engineering\nNational Chiao Tung University\nHsinchuTaiwan, R. O. C\n', 'Chih-Hsien Huang \nDepartment of Photonics and Institute of Electro-Optical Engineering\nNational Chiao Tung University\nHsinchuTaiwan, R. O. C\n', 'Szu-Cheng Cheng \nDepartment of Physics\nChinese Culture University\nTaipeiTaiwan, R. O. C.\n', 'Wen-Feng Hsieh \nDepartment of Photonics and Institute of Electro-Optical Engineering\nNational Chiao Tung University\nHsinchuTaiwan, R. O. C\n\nDepartment of Electro-Optical Engineering\nNational Cheng Kung University\nTainanTaiwan, R. O. C\n'], 'authoraffiliation': ['Department of Photonics and Institute of Electro-Optical Engineering\nNational Chiao Tung University\nHsinchuTaiwan, R. O. C', 'Department of Photonics and Institute of Electro-Optical Engineering\nNational Chiao Tung University\nHsinchuTaiwan, R. O. C', 'Department of Physics\nChinese Culture University\nTaipeiTaiwan, R. O. C.', 'Department of Photonics and Institute of Electro-Optical Engineering\nNational Chiao Tung University\nHsinchuTaiwan, R. O. C', 'Department of Electro-Optical Engineering\nNational Cheng Kung University\nTainanTaiwan, R. O. C'], 'corpusid': 119265507, 'doi': '10.1103/physreva.81.023827', 'github_urls': [], 'n_tokens_mistral': 9100, 'n_tokens_neox': 7605, 'n_words': 5018, 'pdfsha': '7045cdfda77b9d6ac5e71a0d67a4e7c317e75802', 'pdfurls': ['https://arxiv.org/pdf/0906.0743v1.pdf'], 'title': ['Spontaneous emission from a two-level atom in anisotropic one-band photonic crystals: a fractional calculus approach', 'Spontaneous emission from a two-level atom in anisotropic one-band photonic crystals: a fractional calculus approach'], 'venue': []}
arxiv
THE L ∞ -POSITIVITY PRESERVING PROPERTY AND STOCHASTIC COMPLETENESS 22 Dec 2021 Andrea Bisterzo Ludovico Marini THE L ∞ -POSITIVITY PRESERVING PROPERTY AND STOCHASTIC COMPLETENESS 22 Dec 2021 We say that a Riemannian manifold satisfies the L p -positivity preserving property if (−∆ + 1)u ≥ 0 in a distributional sense implies u ≥ 0 for all u ∈ L p . While geodesic completeness of the manifold at hand ensures the L p -positivity preserving property for all p ∈ (1, +∞), when p = +∞ some assumptions are needed. In this paper we show that the L ∞ -positivity preserving property is in fact equivalent to stochastic completeness, i.e., the fact that the minimal heat kernel of the manifold preserves probability. The result is achieved via some monotone approximation results for distributional solutions of −∆ + 1 ≥ 0, which are of independent interest. Introduction Let (M, g) be an n-dimensional Riemannian manifold with Riemannian measure dµ g . In the following ∆ = div ∇ is the (negatively defined) Laplace-Beltrami operator and, unless explicitly stated, all integrals are taken with respect to the Riemannian volume measure dµ g . The aim of this paper is to study qualitative properties for certain solutions of elliptic PDEs involving the following Schrödinger operator (1.1) L := ∆ − 1. We say that u ∈ L 1 loc (M ) solves (−∆ + 1)u ≥ 0 in the sense of distributions if M u(−∆ + 1)ϕ ≥ 0 for all ϕ ∈ C ∞ c (M ) with ϕ ≥ 0. Note that this is equivalent to say that (−∆ + 1)u is a positive Radon measure. If more regularity is assumed, namely u ∈ W 1,2 loc (M ), we talk of a weak solution of (−∆ + 1)u ≥ 0 ifˆM g(∇ϕ, ∇u) + uϕ ≥ 0 for all ϕ ∈ C ∞ c (M ) with ϕ ≥ 0. Finally, if u ∈ C 2 (M ), then (−∆ + 1)u ≥ 0 is intended in a strong, pointwise sense. Naturally, if u ∈ C 2 (M ) is a strong solution of the inequality, it is also a weak and thus distributional solution. We begin with the following definition: The definition was proposed by Güneysu in [13] although the case p = 2 of this property appears in previous works of Kato, [20], and Braverman, Milatovic and Shubin, [4]. In this last paper, the authors proved that the validity of the L 2 -positivity preserving property implies the essential self-adjointness of the Schrödinger operator −∆ + V for all L 2 loc nonnegative potentials V . Recall that −∆+V : C ∞ c (M ) → L 2 (M ) is essentially self-adjoint if it has an unique self-adjoint extension to L 2 (M ) (its closure). On the other hand, the operator −∆ + V is known to be essentially self-adjoint on geodesically complete manifolds, see [29,Corollary 2.9] or [4] and [16] for the case of operators acting on Hermitian vector bundles. This conjecture has remained open for 20 years and has only recently been solved in the positive by Pigola and Veronelli in [25]. For a complete introduction to the topic we refer to the nice survey [14], to Chapter XIV.5 of [15] and Appendix B of [4]. The case p = +∞ of Definition 1.1 is instead related to stochastic completeness. Recall that a manifold is said to be stochastically complete if the Brownian paths on M have almost surely infinite lifetime or, equivalently, if the minimal positive heat kernel associated to the Laplace-Beltrami operator preserves probability. For the scope of this article, however, we shall adopt the following (equivalent) definition, which is more relevant from the point of view of PDEs. Definition 1.2. A Riemannian manifold (M, g) is said to be stochastically complete if the only bounded, non-negative C 2 solution of ∆u ≥ u on M is u ≡ 0. There are countless characterizations of stochastic completeness, a comprehensive account is beyond the scope of this paper, we refer the reader to [9,11,23,24] or the very recent [12]. See also Section 2 below. Stochastic completeness is implied by several geometric, analytic and probabilistic conditions. For instance, stochastic completeness is ensured by conditions on the curvature tensor. In this direction, the most general result is the one of Hsu in [19], a particular case of which states that geodesically complete manifold whose Ricci curvature satisfies Ric(x) ≥ −Cr 2 (x) outside a compact set are in fact stochastically complete. As a matter of fact, the L p -positivity preserving property implies stochastic completeness of the manifold at hand, as it has been observed by Güneysu in [13]. In particular, stochastically incomplete manifolds provide counterexamples to the validity of the L ∞ -positivity preserving property. As an example, take a Cartan-Hadamard manifold whose Ricci curvature diverges at −∞ faster than quadratically, for computations we refer to [21]. In the last years there has been an effort to better understand the L p -positivity preserving property and to find geometric and analytic conditions ensuring its validity. If one takes M = R n with the usual Euclidean metric, the L 2 -positivity preserving property was first proved by Kato,[20], using the theory of operators on tempered distributions. In a Riemannian setting, however, one does not dispose of tempered distributions and it is thus necessary to take other paths. Following an idea of Davies in [4], if the manifold admits a family of smooth cutoff functions with a good control on the Laplacian, it is possible to prove the L p -positivity preserving property. In this direction we mention the results by Braverman, Milatovic and Schubin in [4]; by Güneysu in [13,15]; by Bianchi and Setti in [2]; and by the second author and Veronelli in [21]. Using a completely different strategy, Pigola and Veronelli, [25], were finally able to prove that the L p -positivity preserving property for p ∈ (1, +∞) holds on geodesically complete manifolds, thus verifying that the BMS conjecture is true. Remark 1.3. Without geodesic completeness the L p -positivity preserving property generally fails for every p ∈ [1, +∞]. To see this, take B 1 ⊆ R 2 the Euclidean open ball of radius 1. Then, the radial function u(r) = −r, which belongs to all L p spaces, is a non-positive function which satisfies (−∆ + 1)u ≤ 0. The proof of Pigola and Veronelli uses some new regularity results for non-negative subharmonic distributions to prove that the L p -positivity preserving property is implied by a Liouville-type property for L p -subharmonic distributions. When p = 1, +∞, this property is known to hold on geodesically complete manifolds thanks to a result of Yau, [31]. This strategy, however, fails when p = 1 or p = +∞ since there are known counterexamples to the Liouville-type property. Remark 1.4. To the best of our knowledge, when p = +∞ the most general condition known so far ensuring the validity of the L ∞ -positivity preserving property is the one of Theorem II in [21]. This condition, which requires geodesic completeness and Ric(x) ≥ −Cr 2 (x) outside a compact set, is essentially the celebrated condition of Hsu, [19], for stochastic completeness. The above observation suggests a much closer relation between stochastic completeness and the L ∞ -positivity preserving property. The main result of this article is in fact the following: Theorem A. Let (M, g) be a Riemannian manifold, then M has the L ∞ -positivity preserving property if and only if it is stochastically complete. Theorem A together with the result of Pigola and Veronelli, [25], give the full picture of the L p -positivity preserving property when p ∈ (1, +∞]. When p = 1, the best result we have is the one of the second author with Veronelli, [21,Theorem II], which ensures the L 1 -positivity preserving property if the manifold is complete and the Ricci curvature essentially grows like Ric(x) ≥ −Cr 2 (x) outside of a compact set. Using a construction suggested to us by Veronelli we also prove the following: Theorem B. For every ε > 0, there exists a 2-dimensional Riemannian manifold (M, g) whose Gaussian curvature satisfies K(x) ∼ −Cr(x) 2+ε , such that the L 1 -positivity preserving property fails on M . Here r(x) denotes the Riemannian distance from some fixed pole. Remark 1.5. Theorem B together with Remark 1.3 show that the result of Theorem II in [21] alluded in the above is optimal. Remark 1.6. Using a simple trick introduced in [18], the counterexample in dimension 2 of Theorem B can be used to construct counterexamples to the L 1 -positivity preserving property in arbitrary dimensions n ≥ 2. It suffices to take the product of the 2 dimensional model manifold M with an arbitrary n − 2 dimensional closed Riemannian manifold. Extending the function which provides the counterexample on M to the whole product produces a counterexample in a manifold of dimension n. In order to prove that stochastic completeness implies the L p -positivity preserving property, we show that it is essentially a problem of regularity for the distributional, L ∞ solutions of Lu ≥ 0, where L is defined in (1.1). Using a Brezis-Kato inequality we reduce ourselves to prove the following: Proposition C. Let (M, g) be a Riemannian manifold and let u ∈ L ∞ (M ) satisfying Lu ≥ 0 in the sense of distributions. Then, there exists some w ∈ C ∞ (M ) with sup M w < +∞ such that u ≤ w and Lw ≥ 0 in the strong sense. This latter result follows from a monotone approximation theorem for the distributional solutions of Lu ≥ 0 which is of independent interest. (i) u k ց u pointwise a.e.; (ii) Lu k ≥ 0 for all k; (iii) u k → u in L 1 (Ω); (iv) sup Ω u k ≤ 2 ess sup Ω u. Using a trick due to Protter and Weinberger, [26], it is sufficient to prove a monotone approximation result for the distributional solution of ∆ α v ≥ 0, where ∆ α v := α −2 div(α 2 ∇v) and α is a smooth positive function to be specified later. The monotone approximation for the weighted Laplacian is obtained using a strategy outlined by Bonfiglioli and Lanconelli in [3] together with some mean value representation formulas for the solution of ∆ α v = 0. Theorem D generalizes a result of Pigola and Veronelli in [25] where the monotone approximation was proved only on coordinate charts. If the manifold at hand admits a minimal, positive Green function for the operator ∆ α (i.e. it is α-non-parabolic) and if this Green function vanishes at infinity (i.e. it is strongly α-non-parabolic), as a byproduct of the proof of Theorem D we obtain a global, monotone approximation result. The paper is organized as follows. In Section 2 we study the relation between the L ∞positivity preserving property and stochastic completeness, showing that the former property implies the latter and the converse is true up to a claim which is proved later on. Section 3 is devoted to the monotone approximation results. We first observe that the conclusions of Theorem D can be inferred form an equivalent statement for the operator ∆ α . We prove some mean value representation formulae for α-harmonic functions and show how these can be used to produce a monotone approximating sequence with wanted properties. As a corollary of Theorem D, we obtain the desired claim which concludes the proof of Theorem A. We end the section by observing that if we make some assumptions on the geometry of M , the monotone approximation results have a global nature. Finally, in Section 4 we construct a class of Riemannian manifolds on which the L 1 -positivity preserving property fails thus proving Theorem B. (i) u k ց u pointwise a.e.; (ii) Lu k ≥ 0 for all k; (iii) u k → u in L 1 (M ); (iv) sup M u k ≤ 2 ess sup M u. L ∞ -positivity preserving property and stochastic completeness The aim of this section is to investigate the connection between the L ∞ -positivity preserving property and stochastic completeness. As pointed out in the introduction, there are several possible definitions one can give for stochastic completeness. We cite here the ones relevant to our exposition. (i) for every λ > 0, the only bounded, non-negative C 2 solution of ∆u ≥ λu is u ≡ 0; (ii) for every λ > 0, the only bounded, non-negative C 2 solution of ∆u = λu is u ≡ 0; (iii) the only bounded, non-negative C 2 solution of ∆u = u is u ≡ 0. For a proof of the equivalence we refer to Theorem 6.2 in [9]. Remark 2.1. Note that the regularity required in the above and in Definition 1.2 can be relaxed to C 0 (M ) ∩ W 1,2 loc (M ) see for instance Section 2 of [1]. This fact is a consequence of a stronger version of Theorem 2.6 below. We begin with the following observation due to Güneysu, [13]. Proof. To see this, take u ∈ C 2 (M ) a bounded and non-negative function satisfying ∆u ≥ u. Then, if we set v = −u we have v ∈ L ∞ (M ) (−∆ + 1)v ≥ 0. By the L ∞ -positivity preserving property, we conclude that v ≥ 0, since u is non-negative, this yields v ≡ 0 and hence u ≡ 0. Remark 2.3. It is worthwhile noticing that stochastic completeness is in general unrelated to geodesic completeness. It is possible to find Riemannian manifolds which are geodesically but not stochastically complete such as Cartan-Hadamard manifolds whose Ricci curvature diverges at −∞ faster that quadratically. On the other hand, R n \{0} endowed with the Euclidean metric is stochastically complete but geodesically incomplete. Proposition 2.2 and the above remark explain the failure of the result of Pigola and Veronelli, [25], in the case p = ∞. Since L(−u) ≥ 0 we conclude that Lu − ≥ 0 in the sense of distributions. If u − happened to be a C 2 (M ) function, stochastic completeness would allow us to conclude that u − ≡ 0, hence u ≥ 0. Note that, according to Remark 2.1, u − ∈ C 0 (M ) ∩ W 1,2 loc (M ) would be sufficient. In general, however, this is not the case and, as a matter of fact, it is a stronger requirement than what we actually need. Indeed, if we find w ∈ C 2 (M ) such that sup M w < +∞, 0 ≤ u − ≤ w and Lw ≥ 0, then stochastic completeness applied to w implies that w hence u − are identically zero. The existence of such function w is implied by the following corollary of Theorem D, whose proof is postponed to the next section. Via a compactness argument we use the functions u Ω to construct the function w. The following theorem, proved by Sattinger in [28], also comes into aid as it allows to obtain L-harmonic function from super/sub solutions of Lu = 0. Theorem 2.6. Let u 1 , u 2 ∈ C ∞ (M ) satisfying Lu 1 ≥ 0, Lu 2 ≤ 0, u 1 ≤ u 2 on M . Then, there exists some w ∈ C ∞ (M ) such that u 1 ≤ w ≤ u 2 and Lw = 0. Remark 2.7. Theorem 2.6 is a weaker formulation of a much more general theorem, proved by Ratto, Rigoli and Véron, [27], for a wider class of functions, namely u 1 , u 2 ∈ C 0 (M ) ∩ W 1,2 loc (M ). This result goes under the name of sub and supersolution method or monotone iteration scheme. Note that the results of [27] hold for a larger class of second order elliptic operators. For a survey on the subject, we refer to Heikkilä and Lakshmikantham, [17]. Using the functions constructed locally in Corollary 2.5 together with an exhaustion procedure we obtain the following: Next, take {Ω h } an exhaustion of M by relatively compact sets such that Ω 1 ⋐ Ω 2 ⋐ . . . ⋐ Ω h ⋐ Ω h+1 ⋐ . . . ⋐ M, ∂Ω h is smooth and M = ∪ h Ω h . On each set Ω h we apply Corollary 2.5 and we obtain a sequence of functions u h ∈ C ∞ (Ω h ) such that (1) u ≤ u h ≤ 2c in Ω h ; (2) Lu h ≥ 0 strongly on Ω h . Since Lc ≤ 0, we use Theorem 2.6 on each Ω h to obtain w h ∈ C ∞ (Ω h ) satisfying (1) Lw h = 0; (2) u h ≤ w h ≤ 2c. We conclude by showing that {w h } h is bounded respect to the C ∞ (M )-topology and thus converges, up to a subsequence, to some w ∈ C ∞ (M ). To this end, let K ⊂ M be a compact set and k ∈ N, k ≥ 2. By Schauder estimates for the operator L we have w h C k (K) ≤ A w h L ∞ (K) + Lw h C k−2,α (K) for some α ∈ (0, 1). See for instance Section 6.1 of [8]. In particular there exists a constant C = C(K, n, k) > 0 such that w h C k (K) < C for every h ∈ N. Here ||w h || C k (K) = ||w h || L ∞ (K) + ||∇w h || L ∞ (K) + · · · + ||∇ k w h || L ∞ (K) . Since {w h } h is pre-compact, it converges in the C ∞ (M ) topology up to a subsequence, denoted again with {w h } h . Let w ∈ C ∞ (M ) be the C ∞ limit, we have that u ≤ w ≤ 2c and Lw = 0. This concludes the proof of Theorem A, apart from the proof of Corollary 2.5. Monotone approximation results This section is devoted to the proof of Theorem D. Instead of proving Theorem D directly, we prove an equivalent monotone approximation result for another elliptic differential operator closely related to L. We begin by taking a function α ∈ C ∞ (M ) satisfying (3.1) Lα = 0 α > 0 . The existence of such a function is ensured by [7], and is equivalent to the fact that λ −L 1 > 0. In our case it is easy to see that λ −L 1 ≥ 1. Using α we define the following drifted Laplacian (3.2) ∆ α : u → α −2 div(α 2 ∇u). With a trivial density argument, one has that ∆ α is symmetric in L 2 with respect to the measure α 2 dµ g . Then, using the following idea due to Protter and Weinberger, [26], we establish the relation between ∆ α and L. See also Lemma 2.3 of [25]. Lemma 3.1. If u ∈ L 1 (Ω) with Ω ⋐ M , then (∆ − 1)u ≥ 0 ⇔ ∆ α u α ≥ 0, where both inequalities are intended in the sense of distributions. Proof. Fix 0 ≤ ϕ ∈ C ∞ c (Ω), by direct computation we have α∆ α ϕ α = α −1 div α 2 ∇ ϕ α = α −1 div (α∇ϕ − ϕ∇α) = ∆ϕ − ϕ ∆α α = Lϕ, (3.3) where in the last equation we have used (3.1). Thus, using (3.3) and the symmetry of ∆ α we conclude ∆ α u α , αϕ L 2 =ˆΩ u α ∆ α ϕ α α 2 dµ g =ˆΩ u (∆ − 1)ϕ dµ g = ((∆ − 1)u, ϕ) L 2 . Using Equation (3.3) and setting v = α −1 u, it is possible to obtain Theorem D from an equivalent statement for the operator ∆ α . In this perspective, our goal is to prove the following: } ⊂ C ∞ (Ω) such that: (i) v k ց v pointwise a.e.; (ii) ∆ α v k ≥ 0 for all k; (iii) v k → v in L 1 (Ω); (iv) sup Ω v k ≤ ess sup Ω v. 3.1. Representation formula for α-harmonic functions. We begin by establishing some mean value representation formulae involving the Green function of the operator ∆ α on Ω with Dirichlet boundary conditions. Recall that G : Ω × Ω \ {x = y} → R is a symmetric, L 1 (Ω × Ω) function satisfying the following properties: (a) G ∈ C ∞ (Ω × Ω \ {x = y}) and G(x, y) > 0 for all x, y ∈ Ω with x = y; (b) lim x→y G(x, y) = +∞ and G(x, y) = 0 if x ∈ ∂Ω (or y ∈ ∂Ω); (c) ∆ α G(x, y) = −δ x (y) with respect to α 2 dµ g , that is, ϕ(x) = −ˆΩ G(x, y) ∆ α ϕ(y)α 2 (y)dµ y ∀ϕ ∈ C ∞ C (Ω) . For r > 0 and x ∈ Ω, we define the following set (3.4) B r (x) := y ∈ Ω | G(x, y) > r −1 ∪ {x}. We adopt the convention G(x, x) = +∞ so that B r (x) = y ∈ Ω | G(x, y) > r −1 . Observe that B r (x) ⊂ Ω are open and relatively compact sets, moreover, for almost all r > 0, ∂B r (x) is a smooth hypersurface. This is a consequence of Sard's theorem. In the following, dσ and dµ represent the Riemannian surface and volume measure of ∂B r (x) and B r (x) respectively. Proposition 3.3. For every v ∈ C ∞ (Ω) and almost every r > 0, the following representa- tion formula holds v(x) =ˆ∂ Br (x) v(y)|∇G(x, y)|α 2 (y)dσ y −ˆB r (x) G(x, y) − 1 r ∆ α v(y)α 2 (y)dµ y (3.5) Proof. By the Green identity we have v(x) = −ˆB r (x) G(x, y) ∆ α v(y)α 2 (y)dµ y +ˆ∂ Br(x) G(x, y) ∂v ∂ν (y) − v(y) ∂G ∂ν (x, y) α 2 (y)dσ y . Since ∂G ∂ν = −|∇G|, we obtain v(x) =ˆ∂ Br(x) v(y) ∇G(x, y) α 2 (y)dσ y + 1 rˆ∂ Br(x) ∂v ∂ν (y)α 2 (y)dσ y −ˆB r (x) G(x, y)∆ α v(y)α 2 (y)dµ y =ˆ∂ Br(x) v(y) ∇G(x, y) α 2 (y)dσ y −ˆB r (x) G(x, y) − 1 r ∆ α v(y)α 2 (y)dµ y . In particular, if v ∈ C 2 (Ω) is α-harmonic, i.e. ∆ α u = 0 on Ω, then (3.6) v(x) =ˆ∂ Br (x) |∇G(x, y)| v(y) α 2 (y) dσ y . The formulae (3.6) and (3.5) are a generalization of some standard representation formula for the Laplace-Beltrami operator. See for instance the Appendix of [3], [22] or the very recent [6]. 3.2. Distributional vs. potential α-subharmonic solutions. Before proving the monotone approximation result, we observe that the notion of α-subharmonicity in the distributional sense is closely related to the notion of α-subharmonic solutions in the sense of potential theory. Definition 3.4. We say that an upper semicontinuous function u : Ω → [−∞, +∞) is α-subharmonic in the sense of potential theory on Ω if the following conditions hold (i) {x ∈ Ω | u(x) > −∞} = ∅; (ii) for all V ⋐ Ω and for every h ∈ C 2 (V ) ∩ C 0 (V ) such that ∆ α h = 0 in V with u ≤ h on ∂V , then u ≤ h in V. The key observation, first noted by Sjörgen in [30,Theorem 1] in the Euclidean setting, is that every distributional α-subharmonic function is almost everywhere equal to a function which is α-subharmonic in the sense of potential theory. Note that in [30, Theorem 1], Sjörgen considers a wider class of elliptic differential operators. The drifted Laplace-Beltrami operator falls into that class. More precisely, if v ∈ L 1 (Ω) satisfies ∆ α v ≥ 0 in the sense of distributions, then v is equal almost everywhere to an α-subharmonic function in the sense of potential theory. Naturally, if v has some better regularity property, for example it is continuous, the equality holds everywhere. This fact holds true also in the Riemannian case, we sketch here the proof for clarity of exposition. Recall that for every ϕ ∈ C ∞ c (Ω) we have ϕ(x) = −ˆΩ G(x, y)∆ α ϕ(y) α 2 (y)dµ y . Furthermore, since ∆ α v = dν v is a positive Radon measure, we havê Ω v(x)∆ α ϕ(x) α 2 (x)dµ x =ˆΩ ϕ(x) dν v x for every ϕ ∈ C ∞ c (Ω). The measure dν v is often referred as the ∆ α -Riesz measure of v. By a direct computation we havê Ω v(x)∆ α ϕ(x) α 2 (x)dµ x =ˆΩ ϕ(x) dν v x = −ˆΩˆΩ G(x, y)∆ α ϕ(y) α 2 (y)dµ y dν v x =ˆΩ − ˆΩ G(x, y)dν v x ∆ α ϕ(y)α 2 (y)dµ y , hence,ˆΩ v(y) +ˆΩ G(x, y) dν v x ∆ α ϕ(y)α 2 (y)dµ y = 0, for every 0 ≤ ϕ ∈ C ∞ c (Ω). In other words, the function v +ˆΩ G(x, ·)dν v x is α-harmonic in the sense of distributions. By [30, Theorem 1] of Sjörgen we know that α-harmonic functions are almost everywhere equal to a function which is α-harmonic in the sense of potential theory. When the operator at hand is the Euclidean Laplacian, this result is usually referred as Weyl's lemma. We conclude that (3.7) v a.e. = h −ˆΩ G(x, ·)dν v x , where h is α-harmonic in a strong sense. On the other hand, one can prove that the function (3.8) − G * dν v = −ˆΩ G(x, ·)dν v x is α-subharmonic in the sense of potential theory which concludes the sketch of the proof. For this latter statement, we refer to Section 6 of [3]. Proof of Theorem 3.2. In order to prove Theorem 3.2, we adopt a strategy laid out by Bonfiglioli and Lanconelli in [3], where they obtained some monotone approximation results for a wide class of second order elliptic operators on R n . To do so, we begin by defining the following mean integral operators. If v is an upper semicontinuous function on Ω, x ∈ Ω and r > 0, we set (3.9) m r (v)(x) :=ˆ∂ Br(x) v(y)|∇ y G(x, y)|α 2 (y) dσ y . In particular, if v is an α-subharmonic function in the sense of distributions we have the following results, which are an adaptation to the case or Riemannian manifolds of [3]. Proof. By the observation in the previous section, up to a choice of a good representative, we can assume that v is α-subharmonic in the sense of potential, cf. Definition 3.4. (i) Fix x 0 ∈ Ω and r > 0, consider ϕ ∈ C 0 (∂B r (x 0 )) such that v ≤ ϕ on ∂ B r (x 0 ). Let h : B r (x 0 ) → R be the solution of (3.10) ∆ α h = 0 in B r (x 0 ) h = ϕ on ∂ B r (x 0 ) . Since v is α-subharmonic in the sense of potential, then v ≤ h in B r (x 0 ). By Proposition 3.3 we have (3.11) v(x 0 ) ≤ h(x 0 ) =ˆ∂ B r (x0) ϕ(y)|∇ y G(x 0 , y)|dσ α y where dσ α y = α 2 (y) dσ y . Since v is upper semicontinuous on ∂B r (x 0 ), there exists a sequence {ϕ i } i ⊂ C 0 (∂B r (x 0 )) such that ϕ i (y) ց v(y) almost everywhere on ∂B r (x 0 ). Applying (3.11) to each ϕ i we obtain by Dominated Convergence that v(x 0 ) ≤ˆ∂ Br(x0) v(y)|∇ y G(x 0 , y)|dσ α y = m r (v)(x 0 ). (ii) Fix 0 < s < r, let ϕ and h be as in (i) so that v ≤ h on B r (x 0 ). By Proposition 3.3 we have m s (v)(x 0 ) ≤ˆ∂ Bs(x0) h(y)|∇ y G(x 0 , y)|dσ α y = h(x 0 ) =ˆ∂ B r (x0) ϕ(y)|∇ y G(x 0 , y)|dσ α y . Taking a monotone sequence of continuous functions on the boundary ϕ i ց u and proceeding as above we conclude m s (v)(x 0 ) ≤ˆ∂ B r (x0) ϕ i (y)|∇ y G(x 0 , y)|dσ α y −→ m r (v)(x 0 ). (iii) This property is a consequence of the fact that v is (almost everywhere) equal to an upper semicontinuous function. Fix x 0 ∈ Ω and ε > 0 there exists a small enough neighborhood of x 0 , V (x 0 ), such that . v(y) < v(x 0 ) + ε on V (x 0 ) . Taking for r > 0 small enough, we have m r (v)(x 0 ) ≤ v(x 0 ) + ε. Recall that the function constant to 1 is α-harmonic on Ω. By (i), v(x 0 ) ≤ m r (v)(x 0 ) hence m r (v)(x 0 ) − ε ≤ v(x 0 ) ≤ m r (v)(x 0 ). Letting ε, and thus r go to 0, we obtain desired property. (iv) This last property is a consequence of the decomposition of α-subharmonic functions observed in (3.7). Integrating against |∇G|α 2 both sides of (3.7) we obtain m r (v)(x) = h(x) − m r (G * dν v )(x). The desired property follows from the fact that the mean integral −m r (G * dν v ) is αsubharmonic in the sense of potential. For details we refer to Section 6 of [3]. The next step is to take a convolution of the mean integral functions m r (v) so to obtain smooth functions which produce the desired approximating sequence {v k } k . Proof of Theorem 3.2. Let ϕ ∈ C 1 c ([0, 1]) be a non-negative function with unitary L 1 -norm, we define (3.12) v k (x) := kˆ+ ∞ 0 ϕ(ks) m s (v)(x)ds As shown in [3] the functions defined by (3.12) are eventually smooth. The monotonicity of the approximating sequence follows immediately from the monotonicity of m r (v) with respect to r. Combining this with property (i) of Proposition 3.5 we obtain (i). The proof of (ii) is a consequence of (iii) in Proposition 3.5. Both this proofs are straightforward computations, we refer to [3, Theorem 7.1] for the details. The convergence in L 1 (Ω) follows from (i) and (ii), using the fact that |v k | ≤ max{|v|, |v 1 |} and Dominated Convergence. For the uniform estimate of (iv), it is enough to observe that 1 is an α-harmonic function on Ω and ϕ has unitary L 1 norm, hence, v k (x) = kˆ+ ∞ 0 ϕ(ks) m s (v)(x) ds ≤ ess sup Ω vkˆ+ ∞ 0 ϕ(ks) m s (1)(x) ds = ess sup Ω v. This concludes the proof of Theorem 3.2. Remark 3.6. Note that in the last estimate, one actually has ess sup Ω v k ≤ ess sup B 1/k (x) v ≤ ess sup Ω v. This observation will be crucial later on. 3.4. Proof of Theorem D. Finally, we desume the proof of Theorem D from Theorem 3.2. If {v k } k is the approximating sequence for the function v = u α , we define u k := αv k . By Equation (3.3), {u k } k is an approximating sequence for u as it satisfies (i)− (iii) of Theorem D. The proof is trivial and is therefore omitted. A little more effort is required to show that if sup Ω v k ≤ ess sup Ω v, then sup Ω u k ≤ 2 ess sup Ω u for k large enough. To this end, fix x ∈ Ω. As noted in Remark 3.6 we have u k (x) = α(x)v k (x) ≤ α(x) ess sup B 1/k (x) v ≤ α(x) inf B 1/k α ess sup Ω u. Furthermore, for every y ∈ B 1/k (x) we estimate (3.13) α(x) α(y) ≤ |α(x) − α(y)| α(y) + 1 ≤ r k (x) sup Ω |∇α| inf Ω α + 1 where r k (x) = sup{d(x, z) : z ∈ B 1/k (x)}. Next, we show that the function r k (x) can be uniformly bounded so that (3.13) is bounded above by 2. Lemma 3.7. There exists some k 0 ∈ N such that r k (x) ≤ inf Ω α sup Ω |∇α| =: c ∀x ∈ Ω, ∀k ≥ k 0 . Proof. Suppose by contradiction that there exists a sequence of points {x k } k ⊂ Ω such that r k (x k ) > c for every k ∈ N. By definition of r k (x k ), there exists a sequence of points {y k } k ⊂ B 1/k (x k ) such that d(y k , x k ) > c. Since Ω is relatively compact, up to a subsequence, we can assume that x k → x ∞ ∈ Ω and y k → y ∞ ∈ Ω. Since y k ∈ B 1/k (x k ) we have (3.14) G(x k , y k ) > k → +∞. Note also that the Green function G is smooth and hence continuous on Ω×Ω\{x = y}. Note that since d(x k , y k ) > c, then d(x ∞ , y ∞ ) ≥ c, in particular we deduce that x ∞ ∈ ∂Ω because the Green function G vanishes on the boundary of Ω. If x ∞ ∈ Ω is not on the boundary, fix k ∈ N. By (3.14) and continuity of the Green function we have G(y ∞ , x ∞ ) > k which implies that y ∞ ∈ B 1/k (x ∞ ). In particular we have d(x ∞ , y ∞ ) ≤ r k (x ∞ ) → 0, which is a contradiction since d(x ∞ , y ∞ ) ≥ c. Indeed, for every x ∈ Ω, lim k→+∞ r k (x) = 0. Clearly, r k (x) is a monotone decreasing sequence in k. If its limit is some r 0 = 0 this implies that r k (x) ≥ r 0 for all k. In particular the geodesic ball B r0 (x) is contained in B 1/k (x) for all k ∈ N. This, however, is a contradiction since ∞ k=1 B 1/k (x) = {x}. Thanks to Lemma 3.7, up to taking k large enough, we have α(x) ≤ 2α(y) ∀x ∈ Ω and ∀y ∈ B 1/k (x), hence, u k (x) ≤ α(x) inf B 1/k α ess sup Ω u ≤ 2 ess sup Ω u ∀x ∈ Ω. This concludes the proof of Theorem D. 3.5. Remarks on the global case. A careful analysis of above proofs shows that the monotone approximation results can be obtained globally on the whole manifold M as long as there exists a minimal positive Green function for the operator ∆ α and the super level sets B r (x) are compact. Not all Riemannian manifolds, however, satisfy these conditions. We recall the following in the sense of distributions. If t > t ε , by direct computation we have u ′ (t) = 2(1 + ε)t 1+2ε e t 2+2ε u ′′ (t) = 2(1 + ε)e t 2+2ε 2(1 + ε)t 2+4ε + (1 + 2ε)t 2ε thus ∆U − U = u ′′ (t) + j ′ (t) j(t) u ′ (t) − u(t) = e t 2+2ε 2(1 + ε)εt 2ε − 1 + e t 2+2ε ε ≥ 0. On the other hand, if t < t ε the function U is identically zero, so that ∆U − U ≥ 0 also for t ∈ (0, t ε ). To see that ∆U ≥ U in the sense of distributions on the whole manifold we take 0 ≤ ϕ ∈ C ∞ c (M ) and set M := M \ B tε (0). Then we computê M U (∆ϕ − ϕ) =ˆM U (∆ϕ − ϕ) = −ˆM g(∇ϕ, ∇U ) +ˆ∂ M U ∂ϕ ∂ν −ˆM U ϕ = −ˆM g(∇ϕ, ∇U ) −ˆM U ϕ =ˆM ∆U ϕ −ˆ∂ M ∂U ∂ν ϕ −ˆM U ϕ =ˆM ∆U ϕ +ˆ∂ Bt ε (0) ∂U ∂t ϕ −ˆM U ϕ =ˆM (∆U − U )ϕ +ˆ∂ Bt ε (0) u ′ ϕ ≥ 0. On the other hand we have: M |U |dV g = ω mˆ+ ∞ 0 u(t)j(t)dt =ˆ+ ∞ tε 1 t 1+ε dt < +∞. In conclusion, if we set V = −U we have V ∈ L 1 (M ) and (− ∆ +1)V ≥ 0 but V ≤ 0, which contradicts the validity of the L 1 -positivity preserving property on M . Definition 1. 1 . 1Let p ∈ [1, +∞], we say that (M, g) has the L p -positivity preserving property if every u ∈ L p (M ) satisfying (1.2) (−∆ + 1)u ≥ 0 in the sense of distributions is non-negative a.e. Date: December 23, 2021.The combination of these results lead Braverman, Milatovic and Schubin to formulate the following Conjecture (BMS). If (M, g) is a geodesically complete Riemannian manifold then the L 2 -positivity preserving property holds. Theorem D. Let (M, g) be a Riemannian manifold and let u ∈ L 1 loc (M ) be a solution of Lu ≥ 0 in the sense of distributions. Then for every Ω ⋐ M there exists a sequence {u k } ⊂ C ∞ (Ω) such that: Corollary E. Let (M, g) be a strongly α-non-parabolic Riemannian manifold and let u ∈ L 1 loc (M ) be a solution of Lu ≥ 0 in the sense of distributions. Then there exists a sequence {u k } ⊂ C ∞ (M ) such that: Remark 1 . 7 . 17Results such as Proposition C, Theorem D and Corollary E still hold if the constant 1 in the operator (1.1) is replaced by another positive constant. Actually, negative constants are also allowed as long as −L remains a positive operator. Proposition 2 . 2 . 22If (M, g) has the L ∞ -positivity preserving property, then it is stochastically complete. 2. 1 . 1From stochastic completeness to the L ∞ -positivity preserving property. The goal of this section is to set the ground towards proving the converse of Proposition 2.2.To this end, let (M, g) be a stochastically complete Riemannian manifold and take u ∈ L ∞ (M ) satisfying (−∆ + 1)u ≥ 0 in the sense of distributions. Our purpose is to show that u is non-negative almost everywhere or, equivalently, that the negative part u − = max{0, −u} = (−u) + vanishes a.e.. The next ingredient in our proof is the following Brezis-Kato inequality due toPigola and Veronelli, [25, Proposition 4.1] Theorem 2.4 (Brezis-Kato). Given a Riemannian manifold (M, g), if u ∈ L 1 loc (M ) satisfies Lu ≥ 0 in the sense of distributions, then u + ∈ L 1 loc (M ) and Lu + ≥ 0 in the sense of distributions. Corollary 2. 5 . 5Let (M, g) be a Riemannian manifold and let u ∈ L ∞ (M ) be a distributional solution of Lu ≥ 0. Then, for every relatively compact Ω ⋐ M there exists some u Ω ∈ C ∞ (Ω) which solves Lu Ω ≥ 0 in a strong sense and such that u ≤ u Ω ≤ 2 ess sup Ω u. Theorem 2. 8 . 8Let (M, g) be a Riemannian manifold and let u ∈ L ∞ (M ) satisfying Lu ≥ 0 in the sense of distributions. Then, there exists w ∈ C ∞ (M ) such that u ≤ w, Lw ≥ 0 in a strong sense and sup M w < +∞. Proof. We begin by observing that if u ∈ L ∞ (M ) then, setting c = u L ∞ (M) , we have Lc = −c ≤ 0 on M. Theorem 3 . 2 . 32Let (M, g) be a Riemannian manifold and let v ∈ L 1 loc (M ) be a solution of ∆ α v ≥ 0 in the sense of distributions. Then, for every Ω ⋐ M there exists a sequence {v k Proposition 3 . 5 . 35Given a Riemannian manifold (M, g) and Ω ⋐ M , if v ∈ L 1 (Ω) is α-subharmonic in the sense of distributions, then(i) v(x) ≤ m r (v)(x)for almost every x ∈ Ω and almost every r > 0; (ii) let 0 < s < r then m s (v)(x) ≤ m r (v)(x) almost everywhere in Ω; (iii) for almost every x ∈ Ω we have lim r→0 m r (v)(x) = v(x); (iv) for every r > 0 m r (v) is α-subharmonic in the sense of potential on Ω. Definition 3.8. A Riemannian manifold (M, g) is said to be α-non-parabolic if there exists a minimal positive Green function G for the operator ∆ α . Moreover, if this Green function satisfies(3.15)lim y→∞ G(x, y) = 0, the manifold M is said to be strongly α-non-parabolic.Note that compact Riemannian manifold are always α-parabolic thus we focus on the complete, non-compact case. It is also known that if (M, g) is a geodesically complete, α-non-parabolic manifold, thenis the volume of the geodesic ball of radius t and center x with respect to the measure α 2 dµ g . See for instance Theorem 9.7 of[10]. Furthermore, if we assume a non-negative m-Bakry-Émery Ricci tensor Ric m f := Ric + Hess(f ) − 1 m df ⊗ df ≥ 0 with f = −2 log α, it is possible to prove some Li-Yau type estimates for the heat kernel, see Theorems 5.6 and 5.8 in[5]. Integrating in time these estimates we obtain the following bounds for the Green functiondt.In particular if (3.16) holds true and Ric m f ≥ 0, the previous estimate implies that the manifold at hand is strongly α-non parabolic. It would be interesting to investigate which geometric conditions on the manifold (M, g) imply the existence of a function α such that(3.16)and Ric m f ≥ 0 hold true.A counterexample to the L 1 -positivity preserving propertyThis section is devoted to the proof of Theorem B. Fix ε > 0 and consider the 2dimensional model manifold M = R + × σ S 1 , that is R + ×S 1 with the metric g = dt 2 + σ 2 (t)dθ 2 . Here dθ 2 is the standard round metric on S 1 and σ = σ ε is a C ∞ ((0, +∞)) function satisfyingHere t ε = (2(1 + ε)ε) −1/2ε and the function j is defined asAs a result, outside of a compact set we have the following asymptotic estimate for the Gaussian curvature:= −(1 + ε) 2t 2ε + 4(1 + ε)t 2+4ε + (2 + ε) 1 t 2 g ∼ −4(1 + ε) 2 t 2+4ε g as t → +∞. Next we define the function U (t, θ) = u(t) = (e t 2+2ε − e t 2+2ε ε ) + and prove that it satisfies ∆U ≥ U Maximum principles and geometric applications. L J Alías, P Mastrolia, M Rigoli, SpringerChamAlías, L. J., Mastrolia, P., and Rigoli, M. Maximum principles and geometric applications. Springer Monographs in Mathematics. Springer, Cham, 2016. Laplacian cut-offs, porous and fast diffusion on manifolds and other applications. D Bianchi, A G Setti, Calc. Var. Partial Differential Equations. 57433Bianchi, D., and Setti, A. G. Laplacian cut-offs, porous and fast diffusion on manifolds and other applications. Calc. Var. Partial Differential Equations 57, 1 (2018), Paper No. 4, 33. Subharmonic functions in sub-Riemannian settings. A Bonfiglioli, E Lanconelli, Journal of the European Mathematical Society. 15Bonfiglioli, A., and Lanconelli, E. Subharmonic functions in sub-Riemannian settings. Journal of the European Mathematical Society 15, 2 (2013), 387-441. Essential selfadjointness of Schrödinger-type operators on manifolds. M Braverman, O Milatovich, M Shubin, Uspekhi Mat. Nauk. 57346Braverman, M., Milatovich, O., and Shubin, M. Essential selfadjointness of Schrödinger-type operators on manifolds. Uspekhi Mat. Nauk 57, 4(346) (2002), 3-58. Heat kernel estimates and the essential spectrum on weighted manifolds. N Charalambous, Z Lu, J. Geom. Anal. 25Charalambous, N., and Lu, Z. Heat kernel estimates and the essential spectrum on weighted mani- folds. J. Geom. Anal. 25, 1 (2015), 536-563. On mean value formulas for solutions to second order linear pdes. G Cupini, E Lanconelli, Classe di scienze. 22Annali della Scuola Normale Superiore di PisaCupini, G., and Lanconelli, E. On mean value formulas for solutions to second order linear pdes. Annali della Scuola Normale Superiore di Pisa. Classe di scienze 22, 2 (2021), 777-809. The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature. D Fischer-Colbrie, R Schoen, Communications on Pure and Applied Mathematics. 33Fischer-Colbrie, D., and Schoen, R. The structure of complete stable minimal surfaces in 3- manifolds of non-negative scalar curvature. Communications on Pure and Applied Mathematics 33, 2 (1980), 199-211. Elliptic partial differential equations of second order. D Gilbarg, N S Trudinger, Classics in Mathematics. Springer-VerlagReprint of the 1998 editionGilbarg, D., and Trudinger, N. S. Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. A Grigor&apos;yan, Bull. Amer. Math. Soc. (N.S.). 36Grigor'yan, A. Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Amer. Math. Soc. (N.S.) 36, 2 (1999), 135-249. Heat kernels on weighted manifolds and applications. A Grigor&apos;yan, The ubiquitous heat kernel. Providence, RI398Grigor'yan, A. Heat kernels on weighted manifolds and applications. In The ubiquitous heat kernel, vol. 398 of Contemp. Math. Amer. Math. Soc., Providence, RI, 2006, pp. 93-191. Heat kernel and analysis on manifolds. A Grigor&apos;yan, AMS/IP Studies in Advanced Mathematics. Providence, RI; Boston, MAInternational Press47Grigor'yan, A. Heat kernel and analysis on manifolds, vol. 47 of AMS/IP Studies in Advanced Mathematics. American Mathematical Society, Providence, RI; International Press, Boston, MA, 2009. Nonlinear characterizations of stochastic completeness. G Grillo, K Ishige, M Muratori, J. Math. Pures Appl. 9Grillo, G., Ishige, K., and Muratori, M. Nonlinear characterizations of stochastic completeness. J. Math. Pures Appl. (9) 139 (2020), 63-82. Sequences of Laplacian cut-off functions. B Güneysu, J. Geom. Anal. 26Güneysu, B. Sequences of Laplacian cut-off functions. J. Geom. Anal. 26, 1 (2016), 171-184. The BMS conjecture. B Güneysu, arXiv:1709.07463Ulmer Seminare. 20ArXiv preprintGüneysu, B. The BMS conjecture. Ulmer Seminare 20 (2017), 97-101. ArXiv preprint: arXiv:1709.07463. Covariant Schrödinger semigroups on Riemannian manifolds. B Güneysu, Birkhäuser/Springer264Chamof Operator Theory: Advances and ApplicationsGüneysu, B. Covariant Schrödinger semigroups on Riemannian manifolds, vol. 264 of Operator The- ory: Advances and Applications. Birkhäuser/Springer, Cham, 2017. Path integrals and the essential self-adjointness of differential operators on noncompact manifolds. B Güneysu, O Post, Mathematische Zeitschrift. 275Güneysu, B., and Post, O. Path integrals and the essential self-adjointness of differential operators on noncompact manifolds. Mathematische Zeitschrift 275, 1-2 (2013), 331-348. Monotone iterative techniques for discontinuous nonlinear differential equations. S Heikkilä, V Lakshmikantham, Heikkilä, S., and Lakshmikantham, V. Monotone iterative techniques for discontinuous nonlinear differential equations. Routledge, 2017. Density and non-density of C ∞ c ֒→ W k,p on complete manifolds with curvature bounds. S Honda, L Mari, M Rimoldi, G Veronelli, Nonlinear Anal. 21126Paper No. 112429Honda, S., Mari, L., Rimoldi, M., and Veronelli, G. Density and non-density of C ∞ c ֒→ W k,p on complete manifolds with curvature bounds. Nonlinear Anal. 211 (2021), Paper No. 112429, 26. Heat semigroup on a complete Riemannian manifold. P Hsu, Ann. Probab. 17Hsu, P. Heat semigroup on a complete Riemannian manifold. Ann. Probab. 17, 3 (1989), 1248-1254. Schrödinger operators with singular potentials. T Kato, Israel J. Math. 13Kato, T. Schrödinger operators with singular potentials. Israel J. Math. 13 (1972), 135-148 (1973). Some functional properties on cartan-hadamard manifolds of very negative curvature. L Marini, G Veronelli, arXiv:2105.09024ArXiv preprintMarini, L., and Veronelli, G. Some functional properties on cartan-hadamard manifolds of very negative curvature, 2021. ArXiv preprint: arXiv:2105.09024. Mean value theorems on manifolds. L Ni, Asian J. Math. 11Ni, L. Mean value theorems on manifolds. Asian J. Math. 11, 2 (2007), 277-304. A remark on the maximum principle and stochastic completeness. S Pigola, M Rigoli, A G Setti, Proc. Amer. Math. Soc. 131Pigola, S., Rigoli, M., and Setti, A. G. A remark on the maximum principle and stochastic completeness. Proc. Amer. Math. Soc. 131, 4 (2003), 1283-1288. Maximum principles on Riemannian manifolds and applications. S Pigola, M Rigoli, A G Setti, Mem. Amer. Math. Soc. 17499Pigola, S., Rigoli, M., and Setti, A. G. Maximum principles on Riemannian manifolds and applications. Mem. Amer. Math. Soc. 174, 822 (2005), x+99. . S Pigola, G Veronelli, Positivity, arXiv:2105.14847M. Braverman, O. Milatovic and M. ShubinArXiv preprintPigola, S., and Veronelli, G. L p Positivity Preserving and a conjecture by M. Braverman, O. Milatovic and M. Shubin, 2021. ArXiv preprint: arXiv:2105.14847. Maximum principles in differential equations. M H Protter, H F Weinberger, Springer-VerlagNew YorkCorrected reprint of the 1967 originalProtter, M. H., and Weinberger, H. F. Maximum principles in differential equations. Springer- Verlag, New York, 1984. Corrected reprint of the 1967 original. Scalar curvature and conformal deformation of hyperbolic space. A Ratto, M Rigoli, L Véron, J. Funct. Anal. 121Ratto, A., Rigoli, M., and Véron, L. Scalar curvature and conformal deformation of hyperbolic space. J. Funct. Anal. 121, 1 (1994), 15-77. Monotone methods in nonlinear elliptic and parabolic boundary value problems. D H Sattinger, Indiana University Mathematics Journal. 21Sattinger, D. H. Monotone methods in nonlinear elliptic and parabolic boundary value problems. Indiana University Mathematics Journal 21, 11 (1972), 979-1000. Essential self-adjointness for semi-bounded magnetic schrödinger operators on non-compact manifolds. M Shubin, Journal of Functional Analysis. 186Shubin, M. Essential self-adjointness for semi-bounded magnetic schrödinger operators on non-compact manifolds. Journal of Functional Analysis 186, 1 (2001), 92-116. On the adjoint of an elliptic linear differential operator and its potential theory. P Sjögren, Arkiv för Matematik. 11Sjögren, P. On the adjoint of an elliptic linear differential operator and its potential theory. Arkiv för Matematik 11, 1 (1973), 153-165. Some function-theoretic properties of complete Riemannian manifold and their applications to geometry. S T Yau, Indiana Univ. Math. J. 25Yau, S. T. Some function-theoretic properties of complete Riemannian manifold and their applications to geometry. Indiana Univ. Math. J. 25, 7 (1976), 659-670. Milano Email address: a.bisterzo@campus. unimib.it (L. Marini) Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, Via R. Cozzi55I-20125I-20125. Milano Email address: [email protected](A. Bisterzo) Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano- Bicocca, Via R. Cozzi 55, I-20125, Milano Email address: [email protected] (L. Marini) Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano- Bicocca, Via R. Cozzi 55, I-20125, Milano Email address: [email protected]
{'fraction_non_alphanumeric': 0.08392788818617632, 'fraction_numerical': 0.028493619319844253, 'mean_word_length': 3.587712958182757, 'pattern_counts': {'":': 0, '<': 11, '<?xml version=': 0, '>': 26, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 3, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 9, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'We say that a Riemannian manifold satisfies the L p -positivity preserving property if (−∆ + 1)u ≥ 0 in a distributional sense implies u ≥ 0 for all u ∈ L p . While geodesic completeness of the manifold at hand ensures the L p -positivity preserving property for all p ∈ (1, +∞), when p = +∞ some assumptions are needed. In this paper we show that the L ∞ -positivity preserving property is in fact equivalent to stochastic completeness, i.e., the fact that the minimal heat kernel of the manifold preserves probability. The result is achieved via some monotone approximation results for distributional solutions of −∆ + 1 ≥ 0, which are of independent interest.', 'arxivid': '2112.11774', 'author': ['Andrea Bisterzo ', 'Ludovico Marini ', 'Andrea Bisterzo ', 'Ludovico Marini '], 'authoraffiliation': [], 'corpusid': 245385389, 'doi': '10.1007/s11118-022-10041-w', 'github_urls': [], 'n_tokens_mistral': 16168, 'n_tokens_neox': 13871, 'n_words': 8252, 'pdfsha': '5044ede2ce20b5222bffd267f1231b63cb0cc7f2', 'pdfurls': ['https://export.arxiv.org/pdf/2112.11774v1.pdf'], 'title': ['THE L ∞ -POSITIVITY PRESERVING PROPERTY AND STOCHASTIC COMPLETENESS', 'THE L ∞ -POSITIVITY PRESERVING PROPERTY AND STOCHASTIC COMPLETENESS', 'THE L ∞ -POSITIVITY PRESERVING PROPERTY AND STOCHASTIC COMPLETENESS', 'THE L ∞ -POSITIVITY PRESERVING PROPERTY AND STOCHASTIC COMPLETENESS'], 'venue': []}
arxiv
EXPANSIONS OF ITERATED STRATONOVICH STOCHASTIC INTEGRALS FROM THE TAYLOR-STRATONOVICH EXPANSION BASED ON MULTIPLE TRIGONOMETRIC FOURIER SERIES. COMPARISON WITH THE MILSTEIN EXPANSION Sep 2022 Dmitriy F Kuznetsov EXPANSIONS OF ITERATED STRATONOVICH STOCHASTIC INTEGRALS FROM THE TAYLOR-STRATONOVICH EXPANSION BASED ON MULTIPLE TRIGONOMETRIC FOURIER SERIES. COMPARISON WITH THE MILSTEIN EXPANSION Sep 2022 The article is devoted to comparison of the Milstein expansion of iterated Stratonovich stochastic integrals with the method of expansion of iterated stochastic integrals based on generalized multiple Fourier series. We consider the practical material connected with the expansions of iterated Stratonovich stochastic integrals from the Taylor-Stratonovich expansion based on multiple trigonometric Fourier series. The comparison of effectiveness of the Fourier-Legendre series as well as the trigonomertic Fourier series for expansions of iterated Stratonovich stochastic integrals is considered.(i = 1, . . . , m) of this process are independent. Consider an Ito stochastic differential equation (SDE) in the integral form Mathematics Subject Classification: 60H05, 60H10, 42B05, 42C10. (1) x t = x 0 + t 0 a(x τ , τ )dτ + t 0 B(x τ , τ )df τ , x 0 = x(0, ω). Here x t is some n-dimensional stochastic process satisfying the equation (1). The nonrandom functions a : R n × [0, T ] → R n , B : R n × [0, T ] → R n×m guarantee the existence and uniqueness up to stochastic equivalence of a solution of the equation (1) [1]. The second integral on the right-hand side of (1) is interpreted as an Ito stochastic integral. Let x 0 be an n-dimensional random variable, which is F 0 -measurable and M |x 0 | 2 < ∞ (M denotes a mathematical expectation). We assume that x 0 and f t − f 0 are independent when t > 0. It is well known that one of the effective approaches to the numerical integration of Ito SDEs is an approach based on the Taylor-Ito and Taylor-Stratonovich expansions [2]- [7]. The most important feature of such expansions is a presence in them of the so-called iterated Ito and Stratonovich stochastic integrals, which play the key role for solving the problem of numerical integration of Ito SDEs and have the following form (2) J[ψ (k) ] T,t = T t ψ k (t k ) . . . t2 t ψ 1 (t 1 )dw (i1) t1 . . . dw (i k ) t k ,(3)J * [ψ (k) ] T,t = * t T ψ k (t k ) . . . * t t2 ψ 1 (t 1 )dw (i1) t1 . . . dw (i k ) t k , where every ψ l (τ ) (l = 1, . . . , k) is a nonrandom function on [t, T ], w and * denote Ito and Stratonovich stochastic integrals, respectively (in this paper, we use the definition of the Stratonovich stochastic integral from [3]). Note that ψ l (τ ) ≡ 1 (l = 1, . . . , k) and i 1 , . . . , i k = 0, 1, . . . , m for the classical Taylor-Ito and Taylor-Stratonovich expansions [2]- [7] and ψ l (τ ) ≡ (t − τ ) q l (l = 1, . . . , k; q 1 , . . . , q k = 0, 1, 2, . . .) and i 1 , . . . , i k = 1, . . . , m for the unified Taylor-Ito and Taylor-Stratonovich expansions [8]- [24]. Milstein Expansion and Method of Generatized Multiple Fourier Series Milstein G.N. proposed [2] (1988) an approach to the expansion of iterated stochastic integrals based on the trigonometric Fourier expansion of the Brownian bridge process (version of the so-called Karhunen-Loeve expansion). Let us consider the Brownian bridge process [2] (4) f t − t ∆ f ∆ , t ∈ [0, ∆], ∆ > 0, where f t is a standard m-dimensional Wiener process with independent components f Consider the componentwise Karhunen-Loeve expansion of the process (4) [2] (5) f (i) t − t ∆ f (i) ∆ = 1 2 a i,0 + ∞ r=1 a i,r cos 2πrt ∆ + b i,r sin 2πrt ∆ converging in the mean-square sense, where a i,r = 2 ∆ ∆ 0 f (i) s − s ∆ f (i) ∆ cos 2πrs ∆ ds, b i,r = 2 ∆ ∆ 0 f (i) s − s ∆ f (i) ∆ sin 2πrs ∆ ds, where r = 0, 1, . . . ; i = 1, . . . , m. It is easy to demonstrate [2] that the random variables a i,r , b i,r are Gaussian ones and they satisfy the following relations M {a i,r b i,r } = M {a i,r b i,k } = 0, M {a i,r a i,k } = M {b i,r b i,k } = 0, M {a i1,r a i2,r } = M {b i1,r b i2,r } = 0, M a 2 i,r = M b 2 i,r = ∆ 2π 2 r 2 , where i, i 1 , i 2 = 1, . . . , m; r = k; i 1 = i 2 . According to (5), we have (6) f (i) t = f (i) ∆ t ∆ + 1 2 a i,0 + ∞ r=1 a i,r cos 2πrt ∆ + b i,r sin 2πrt ∆ , where the series converges in the mean-square sense. Note that the trigonometric functions are the eigenfunctions of the covariance operator of the Brownian bridge process. That is why the basis functions are the trigonometric functions in the considered approach. In [2] Milstein G.N. proposed to expand (2) or (3) (for the case k = 2 and ψ 1 (s), ψ 2 (s) ≡ 1) into iterated series of products of standard Gaussian random variables by representing the Wiener process as the series (6). To obtain the Milstein expansion of (2) or (3), the truncated expansions (6) of components of the Wiener process f s must be iteratively substituted in the single integrals, and the integrals must be calculated, starting from the innermost integral. This is a complicated procedure that obviously does not lead to a general expansion of (2) or (3) valid for an arbitrary multiplicity k. For this reason, only expansions of simplest single, double, and triple integrals (2), (3) were obtained (see [2]- [7]). At that, in [2], [7] the case ψ 1 (s), ψ 2 (s) ≡ 1 and i 1 , i 2 = 0, 1, . . . , m is considered. In [3]- [6] the attempt to consider the case ψ 1 (s), ψ 2 (s), ψ 3 (s) ≡ 1 and i 1 , i 2 , i 3 = 0, 1, . . . , m is realized. It should be noted that the authors of the works [3] (Sect. 5.8, pp. 202-204), [4] (pp. 82-84), [5] (pp. 438-439), [6] (pp. 263-264) use the Wong-Zakai approximation [38]- [40] (without rigorous proof) within the frames of the Milstein approach [2] based on the series expansion of the Brownian bridge process. See discussion in Sect. 7 of this paper for details. Let us consider an another approach to the expansion of iterated stochastic integrals [10]- [37], which is reffered to as the method of generalized multiple Fourier series. Suppose that every ψ l (τ ) (l = 1, . . . , k) is a nonrandom function from the space L 2 ([t, T ]). Define the following function on the hypercube [t, T ] k (7) K(t 1 , . . . , t k ) =          ψ 1 (t 1 ) . . . ψ k (t k ), t 1 < . . . < t k 0, otherwise = k l=1 ψ l (t l ) k−1 l=1 1 {t l <t l+1 } , where t 1 , . . . , t k ∈ [t, T ] (k ≥ 2) and K(t 1 ) ≡ ψ 1 (t 1 ) for t 1 ∈ [t, T ]. Here 1 A denotes the indicator of the set A. Suppose that {φ j (x)} ∞ j=0 is a complete orthonormal system of functions in the space L 2 ([t, T ]). The function K(t 1 , . . . , t k ) belongs to the space L 2 ([t, T ] k ). At this situation it is well known that the generalized multiple Fourier series of K(t 1 , . . . , t k ) ∈ L 2 ([t, T ] k ) is converging to K(t 1 , . . . , t k ) in the hypercube [t, T ] k in the mean-square sense, i.e. (8) lim p1,...,p k →∞ K(t 1 , . . . , t k ) − p1 j1=0 . . . p k j k =0 C j k ...j1 k l=1 φ j l (t l ) L2([t,T ] k ) = 0, where (9) C j k ...j1 = [t,T ] k K(t 1 , . . . , t k ) k l=1 φ j l (t l )dt 1 . . . dt k is the Fourier coefficient and f L2([t,T ] k ) =    [t,T ] k f 2 (t 1 , . . . , t k )dt 1 . . . dt k    1/2 . Consider the partition {τ j } N j=0 of [t, T ] such that (10) t = τ 0 < . . . < τ N = T, ∆ N = max 0≤j≤N −1 ∆τ j → 0 if N → ∞, ∆τ j = τ j+1 − τ j . Theorem 1 [10] (2006), [11]- [37]. Suppose that every ψ l (τ ) (l = 1, . . . , k) is a continuous nonrandom function on [t, T ] and {φ j (x)} ∞ j=0 is a complete orthonormal system of continuous functions in the space L 2 ([t, T ]). Then J[ψ (k) ] T,t = l.i.m. p1,...,p k →∞ p1 j1=0 . . . p k j k =0 C j k ...j1 k l=1 ζ (i l ) j l − (11) − l.i.m. N →∞ (l1,...,l k )∈G k φ j1 (τ l1 )∆w (i1) τ l 1 . . . φ j k (τ l k )∆w (i k ) τ l k , where G k = H k \L k , H k = {(l 1 , . . . , l k ) : l 1 , . . . , l k = 0, 1, . . . , N − 1}, L k = {(l 1 , . . . , l k ) : l 1 , . . . , l k = 0, 1, . . . , N − 1; l g = l r (g = r); g, r = 1, . . . , k}, l.i.m. is a limit in the mean-square sense, i 1 , . . . , i k = 0, 1, . . . , m, (12) ζ (i) j = T t φ j (s)dw (i) s are independent standard Gaussian random variables for various i or j (if i = 0), C j k ...j1 is the Fourier coefficient (9), ∆w (i) τj = w (i) τj+1 − w (i) τj (i = 0, 1, . . . , m), {τ j } N j=0 is the partition of [t, T ], which satisfies the condition (10). In order to evaluate the significance of Theorem 1 for practice we will demonstrate its transformed particular cases for k = 1, . . . , 6 [10]- [37] (13) J[ψ (1) ] T,t = l.i.m. p1→∞ p1 j1=0 C j1 ζ (i1) j1 ,(14) J[ψ (2) ] T,t = l.i.m. p1,p2→∞ p1 j1=0 p2 j2=0 C j2j1 ζ (i1) j1 ζ (i2) j2 − 1 {i1=i2 =0} 1 {j1=j2} , J[ψ (3) ] T,t = l.i.m. p1,...,p3→∞ p1 j1=0 p2 j2=0 p3 j3=0 C j3j2j1 ζ (i1) j1 ζ (i2) j2 ζ (i3) j3 − (15) −1 {i1=i2 =0} 1 {j1=j2} ζ (i3) j3 − 1 {i2=i3 =0} 1 {j2=j3} ζ (i1) j1 − 1 {i1=i3 =0} 1 {j1=j3} ζ (i2) j2 , J[ψ (4) ] T,t = l.i.m. p1,...,p4→∞ p1 j1=0 . . . p4 j4=0 C j4...j1 4 l=1 ζ (i l ) j l − −1 {i1=i2 =0} 1 {j1=j2} ζ (i3) j3 ζ (i4) j4 − 1 {i1=i3 =0} 1 {j1=j3} ζ (i2) j2 ζ (i4) j4 − −1 {i1=i4 =0} 1 {j1=j4} ζ (i2) j2 ζ (i3) j3 − 1 {i2=i3 =0} 1 {j2=j3} ζ (i1) j1 ζ (i4) j4 − −1 {i2=i4 =0} 1 {j2=j4} ζ (i1) j1 ζ (i3) j3 − 1 {i3=i4 =0} 1 {j3=j4} ζ (i1) j1 ζ (i2) j2 + +1 {i1=i2 =0} 1 {j1=j2} 1 {i3=i4 =0} 1 {j3=j4} + +1 {i1=i3 =0} 1 {j1=j3} 1 {i2=i4 =0} 1 {j2=j4} + (16) + 1 {i1=i4 =0} 1 {j1=j4} 1 {i2=i3 =0} 1 {j2=j3} , J[ψ (5) ] T,t = l.i.m. p1,...,p5→∞ p1 j1=0 . . . p5 j5=0 C j5...j1 5 l=1 ζ (i l ) j l − −1 {i1=i2 =0} 1 {j1=j2} ζ (i3) j3 ζ (i4) j4 ζ (i5) j5 − 1 {i1=i3 =0} 1 {j1=j3} ζ (i2) j2 ζ (i4) j4 ζ (i5) j5 − −1 {i1=i4 =0} 1 {j1=j4} ζ (i2) j2 ζ (i3) j3 ζ (i5) j5 − 1 {i1=i5 =0} 1 {j1=j5} ζ (i2) j2 ζ (i3) j3 ζ (i4) j4 − −1 {i2=i3 =0} 1 {j2=j3} ζ (i1) j1 ζ (i4) j4 ζ (i5) j5 − 1 {i2=i4 =0} 1 {j2=j4} ζ (i1) j1 ζ (i3) j3 ζ (i5) j5 − −1 {i2=i5 =0} 1 {j2=j5} ζ (i1) j1 ζ (i3) j3 ζ (i4) j4 − 1 {i3=i4 =0} 1 {j3=j4} ζ (i1) j1 ζ (i2) j2 ζ (i5) j5 − −1 {i3=i5 =0} 1 {j3=j5} ζ (i1) j1 ζ (i2) j2 ζ (i4) j4 − 1 {i4=i5 =0} 1 {j4=j5} ζ (i1) j1 ζ (i2) j2 ζ (i3) j3 + +1 {i1=i2 =0} 1 {j1=j2} 1 {i3=i4 =0} 1 {j3=j4} ζ (i5) j5 + 1 {i1=i2 =0} 1 {j1=j2} 1 {i3=i5 =0} 1 {j3=j5} ζ (i4) j4 + +1 {i1=i2 =0} 1 {j1=j2} 1 {i4=i5 =0} 1 {j4=j5} ζ (i3) j3 + 1 {i1=i3 =0} 1 {j1=j3} 1 {i2=i4 =0} 1 {j2=j4} ζ (i5) j5 + +1 {i1=i3 =0} 1 {j1=j3} 1 {i2=i5 =0} 1 {j2=j5} ζ (i4) j4 + 1 {i1=i3 =0} 1 {j1=j3} 1 {i4=i5 =0} 1 {j4=j5} ζ (i2) j2 + +1 {i1=i4 =0} 1 {j1=j4} 1 {i2=i3 =0} 1 {j2=j3} ζ (i5) j5 + 1 {i1=i4 =0} 1 {j1=j4} 1 {i2=i5 =0} 1 {j2=j5} ζ (i3) j3 + +1 {i1=i4 =0} 1 {j1=j4} 1 {i3=i5 =0} 1 {j3=j5} ζ (i2) j2 + 1 {i1=i5 =0} 1 {j1=j5} 1 {i2=i3 =0} 1 {j2=j3} ζ (i4) j4 + +1 {i1=i5 =0} 1 {j1=j5} 1 {i2=i4 =0} 1 {j2=j4} ζ (i3) j3 + 1 {i1=i5 =0} 1 {j1=j5} 1 {i3=i4 =0} 1 {j3=j4} ζ (i2) j2 + +1 {i2=i3 =0} 1 {j2=j3} 1 {i4=i5 =0} 1 {j4=j5} ζ (i1) j1 + 1 {i2=i4 =0} 1 {j2=j4} 1 {i3=i5 =0} 1 {j3=j5} ζ (i1) j1 + (17) + 1 {i2=i5 =0} 1 {j2=j5} 1 {i3=i4 =0} 1 {j3=j4} ζ (i1) j1 , J[ψ (6) ] T,t = l.i.m. j l − −1 {i1=i6 =0} 1 {j1=j6} ζ (i2) j2 ζ (i3) j3 ζ (i4) j4 ζ (i5) j5 − 1 {i2=i6 =0} 1 {j2=j6} ζ (i1) j1 ζ (i3) j3 ζ (i4) j4 ζ (i5) j5 − −1 {i3=i6 =0} 1 {j3=j6} ζ (i1) j1 ζ (i2) j2 ζ (i4) j4 ζ (i5) j5 − 1 {i4=i6 =0} 1 {j4=j6} ζ (i1) j1 ζ (i2) j2 ζ (i3) j3 ζ (i5) j5 − −1 {i5=i6 =0} 1 {j5=j6} ζ (i1) j1 ζ (i2) j2 ζ (i3) j3 ζ (i4) j4 − 1 {i1=i2 =0} 1 {j1=j2} ζ (i3) j3 ζ (i4) j4 ζ (i5) j5 ζ (i6) j6 − −1 {i1=i3 =0} 1 {j1=j3} ζ (i2) j2 ζ (i4) j4 ζ (i5) j5 ζ (i6) j6 − 1 {i1=i4 =0} 1 {j1=j4} ζ (i2) j2 ζ (i3) j3 ζ (i5) j5 ζ (i6) j6 − −1 {i1=i5 =0} 1 {j1=j5} ζ (i2) j2 ζ (i3) j3 ζ (i4) j4 ζ (i6) j6 − 1 {i2=i3 =0} 1 {j2=j3} ζ (i1) j1 ζ (i4) j4 ζ (i5) j5 ζ (i6) j6 − −1 {i2=i4 =0} 1 {j2=j4} ζ (i1) j1 ζ (i3) j3 ζ (i5) j5 ζ (i6) j6 − 1 {i2=i5 =0} 1 {j2=j5} ζ (i1) j1 ζ (i3) j3 ζ (i4) j4 ζ (i6) j6 − −1 {i3=i4 =0} 1 {j3=j4} ζ (i1) j1 ζ (i2) j2 ζ (i5) j5 ζ (i6) j6 − 1 {i3=i5 =0} 1 {j3=j5} ζ (i1) j1 ζ (i2) j2 ζ (i4) j4 ζ (i6) j6 − −1 {i4=i5 =0} 1 {j4=j5} ζ (i1) j1 ζ (i2) j2 ζ (i3) j3 ζ (i6) j6 + +1 {i1=i2 =0} 1 {j1=j2} 1 {i3=i4 =0} 1 {j3=j4} ζ (i5) j5 ζ (i6) j6 + 1 {i1=i2 =0} 1 {j1=j2} 1 {i3=i5 =0} 1 {j3=j5} ζ (i4) j4 ζ (i6) j6 + +1 {i1=i2 =0} 1 {j1=j2} 1 {i4=i5 =0} 1 {j4=j5} ζ (i3) j3 ζ (i6) j6 + 1 {i1=i3 =0} 1 {j1=j3} 1 {i2=i4 =0} 1 {j2=j4} ζ (i5) j5 ζ (i6) j6 + +1 {i1=i3 =0} 1 {j1=j3} 1 {i2=i5 =0} 1 {j2=j5} ζ (i4) j4 ζ (i6) j6 + 1 {i1=i3 =0} 1 {j1=j3} 1 {i4=i5 =0} 1 {j4=j5} ζ (i2) j2 ζ (i6) j6 + +1 {i1=i4 =0} 1 {j1=j4} 1 {i2=i3 =0} 1 {j2=j3} ζ (i5) j5 ζ (i6) j6 + 1 {i1=i4 =0} 1 {j1=j4} 1 {i2=i5 =0} 1 {j2=j5} ζ (i3) j3 ζ (i6) j6 + +1 {i1=i4 =0} 1 {j1=j4} 1 {i3=i5 =0} 1 {j3=j5} ζ (i2) j2 ζ (i6) j6 + 1 {i1=i5 =0} 1 {j1=j5} 1 {i2=i3 =0} 1 {j2=j3} ζ (i4) j4 ζ (i6) j6 + +1 {i1=i5 =0} 1 {j1=j5} 1 {i2=i4 =0} 1 {j2=j4} ζ (i3) j3 ζ (i6) j6 + 1 {i1=i5 =0} 1 {j1=j5} 1 {i3=i4 =0} 1 {j3=j4} ζ (i2) j2 ζ (i6) j6 + +1 {i2=i3 =0} 1 {j2=j3} 1 {i4=i5 =0} 1 {j4=j5} ζ (i1) j1 ζ (i6) j6 + 1 {i2=i4 =0} 1 {j2=j4} 1 {i3=i5 =0} 1 {j3=j5} ζ (i1) j1 ζ (i6) j6 + +1 {i2=i5 =0} 1 {j2=j5} 1 {i3=i4 =0} 1 {j3=j4} ζ (i1) j1 ζ (i6) j6 + 1 {i6=i1 =0} 1 {j6=j1} 1 {i3=i4 =0} 1 {j3=j4} ζ (i2) j2 ζ (i5) j5 + +1 {i6=i1 =0} 1 {j6=j1} 1 {i3=i5 =0} 1 {j3=j5} ζ (i2) j2 ζ (i4) j4 + 1 {i6=i1 =0} 1 {j6=j1} 1 {i2=i5 =0} 1 {j2=j5} ζ (i3) j3 ζ (i4) j4 + +1 {i6=i1 =0} 1 {j6=j1} 1 {i2=i4 =0} 1 {j2=j4} ζ (i3) j3 ζ (i5) j5 + 1 {i6=i1 =0} 1 {j6=j1} 1 {i4=i5 =0} 1 {j4=j5} ζ (i2) j2 ζ (i3) j3 + +1 {i6=i1 =0} 1 {j6=j1} 1 {i2=i3 =0} 1 {j2=j3} ζ (i4) j4 ζ (i5) j5 + 1 {i6=i2 =0} 1 {j6=j2} 1 {i3=i5 =0} 1 {j3=j5} ζ (i1) j1 ζ (i4) j4 + +1 {i6=i2 =0} 1 {j6=j2} 1 {i4=i5 =0} 1 {j4=j5} ζ (i1) j1 ζ (i3) j3 + 1 {i6=i2 =0} 1 {j6=j2} 1 {i3=i4 =0} 1 {j3=j4} ζ (i1) j1 ζ (i5) j5 + +1 {i6=i2 =0} 1 {j6=j2} 1 {i1=i5 =0} 1 {j1=j5} ζ (i3) j3 ζ (i4) j4 + 1 {i6=i2 =0} 1 {j6=j2} 1 {i1=i4 =0} 1 {j1=j4} ζ (i3) j3 ζ (i5) j5 + +1 {i6=i2 =0} 1 {j6=j2} 1 {i1=i3 =0} 1 {j1=j3} ζ (i4) j4 ζ (i5) j5 + 1 {i6=i3 =0} 1 {j6=j3} 1 {i2=i5 =0} 1 {j2=j5} ζ (i1) j1 ζ (i4) j4 + +1 {i6=i3 =0} 1 {j6=j3} 1 {i4=i5 =0} 1 {j4=j5} ζ (i1) j1 ζ (i2) j2 + 1 {i6=i3 =0} 1 {j6=j3} 1 {i2=i4 =0} 1 {j2=j4} ζ (i1) j1 ζ (i5) j5 + +1 {i6=i3 =0} 1 {j6=j3} 1 {i1=i5 =0} 1 {j1=j5} ζ (i2) j2 ζ (i4) j4 + 1 {i6=i3 =0} 1 {j6=j3} 1 {i1=i4 =0} 1 {j1=j4} ζ (i2) j2 ζ (i5) j5 + +1 {i6=i3 =0} 1 {j6=j3} 1 {i1=i2 =0} 1 {j1=j2} ζ (i4) j4 ζ (i5) j5 + 1 {i6=i4 =0} 1 {j6=j4} 1 {i3=i5 =0} 1 {j3=j5} ζ (i1) j1 ζ (i2) j2 + +1 {i6=i4 =0} 1 {j6=j4} 1 {i2=i5 =0} 1 {j2=j5} ζ (i1) j1 ζ (i3) j3 + 1 {i6=i4 =0} 1 {j6=j4} 1 {i2=i3 =0} 1 {j2=j3} ζ (i1) j1 ζ (i5) j5 + +1 {i6=i4 =0} 1 {j6=j4} 1 {i1=i5 =0} 1 {j1=j5} ζ (i2) j2 ζ (i3) j3 + 1 {i6=i4 =0} 1 {j6=j4} 1 {i1=i3 =0} 1 {j1=j3} ζ (i2) j2 ζ (i5) j5 + +1 {i6=i4 =0} 1 {j6=j4} 1 {i1=i2 =0} 1 {j1=j2} ζ (i3) j3 ζ (i5) j5 + 1 {i6=i5 =0} 1 {j6=j5} 1 {i3=i4 =0} 1 {j3=j4} ζ (i1) j1 ζ (i2) j2 + +1 {i6=i5 =0} 1 {j6=j5} 1 {i2=i4 =0} 1 {j2=j4} ζ (i1) j1 ζ (i3) j3 + 1 {i6=i5 =0} 1 {j6=j5} 1 {i2=i3 =0} 1 {j2=j3} ζ (i1) j1 ζ (i4) j4 + +1 {i6=i5 =0} 1 {j6=j5} 1 {i1=i4 =0} 1 {j1=j4} ζ (i2) j2 ζ (i3) j3 + 1 {i6=i5 =0} 1 {j6=j5} 1 {i1=i3 =0} 1 {j1=j3} ζ (i2) j2 ζ (i4) j4 + +1 {i6=i5 =0} 1 {j6=j5} 1 {i1=i2 =0} 1 {j1=j2} ζ (i3) j3 ζ (i4) j4 − −1 {i6=i1 =0} 1 {j6=j1} 1 {i2=i5 =0} 1 {j2=j5} 1 {i3=i4 =0} 1 {j3=j4} − −1 {i6=i1 =0} 1 {j6=j1} 1 {i2=i4 =0} 1 {j2=j4} 1 {i3=i5 =0} 1 {j3=j5} − −1 {i6=i1 =0} 1 {j6=j1} 1 {i2=i3 =0} 1 {j2=j3} 1 {i4=i5 =0} 1 {j4=j5} − −1 {i6=i2 =0} 1 {j6=j2} 1 {i1=i5 =0} 1 {j1=j5} 1 {i3=i4 =0} 1 {j3=j4} − −1 {i6=i2 =0} 1 {j6=j2} 1 {i1=i4 =0} 1 {j1=j4} 1 {i3=i5 =0} 1 {j3=j5} − −1 {i6=i2 =0} 1 {j6=j2} 1 {i1=i3 =0} 1 {j1=j3} 1 {i4=i5 =0} 1 {j4=j5} − −1 {i6=i3 =0} 1 {j6=j3} 1 {i1=i5 =0} 1 {j1=j5} 1 {i2=i4 =0} 1 {j2=j4} − −1 {i6=i3 =0} 1 {j6=j3} 1 {i1=i4 =0} 1 {j1=j4} 1 {i2=i5 =0} 1 {j2=j5} − −1 {i3=i6 =0} 1 {j3=j6} 1 {i1=i2 =0} 1 {j1=j2} 1 {i4=i5 =0} 1 {j4=j5} − −1 {i6=i4 =0} 1 {j6=j4} 1 {i1=i5 =0} 1 {j1=j5} 1 {i2=i3 =0} 1 {j2=j3} − −1 {i6=i4 =0} 1 {j6=j4} 1 {i1=i3 =0} 1 {j1=j3} 1 {i2=i5 =0} 1 {j2=j5} − −1 {i6=i4 =0} 1 {j6=j4} 1 {i1=i2 =0} 1 {j1=j2} 1 {i3=i5 =0} 1 {j3=j5} − −1 {i6=i5 =0} 1 {j6=j5} 1 {i1=i4 =0} 1 {j1=j4} 1 {i2=i3 =0} 1 {j2=j3} − −1 {i6=i5 =0} 1 {j6=j5} 1 {i1=i2 =0} 1 {j1=j2} 1 {i3=i4 =0} 1 {j3=j4} − (18) −1 {i6=i5 =0} 1 {j6=j5} 1 {i1=i3 =0} 1 {j1=j3} 1 {i2=i4 =0} 1 {j2=j4} , where 1 A is the indicator of the set A. For further consideration, let us consider the generalization of formulas (13)- (18) for the case of an arbitrary multiplicity k (k ∈ N) of the iterated Ito stochastic integral J[ψ (k) ] T,t defined by (2). In order to do this, let us introduce some notations. Consider the unordered set {1, 2, . . . , k} and separate it into two parts: the first part consists of r unordered pairs (sequence order of these pairs is also unimportant) and the second one consists of the remaining k − 2r numbers. So, we have (19) ({{g 1 , g 2 }, . . . , {g 2r−1 , g 2r } part 1 }, {q 1 , . . . , q k−2r part 2 }), where {g 1 , g 2 , . . . , g 2r−1 , g 2r , q 1 , . . . , q k−2r } = {1, 2, . . . , k}, braces mean an unordered set, and parentheses mean an ordered set. We will say that (19) is a partition and consider the sum with respect to all possible partitions (20) ({{g 1 ,g 2 },...,{g 2r−1 ,g 2r }},{q 1 ,...,q k−2r }) {g 1 ,g 2 ,...,g 2r−1 ,g 2r ,q 1 ,...,q k−2r }={1,2,...,k} a g1g2,...,g2r−1g2r ,q1...q k−2r . Below there are several examples of sums in the form (20) ({g 1 ,g 2 }) {g 1 ,g 2 }={1,2} a g1g2 = a 12 , ({{g 1 ,g 2 },{g 3 ,g 4 }}) {g 1 ,g 2 ,g 3 ,g 4 }={1,2,3,4} a g1g2g3g4 = a 1234 + a 1324 + a 2314 , ({g 1 ,g 2 },{q 1 ,q 2 }) {g 1 ,g 2 ,q 1 ,q 2 }={1,2,3,4} a g1g2,q1q2 = = a 12,34 + a 13,24 + a 14,23 + a 23,14 + a 24,13 + a 34,12 , ({g 1 ,g 2 },{q 1 ,q 2 ,q 3 }) {g 1 ,g 2 ,q 1 ,q 2 ,q 3 }={1,2,3,4,J[ψ (k) ] T,t = l.i.m. p1,...,p k →∞ p1 j1=0 . . . p k j k =0 C j k ...j1 k l=1 ζ (i l ) j l + [k/2] r=1 (−1) r × (21) × ({{g 1 ,g 2 },...,{g 2r−1 ,g 2r }},{q 1 ,...,q k−2r }) {g 1 ,g 2 ,...,g 2r−1 ,g 2r ,q 1 ,...,q k−2r }={1,2,...,k} r s=1 1 {ig 2s−1 = ig 2s =0} 1 {jg 2s−1 = jg 2s } k−2r l=1 ζ (iq l ) jq l , where [x] is an integer part of a real number x; another notations are the same as in Theorem 1. In particular, from (21) for k = 5 we obtain J[ψ (5) ] T,t = l.i.m. p1,...,p5→∞ p1 j1=0 . . . p5 j5=0 C j5...j1 5 l=1 ζ (i l ) j l − − ({g 1 ,g 2 },{q 1 ,q 2 ,q 3 }) {g 1 ,g 2 ,q 1 ,q 2 ,q 3 }={1,2,3,4,5} 1 {ig 1 = ig 2 =0} 1 {jg 1 = jg 2 } 3 l=1 ζ (iq l ) jq l + + ({{g 1 ,g 2 },{g 3 ,g 4 }},{q 1 }) {g 1 ,g 2 ,g 3 ,g 4 ,q 1 }={1,2,3,4,5} 1 {ig 1 = ig 2 =0} 1 {jg 1 = jg 2 } 1 {ig 3 = ig 4 =0} 1 {jg 3 = jg 4 } ζ (iq 1 ) jq 1 . The last equality obviously agrees with (17). Let us consider a generalization of Theorem 1 for the case of an arbitrary complete orthonormal systems of functions in the space L 2 ([t, T ]) and ψ 1 (τ ), . . . , ψ k (τ ) ∈ L 2 ([t, T ]). Theorem 2 [22] (Sect. 1.11), [29] (Sect. 15). Suppose that ψ 1 (τ ), . . . , ψ k (τ ) ∈ L 2 ([t, T ]) and {φ j (x)} ∞ j=0 is an arbitrary complete orthonormal system of functions in the space L 2 ([t, T ]). Then the following expansion J[ψ (k) ] T,t = l.i.m. p1,...,p k →∞ p1 j1=0 . . . p k j k =0 C j k ...j1 k l=1 ζ (i l ) j l + [k/2] r=1 (−1) r × (22) × ({{g 1 ,g 2 },...,{g 2r−1 ,g 2r }},{q 1 ,...,q k−2r }) {g 1 ,g 2 ,...,g 2r−1 ,g 2r ,q 1 ,...,q k−2r }={1,2,...,k} r s=1 1 {ig 2s−1 = ig 2s =0} 1 {jg 2s−1 = jg 2s } k−2r l=1 ζ (iq l ) jq l converging in the mean-square sense is valid, where [x] is an integer part of a real number x; another notations are the same as in Theorem 1. It should be noted that an analogue of Theorem 2 was considered in [41]. Note that we use another notations [22] (Sect. 1.11), [29] (Sect. 15) in comparison with [41]. Moreover, the proof of an analogue of Theorem 2 from [41] is somewhat different from the proof given in [22] (Sect. 1.11), [29] (Sect. 15). Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicities 2 to 6 In a number of works of the author [15]- [24], [30] Theorems 1, 2 have been adapted for the iterated Stratonovich stochastic integrals (3) of multiplicities 2 to 6. Let us first present some old results as the following theorem. Theorem 3 [15]- [24], [30]. Suppose that {φ j (x)} ∞ j=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space Recently, a new approach to the expansion and mean-square approximation of iterated Stratonovich stochastic integrals has been obtained [22] (Sect. 2.10-2.16), [30] (Sect. [13][14][15][16][17][18][19], [33] (Sect. 7-13), [34] (Sect. 5-11), [54] (Sect. 4-9), [55]. Let us formulate four theorems that were obtained using this approach. L 2 ([t, T ]). At the same time ψ 2 (τ ) is a continu- ously differentiable function on [t, T ] and ψ 1 (τ ), ψ 3 (τ ) are twice continuously differentiable functions on [t, T ]. Then (23) J * [ψ (2) ] T,t = l.i.m. p1,p2→∞ p1 j1=0 p2 j2=0 C j2j1 ζ (i1) j1 ζ (i2) j2 (i 1 , i 2 = 1, . . . , m),(24)J * [ψ (3) ] T,t = l.i.m. p1,p2,p3→∞ p1 j1=0 p2 j2=0 p3 j3=0 C j3j2j1 ζ (i1) j1 ζ (i2) j2 ζ (i3) j3 (i 1 , i 2 , i 3 = 0, 1, . . . , m),(25)J * [ψ (3) ] T,t = l.i.m. p→∞ p j1,j2,j3=0 C j3j2j1 ζ (i1) j1 ζ (i2) j2 ζ (i3) j3 (i 1 , i 2 , i 3 = 1, . . . , m),(26)J * [ψ (4) ] T,t = l.i.m. p→∞ p j1,j2,j3,j4=0 C j4j3j2j1 ζ (i1) j1 ζ (i2) j2 ζ (i3) j3 ζ (i4) j4 (i 1 , i 2 , i 3 , i 4 = 0, 1, . . . , m), where J * [ψ (k) ] T, Theorem 4 [22], [30], [33], [34], [54]. Suppose that {φ j (x)} ∞ j=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L 2 ([t, T ]). Furthermore, let ψ 1 (τ ), ψ 2 (τ ), ψ 3 (τ ) are continuously differentiable nonrandom functions on [t, T ]. Then, for the iterated Stratonovich stochastic integral of third multiplicity J * [ψ (3) ] T,t = * t T ψ 3 (t 3 ) * t t3 ψ 2 (t 2 ) * t t2 ψ 1 (t 1 )dw (i1) t1 dw (i2) t2 dw (i3) t3 (i 1 , i 2 , i 3 = 0, 1, . . . , m) the following relations (27) J * [ψ (3) ] T,t = l.i.m. p→∞ p j1,j2,j3=0 C j3j2j1 ζ (i1) j1 ζ (i2) j2 ζ (i3) j3 ,(28)M        J * [ψ (3) ] T,t − p j1,j2,j3=0 C j3j2j1 ζ (i1) j1 ζ (i2) j2 ζ (i3) j3   2      ≤ C p are fulfilled, where i 1 , i 2 , i 3 = 0, 1, . . . , m in (27) and i 1 , i 2 , i 3 = 1, . . . , m in (28), constant C is independent of p, C j3j2j1 = T t ψ 3 (t 3 )φ j3 (t 3 ) t3 t ψ 2 (t 2 )φ j2 (t 2 ) t2 t ψ 1 (t 1 )φ j1 (t 1 )dt 1 dt 2 dt 3 and ζ (i) j = T t φ j (τ )df (i) τ are independent standard Gaussian random variables for various i or j (in the case when i = 0); another notations are the same as in Theorems 1, 2. Theorem 5 [22], [30], [33], [34], [54]. Let {φ j (x)} ∞ j=0 be a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L 2 ([t, T ]). Furthermore, let ψ 1 (τ ), . . . , ψ 4 (τ ) be continuously differentiable nonrandom functions on [t, T ]. Then, for the iterated Stratonovich stochastic integral of fourth multiplicity (29) J * [ψ (4) ] T,t = * t T ψ 4 (t 4 ) * t t4 ψ 3 (t 3 ) * t t3 ψ 2 (t 2 ) * t t2 ψ 1 (t 1 )dw (i1) t1 dw (i2) t2 dw (i3) t3 dw (i4) t4 the following relations (29), (30) and i 1 , . . . , i 4 = 1, . . . , m in (31), constant C does not depend on p, ε is an arbitrary small positive real number for the case of complete orthonormal system of Legendre polynomials in the space L 2 ([t, T ]) and ε = 0 for the case of complete orthonormal system of trigonometric functions in the space L 2 ([t, T ]), (30) J * [ψ (4) ] T,t = l.i.m. p→∞ p j1,j2,j3,j4=0 C j4j3j2j1 ζ (i1) j1 ζ (i2) j2 ζ (i3) j3 ζ (i4) j4 ,(31)M        J * [ψ (4) ] T,t − p j1,j2,j3,j4=0 C j4j3j2j1 ζ (i1) j1 ζ (i2) j2 ζ (i3) j3 ζ (i4) j4   2      ≤ C p 1−ε are fulfilled, where i 1 , . . . , i 4 = 0, 1, . . . , m inC j4j3j2j1 = = T t ψ 4 (t 4 )φ j4 (t 4 ) t4 t ψ 3 (t 3 )φ j3 (t 3 ) t3 t ψ 2 (t 2 )φ j2 (t 2 ) t2 t ψ 1 (t 1 )φ j1 (t 1 )dt 1 dt 2 dt 3 dt 4 ; another notations are the same as in Theorem 4. Theorem 6 [22], [30], [33], [34], [54]. Assume that {φ j (x)} ∞ j=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L 2 ([t, T ]) and ψ 1 (τ ), . . . , ψ 5 (τ ) are continuously differentiable nonrandom functions on [t, T ]. Then, for the iterated Stratonovich stochastic integral of fifth multiplicity (32) J * [ψ (5) ] T,t = * t T ψ 5 (t 5 ) . . . * t t2 ψ 1 (t 1 )dw (i1) t1 . . . dw (i5) t5 the following relations (32) another notations are the same as in Theorems 4, 5. (33) J * [ψ (5) ] T,t = l.i.m. p→∞ p j1,...,j5=0 C j5...j1 ζ (i1) j1 . . . ζ (i5) j5 ,(34)M        J * [ψ (5) ] T,t − p j1,...,j5=0 C j5...j1 ζ (i1) j1 . . . ζ (i5) j5   2      ≤ C p 1−ε are fulfilled, where i 1 , . . . , i 5 = 0, 1, . . . , m in Theorem 7 [22], [30], [33], [34], [55]. Suppose that {φ j (x)} ∞ j=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L 2 ([t, T ]). Then, for the iterated Stratonovich stochastic integral of sixth multiplicity (35) J * (i1...i6) T,t = * t T . . . * t t2 dw (i1) t1 . . . dw (i6) t6 the following expansion J * (i1...i6) T,t = l.i.m. p→∞ p j1,...,j6=0 C j6...j1 ζ (i1) j1 . . . ζ (i6) j6 that converges in the mean-square sense is valid, where i 1 , . . . , i 6 = 0, 1, . . . , m, another notations are the same as in Theorems 4-6. C j6...j1 = T t φ j6 (t 6 ) . . . 4. Exact Calculation of the Mean-Square Error in Theorems 1, 2 Theorems 1 and 2 allow us to accurately calculate the mean-square approximation error for iterated Ito stochastic integrals (see Theorem 8 below). Assume that J[ψ (k) ] p1...p k T,t is the approximation of (2), which is the expression on the right-hand side of (22) before passing to the limit J[ψ (k) ] p1...p k T,t = p1 j1=0 . . . p k j k =0 C j k ...j1 k l=1 ζ (i l ) j l + [k/2]1 {ig 2s−1 = ig 2s =0} 1 {jg 2s−1 = jg 2s } k−2r l=1 ζ (iq l ) jq l , where [x] is an integer part of a real number x; another notations are the same as in Theorems 1, 2. Let us denote E p1,...,p k k def = M J[ψ (k) ] T,t − J[ψ (k) ] p1,...,p k T,t 2 , E p1,...,p k k def = E p k if p 1 = . . . = p k = p, I k def = K 2 L2([t,T ] k ) = [t,T ] k K 2 (t 1 , . . . , t k )dt 1 . . . dt k . In [10]- [24], [29] it was shown that (36) E p1,...,p k k ≤ k!   I k − p1 j1=0 . . . p k j k =0 C 2 j k ...j1   if i 1 , . . . , i k = 1, . . . , m and 0 < T − t < ∞ or i 1 , . . . , i k = 0, 1, . . . , m and 0 < T − t < 1. Moreover, in [12]- [24], [29] the following estimate M J[ψ (k) ] T,t − J[ψ (k) ] p1,...,p k T,t 2n ≤ (37) ≤ (k!) 2n (n(2n − 1)) n(k−1) (2n − 1)!!   I k − p1 j1=0 . . . p k j k =0 C 2 j k ...j1   n is obtained, where n ∈ N. The value E p k can be calculated exactly. Theorem 8 [22] (Sect. 1.12), [35] (Sect. 6). Suppose that {φ j (x)} ∞ j=0 is an arbitrary complete orthonormal system of functions in the space L 2 ([t, T ]) and ψ 1 (τ ), . . . , ψ k (τ ) ∈ L 2 ([t, T ]), i 1 , . . . , i k = 1, . . . , m. Then (38) E p k = I k − p j1,...,j k =0 C j k ...j1 M    J[ψ (k) ] T,t (j1,...,j k ) T t φ j k (t k ) . . . t2 t φ j1 (t 1 )df (i1) t1 . . . df (i k ) t k    , where i 1 , . . . , i k = 1, . . . , m; the expression (j1,...,j k ) means the sum with respect to all possible permutations (j 1 , . . . , j k ). At the same time if j r swapped with j q in the permutation (j 1 , . . . , j k ), then i r swapped with i q in the permutation (i 1 , . . . , i k ); another notations are the same as in Theorems 1, 2. Note that M    J[ψ (k) ] T,t T t φ j k (t k ) . . . t2 t φ j1 (t 1 )df (i1) t1 . . . df (i k ) t k    = C j k ...j1 . Then from Theorem 8 for pairwise different i 1 , . . . , i k and for i 1 = . . . = i k we obtain E p k = I k − p j1,...,j k =0 C 2 j k ...j1 , E p k = I k − p j1,...,j k =0 C j k ...j1 (j1,...,j k ) C j k ...j1 . Consider some examples of the application of Theorem 8 (i 1 , i 2 , i 3 = 1, . . . , m) E p 2 = I 2 − p j1,j2=0 C 2 j2j1 − p j1,j2=0 C j2j1 C j1j2 (i 1 = i 2 ), E p 3 = I 3 − p j3,j2,j1=0 C 2 j3j2j1 − p j3,j2,j1=0 C j3j1j2 C j3j2j1 (i 1 = i 2 = i 3 ), E p 3 = I 3 − p j3,j2,j1=0 C 2 j3j2j1 − p j3,j2,j1=0 C j2j3j1 C j3j2j1 (i 1 = i 2 = i 3 ), E p 3 = I 3 − p j3,j2,j1=0 C 2 j3j2j1 − p j3,j2,j1=0 C j3j2j1 C j1j2j3 (i 1 = i 3 = i 2 ), E p 4 = I 4 − p j1,...,j4=0 C j4...j1 (j1,j2) C j4...j1 (i 1 = i 2 = i 3 , i 4 ; i 3 = i 4 ), E p 4 = I 4 − p j1,...,j4=0 C j4...j1 (j1,j3) C j4...j1 (i 1 = i 3 = i 2 , i 4 ; i 2 = i 4 ), E p 4 = I 4 − p j1,...,j4=0 C j4...j1 (j1,j4) C j4...j1 (i 1 = i 4 = i 2 , i 3 ; i 2 = i 3 ), E p 4 = I 4 − p j1,...,j4=0 C j4...j1 (j2,j3) C j4...j1 (i 2 = i 3 = i 1 , i 4 ; i 1 = i 4 ), E p 4 = I 4 − p j1,...,j4=0 C j4...j1 (j2,j4) C j4...j1 (i 2 = i 4 = i 1 , i 3 ; i 1 = i 3 ), E p 4 = I 4 − p j1,...,j4=0 C j4...j1 (j3,j4) C j4...j1 (i 3 = i 4 = i 1 , i 2 ; i 1 = i 2 ), E p 4 = I 4 − p j1,...,j4=0 C j4...j1 (j1,j2,j3) C j4...j1 (i 1 = i 2 = i 3 = i 4 ), E p 4 = I 4 − p j1,...,j4=0 C j4...j1 (j2,j3,j4) C j4...j1 (i 2 = i 3 = i 4 = i 1 ), E p 4 = I 4 − p j1,...,j4=0 C j4...j1 (j1,j2,j4) C j4...j1 (i 1 = i 2 = i 4 = i 3 ), E p 4 = I 4 − p j1,...,j4=0 C j4...j1 (j1,j3,j4) C j4...j1 (i 1 = i 3 = i 4 = i 2 ), E p 4 = I 4 − p j1,...,j4=0 C j4...j1 (j1,j2) (j3,j4) C j4...j1 (i 1 = i 2 = i 3 = i 4 ), E p 4 = I 4 − p j1,...,j4=0 C j4...j1 (j1,j3) (j2,j4) C j4...j1 (i 1 = i 3 = i 2 = i 4 ), E p 4 = I 4 − p j1,...,j4=0 C j4...j1 (j1,j4) (j2,j3) C j4...j1 (i 1 = i 4 = i 2 = i 3 ), E p 5 = I 5 − p j1,...,j5=0 C j5...j1 (j2,j4) (j3,j5) C j5...j1 (i 1 = i 2 = i 4 = i 3 = i 5 = i 1 ). Some Technical Problems of the Milstein Approach Let us denote (39) I * (i1...i k ) (l1...l k )T,t = * t T (t − t k ) l k . . . * t t2 (t − t 1 ) l1 df (i1) t1 . . . df (i k ) t k , where i 1 , . . . , i k = 1, . . . , m; l 1 , . . . , l k = 0, 1, . . . . Consider the Milstein expansions for the simplest iterated Stratonovich stochastic integrals (39) (40) I * (i1) (0)T,t = √ T − tζ (i1) 0 ,(41)I * (i1) (1)T,t = − (T − t) 3/2 2 ζ (i1) 0 − √ 2 π ∞ r=1 1 r ζ (i1) 2r−1 , I * (i1i2) (00)T,t = 1 2 (T − t) ζ (i1) 0 ζ (i2) 0 + 1 π ∞ r=1 1 r ζ (i1) 2r ζ (i2) 2r−1 − ζ (i1) 2r−1 ζ (i2) 2r + (42) + √ 2 ζ (i1) 2r−1 ζ (i2) 0 − ζ (i1) 0 ζ (i2) 2r−1 ,(43)I * (i1) (2)T,t = (T − t) 5/2 1 3 ζ (i1) 0 + 1 √ 2π 2 ∞ r=1 1 r 2 ζ (i1) 2r − 1 √ 2π ∞ r=1 1 r ζ (i1) 2r−1 , where i 1 , i 2 = 1, . . . , m; ζ (i) j = T t φ j (s)df (i)(45) I * (i1)q (1)T,t = − (T − t) 3/2 2 ζ (i1) 0 − √ 2 π q r=1 1 r ζ (i1) 2r−1 + √ α q ξ (i1) q , I * (i1i2)q (00)T,t = 1 2 (T − t) ζ (i1) 0 ζ (i2) 0 + 1 π q r=1 1 r ζ (i1) 2r ζ (i2) 2r−1 − ζ (i1) 2r−1 ζ (i2) 2r + (46) + √ 2 ζ (i1) 2r−1 ζ (i2) 0 − ζ (i1) 0 ζ (i2) 2r−1 + √ 2 π √ α q ξ (i1) q ζ (i2) 0 − ζ (i1) 0 ξ (i2) q , where (47) ξ (i) q = 1 √ α q ∞ r=q+1 1 r ζ (i) 2r−1 , α q = π 2 6 − q r=1 1 r 2 , where ζ The approximation I * (i1)q (2)T,t , which corresponds to (45), (46) has the form [3] I * (i1)q (2)T,t = (T − t) 5/2 1 3 ζ (i1) 0 + 1 √ 2π 2 q r=1 1 r 2 ζ (i1) 2r + β q µ (i1) q − (48) − 1 √ 2π q r=1 1 r ζ (i1) 2r−1 + √ α q ξ (i1) q , where ξ (i) q , α q has the form (47) and µ (i) q = 1 β q ∞ r=q+1 1 r 2 ζ (i) 2r , β q = π 4 90 − q r=1 1 r 4 , φ j (s) is defined by (44); ζ (i) 0 , ζ (i) 2r , ζ (i) 2r−1 , ξ (i) q , µI * (i1) (0)T,t = √ T − tζ (i1) 0 ,(50)I * (i1) (1)T,t = − (T − t) 3/2 2 ζ (i1) 0 + 1 √ 3 ζ (i1) 1 ,(51)I * (i1) (2)T,t = (T − t) 5/2 3 ζ (i1) 0 + √ 3 2 ζ (i1) 1 + 1 2 √ 5 ζ (i1) 2 ,(52)I * (i1i2) (00)T,t = T − t 2 ζ (i1) 0 ζ (i2) 0 + ∞ i=1 1 √ 4i 2 − 1 ζ (i1) i−1 ζ (i2) i − ζ (i1) i ζ (i2) i−1 , ζ (i) j = T t φ j (s)df (i) s are independent standard Gaussian random variables for various i or j, where (53) φ j (x) = 2j + 1 T − t P j x − T + t 2 2 T − t ; j = 0, 1, 2, . . . , where P j (x) is the Legendre polynomial. It is not difficult to see that the expansions (50), (51) are much simpler than the expansions (45), (48). Obviously that the Milstein approach [2] leads to iterated series (iterated application of the operation of limit transitions) in contradiction to multiple series (the operation of limit transition is implemented only once) from Theorems 1-7. For the case of simplest stochastic integral I * (i1i2) (00)T,t of second multiplicity this problem was avoided as we saw earlier. However, the situation is not the same for the simplest iterated stochastic integral I * (i1i2i3) (000)T,t of third multiplicity. Let us denote J * (i1...i k ) (λ1...λ k )T,t = * t T . . . * t t2 dw (i1) t1 . . . dw (i k ) t k ,J * (i1i2i3) (111)∆,0 = 1 ∆ J * (i1) (1)∆,0 J * (0i2i3) (011)∆,0 + 1 2 a i1,0 J * (i2i3) (11)∆,0 + 1 2π b i1 J (i2) (1)∆,0 J * (i3) (1)∆,0 − (54) − ∆J * (i2) (1)∆,0 B i1i3 + ∆J * (i3) (1)∆,0 1 2 A i1i2 − C i2i1 + ∆ 3/2 D i1i2i3 , where 1 π ∆J * (i3) (1)∆,0 b i2 + +∆ 2 B i2i3 − 1 4 ∆a i3,0 J * (i2) (1)∆,0 + 1 2π ∆b i3 J * (i2) (1)∆,0 + ∆ 2 C i2i3 + 1 2 ∆ 2 A i2i3 , A i2i3 = π ∆ ∞ r=1 r (a i2,r b i3,r − b i2,r a i3,r ) , C i2i3 = − 1 ∆ ∞ l=1 ∞ r=1(r =l) r r 2 − l 2 (ra i2,r a i3,l + lb i2,r b i3,l ) , B i2i3 = 1 2∆ ∞ r=1 (a i2,r a i3,r + b i2,r b i3,r ) , b i = ∞ r=1 1 r b i,r , D i1i2i3 = − π 2∆ 3/2 ∞ l=1 ∞ r=1 l a i2,l (a i3,l+r b i1,r − a i1,r b i3,l+r ) + +b i2,l (a i1,r a i3,r+l + b i1,r b i3,l+r ) + + π 2∆ 3/2 ∞ l=1 l−1 r=1 l a i2,l (a i1,r b i3,l−r + a i3,l−r b i1,r ) − −b i2,l (a i1,r a i3,l−r − b i1,r b i3,l−r ) + + π 2∆ 3/2 ∞ l=1 ∞ r=l+1 l a i2,l (a i3,r−l b i1,r − a i1,r b i3,r−l ) + +b i2,l (a i1,r a i3,r−l + b i1,r b i3,r−l ) . From the form of expansion (54) and expansion of the stochastic integral J * (0i2i3) (011)∆,0 we can conclude that they include iterated (double) series. Moreover, for approximation of the considered stochastic integral J * (i1i2i3) (111)∆,0 in the works [3] (Sect. 5.8, pp. 202-204), [4] (pp. 82-84), [5] (pp. 438-439), [6] (pp. 263-264) it is proposed to put upper limits of summation by equal q (on the base of the Wong-Zakai approximation [38]- [40], but without rigorous proof; also see discussion in Sect. 7). For example, the value D i1i2i3 is approximated in [3] D (q) i1i2i3 = − π 2∆ 3/2 q l=1 q r=1 l a i2,l (a i3,l+r b i1,r − a i1,r b i3,l+r ) + +b i2,l (a i1,r a i3,r+l + b i1,r b i3,l+r ) + + π 2∆ 3/2 q l=1 l−1 r=1 l a i2,l (a i1,r b i3,l−r + a i3,l−r b i1,r ) − −b i2,l (a i1,r a i3,l−r − b i1,r b i3,l−r ) + + π 2∆ 3/2 q l=1 2q r=l+1 l a i2,l (a i3,r−l b i1,r − a i1,r b i3,r−l ) + +b i2,l (a i1,r a i3,r−l + b i1,r b i3,r−l ) . Obviously, we can avoid this problem (iterated application of the operation of limit transition) using the method based on Theorems 1-7. If we prove that the terms of the expansion (54) coincide with the terms of its analogue obtained using Theorems 1-3 (this fact is proved in [10]- [24] for the simplest stochastic integrals I * (i1) (1)T,t , I * (i1i2) (00)T,t of first and second multiplicity), then we can replace the iterated (double) series in (54) by the multiple ones, as in Theorems 1-3 (as was made formally in [3]- [6]). However, it requires a separate argumentation. Approximation of Specific Iterated Stochastic Integrals of Multiplicities 1 to 3 Using Theorem 3 and Trigonometric System of Functions In [10]- [24] on the base of Theorems 1-3 the author of this paper obtained the following expansions of the iterated Stratonovich stochastic integrals (39) (independently from the papers [2]- [7] excepting the method in which additional random variables ξ (i) q and µ (i) q ) are introduced) (55) I * (i1) (0)T,t = √ T − tζ (i1) 0 , (56) I * (i1)q (1)T,t = − (T − t) 3/2 2 ζ (i1) 0 − √ 2 π q r=1 1 r ζ (i1) 2r−1 + √ α q ξ (i1) q , I * (i1i2)q (00)T,t = 1 2 (T − t) ζ (i1) 0 ζ (i2) 0 + 1 π q r=1 1 r ζ (i1) 2r ζ (i2) 2r−1 − ζ (i1) 2r−1 ζ (i2) 2r + (57) + √ 2 ζ (i1) 2r−1 ζ (i2) 0 − ζ (i1) 0 ζ (i2) 2r−1 + √ 2 π √ α q ξ (i1) q ζ (i2) 0 − ζ (i1) 0 ξ (i2) q , I * (i1i2i3)q (000)T,t = (T − t) 3/2 1 6 ζ (i1) 0 ζ (i2) 0 ζ (i3) 0 + √ α q 2 √ 2π ξ (i1) q ζ (i2) 0 ζ (i3) 0 − ξ (i3) q ζ (i2) 0 ζ (i1) 0 + + 1 2 √ 2π 2 β q µ (i1) q ζ (i2) 0 ζ (i3) 0 − 2µ (i2) q ζ (i1) 0 ζ (i3) 0 + µ (i3) q ζ (i1) 0 ζ (i2) 0 + + 1 2 √ 2 q r=1 1 πr ζ (i1) 2r−1 ζ (i2) 0 ζ (i3) 0 − ζ (i3) 2r−1 ζ (i2) 0 ζ (i1) 0 + + 1 π 2 r 2 ζ (i1) 2r ζ (i2) 0 ζ (i3) 0 − 2ζ (i2) 2r ζ (i3) 0 ζ (i1) 0 + ζ (i3) 2r ζ (i2) 0 ζ (i1) 0 + + q r=1 1 4πr ζ (i1) 2r ζ (i2) 2r−1 ζ (i3) 0 − ζ (i1) 2r−1 ζ (i2) 2r ζ (i3) 0 − ζ (i2) 2r−1 ζ (i3) 2r ζ (i1) 0 + ζ (i3) 2r−1 ζ (i2) 2r ζ (i1) 0 + + 1 8π 2 r 2 3ζ (i1) 2r−1 ζ (i2) 2r−1 ζ (i3) 0 + ζ (i1) 2r ζ (i2) 2r ζ (i3) 0 − 6ζ (i1) 2r−1 ζ (i3) 2r−1 ζ (i2) 0 + (58) +3ζ (i2) 2r−1 ζ (i3) 2r−1 ζ (i1) 0 − 2ζ (i1) 2r ζ (i3) 2r ζ (i2) 0 + ζ (i3) 2r ζ (i2) 2r ζ (i1) 0 +D (i1i2i3)q T,t , where D (i1i2i3)q T,t = 1 2π 2 q r,l=1 r =l 1 r 2 − l 2 ζ (i1) 2r ζ (i2) 2l ζ (i3) 0 − ζ (i2) 2r ζ (i1) 0 ζ (i3) 2l + + r l ζ (i1) 2r−1 ζ (i2) 2l−1 ζ (i3) 0 − l r ζ (i1) 0 ζ (i2) 2r−1 ζ (i3) 2l−1 − 1 rl ζ (i1) 2r−1 ζ (i2) 0 ζ (i3) 2l−1 + + 1 4 √ 2π 2 q r,m=1 2 rm −ζ (i1) 2r−1 ζ (i2) 2m−1 ζ (i3) 2m + ζ (i1) 2r−1 ζ (i2) 2r ζ (i3) 2m−1 + +ζ (i1) 2r−1 ζ (i2) 2m ζ (i3) 2m−1 − ζ (i1) 2r ζ (i2) 2r−1 ζ (i3) 2m−1 + + 1 m(r + m) −ζ (i1) 2(m+r) ζ (i2) 2r ζ (i3) 2m − ζ (i1) 2(m+r)−1 ζ (i2) 2r−1 ζ (i3) 2m − −ζ (i1) 2(m+r)−1 ζ (i2) 2r ζ (i3) 2m−1 + ζ (i1) 2(m+r) ζ (i2) 2r−1 ζ (i3) 2m−1 + + q m=1 q l=m+1 1 m(l − m) ζ (i1) 2(l−m) ζ (i2) 2l ζ (i3) 2m + ζ (i1) 2(l−m)−1 ζ (i2) 2l−1 ζ (i3) 2m − −ζ (i1) 2(l−m)−1 ζ (i2) 2l ζ (i3) 2m−1 + ζ (i1) 2(l−m) ζ (i2) 2l−1 ζ (i3) 2m−1 + + 1 l(l − m) −ζ (i1) 2(l−m) ζ (i2) 2m ζ (i3) 2l + ζ (i1) 2(l−m)−1 ζ (i2) 2m−1 ζ (i3) 2l − −ζ (i1) 2(l−m)−1 ζ (i2) 2m ζ (i3) 2l−1 − ζ (i1) 2(l−m) ζ (i2) 2m−1 ζ (i3) 2l−1 , I * (i1i2)q (10)T,t = −(T − t) 2 1 6 ζ (i1) 0 ζ (i2) 0 − 1 2 √ 2π √ α q ξ (i2) q ζ (i1) 0 + + 1 2 √ 2π 2 β q µ (i2) q ζ (i1) 0 − 2µ (i1) q ζ (i2) 0 + + 1 2 √ 2 q r=1 − 1 πr ζ (i2) 2r−1 ζ (i1) 0 + 1 π 2 r 2 ζ (i2) 2r ζ (i1) 0 − 2ζ (i1) 2r ζ (i2) 0 − − 1 2π 2 q r,l=1 r =l 1 r 2 − l 2 ζ (i1) 2r ζ (i2) 2l + l r ζ (i1) 2r−1 ζ (i2) 2l−1 + (59) + q r=1 1 4πr ζ (i1) 2r ζ (i2) 2r−1 − ζ (i1) 2r−1 ζ (i2) 2r + 1 8π 2 r 2 3ζ (i1) 2r−1 ζ (i2) 2r−1 + ζ (i2) 2r ζ (i1) 2r , I * (i1i2)q (01)T,t = (T − t) 2 − 1 3 ζ (i1) 0 ζ (i2) 0 − 1 2 √ 2π √ α q ξ (i1) q ζ (i2) 0 − 2ξ (i2) q ζ (i1) 0 + + 1 2 √ 2π 2 β q µ (i1) q ζ (i2) 0 − 2µ (i2) q ζ (i1) 0 − − 1 2 √ 2 q r=1 1 πr ζ (i1) 2r−1 ζ (i2) 0 − 2ζ (i2) 2r−1 ζ (i1) 0 − 1 π 2 r 2 ζ (i1) 2r ζ (i2) 0 − 2ζ (i2) 2r ζ (i1) 0 + + 1 2π 2 q r,l=1 r =l 1 r 2 − l 2 r l ζ (i1) 2r−1 ζ (i2) 2l−1 + ζ (i1) 2r ζ (i2) 2l − (60) − q r=1 1 4πr ζ (i1) 2r ζ (i2) 2r−1 − ζ (i1) 2r−1 ζ (i2) 2r − 1 8π 2 r 2 3ζ (i1) 2r−1 ζ (i2) 2r−1 + ζ (i1) 2r ζ (i2) 2r , I * (i1)q (2)T,t = (T − t) 5/2 1 3 ζ (i1) 0 + 1 √ 2π 2 q r=1 1 r 2 ζ (i1) 2r + β q µ (i1) q − (61) − 1 √ 2π q r=1 1 r ζ (i1) 2r−1 + √ α q ξ (i1) q , where ξ (i) q = 1 √ α q ∞ r=q+1 1 r ζ (i) 2r−1 , α q = π 2 6 − q r=1 1 r 2 , µ (i) q = 1 β q ∞ r=q+1 1 r 2 ζ (i) 2r , β q = π 4 90 − q r=1 1 r 4 , ζ (i) j = T t φ j (s)df (i) s , where φ j (s) has the form (44); ζ (i) 0 , ζ (i) 2r , ζ (i) 2r−1 , ξ (i) q , µ (i) q ; r = 1, . . . , q; i = 1, . . . , m are independent standard Gaussian random variables; i 1 , i 2 , i 3 = 1, . . . , m. Note that from (59), (60) it follows that (62) ∞ j=0 C 10 jj = ∞ j=0 C 01 jj = − (T − t) 2 4 , where C 10 jj = T t φ j (x) x t φ j (y)(t − y)dydx, C 01 jj = T t φ j (x)(t − x) x t φ j (y)dydx. The formulas (62) are particular cases of the more general relation, which we applied for the proof of Theorem 3 for the case k = 2 (see [15]- [24]). Let us consider the mean-square errors of approximations (57)-(60). From the relations (57)-(60) when i 1 = i 2 , i 2 = i 3 , i 1 = i 3 we obtain by direct calculation 1 (63) M I * (i1i2) (00)T,t − I * (i1i2)q (00)T,t 2 = (T − t) 2 2π 2 π 2 6 − q r=1 1 r 2 , M I * (i1i2i3) (000)T,t − I * (i1i2i3)q (000)T,t 2 = (T − t) 3 1 4π 2 π 2 6 − q r=1 1 r 2 + (64) + 55 32π 4 π 4 90 − q r=1 1 r 4 + 1 4π 4 ∞ r,l=1 r =l − q r,l=1 r =l 5l 4 + 4r 4 − 3l 2 r 2 r 2 l 2 (r 2 − l 2 ) 2 , M I * (i1i2) (01)T,t − I * (i1i2)q (01)T,t 2 = (T − t) 4 1 8π 2 π 2 6 − q r=1 1 r 2 + (65) + 5 32π 4 π 4 90 − q r=1 1 r 4 + 1 4π 4 ∞ k,l=1 k =l − q k,l=1 k =l l 2 + k 2 k 2 (l 2 − k 2 ) 2 , M I * (i1i2)r 4 + 1 4π 4 ∞ k,l=1 k =l − q k,l=1 k =l l 2 + k 2 l 2 (l 2 − k 2 ) 2 . It is easy to demonstrate that the relations (64), (65), and (66) can be represented using Theorem 8 in the following form M I * (i1i2i3) (000)T,t − I * (i1i2i3)q (000)T,t 2 = (T − t) 3 4 45 − 1 4π 2 q r=1 1 r 2 − (67) − 55 32π 4 q r=1 1 r 4 − 1 4π 4 q r,l=1 r =l 5l 4 + 4r 4 − 3r 2 l 2 r 2 l 2 (r 2 − l 2 ) 2 , M I * (i1i2) (10)T,t − I * (i1i2)q (10)T,t 2 = (T − t) 4 4 1 9 − 1 2π 2 q r=1 1 r 2 − (68) − 5 8π 4 q r=1 1 r 4 − 1 π 4 q k,l=1 k =l k 2 + l 2 l 2 (l 2 − k 2 ) 2 , M I * (i1i2) (01)T,t − I * (i1i2)q (01)T,t 2 = (T − t) 4 4 1 9 − 1 2π 2 q r=1 1 r 2 − (69) − 5 8π 4 q r=1 1 r 4 − 1 π 4 q k,l=1 k =l l 2 + k 2 k 2 (l 2 − k 2 ) 2 . Comparing (67)-(69) and (64)-(66), we obtain (70) ∞ k,l=1 k =l l 2 + k 2 k 2 (l 2 − k 2 ) 2 = ∞ k,l=1 k =l l 2 + k 2 l 2 (l 2 − k 2 ) 2 = π 4 48 ,(71)∞ r,l=1 r =l 5l 4 + 4r 4 − 3r 2 l 2 r 2 l 2 (r 2 − l 2 ) 2 = 9π 4 80 . Let us consider approximations of the stochastic integrals I * (i1i1) (10)T,t , I * (i1i1) (01)T,t and conditions for selecting the number q using the trigonometric system of functions I * (i1i1)q (10)T,t = −(T − t) 2 1 6 ζ (i1) 0 2 − 1 2 √ 2π √ α q ξ (i1) q ζ (i1) 0 − − 1 2 √ 2π 2 β q µ (i1) q ζ (i1) 0 − 1 2 √ 2 q r=1 1 πr ζ (i1) 2r−1 ζ (i1) 0 + 1 π 2 r 2 ζ (i1) 2r ζ (i1) 0 − − 1 2π 2 q r,l=1 r =l 1 r 2 − l 2 ζ (i1) 2r ζ (i1) 2l + l r ζ (i1) 2r−1 ζ (i1) 2l−1 + + 1 8π 2 q r=1 1 r 2 3 ζ (i1) 2r−1 2 + ζ (i1) 2r 2 , I * (i1i1)q (01)T,t = (T − t) 2 − 1 3 ζ (i1) 0 2 + 1 2 √ 2π √ α q ξ (i1) q ζ (i1) 0 − − 1 2 √ 2π 2 β q µ (i1) q ζ (i1) 0 + 1 2 √ 2 q r=1 1 πr ζ (i1) 2r−1 ζ (i1) 0 − 1 π 2 r 2 ζ (i1) 2r ζ (i1) 0 + + 1 2π 2 q r,l=1 r =l 1 r 2 − l 2 ζ (i1) 2r ζ (i1) 2l + r l ζ (i1) 2r−1 ζ (i1) 2l−1 + + 1 8π 2 q r=1 1 r 2 3 ζ (i1) 2r−1 2 + ζ (i1) 2r 2 . Then, we obtain M I * (i1i1) + 1 π 4 ∞ k,l=1 k =l − q k,l=1 k =l l 2 + k 2 k 2 (l 2 − k 2 ) 2 . Using (70) In Tables 1-3, we confirm numerically the formulas (67)-(69), (73) for various values q. In Tables 1-3, the number ε means the right-hand sides of the mentioned formulas. l 2 + k 2 k 2 (l 2 − k 2 ) 2 . The formulas (70), (71) appear to be interesting. Let us confirm numerically their correctness in Tables 4 and 5 (the number ε q is the absolute deviation of multiple partial sums with the upper limit of summation q for the series (70), (71) from the right-hand sides of the formulas (70), (71); convergence of multiple series is regarded here when p 1 = p 2 = q → ∞, which is acceptable according to Theorems 1,2). Using the trigonometric system of functions, let us consider the approximations of iterated stochastic integrals of the following form J * (i1...i k ) (λ1...λ k )T,t = * t T . . . * t t2 dw (i1) t1 . . . dw (i k ) t k ,(74) ζ (i) j = T t φ j (s)dw (i) s and i 1 , i 2 , i 3 = 0, 1, . . . , m. Since T t φ j (s)dw (0) s =      √ T − t if j = 0 0 if j = 0 , then it is easy to get from (57) and (58), considering that in these equalities ζ (i) j has the form (74) and i 1 , i 2 , i 3 = 0, 1, . . . , m, the following family of formulas [38], [39], it was shown that under the special conditions and for some types of approximations of the Wiener process the answere is affirmative with one peculiarity: the convergence takes place to the iterated Stratonovich stochastic integrals and solutions of Stratonovich SDEs and not to iterated Ito stochastic integrals and solutions of Ito SDEs. The piecewise linear approximation as well as the regularization by convolution [38]- [40] relate the mentioned types of approximations of the Wiener process. The above approximation of stochastic integrals and solutions of SDEs is often called the Wong-Zakai approximation. J (i10)q (10)T,t = 1 2 (T − t) 3/2 ζ (i1) 0 + √ 2 π q r=1 1 r ζ (i1) 2r−1 + √ α q ξ (i1) q , J (0i2)q (01)T,t = 1 2 (T − t) 3/2 ζ (i2) 0 − √ 2 π q r=1 1 r ζ (i2) 2r−1 + √ α q ξ (i2) q , J (00i3)q (001)T,t = (T − t) 5/2 1 6 ζ (i3) 0 + 1 2 √ 2π 2 q r=1 1 r 2 ζ (i3) 2r + β q µ (i3) q − − 1 2 √ 2π q r=1 1 r ζ (i3) 2r−1 + √ α q ξ (i3) q , J (0i20)q (010)T,t = (T − t) 5/2 1 6 ζ (i2) 0 − 1 √ 2π 2 q r=1 1 r 2 ζ (i2) 2r + β q µ (i2) q , J (i100)q (100)T,t = (T − t) 5/2 1 6 ζ (i1) 0 + 1 2 √ 2π 2 q r=1 1 r 2 ζ (i1) 2r + β q µ (i1) q + + 1 2 √ 2π q r=1 1 r ζ (i1) 2r−1 + √ α q ξ (i1) q , J * (0i2i3)q (011)T,t = (T − t) 2 1 6 ζ (i2) 0 ζ (i3) 0 − 1 2 √ 2π √ α q ξ (i3) q ζ (i2) 0 + + 1 2 √ 2π 2 β q µ (i3) q ζ (i2) 0 − 2µ (i2) q ζ (i3) 0 + + 1 2 √ 2 q r=1 − 1 πr ζ (i3) 2r−1 ζ (i2) 0 + 1 π 2 r 2 ζ (i3) 2r ζ (i2) 0 − 2ζ (i2) 2r ζ (i3) 0 − − 1 2π 2 q r,l=1 r =l 1 r 2 − l 2 ζ (i2) 2r ζ (i3) 2l + l r ζ (i2) 2r−1 ζ (i3) 2l−1 + + q r=1 1 4πr ζ (i2) 2r ζ (i3) 2r−1 − ζ (i2) 2r−1 ζ (i3) 2r + (75) + 1 8π 2 r 2 3ζ (i2) 2r−1 ζ (i3) 2r−1 + ζ (i3) 2r ζ (i2) 2r , J * (i1i20)q (110)T,t = (T − t) 2 1 6 ζ (i1) 0 ζ (i2) 0 + 1 2 √ 2π √ α q ξ (i1) q ζ (i2) 0 + + 1 2 √ 2π 2 β q µ (i1) q ζ (i2) 0 − 2µ (i2) q ζ (i1) 0 + + 1 2 √ 2 q r=1 1 πr ζ (i1) 2r−1 ζ (i2) 0 + 1 π 2 r 2 ζ (i1) 2r ζ (i2) 0 − 2ζ (i2) 2r ζ (i1) 0 + + 1 2π 2 q r,l=1 r =l 1 r 2 − l 2 r l ζ (i1) 2r−1 ζ (i2) 2l−1 + ζ (i1) 2r ζ (i2) 2l + + q r=1 1 4πr ζ (i2) 2r−1 ζ (i1) 2r − ζ (i1) 2r−1 ζ (i2) 2r + + 1 8π 2 r 2 3ζ (i1) 2r−1 ζ (i2) 2r−1 + ζ (i1) 2r ζ (i2) 2r , J * (i10i3)q (101)T,t = (T − t) 2 1 6 ζ (i1) 0 ζ (i3) 0 + 1 2 √ 2π √ α q ξ (i1) q ζ (i3) 0 − ξ (i3) q ζ (i1) 0 + + 1 2 √ 2π 2 β q µ (i1) q ζ (i3) 0 + µ (i3) q ζ (i1) 0 + + 1 2 √ 2 q r=1 1 πr ζ (i1) 2r−1 ζ (i3) 0 − ζ (i3) 2r−1 ζ (i1) 0 + + 1 π 2 r 2 ζ (i1) 2r ζ (i3) 0 + ζ (i3) 2r ζ (i1) 0 − 1 2π 2 q r,l=1 r =l 1 rl ζ (i1) 2r−1 ζ (i3) 2l−1 − − q r=1 1 4π 2 r 2 3ζ (i1) 2r−1 ζ (i3) 2r−1 + ζ (i1) 2r ζ (i3) 2r Let f s , s ∈ [0, T ] be an m-dimensional standard Wiener process with independent components f (i) s , i = 1, . . . , m. It is well known that the following representation takes place [42], [43] (76) f (i) τ − f (i) t = ∞ j=0 τ t φ j (s)ds ζ (i) j , ζ (i) j = T t φ j (s)df (i) s , where τ ∈ [t, T ], t ≥ 0, {φ j (x)} ∞ j=0 is an arbitrary complete orthonormal system of functions in the space L 2 ([t, T ]), and ζ (i) j are independent standard Gaussian random variables for various i or j. Moreover, the series (76) converges for any τ ∈ [t, T ] in the mean-square sense. Let f (i)p τ − f (i)p t be the mean-square approximation of the process f (i) τ − f (i) t , which has the following form (77) f (i)p τ − f (i)p t = p j=0 τ t φ j (s)ds ζ (i) j . From (77) we obtain (78) df (i)p τ = p j=0 φ j (τ )ζ (i) j dτ. Consider the following iterated Riemann-Stieltjes integral (79) T t ψ k (t k ) . . . t2 t ψ 1 (t 1 )dw (i1)p1 t1 . . . dw (i k )p k t k , where p 1 , . . . , p k ∈ N, i 1 , . . . , i k = 0, 1, . . . , m, (80) dw (i)p τ =        df (i)ψ 1 (t 1 )dw (i1)p1 t1 . . . dw (i k )p k t k = p1 j1=0 . . . p k j k =0 C j k ...j1 k l=1 ζ (i l ) j l , where ζ (i) j = T t φ j (s)dw (i)C j k ...j1 = T t ψ k (t k )φ j k (t k ) . . . t2 t ψ 1 (t 1 )φ j1 (t 1 )dt 1 . . . dt k is the Fourier coefficient. To best of our knowledge [38]- [40] the approximations of the Wiener process in the Wong-Zakai approximation must satisfy fairly strong restrictions [40] (see Definition 7.1, pp. 480-481). Moreover, approximations of the Wiener process that are similar to (77) were not considered in [38], [39] (also see [40], Theorems 7.1, 7.2). Therefore, the proof of analogs of Theorems 7.1 and 7.2 [40] for approximations of the Wiener process based on its series expansion (76) should be carried out separately. Thus, the mean-square convergence of the right-hand side of (81) to the iterated Stratonovich stochastic integral (3) does not follow from the results of the papers [38], [39] (also see [40], Theorems 7.1, 7.2). From the other hand, Theorems 1-7 from this paper can be considered as the proof of the Wong-Zakai approximation for the iterated Stratonovich stochastic integrals (3) of multiplicities 1 to 6 based on the approximation (77) of the Wiener process. At that, the iterated Riemann-Stieltjes integrals (79) converge (according to Theorems 1-7) to the appropriate iterated Stratonovich stochastic integrals (3). Recall that {φ j (x)} ∞ j=0 (see (76), (77), and Theorems 3-7) is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L 2 ([t, T ]). To illustrate the above reasoning, consider two examples for the case k = 2, ψ 1 (s), ψ 2 (s) ≡ 1; i 1 , i 2 = 1, . . . , m. The first example relates to the piecewise linear approximation of the multidimensional Wiener process (these approximations were considered in [38]- [40]). Let b As we noted above, approximations of the Wiener process that are similar to (77) were not considered in [38], [39] (also see Theorems 7.1, 7.2 in [40]). Furthermore, the extension of the results of Theorems 7.1 and 7.2 [40] to the case under consideration is not obvious. ∆ (t) = f (i) k∆ + t − k∆ ∆ ∆f (i) k∆ , where ∆f (i) k∆ = f (i) (k+1)∆ − f On the other hand, we can apply the theory built in Chapters 1 and 2 of the monographs [22]- [24]. More precisely, using Theorem 3, we obtain from (86) the desired result l.i.m. From the other hand, by Theorems 1, 2 (see (14)) for the case k = 2 we obtain from (86) the following relation l.i.m. C j2j1 ζ (i1) j1 ζ (i2) j2 − 1 {i1=i2} 1 {j1=j2} + 1 {i1=i2} ∞ j1=0 C j1j1 = (88) = T 0 s 0 df (i1) τ df (i2) s + 1 {i1=i2} ∞ j1=0 C j1j1 . Since ∞ j1=0 C j1j1 = 1 2 ∞ j1=0   T 0 φ j (τ )dτ   2 = 1 2   T 0 φ 0 (τ )dτ   2 = 1 2 T 0 ds, then from (88) we obtain (87). τ = τ ; i 1 , . . . , i k = 0, 1, . . . , m; t (i = 1, . . . , m). t is defined by (3) and ψ l (τ ) ≡ 1 (l = 1, . . . , 4) in (24),(26); another notations are the same as in Theorems 1, 2. , (33) and i 1 , . . . , i 5 = 1, . . . , m in (34), constant C is independent of p, ε is an arbitrary small positive real number for the case of complete orthonormal system of Legendre polynomials in the space L 2 ([t, T ]) and ε = 0 for the case of complete orthonormal system of trigonometric functions in the space L 2 ([t, T ]), C j5...j1 = T t ψ 5 (t 5 )φ j5 (t 5 ) . . . (t 1 )φ j1 (t 1 )dt 1 . . . dt 5 ; φ j1 (t 1 )dt 1 . . . dt 6 ; 1 ,g 2 },...,{g 2r−1 ,g 2r }},{q 1 ,...,q k−2r }) {g 1 ,g 2 ,...,g 2r−1 ,g 2r ,q 1 ,...,q k−2r }={1,2,...,k} r s=1 (2πr(s − t)/(T − t)) when j = 2r − 1√ 2cos(2πr(s − t)/(T − t)) when j = 2r, .N. proposed[2] the following mean-square approximations on the base of the expansions (41),(42) q ; r = 1, . . . , q; i = 1, . . . , m are independent standard Gaussian random variables. q ; r = 1, . . . , q; i = 1, . . . , m are independent standard Gaussian random variables; i = 1, . . . , m.Nevetheless, the expansions (45),(48) are too complex for the approximation of two Gaussian random variables I * (i1) ( 1 ) 1T,t , I * (i1) ( 2 ) 2T,t . Using Theorems 1-3 and complete orthonormal system of Legendre polynomials in the space L 2 ([t, T ]), we obtain for i 1 , i 2 = 1, . . . , m [10]-[37] (49) where λ l = 1 if i l = 1 ,τ 1. . . , m and λ l = 0 if i l = 0; l = 1, . . . , k (w = τ ). Consider the expansion of iterated Stratonovich stochastic integral of third multiplicity obtained in [3]-[6] by the Milstein approach where λ l = 1 if i l = 1, . . . , m and λ l = 0 if i l = 0; l = 1, . . . , k (w J * (i1i2) (λ1λ2)T,t , J * (i1i2i3) (λ1λ2λ3)T,tare defined by the right-hand sides of the formulas (57), (58), where it is necessary to take . 7 .s 7Theorems 1-7 from Point of View of the Wong-Zakai Approximation The iterated Ito stochastic integrals and solutions of Ito SDEs are complex and important functionals from the independent components f (i) s , i = 1, . . . , m of the multidimensional Wiener process f s , s ∈ [0, T ]. Let f (i)p s , p ∈ N be some approximation of f (i) s , i = 1, . . . , m. Suppose that f , i = 1, . . . , m if p → ∞ in some sense and has differentiable sample trajectories. A natural question arises: if we replace f = 1, . . . , m in the functionals mentioned above, will the resulting functionals converge to the original functionals from the components f (i) s , i = 1, . . . , m of the multidimentional Wiener process f s ? The answere to this question is negative in the general case. However, in the pioneering works of Wong E. and Zakai M. (t k ) . . . s are independent standard Gaussian random variables for various i or j (in the case when i = 0 ∆ (t), t ∈ [0, T ] be the piecewise linear approximation of the ith component f (i) t of the multidimensional standard Wiener process f t , t ∈ [0, T ] with independent components f ∆ (s), i 1 , i 2 = 1, . . . , m.Using (82) and additive property of the Riemann-Stieltjes integral, we can write w.Using (83), it is not difficult to show that → 0 if N → ∞ (N ∆ = T ).Obviously, (84) agrees with Theorem 7.1 (see[40], p. 486). The next example relates to the approximation (77) of the Wiener process based on its series expansion (76) for t = 0, where {φ j (x)} ∞ j=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L 2 ([0, T ]).Consider the following iterated Riemann-(τ )dτ ds is the Fourier coefficient; another notations are the same as in (81). 5} a 5}g1g2,q1q2q3 = = a 12,345 + a 13,245 + a 14,235 + a 15,234 + a 23,145 + a 24,135 + +a 25,134 + a 34,125 + a 35,124 + a 45,123 ,({{g 1 ,g 2 },{g 3 ,g 4 }},{q 1 }) {g 1 ,g 2 ,g 3 ,g 4 ,q 1 }={1,2,3,4,5} a g1g2,g3g4,q1 = = a 12,34,5 + a 13,24,5 + a 14,23,5 + a 12,35,4 + a 13,25,4 + a 15,23,4 + +a 12,54,3 + a 15,24,3 + a 14,25,3 + a 15,34,2 + a 13,54,2 + a 14,53,2 + +a 52,34,1 + a 53,24,1 + a 54,23,1 . Now we can write (11) as Table 1 . 1Confirmation of the formula (67)ε/(T − t) 3 0.0459 0.0072 7.5722 · 10 −4 7.5973 · 10 −5 7.5990 · 10 −6 q 1 10 100 1000 10000 Table 2. Confirmation of the formulas (68), (69) 4ε/(T − t) 4 0.0540 0.0082 8.4261 · 10 −4 8.4429 · 10 −5 8.4435 · 10 −6 q 1 10 100 1000 10000 Table 3 . 3Confirmationof the formula (73) Table 5 . 5Confirmation of the formula (71)ε q 10.9585 1.8836 0.1968 0.0197 0.0020 q 1 10 100 1000 10000 I I Gihman, A V Skorochod, Stochastic Differential Equations and its Applications. Naukova Dumka. Kiev354Gihman I.I., Skorochod A.V. Stochastic Differential Equations and its Applications. Naukova Dumka, Kiev, 1982. 354 pp. Numerical Integration of Stochastic Differential Equations. G N Milstein, Ural University Press225SverdlovskMilstein G.N. Numerical Integration of Stochastic Differential Equations. Ural University Press, Sverdlovsk, 1988. 225 pp. Numerical Solution of Stochastic Differential Equations. P E Kloeden, E Platen, Springer632BerlinKloeden P.E., Platen E. Numerical Solution of Stochastic Differential Equations. Springer, Berlin, 1995. 632 pp. Numerical Solution of SDE Through Computer Experiments. P E Kloeden, E Platen, H Schurz, Springer292BerlinKloeden P.E., Platen E., Schurz H. Numerical Solution of SDE Through Computer Experiments. Springer, Berlin, 1994. 292 pp. The approximation of multiple stochastic integrals. P E Kloeden, E Platen, I W Wright, Stochastic Analysis and Applications. 10Kloeden P.E., Platen E., Wright I.W. The approximation of multiple stochastic integrals. Stochastic Analysis and Applications. 10, 4 (1992), 431-441. Numerical Solution of Stochastic Differential Equations with Jumps in Finance. E Platen, N Bruti-Liberati, Springer868Berlin, HeidelbergPlaten, E., Bruti-Liberati, N. Numerical Solution of Stochastic Differential Equations with Jumps in Finance. Springer, Berlin, Heidelberg, 2010. 868 pp. G N Milstein, M Tretyakov, Stochastic Numerics for Mathematical Physics. BerlinSpringer616Milstein G.N., Tretyakov M.V. Stochastic Numerics for Mathematical Physics. Springer, Berlin, 2004. 616 pp. The unified Taylor-Ito expansion. O Kulchitskiy, Yu, D F Kuznetsov, 10.1007/BF02673635J. Math. Sci. (N. Y.). 992Kulchitskiy O.Yu., Kuznetsov D.F. The unified Taylor-Ito expansion. J. Math. Sci. (N. Y.), 99, 2 (2000), 1130- 1140. DOI: http://doi.org/10.1007/BF02673635 New representations of the Taylor-Stratonovich expansion. D F Kuznetsov, 10.1023/A:1026138522239J. Math. Sci. (N. Y.). 118Kuznetsov D.F. New representations of the Taylor-Stratonovich expansion. J. Math. Sci. (N. Y.), 118, 6 (2003), 5586-5596. DOI: http://doi.org/10.1023/A:1026138522239 Numerical Integration of Stochastic Differential Equations. 2. D F Kuznetsov, In RussianKuznetsov D.F. Numerical Integration of Stochastic Differential Equations. 2. [In Russian]. . 10.18720/SPBPU/2/s17-2275-7422-1191-0764Saint-PetersburgPolytechnical University Publishing HousePolytechnical Univer- sity Publishing House, Saint-Petersburg, 2006, 764 pp. DOI: http://doi.org/10.18720/SPBPU/2/s17-227 Avail- able at: http://www.sde-kuznetsov.spb.ru/06.pdf (ISBN 5-7422-1191-0) Stochastic Differential Equations: Theory and Practice of Numerical Solution. With MatLab Program, 1st Edition. D F Kuznetsov, In RussianKuznetsov D.F. Stochastic Differential Equations: Theory and Practice of Numerical Solution. With MatLab Program, 1st Edition. [In Russian]. . 10.18720/SPBPU/2/s17-2285-7422-1394-8778Polytechnical University Publishing House, Saint-PetersburgPolytechnical University Publishing House, Saint-Petersburg, 2007, 778 pp. DOI: http://doi.org/10.18720/SPBPU/2/s17-228 Available at: http://www.sde-kuznetsov.spb.ru/07b.pdf (ISBN 5-7422-1394-8) Stochastic Differential Equations: Theory and Practice of Numerical Solution. With MatLab Programs. D F Kuznetsov, 2nd Edition. [In RussianKuznetsov D.F. Stochastic Differential Equations: Theory and Practice of Numerical Solution. With MatLab Pro- grams, 2nd Edition. [In Russian]. . 10.18720/SPBPU/2/s17-229770Polytechnical University Publishing House, Saint-PetersburgPolytechnical University Publishing House, Saint-Petersburg, 2007, XXXII+770 pp. DOI: http://doi.org/10.18720/SPBPU/2/s17-229 Available at: http://www.sde-kuznetsov.spb.ru/07a.pdf (ISBN 5-7422-1439-1) Stochastic Differential Equations: Theory and Practice of Numerical Solution. With MatLab Programs. D F Kuznetsov, 3rd Edition. [In RussianKuznetsov D.F. Stochastic Differential Equations: Theory and Practice of Numerical Solution. With MatLab Pro- grams, 3rd Edition. [In Russian]. . 10.18720/SPBPU/2/s17-230978-5-7422-2132-6768Polytechnical University Publishing House, Saint-PetersburgPolytechnical University Publishing House, Saint-Petersburg, 2009, XXXIV+768 pp. DOI: http://doi.org/10.18720/SPBPU/2/s17-230 Available at: http://www.sde-kuznetsov.spb.ru/09.pdf (ISBN 978-5-7422-2132-6) Stochastic Differential Equations: Theory and Practice of Numerical Solution. With MatLab Programs. D F Kuznetsov, 4th Edition. [In RussianKuznetsov D.F. Stochastic Differential Equations: Theory and Practice of Numerical Solution. With MatLab Pro- grams. 4th Edition. [In Russian]. . 10.18720/SPBPU/2/s17-231978-5-7422-2448-8786Polytechnical University Publishing House, Saint-PetersburgPolytechnical University Publishing House, Saint-Petersburg, 2010, XXX+786 pp. DOI: http://doi.org/10.18720/SPBPU/2/s17-231 Available at: http://www.sde-kuznetsov.spb.ru/10.pdf (ISBN 978-5-7422-2448-8) Multiple Stochastic Ito and Stratonovich Integrals and Multiple Fourier Series. D F Kuznetsov, In RussianKuznetsov D.F. Multiple Stochastic Ito and Stratonovich Integrals and Multiple Fourier Series. [In Russian]. 10.18720/SPBPU/2/z17-71817-2172 (online)Differential Equations and Control Processes. 3A.1-A.257Electronic Journal "Differential Equations and Control Processes" ISSN 1817-2172 (online), 3 (2010), A.1-A.257. DOI: http://doi.org/10.18720/SPBPU/2/z17-7 Available at: http://diffjournal.spbu.ru/EN/numbers/2010.3/article.2.1.html Strong Approximation of Multiple Ito and Stratonovich Stochastic Integrals: Multiple Fourier Series Approach. 1st Edition. D F Kuznetsov, In EnglishKuznetsov D.F. Strong Approximation of Multiple Ito and Stratonovich Stochastic Integrals: Multiple Fourier Series Approach. 1st Edition. [In English]. . 10.18720/SPBPU/2/s17-232978-5-7422-2988-9250Polytechnical University Publishing House, Saint-PetersburgPolytechnical University Publishing House, Saint-Petersburg, 2011, 250 pp. DOI: http://doi.org/10.18720/SPBPU/2/s17-232 Available at: http://www.sde-kuznetsov.spb.ru/11b.pdf (ISBN 978-5-7422-2988-9) Strong Approximation of Multiple Ito and Stratonovich Stochastic Integrals: Multiple Fourier Series Approach. D F Kuznetsov, 2nd Edition.. In EnglishKuznetsov D.F. Strong Approximation of Multiple Ito and Stratonovich Stochastic Integrals: Multiple Fourier Series Approach. 2nd Edition. [In English]. . 10.18720/SPBPU/2/s17-233978-5-7422-3162-2284Polytechnical University Publishing House, Saint-PetersburgPolytechnical University Publishing House, Saint-Petersburg, 2011, 284 pp. DOI: http://doi.org/10.18720/SPBPU/2/s17-233 Available at: http://www.sde-kuznetsov.spb.ru/11a.pdf (ISBN 978-5-7422-3162-2) Multiple Ito and Stratonovich Stochastic Integrals: Approximations, Properties, Formulas. D F Kuznetsov, In EnglishKuznetsov D.F. Multiple Ito and Stratonovich Stochastic Integrals: Approximations, Properties, Formulas. [In English]. . 10.18720/SPBPU/2/s17-234382Polytechnical University Publishing House, Saint-PetersburgPolytechnical University Publishing House, Saint-Petersburg, 2013, 382 pp. DOI: http://doi.org/10.18720/SPBPU/2/s17-234 Multiple Ito and Stratonovich Stochastic Integrals: Fourier-Legendre and Trigonometric Expansions, Approximations, Formulas. D F Kuznetsov, 10.18720/SPBPU/2/z17-3A.1-A.385Electronic Journal "Differential Equations and Control Processes. 1In EnglishKuznetsov D.F. Multiple Ito and Stratonovich Stochastic Integrals: Fourier-Legendre and Trigonometric Expan- sions, Approximations, Formulas. [In English]. Electronic Journal "Differential Equations and Control Processes" ISSN 1817-2172 (online), 1 (2017), A.1-A.385. DOI: http://doi.org/10.18720/SPBPU/2/z17-3 D F Kuznetsov, 10.18720/SPBPU/2/z17-41817-2172 (onlineStochastic Differential Equations: Theory and Practice of Numerical Solution. With Programs on MATLAB. 2A.1-A.1000Electronic Journal "Differential Equations and Control ProcessesKuznetsov D.F. Stochastic Differential Equations: Theory and Practice of Numerical Solution. With Programs on MATLAB, 5th Edition. [In Russian]. Electronic Journal "Differential Equations and Control Processes" ISSN 1817-2172 (online), 2 (2017), A.1-A.1000. DOI: http://doi.org/10.18720/SPBPU/2/z17-4 Available at: http://diffjournal.spbu.ru/EN/numbers/2017.2/article.2.1.html Stochastic Differential Equations: Theory and Practice of Numerical Solution. With MATLAB Programs, 6th Edition. D F Kuznetsov, A.1-A.1073Electronic Journal "Differential Equations and Control Processes. 4In RussianKuznetsov D.F. Stochastic Differential Equations: Theory and Practice of Numerical Solution. With MATLAB Programs, 6th Edition. [In Russian]. Electronic Journal "Differential Equations and Control Processes" ISSN 1817-2172 (online), 4 (2018), A.1-A.1073. Available at: http://diffjournal.spbu.ru/EN/numbers/2018.4/article.2.1.html Strong Approximation of Iterated Ito and Stratonovich Stochastic Integrals Based on Generalized Multiple Fourier Series. Application to Numerical Solution of Ito SDEs and Semilinear SPDEs. D F Kuznetsov, arXiv:2003.14184math.PR]. 2022, 923 pp. [In EnglishKuznetsov D.F. Strong Approximation of Iterated Ito and Stratonovich Stochastic Integrals Based on Generalized Multiple Fourier Series. Application to Numerical Solution of Ito SDEs and Semilinear SPDEs. arXiv:2003.14184 [math.PR]. 2022, 923 pp. [In English]. Approximation of Iterated Itô and Stratonovich Stochastic Integrals Based on Generalized Multiple Fourier Series. Application to Numerical Solution of Itô SDEs and Semilinear SPDEs. D F Kuznetsov, Strong, In EnglishKuznetsov D.F. Strong Approximation of Iterated Itô and Stratonovich Stochastic Integrals Based on Generalized Multiple Fourier Series. Application to Numerical Solution of Itô SDEs and Semilinear SPDEs. [In English]. 1817-2172Differential Equations and Control Processes. 4A.1-A.606. Available atElectronic Journal "Differential Equations and Control Processes" ISSN 1817-2172 (online), 4 (2020), A.1-A.606. Available at: http://diffjournal.spbu.ru/EN/numbers/2020.4/article.1.8.html Mean-Square Approximation of Iterated Itô and Stratonovich Stochastic Integrals Based on Generalized Multiple Fourier Series. D F Kuznetsov, Electronic Journal "Differential Equations and Control Processes. Application to Numerical Integration of Itô SDEs and Semilinear SPDEs. In English. ISSN 1817-2172 (online), 4 (2021), A.1-A.788. Available atKuznetsov D.F. Mean-Square Approximation of Iterated Itô and Stratonovich Stochastic Integrals Based on Generalized Multiple Fourier Series. Application to Numerical Integration of Itô SDEs and Semilinear SPDEs. [In English]. Electronic Journal "Differential Equations and Control Processes" ISSN 1817-2172 (online), 4 (2021), A.1-A.788. Available at: http://diffjournal.spbu.ru/EN/numbers/2021.4/article.1.9.html Development and application of the Fourier method for the numerical solution of Ito stochastic differential equations. D F Kuznetsov, 10.1134/S0965542518070096Computational Mathematics and Mathematical Physics. 58Kuznetsov D.F. Development and application of the Fourier method for the numerical solution of Ito stochastic differential equations. [In English]. Computational Mathematics and Mathematical Physics, 58, 7 (2018), 1058- 1070. DOI: http://doi.org/10.1134/S0965542518070096 On numerical modeling of the multidimensional dynamic systems under random perturbations with the 1.5 and 2.0 orders of strong convergence. Automation and Remote Control. D F Kuznetsov, 10.1134/S000511791807005679Kuznetsov D.F. On numerical modeling of the multidimensional dynamic systems under random perturbations with the 1.5 and 2.0 orders of strong convergence. Automation and Remote Control, 79, 7 (2018), 1240-1254. DOI: http://doi.org/10.1134/S0005117918070056 On numerical modeling of the multidimentional dynamic systems under random perturbations with the 2.5 order of strong convergence. Automation and Remote Control. D F Kuznetsov, 10.1134/S000511791905006080Kuznetsov D.F. On numerical modeling of the multidimentional dynamic systems under random perturba- tions with the 2.5 order of strong convergence. Automation and Remote Control, 80, 5 (2019), 867-881. DOI: http://doi.org/10.1134/S0005117919050060 Comparative analysis of the efficiency of application of Legendre polynomials and trigonometric functions to the numerical integration of Itô stochastic differential equations. D F Kuznetsov, 10.1134/S0965542519080116Computational Mathematics and Mathematical Physics. 59In EnglishKuznetsov D.F. Comparative analysis of the efficiency of application of Legendre polynomials and trigonometric functions to the numerical integration of Itô stochastic differential equations. [In English]. Computational Math- ematics and Mathematical Physics, 59, 8 (2019), 1236-1250. DOI: http://doi.org/10.1134/S0965542519080116 Expansion of iterated Ito stochastic integrals of arbitrary multiplicity based on generalized multiple Fourier series converging in the mean. D F Kuznetsov, arXiv:1712.09746math.PR]. 2022, 111 pp. [in EnglishKuznetsov D.F. Expansion of iterated Ito stochastic integrals of arbitrary multiplicity based on generalized multiple Fourier series converging in the mean. arXiv:1712.09746 [math.PR]. 2022, 111 pp. [in English]. Expansions of iterated Stratonovich stochastic integrals based on generalized multiple Fourier series: multiplicities 1 to 6 and beyond. D F Kuznetsov, arXiv:1712.09516math.PR]. 2022, 204 pp. [In EnglishKuznetsov D.F. Expansions of iterated Stratonovich stochastic integrals based on generalized multiple Fourier series: multiplicities 1 to 6 and beyond. arXiv:1712.09516 [math.PR]. 2022, 204 pp. [In English]. Mean-square approximation of iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 from the Taylor-Ito and Taylor-Stratonovich expansions using Legendre polynomials. D F Kuznetsov, arXiv:1801.00231math.PR]. 2019, 106 pp. [In EnglishKuznetsov D.F. Mean-square approximation of iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 from the Taylor-Ito and Taylor-Stratonovich expansions using Legendre polynomials. arXiv:1801.00231 [math.PR]. 2019, 106 pp. [In English]. Strong numerical methods of orders 2.0, 2.5, and 3.0 for Ito stochastic differential equations based on the unified stochastic Taylor expansions and multiple Fourier-Legendre series. D F Kuznetsov, arXiv:1807.0219044math.PR. In EnglishKuznetsov D.F. Strong numerical methods of orders 2.0, 2.5, and 3.0 for Ito stochastic differential equations based on the unified stochastic Taylor expansions and multiple Fourier-Legendre series, arXiv:1807.02190 [math.PR]. 2018, 44 pp. [In English]. The hypotheses on expansions of iterated Stratonovich stochastic integrals of arbitrary multiplicity and their partial proof. D F Kuznetsov, arXiv:1801.03195math.PR]. 2022, 138 pp. [in EnglishKuznetsov D.F. The hypotheses on expansions of iterated Stratonovich stochastic integrals of arbitrary multi- plicity and their partial proof. arXiv:1801.03195 [math.PR]. 2022, 138 pp. [in English]. Expansion of iterated Stratonovich stochastic integrals of fifth and sixth multiplicity based on generalized multiple Fourier series. D F Kuznetsov, arXiv:1802.00643math.PR]. 2022, 129 pp. [In EnglishKuznetsov D.F. Expansion of iterated Stratonovich stochastic integrals of fifth and sixth multiplicity based on generalized multiple Fourier series. arXiv:1802.00643 [math.PR]. 2022, 129 pp. [In English]. Exact calculation of the mean-square error in the method of approximation of iterated Ito stochastic integrals based on generalized multiple Fourier series. D F Kuznetsov, arXiv:1801.0107968ppIn English. math.PRKuznetsov D.F. Exact calculation of the mean-square error in the method of approximation of iterated Ito stochastic integrals based on generalized multiple Fourier series. [In English]. arXiv:1801.01079 [math.PR]. 2019, 68 pp. Comparative analysis of the efficiency of application of Legendre polynomials and trigonometric functions to the numerical integration of Ito stochastic differential equations. D F Kuznetsov, arXiv:1901.0234540math.GM. In EnglishKuznetsov D.F. Comparative analysis of the efficiency of application of Legendre polynomials and trigonometric functions to the numerical integration of Ito stochastic differential equations. arXiv:1901.02345 [math.GM], 2019, 40 pp. [In English]. Application of the method of approximation of iterated Ito stochastic integrals based on generalized multiple Fourier series to the high-order strong numerical methods for non-commutative semilinear stochastic partial differential equations. D F Kuznetsov, arXiv:1905.0372441math.GM. In EnglishKuznetsov D.F. Application of the method of approximation of iterated Ito stochastic integrals based on general- ized multiple Fourier series to the high-order strong numerical methods for non-commutative semilinear stochastic partial differential equations. arXiv:1905.03724 [math.GM], 2019, 41 pp. [In English]. On the convergence of ordinary integrals to stochastic integrals. E Wong, M Zakai, Ann. Math. Stat. 5Wong E., Zakai M. On the convergence of ordinary integrals to stochastic integrals. Ann. Math. Stat., 5, 36 (1965), 1560-1564. On the relation between ordinary and stochastic differential equations. E Wong, M Zakai, Int. J. Eng. Sci. 3Wong E., Zakai M. On the relation between ordinary and stochastic differential equations. Int. J. Eng. Sci., 3 (1965), 213-229. N Ikeda, S Watanabe, Stochastic Differential Equations and Diffusion Processes. Amsterdam, Oxford, New-YorkNorth-Holland Publishing Company5552nd EditionIkeda N., Watanabe S. Stochastic Differential Equations and Diffusion Processes. 2nd Edition. North-Holland Publishing Company, Amsterdam, Oxford, New-York, 1989. 555 pp. Orthogonal expansion of multiple Itô stochastic integrals. K A Rybakov, 1817-2172 (onlineElectronic Journal "Differential Equations and Control Processes. 3Rybakov K.A. Orthogonal expansion of multiple Itô stochastic integrals. Electronic Journal "Differ- ential Equations and Control Processes" ISSN 1817-2172 (online), 3 (2021), 109-140. Available at: http://diffjournal.spbu.ru/EN/numbers/2021.3/article.1.8.html Statistics of Stochastic Processes: Nonlinear Filtering and Related Problems. R Liptser, Sh, A Shirjaev, In RussianLiptser R.Sh., Shirjaev A.N. Statistics of Stochastic Processes: Nonlinear Filtering and Related Problems. [In Russian]. . Nauka Moscow, 696Moscow, Nauka, 1974. 696 pp. Wiener chaos expansion and numerical solutions of stochastic partial differential equations. W Luo, California Inst. of Technology. 225ppPhD thesisLuo W. Wiener chaos expansion and numerical solutions of stochastic partial differential equations. PhD thesis, California Inst. of Technology, 2006, 225 pp. Implementation of strong numerical methods of orders 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 for Ito SDEs with non-commutative noise based on the unified Taylor-Ito and Taylor-Stratonovich expansions and multiple Fourier-Legendre series. M D Kuznetsov, D F Kuznetsov, arXiv:2009.14011343math.PR. In EnglishKuznetsov M.D., Kuznetsov D.F. Implementation of strong numerical methods of orders 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 for Ito SDEs with non-commutative noise based on the unified Taylor-Ito and Taylor-Stratonovich expansions and multiple Fourier-Legendre series. arXiv:2009.14011 [math.PR], 2020, 343 pp. [In English]. Application of multiple Fourier-Legendre series to the implementation of strong exponential Milstein and Wagner-Platen methods for non-commutative semilinear SPDEs. D F Kuznetsov, Proceedings of the XIII International Conference on Applied Mathematics and Mechanics in the Aerospace Industry AMMAI-2020. the XIII International Conference on Applied Mathematics and Mechanics in the Aerospace Industry AMMAI-2020Crimea, Alushta; MoscowMAIIn EnglishKuznetsov D.F. Application of multiple Fourier-Legendre series to the implementation of strong exponential Milstein and Wagner-Platen methods for non-commutative semilinear SPDEs. [In English]. Proceedings of the XIII International Conference on Applied Mathematics and Mechanics in the Aerospace Industry AMMAI-2020 (Crimea, Alushta, 6-13 September, 2020), MAI, Moscow, 2020, pp. 451-453. Available at: http://www.sde-kuznetsov.spb.ru/20e.pdf Application of the method of approximation of iterated stochastic Ito integrals based on generalized multiple Fourier series to the high-order strong numerical methods for non-commutative semilinear stochastic partial differential equations. D F Kuznetsov, 1817-2172 (online)Electronic Journal "Differential Equations and Control Processes. 3In EnglishKuznetsov D.F. Application of the method of approximation of iterated stochastic Ito integrals based on general- ized multiple Fourier series to the high-order strong numerical methods for non-commutative semilinear stochastic partial differential equations. [In English]. Electronic Journal "Differential Equations and Control Processes" ISSN 1817-2172 (online), 3 (2019), 18-62. Available at: http://diffjournal.spbu.ru/EN/numbers/2019.3/article.1.2.html The proof of convergence with probability 1 in the method of expansion of iterated Ito stochastic integrals based on generalized multiple Fourier series. D F Kuznetsov, 1817-2172Electronic Journal "Differential Equations and Control Processes. 2In EnglishKuznetsov D.F. The proof of convergence with probability 1 in the method of expansion of iterated Ito stochastic integrals based on generalized multiple Fourier series. [In English]. Electronic Journal "Differential Equations and Control Processes" ISSN 1817-2172 (online), 2 (2020), 89-117. Available at: http://diffjournal.spbu.ru/RU/numbers/2020.2/article.1.6.html Application of multiple Fourier-Legendre series to strong exponential Milstein and Wagner-Platen methods for non-commutative semilinear stochastic partial differential equations. D F Kuznetsov, 1817-2172Electronic Journal "Differential Equations and Control Processes. 3In EnglishKuznetsov D.F. Application of multiple Fourier-Legendre series to strong exponential Milstein and Wagner- Platen methods for non-commutative semilinear stochastic partial differential equations. [In English]. Electronic Journal "Differential Equations and Control Processes" ISSN 1817-2172 (online), 3 (2020), 129-162. Available at: http://diffjournal.spbu.ru/RU/numbers/2020.3/article.1.6.html Optimization of the mean-square approximation procedures for iterated Ito stochastic integrals of multiplicities 1 to 5 from the unified Taylor-Ito expansion based on multiple Fourier-Legendre series. M D Kuznetsov, D F Kuznetsov, arXiv:2010.1356463math.PR. In EnglishKuznetsov M.D., Kuznetsov D.F. Optimization of the mean-square approximation procedures for iterated Ito stochastic integrals of multiplicities 1 to 5 from the unified Taylor-Ito expansion based on multiple Fourier- Legendre series arXiv:2010.13564 [math.PR], 2020, 63 pp. [In English]. Optimization of the mean-square approximation procedures for iterated Ito stochastic integrals based on multiple Fourier-Legendre series. D F Kuznetsov, M D Kuznetsov, 10.1088/1742-6596/1925/1/012010Journal of Physics: Conference Series. 192512010In EnglishKuznetsov D.F., Kuznetsov M.D. Optimization of the mean-square approximation procedures for iterated Ito stochastic integrals based on multiple Fourier-Legendre series. [In English]. Journal of Physics: Conference Series, Vol. 1925 (2021), article id: 012010, 12 pp. DOI: http://doi.org/10.1088/1742-6596/1925/1/012010 SDE-MATH: A software package for the implementation of strong high-order numerical methods for Ito SDEs with multidimensional non-commutative noise based on multiple Fourier-Legendre series. M D Kuznetsov, D F Kuznetsov, 1817-2172 (onlineElectronic Journal "Differential Equations and Control Processes. In EnglishKuznetsov M.D., Kuznetsov D.F. SDE-MATH: A software package for the implementation of strong high-order nu- merical methods for Ito SDEs with multidimensional non-commutative noise based on multiple Fourier-Legendre series. [In English]. Electronic Journal "Differential Equations and Control Processes" ISSN 1817-2172 (online), 1 (2021), 93-422. Available at: http://diffjournal.spbu.ru/EN/numbers/2021.1/article.1.5.html Mean-square approximation of iterated stochastic integrals from strong exponential Milstein and Wagner-Platen methods for non-commutative semilinear SPDEs based on multiple Fourier-Legendre series. D F Kuznetsov, M D Kuznetsov, 10.1007/978-3-030-83266-7_2Recent Developments in Stochastic Methods and Applications. ICSM-5 2020. Springer Proceedings in Mathematics & Statistics. Shiryaev, A.N., Samouylov, K.E., Kozyrev, D.V. SpringerCham371Kuznetsov D.F., Kuznetsov M.D. Mean-square approximation of iterated stochastic integrals from strong expo- nential Milstein and Wagner-Platen methods for non-commutative semilinear SPDEs based on multiple Fourier- Legendre series. Recent Developments in Stochastic Methods and Applications. ICSM-5 2020. Springer Proceed- ings in Mathematics & Statistics, vol 371, Eds. Shiryaev, A.N., Samouylov, K.E., Kozyrev, D.V. Springer, Cham, 2021, pp. 17-32. DOI: http://doi.org/10.1007/978-3-030-83266-7 2 Application of multiple Fourier-Legendre series to implementation of strong exponential Milstein and Wagner-Platen methods for non-commutative semilinear stochastic partial differential equations. D F Kuznetsov, arXiv:1912.0261232math.PR. In EnglishKuznetsov D.F. Application of multiple Fourier-Legendre series to implementation of strong exponential Mil- stein and Wagner-Platen methods for non-commutative semilinear stochastic partial differential equations. arXiv:1912.02612 [math.PR], 2019, 32 pp. [In English]. A new approach to the series expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity with respect to components of the multidimensional Wiener process. D F Kuznetsov, 1817-2172Electronic Journal "Differential Equations and Control Processes. 2In EnglishKuznetsov D.F. A new approach to the series expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity with respect to components of the multidimensional Wiener process. [In English]. Electronic Journal "Differential Equations and Control Processes" ISSN 1817-2172 (online), 2 (2022), 83-186. Available at: http://diffjournal.spbu.ru/EN/numbers/2022.2/article.1.6.html A new approach to the series expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity with respect to components of the multidimensional Wiener process. II. D F Kuznetsov, 1817-2172 (onlineElectronic Journal "Differential Equations and Control Processes. 4In English. To appear. Available atKuznetsov D.F. A new approach to the series expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity with respect to components of the multidimensional Wiener process. II. [In English]. Electronic Journal "Differential Equations and Control Processes" ISSN 1817-2172 (online), 4 (2022). To appear. Available at: http://diffjournal.spbu.ru/EN/collection.html . Polytechnicheskaya ul. 29Dmitriy Feliksovich Kuznetsov Peter the Great Saint-Petersburg Polytechnic UniversityRussia Email address: sde [email protected] Feliksovich Kuznetsov Peter the Great Saint-Petersburg Polytechnic University, Polytechnicheskaya ul., 29, 195251, Saint-Petersburg, Russia Email address: sde [email protected]
{'fraction_non_alphanumeric': 0.14385053722415725, 'fraction_numerical': 0.10600484917108713, 'mean_word_length': 3.0943310657596372, 'pattern_counts': {'":': 0, '<': 10, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 8, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 174, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'The article is devoted to comparison of the Milstein expansion of iterated Stratonovich stochastic integrals with the method of expansion of iterated stochastic integrals based on generalized multiple Fourier series. We consider the practical material connected with the expansions of iterated Stratonovich stochastic integrals from the Taylor-Stratonovich expansion based on multiple trigonometric Fourier series. The comparison of effectiveness of the Fourier-Legendre series as well as the trigonomertic Fourier series for expansions of iterated Stratonovich stochastic integrals is considered.(i = 1, . . . , m) of this process are independent. Consider an Ito stochastic differential equation (SDE) in the integral form Mathematics Subject Classification: 60H05, 60H10, 42B05, 42C10.', 'arxivid': '1801.08862', 'author': ['Dmitriy F Kuznetsov '], 'authoraffiliation': [], 'corpusid': 56284920, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 40611, 'n_tokens_neox': 34720, 'n_words': 15189, 'pdfsha': 'b5012a144d592455bd77b9e27afe3edfcf4d3eb7', 'pdfurls': ['https://export.arxiv.org/pdf/1801.08862v9.pdf'], 'title': ['EXPANSIONS OF ITERATED STRATONOVICH STOCHASTIC INTEGRALS FROM THE TAYLOR-STRATONOVICH EXPANSION BASED ON MULTIPLE TRIGONOMETRIC FOURIER SERIES. COMPARISON WITH THE MILSTEIN EXPANSION', 'EXPANSIONS OF ITERATED STRATONOVICH STOCHASTIC INTEGRALS FROM THE TAYLOR-STRATONOVICH EXPANSION BASED ON MULTIPLE TRIGONOMETRIC FOURIER SERIES. COMPARISON WITH THE MILSTEIN EXPANSION'], 'venue': []}
arxiv
UNSSOR: Unsupervised Neural Speech Separation by Leveraging Over-determined Training Mixtures Zhong-Qiu Wang [email protected] Language Technologies Institute Carnegie Mellon University PittsburghUSA Shinji Watanabe Language Technologies Institute Carnegie Mellon University PittsburghUSA UNSSOR: Unsupervised Neural Speech Separation by Leveraging Over-determined Training Mixtures In reverberant conditions with multiple concurrent speakers, each microphone acquires a mixture signal of multiple speakers at a different location. In overdetermined conditions where the microphones out-number speakers, we can narrow down the solutions to speaker images and realize unsupervised speech separation by leveraging each mixture signal as a constraint (i.e., the estimated speaker images at a microphone should add up to the mixture). Equipped with this insight, we propose UNSSOR, an algorithm for unsupervised neural speech separation by leveraging over-determined training mixtures. At each training step, we feed an input mixture to a deep neural network (DNN) to produce an intermediate estimate for each speaker, linearly filter the estimates, and optimize a loss so that, at each microphone, the filtered estimates of all the speakers can add up to the mixture to satisfy the above constraint. We show that this loss can promote unsupervised separation of speakers. The linear filters are computed in each sub-band based on the mixture and DNN estimates through the forward convolutive prediction (FCP) algorithm. To address the frequency permutation problem incurred by using sub-band FCP, a loss term based on minimizing intra-source magnitude scattering is proposed. Although UNSSOR requires over-determined training mixtures, we can train DNNs to achieve under-determined separation (e.g., unsupervised monaural speech separation). Evaluation results on two-speaker separation in reverberant conditions show the effectiveness and potential of UNSSOR. Preprint. Under review. arXiv:2305.20054v1 [cs.SD] 31 May 2023objects (e.g., clean speaker signals) are, as there are infinite solutions where in each solution the estimated sources can sum up to the mixture. Supposing that the separation model does not separate well and outputs a clean speaker signal plus some competing speech, noise or reverberation, would this output be viewed as a desired sound object? This is clear to humans, clear to supervised learning based models (by comparing the outputs with training labels), but not really clear to an unsupervised model. On the other hand, many studies[3][4][5][6][7][8][9][10][11][12][13][14]have observed that deep learning based supervised learning can achieve remarkable separation. In other words, with proper supervision, modern DNNs are capable of separating mixed speakers, but, in an unsupervised setup, there lacks an accurate supervision to unleash this capability. The key to successful unsupervised neural separation, we believe, is designing a clever supervision that can inform the model what desired sound objects are, and penalize the model if its outputs are not good and reward otherwise.Our insight is that, in multi-microphone over-determined conditions where the microphones outnumber speakers, the ill-posed problem can be turned into a well-posed one, where a unique solution to the speakers exists (up to speaker permutation). This well-posed property (that a unique solution exists) can be leveraged as a supervison (or regularizer) to design loss functions that could inform the unsupervised separation model what desired sound objects are and promote separation of speakers.Equipped with this insight, we perform unsupervised neural speech separation by leveraging multimicrophone over-determined training mixtures. Our DNNs can be trained directly on over-determined mixtures to realize over-and under-determined separation. The proposed algorithm, named UNSSOR, obtains strong separation performance on two-speaker separation. Our contributions include:• We enforce a linear-filter constraint between each speaker's reverberant images at each microphone pair, turning the ill-posed problem into a well-posed one that can promote separation of speakers. • We formulate unsupervised neural speech separation as a blind deconvolution problem, where both the speaker images and linear filters need to be estimated. We design loss functions motivated by the blind deconvolution problem, and propose a DNN approach to optimize the loss functions, where the speaker images are estimated via DNNs and the linear filters are estimated via a sub-band linear prediction algorithm named FCP [17] based on the mixture and DNN estimates. • We propose a loss term, which minimizes a measure named intra-source magnitude scattering, to address the frequency permutation problem incurred when using sub-band FCP. • Based on over-determined training mixtures, UNSSOR can be trained to perform under-determined separation (e.g., monaural unsupervised speech separation). Introduction In many machine learning and artificial intelligence applications, sensors, while recording, usually capture a mixture of desired and undesired signals. One example is the cocktail party problem (or speech separation) [1,2], where, given a recorded mixture of the concurrent speech by multiple speakers, the task is to separate the mixture to individual speaker signals. Speech separation [3] has been dramatically advanced by deep learning, since deep clustering [4] and permutation invariant training (PIT) [5] solved the label permutation problem. They (and their subsequent studies [6][7][8][9][10][11][12][13][14]) are based on supervised learning, requiring paired clean speech and its corrupted signal generated via simulation, where clean speech is mixed with, for example, various noises and competing speakers at diverse energy and reverberation levels in simulated rooms [3]. The clean speech can provide an accurate, sample-level supervision for model training. Such simulated data, however, may not match the distribution of real-recorded test data in the target domain, and the resulting supervised learning based models would have generalization issues [15,16]. How to train unsupervised neural speech separation systems on unlabelled target-domain mixtures is hence an important problem to study. Training unsupervised speech separation models directly on monaural mixtures is an ill-posed task [2], since there is only one mixture signal observed but multiple speaker signals to reconstruct. The separation model would lack an accurate supervision (or regularizer) to figure out what desired sound 2 Related work Various unsupervised neural separation algorithms, which do not require labelled mixtures, have been proposed. The most notable one is mixture invariant training (MixIT) [18][19][20][21][22], which first synthesizes training mixtures, each by mixing two existing mixtures, and then trains a DNN to separate the resulting mixed mixture to underlying sources such that the separated sources can be partitioned into two groups and the separated sources in each group can sum up to one of the two existing mixtures (used for mixing). Care needs to be taken when synthesizing mixture of mixtures. First, the sources in an existing mixture could have similar characteristics (e.g., similar reverberation patterns as the sources in an existing mixture are recorded in the same room) that are informative about which sources belong to the same existing mixture, and this would prevent MixIT from separating the sources [18,23]. Second, it is unclear how to mix existing multi-channel mixtures, which are usually recorded by devices with different microphone geometry and number of microphones. Third, mixing existing mixtures with different reverberation characteristics would create unrealistic mixtures. UNSSOR avoids the above issues by training unsupervised neural separation models directly on existing mixtures rather than on synthesized mixture of mixtures. An earlier study related to this direction is the reverberation as supervsion (RAS) algorithm [24], which addresses monaural twospeaker separation given binaural (two-channel) training mixtures. RAS performs magnitude-domain monaural separation directly on the left-ear mixture and then linearly filters the estimates through time-domain Wiener filtering so that the filtered estimates can approximate the right-ear mixture. RAS essentially does monaural separation and is effective at separating speakers in a semi-supervised learning setup, where a supervised PIT-based model is first trained and then used to boot-start unsupervised training. It however fails completely in fully-unsupervised setup [24], unlike UNSSOR. Conventional algorithms such as independent component analysis [25][26][27][28][29], independent vector analysis (IVA) [29][30][31][32] and spatial clustering [33][34][35][36] can perform unsupervised separation directly on existing mixtures. They perform separation based on a single test mixture at hand and are not designed to learn speech patterns from large training data, while UNSSOR leverages DNNs to model speech patterns through unsupervised learning, which could result in better separation. Another difference is that UNSSOR can be configured for monarual, under-determined separation, while ICA, IVA and spatial clustering cannot. There are studies [37] training DNNs to approximate pseudo-labels produced by spatial clustering. Their performance is however limited by that of spatial clustering. Problem formulation Given a P -microphone mixture with C speakers in reverberation conditions, the physical model can be formulated using a system of linear equations in the short-time Fourier transform (STFT) domain: Y p (t, f ) = C c=1 X p (c, t, f ) + ε p (t, f ), for p ∈ {1, . . . , P },(1) where t indexes T frames, f indexes F frames, and at microphone p, time t and frequency f , Y p (t, f ), X p (c, t, f ) and ε p (t, f ) ∈ C respectively denote the STFT coefficients of the mixture, reverberant image of speaker c, and non-speech signals. In the rest of this paper, we refer to the corresponding spectrogram when dropping the index c, p, t or f . We assume that ε is weak and stationary (e.g., a time-invariant Gaussian noise or simply modelling errors). Without loss of generality, we designate microphone 1 as the reference microphone. Our goal is to, in an unsupervised way, estimate each speaker's image at the reference microphone (i.e., X 1 (c) for each speaker c) given the input mixture. We do not aim at dereverberation, instead targeting at maintaining the reverberation of each speaker. Unsupervised separation based only on the observed mixture is difficult. There are infinite solutions to the above linear system where there are T × F × P equations (we have a mixture observation for each Y p (t, f )) but T × F × P × C unknowns (we have one unknown for each X p (c, t, f )). Our insight is that the number of unknowns can be dramatically reduced, if we enforce constraints to the speaker images at different microphones. Since X 1 (c) and X p (c) are both convolved versions of the dry signal of speaker c, there exists a linear filter between them such that convolving X 1 (c) with the filter would reproduce X p (c). This convolutive relationship is a physical constraint, which can be leveraged to reduce the number of unknowns. Specifically, we formulate (1) as Y 1 (t, f ) = C c=1 X 1 (c, t, f ) + ε 1 (t, f ), Y p (t, f ) = C c=1 g p (c, f ) H X 1 (c, t, f ) + ε p (t, f ), for p ∈ {2, . . . , P },(2) where X 1 (c, t, f ) = [X 1 (c, t − A, f ), . . . , X 1 (c, t, f ), . . . , X 1 (c, t + B, f )] T ∈ C A+1+B stacks a window of E = A + 1 + B T-F units, g p (c, f ) ∈ C E is the relative room impulse response (relative RIR) relating X 1 (c) to X p (c), and (·) H computes Hermittan. Note that g p (c, f ) is very short (i.e., E is small) if microphone 1 and p are placed close to each other, which is the case for compact arrays. An implication of this constraint is that the number of unknowns is reduced from T × F × P × C to T × F × C + F × (P − 1) × E × C 1 , which can be smaller than the number of equations (i.e., T × F × P ) when P > C (i.e., over-determined conditions) and when T is sufficiently large (i.e., the input mixture is reasonably long). In other words, this formulation suggests that (1) there exists a solution for separation, which is most consistent with the above linear system; and (2) in over-determined cases, it is possible to estimate the speaker images in an unsupervised way. As ε is assumed weak, time-invariant and Gaussian, one way to find the solution is to comptue an estimate that is most consistent with the linear system in (2): argmin g·(·,·),X1(·,·,·) t,f Y 1 (t, f ) − C c=1 X 1 (c, t, f ) 2 + P p=2 t,f Y p (t, f ) − C c=1 g p (c, f ) H X 1 (c, t, f ) 2 . (3) This is a blind deconvolution problem [38], which is non-convex in nature and difficult to be solved if no prior knowledge is assumed about the relative RIRs or the speaker images, because both of them are unknown. In the next section, we propose a DNN-based approach, which can model speech patterns through unsupervised learning (and hence model speech priors), to tackle this problem. Figure 1: Illustration of UNSSOR (assuming P > C). FCP [17] is then performed onẐ(c) at each microphone p to compute a linear-filtering result, denoted asX FCP p (c), which, we will describe, is essentially an estimate of the speaker image X p (c). After that, two loss functions are computed and combined for DNN training. This section describes the DNN configuration, loss functions, FCP filtering, and an extension for monaural separation. Microphone ! … Microphone " DNN multi-channel input: ! & , … , ! ! , … , ! ' or monaural input: ! & " #(%) " #(+) " # 1 FCP FCP FCP , 3 ! ()* 1 , 3 ! ()* % , 3 ! ()* + ⨁ , ! ! ℒ "),! ! ! ℒ .45/6.6 ⨁ … … … … ℒ #+"+,! Method DNN configurations The intermediate estimateẐ(c) for each speaker c is obtained via complex spectral mapping [14,39], where we stack the real and imaginary (RI) parts of the input mixture as features for the DNN to predict the RI parts ofẐ(c). For the DNN architecture, we employ TF-GridNet [14], which obtains strong results on supervised speech separation benchmarks. See Appendix F for more DNN details. Mixture-consistency loss on filtered estimates Following (3), we propose mixture consistency (MC) loss, which is computed by filtering the DNN estimateẐ(c) of each speaker c to approximate the P -channel input mixture: L MC = α 1 t,f F(Y 1 (t, f ), C c=1Ẑ (c, t, f )) + P p=2 α p t,f F(Y p (t, f ), C c=1ĝ p (c, f ) H Ẑ (c, t, f )). (4) Ẑ (c, t, f ) stacks a window of T-F units aroundẐ(c, t, f ), andĝ p (c, f ) is an estimated relative RIR computed based onẐ(c, ·, f ) and the mixture Y p (·, f ) through FCP [17]. Both of them will be described in the next sub-section. α p ∈ R is a weighting term for microphone p. Following [14], F(·, ·) computes an absolute loss on the estimated RI components and their magnitude: F Y p (t, f ),Ŷ p (t, f ) = 1 t ′ ,f ′ |Y p (t ′ , f ′ )| Re(Y p (t, f )) − Re(Ŷ p (t, f )) + Im(Y p (t, f )) − Im(Ŷ p (t, f )) + |Y p (t, f )|−|Ŷ p (t, f )| ,(5) where Re(·) and Im(·) respectively extract RI components and |·| computes magnitude. The term 1/ t ′ ,f ′ |Y p (t ′ , f ′ )| balances the losses at different microphones and across training mixtures. According to the discussion in Section 3, minimizing L MC would encourage separation of speakers. We give an illustration of loss surface of L MC in Appendix B. FCP for relative RIR estimation To compute L MC , we need to first estimate each of the relative RIRs,ĝ p (c, f ). In [17], FCP is proposed to estimate the relative RIR relating direct-path signal to reverberant image for speech dereverberation. In this study, we employ FCP to estimate the relative RIR relatingẐ(c) to the speaker image captured at each microphone p (i.e., X p (c)). Assuming speakers are non-moving, we estimate relative RIRs by solving the following problem: g p (c, f ) = argmin gp(c,f ) t 1 λ p (c, t, f ) |Y p (t, f ) − g p (c, f ) H Ẑ (c, t, f )| 2 ,(6) where g p (c, f ) ∈ C I+1+J is a K-tap (with K = I + 1 + J) time-invariant FCP filter, Ẑ (c, t, f ) = [Ẑ(c, t − I, f ), . . . ,Ẑ(c, t, f ), . . . ,Ẑ(c, t + J, f )] T ∈ C K stacks I past and J future T-F units with the current one. Since the actual number of filter taps (i.e., A and B defined in the text below (2)) is unknown, we set them to I and J, both of which are hyper-parameters to tune.λ p (c, t, f ) is a weighting term balancing the importance of each T-F unit. Following [17], we define it aŝ λ p (c, t, f ) = ξ max( 1 P P p ′ =1 |Y p ′ | 2 ) + |Y p (t, f )| 2 , where ξ (= 10 −4 in this study) is used to floor the weighting term and max(·) extracts the maximum value of a spectrogram. (6) is a weighted linear regression problem, where a closed-form solution can be readily computed: g p (c, f ) = t 1 λ p (c, t, f ) Ẑ (c, t, f ) Ẑ (c, t, f ) H −1 t 1 λ p (c, t, f ) Ẑ (c, t, f )(Y p (t, f )) * ,(7) where (·) * computes complex conjugate. We then plugĝ p (c, f ) into (4) and compute the loss. Although in (6) we linearly filterẐ(c) to approximate Y p , earlier studies [17] suggest that the resultinĝ g p (c, f ) H Ẑ (c, t, f ) would be an estimate of X p (c, t, f ), ifẐ(c) is reasonably accurate (see Appendix C for the derivation). We name the speaker image estimated this way as FCP-estimated image: X FCP p (c, t, f ) =ĝ p (c, f ) H Ẑ (c, t, f ).(8) It is therefore reasonable to sum up the FCP-estimated images of all the speakers and define a loss between the summation and Y p as in (4). Time alignment issues and alternative loss functions In (4), we do not filter the DNN estimates when computing the loss on the first (reference) microphone. We expect this to result in aẐ(c) time-aligned with the speaker image X 1 (c) (i.e.,Ẑ(c) is an estimate of X 1 (c)). Since the reference microphone may not be the microphone closest to speaker c, it is best to use non-causal filters when filteringẐ(c) to approximate the reverberant image X p (c) at non-reference microphones that are closer to source c than the reference microphone, and instead use causal filters for non-reference microphones that are farther. 2 Since estimating which non-reference microphones are closer or farther to a source than the reference microphone is not an easy task and doing this would complicates our system, we can just choose to use non-causal filters for all the non-reference microphones. This could, however, limit the DNN's capability at separating the speakers, because the relative RIRs for some non-reference microphones (farther to source c than the reference microphone) are causal, and it may not be a good idea to assume non-causal filters. To address this issue, we make a simple modification to the loss function in (4): L MC = P p=1 α p L MC,p = P p=1 α p t,f F Y p (t, f ), C c=1ĝ p (c, f ) H Ẑ (c, t, f ) ,(9) where the difference is that we also filter the DNN estimates when computing the loss on the reference microphone, and we constrainĝ p (c, f ) to be causal and that Ẑ (c, t, f ) only stacks current and past frames. This way, the resultingẐ(c) would not be time-aligned with the revererberant image captured at the reference microphone (i.e., X 1 (c)) or any other non-reference microphones. Because of the causal filtering,Ẑ(c) would be more like an estimate of the reverberant image captured by a virtual microphone that is closer to speaker c than all the P microphones. It would contain less reverberation of speaker c than any of the speaker images captured by the P microphones due to the causal filtering. To produce an estimate that is time-aligned with the reverberant image at a microphone (e.g., X p (c)), we use the FCP-estimated image computed in (8) (i.e.,X FCP p (c)) as the output. Addressing frequency permutation problem In (4) and (9), FCP is performed in each frequency independently from the others. Even though the speakers are separated at each frequency, the separation results of the same speaker at different frequencies may however not be grouped into the same output spectrogram (see an example in Appendix D). This is known as the frequency permutation problem [29], which has been studied for decades in frequency-domain blind source separation algorithms such as frequency-domain independent component analysis [25][26][27][28][29] and spatial clustering [33][34][35][36]. Popular solutions for frequency alignment are designed by leveraging cross-frequency correlation of spectral patterns [35,41] and direction-of-arrival estimation [42]. However, these solutions are often empirical and have a complicated design. They can be used to post-process DNN estimates for frequency alignment, but it is not easy to integrate them with UNSSOR for joint training. This section proposes a loss term, with which the trained DNN can learn to produce target estimates without frequency permutation. To deal with frequency permutation, IVA [30][31][32] assumes that, at each frame, the de-mixed outputs at all the frequencies follow a complex Gaussian distribution with a shared variance term across frequencies: w(c, f ) H Y(t, f ) ∼ N (0, D(t, c)), where w(c, f ) ∈ C P is the de-mixing weight vector (in a time-invariant de-mixing matrix) for speaker c at frequency f , and D(t, c) ∈ R is the shared variance term, assumed time-variant. When maximum likelihood estimation is performed to estimate the de-mixing matrix, the variance term shared across all the frequencies is found very effective at solving the frequency permutation problem [29][30][31][32]. Motivated by IVA, we design the following loss term, named intra-source magnitude scattering (ISMS), to alleviate the frequency permutation problem in DNN outputs: L ISMS = P p=1 α p L ISMS,p = P p=1 α p t 1 C C c=1 var log(|X FCP p (c, t, ·)|) t var log(|Y p (t, ·)|) ,(10) whereX FCP p is computed via (8),X FCP p (c, t, ·) ∈ C F , and var(·) computes the variance of the values in a vector. At each frame, we essentially want the the magnitudes of the estimated spectrogram of each speaker (i.e.,X FCP p (c, t, ·)) to have a small intra-source variance. The rationale is that, when frequency permutation happens,X FCP p (c, t, ·) would contain multiple sources, and the resulting variance would be larger than that computed whenX FCP p (c, t, ·) contains only one source. L ISMS echoes IVA's idea of assuming a shared variance term across all the frequencies. If the ratio in (10) becomes smaller, it indicates that the magnitudes ofX FCP p (c, t, ·) are more centered around their mean. This is similar to optimizing the likelihood ofX FCP p (c, t, ·) under a Gaussian distribution with a variance term shared across all the frequencies. In (10), a logarithmic compression is applied, since log-compressed magnitudes better follow Gaussian distribution than raw magnitudes [43]. We combine L ISMS with L MC in (9) for DNN training, using a weighting term γ ∈ R: L MC+ISMS = L MC + γ × L ISMS .(11) 4.6 Training UNSSOR for monaural unsupervised separation UNSSOR can be trained for monaural unsupervised separation by only feeding the mixture at the reference microphone to the DNN but still computing the loss on multiple microphones. Fig. 1 illustrates the idea. At run time, the trained system performs monaural under-determined separation, and multi-microphone over-determined mixtures are only required for DNN training. The loss computed at multiple microphones could guide the DNN to exploit monaural spectro-temporal patterns for separation, even in an unsupervised setup. Experimental setup We validate the proposed algorithms on two-speaker separation in reverberant conditions based on the six-channel SMS-WSJ dataset [44] (see Appendix A for its details). This section describes the baseline systems and evaluation metrics. See Appendix F for miscellaneous system and DNN setup. Baselines The baselines include conventional unsupervised separation algorithms, an improved version of RAS, and supervised learning based models. We include spatial clustering [34][35][36] for comparison. We use a public implementation [45], which leverages complex angular-central Gaussian mixture models [36] for sub-band spatial clustering and exploits inter-frequency correlation of cluster posteriors [34,41] for frequency alignment. The number of sources is set to three, one of which is used for garbage collection, following [32]. After obtaining the estimates, we discard the one with the lowest energy. The STFT window size is tuned to 128 ms and hop size to 16 ms. We include IVA [29,32] for comparison. We use the public implementations provided by the torchiva toolkit [46]. We use the default spherical Laplacian model to model source distribution. In overdetermined cases, the number of sources is set to three and we discard the estimate with the lowest energy, similarly to the setup in the spatial clustering baseline. The STFT window size is tuned to 256 ms and hop size to 32 ms. We propose a novel variant of the RAS algorithm [24] for comparison. Appendix E discusses the differences between UNSSOR and RAS. Since RAS cannot achieve unsupervised separation, we improve it by computing loss on multi-microphone mixtures, and name the new algorithm as improved RAS (iRAS). We employ the time-domain Wiener filtering (WF) technique in [24] to filter re-synthesized time-domain estimatesẑ(c) = iSTFT(Ẑ(c)), whereẐ(c) is produced by TF-GridNet. The loss is defined as: L iRAS = P p=1 α p L iRAS,p = P p=1 α p 1 ∥y p ∥ 1 y p − C c=1ĥ p (c) * ẑ(c) 1 ,(12) with * denoting linear convolution, y p the time-domain mixture at microphone p, andĥ p (c) a time-domain Wiener filter computed by solving the following problem: h p (c) = argmin hp(c) y p − h p (c) * ẑ(c) 2 2 ,(13) which has a closed-form solution. The separation result is computed asx WF p (c) =ĥ p (c) * ẑ(c). Following [24], we use 512 filter taps, and filter the future 100, the current, and the past 411 samples (i.e., non-causal filtering). We can also filter the current and the past 511 samples (i.e., causal filtering), and experiment with a filter length (in time) same as the length of FCP filters (see Appendix G)). We report the result of using supervised learning, where PIT [5] is used to address the permutation problem. This result can be viewed as a performance upper bound of unsupervised separation. We use the same DNN and training configurations as those in UNSSOR for a fair comparison. Evaluation metrics We designate the first microphone as the reference microphone, and use the time-domain signal corresponding to X 1 (c) of each speaker c for metric computation. The evaluation metrics include signal-to-distortion ratio (SDR) [47], scale-invariant SDR (SI-SDR) [48], perceptual evaluation of speech quality (PESQ) [49], and extended short-time objective intelligibility (eSTOI) [50]. 6 Evaluation results 6.1 Effectiveness of UNSSOR at promoting separation Table 1 and 2 respectively report the results of using six-and three-microphone input and loss. After hyper-parameter tuning, in default we use the loss in (9) for DNN training, set I = 19 and J = 0 (defined below (6)) for FCP (i.e., causal FCP filtering with 20 taps), and set α p = 1 (meaning no weighting is applied for different microphones). For the 3-microphone case, we use the mixtures at the first, third, and fifth microphones for training and testing. In both tables, from row 1a we notice that UNSSOR produces reasonable separation of speakers, improving the SDR from 0.1 to, for example, 12.5 dB in Table 1, but its output suffers from the frequency permutation problem (see Appendix D for an example). In row 1c, we use oracle target speech to obtain oracle frequency alignment and observe much better results over 1a. This shows the effectiveness of L MC at promoting separation of speakers and the severity of the frequency permutation problem. In row 1b, we use a frequency alignment algorithm (same as that used in the spatial clustering baseline) [34,41] to post-process the separation results of 1a. This algorithm leads to impressive frequency alignment (see 1b vs. 1c), but it is empirical and has a complicated design. Effectiveness of including intra-source magnitude scattering loss We train DNNs using L MC+ISMS defined in (11). In each case (i.e, six-and three-microphone), we separately tune the weighting term γ in (11) based on the validation set. In both table, comparing row 2a-2c with 1a-1c, we observe that including L ISMS is very effective at dealing with the frequency permutation problem, yielding almost the same performance as using oracle frequency alignment. Table 3 and 4 use the mixture only at the reference microphone 1 as the network input, while computing the loss respectively on three and six microphones. We tune J to 1 (i.e., non-causal FCP filter), considering that, for a specific target speaker, the reference microphone may not be the microphone closest to that speaker. 3 We still set the microphone weight α p to 1.0 for non-reference microphones (i.e., when p ̸ = 1), but tune α 1 to a smaller value based on the validation set. Without using a smaller α 1 , we found that the DNN easily overfits to microphone 1, as we use the mixture at microphone 1 as the only input and compute the MC loss also on the mixture at microphone 1. The DNN can just optimize L MC,p to zero at microphone 1 and not optimize that at other microphones. Results of training UNSSOR for monaural unsupervised separation From row 1a of both tables, strong performance is observed in this under-determined setup, indicating that the multi-microphone loss can inform the DNN what desired target sound objects are and the DNN can learn to model spectral patterns in speech for unsupervised separation. Comparison with other methods In Table 1-4, we compare the performance of UNSSOR with spatial clustering, IVA, iRAS, and supervised PIT-based models. In Appendix G, we compare UNSSOR and iRAS when they use the same filter length (in time). UNSSOR shows better performance than previous unsupervised separation models that can be performed or trained directly on mixtures. It is worse than supervised PIT but the performance is reasonably strong. For example, in row 2a of Conclusion We have proposed UNSSOR for unsupervised neural speech separation. We show that it is possible to train unsupervised models directly on mixtures, if the mixtures are over-determined. We have proposed mixture-consistency loss functions, which leverage multi-microphone mixtures as constraints, to promote separation of speakers. We find that minimizing ISMS can alleviate the frequency permutation problem. Although UNSSOR requires over-determined training mixtures, it can be trained to perform under-determined unsupervised separation. Future research will combine UNSSOR with semi-supervised learning, evaluate it on real data recorded in noisy-reverberant conditions, and address the limitation of our current system (see Appendix H). In closing, we emphasize that the key scientific contribution of this paper is that the over-determined property afforded by having more microphones than speakers can narrow down the solutions to the underlying sources, and this property can be leveraged to design a supervision to train DNNs to model speech patterns via unsupervised learning and realize unsupervised separation. This meta-idea, we believe, would motivate the design of many algorithms in future research on neural source separation. Appendix This appendix is organized as follows. Appendix A describes the SMS-WSJ dataset, B visualizes the surface of the proposed mixture-consistency loss to show that minimizing the loss can promote separation of speakers, C provides the derivation that FCP can be used to approximate speaker images, D illustrates the frequency permutation problem, E discusses differences between UNSSOR and RAS, F presents miscellaneous system and DNN configurations, G experiments with alternative filters taps for iRAS and UNSSOR, and H describes the limitation of our current system. A SMS-WSJ dataset SMS-WSJ [44] is a popular corpus for evaluating two-speaker separation algorithms in reverberant conditions. The clean speech is sampled from the WSJ0 and WSJ1 datasets. The corpus contains 33,561 (∼87.4 h), 982 (∼2.5 h), and 1,332 (∼3.4 h) two-speaker mixtures respectively for training, validation, and testing. The simulated microphone array has six microphones arranged uniformly on a circle with a diameter of 20 cm. For each mixture, the speaker-to-array distance is drawn from the range [1.0, 2.0] m, and the reverberation time (T60) is sampled from [0.2, 0.5] s. A weak white noise is added to simulate microphone self-noises, and the energy level between the sum of the reverberant speech signals and the noise is sampled from the range [20,30] dB. The sampling rate is 8 kHz. (4) based on a six-channel noisy-reverberant two-speaker mixture sampled from the SMS-WSJ dataset (see Section A for the dataset details). Let C = 2 and suppose that the DNN estimates arê B Visualization of loss values Z(1) = µ × X 1 (1) + ν × X 1 (2) + ε 1 /2 Z(2) = (1 − µ) × X 1 (1) + (1 − ν) × X 1 (2) + ε 1 /2,(14) where µ and ν ∈ R are bounded in the range [0, 1]. Essentially, we use µ and ν to mimic the cases that the DNN produces (1) good separation (i.e., when µ ≈ 1 and ν ≈ 0, or µ ≈ 0 and ν ≈ 1); and (2) bad separation (i.e., when µ and ν are both away from 0 and 1, meaning that each estimate contains multiple speakers, and when µ ≈ 0 and ν ≈ 0, or µ ≈ 1 and ν ≈ 1, meaning that the two speakers are merged into one output and the other output does not contain any speakers). Fig. 2 enumerates µ and ν and plots the resulting separation results against the loss value of (4). We can see the loss values are smallest when µ ≈ 1 and ν ≈ 0 or when µ ≈ 0 and ν ≈ 1 (i.e., when the speakers are successfully separated), and clearly larger otherwise. This indicates that minimizing the proposed loss function can encourage separation. C Effectiveness of FCP at approximating speaker images Following the derivation in [17], let us define the mixture as Y p = X p (c) + V p (c), where V p (c) consists of the signals of all the sources but c. We can formulate (6) as argmin gp(c,f ) t 1 λ p (c, t, f ) |X p (c, t, f ) + V p (c, t, f ) − g p (c, f ) H Ẑ (c, t, f )| 2 ≈ argmin gp(c,f ) t 1 λ p (c, t, f ) |X p (c, t, f ) − g p (c, f ) H Ẑ (c, t, f )| 2 +|V p (c, t, f )| 2 = argmin gp(c,f ) t 1 λ p (c, t, f ) |X p (c, t, f ) − g p (c, f ) H Ẑ (c, t, f )| 2 ,(15) where the derivation from the first row to the second is based on the assumption that, as the DNN training continues,Ẑ(c) would become more and more accurate so that, after some epochs, it can become uncorrelated (or little correlated) with V p (c), meaning that the cross-term would be small: t 1 λ p (c, t, f ) X p (c, t, f ) − g p (c, f ) H Ẑ (c, t, f ) H V p (c, t, f ) ≈ 0.(16) From the third row of (15), the resultingĝ p (c, f ) H Ẑ (c, t, f ) would approximate X p (c, t, f ). D Illustration of frequency permutation problem We use three-channel input and loss for DNN training. Fig. 3(a) and (b) show an example separation result of the model trained with L MC in (9). Comparing the separated speech in Fig. 3 E Differences from RAS In many aspects, UNSSOR differs from RAS [24], which deals with monaural two-speaker separation given binaural (two-channel) training mixtures. RAS first performs DNN-based monaural separation on the left-ear mixture in the magnitude T-F domain, then linearly filters the DNN estimates at the left ear through time-domain Wiener filtering such that the filtered estimate can approximate the right-ear mixture, and the training loss is computed between the summation of the filtered DNN estimates and the right-ear mixture. Besides differences in, for example, the DNN architectures, linear filtering algorithms, how phase estimation (important for estimating relative RIRs) is handled, how frequency permutation problem is dealt with, and training data curation (difficult training examples need to be removed to train RAS), the key difference is that RAS fails to be trained in an unsupervised way [24]. It still needs labelled mixtures so that a supervised PIT-based model can be trained and then used as the starting point for their unsupervised algorithm. We think that the ineffectiveness of RAS in fully-unsupervised setup is likely because the loss is computed only on the right-ear mixture. Following our analysis in Section 3, in RAS there are N × 1 equations (because the loss is only computed on the right-ear time-domain signal, assumed N -sample) but N × C + (2 − 1) × 512 × C unknowns (where the 512 term is because the filter is assumed 512-tap in [24], and the (2 − 1) term is because there is only one filter for each speaker in the binaural setup, i.e., only one non-reference microphone). This is an ill-posed problem, not likely to be solved via the current RAS algorithm. F Miscellaneous system and DNN setup For STFT, the window size is 32 ms, the hop size is 8 ms, and the square-root Hann window is used as the analysis window. For 8 kHz sampling rate, a 256-point discrete Fourier transform is applied to extract 129-dimensional complex STFT spectra at each frame. Our DNN architecture is TF-GridNet [14]. Using the symbols defined in Table I In each epoch, we sample an l-second mixture segment from each training mixture for model training. If a mixture is shorter than l seconds, we pad zeros in the front rather than in the end, since padding in the end would result in a mixture that has abrupt stop of reverberation, which is not realistic and would be detrimental to FCP-based relative RIR estimation. In comparison, padding zeros in the front can avoid this problem. If a mixture is longer than l seconds, we randomly pick an l-second segment. In default, l is set to four seconds. We normalize the sample variance of each sampled mixture segment to 1.0, before feeding them to DNN for training. Adam (with the default setup in Pytorch v1.9) is used as the optimizer. The L 2 norm for gradient clipping is set to 1.0. The learning rate starts from 10 −3 and is halved if the validation loss is not improved in two epochs. We terminate training once the learning rate is reduced to 6.25 × 10 −5 . The batch size is set to four, with each segment being 4-second long. For each model, an Nvidia A100 40GB GPU is used for training, and the model converges in three to four days. We sweep γ in (11) based on the set of {0.02, 0.04, 0.06, 0.1, 0.3, 1.0}. The microphone weight α p in (4), (9), (10) and (12) is set to 1.0 in default for all microphones (i.e., no weighting), and, in the monaural input case (e.g., in Table 3 and 4), we sweep α 1 at the reference microphone 1 based on the set of {1/2, 1/3, 1/4, 1/5, 1/6, 1/8, 1/10} to alleviate overfitting to microphone 1. G Alternative filter length for iRAS and UNSSOR We also provide the results of iRAS that uses the same filter length (in seconds) as that in UNSSOR. Given 8 kHz sampling rate, 32 ms window size, 8 ms hop size, and K = I + 1 + J (defined below (6)) filter taps in FCP, we use M = ((K − 1) × 8 + 32)/1000 × 8000 filter taps for eachĥ p (c) in (12), and configureĥ p (c) to filter the past M − 8/1000 × 8000 − 1, the current, and the future 64 (= 8/1000 × 8000) samples. We filter the future 8 ms of samples, because, in the STFT case, the hop size is 8 ms. We report the results in Table 5-8, each respectively corresponding to the results in Table 1-4. We observe that the performance is not as good as that of UNSSOR. In Fig. 4, we make further comparisons of using different filter taps between UNSSOR and iRAS, following the experimental setup in the previous paragraph. Fig. 4(a) uses six-microphone input, Fig. 4(b) uses monaural input, and both of them use the six-microphone L MC+ISMS (or L iRAS ) loss. For UNSSOR, in Fig. 4(a) we set J = 0 and sweep K ∈ {5, 9, 13, 17, 20, 25} and in Fig. 4(b) we set J = 1 and sweep K ∈ {5, 9, 13, 17, 21, 25}. For iRAS, we configure the filter to always filter the future 64 samples, and in Fig. 4(a) we sweep the filter taps M ∈ {128, 256, 384, 512, 768, 1024, 1280, 1472} and in Fig. 4(b) we sweep M ∈ {128, 256, 384, 512, 768, 1024, 1280, 1536}. Notice that in Fig. 4, the filter taps M in iRAS and K in UNSSOR are vertically corresponding to each other. That is, some swepted filter taps M are computed based on the swepted K (e.g., for M = 1024 and K = 13, we have 1024 = ((13 − 1) × 8 + 32)/1000 × 8000) so that they can result in the same filter length in time. From the figures, we observe that the best filter length (in seconds) is different for UNSSOR and iRAS, and the best performance of UNSSOR is higher than that of iRAS. We point out that using a very long filter (i.e., a large M ) for time-domain Wiener filtering would prevent separation, since, in that case, the filter would have a large degrees of freedom to filterẑ(c) to fit y p very well and obtain a small L iRAS (see (13)), even thoughẑ(c) is not a good separation result. From Fig. 4(a), it can be observed that a very large M (e.g., 1472) does not yield good separation; and in addition, a very small M (e.g., 128) is also not good, as the linear filter could be just too short to fit the mixture y p well. Similar trend can also be observed in the results of UNSSOR. For example, in Fig. 4(a), setting K to 5 and 25 produces worse performance than to 20. We can see that the filter length is an important hyper-parameter to tune. We emphasize that, to compute the closed-form solution of time-domain Wiener filtering (see (13)), we need to invert a big M × M matrix for each mixture, while in FCP (see (6)), we only need to invert a much smaller K × K matrix for each of the F frequencies. Using FCP is clearly less computationally-expensive, given that the time complexity of matrix inversion is typically O(n 3 ). This also indicates that if the same amount of computation is required for linear filtering, FCP can use a much longer filter (in time) than time-domain Wiener filtering. H Limitations Our study shows the strong potential of UNSSOR for unsupervised speech separation. There are, however, several weaknesses we need to address in future research. First, we assume that sources are directional point sources, and diffuse sources are not considered. Second, we assume that sources are non-moving within each utterance so we can use time-invariant FCP filters. Third, we assume that the number of sources is known. Although these assumptions are also commonly made in many algorithms such as IVA, spatial clustering, RAS and iRAS, they need to be addressed to realize more practical and robust speech separation systems. Fig. 1 1illustrates the proposed system. The DNN takes in the mixture at all the P microphones or at the reference microphone 1 as input and produces an intermediate estimateẐ(c) for each speaker c. Fig. 2 2visualizes the values of L MC in Figure 2 : 2Loss values of LMC in (4) against hypothesized separation outputs generated by using various µ and ν (see Appendix B for detailed setup). Best viewed in color. (a) and (b) with the clean speech in (c) and (d), we can see that the separated speech suffers from the frequency permutation problem approximately in the range [1.6, 2.9] kHz. Fig. 3(e) and (f) show the separation result of the model trained with L MC+ISMS in(11), which effectively addresses the frequency permutation problem. Figure 3 : 3Example spectrograms of (a)-(b): FCP-estimated speaker image 1 and 2 with SDR scores of 8.7 and 7.7 dB (using LMC in (9) for training); (c)-(d): oracle speaker image 1 and 2; and (e)-(f): FCP-estimated speaker image 1 and 2 with SDR scores of 17.1 and 16.8 dB (using LMC+ISMS in (11) for training). The blue rectangles in (a) and (b) mark the region with frequency permutation. The mixture SDR scores of the two speakers are respectively 0.2 and −0.1 dB. Best viewed in color. of [14], we set its hyper-parameters to D = 48, B = 4, I = 4, J = 1, H = 192, L = 4 and E = 4 for 8 kHz sampling rate. Please do not confuse the symbols in TF-GridNet with the ones in this paper. Figure 4 : 4Averaged SDR (dB) results of UNSSOR and iRAS using various filter taps in cases of using (a) 6-channel input and loss; and (b) 1-channel input and 6-channel loss on SMS-WSJ. Vertically-corresponded M and K mean that the filter lengths in time are the same. Best viewed in color. Table 1 1Averaged results of 2-speaker separation on SMS-WSJ (6-channel input and loss).Val. set Test set Table 2 2Averaged results of 2-speaker separation on SMS-WSJ (3-channel input and loss).Val. set Test set Table 3 3Averaged results of 2-speaker separation on SMS-WSJ (1-channel input and 6-channel loss).Val. set Test set Table 4 4Averaged results of 2-speaker separation on SMS-WSJ (1-channel input and 3-channel loss).Val. set Test set Table 2 , 2UNSSOR obtains 15.4 dB SDR on the test set, which is close to the 16.8 dB result obtained by supervised PIT in 4a. Table 5 5Supplementary averaged results of 2-speaker separation on SMS-WSJ (6-channel input and loss).Val. set Test set Table 6 6Supplementary averaged results of 2-speaker separation on SMS-WSJ (3-channel input and loss).Val. set Test set Table 7 7Supplementary averaged results of 2-speaker separation on SMS-WSJ (1-channel input and 6-channel loss).Val. set Test set Table 8 8Supplementary averaged results of 2-speaker separation on SMS-WSJ (1-channel input and 3-channel loss). 2c iRAS w/ non-causal 1536-tap filters (64 future taps) --M in iRAS (filter future 64 samples)Val. set Test set T × F × C is because there is one unknown for each X1(c, t, f ), and F × (P − 1) × E × C is because gp(c, f ) is E-tap and we have one such filter for each of P − 1 microphone pairs for each frequency and speaker. Note that the relative RIR relating a signal to its delayed version is causal and the relative RIR relating a signal to its advanced version is non-causal[40]. We do not need many future taps, considering that the hop size is 8 ms in our system and the microphone array in SMS-WSJ is a compact array with a diameter of 20 cm. In air, sound would travel 340 × 0.008 = 2.72 meters in 8 ms if its speed is 340 meters per second. This distance is far larger than the array aperture size. Acknowledgments and Disclosure of FundingWe would like to thank Dr. Robin Scheibler at LINE Corporation for constructive discussions on IVA. This research is part of the Delta research computing project, which is supported by the National Science Foundation (award OCI 2005572) and the State of Illinois. Delta is a joint effort of the University of Illinois at Urbana-Champaign and its National Center for Supercomputing Applications. We also gratefully acknowledge the support of NVIDIA Corporation with the donation of the RTX 8000 GPUs used in this research. Some Experiments on the Recognition of Speech, with One and with Two Ears. C E Cherry, Journal of the Acoustical Society of America. 255C. E. Cherry, "Some Experiments on the Recognition of Speech, with One and with Two Ears," Journal of the Acoustical Society of America, vol. 25, no. 5, pp. 975-979, 1953. The Cocktail Party Problem. J H Mcdermott, Current Biology. 1922J. H. McDermott, "The Cocktail Party Problem," Current Biology, vol. 19, no. 22, pp. 1024-1027, 2009. Supervised Speech Separation Based on Deep Learning: An Overview. D Wang, J Chen, Speech, and Language Processing. 26D. Wang and J. Chen, "Supervised Speech Separation Based on Deep Learning: An Overview," IEEE/ACM Transactions on Audio, Speech, and Language Processing, vol. 26, no. 10, pp. 1702-1726, 2018. Deep Clustering: Discriminative Embeddings for Segmentation and Separation. J R Hershey, Z Chen, J Le Roux, S Watanabe, Proc. ICASSP. ICASSPJ. R. Hershey, Z. Chen, J. Le Roux, and S. Watanabe, "Deep Clustering: Discriminative Embeddings for Segmentation and Separation," in Proc. ICASSP, 2016, pp. 31-35. Multitalker Speech Separation with Utterance-Level Permutation Invariant Training of Deep Recurrent Neural Networks. M Kolbaek, D Yu, Z.-H Tan, J Jensen, IEEE/ACM Transactions on Audio, Speech, and Language Processing. 2510M. Kolbaek, D. Yu, Z.-H. Tan, and J. Jensen, "Multitalker Speech Separation with Utterance-Level Permutation Invariant Training of Deep Recurrent Neural Networks," IEEE/ACM Transactions on Audio, Speech, and Language Processing, vol. 25, no. 10, pp. 1901-1913, 2017. Divide and Conquer: A Deep CASA Approach to Talker-Independent Monaural Speaker Separation. Y Liu, D Wang, Speech, and Language Processing. 27Y. Liu and D. Wang, "Divide and Conquer: A Deep CASA Approach to Talker-Independent Monaural Speaker Separation," IEEE/ACM Transactions on Audio, Speech, and Language Processing, vol. 27, no. 12, pp. 2092-2102, 2019. Conv-TasNet: Surpassing Ideal Time-Frequency Magnitude Masking for Speech Separation. Y Luo, N Mesgarani, Speech, and Language Processing. 27Y. Luo and N. Mesgarani, "Conv-TasNet: Surpassing Ideal Time-Frequency Magnitude Masking for Speech Separation," IEEE/ACM Transactions on Audio, Speech, and Language Processing, vol. 27, no. 8, pp. 1256-1266, 2019. PHASEN: A Phase-and-Harmonics-Aware Speech Enhancement Network. D Yin, C Luo, Z Xiong, W Zeng, Proc. AAAI, 2020. AAAI, 2020D. Yin, C. Luo, Z. Xiong, and W. Zeng, "PHASEN: A Phase-and-Harmonics-Aware Speech Enhancement Network," in Proc. AAAI, 2020, pp. 9458-9465. The Cone of Silence: Speech Separation by Localization. T Jenrungrot, V Jayaram, S Seitz, I Kemelmacher-Shlizerman, Proc. NeurIPS. NeurIPST. Jenrungrot, V. Jayaram, S. Seitz, and I. Kemelmacher-Shlizerman, "The Cone of Silence: Speech Separation by Localization," in Proc. NeurIPS, 2020. Joint Time-Frequency and Time Domain Learning for Speech Enhancement. C Tang, C Luo, Z Zhao, W Xie, W Zeng, Proc. IJCAI. IJCAIC. Tang, C. Luo, Z. Zhao, W. Xie, and W. Zeng, "Joint Time-Frequency and Time Domain Learning for Speech Enhancement," in Proc. IJCAI, 2020, pp. 3816-3822. Voice Separation with An Unknown Number of Multiple Speakers. E Nachmani, Y Adi, L Wolf, Proc. ICML, 2020. ICML, 2020E. Nachmani, Y. Adi, and L. Wolf, "Voice Separation with An Unknown Number of Multiple Speakers," in Proc. ICML, 2020, pp. 7121-7132. Wavesplit: End-to-End Speech Separation by Speaker Clustering. N Zeghidour, D Grangier, IEEE/ACM Transactions on Audio, Speech, and Language Processing. 29N. Zeghidour and D. Grangier, "Wavesplit: End-to-End Speech Separation by Speaker Clustering," IEEE/ACM Transactions on Audio, Speech, and Language Processing, vol. 29, pp. 2840-2849, 2021. SFSRNet: Super-Resolution for Single-Channel Audio Source Separation. J Rixen, M Renz, Proc. AAAI. AAAIJ. Rixen and M. Renz, "SFSRNet: Super-Resolution for Single-Channel Audio Source Separation," in Proc. AAAI, 2022. Z.-Q Wang, S Cornell, S Choi, Y Lee, B.-Y. Kim, S Watanabe, arXiv:2211.12433TF-GridNet: Integrating Full-and Sub-Band Modeling for Speech Separation. arXiv preprintZ.-Q. Wang, S. Cornell, S. Choi, Y. Lee, B.-Y. Kim, and S. Watanabe, "TF-GridNet: Integrating Full-and Sub-Band Modeling for Speech Separation," arXiv preprint arXiv:2211.12433, 2022. On Cross-Corpus Generalization of Deep Learning Based Speech Enhancement. A Pandey, D Wang, IEEE/ACM Transactions on Audio, Speech, and Language Processing. 28A. Pandey and D. Wang, "On Cross-Corpus Generalization of Deep Learning Based Speech Enhancement," IEEE/ACM Transactions on Audio, Speech, and Language Processing, vol. 28, pp. 2489-2499, 2020. Closing The Gap Between Time-Domain Multi-Channel Speech Enhancement on Real and Simulation Conditions. W Zhang, J Shi, C Li, S Watanabe, Y Qian, Proc. WASPAA, 2021. WASPAA, 2021W. Zhang, J. Shi, C. Li, S. Watanabe, and Y. Qian, "Closing The Gap Between Time-Domain Multi-Channel Speech Enhancement on Real and Simulation Conditions," in Proc. WASPAA, 2021, pp. 146-150. Convolutive Prediction for Monaural Speech Dereverberation and Noisy-Reverberant Speaker Separation. Z.-Q Wang, G Wichern, J Le Roux, IEEE/ACM Transactions on Audio, Speech, and Language Processing. 29Z.-Q. Wang, G. Wichern, and J. Le Roux, "Convolutive Prediction for Monaural Speech Dereverberation and Noisy-Reverberant Speaker Separation," IEEE/ACM Transactions on Audio, Speech, and Language Processing, vol. 29, pp. 3476-3490, 2021. Unsupervised Sound Separation using Mixture Invariant Training. S Wisdom, E Tzinis, H Erdogan, R J Weiss, K Wilson, J R Hershey, Proc. NeurIPS. NeurIPSS. Wisdom, E. Tzinis, H. Erdogan, R. J. Weiss, K. Wilson, and J. R. Hershey, "Unsupervised Sound Separation using Mixture Invariant Training," in Proc. NeurIPS, 2020. Into the Wild with AudioScope: Unsupervised Audio-Visual Separation of On-Screen Sounds. E Tzinis, S Wisdom, A Jansen, S Hershey, T Remez, D P W Ellis, J R Hershey, Proc. ICLR. ICLRE. Tzinis, S. Wisdom, A. Jansen, S. Hershey, T. Remez, D. P. W. Ellis, and J. R. Hershey, "Into the Wild with AudioScope: Unsupervised Audio-Visual Separation of On-Screen Sounds," in Proc. ICLR, 2021. RemixIT: Continual Self-Training of Speech Enhancement Models via Bootstrapped Remixing. E Tzinis, Y Adi, V K Ithapu, B Xu, P Smaragdis, A Kumar, IEEE Journal of Selected Topics in Signal Processing. 166E. Tzinis, Y. Adi, V. K. Ithapu, B. Xu, P. Smaragdis, and A. Kumar, "RemixIT: Continual Self-Training of Speech Enhancement Models via Bootstrapped Remixing," IEEE Journal of Selected Topics in Signal Processing, vol. 16, no. 6, pp. 1329-1341, 2022. AudioScopeV2: Audio-Visual Attention Architectures for Calibrated Open-Domain On-Screen Sound Separation. E Tzinis, S Wisdom, T Remez, J R Hershey, Proc. ECCV, 2022. ECCV, 2022E. Tzinis, S. Wisdom, T. Remez, and J. R. Hershey, "AudioScopeV2: Audio-Visual Attention Architectures for Calibrated Open-Domain On-Screen Sound Separation," in Proc. ECCV, 2022, pp. 368-385. Self-Remixing: Unsupervised Speech Separation via Separation and Remixing. K Saijo, T Ogawa, Proc. ICASSP. ICASSPK. Saijo and T. Ogawa, "Self-Remixing: Unsupervised Speech Separation via Separation and Remixing," in Proc. ICASSP, 2023. Adapting Speech Separation To Real-World Meetings Using Mixture Invariant Training. A Sivaraman, S Wisdom, H Erdogan, J R Hershey, Proc. ICASSP, 2022. ICASSP, 2022A. Sivaraman, S. Wisdom, H. Erdogan, and J. R. Hershey, "Adapting Speech Separation To Real-World Meetings Using Mixture Invariant Training," in Proc. ICASSP, 2022, pp. 686-690. Reverberation as Supervision for Speech Separation. R Aralikatti, C Boeddeker, G Wichern, A S Subramanian, J Le Roux, Proc. ICASSP. ICASSPR. Aralikatti, C. Boeddeker, G. Wichern, A. S. Subramanian, and J. Le Roux, "Reverberation as Supervision for Speech Separation," in Proc. ICASSP, 2023. A New Learning Algorithm for Blind Signal Separation. S.-I Amari, A Cichocki, H Yang, Proc. NeurIPS. NeurIPS8S.-I. Amari, A. Cichocki, and H. Yang, "A New Learning Algorithm for Blind Signal Separation," in Proc. NeurIPS, vol. 8, 1996, pp. 757-763. Independent Component Analysis: Algorithms and Applications. A Hyvärinen, E Oja, Neural Networks. 454-5A. Hyvärinen and E. Oja, "Independent Component Analysis: Algorithms and Applications," in Neural Networks, vol. 45, no. 4-5, 2000, pp. 411-430. Blind Source Separation and Independent Component Analysis. C G Puntonet, E W Lang, Neurocomputing. 61C. G. Puntonet and E. W. Lang, "Blind Source Separation and Independent Component Analysis," Neuro- computing, vol. 6, no. 1, pp. 1-57, 2006. Diversity in Independent Component and Vector Analyses. T Adali, M Anderson, G.-S Fu, IEEE Signal Processing Magazine. 313T. Adali, M. Anderson, and G.-s. Fu, "Diversity in Independent Component and Vector Analyses," IEEE Signal Processing Magazine, vol. 31, no. 3, pp. 18-33, 2014. A Review of Blind Source Separation Methods: Two Converging Routes to ILRMA Originating from ICA and NMF. H Sawada, N Ono, H Kameoka, D Kitamura, H Saruwatari, APSIPA Transactions on Signal and Information Processing. 8H. Sawada, N. Ono, H. Kameoka, D. Kitamura, and H. Saruwatari, "A Review of Blind Source Separation Methods: Two Converging Routes to ILRMA Originating from ICA and NMF," APSIPA Transactions on Signal and Information Processing, vol. 8, pp. 1-14, 2019. Independent Vector Analysis: An Extension of ICA to Multivariate Components. T Kim, T Eltoft, T.-W Lee, International Conference on Independent Component Analysis and Signal Separation. T. Kim, T. Eltoft, and T.-W. Lee, "Independent Vector Analysis: An Extension of ICA to Multivariate Components," in International Conference on Independent Component Analysis and Signal Separation, 2006, pp. 165-172. Solution of Permutation Problem in Frequency Domain ICA, using Multivariate Probability Density Functions. A Hiroe, International Conference on Independent Component Analysis and Signal Separation. A. Hiroe, "Solution of Permutation Problem in Frequency Domain ICA, using Multivariate Probability Den- sity Functions," in International Conference on Independent Component Analysis and Signal Separation, 2006, pp. 601-608. A Comparison and Combination of Unsupervised Blind Source Separation Techniques. C Boeddeker, F Rautenberg, R Haeb-Umbach, ITG Conference on Speech Communication. C. Boeddeker, F. Rautenberg, and R. Haeb-Umbach, "A Comparison and Combination of Unsupervised Blind Source Separation Techniques," in ITG Conference on Speech Communication, 2021, pp. 129-133. The DUET Blind Source Separation Algorithm. S Rickard, Blind Speech Separation. S. Rickard, "The DUET Blind Source Separation Algorithm," in Blind Speech Separation, 2007, pp. 217-237. Blind Speech Separation Employing Directional Statistics in An Expectation Maximization Framework. D H T Vu, R Haeb-Umbach, Proc. ICASSP. ICASSPD. H. T. Vu and R. Haeb-Umbach, "Blind Speech Separation Employing Directional Statistics in An Expectation Maximization Framework," in Proc. ICASSP, 2010, pp. 241-244. Underdetermined Convolutive Blind Source Separation via Frequency Bin-Wise Clustering and Permutation Alignment. H Sawada, S Araki, S Makino, IEEE Transactions Audio, Speech, and Language Processing. 193H. Sawada, S. Araki, and S. Makino, "Underdetermined Convolutive Blind Source Separation via Frequency Bin-Wise Clustering and Permutation Alignment," IEEE Transactions Audio, Speech, and Language Processing, vol. 19, no. 3, pp. 516-527, 2011. Complex Angular Central Gaussian Mixture Model for Directional Statistics in Mask-Based Microphone Array Signal Processing. N Ito, S Araki, T Nakatani, Proc. EUSIPCO. EUSIPCON. Ito, S. Araki, and T. Nakatani, "Complex Angular Central Gaussian Mixture Model for Directional Statistics in Mask-Based Microphone Array Signal Processing," in Proc. EUSIPCO, 2016, pp. 1153-1157. Unsupervised Deep Clustering for Source Separation: Direct Learning from Mixtures Using Spatial Information. E Tzinis, S Venkataramani, P Smaragdis, Proc. ICASSP. ICASSPE. Tzinis, S. Venkataramani, and P. Smaragdis, "Unsupervised Deep Clustering for Source Separation: Direct Learning from Mixtures Using Spatial Information," in Proc. ICASSP, 2019, pp. 81-85. Understanding Blind Deconvolution Algorithms. A Levin, Y Weiss, F Durand, W T Freeman, IEEE Transactions on Pattern Analysis and Machine Intelligence. 3312A. Levin, Y. Weiss, F. Durand, and W. T. Freeman, "Understanding Blind Deconvolution Algorithms," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 33, no. 12, pp. 2354-2367, 2011. Multi-Microphone Complex Spectral Mapping for Utterance-Wise and Continuous Speech Separation. Z.-Q Wang, P Wang, D Wang, IEEE/ACM Transactions on Audio, Speech, and Language Processing. 29Z.-Q. Wang, P. Wang, and D. Wang, "Multi-Microphone Complex Spectral Mapping for Utterance-Wise and Continuous Speech Separation," IEEE/ACM Transactions on Audio, Speech, and Language Processing, vol. 29, pp. 2001-2014, 2021. Discrete-Time Signal Processing. A V Oppenheim, Pearson Education IndiaA. V. Oppenheim, Discrete-Time Signal Processing. Pearson Education India, 1999. A Two-Stage Frequency-Domain Blind Source Separation Method for Underdetermined Convolutive Mixtures. H Sawada, S Araki, S Makino, Proc. WASPAA. WASPAAH. Sawada, S. Araki, and S. Makino, "A Two-Stage Frequency-Domain Blind Source Separation Method for Underdetermined Convolutive Mixtures," in Proc. WASPAA, 2007, pp. 139-142. A Robust and Precise Method for Solving The Permutation Problem of Frequency-Domain Blind Source Separation. H Sawada, R Mukai, S Araki, S Makino, IEEE Transactions on Speech and Audio Processing. 125H. Sawada, R. Mukai, S. Araki, and S. Makino, "A Robust and Precise Method for Solving The Permutation Problem of Frequency-Domain Blind Source Separation," IEEE Transactions on Speech and Audio Processing, vol. 12, no. 5, pp. 530-538, 2004. A Consolidated Perspective on Multi-Microphone Speech Enhancement and Source Separation. S Gannot, E Vincent, S Markovich-Golan, A Ozerov, IEEE/ACM Transactions on Audio, Speech, and Language Processing. 25S. Gannot, E. Vincent, S. Markovich-Golan, and A. Ozerov, "A Consolidated Perspective on Multi- Microphone Speech Enhancement and Source Separation," IEEE/ACM Transactions on Audio, Speech, and Language Processing, vol. 25, pp. 692-730, 2017. SMS-WSJ: Database, Performance Measures, and Baseline Recipe for Multi-Channel Source Separation and Recognition. L Drude, J Heitkaemper, C Boeddeker, R Haeb-Umbach, arXiv:1910.13934in arXiv preprintL. Drude, J. Heitkaemper, C. Boeddeker, and R. Haeb-Umbach, "SMS-WSJ: Database, Performance Measures, and Baseline Recipe for Multi-Channel Source Separation and Recognition," in arXiv preprint arXiv:1910.13934, 2019. . C Boeddeker, C. Boeddeker, "https://github.com/fgnt/pb_bss/blob/master/examples/mixture_model_example.ipynb," 2019. . R Scheibler, K Saijo, 2022R. Scheibler and K. Saijo, "https://github.com/fakufaku/torchiva," 2022. Performance Measurement in Blind Audio Source Separation. E Vincent, R Gribonval, C Févotte, IEEE Transactions Audio, Speech, and Language Processing. 14E. Vincent, R. Gribonval, and C. Févotte, "Performance Measurement in Blind Audio Source Separation," IEEE Transactions Audio, Speech, and Language Processing, vol. 14, no. 4, pp. 1462-1469, 2006. SDR -Half-Baked or Well Done?. J Le Roux, S Wisdom, H Erdogan, J R Hershey, Proc. ICASSP. ICASSPJ. Le Roux, S. Wisdom, H. Erdogan, and J. R. Hershey, "SDR -Half-Baked or Well Done?" in Proc. ICASSP, 2019, pp. 626-630. Perceptual Evaluation of Speech Quality (PESQ)-A New Method for Speech Quality Assessment of Telephone Networks and Codecs. A Rix, J Beerends, M Hollier, A Hekstra, Proc. ICASSP. ICASSP2A. Rix, J. Beerends, M. Hollier, and A. Hekstra, "Perceptual Evaluation of Speech Quality (PESQ)-A New Method for Speech Quality Assessment of Telephone Networks and Codecs," in Proc. ICASSP, vol. 2, 2001, pp. 749-752. An Algorithm for Intelligibility Prediction of Time-Frequency Weighted Noisy Speech. C H Taal, R C Hendriks, R Heusdens, J Jensen, IEEE Transactions Audio, Speech, and Language Processing. 197C. H. Taal, R. C. Hendriks, R. Heusdens, and J. Jensen, "An Algorithm for Intelligibility Prediction of Time-Frequency Weighted Noisy Speech," IEEE Transactions Audio, Speech, and Language Processing, vol. 19, no. 7, pp. 2125-2136, 2011.
{'fraction_non_alphanumeric': 0.06591574111946638, 'fraction_numerical': 0.02510776201113803, 'mean_word_length': 4.268711147948611, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 2, 'https://': 2, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 6, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': "In reverberant conditions with multiple concurrent speakers, each microphone acquires a mixture signal of multiple speakers at a different location. In overdetermined conditions where the microphones out-number speakers, we can narrow down the solutions to speaker images and realize unsupervised speech separation by leveraging each mixture signal as a constraint (i.e., the estimated speaker images at a microphone should add up to the mixture). Equipped with this insight, we propose UNSSOR, an algorithm for unsupervised neural speech separation by leveraging over-determined training mixtures. At each training step, we feed an input mixture to a deep neural network (DNN) to produce an intermediate estimate for each speaker, linearly filter the estimates, and optimize a loss so that, at each microphone, the filtered estimates of all the speakers can add up to the mixture to satisfy the above constraint. We show that this loss can promote unsupervised separation of speakers. The linear filters are computed in each sub-band based on the mixture and DNN estimates through the forward convolutive prediction (FCP) algorithm. To address the frequency permutation problem incurred by using sub-band FCP, a loss term based on minimizing intra-source magnitude scattering is proposed. Although UNSSOR requires over-determined training mixtures, we can train DNNs to achieve under-determined separation (e.g., unsupervised monaural speech separation). Evaluation results on two-speaker separation in reverberant conditions show the effectiveness and potential of UNSSOR. Preprint. Under review. arXiv:2305.20054v1 [cs.SD] 31 May 2023objects (e.g., clean speaker signals) are, as there are infinite solutions where in each solution the estimated sources can sum up to the mixture. Supposing that the separation model does not separate well and outputs a clean speaker signal plus some competing speech, noise or reverberation, would this output be viewed as a desired sound object? This is clear to humans, clear to supervised learning based models (by comparing the outputs with training labels), but not really clear to an unsupervised model. On the other hand, many studies[3][4][5][6][7][8][9][10][11][12][13][14]have observed that deep learning based supervised learning can achieve remarkable separation. In other words, with proper supervision, modern DNNs are capable of separating mixed speakers, but, in an unsupervised setup, there lacks an accurate supervision to unleash this capability. The key to successful unsupervised neural separation, we believe, is designing a clever supervision that can inform the model what desired sound objects are, and penalize the model if its outputs are not good and reward otherwise.Our insight is that, in multi-microphone over-determined conditions where the microphones outnumber speakers, the ill-posed problem can be turned into a well-posed one, where a unique solution to the speakers exists (up to speaker permutation). This well-posed property (that a unique solution exists) can be leveraged as a supervison (or regularizer) to design loss functions that could inform the unsupervised separation model what desired sound objects are and promote separation of speakers.Equipped with this insight, we perform unsupervised neural speech separation by leveraging multimicrophone over-determined training mixtures. Our DNNs can be trained directly on over-determined mixtures to realize over-and under-determined separation. The proposed algorithm, named UNSSOR, obtains strong separation performance on two-speaker separation. Our contributions include:• We enforce a linear-filter constraint between each speaker's reverberant images at each microphone pair, turning the ill-posed problem into a well-posed one that can promote separation of speakers. • We formulate unsupervised neural speech separation as a blind deconvolution problem, where both the speaker images and linear filters need to be estimated. We design loss functions motivated by the blind deconvolution problem, and propose a DNN approach to optimize the loss functions, where the speaker images are estimated via DNNs and the linear filters are estimated via a sub-band linear prediction algorithm named FCP [17] based on the mixture and DNN estimates. • We propose a loss term, which minimizes a measure named intra-source magnitude scattering, to address the frequency permutation problem incurred when using sub-band FCP. • Based on over-determined training mixtures, UNSSOR can be trained to perform under-determined separation (e.g., monaural unsupervised speech separation).", 'arxivid': '2305.20054', 'author': ['Zhong-Qiu Wang [email protected] \nLanguage Technologies Institute\nCarnegie Mellon University\nPittsburghUSA\n', 'Shinji Watanabe \nLanguage Technologies Institute\nCarnegie Mellon University\nPittsburghUSA\n'], 'authoraffiliation': ['Language Technologies Institute\nCarnegie Mellon University\nPittsburghUSA', 'Language Technologies Institute\nCarnegie Mellon University\nPittsburghUSA'], 'corpusid': 258987929, 'doi': '10.48550/arxiv.2305.20054', 'github_urls': ['https://github.com/fgnt/pb_bss/blob/master/examples/mixture_model_example.ipynb,"', 'https://github.com/fakufaku/torchiva,"'], 'n_tokens_mistral': 19646, 'n_tokens_neox': 17119, 'n_words': 10225, 'pdfsha': '68020c732975f09569060b7fbc484229d9b96d77', 'pdfurls': ['https://export.arxiv.org/pdf/2305.20054v1.pdf'], 'title': ['UNSSOR: Unsupervised Neural Speech Separation by Leveraging Over-determined Training Mixtures', 'UNSSOR: Unsupervised Neural Speech Separation by Leveraging Over-determined Training Mixtures'], 'venue': []}
arxiv
Computational study of defect complexes in β-LiGaO 2 and their relation to the donor-acceptor-pair recombination Klichchupong Dabsamut Adisak Boonchun Walter R L Lambrecht Department of Physics Faculty of Science Department of Physics Kasetsart University 10900BangkokThailand Case Western Reserve University 10900 Euclid Avenue44106-7079ClevelandOhioUSA Computational study of defect complexes in β-LiGaO 2 and their relation to the donor-acceptor-pair recombination Hybrid functional calculations are presented for defects in LiGaO2 with the fraction of non-local exchange adjusted to reproduce the recently reported exciton gap of 6.0 eV. We study how the defect transition levels of the main native defects change with respect to the band edges compared to earlier calculations which assumed a smaller band gap near 5.1 eV. In addition, we consider defect complexes formed by combining the main native donor GaLi with the main acceptors, VLi and LiGa antisites as function of their relative position. These results are used to tentatively identify the photoluminescence bands previous assigned to donor-acceptor-pair recombination. I. INTRODUCTION Lithium gallate (β-LiGaO 2 ) has recently been proposed as a potential ultra-wide-band-gap (UWBG) semiconductor. Its crystal structure, reported by Marezio [1], is a cation-ordered wurtzite-derived structure with space group P na2 1 . Large heteroepitaxial bulk single crystals, grown by the Czochralski method have been reported [2,3] and most of this effort was motivated by the need for lattice matched substrates for heteroepitaxial growth of GaN [2,4,5]. Its thermal properties are of interest in this context and were reported by by Weise and Neumann [6] and Neumann et al. [7]. It can also be used as substrate for heteroepitaxial growth of ZnO [8]. In fact, ZnO can be considered as the wurtzite parent compound of LiGaO 2 , which conceptually is obtained by replacing the group-II Zn ion by alternating group-I Li and group-III Ga ions, thereby locally maintaining the octet-rule if each O is surrounded by two Li and two Ga ions. It can thus be alloyed with ZnO [9,10] and other ternary I-III-O 2 oxides like CuGaO 2 [11], which opens the way toward band gap tuning. Until recently LiGaO 2 was primarily considered as an optical material or substrate material for growth of other semiconductors. Its elastic, phonon and piezoelectric properties were calculated using density functional theory (DFT) by Boonchun and Lambrecht [12]. It electronic structure was calculated at the DFT level using the modified Becke-Johnson (mBJ) exchange-correlation [13,14] functional by Johnson et al. [15]. Its optical gap was obtained from absorption measurements [3,16] and a combination of X-ray absorption and emission spectroscopies [15] and found generally to be about 5.3-5.6 eV. Recently, quasiparticle self-consistent (QS)GW (where G is the one-electron Green's function and W the screened Coulomb interaction) calculations were per- * [email protected] formed by Radha et al. [17] for various crystal structures of LiGaO 2 following an earlier QSGW calculation by Boonchun et al. [18]. The possibility of doping the material and thereby making a functional semiconductor was first suggested by Boonchun and Lambrecht [18] but in this study only small cells with unrealistically high dopant concentrations were considered. Its native defects were recently studied using first-principles calculations [19]. The opportunities for ntype and p-type doping were studied in Ref. [20]. It was predicted that Si and Ge would be shallow donors, while Sn would be a deep donor. However, p-type doping by N on O or by Zn on Ga site were found to be less promising because of deep levels and site competition with Zn on Li donors. Doping by various diatomic molecules was also investigated but not found to lead to p-type doping [21]. Electron paramagnetic resonance of Li and Ga vacancies were reported by Lenyk et al. [22] and analyzed computationally by Skachkov et al. [23]. Recently, the infrared (phonon related) as well as visible ultraviolet (interband transition related) optical properties were studied by reflectivity, transmission and spectroscopic reflectivity by Tumėnas et al. [24] and indicated sharp excitons near 6.0 eV. Luminescence properties were studied by Trinkler et al. [25,26] and the photoluminescence excitation (PLE) spectroscopy confirmed the presence of sharp free excitons near 6.0 eV. The anisotropic splitting of these excitons, reported in [26] reflects the valence band splitting, characteristic of the orthorhombic symmetry of the crystal and is in good agreement with the recent computational study by Radha et al. [17]. This much larger optical exciton gap than previously accepted led one of us to re-examine the convergence of the QSGW calculations and to also study the excitons by means of the Bethe-Salpeter-Equation method and not only found resuls in close agreement for the exciton gap near 6.0 eV but also found a large exciton binding energy of about 0.7 eV. [27] Furthermore, a series of excited state excitons were revealed in this work. In view of this larger gap we here present new calculations arXiv:2302.09206v1 [cond-mat.mtrl-sci] 18 Feb 2023 of some of the primary defects found earlier but with the hybrid functional fraction of exchange adjusted to the larger gap. Furthermore the work of Trinkler et al. [25] assigned the primary photoluminescence peaks to donor acceptor pair recombination. This inspired us to consider defect complexes combining donors and acceptors. Trinkler et al. [25] assigned a peak in luminescence centered at about 4.43 eV and from which the PLE gave sharp peaks near 6.0 eV to a donor acceptor pair (DAP) recombination based on its blue shift under higher power excitation, and temporal behavior. Our prior study [19] of defects identified the Ga Li as the primary native defect donor and V Li and Li Ga as the most likely acceptors with the second one having slightly higher acceptor binding energy. However, those calculations were done with a hybrid functional scheme in which the non-local exchange fraction α was set to 0.25 which gave a band gap of 5.1 eV. In view of the now established significantly larger gap, we repeated these calculations for a larger fraction of exchange (α = 0.38) tuning the gap to 6.09 eV and also considered close pairs of donors and acceptors in a complex. II. COMPUTATIONAL METHOD Our calculations were performed using the VASP package [28,29] using the projector-augmented wave (PAW) method [30] and the Heid-Scuseria-Ernzerhof (HSE) hybrid exchange correlation functional [31,32] with screening length of 10Å and fraction of non-local exchange chosen α = 0.38 chosen to adjust the band gap. Other aspects of our computational method are the same as in Refs. [19][20][21]. While the 6.09 eV corresponds strictly to the exciton gap and not to the quasiparticle gap one may view these calculations as a guide for how the defect levels behave with the gap and assume that excitonic effects are pertaining to the defect levels as well. In other words, donor or acceptor bound excitons are assumed to have similar exciton binding energy as the free exciton. III. RESULTS AND DISCUSSION The results for the main defects of interest here are shown in Fig. 1 and the transition levels are summarized in Table I. It is important to note that compared to our previous study of native defects [19] both donor and acceptor levels moved deeper into the gap when using a larger fraction of exchange and hence larger gap. In that study, the gap was 5.1 eV and the V Li (0/−) level was located at 1.03 eV above the VBM, the Li Ga (0/−) acceptor at 1.55 eV above the VBM and the Ga Li (2 + /0) deep donor level at 0.74 eV below the CBM. However, the difference between donor and acceptor levels were close to what we find here. As in our previous work we find the Ga Li to be a deep double donor with a 2 + /0 transition. This implies a socalled negative U center in which the q = +1 state has higher energy than the neutral or 2+ state for any Fermi level position. However, in photoluminescence, we are not dealing with equilibrium but with a situation where electrons are excited to the conduction band and then relax to become trapped on the donors, which initially were in the q = +2 state. Thus some donors will occur in the q = +1 state and may then recombine with holes before they trap a second electron and become neutral. It is therefore also important to locate the 2 + /+ level. We find the 2 + /0 to lie at 1.01 eV below the CBM and the 2 + /1+ at 5.26 eV (above VBM) or 0.83 eV below the CBM and the 1 + /0 at VBM+4.89 or CBM−1.2 eV. Meanwhile the V Li vacancy 0/− level lies at about 1.63 eV above the VBM. The energy difference between these remote and isolated donor and acceptors is thus 3.45 eV or 3.63 eV depending on whether we use the 2 + /0 or 2 + /+ level of the Ga Li . On the other hand, in a DAP transition the photon energy is given by ω = E g − E D − E A + e 2 /εR,(1) where R is the distance between the donor and acceptor and ε the dielectric constant. This is because in the final state after recombination, the donor and acceptor find themselves in ionized states and experience a Coulomb attraction. This energy gain is transferred to the photon. Usually, for shallow donors and acceptors one uses the static dielectric constant which includes lattice screening. Here, however the donor and acceptor are both quite deep and hence the binding energies may involve only electronic screening. In other words, the electron and hole involved in the recombination move too fast around their donor and acceptor for the lattice vibrations to participate in the screening of their Coulomb interaction. A full calculation of the DAP spectrum is rather complex as it would involve the random probability distribtion of the donors and acceptors relative to each other and the overlap of their wave functions for each separation, which would determine their recombination probability. Instead we start from the experimental value of the peak position and estimate which distance this would correspond to. To explain the DAP peak at 4.4 eV we need to assume the e 2 /εR ≈ 0.8 eV if we assume the donors make a transition from + to 2+. Using a isotropically averaged dielectric high-frequency constant [24,27] of 3.0, this suggests a distance R ≈ 6.0Å between the donor and acceptor corresponding to the peak value of the DAP band. The spectrum of this DAP is asymmetrically stretched with a tail toward lower recombination energies which would correspond to more distant pairs. This DAP distance of 6.0Å is slightly larger than third nearest neighbor V Li − Ga Li pairs. The closest distance between them is 3.10Å. Trinkler et al. [25] proposed a randomly placed DAP for the 4.43 eV band and assign it to tunnel recombination. Keeping in mind that with such rather deep acceptor and donors the overlap of their wave functions which would allow tunneling would be rather small for remote DAP. A distance of ∼ 6Å is thus not unreasonable for the peak position. We can see that the Li Ga acceptor level is 0.44 eV deeper than the V Li . Hence with similar assumptions, this DAP would then occur at about 3.9 eV if we assume a similar distance between donor an acceptor corresponding to the peak of the DAP, as for the V Li . Hence these acceptors could possibly account for the second observed luminescence band peaked at 3.76 eV. Trinkler et al. [25] also suggested that the 4.43 eV could in part also result from a fast decay related to a free electron to acceptor (eA) process. With our calculated acceptor levels of V Li at 1.63 and Li Ga at 2.06 eV above the VBM and the gap of 6.09 eV, this would correspond to 4.46 and 4.03 eV, which are indeed also close to the observed PL band peak positions. We here also explicitly consider donor acceptor complexes at various relative positions from each other. The bottom part of Fig. 1 shows that there is a net binding energy between the donor and acceptor in the DAP complex over a considerable range of Fermi energies which is defined by Equation 2 in [33], or E B = −E f (DAP ) + E f (D) + E f (A) where the DAP complex and the donor D and acceptor A formation energies are all taken at the same Fermi-level position. Considering the defect levels of the nearest neighbor complex in Fig. 1, we first note that its transition levels are closely related to the individual donor and acceptor ones. When the acceptor is in the neutral state and the donor in the 2+ state, for Fermi level positions close to the VBM, the complex is obviously in a 2+ state. Now, first the acceptor goes to the q = −1 state and the complex than goes to a +1 state. Since the V Li only occurs in 0 and −1 states, the next transition happens for the donor part of the complex Ga Li transitioning from 2+ to 0. At that point the complex will thus go from +1 to −1. However, we see from Fig.1 that the 2 + /+ and +/− transitions of the complex are pushed closer to their respective band edges. This must result from the defect levels in the complex interacting with each other. Forming bonding and antibonding states between the donor and acceptor wave functions as in a molecule will push these states farther apart. This means that in a complex the E D and E A binding energies of the neutral DAP before its recombination are reduced, but this reduction will decrease the farther the D and A are apart. The net result is that E D −E A is increased. For the closest distance of D and A the donor and acceptor energies are about 0.5 eV below the CBM and 1 eV above the VBM giving a DAP energy of 4.5 eV in the initial state but the Coulomb energy is then about 1.3 eV so the DAP photon energy would be about 5.8 eV. This is above the maximum of the DAP band, which is near 5 eV, indicating that such close DAPs are unlikely. On the other hand, transitions with energies larger that the peak energy up to 5 eV could be partially accounted for by slightly more remote DAPs. In fact for the DAP at 9.75Å apart (shown in Fig. 2) the DAP photon energy using the D and A transition levels from Fig. 2 and a Coulomb energy at that distance we obtain 4.7 eV, which is still within this DAP band. In Fig. 2 we show the transition levels of various V Li − Ga Li complexes with different distances between the donor and acceptor. We can see that for the larger distances, we can now separately see the +/0 and 0/transitions of the donor part. However, it is also clear that the larger the distance the larger the donor binding energies become. On the acceptor side, the same trend is not as clear although still present. This indicates that the interaction between donor and acceptor levels in the complex does not only depend on their relative distance but also on their relative orientation. In other words, anisotropy of the acceptor-like wave function plays a role. IV. CONCLUSIONS In summary, for the DAP recombination band peaked at 4.43 eV observed in [25], we can tentatively assign the donor to be the native defect Ga Li donor and the V Li as the acceptor. The same donor but the Li Ga acceptor could then explain the slightly lower energy 3.76 eV DAP PL bands. As already suggested by Trinkler et al. [25] these PL bands would indeed also be consistent with having a contribution from free-electron to acceptor recombinations. Furthermore, we find that for relatively close DAP distances, there is a considerable repulsion of the neutral donor and acceptor levels moving them closer to the band edges in the initial excited state before the recombination. FIG. 1 . 1Top panel : Donor and acceptor transition levels and their nearest neighbor complexes. Bottom panel : Complex defects binding energy. Left: VLi and right LiGa acceptor combined with the same donor GaLi FIG. 2 . 2Transition levels of VLi − GaLi complexes for various distances between the donor and acceptor parts of the complex. TABLE I . IDefect transition levels ε(q, q ) in eV with respect to the VBM.Defect q, q ε(q, q ) GaLi (2+/1+) 5.26 (1+/0) 4.89 (2+/0) 5.08 VLi (0/1-) 1.63 LiGa (0/1-) 2.06 (1-/2-) 2.87 GaLi+VLi (2+/1+) 0.98 (1+/1-) 5.63 GaLi+LiGa (2+/1+) 1.32 (1+/0) 1.84 CONFLICT OF INTERESTThe authors have no conflicts to disclose. The crystal structure of LiGaO2. M Marezio, 10.1107/S0365110X65001068Acta Crystallographica. 18481M. Marezio, The crystal structure of LiGaO2, Acta Crys- tallographica 18, 481 (1965). Single-crystal growth of LiGaO2 for a substrate of GaN thin films. T Ishii, Y Tazoh, S Miyazawa, 10.1016/S0022-0248(97)00510-1J. Crystal Growth. 186409T. Ishii, Y. Tazoh, and S. Miyazawa, Single-crystal growth of LiGaO2 for a substrate of GaN thin films, J. Crystal Growth 186, 409 (1998). Growth and characterization of β-LiGaO2 single crystal. C Chen, C.-A Li, S.-H Yu, M M Chou, 10.1016/j.jcrysgro.2014.06.040Journal of Crystal Growth. 402325C. Chen, C.-A. Li, S.-H. Yu, and M. M. Chou, Growth and characterization of β-LiGaO2 single crystal, Journal of Crystal Growth 402, 325 (2014). Heat dissipation in high-power GaN electronics on thermally resistive substrates. A Christensen, W A Doolittle, S Graham, 10.1109/TED.2005.851815IEEE Transactions on Electron Devices. 521683A. Christensen, W. A. Doolittle, and S. Graham, Heat dissipation in high-power GaN electronics on thermally resistive substrates, IEEE Transactions on Electron De- vices 52, 1683 (2005). Growth of GaN on lithium gallate substrates for development of a GaN thin compliant substrate. W A Doolittle, T Kropewnicki, C Carter-Coman, S Stock, P Kohl, N M Jokerst, R A Metzger, S Kang, K K Lee, G May, A S Brown, 10.1116/1.590005Journal of Vacuum Science & Technology B: Microelectronics and Nanometer Structures Processing. 161300Measurement, and PhenomenaW. A. Doolittle, T. Kropewnicki, C. Carter-Coman, S. Stock, P. Kohl, N. M. Jokerst, R. A. Metzger, S. Kang, K. K. Lee, G. May, and A. S. Brown, Growth of GaN on lithium gallate substrates for development of a GaN thin compliant substrate, Journal of Vacuum Science & Technology B: Microelectronics and Nanometer Struc- tures Processing, Measurement, and Phenomena 16, 1300 (1998). S Weise, H Neumann, 10.1002/crat.2170310523Thermal Analysis of LiGaO2 and NaGaO2. 31659S. Weise and H. Neumann, Thermal Analysis of LiGaO2 and NaGaO2, Crystal Research and Technology 31, 659 (1996). Heat capacity and thermodynamic properties of LiGaO2 from 180 to 700 K. H Neumann, G Kühn, H Neels, E Nowak, E Pirl, 10.1002/crat.2170220322Crystal Research and Technology. 22413H. Neumann, G. Kühn, H. Neels, E. Nowak, and E. Pirl, Heat capacity and thermodynamic properties of LiGaO2 from 180 to 700 K, Crystal Research and Technology 22, 413 (1987). I Ohkubo, C Hirose, K Tamura, J Nishii, H Saito, H Koinuma, P Ahemt, T Chikyow, T Ishii, S Miyazawa, Y Segawa, T Fukumura, M Kawasaki, 10.1063/1.1512311Heteroepitaxial growth of β-LiGaO2 thin films on ZnO. 925587I. Ohkubo, C. Hirose, K. Tamura, J. Nishii, H. Saito, H. Koinuma, P. Ahemt, T. Chikyow, T. Ishii, S. Miyazawa, Y. Segawa, T. Fukumura, and M. Kawasaki, Heteroepitaxial growth of β-LiGaO2 thin films on ZnO, Journal of Applied Physics 92, 5587 (2002). Wide band gap semiconductor alloy: x(LiGaO2)1/2-(1-x)ZnO. T Omata, K Tanaka, A Tazuke, K Nose, S Otsuka-Yao-Matsuo, 10.1063/1.2903906J. Appl. Phys. 10383706T. Omata, K. Tanaka, A. Tazuke, K. Nose, and S. Otsuka-Yao-Matsuo, Wide band gap semiconductor al- loy: x(LiGaO2)1/2-(1-x)ZnO, J. Appl. Phys. 103, 083706 (2008). Wurtzite-Derived Wide Band Gap Oxide. T Omata, M Kita, K Nose, K Tachibana, S Otsuka-Yao-Matsuo, 10.1143/jjap.50.031102Jpn. J. Appl. Phys. 2431102T. Omata, M. Kita, K. Nose, K. Tachibana, and S. Otsuka-Yao-Matsuo, Zn2LiGaO4, Wurtzite-Derived Wide Band Gap Oxide, Jpn. J. Appl. Phys. 50, 031102 (2011). Tunable Direct Band Gap of β-CuGaO2 and β-LiGaO2 Solid Solutions in the Full Visible Range. I Suzuki, Y Mizuno, T Omata, 10.1021/acs.inorgchem.8b03370Inorganic Chemistry. 584262I. Suzuki, Y. Mizuno, and T. Omata, Tunable Direct Band Gap of β-CuGaO2 and β-LiGaO2 Solid Solutions in the Full Visible Range, Inorganic Chemistry 58, 4262 (2019). First-principles study of the elasticity, piezoelectricity, and vibrational modes in LiGaO2 compared with ZnO and GaN. A Boonchun, W R L Lambrecht, 10.1103/PhysRevB.81.235214Phys. Rev. B. 81235214A. Boonchun and W. R. L. Lambrecht, First-principles study of the elasticity, piezoelectricity, and vibrational modes in LiGaO2 compared with ZnO and GaN, Phys. Rev. B 81, 235214 (2010). A simple effective potential for exchange. A D Becke, E R Johnson, 10.1063/1.2213970The Journal of Chemical Physics. 124221101A. D. Becke and E. R. Johnson, A simple effective poten- tial for exchange, The Journal of Chemical Physics 124, 221101 (2006). Accurate Band Gaps of Semiconductors and Insulators with a Semilocal Exchange-Correlation Potential. F Tran, P Blaha, 10.1103/PhysRevLett.102.226401Phys. Rev. Lett. 102226401F. Tran and P. Blaha, Accurate Band Gaps of Semi- conductors and Insulators with a Semilocal Exchange- Correlation Potential, Phys. Rev. Lett. 102, 226401 (2009). The electronic structure of lithium metagallate. N W Johnson, J A Mcleod, A Moewes, 10.1088/0953-8984/23/44/445501Journal of Physics: Condensed Matter. 23445501N. W. Johnson, J. A. McLeod, and A. Moewes, The electronic structure of lithium metagallate, Journal of Physics: Condensed Matter 23, 445501 (2011). Chemical alteration of the native oxide layer on LiGaO2(001) by exposure to hyperthermal atomic hydrogen. J T Wolan, G B Hoflund, 10.1116/1.581495J. Vac. Sci. Tech. A. 163414J. T. Wolan and G. B. Hoflund, Chemical alteration of the native oxide layer on LiGaO2(001) by exposure to hyperthermal atomic hydrogen, J. Vac. Sci. Tech. A 16, 3414 (1998). Quasiparticle self-consistent GW band structures and high-pressure phase transitions of LiGaO2 and NaGaO2. S K Radha, A Ratnaparkhe, W R L Lambrecht, 10.1103/PhysRevB.103.045201Phys. Rev. B. 10345201S. K. Radha, A. Ratnaparkhe, and W. R. L. Lambrecht, Quasiparticle self-consistent GW band structures and high-pressure phase transitions of LiGaO2 and NaGaO2, Phys. Rev. B 103, 045201 (2021). Electronic structure, doping, and lattice dynamics of LiGaO2, in Oxidebased Materials and Devices II. A Boonchun, W R L Lambrecht, 10.1117/12.879320International Society for Optics and Photonics. F. H. Terani, D. C. Look, and D. J. Rogers7940SPIEProceedings of SPIEA. Boonchun and W. R. L. Lambrecht, Electronic struc- ture, doping, and lattice dynamics of LiGaO2, in Oxide- based Materials and Devices II , Proceedings of SPIE, Vol. 7940, edited by F. H. Terani, D. C. Look, and D. J. Rogers, International Society for Optics and Photonics (SPIE, 2011) pp. 129-134. First-principles study of point defects in LiGaO2. A Boonchun, K Dabsamut, W R L Lambrecht, 10.1063/1.5126028Journal of Applied Physics. 126155703A. Boonchun, K. Dabsamut, and W. R. L. Lambrecht, First-principles study of point defects in LiGaO2, Journal of Applied Physics 126, 155703 (2019). First-principles study of n-and p-type doping opportunities in LiGaO2. K Dabsamut, A Boonchun, W R L Lambrecht, http:/iopscience.iop.org/10.1088/1361-6463/ab8514Journal of Physics D: Applied Physics. K. Dabsamut, A. Boonchun, and W. R. L. Lambrecht, First-principles study of n-and p-type doping opportu- nities in LiGaO2, Journal of Physics D: Applied Physics (2020). NO, and O2 molecules in LiGaO2 in both Ga and Li sites and their relation to the vacancies. K Dabsamut, A Boonchun, W R L Lambrecht, 10.1063/5.0085366Journal of Applied Physics. 2145705K. Dabsamut, A. Boonchun, and W. R. L. Lambrecht, N2, NO, and O2 molecules in LiGaO2 in both Ga and Li sites and their relation to the vacancies, Journal of Applied Physics 131, 145705 (2022). Lithium and gallium vacancies in LiGaO2 crystals. C A Lenyk, M S Holston, B E Kananen, L E Halliburton, N C Giles, 10.1063/1.5050532Journal of Applied Physics. 124135702C. A. Lenyk, M. S. Holston, B. E. Kananen, L. E. Hal- liburton, and N. C. Giles, Lithium and gallium vacan- cies in LiGaO2 crystals, Journal of Applied Physics 124, 135702 (2018). Computational study of electron paramagnetic resonance spectra for Li and Ga vacancies in LiGaO2. D Skachkov, W R L Lambrecht, K Dabsamut, A Boonchun, 10.1088/1361-6463/ab6f1cJournal of Physics D: Applied Physics. 53D. Skachkov, W. R. L. Lambrecht, K. Dabsamut, and A. Boonchun, Computational study of electron para- magnetic resonance spectra for Li and Ga vacancies in LiGaO2, Journal of Physics D: Applied Physics 53, 17LT01 (2020). Optical properties of lithium gallium oxide. S Tumėnas, P Mackonis, R Nedzinskas, L Trinkler, B Berzina, V Korsaks, L Chang, M Chou, 10.1016/j.apsusc.2017.01.098Applied Surface Science. 421837S. Tumėnas, P. Mackonis, R. Nedzinskas, L. Trinkler, B. Berzina, V. Korsaks, L. Chang, and M. Chou, Opti- cal properties of lithium gallium oxide, Applied Surface Science 421, 837 (2017). Luminescence properties of LiGaO2 crystal. L Trinkler, A Trukhin, B Berzina, V Korsaks, P Ščajev, R Nedzinskas, S Tumėnas, M Chou, L Chang, C.-A Li, 10.1016/j.optmat.2016.11.012Optical Materials. 69449L. Trinkler, A. Trukhin, B. Berzina, V. Korsaks, P.Ščajev, R. Nedzinskas, S. Tumėnas, M. Chou, L. Chang, and C.-A. Li, Luminescence properties of LiGaO2 crystal, Optical Materials 69, 449 (2017). Anisotropic photoluminescence of β-LiGaO2 crystal. L Trinkler, V Pankratov, A Trukhin, B Berzina, M Chou, L Chang, 10.1016/j.optmat.2022.112856Optical Materials. 132112856L. Trinkler, V. Pankratov, A. Trukhin, B. Berzina, M. Chou, and L. Chang, Anisotropic photoluminescence of β-LiGaO2 crystal, Optical Materials 132, 112856 (2022). Improved quasiparticle self-consistent electronic band structure and excitons in β-LiGaO2. N Dadkhah, W R L Lambrecht, D Pashov, M Van Schilfgaarde, 10.48550/ARXIV.2302.03150aeXiv.2302.03150N. Dadkhah, W. R. L. Lambrecht, D. Pashov, and M. van Schilfgaarde, Improved quasiparticle self-consistent elec- tronic band structure and excitons in β-LiGaO2 (2023), aeXiv.2302.03150. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. G Kresse, J Furthmüller, Physical Review B. 5411169G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Physical Review B 54, 11169 (1996). Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. G Kresse, J Furthmüller, Computational materials science. 615G. Kresse and J. Furthmüller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Computational materials science 6, 15 (1996). Projector augmented-wave method. P E Blöchl, Physical Review B. 5017953P. E. Blöchl, Projector augmented-wave method, Physi- cal Review B 50, 17953 (1994). Hybrid functionals based on a screened Coulomb potential. J Heyd, G E Scuseria, M Ernzerhof, 10.1063/1.1564060J. Chem. Phys. 1188207J. Heyd, G. E. Scuseria, and M. Ernzerhof, Hybrid func- tionals based on a screened Coulomb potential, J. Chem. Phys. 118, 8207 (2003). Hybrid functionals based on a screened Coulomb potential. J Heyd, G E Scuseria, M Ernzerhof, Erratum , 10.1063/1.2204597J. Chem. Phys. 118219906J. Chem. Phys.J. Heyd, G. E. Scuseria, and M. Ernzerhof, Erratum: "Hybrid functionals based on a screened Coulomb poten- tial" [J. Chem. Phys. 118, 8207 (2003)], J. Chem. Phys. 124, 219906 (2006). Van de Walle, Firstprinciples calculations for point defects in solids. C Freysoldt, B Grabowski, T Hickel, J Neugebauer, G Kresse, A Janotti, C G , Rev. Modern Phys. 86253C. Freysoldt, B. Grabowski, T. Hickel, J. Neugebauer, G. Kresse, A. Janotti, and C. G. Van de Walle, First- principles calculations for point defects in solids, Rev. Modern Phys. 86, 253 (2014).
{'fraction_non_alphanumeric': 0.056675242995997716, 'fraction_numerical': 0.052994568324757005, 'mean_word_length': 4.159476401179941, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'Hybrid functional calculations are presented for defects in LiGaO2 with the fraction of non-local exchange adjusted to reproduce the recently reported exciton gap of 6.0 eV. We study how the defect transition levels of the main native defects change with respect to the band edges compared to earlier calculations which assumed a smaller band gap near 5.1 eV. In addition, we consider defect complexes formed by combining the main native donor GaLi with the main acceptors, VLi and LiGa antisites as function of their relative position. These results are used to tentatively identify the photoluminescence bands previous assigned to donor-acceptor-pair recombination.', 'arxivid': '2302.09206', 'author': ['Klichchupong Dabsamut ', 'Adisak Boonchun ', 'Walter R L Lambrecht ', '\nDepartment of Physics\nFaculty of Science\nDepartment of Physics\nKasetsart University\n10900BangkokThailand\n', '\nCase Western Reserve University\n10900 Euclid Avenue44106-7079ClevelandOhioUSA\n'], 'authoraffiliation': ['Department of Physics\nFaculty of Science\nDepartment of Physics\nKasetsart University\n10900BangkokThailand', 'Case Western Reserve University\n10900 Euclid Avenue44106-7079ClevelandOhioUSA'], 'corpusid': 257038723, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 9553, 'n_tokens_neox': 7841, 'n_words': 4454, 'pdfsha': 'af010bf9ad4a9274dbdd2e966266a8b1b73031c9', 'pdfurls': ['https://export.arxiv.org/pdf/2302.09206v1.pdf'], 'title': ['Computational study of defect complexes in β-LiGaO 2 and their relation to the donor-acceptor-pair recombination', 'Computational study of defect complexes in β-LiGaO 2 and their relation to the donor-acceptor-pair recombination'], 'venue': []}
arxiv
Diffusion MRI microstructure models with in vivo human brain Connectome data: results from a multi-group comparison TE |G| b Nr (ms) (ms) (mT/m) (s/mm 2 ) Nr (ms) (ms) (mT/m) (s/mm 2 ) Uran Ferizi [email protected] Centre for Medical Image Computing Department of Computer Science University College London UK Department of Radiology New York University School of Medicine USA Department of Neuroinflammation Institute of Neurology University College London UK Benoit Scherrer Computational Radiology Laboratory Boston Children's Hosp Harvard University USA Torben Schneider Department of Neuroinflammation Institute of Neurology University College London UK Philips Healthcare GuildfordSurreyUK Mohammad Alipoor Chalmers University of Technology GothenburgSweden Odin Eufracio Centro de Investigacion en Matematicas AC GuanajuatoMexico Rutger H J Fick Athena Project-Team INRIA Sophia Antipolis -Méditerranée France Rachid Deriche Athena Project-Team INRIA Sophia Antipolis -Méditerranée France Markus Nilsson Lund University Bioimaging Center Lund University Sweden Ana K Loya-Olivas Centro de Investigacion en Matematicas AC GuanajuatoMexico Mariano Rivera Centro de Investigacion en Matematicas AC GuanajuatoMexico Dirk H J Poot Erasmus Medical Center Delft University of Technology the Netherlands Alonso Ramirez-Manzanares Centro de Investigacion en Matematicas AC GuanajuatoMexico Jose L Marroquin Centro de Investigacion en Matematicas AC GuanajuatoMexico Ariel Rokem eScience Institute University of Washington USA Center for Cognitive and Neurobiological Imaging Stanford University USA Christian Pötter Center for Cognitive and Neurobiological Imaging Stanford University USA Robert F Dougherty Center for Cognitive and Neurobiological Imaging Stanford University USA Ken Sakaie Imaging Institute The Cleveland Clinic ClevelandUSA Claudia Wheeler-Kingshott Department of Neuroinflammation Institute of Neurology University College London UK Simon K Warfield Computational Radiology Laboratory Boston Children's Hosp Harvard University USA Thomas Witzel A.A. Martinos Center for Biomedical Imaging MGH Harvard University USA Center for Biomedical Imaging Department of Radiology Correspondence Uran Ferizi PhD New York University Langone Medical Center 660 First Avenue, 4th floor10016New YorkNYUSA Lawrence L Wald A.A. Martinos Center for Biomedical Imaging MGH Harvard University USA Center for Biomedical Imaging Department of Radiology Correspondence Uran Ferizi PhD New York University Langone Medical Center 660 First Avenue, 4th floor10016New YorkNYUSA José G Raya Department of Radiology New York University School of Medicine USA Daniel C Alexander Centre for Medical Image Computing Department of Computer Science University College London UK Diffusion MRI microstructure models with in vivo human brain Connectome data: results from a multi-group comparison TE |G| b Nr (ms) (ms) (mT/m) (s/mm 2 ) Nr (ms) (ms) (mT/m) (s/mm 2 ) 10.1002/nbm.3734Received: 6 November 2016 Revised: 1 March 2017 Accepted: 27 March 2017R E S E A R C H A R T I C L E Funding information The (UK) Engineering and Physical Sciences Research Council (EPSRC), Grant/Award Number: EP/G007748, EP/L022680/1, EP/I027084/01, EP/M020533/1 and EP/N018702/1; The (US) National Institute of Arthritis and Musculoskeletal and Skin Diseases (NIAMS) of the National Institute of Health (NIH), Grant/ signal prediction in two regions: the genu in the corpus callosum, where the fibres are relatively straight and parallel, and the fornix, where the . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. 1 of 23 2 of 23 3 of 23Abbreviations used: CT, computerized tomographyCV, cross-validationDBF, diffusion basis functionDF, diffusion functionDT, diffusion tensorDTI, diffusion tensor imagingDW, diffusion-weightedEN, elastic netISBI, International Symposium on Biomedical ImagingLASADD, Linear Acceleration of Sparse and Adaptive Diffusion DictionaryLS, least-squaresMRI, magnetic resonance imagingPDD, principal diffusion directionsRSI, restriction spectrum imagingROI, region of interestSFM, sparse fascicle modelSNR, signal-to-noise ratioSSE, sum of squared errorsTE, echo timeWM, white matter A large number of mathematical models have been proposed to describe the measured signal in diffusion-weighted (DW) magnetic resonance imaging (MRI). However, model comparison to date focuses only on specific subclasses, e.g. compartment models or signal models, and little or no information is available in the literature on how performance varies among the different types of models. To address this deficiency, we organized the 'White Matter Modeling Challenge' during the International Symposium on Biomedical Imaging (ISBI) 2015 conference. This competition aimed to compare a range of different kinds of models in their ability to explain a large range of measurable in vivo DW human brain data. Specifically, we assessed the ability of models to predict the DW signal accurately for new diffusion gradients and b values. We did not evaluate the accuracy of estimated model parameters, as a ground truth is hard to obtain. We used the Connectome scanner at the Massachusetts General Hospital, using gradient strengths of up to 300 mT/m and a broad set of diffusion times. We focused on assessing the DW This work builds on the statistical modelling of the apparent diffusion coefficient 72 and tackles the modelling of axon fibre dispersion in single15,73and multiple fibre bundle cases. The method estimates empirically (rather than imposes) the distribution of tissue properties (axon radius, parallel diffusion, etc.), as well as the orientational distribution of the bundles. The general framework is as follows:• estimation of mean principal diffusion directions (PDD) per axon bundle;• selection of a dense set of orientationally focused basis directions that capture the discrete non-parametric fibre dispersion;• design of a dictionary of intra/extracellular synthetic DW signals, which are precomputed along the basis directions (see the DBF method in Ramirez-Manzanares et al. 74 );• computation of the size compartments per diffusion atom of the dictionary (model fitting). configuration of fibres is more complex. The challenge participants had access to three-quarters of the dataset and their models were ranked on their ability to predict the remaining unseen quarter of the data. The challenge provided a unique opportunity for a quantitative comparison of diverse methods from multiple groups worldwide. The comparison of the challenge entries reveals interesting trends that could potentially influence the next generation of diffusion-based quantitative MRI techniques. The first is that signal models do not necessarily outperform tissue models; in fact, of those tested, tissue models rank highest on average. The second is that assuming a non-Gaussian (rather than purely Gaussian) noise model provides little improvement in prediction of unseen data, although it is possible that this may still have a beneficial effect on estimated parameter values. The third is that preprocessing the training data, here by omitting signal outliers, and using signal-predicting strategies, such as bootstrapping or cross-validation, could benefit the model fitting. The analysis in this study provides a benchmark for other models and the data remain available to build up a more complete comparison in the future. KEYWORDS brain microstructure, Connectome, diffusion MRI, fornix, genu, model selection INTRODUCTION Diffusion-weighted (DW) magnetic resonance imaging (MRI) can provide unique insights into the microstructure of living tissue and is increasingly used to study the microanatomy and development of normal functioning tissue as well as its pathology. Mathematical models for analysis and interpretation have been crucial for the development and translation of DW-MRI. Even though diffusion tensor imaging (DTI), 1 which is based on a simple Gaussian model of the DW-MRI signal, has shown promise in clinical applications, 2 e.g. Alzheimer's disease, 3 multiple sclerosis 4 or brain tumors, 5 a much wider variety of DW-MRI models has been proposed to extract more information from the DW signal. Models generally fall between two extremes: 'models of the tissue' and 'models of the signal' . Models of the tissue 6-17 describe the underlying tissue microstructure in each voxel explicitly with a multi-compartment approach. 18,19,20 Models of the signal focus on describing the DW signal attenuation without describing the underlying tissue composition that gives rise to the signal explicitly. [21][22][23][24][25][26][27][28][29] Other approaches fall between these two classes and include some features of the tissue, such as the distribution of fibre orientations, but often describe the signal from individual fibres without modelling the fibre composition explicitly. [30][31][32][33][34][35][36][37][38][39][40] Despite this explosion of DW-MRI models, a broad comparison on a common dataset and within a common evaluation framework is lacking, so little is understood about which models are more plausible representations or explanations of the signal. Panagiotaki et al. 18 established a taxonomy of diffusion compartment models and compared 47 of them using data from the fixed corpus callosum of a rat acquired on a pre-clinical system. Later, Ferizi et al. 39 performed a similar experiment using data from a live human subject, while Ferizi et al. 41,42 explored a different class of models, which aim to capture fibre dispersion. Rokem et al. 43 compared two classes of models using cross-validation and test-retest accuracy. All these studies 18,43,44 aim to evaluate variations with specific classes of models with all other variables of the parameter estimation pipeline (i.e. noise model, fitting routine, etc.) fixed. While this provides fundamental insight into which compartments are important in compartment models, questions remain about the broader landscape of models; in particular, which classes of models explain the signal best and how strongly performance depends on the choice of parameter-estimation procedure. Publicly organized challenges provide a unique opportunity to bring a research community together to gain a quantitative and unbiased comparison of a diverse set of methods applicable to a particular data-processing task. Such publicly organized challenges have helped to establish a common ground for the evaluation of competing methods in a variety of imaging-related tasks, e.g. in brain MR image registration 45 and segmentation. 46 In DW-MRI, public challenges have focused on recovering synthetic intra-voxel fibre configurations 47 or evaluating tractography techniques 48,50 and have been very successful at driving research and translation forward. Another interesting comparison of reconstruction methods using DW-MRI data was based on the signal acquired from a physical phantom. 49 Here we report on such a community-wide challenge to model the variation of DW-MRI signals at the voxel level in the in vivo human brain. Modelling the diffusion signal is a key step in realizing practical and reliable quantitative imaging techniques based on diffusion MRI. The challenge in the area is to extract the salient features from the diffusion signal and relate them to the principal features of the underlying tissue (e.g. in the case of brain white matter (WM) the fibre orientation, axonal packing and axonal size). Three distinct questions arise. i. Given the richest possible dataset that samples the space of achievable measurements as widely as possible, which mathematical model can capture best the intrinsic variation of the acquired signal? *Uran Ferizi, Benoit Scherrer and Torben Schneider joint first co-authors. ii. Which tissue features can be derived from the model? iii. What subset of those features can we estimate from limited acquisition time on a standard clinical scanner and what dataset best supports such estimates? The intuition gained from (i) is generalizable over a wide range of applications, while (ii) and (iii) are highly dependent on the MRI study design and the available hardware. Therefore, our challenge focuses on question (i), as an understanding of (i) is necessary to inform (ii) and (iii). To that end, we acquire the richest possible dataset using the most powerful hardware available and the most motivated subject available (UF). Specifically, we use the Connectome scanner, 51 which is unique among human scanners in having 300 mT/m gradients, rather than 40 mT/m as is typical of state-of-the-art human scanners. Preclinical work by Dyrby et al. 13 highlights the benefits of such strong gradients and the first results from the Connectome scanner 42,[52][53][54] are now starting to verify those findings. This kind of model comparison, based on prediction error, is a common and crucial part of the development of any statistical model-based estimation applications. Burnham and Anderson 55 explain how and why such comparisons should be performed to reject models that are theoretically plausible but not supported by the data. To that end, we used a uniquely rich dataset acquired on the Connectome system 42 composed of around 5000 points in q space with, for each shell, a unique combination of gradient strength, diffusion time, pulse width and echo time. This offers the opportunity for the comparison of the many different types of models within a common framework, over a very wide range of measurement space. Using this rich dataset, we organized the White Matter Modeling challenge, held during the 2015 International Symposium on Biomedical Imaging (ISBI) in New York. The goal of the challenge was to evaluate and compare the models in two different tissue configurations that are common in the brain: (1) a WM region of interest where fibres are relatively straight and parallel, specifically the genu of the corpus callosum; and (2) a region in which the fibre configuration is more complex, specifically the fornix. Challenge participants had access to three-quarters of each whole dataset; the participating models were evaluated on how well they predicted the remaining 'unseen' part of the data. As announced before the challenge, the final ranking was based exclusively on the performance on the genu data. In this article, however, we include results from both the genu and the fornix. The article is organized as follows. We first describe in section 2 the experimental protocol, data post-processing and preparation of the training and testing data for the challenge. We then present the methods for ranking the models and tabulate the various models involved in the competition succinctly. We report the challenge results in section 3 and discuss these results in section 4; a more detailed description of the models follows in the Appendix. MATERIAL AND METHODS The complete experiment protocol One healthy volunteer was scanned over two non-stop 4 h sessions. The imaged volume comprised twenty 4 mm thick whole-brain sagittal slices covering the corpus callosum left-right. The image size was 110 × 110 and the in-plane resolution 2 × 2 mm 2 . 45 unique and evenly distributed diffusion directions (taken from http://www.camino.org.uk) were acquired for each shell, with both positive and negative polarities; these directions were the same in each shell. We also included 10 interleaved b = 0 measurements, leading to a total of 100 measurements per shell. Each shell had a unique combination of Δ = {22, 40, 60, 80, 100, 120} ms, = {3, 8} ms and |G| = {60, 100, 200, 300} mT/m (see Table 1). The measurements were randomized within each shell, whereas the gradient strengths were interleaved. We inspected the images visually and did not observe any obvious shifts from gradient heating. The minimum possible echo time (TE) for each gradient duration and diffusion time combination was chosen to enhance signal-to-noise ratio (SNR) and potential estimation of compartment-specific relaxation constants. The SNR of b = 0 images was 35 at TE = 49 ms and 6 at TE = 152 ms. The SNR was computed by assessing the signal mean and noise variance across the selected WM voxels on multiple b = 0 images. In both cases these estimates matched reasonably well. More details about the acquisition protocol can be found in Post-processing All post-processing was performed using Software Library (FSL). 56 The DW images were corrected for eddy current distortions separately for each combination of and Δ using FSL's Eddy module (www.fmrib.ox.ac.uk/fsl/eddy) with its default settings. The images were then co-registered using FSL's Fnirt package. As the 48 shells were acquired across a wide range of TEs, over two days, we chose to proceed in two steps. First, within each quarter of the dataset (different day, different ) we registered all the b = 0 images together. We then applied these transformations to their intermediary DW images, using a trilinear resampling interpolation. The second stage involved co-registering the four different quarters. To help the co-registration, especially between the two days images that required some through-plane adjustment as well, we omitted areas of considerable eddy-current distortions by reducing the number of slices from 20 to 5 (i.e. leaving two images either side of the mid-sagittal plane) and reducing the in-plane image size to 75 × 80. Note. We provide signal for the parts of protocol marked in black. In red is the protocol for which the signal needs to be predicted. FIGURE 1 We only consider two ROIs, each containing six voxels from the genu in the corpus callosum, where the fibres are approximately straight and parallel, and from the fornix, where the configuration of fibres is more complex Training and testing data The data for this work originated from two regions of interest (ROIs), each containing 6 voxels (see Figure 1). The first ROI was selected in the middle of the genu in the corpus callosum, where the fibres are mostly straight and coherent. The second ROI's fibre configuration is more complex: it lies in the body of fornix, where two bundles of fibres bend and bifurcate. The dataset was split into two parts: the training dataset and the testing dataset. The training dataset was fully available for the challenge participants. The testing dataset was retained by one of the organizers (UF). The DW signal of the training dataset (36 shells, with acquisition parameters shown in black in Table 1) was provided together with the gradient scheme on the challenge website (http://cmic.cs.ucl.ac.uk/wmmchallenge/). This data was used by the participants to estimate their DW-MRI model parameters. The signal attenuation in the testing dataset (12 shells, with acquisition parameters shown in red in Table 1) was kept unseen. It contained one shell, chosen at random, from each TE-specific set of four shells (i.e of the same combination of and Δ). The challenge participants were then asked to predict the signal for the corresponding gradient scheme. They were free to use as much or as little of the training data provided as they wished to predict the signal of the test dataset for the six voxels in each ROI. Figure 2 shows the DW signal attenuation for each shell in the genu dataset, with stars in the legend indicating which shells were left out for testing. In this plot, a small number of data appear as 'outliers' (two such data are shown with arrows in the bottom-left subplot of Figure 2). Specifically, we counted about 10 of them among more than 4812 measurements, most of them being in the b = 300 s/mm 2 shell. Since these outliers appear to be specific to the b = 300 s/mm 2 shell and are not in other shells with similar b value, we attribute them to a momentary twitching of the subject rather than more systematic effects, such as perfusion. Similarly, Figure 3 shows the signal for the fornix region, with the signal over the six voxels averaged out. Model ranking Models were evaluated and ranked based on their ability to predict the unseen DW signal accurately. Specifically, the metric used was the sum of square differences between the hidden signal and the predicted signal, corrected for Rician noise: 57 SSE = 1 N N ∑ i=1 (S i − √ S 2 i + 2 ) 2 2 (1) where N is the number of measurements,S i is the ith measured signal, S i its prediction from the model and the noise standard deviation. Competing models Here we give a short summary of the competing models. Additionally, Table 2 provides a summary of their key characteristics. More details are included in the Appendix. • Ramirez-Manzanares: a dictionary-based technique that accounts for multiple fibre bundles and models the distribution of tissue properties (axon radius, parallel diffusivity) and the orientation dispersion of fibres. • Nilsson: a multi-compartment model that models isotropic, hindered and restricted diffusion and accounts for varying (T 1 , T 2 ) relaxation times for each compartment. 58 • Scherrer a multi-compartment model in which each compartment is modelled by a statistical distribution of 3-D tensors. 16 • Ferizi 1 and Ferizi 2 : two three-compartment models that account for varying T 2 relaxation times for each compartment. As regards the intracellular compartment, Ferizi 1 models the orientation dispersion by using dispersed sticks as one compartment; Ferizi 2 uses a single radius cylinder instead. 42 • Poot: a three-compartment model comprising an isotropic diffusion compartment, a tensor compartment and a model-free compartment in which an Apparent Diffusion Coefficient (ADC) is estimated for each direction independently. T 2 relaxation times are also estimated for each compartment. 59 • Rokem: a combination of the sparse fascicle model 43 with restriction spectrum imaging 60 that describes the signal arising from a multi-compartment model in a densely sampled spherical grid, using L1 regularization to enforce sparsity. • Eufracio: an extension of the Diffusion Basis Function (DBF) model that accounts for multiple b-value shells. • Loya-Olivas 1 and Loya-Olivas 2 : two models based on the Linear Acceleration of Sparse and Adaptive Diffusion Dictionary (LASADD) technique. Loya-Olivas 1 uses the DBF signal model, while Loya-Olivas 2 uses a three-compartment tissue model. The optimization uses linearized signal models to speed up computation and sparseness constraints to regularize. • Alipoor: a model of fourth-order tensors, corrected for T 2 -relaxation across different shells. A robust LS fitting was applied to mitigate influence of outliers. • Sakaie: a two-compartment model of restricted and hindered diffusion with angular variation. A simple exclusion scheme based on the b = 0 signal intensity was applied to remove outliers. • Fick: a spatio-temporal signal model to represent 3-D diffusion signal simultaneously over varying diffusion time. Laplacian regularization was applied during the fitting. 61 • Rivera: a regularized linear regression model of diffusion encoding variables. This is intentionally built as a simplistic model to be used as a baseline for model comparison. While the challenge organizers also had competing models (Ferizi 1 , Ferizi 2 and Scherrer), only Ferizi had access to the hidden data. The hidden data were never used to tune the results of his models. Diffusion-weighted signal from the genu ROI, averaged over the six voxels. Across each column and row, the signal pertains to one of the gradient strengths or pulse times used; in each subplot, the six shells shown in different colours are Δ-specific, increasing in value (22,40,60,80, 100, 120 ms) from top to bottom. Inside the legend, the b value is in s/mm 2 units; here, the HARDI shells kept for testing are those marked with a star; the remaining shells comprise the training data. On the x-axis is the cosine of the angle between the applied diffusion gradient vector G and the fibre direction n. Some models in this study omit data outliers; two such data points are shown in the bottom-left subplot with vertical arrows -obviously each model has its own criteria for determining the outliers Figure 4 shows the averaged prediction error in each ROI (top subplot is for the genu, bottom subplot is for the fornix) and the corresponding overall ranking of the participating models in the challenge. The first six models in the genu ranking performed similarly, each higher ranked model marginally improving on the prediction error. The prediction error clearly increased at a higher rate for the subsequent models. In the fornix dataset, the prediction error was higher than in the genu. For both datasets, the first six models were the same, albeit permuted. Most of the models performed bilities; some models perform better within a given b-value range but are penalized more in another. Across the models, as the figure shows, the ranking between models was dominated by the signal prediction accuracy for b values between 750 and 1400s/mm 2 ; specifically, the shell that has the largest weight on this error is the b = 1100 s/mm 2 one. The top-ranking models, nevertheless, were better at predicting the signal for higher b-value images as well. The prediction performance of lower b-value images (<750s/mm 2 ) in the genu was less consistent across ranks. For example, the models of Rokem and Sakaie outperformed most of the higher ranking models in this low b-value range. The fornix is a more complex region than the genu, hence the performance across the shells is less consistent. In the fornix, the prediction errors were generally larger than in the genu across all b values for all models, except Rivera's, which showed the opposite effect. The prediction errors of the b = 0 images were also larger than in the genu, especially for the highly ranked models of Poot and Ferizi. The prediction errors in other b-value shells followed the overall ranking of the models more closely. Figure 5 shows the prediction error for each voxel independently. In the genu plot, the best performing models had high consistency of low prediction errors across all individual voxels. These were followed by the models with consistent larger prediction error in all voxels. Most of the lowest ranking models not only had largest prediction errors, they also showed large variations in prediction performance. For example, while the model of Loya-Olivas 2 was competitive in voxel 5, it ranked low due to large prediction errors in voxels 4 and 6. The results in the fornix show a lower consistency of prediction errors between the voxels than in the genu. Specifically, two voxels (3 and 4) showed substantially larger prediction errors and were likely responsible for much of the overall ranking. RESULTS Finally, we report in Figures 6 -8 and 9 an illustration of the quality of fit of each model to four representative shells, including the b=1100s/mm 2 shell mentioned above; Figures 6 and 7 concern the genu data and Figures 8 and 9 are for fornix data. DISCUSSION The challenge set out to compare the ability of various kinds of models to predict the diffusion MR signal from WM over a very wide range of mea- Despite their success, intense debate continues in the field about applicability of different models and fitting routines. 64,65 The insights from this challenge provide key pointers to the important features of the next-generation of front-line imaging techniques of this type. Moreover, the data and evaluation routines remain available to form the basis of an expanding ranking of models and fitting routines and a benchmark for future model development. Main conclusions The first insight is on the type of model used. Signal models do not necessarily outrank tissue models; indeed, using our dataset, models of the signal (Alipoor, Sakaie, Fick, Rivera) ranked on average lower than models of the tissues, despite their theoretical ability to offer more flexibility in describing the raw signal. This is quite surprising, as the current perception within the field is that, generally, we can capture the signal variation much better through a functional description of the signal (signal models) rather than via a biophysical model of the tissue (tissue models). The former generally consist of bases of arbitrary complexity, whereas the latter are generally very parsimonious models that rely on extremely crude descriptions of tissue (e.g. white matter as parallel impermeable cylinders). The results suggest that the flexibility of signal models can rapidly lead to overfitting. However, the tissue models can explain the signal relatively well even with just a few parameters (compare the quality-of-fit plots of the Rivera model in Figure 7 with the signal prediction of the top models in Figure 6: the higher the b value, the worse the prediction of the linear signal model). Certain underlying assumptions may cause the signal models to perform less well than expected. For example, they are often designed to work with data with a single diffusion time and do not generalize naturally to incorporate the additional dimension (although see Fick et al. 61 The third observation is about removing signal outliers. Five of the eleven models preprocessed the training data by clearing out outliers, including the top two models. We tried this procedure with two good models that did not use such a procedure, Ferizi 1 and Ferizi 2 , and observed that it did not affect the ranking, though it did improve the prediction error marginally. This is understandable, considering the relatively little weight FIGURE 7 Genu signal for the second group of 14 models. Raw testing data are shown by blue stars, while red circles denote the model-predicted data. The x-axis is the cosine of the angle between G and n these apparent outliers have on the total number of measurements (10 points from a 4812-strong dataset). Additionally, specific strategies for predicting the signal, e.g. bootstrapping or cross-validation, as used by the top two models of Ramirez-Manzanares and Nilsson, may also help the model ranking. Limitations and future directions Although this challenge provides several new insights into the choice of model and fitting procedure for diffusion-based quantitative imaging tools, it has a number of limitations that future challenges might be designed to address. One limitation of the study is that we use a very rich acquisition FIGURE 8 Fornix signal for the group consisting of the best 7 from 14 models. We show only four (of twelve) representative shells; these are shown by blue stars, while red circles denote model-predicted data. The best models are listed first. The x-axis is the cosine of the angle between G and n protocol that is not representative of common or clinical acquisition protocols. In particular, we cover a very wide range of b values and the data acquisition (protocol) we use consists of many TEs, unlike many other multi-shell diffusion datasets that use a fixed TE. As stated in the Introduction, our intention is to sample the measurement space as widely as possible to support the most informative models possible. Varying the TE makes it possible to probe compartment-specific T 2 (the decay of which Ferizi et al. 42 finds to be monoexponential at the voxel level), an investigation that would be impossible with a single TE. However, the good performance of DIAMOND also shows that a model with fixed and Δ can still capture the signal variation in multi-TE datasets and that, while the majority of the full data was ignored in each of the reconstructions, its prediction error compared favourably with other techniques. Fornix signal for the second group of 14 models. Raw testing data are shown by blue stars, while red circles denote the model-predicted data. The x-axis is the cosine of the angle between G and n We use the unique human Connectome scanner 51 to acquire a dataset with gradients of up to 300mT/m, which is not readily available in most current MR machines. However, previous preclinical work by Dyrby et al. 13 suggests that high diffusion gradients enrich the signal, which helps model fitting and comparison. Future challenges might be designed that focus on explaining the signal and estimating parameters from data more typical of clinical acquisitions. Assessing the prediction performance on unseen data as in this challenge is different from assessing the fitting error: it implicitly penalizes models that overfit the data. However, since most of the missing shells lie in between other shells (in terms of b values, TEs, etc.), the quality of signal extrapolation was not assessed. We get a glimpse of this from Figure 4, where the SSE is unevenly distributed between the b values. Here, the shell that bore the largest error is the b = 1100 s/mm 2 one; see also Figures 6 and 7. Of all 'unseen' shells, this shell combines the lowest Δ and highest |G|, placing it on the edge of the range of the measurement space sampled. Such a b-value shell combines high signal magnitude with high sensitivity, i.e. the gradient of signal against b-value is highest in this range, which makes it hard to predict. (We stress that this observation is in the context of the wider multi-shell acquisition, and is not to be seen in isolation for its potential impact on single-shell acquisition methods.) On the other hand, the variability of prediction errors in the b < 750 s/mm 2 range could arise from the varying sensitivity of different models to the free water component, which is challenging to estimate as it can easily be confounded with hindered water, or physiological effects, which are mostly observable in this low b-value range. Future work can take this further, by selecting unseen shells outside the min-max range of experimental parameters. This is likely to penalize more complex models that overfit the data even more strongly. We did not take into account the computational demand of each model, and this might limit the generalization of the results. Models that use bootstrapping generally have a higher computational burden and may not be feasible for large datasets, e.g. whole brain coverage. The dataset used in this challenge is specific to one subject who underwent a long-duration acquisition, which adds to the question of generalizability. The subsequent preprocessing of the data is also a factor to bear in mind: the registration of two 4h datasets, across such a broad range of echo times, poses its own challenges for certain non-homogenous regions in the brain, such as the fornix (compared with, for example, the relatively large genu). Thus the results may be somewhat subject-specific and may be affected by residual alignment errors. Another limitation is that we only look at isolated voxels inside the corpus callosum and the fornix. Questions still remain about which models are viable even in the most coherent areas of the brain with the simplest geometry, so we believe our focused challenge on well-defined areas is an informative first step necessary before extending the idea to the whole of the white matter, which would make for an extremely complex challenge. We note, however, recent work by Ghosh et al. 66 that illustrates such an approach with Human Connectome Project (HCP) data. We focused here on comparing models based on their ability to predict unseen data. Although models that reflect true underlying tissue structure should explain the data well, we cannot infer in general that models that predict unseen data better are mechanistically closer to the tissue than those that do not. As we discuss in the Introduction, the main power of evaluating models in terms of prediction error is to reject models that cannot explain the data. Thus, while the identification of parsimonious models that explain the data certainly has great benefit, further validation is necessary through comparison of the parameters that they estimate with independent measurements, e.g. obtained through microscopy (our challenge makes no attempt to assess the integrity of parameter estimates themselves, but future challenges might use such performance criteria). Models can be evaluated to some extent by sanity checking the realism of their fitted parameter values, as in for example Jelescu et al. 64 or Burcaw et al. 67 However, obtaining accurate ground-truth values for quantitative evaluation remains a hard and yet unsolved problem for diffusion MRI in general. In particular, histology can only roughly approximate the in vivo ground truth and introduces its own set of challenges in sample preparation, acquisition and biophysical interpretation. 12,13,65,[68][69][70][71] This challenge highlights the need for improved model comparison and validation methods. CONCLUSION Challenges such as this have great value in bringing the community together and provide an unbiased comparison of wide-ranging solutions to key data-processing problems. They raise new insights and ideas, motivating more directed future studies. The data are publicly available for others to use, with more details of the dataset given on the Challenge website at http://cmic.cs.ucl.ac.uk/wmmchallenge/. On this website, an up-to-date ranking of the models will be available, where additional models can be added after the publication of the article and where the community will be able to evaluate further the impact of noise correction, compartment-specific T 2 estimation, inter-class model assumptions, e.g. tissue versus signal models, or indeed intra-class model assumptions, e.g. whether cylinders or sticks are optimal models for the given dataset. 42 This will provide an important benchmark for future models and parameter estimation routines. Three modifications were performed to this very general model. First, to accommodate for potential bias in the b 0 images (which was the case for fornix data, where deviations of up to 20 was observed), the prediction for b 0 data was obtained from the median of all signals acquired with identical TE instead of from Equation A2. Second, opposite direction acquisitions were rescaled by a free model parameter, in order to allow for potential gradient instabilities inducing differences between the directions and their opposite directions. Third, models were generated dynamically during fitting by randomly selecting up to four hindered components and up to three restricted components. One isotropic component was always included. The model was first fitted to half of the diffusion-weighted data (randomly selected), after which outliers were rejected (> 2.5 ). Thereafter a second fit was performed. Both fit steps assumed Gaussian noise and utilized the 'lsqcurvefit' function in Matlab. The procedure was repeated 100 times for different randomly generated models. To prepare for submission of the results, only the models that best predicted the hidden half of the data were selected, after which the median of the selected predictions was used for the final prediction. A.1.3 Scherrer (Harvard, USA): distribution of anisotropic microstructural environments in diffusion compartment imaging (DIAMOND) DIAMOND models the set of tissue compartments in each voxel by a finite sum of unimodal continuous distributions of diffusion tensors. This corresponds to a hybrid tissue model that combines biophysical and statistical modelling. As described by Scherrer et al., 16 the DW signal S k for a gradient vector g k and b value b k is modelled by S k = S 0 [ N ∑ j=0 f j ( 1 + b k g T k D 0 j g k j ) − j ] where S 0 is the non-attenuated signal, N is the number of compartments, f j the relative fraction of occupancy of the jth compartment and j and D 0 j are respectively the concentration and expectation of the jth continuous tensor distribution. DIAMOND enables assessment of compartment-specific diffusion characteristics such as the compartment FA (cFA), compartment RD (cRD) and compartment MD (cMD). It also provides a novel measure of microstructural heterogeneity for each compartment. The estimation of a continuous distribution of diffusion tensors requires DW data acquired with the same timing parameters and Δ. 16 To compare DIAMOND with other models in this dataset, we fitted one DIAMOND model separately for each { , Δ} group (i.e. for each TE group), leading to 12 DIAMOND models. One shell was missing in each TE group; we predicted its signal using the corresponding DIAMOND model. The model estimation was achieved as follows. We first computed the mean and standard deviation of S 0 ( S 0 and S 0 ) within each TE group and discarded DW signals with intensity larger than S 0 + 3 S 0 (simple artefact correction). We then estimated DIAMOND parameters as described in Scherrer et al., 16 considering Gaussian noise and cylindrical anisotropic compartments. For the genu, we considered a model with one freely diffusing and one anisotropic compartment; for the fornix, we considered a model with one freely diffusing compartment and two anisotropic compartments. A.1.4 Ferizi_1 and Ferizi_2 (UCL, England) This submission uses two three-compartment models, as described in previous studies. 39,41 These models consist of (1) either a Bingham distribution of sticks or a cylinder for the intracellular compartment; (2) a diffusion tensor for the extracellular compartment; (3) an isotropic cerebrospinal fluid (CSF) compartment. The T 2 relaxation element is fitted beforehand to the (variable echo time) b = 0 measurements. The signal model is as follows: S =S 0 [ f i exp ( − TE T i 2 ) S i + f e exp ( − TE T e 2 ) S e + f c exp ( − TE T c 2 ) S c ](A3) where f i , f e and f c are the weights of the intracellular, extracellular and third normalized compartment signals S i , S e and S c , respectively; the values of compartmental T 2 are indexed similarly;S 0 is the proton density signal (which is TE-independent and obtained from fitting to the b = 0 signal). These models emerged from previous studies. 41,44 Here, however, a single white matter T 2 and separate compartmental diffusivities are additionally fitted. There is a two-stage model fitting procedure. The first step estimates the T 2 decay rate of tissue separately in each voxel, by fitting a bi-exponential A.1.5 Poot (Erasmus, the Netherlands) This submission uses a three-compartment model, with for each compartment a different complexity of the diffusion model and an individual T 2 value. This model was developed specifically for the ISBI WM challenge and is the result of iteratively visualizing different projections of the residuals and trying to infer the maximum complexity that the rich data supports. The first compartment models isotropic diffusion and, through the initialization procedure, it captures the fast diffusion components. The second compartment is modelled by a second-order (diffusion) tensor and models intermediate diffusion strengths. The third compartment is model-free, as the ADC is estimated for each direction independently. Each compartment additionally has an individual T 2 value and signal intensity at b = 0, TE = 0 (which could easily be translated into volume fractions). Hence, the complete model of a voxel in image j is given by S j ( ) = 3 ∑ i=1 A i e −TE j R 2,i e −b j ADC j,i = 3 ∑ i=1 e M i,j(A4) where S j is the predicted signal intensity of image j, A i is the non-diffusion weighted signal intensity of compartment i at zero TE, TE is the echo time, R 2 is the reciprocal of the T 2 relaxation time of compartment i, b = (Δ− ∕3) 2 |G| 2 2 , with = 42.5781 MHz/T, ADC j,1 = c, ADC j,2 = g T j Dg j , ADC j,3 = dh T j , where d is a vector with the ADC value of each orientation group and h j is a vector that selects the orientation group to which image j belongs (90 groups in total). Note that h j has at most one non-zero element and that element has a value of one. As displayed in the rightmost part of Equation value on the signal. These are fitted as a log(TE)-dependent decay with a low-order polynomial function and a b-value-dependent multi-exponential decay (also including an offset to account for the Rician noise floor). The residuals from the isotropic component are then deconvolved with the perturbations in the signal due to a set of fascicle kernels, each modelled as a radially symmetric ( 2 = 3 ) diffusion tensor. The putative kernels are distributed in a dense sampling grid on the sphere. Furthermore, restriction spectrum imaging (RSI) 60 which applies a tunable combination of L1 and L2 regularization on the weights of the fascicle kernels. We used elements of the SFM implemented in the dipy software library 80 and the EN implemented in scikit-learn. 81 In addition, to account for differences in SNR, we implemented a weighted least-squares strategy, whereby each signal's contribution to the fit was weighted by its TE, as well as the gradient strength used. EN has two tuning parameters, determining (1) the ratio of L1-to-L2 regularization and (2) the weight of the regularization relative to the least-squares fit to the signal. To find the proper values of these parameters, we employed k-fold cross-validation, 43 leaving out one shell of measurement in each iteration for cross-validation. We determined that the tuning parameters with the lowest Least Squares Error (LSE) 18 provide an almost-even balance of L1 and L2 penalty with weak overall regularization. Because of the combination of a dense sampling grid (362 points distributed on the sphere) and multiple restriction kernels (45 per sampling point), the maximal number of parameters for the model is approximately 16 300, more than the number of data points. However, because regularization is employed, the effective number of parameters is much smaller, resulting in an active set of approximately 20 regressors. 82 We have made the code to reproduce our results fully available at https://arokem.github.io/ISBI2015. A.1.7 Eufracio (CIMAT, Mexico): diffusion basis functions for multi-shell scheme This model is based on the Diffusion Basis Functions (DBF) model, 74 a discrete version of the Gaussian Mixture Model for the sphere: ŝ i = ∑ m j=1 j ij + , withŝ i = s i ∕s 0 , ij = exp ( − bq T i T j q i ) and T j = ( 1 v j v T j + 2 I). The DBF model is reformulated by substituting ij and T j : ŝ i = ∑ m j=1 j exp ( −b i 2 g T i g i ) exp ( −b i 1 (v T j g i ) 2 ) + . The first exponential can be defined as a scale factor that depends on the b values, i = exp(−b i 2 q T i q i ). In this way, the i factors are associated with different b values, so the new model includes information for multi-shell schemes. The coefficients and the shell scale factor are computed by solving the optimization problem: min , c f( , c ; , ) = ||BΦ − S|| 2 2 + || || 1 + || 0 c − c || 2 2 s.t. 1 T = 1, ⩾ 0 (A5) where B = diag( c ), c = 1 #C ∑ i∈C exp ( − b î2 (q T i q i ) A.1.8 Loya-Olivas_1 and Loya-Olivas_2 (CIMAT, Mexico): Linear Acceleration of Sparse and Adaptive Diffusion Dictionary (LASADD) LASADD is a multi-tensor based technique to adapt dynamically the diffusion functions (DFs) dictionary to a DW-MRI signal. 83,84 The method changes the size and orientation of relevant diffusion tensors (DTs). The optimization algorithm uses a special DT expression and assumptions to reduce the computational cost. The one-compartment version (LASADD-1C) is based on a DBF multi-tensor model: 74 s * i = ∑ n j=1 j i,j , where s * i = s i ∕s 0i , i,j = exp ( −b i g T i T j g i ) , j > 0 and ∑ n j=1 j = 1. LASADD expresses the DT as T j = 1j v j v T j + 2j I,(A6) where {1,2} j are scalars associated with the eigenvalues, v j is the principal diffusion direction (PDD) and I is the identity matrix. [1,9] × 10 −4 mm 2 ∕s and the LASSO regularization parameter (equals 1.7) was tuned by hand such as to provide the minimum error. The best multi-tensorial model for both algorithms was used for each voxel to predict the corresponding unseen data. A.2 Signal models A.2.1 Alipoor (Chalmers, Sweden) The diffsuion MRI signal is modelled as a fourth-order symmetric tensor as proposed by Özarslan and Mareci. 89 where S(g i ) is the measured signal when the diffusion sensitizing gradient is applied in the direction g i , S 0 is the observed signal in the absence of such a gradient, b is the diffusion-weighting factor and t ∈ R 15 contains the distinct entries of a fourth-order symmetric tensor. Note that d(g i , t) = d(g i ) is used for simplification. Given measurements in N > 15 different directions, the least-squares (LS) estimate of the diffusion tensor ist = (G T G) −1 G T y, where G is an N × 15 matrix defined by G = [a 1 a 2 · · · a N ] T and y i = −b −1 ln(S(g i )∕S 0 ). We use the weighted LS tensor estimation method in Alipoor et al. 90 to mitigate the influence of outliers. To estimate the diffusion signal for a given acquisition protocol with TE = TE x , b = b x and = x , the two non-diffusion-weighted measurements with the closest TE to TE x (among measurements with = x ) are used to estimate T 2 and S 0 for each voxel. Then, data from the closest shell to b x (among shells with = x ) are used to estimate the tensor describing the underlying structure. A.2.2 Sakaie-Tatsuoka-Ghosh (Cleveland, USA): an empirical approach As the extent of q space in the dataset is unusually comprehensive, we chose a simple, generic approach to gain intuition. Visual inspection suggested use of a restricted and hindered component, each with angular variation: S i = A TE i (fR i + (1 − f) exp(−b i D i )) (A8) where S i is the predicted signal for a signal acquired with TE i , b i . A TE i is the median signal at a given TE with no diffusion weighting. Fitted parameters are f, the volume fraction of R i , the restricted component, and D i , the diffusivity. R i and D i are modelled as spherical harmonics with real, antipodal symmetry 91 with maximum degree 4. The model has 31 fit parameters for each voxel. Data were fitted using using a non-linear least-squares algorithm (lsqcurvefit, MATLAB). Prior to the fit, data points with non-zero bvalue that had a signal higher than the median of the b = 0 signal plus 1.4826 times the median absolute deviation were excluded. Shells with a normalized median signal smaller than that of shells with lower bvalues were also excluded. Normalization was performed by dividing by the median of the b = 0 signal with the same TE. A.2.3 Fick (INRIA, France): a spatio-temporal functional basis to represent the diffusion MRI signal We use our recently proposed spatio-temporal (3D+t) functional basis 61 to represent the diffusion MRI signal simultaneously over three-dimensional wave vector q and diffusion time . Based on Callaghan's theoretical model of spatio-temporal diffusion in pores, 92 our basis represents the 3D+t diffusion signal attenuation E(q, ) as a product of a spatial and temporal functional basis as E(q, ) = N max ∑ N=0 ∑ {jlm} O max ∑ o=0 c {jlmo} S jlm (q, u s )T o ( , u t )(A9) where T o is our temporal basis with basis order o and S jlm is the spatial isotropic MAP-MRI basis 29 with radial and angular basis orders j, l and m. Here, N max and O max are the maximum spatial and temporal order of the bases, which can be chosen independently. We formulate the bases themselves as S jlm (q, u s ) = √ 4πi −l ( 2π 2 u 2 s q 2 ) l∕2 e −2π 2 u 2 s q 2 × L l+1∕2 j−1 ( 4π 2 u 2 s q 2 ) Y m l (u) T o ( , u t ) = exp(−u t ∕2)L o (u t )(A10) with u s and u t the spatial and temporal scaling functions, Y m l the spherical harmonics and L o a Laguerre polynomial. We calculate the spatial scaling u s by fitting an isotropic tensor to the TE-normalized signal attenuation E(q, ·) for all q. Similarly, we compute u t by fitting an exponential e −u t ∕2 to E(·, ) FIGURE 2 2FIGURE 2 Diffusion-weighted signal from the genu ROI, averaged over the six voxels. Across each column and row, the signal pertains to one of the gradient strengths or pulse times used; in each subplot, the six shells shown in different colours are Δ-specific, increasing in value (22, 40, 60, 80, 100, 120 ms) from top to bottom. Inside the legend, the b value is in s/mm 2 units; here, the HARDI shells kept for testing are those marked with a star; the remaining shells comprise the training data. On the x-axis is the cosine of the angle between the applied diffusion gradient vector G and the fibre direction n. Some models in this study omit data outliers; two such data points are shown in the bottom-left subplot with vertical arrows -obviously each model has its own criteria for determining the outliers FIGURE 3 3Diffusion-weighted signal from the fornix ROI, averaged over the six voxels. The legend's b value is in s/mm 2 units. Testing shells are marked with a star. On the x-axis is the cosine of the angle between the applied diffusion gradient vector G and the fibre direction n Figure 4 4also details the prediction error for different ranges of b values in the unseen dataset. Models inevitably vary in their prediction capa- surement parameters -exploring the boundaries of possible future quantitative diffusion MR techniques. The 14 challenge entries were a good representation of the many available models that are proposed in the literature. The acquired data aimed to cover the broadest spectrum of experimental parameters possible. The participating models use a variety of fitting routines and modelling assumptions, providing additional insight into the effects of algorithmic and modelling choices during parameter estimation. Although the set of methods tested is not sufficient to make a full FIGURE 4 4Overall ranking of models by sum-of-squared-errors (SSE) metric over all voxels in genu (top) and fornix (bottom) ROIs. The colors represent different ranges of b-value shells comparison of each independent feature (diffusion model, noise model, fitting routine, etc.) and the number of combinations prohibits an exhaustive comparison, the results of the challenge do reveal some important trends. In contrast with earlier model comparisons, 18,43,44 the results provide new insight into which broad classes of model explain the signal best and what features of the estimation procedure are important. This information is very timely, as recent model-based diffusion MRI techniques, such as NODDI, 15 SMT, 17,40 DIAMOND, 16 DKI 62 and LEMONADE, 63 are starting to become widely adopted in clinical studies and trials. FIGURE 5 5Sum-of-squared-errors (SSE) per voxel for each model in genu and fornix. The size of rectangles represent the SSE value per voxel for some steps towards generalization). Many of the tissue models, on the other hand, naturally account for finite , varying diffusion times and gradient strength (e.g. the Ramirez-Manzanares, Nilsson and Ferizi models in our collection). We cannot draw any conclusion about the benefits of an adjustable number of parameters in a model, because of the limited number of models in our study that do this and because the models differ in a range of other aspects.The second insight concerns the choice of noise modelling. Despite the fact that SNR at b = 0 and TE = 152 ms falls to about 6, use of the Rician noise model does not appear to be a significant benefit in predicting unseen signal; here, however, we do not investigate the effect on estimated FIGURE 6 6Genu signal for the group consisting of the best seven from 14 models. We show only four (of twelve) representative shells; these are shown by blue stars, while red circles denote the model-predicted data. The best models are listed first. The x-axis is the cosine of the angle between G and n model parameters, which may still benefit from the more accurate noise model. In this challenge, most participants used non-linear least-squares or maximum-likelihood optimization. Additional regularization of the objective function (Eufracio & Rivera/Lasso, Rokem/Elastic Net, Fick/Laplacian) appeared to have little benefit over non-regularized optimization. FIGURE 9 9FIGURE 9 Fornix signal for the second group of 14 models. Raw testing data are shown by blue stars, while red circles denote the model-predicted data. The x-axis is the cosine of the angle between G and n B = b⃗ n ⊗2 and b = ( g) 2 t d . The diffusion time t d was corrected for rise times ( ) according to t d = Δ− ∕3 + 3 ∕30 2 − 2 ∕6 . Each component was also described by a weight (w i ) and relaxation times (T1 i and T2 i ). The model featured three types of component, with either isotropic, hindered or restricted diffusion. Diffusion in the isotropic component was modelled by a single diffusion coefficient. The hindered and restricted components were modelled by cylinder-symmetric tensors described by axial and radial diffusivities together with the polar and azimuth angles. In the restricted component, the apparent diffusion coefficient of the radial component depended on and Δ, as well as on the cylinder radius, according to van Gelderen et al.78 model to the b = 0 intensity as a function of TE, in which one component is from tissue and the other from CSF. A preliminary analysis of voxels fully inside WM regions shows no significant departure from mono-exponential decay; equal T 2 values are then assumed within the intra and extracellular compartments. When fitting the bi-exponential model, the value of T 2 in CSF is fixed to 1000 ms (a more precise value of CSF is unlikely to be estimated with this protocol). Thus, for each voxel, the volume fraction of CSF,S 0 and T 2 of the tissue are estimated. These three estimates are then fixed for all the subsequent model fits. Then, each model is fitted using the Levenberg-Marquardt algorithm with an offset Gaussian noise model. The model parameters obtained were similar to earlier estimates obtained using the full dataset, 42 differing by between 5-10% from the original. A4, the model can be written as a multiplication of matrices M i , containing all rows M i,j , with = [ln A 1 , R 2,1 , c, ln A 2 , R 2,2 , D 11 , D 12 , D 13 , D 22 , D 23 , D 33 , ln A 3 ,R 2,3 , d]T , which combines all 103 parameters into a single parameter vector. All parameters are estimated simultaneously from the 3311 measurements provided per voxel by a maximum-likelihood estimator that assumes a Rician distribution of the measurements and simultaneously optimizes the noise level.59 Finally, the signal intensities of the 'unseen' data are predicted by substituting the estimate into Equation A4.A.1.6 Rokem (Standford, USA): a restriction-spectrum sparse fascicle model (RS-SFM)The sparse fascicle model (SFM)43 is a member of the large family of models that account for the diffusion MRI signal in the white matter as a combination of signals due to compartments corresponding to different axonal fibre populations (fascicles) and other parts of the tissue. Model fitting proceeds in two steps. First, an isotropic component is fitted. We model the effects of both the measurement echo time (TE) and the measurement b ) and C is the set of indices grouped by different b values (#C is the number of elements in it). The regularization term weighted by demands sparseness and the term weighted by prevents overfitting. The problem in Equation A5 is solved in three steps. First, the active atoms are predicted ( i > 0) with̃= argmin f( , c ; , ). Second, the active atoms are corrected with = argmin { i }∶̃i>0 f( , c ; 0, ). Finally, the factors c are updated with c = argmin c f( , c ; , ). To solve each step, the active sets algorithm for quadratic programming is used. To train the model for the WMM'15 data, Equation A5 is solved for each voxel with the training data to find the optimal weights j and scale factors c that best reproduce the training data. For this challenge, the c factors are grouped by the 36 training shells and the method parameters are set by hand: = 0.5, = 0.02, 1 = 9.5 × 10 −4 and 2 = 5 × 10 −5 . To predict the unseen signal at each voxel, the reformulated model is used with the optimal weights j and the 12 scale factors for the unseen c are calculated by interpolation with the 36 optimal c of the training data. Let g i = [x i y i z i ] and a i ] T be a gradient encoding direction and the corresponding design vector, respectively. The diffusion signal is then described byS(g i ) = S 0 exp ( −TE T 2 ) exp(−bt T a i ) Abbreviations used: CT, computerized tomography; CV, cross-validation; DBF, diffusion basis function; DF, diffusion function; DT, diffusion tensor; DTI, diffusion tensor imaging; DW, diffusion-weighted; EN, elastic net; ISBI, International Symposium on Biomedical Imaging; LASADD, Linear Acceleration of Sparse and Adaptive Diffusion Dictionary; LS, least-squares; MRI, magnetic resonance imaging; PDD, principal diffusion directions; RSI, restriction spectrum imaging; ROI, region of interest; SFM, sparse fascicle model; SNR, signal-to-noise ratio; SSE, sum of squared errors; TE, echo time; WM, white matter. TABLE 1 The 1scanning protocol used, acquired in ∼8 hours over two non-stop sessions. The protocol has 48 shells, each with 45 unique gradient directions ('blip-up-blip-down') Acquisition Protocol = 3ms = 8ms Δ TE |G| b Δ TE |G| b Nr (ms) (ms) (mT/m) (s/mm 2 ) Nr (ms) (ms) (mT/m) (s/mm 2 ) 1 22 49 61 50 25 22 58 58 300 2 22 49 86 100 26 22 58 95 800 3 22 49 192 500 27 22 58 190 3,200 4 22 49 285 1,100 28 22 58 275 6,700 5 40 67 63 100 29 40 72 59 600 6 40 67 100 250 30 40 72 100 1,700 7 40 67 200 1,000 31 40 72 200 6,850 8 40 67 289 2,100 32 40 72 292 14,550 9 60 87 63 150 33 60 92 34 300 10 60 87 103 400 34 60 92 100 2,650 11 60 87 199 1,500 35 60 92 200 10,500 12 60 87 290 3,200 36 60 92 292 22,350 13 80 107 63 200 37 80 112 61 1,300 14 80 107 99 500 38 80 112 100 3,550 15 80 107 201 2,050 39 80 112 200 14,150 16 80 107 291 4,300 40 80 112 292 30,200 17 100 127 63 250 41 100 132 60 1,600 18 100 127 101 650 42 100 132 100 4,450 19 100 127 200 2,550 43 100 132 200 17,850 20 100 127 291 5,400 44 100 132 292 38,050 21 120 147 63 300 45 120 152 60 1,950 22 120 147 99 750 46 120 152 100 5,350 23 120 147 199 3,050 47 120 152 200 21,500 24 120 147 291 6,500 48 120 152 292 45,900 TABLE 2 2Summary of the various diffusion models evaluated. Tissue models are models that include an explicit description of the underlying tissue microstructure with a multi-compartment approach. In contrast, signal models focus on describing the DW signal attenuation without explicitly describing the underlying tissue and instead correspond to a 'signal processing' approach Abbreviations: LS=least-squares, LM=Levenberg-Marquardt, CV=cross-validation, reg=regularized similarly in terms of ranking in both genu and fornix cases, i.e. Nilsson (second in genu/first in fornix), Scherrer (third/second) and Ferizi_2 (fourth/fourth). Others performed significantly better in one of the cases, with Ramirez-Manzanares (first/sixth) being the most notable.Type of Nb of free param. Models effect Noise Optimization Outliers Special signal model (genu/fornix) of and Δ assumption algorithm strategy prediction strategy R-Manzanares Tissue N/A Yes Gaussian weighted-LS Yes CV bootstrapping Nilsson Tissue < 12/12 Yes Gaussian LM Yes CV Scherrer Tissue 10/16 No Gaussian Bobyqa Yes No Ferizi_1 Tissue < 12/12 Yes approx.-Rician LM No No Ferizi_2 Tissue < 10/10 Yes approx.-Rician LM No No Alipoor Signal 17/17 No Gaussian weighted-LS Yes No Sakaie Signal N/A No Gaussian nonlinear-LS Yes No Rokem Tissue ∼20 No Gaussian Elastic net No CV + Noise floor Eufracio Tissue 7/7 No Gaussian bounded-LS No No Lasso, Ridge Loya-Olivas_1 Tissue 11 No Gaussian bounded-LS No No & Lasso Loya-Olivas_2 Tissue 11 No Gaussian bounded-LS No No Poot Signal 103 No Rician LM-like No No Fick Signal 475 Yes Gaussian Laplacian-reg-LS No partial-CV Rivera Signal 23 Yes Gaussian Weighted Lasso Yes CV is used to extend the model, by adding a range of fascicle kernels at each sampling point, with different axial and radial diffusivities, capturing diffusion at different scales. To restrict the number of anisotropic components (fascicles) in each voxel and to prevent overfitting, the RS-SFM model employs the Elastic N et algorithm (EN),79 The algorithm iterates three steps, like Aranda et al.:85,86 Predict, Correct, and Generate, until convergence. Prediction selects the relevant DFs using LASSO to regulate the number to choose. Correction adjusts the volume fraction, size and orientation of the DTs. Taking advantage of the DT expression and Taylor first-order series approximation of the exponential, the optimizations are reduced to bounded least-squares problems, which are solved by a Projected Gauss-Seidel scheme. Generation controls the overestimation of fibres by adding to the basis the DTs resulting from combining two and three DFs for the new iteration.An extra refinement to the computed results, named LASADD-3C, splits each detected DF into three compartments: 87 intracellular (IC), extracellular (EC) and CSF. The multi-tensor model is s * i = CSF = 1. The i,j models the directional IC compartment diffusion for each fibre bundle using T IC j = 0j v j v T j . The EC compartment with hindered diffusion uses the representation (A6) for i,j . The isotropic diffusion i uses T CSF = 3 I. This stage keeps the PDDs fixed and only adjusts the 's and 's of the three compartments.The parameters of the models were estimated using the training dataset: the b values using the equation byStejskal and Tanner 88 and the S 0 values as the median of the gradient-free signals with equal echo time per voxel. The initial basis comprises 33 PDDs distributed in the unitary sphere. The bounds {0,1} ∈ [1, 39] × 10 −4 and {2,3} ∈∑ n j=1 IC j i,j + ∑ n j=1 EC j i,j + CSF i with ∑ n j=1 ( IC j + EC j ) + ACKNOWLEDGEMENTSResearch reported in this manuscript was supported by the following organizations. EPSRC supported this work through grants EP/G007748, EP/L022680/1, EP/I027084/01, EP/M020533/1 and EP/N018702/1. U. The content is solely the responsibility of the authors and does not necessarily represent the official views of the funding bodies (EPSRC or NIH).Lastly, we thank the ISBI 2015 challenge organizers, in particular Stephen R. Aylward (Kitware Inc., USA) and Badri Roysam (University of Houston, USA).The PDDs are estimated from the Diffusion Tensor (DT) (single bundle case) and DBF74(complex structure cases). The 120 orientations closest to the PDDs are selected from a set of 1000 evenly distributed orientations. The intra-axonal signals S i are precomputed from the model in Van Gelderen et al.75for restricted diffusion within a cylinder with radius R = 1, 2, … , 10 m and parallel diffusion d || = 1, 1.1, … , 2.1 m 2 /ms. The extra-axonal signals are generated as follows: S e from zeppelins with combinations of parallel and radial diffusion, d || = 1, 1.1, … , 2.5 m 2 /ms and d ⟂ = 2, 3, … , 8 m 2 /ms; isotropic diffusion compartment signals S iso i = exp −q d iso for d iso = 2, 2.1, 2.2, … , 4 m 2 /ms; and the dot signal, which takes into account static proton density. The values of the dictionary atoms above were tuned by cross-validation.76The size compartments ⩾ 0 computed in the weighted non-negative LS formulation,indicate the atoms that explain the signal; the W weights are proportional to SNR. Overfitting is reduced by a bootstrap 77 procedure.The cross-validation experiments indicate that the reconstructions given by the robust fitting of this rich multi-compartment diffusion dictionary allow us to predict accurately non-acquired MR signals for different machine protocols. This is of most interest in the development of methods able to detect the complex microstructure heterogeneity associated with the different compartments within the voxels. The atoms with coefficients > 0 depict the empirical distributions and their orientations indicate non-parametrical bundle-dispersion configurations (such as fanning or radially symmetrical). The recovered distributions reveal, for instance, an axon radius of between 1 and 4 m. One should take into account, however, that, since the heterogeneous intra/extra-axonal T 2 relaxation feature is not modelled explicitly, the method may compensate for T 2 variations by using, for instance, large isotropic d iso coefficients to fit the signal accurately. For this reason, a direct interpretation of the fitted parameters may be misleading. The use of more specific models is a part of ongoing work.A.1.2 Nilsson (Lund, Sweden): multi-compartment model outlier rejection and separate fitting of b 0 dataThis multiple compartment model was developed specifically for the ISBI WM challenge and built up by relaxation-weighted and time-dependent diffusion tensors according toA.2.4 Rivera (CIMAT, Mexico): baseline method: robust regressionWe regard this very simplistic model as a baseline for other model-based methods. It assumes as little information as possible from the diffusion signal. The vector of independent variables iscontaining the gradient strength g, the echo time TE and the b value b. Given signal s i , we then estimate the parameters of the linear regression model:where ∈ R 23 is the unknown vector of coefficients, is the residual error andis the matrix design (x| 2 is obtained from squaring each element of the matrix x). To account for outliers, we estimate with a weighted (robust) least-squares approach using the Lasso regularization:where W 0 is the identity matrix and each subsequent W is computed viawith outlier weighting in t+1through , an arbitrary parameter that controls the outlier sensitivity. The protocol weight v t+1computes a confidence factor for the complete protocol.Equations A12 and A13 are iterated three times. The final estimated signal is computed using (A11), using the protocol of the unseen signal. Estimation of the effective self-diffusion tensor from the NMR spin echo. P J Basser, J Mattiello, D Lebihan, J Magn Reson, Ser B. 1033Basser PJ, Mattiello J, LeBihan D. Estimation of the effective self-diffusion tensor from the NMR spin echo. J Magn Reson, Ser B. 1994;103(3):247-254. Diffusion tensor imaging (DTI)-based white matter mapping in brain research: A review. Y Assaf, O Pasternak, J Mol Neurosci: MN. 341Assaf Y, Pasternak O. Diffusion tensor imaging (DTI)-based white matter mapping in brain research: A review. J Mol Neurosci: MN. 2008;34(1):51-61. Loss of connectivity in Alzheimer's disease: An evaluation of white matter tract integrity with colour coded MR diffusion tensor imaging. S E Rose, F Chen, J B Chalk, J Neurol, Neurosurg Psychiatry. 694Rose SE, Chen F, Chalk JB, et al. Loss of connectivity in Alzheimer's disease: An evaluation of white matter tract integrity with colour coded MR diffusion tensor imaging. J Neurol, Neurosurg Psychiatry. 2000;69(4):528-530. The pathogenesis of lesions and normal-appearing white matter changes in multiple sclerosis. D Werring, D Brassat, A Droogan, Brain. 1238Werring D, Brassat D, Droogan A, et al. The pathogenesis of lesions and normal-appearing white matter changes in multiple sclerosis. Brain. 2000;123(8):1667-1676. Improved delineation of glioma margins and regions of infiltration with the use of diffusion tensor imaging: An image-guided biopsy study. S Price, Jena R Burnet, N , Amer J Neuroradiol. 279Price S, Jena R, Burnet N, et al. Improved delineation of glioma margins and regions of infiltration with the use of diffusion tensor imaging: An image-guided biopsy study. Amer J Neuroradiol. 2006;27(9):1969-1974. Diffusion propagator as a probe of the structure of porous media. P P Mitra, P N Sen, L M Schwartz, Le Doussal, P , Phys Rev Lett. 68243555Mitra PP, Sen PN, Schwartz LM, Le Doussal P. Diffusion propagator as a probe of the structure of porous media. Phys Rev Lett. 1992;68(24):3555. Characterization and propagation of uncertainty in diffusion-weighted MR imaging. T Behrens, M Woolrich, M Jenkinson, Magn Reson Med. 505Behrens T, Woolrich M, Jenkinson M, et al. Characterization and propagation of uncertainty in diffusion-weighted MR imaging. Magn Reson Med. 2003;50(5):1077-1088. Probabilistic diffusion tractography with multiple fibre orientations: What can we gain?. T Behrens, H J Berg, S Jbabdi, M Rushworth, M Woolrich, NeuroImage. 341Behrens T, Berg HJ, Jbabdi S, Rushworth M, Woolrich M. Probabilistic diffusion tractography with multiple fibre orientations: What can we gain?NeuroImage. 2007;34(1):144-155. Composite hindered and restricted model of diffusion (CHARMED) MR imaging of the human brain. Y Assaf, P J Basser, NeuroImage. 271Assaf Y, Basser PJ. Composite hindered and restricted model of diffusion (CHARMED) MR imaging of the human brain. NeuroImage. 2005;27(1):48-58. Modeling dendrite density from magnetic resonance diffusion measurements. S N Jespersen, C D Kroenke, L Østergaard, Jjh Ackerman, D A Yablonskiy, Neuroimage. 344Jespersen SN, Kroenke CD, Østergaard L, Ackerman JJH, Yablonskiy DA, Modeling dendrite density from magnetic resonance diffusion measurements. Neuroimage. 2007;34(4):1473-1486. Neurite density from magnetic resonance diffusion measurements at ultrahigh field: Comparison with light microscopy and electron microscopy. S N Jespersen, C R Bjarkam, J R Nyengaard, Neuroimage. 491Jespersen SN, Bjarkam CR, Nyengaard JR, et al. Neurite density from magnetic resonance diffusion measurements at ultrahigh field: Comparison with light microscopy and electron microscopy. Neuroimage. 2010;49(1):205-216. Orientationally invariant indices of axon diameter and density from diffusion MRI. D C Alexander, P L Hubbard, M G Hall, NeuroImage. 524Alexander DC, Hubbard PL, Hall MG, et al. Orientationally invariant indices of axon diameter and density from diffusion MRI. NeuroImage. 2010;52(4):1374-1389. Contrast and stability of the axon diameter index from microstructure imaging with diffusion MRI. T B Dyrby, M G Hall, M Ptito, Magn Reson Med. 703Dyrby TB, Hall MG, Ptito M, et al. Contrast and stability of the axon diameter index from microstructure imaging with diffusion MRI. Magn Reson Med. 2013;70(3):711-721. Ball and rackets: Inferring fiber fanning from diffusion-weighted MRI. S N Sotiropoulos, T E Behrens, S Jbabdi, NeuroImage. 602Sotiropoulos SN, Behrens TE, Jbabdi S. Ball and rackets: Inferring fiber fanning from diffusion-weighted MRI. NeuroImage. 2012;60(2):1412-1425. NODDI: Practical in vivo neurite orientation dispersion and density imaging of the human brain. H Zhang, T Schneider, C A Wheeler-Kingshott, Alexander Dc, NeuroImage. 614Zhang H, Schneider T, Wheeler-Kingshott CA, Alexander DC. NODDI: Practical in vivo neurite orientation dispersion and density imaging of the human brain. NeuroImage. 2012;61(4):1000-1016. Characterizing brain tissue by assessment of the distribution of anisotropic microstructural environments in diffusion-compartment imaging (DIAMOND). B Scherrer, A Schwartzman, M Taquet, M Sahin, S P Prabhu, S K Warfield, 10.1002/mrm.25912Magn Reson Med. 763Scherrer B, Schwartzman A, Taquet M, Sahin M, Prabhu SP, Warfield SK. Characterizing brain tissue by assessment of the distribution of anisotropic microstructural environments in diffusion-compartment imaging (DIAMOND). Magn Reson Med. 2016;76(3):963-977. https://doi.org/10.1002/mrm. 25912 Multi-compartment microscopic diffusion imaging. E Kaden, N D Kelm, R P Carson, M D Does, D C Alexander, NeuroImage. 139Kaden E, Kelm ND, Carson RP, Does MD, Alexander DC. Multi-compartment microscopic diffusion imaging. NeuroImage. 2016;139:346-359. Compartment models of the diffusion MR signal in brain white matter: A taxonomy and comparison. E Panagiotaki, T Schneider, B Siow, M G Hall, M F Lythgoe, D C Alexander, NeuroImage. 593Panagiotaki E, Schneider T, Siow B, Hall MG, Lythgoe MF, Alexander DC. Compartment models of the diffusion MR signal in brain white matter: A taxonomy and comparison. NeuroImage. 2012;59(3):2241-2254. An analytical model of restricted diffusion in bovine optic nerve. G J Stanisz, G A Wright, R M Henkelman, A Szafer, Magn Reson Med. 371Stanisz GJ, Wright GA, Henkelman RM, Szafer A. An analytical model of restricted diffusion in bovine optic nerve. Magn Reson Med. 1997;37(1):103-111. The role of tissue microstructure and water exchange in biophysical modelling of diffusion in white matter. M Nilsson, D Van Westen, F Ståhlberg, P C Sundgren, J Lätt, Magn Reson Mater Phys, Biol Med. 264Nilsson M, van Westen D, Ståhlberg F, Sundgren PC, Lätt J. The role of tissue microstructure and water exchange in biophysical modelling of diffusion in white matter. Magn Reson Mater Phys, Biol Med. 2013;26(4):345-370. Generalized diffusion tensor imaging (GDTI): A method for characterizing and imaging diffusion anisotropy caused by non-Gaussian diffusion. C Liu, R Bammer, M E Moseley, Isr J Chem. 431-2Liu C, Bammer R, Moseley ME. Generalized diffusion tensor imaging (GDTI): A method for characterizing and imaging diffusion anisotropy caused by non-Gaussian diffusion. Isr J Chem. 2003;43(1-2):145-154. Q-ball imaging. D S Tuch, Magn Reson Med. 526Tuch DS. Q-ball imaging. Magn Reson Med. 2004;52(6):1358-1372. Diffusional kurtosis imaging: The quantification of non-Gaussian water diffusion by means of magnetic resonance imaging. J H Jensen, J A Helpern, A Ramani, H Lu, K Kaczynski, Magn Reson Med. 536Jensen JH, Helpern JA, Ramani A, Lu H, Kaczynski K. Diffusional kurtosis imaging: The quantification of non-Gaussian water diffusion by means of magnetic resonance imaging. Magn Reson Med. 2005;53(6):1432-1440. Regularized, fast, and robust analytical Q-ball imaging. M Descoteaux, E Angelino, S Fitzgibbons, R Deriche, Magn Reson Med. 583Descoteaux M, Angelino E, Fitzgibbons S, Deriche R. Regularized, fast, and robust analytical Q-ball imaging. Magn Reson Med. 2007;58(3):497-510. Efficient and robust computation of pdf features from diffusion MR signal. H E Assemlal, D Tschumperlé, L Brun, Med Image Anal. 135Assemlal HE, Tschumperlé D, Brun L. Efficient and robust computation of pdf features from diffusion MR signal. Med Image Anal. 2009;13(5):715-729. Theoretical models of the diffusion weighted MR signal. D A Yablonskiy, A L Sukstanskii, NMR Biomed. 237Yablonskiy DA, Sukstanskii AL. Theoretical models of the diffusion weighted MR signal. NMR Biomed. 2010;23(7):661-681. The cumulant expansion: An overarching mathematical framework for understanding diffusion NMR. V G Kiselev, Derek KJ, ed. Diffusion MRI: Theory, Methods, and Applications. Kiselev VG. The cumulant expansion: An overarching mathematical framework for understanding diffusion NMR. In: Derek KJ, ed. Diffusion MRI: Theory, Methods, and Applications. OUP USA; 2011:152-168. Simple harmonic oscillator based estimation and reconstruction for one-dimensional q-space MR. E Özarslan, C Koay, P Basser, Proc. Intl. Soc. Mag. Reson. Med. 1635Özarslan E, Koay C, Basser P. Simple harmonic oscillator based estimation and reconstruction for one-dimensional q-space MR. In: Proc. Intl. Soc. Mag. Reson. Med, Vol. 16; 2008:35. Mean apparent propagator (MAP) MRI: A novel diffusion imaging method for mapping tissue microstructure. E Özarslan, C G Koay, T M Shepherd, NeuroImage. 78Özarslan E, Koay CG, Shepherd TM, et al. Mean apparent propagator (MAP) MRI: A novel diffusion imaging method for mapping tissue microstructure. NeuroImage. 2013;78:16-32. Direct estimation of the fiber orientation density function from diffusion-weighted MRI data using spherical deconvolution. J D Tournier, F Calamante, D G Gadian, A Connelly, Neuroimage. 233Tournier JD, Calamante F, Gadian DG, Connelly A. Direct estimation of the fiber orientation density function from diffusion-weighted MRI data using spherical deconvolution. Neuroimage. 2004;23(3):1176-1185. Maximum entropy spherical deconvolution for diffusion MRI. D C Alexander, Biennial International Conference on Information Processing in Medical Imaging: 19th International Conference, IPMI 2005. Glenwood Springs, CO, USA. BerlinSpringerAlexander DC. Maximum entropy spherical deconvolution for diffusion MRI. Biennial International Conference on Information Processing in Medical Imaging: 19th International Conference, IPMI 2005. Glenwood Springs, CO, USA. Berlin: Springer; 2005:76-87. Measurement of fiber orientation distributions using high angular resolution diffusion imaging. A W Anderson, Magn Reson Med. 545Anderson AW. Measurement of fiber orientation distributions using high angular resolution diffusion imaging. Magn Reson Med. 2005;54(5):1194-1206. Multi-fiber reconstruction from diffusion MRI using mixture of Wisharts and sparse deconvolution. Information Processing in Medical Imaging. B Jian, B C Vemuri, SpringerBerlinJian B, Vemuri BC. Multi-fiber reconstruction from diffusion MRI using mixture of Wisharts and sparse deconvolution. Information Processing in Medical Imaging. Berlin: Springer; 2007:384-395. A novel tensor distribution model for the diffusion-weighted MR signal. B Jian, B C Vemuri, E Özarslan, P R Carney, T H Mareci, NeuroImage. 371Jian B, Vemuri BC, Özarslan E, Carney PR, Mareci TH. A novel tensor distribution model for the diffusion-weighted MR signal. NeuroImage. 2007;37(1):164-176. An objective method for regularization of fiber orientation distributions derived from diffusion-weighted MRI. K E Sakaie, M J Lowe, NeuroImage. 341Sakaie KE, Lowe MJ. An objective method for regularization of fiber orientation distributions derived from diffusion-weighted MRI. NeuroImage. 2007;34(1):169-176. A model-based deconvolution approach to solve fiber crossing in diffusion-weighted MR imaging. F Dell&apos; Acqua, G Rizzo, P Scifo, R A Clarke, G Scotti, F Fazio, IEEE Trans Biomed Eng. 543Dell' Acqua F, Rizzo G, Scifo P, Clarke RA, Scotti G, Fazio F. A model-based deconvolution approach to solve fiber crossing in diffusion-weighted MR imaging. IEEE Trans Biomed Eng. 2007;54(3):462-472. Model-based analysis of multishell diffusion MR data for tractography: How to get over fitting problems. S Jbabdi, S Sotiropoulos, A Savio, M Graña, T Behrens, Magn Reson Med. 686Jbabdi S, Sotiropoulos S, Savio A, Graña M, Behrens T. Model-based analysis of multishell diffusion MR data for tractography: How to get over fitting problems. Magn Reson Med. 2012;68(6):1846-1855. Multi-shell diffusion signal recovery from sparse measurements. Y Rathi, O Michailovich, F Laun, K Setsompop, P E Grant, C F Westin, Med Image Anal. 187Rathi Y, Michailovich O, Laun F, Setsompop K, Grant PE, Westin CF. Multi-shell diffusion signal recovery from sparse measurements. Med Image Anal. 2014;18(7):1143-1156. A ranking of diffusion MRI compartment models with in vivo human brain data. U Ferizi, T Schneider, E Panagiotaki, Magn Reson Med. 726Ferizi U, Schneider T, Panagiotaki E, et al. A ranking of diffusion MRI compartment models with in vivo human brain data. Magn Reson Med. 2014;72(6):1785-1792. Quantitative mapping of the per-axon diffusion coefficients in brain white matter. E Kaden, F Kruggel, D C Alexander, 10.1002/mrm.25734Magn Reson Med. 754Kaden E, Kruggel F, Alexander DC. Quantitative mapping of the per-axon diffusion coefficients in brain white matter. Magn Reson Med. 2016;75(4):1752-1763. https://doi.org/10.1002/mrm.25734 The importance of being dispersed: A ranking of diffusion MRI models for fibre dispersion using in vivo human brain data. U Ferizi, T Schneider, M Tariq, C Wheeler-Kingshott, H Zhang, Alexander D , Medical Image Computing and Computer-Assisted Intervention -MICCAI. Mori K, Sakuma I, Sato Y, Barillot C, Navab N8149SpringerLecture Notes in Computer ScienceFerizi U, Schneider T, Tariq M, Wheeler-Kingshott C, Zhang H, Alexander D. The importance of being dispersed: A ranking of diffusion MRI models for fibre dispersion using in vivo human brain data. In: Mori K, Sakuma I, Sato Y, Barillot C, Navab N, eds. Medical Image Computing and Computer-Assisted Intervention -MICCAI 2013, Lecture Notes in Computer Science, vol. 8149. Berlin: Springer; 2013:74-81. White matter compartment models for in vivo diffusion MRI at 300 mT/m. U Ferizi, T Schneider, H Zhang, C A Wheeler-Kingshott, Alexander Dc, 10.1016/j.neuroimage.2015.06.027Neuroimage. 118Ferizi U, Schneider T, Zhang H, Wheeler-Kingshott CA, Alexander DC. White matter compartment models for in vivo diffusion MRI at 300 mT/m. Neuroimage. 2015;118:468-483. https://doi.org/10.1016/j.neuroimage.2015.06.027 Evaluating the accuracy of diffusion MRI models in white matter. A Rokem, J D Yeatman, F Pestilli, PLoS One. 104123272Rokem A, Yeatman JD, Pestilli F, et al. Evaluating the accuracy of diffusion MRI models in white matter. PLoS One. 2015;10(4):e0123272. Compartment Models and Model Selection for in-vivo Diffusion-MRI of Human Brain White Matter. U Ferizi, London; University College LondonPhD thesisFerizi U. Compartment Models and Model Selection for in-vivo Diffusion-MRI of Human Brain White Matter [PhD thesis]. London: University College London; 2014. Evaluation of 14 nonlinear deformation algorithms applied to human brain MRI registration. A Klein, J Andersson, B A Ardekani, NeuroImage. 463Klein A, Andersson J, Ardekani BA, et al. Evaluation of 14 nonlinear deformation algorithms applied to human brain MRI registration. NeuroImage. 2009;46(3):786-802. MRBrainS challenge: Online evaluation framework for brain image segmentation in 3T MRI scans. A Mendrik, K Vincken, H Kuijf, 10.1155/2015/813696Comput Intell Neurosci. 813696Mendrik A, Vincken K, Kuijf H, et al. MRBrainS challenge: Online evaluation framework for brain image segmentation in 3T MRI scans. Comput Intell Neurosci. 2015;2015:813696. https://doi.org/10.1155/2015/813696 Quantitative comparison of reconstruction methods for intra-voxel fiber recovery from diffusion MRI. A Daducci, E J Canales-Rodriguez, M Descoteaux, IEEE Trans Med Imaging. 332Daducci A, Canales-Rodriguez EJ, Descoteaux M, et al. Quantitative comparison of reconstruction methods for intra-voxel fiber recovery from diffusion MRI. IEEE Trans Med Imaging. 2014;33(2):384-399. Quantitative evaluation of 10 tractography algorithms on a realistic diffusion MR phantom. P Fillard, M Descoteaux, A Goh, Neuroimage. 561Fillard P, Descoteaux M, Goh A, et al. Quantitative evaluation of 10 tractography algorithms on a realistic diffusion MR phantom. Neuroimage. 2011;56(1):220-234. Sparse reconstruction challenge for diffusion MRI: Validation on a physical phantom to determine which acquisition scheme and analysis method to use?. L Ning, F Laun, Y Gur, Med Image Anal. 261Ning L, Laun F, Gur Y, et al. Sparse reconstruction challenge for diffusion MRI: Validation on a physical phantom to determine which acquisition scheme and analysis method to use? Med Image Anal. 2015;26(1):316-331. The DTI challenge: Toward standardized evaluation of diffusion tensor imaging tractography for neurosurgery. S Pujol, W Wells, C Pierpaoli, J Neuroimaging. 256Pujol S, Wells W, Pierpaoli C, et al. The DTI challenge: Toward standardized evaluation of diffusion tensor imaging tractography for neurosurgery. J Neuroimaging. 2015;25(6):875-882. Pushing the limits of in vivo diffusion MRI for the Human Connectome project. K Setsompop, R Kimmlingen, E Eberlein, NeuroImage. 80Setsompop K, Kimmlingen R, Eberlein E, et al. Pushing the limits of in vivo diffusion MRI for the Human Connectome project. NeuroImage. 2013;80:220-233. The human connectome project and beyond: Initial applications of 300 mT/m gradients. J A Mcnab, B L Edlow, T Witzel, NeuroImage. 80McNab JA, Edlow BL, Witzel T, et al. The human connectome project and beyond: Initial applications of 300 mT/m gradients. NeuroImage. 2013;80:234-245. In vivo estimation of axon diameter in the human spinal cord using 300 mT/m gradients. T Duval, J Mcnab, K Setsompop, International Society of Magnetic Resonance in Medicine (ISMRM) 24th Annual Meeting & Exhibition. Milan, ItalyDuval T, McNab J, Setsompop K, et al. In vivo estimation of axon diameter in the human spinal cord using 300 mT/m gradients. International Society of Magnetic Resonance in Medicine (ISMRM) 24th Annual Meeting & Exhibition. Milan, Italy; 2014. The impact of gradient strength on in vivo diffusion MRI estimates of axon diameter. S Y Huang, A Nummenmaa, T Witzel, NeuroImage. 106Huang SY, Nummenmaa A, Witzel T, et al. The impact of gradient strength on in vivo diffusion MRI estimates of axon diameter. NeuroImage. 2015;106:464-472. Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach. K P Burnham, D R Anderson, SpringerBerlinBurnham KP, Anderson DR. Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach. Berlin: Springer; 2002. Improved optimization for the robust and accurate linear registration and motion correction of brain images. M Jenkinson, P Bannister, M Brady, S Smith, NeuroImage. 172Jenkinson M, Bannister P, Brady M, Smith S. Improved optimization for the robust and accurate linear registration and motion correction of brain images. NeuroImage. 2002;17(2):825-841. Squashing peanuts and smashing pumpkins': How noise distorts diffusion-weighted MR data. D K Jones, P J Basser, Magn Reson Med. 525Jones DK, Basser PJ. Squashing peanuts and smashing pumpkins': How noise distorts diffusion-weighted MR data. Magn Reson Med. 2004;52(5): 979-993. The importance of axonal undulation in diffusion MR measurements: A Monte Carlo simulation study. M Nilsson, J Lätt, F Ståhlberg, D Westen, H Hagslätt, NMR Biomed. 255Nilsson M, Lätt J, Ståhlberg F, Westen D, Hagslätt H. The importance of axonal undulation in diffusion MR measurements: A Monte Carlo simulation study. NMR Biomed. 2012;25(5):795-805. Detecting statistically significant differences in quantitative MRI experiments, applied to diffusion tensor imaging. D Poot, S Klein, IEEE Trans Med Imaging. 345Poot D, Klein S. Detecting statistically significant differences in quantitative MRI experiments, applied to diffusion tensor imaging. IEEE Trans Med Imaging. 2015;34(5):1164-1176. Probing tissue microstructure with restriction spectrum imaging: Histological and theoretical validation. N S White, T B Leergaard, D &apos; Arceuil, H Bjaalie, J G Dale, A M , Hum Brain Mapp. 342White NS, Leergaard TB, D' Arceuil H, Bjaalie JG, Dale AM. Probing tissue microstructure with restriction spectrum imaging: Histological and theoretical validation. Hum Brain Mapp. 2013;34(2):327-346. A unifying framework for spatial and temporal diffusion in diffusion MRI. R Fick, D Wassermann, M Pizzolato, R Deriche, International Conference on Information Processing in Medical Imaging. BerlinSpringer2015Fick R, Wassermann D, Pizzolato M, Deriche R. A unifying framework for spatial and temporal diffusion in diffusion MRI. International Conference on Information Processing in Medical Imaging. Berlin: Springer; 2015:167-178. White matter characterization with diffusional kurtosis imaging. E Fieremans, J H Jensen, J A Helpern, Neuroimage. 581Fieremans E, Jensen JH, Helpern JA. White matter characterization with diffusional kurtosis imaging. Neuroimage. 2011;58(1):177-188. Mapping orientational and microstructural metrics of neuronal integrity with in vivo diffusion MRI. D S Novikov, J Veraart, I O Jelescu, E Fieremans, arXiv:1609.09144arXiv preprintNovikov DS, Veraart J, Jelescu IO, Fieremans E. Mapping orientational and microstructural metrics of neuronal integrity with in vivo diffusion MRI. arXiv preprint arXiv:1609.09144. 2016 Sep 28. One diffusion acquisition and different white matter models: How does microstructure change in human early development based on WMTI and NODDI?. I O Jelescu, J Veraart, V Adisetiyo, S S Milla, D S Novikov, E Fieremans, NeuroImage. 107Jelescu IO, Veraart J, Adisetiyo V, Milla SS, Novikov DS, Fieremans E. One diffusion acquisition and different white matter models: How does microstruc- ture change in human early development based on WMTI and NODDI? NeuroImage. 2015;107:242-256. Degeneracy in model parameter estimation for multi-compartmental diffusion in neuronal tissue. I O Jelescu, J Veraart, E Fieremans, D S Novikov, NMR Biomed. 291Jelescu IO, Veraart J, Fieremans E, Novikov DS. Degeneracy in model parameter estimation for multi-compartmental diffusion in neuronal tissue. NMR Biomed. 2016;29(1):33-47. To be dispersed or not to be dispersed: A study using HCP data. A Ghosh, H Zhang, D C Alexander, International Society of Magnetic Resonance in Medicine (ISMRM) 24th Annual Meeting & Exhibition. SingaporeGhosh A, Zhang H, Alexander DC. To be dispersed or not to be dispersed: A study using HCP data. International Society of Magnetic Resonance in Medicine (ISMRM) 24th Annual Meeting & Exhibition. Singapore; 2016. Mesoscopic structure of neuronal tracts from time-dependent diffusion. L M Burcaw, E Fieremans, D S Novikov, NeuroImage. 114Burcaw LM, Fieremans E, Novikov DS. Mesoscopic structure of neuronal tracts from time-dependent diffusion. NeuroImage. 2015;114:18-37. In vivo measurement of axon diameter distribution in the corpus callosum of rat brain. D Barazany, P J Basser, Y Assaf, Brain. 1325Barazany D, Basser PJ, Assaf Y. In vivo measurement of axon diameter distribution in the corpus callosum of rat brain. Brain. 2009;132(5):1210-1220. In vivo observation and biophysical interpretation of time-dependent diffusion in human white matter. E Fieremans, L M Burcaw, H H Lee, G Lemberskiy, J Veraart, D S Novikov, NeuroImage. 129Fieremans E, Burcaw LM, Lee HH, Lemberskiy G, Veraart J, Novikov DS. In vivo observation and biophysical interpretation of time-dependent diffusion in human white matter. NeuroImage. 2016;129:414-427. Diffusion kurtosis imaging probes cortical alterations and white matter pathology following cuprizone induced demyelination and spontaneous remyelination. C Guglielmetti, J Veraart, E Roelant, Neuroimage. 125Guglielmetti C, Veraart J, Roelant E, et al. Diffusion kurtosis imaging probes cortical alterations and white matter pathology following cuprizone induced demyelination and spontaneous remyelination. Neuroimage. 2016;125:363-377. Evaluation of diffusion kurtosis imaging in ex vivo hypomyelinated mouse brains. N D Kelm, K L West, R P Carson, D F Gochberg, K C Ess, M D Does, NeuroImage. 124Kelm ND, West KL, Carson RP, Gochberg DF, Ess KC, Does MD. Evaluation of diffusion kurtosis imaging in ex vivo hypomyelinated mouse brains. NeuroImage. 2016;124:612-626. Statistical model for diffusion attenuated MR signal. D A Yablonskiy, L G Bretthorst, J J Ackerman, Magn Reson Med. 504Yablonskiy DA, Bretthorst LG, Ackerman JJ. Statistical model for diffusion attenuated MR signal. Magn Reson Med. 2003;50(4):664-669. Quantitative estimation of 3-d fiber course in gross histological sections of the human brain using polarized light. H Axer, M Axer, T Krings, Graf Diedrich V, D Keyserlingk, J Neurosci Methods. 1051Axer H, Axer M, Krings T, Graf Diedrich v, Keyserlingk D. Quantitative estimation of 3-d fiber course in gross histological sections of the human brain using polarized light. J Neurosci Methods. 2001;105(1):121-131. Diffusion basis functions decomposition for estimating white matter intravoxel fiber geometry. A Ramirez-Manzanares, M Rivera, B C Vemuri, P Carney, IEEE Trans Med Imaging. 268Ramirez-Manzanares A, Rivera M, Vemuri BC, Carney P. Diffusion basis functions decomposition for estimating white matter intravoxel fiber geometry. IEEE Trans Med Imaging. 2007;26(8):1091-1102. Evaluation of restricted diffusion in cylinders. phosphocreatine in rabbit leg muscle. P Van Gelderen, D Despres, P Van Zijl, C T Moonen, J Magn Reson, Ser B. 1033Van Gelderen P, DesPres D, van Zijl P, Moonen CT. Evaluation of restricted diffusion in cylinders. phosphocreatine in rabbit leg muscle. J Magn Reson, Ser B. 1994;103(3):255-260. Cross-validatory choice and assessment of statistical predictions. M Stone, J R Stat Soc Ser B (Methodological). 1Stone M. Cross-validatory choice and assessment of statistical predictions. J R Stat Soc Ser B (Methodological). 1974;1:111-147. Bootstrap methods: Another look at the jackknife. B Efron, Ann Stat. 1979Efron B. Bootstrap methods: Another look at the jackknife. Ann Stat. 1979:1-26. Evaluation of restricted diffusion in cylinders. phosphocreatine in rabbit leg muscle. P Van Gelderen, D Despres, P Vanzijl, C Moonen, J Magn Reson, Ser B. 1033van Gelderen P, DesPres D, Vanzijl P, Moonen C. Evaluation of restricted diffusion in cylinders. phosphocreatine in rabbit leg muscle. J Magn Reson, Ser B. 1994;103(3):255-260. Regularization and variable selection via the elastic net. H Zou, T Hastie, J R Stat Soc Ser B Stat Methodol. 672Zou H, Hastie T. Regularization and variable selection via the elastic net. J R Stat Soc Ser B Stat Methodol. 2005;67(2):301-320. DIPY, a library for the analysis of diffusion MRI data. E Garyfallidis, M Brett, B Amirbekian, Front Neuroinform. 88Garyfallidis E, Brett M, Amirbekian B, et al. DIPY, a library for the analysis of diffusion MRI data. Front Neuroinform. 2014;8:8. SCIKIT-LEARN: Machine learning in Python. F Pedregosa, G Varoquaux, A Gramfort, J Mach Learning Res. 12Pedregosa F, Varoquaux G, Gramfort A, et al. SCIKIT-LEARN: Machine learning in Python. J Mach Learning Res. 2011;12:2825-2830. On the 'degrees of freedom' of the lasso. H Zou, T Hastie, R Tibshirani, Ann Stat. 355Zou H, Hastie T, Tibshirani R, et al. On the 'degrees of freedom' of the lasso. Ann Stat. 2007;35(5):2173-2192. LASADD: linear acceleration method for adapting diffusion dictionaries. A Loya-Olivas, R Aranda, M Rivera, International Society of Magnetic Resonance in Medicine (ISMRM) 23rd Annual Meeting and Exhibition. Toronto, Ontario, CanadaLoya-Olivas A, Aranda R, Rivera M. LASADD: linear acceleration method for adapting diffusion dictionaries. In: International Society of Magnetic Resonance in Medicine (ISMRM) 23rd Annual Meeting and Exhibition; 2015; Toronto, Ontario, Canada. Algoritmos para solución de modelos no lineales en DW-MRI. A K Loya, Master's thesisLoya AK. Algoritmos para solución de modelos no lineales en DW-MRI [Master's thesis]. Centro de Investigación en Matemáticas. Mexico Guanajuato, Guanajuato, Mexico: Centro de Investigación en Matemáticas; 2015. Recovering detailed intra-voxel white matter structure by using an adaptive diffusion dictionary. R Aranda, A Ramirez-Manzanares, M Rivera, International Society of Magnetic Resonance in Medicine (ISMRM) 23rd Annual Meeting and Exhibition. Toronto, Ontario, CanadaAranda R, Ramirez-Manzanares A, Rivera M. Recovering detailed intra-voxel white matter structure by using an adaptive diffusion dictionary. In: International Society of Magnetic Resonance in Medicine (ISMRM) 23rd Annual Meeting and Exhibition; Toronto, Ontario, Canada:2015. Sparse and Adaptive Diffusion Dictionary (SADD) for recovering intra-voxel white matter structure. R Aranda, A Ramirez-Manzanares, M Rivera, Med Image Anal. 261Aranda R, Ramirez-Manzanares A, Rivera M. Sparse and Adaptive Diffusion Dictionary (SADD) for recovering intra-voxel white matter structure. Med Image Anal. 2015;26(1):243-255. Microtrack: An algorithm for concurrent projectome and microstructure estimation. A J Sherbondy, M C Rowe, D C Alexander, Medical Image Computing and Computer-Assisted Intervention -MICCAI. 6361SpringerLecture Notes in Computer ScienceSherbondy AJ, Rowe MC, Alexander DC. Microtrack: An algorithm for concurrent projectome and microstructure estimation. Medical Image Computing and Computer-Assisted Intervention -MICCAI 2010, Lecture Notes in Computer Science, vol. 6361. Berlin: Springer; 2010:183-190. Spin diffusion measurements: Spin echoes in the presence of a time-dependent field gradient. E O Stejskal, J E Tanner, 10.1063/1.1695690J Chem Phys. 421Stejskal EO, Tanner JE. Spin diffusion measurements: Spin echoes in the presence of a time-dependent field gradient. J Chem Phys. 1965;42(1):288-292. https://doi.org/10.1063/1.1695690 Generalized diffusion tensor imaging and analytical relationships between diffusion tensor imaging and high angular resolution diffusion imaging. E Özarslan, T H Mareci, Magn Reson Med. 505Özarslan E, Mareci TH. Generalized diffusion tensor imaging and analytical relationships between diffusion tensor imaging and high angular resolution diffusion imaging. Magn Reson Med. 2003;50(5):955-965. On high order tensor-based diffusivity profile estimation. M Alipoor, I Y Gu, A J Mehnert, Y Lilja, D Nilsson, 35th Annual International Conference of the IEEE. Osaka, Japan2013Alipoor M, Gu IY, Mehnert AJ, Lilja Y, Nilsson D. On high order tensor-based diffusivity profile estimation. Engineering in Medicine and Biology Society (EMBC), 2013 35th Annual International Conference of the IEEE. Osaka, Japan; 2013:93-96. Detection and modeling of non-Gaussian apparent diffusion coefficient profiles in human brain data. D Alexander, G Barker, S Arridge, Magn Reson Med. 482Alexander D, Barker G, Arridge S. Detection and modeling of non-Gaussian apparent diffusion coefficient profiles in human brain data. Magn Reson Med. 2002;48(2):331-340. Pulsed-gradient spin-echo NMR for planar, cylindrical, and spherical pores under conditions of wall relaxation. P T Callaghan, J Magn Reson, Ser A. 1131Callaghan PT. Pulsed-gradient spin-echo NMR for planar, cylindrical, and spherical pores under conditions of wall relaxation. J Magn Reson, Ser A. 1995;113(1):53-59. We fit our basis using Laplacian-regularized least squares in the following steps. P Craven, G Wahba, Numer Math. 314Smoothing noisy data with spline functions. We first denote Ξ i (q, , u s , u t ) = S jlm(i) (q, u s )T o(i) ( , u tCraven P, Wahba G. Smoothing noisy data with spline functions. Numer Math. 1978;31(4):377-403. for all . We fit our basis using Laplacian-regularized least squares in the following steps. We first denote Ξ i (q, , u s , u t ) = S jlm(i) (q, u s )T o(i) ( , u t ) with with N coef the number of fitted coefficients. We then construct a design matrix Q ∈ R N data ×N coef with Q ik = S N i (A. ∈ {1 … N Coef, } , q k )T o i ( k , u ti ∈ {1 … N coef }, with N coef the number of fitted coefficients. We then construct a design matrix Q ∈ R N data ×N coef with Q ik = S N i (A, q k )T o i ( k , u t ). The signal is then fitted as c = argmin c ||y − Qc|| 2 + U(c) with y the measured signal, c the fitted coefficients and the weight for our analytic. The signal is then fitted as c = argmin c ||y − Qc|| 2 + U(c) with y the measured signal, c the fitted coefficients and the weight for our analytic
{'fraction_non_alphanumeric': 0.04570585000144731, 'fraction_numerical': 0.03226522322246987, 'mean_word_length': 4.410984650725697, 'pattern_counts': {'":': 0, '<': 5, '<?xml version=': 0, '>': 6, 'https://': 6, 'lorem ipsum': 0, 'www.': 2, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 51, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': "A large number of mathematical models have been proposed to describe the measured signal in diffusion-weighted (DW) magnetic resonance imaging (MRI). However, model comparison to date focuses only on specific subclasses, e.g. compartment models or signal models, and little or no information is available in the literature on how performance varies among the different types of models. To address this deficiency, we organized the 'White Matter Modeling Challenge' during the International Symposium on Biomedical Imaging (ISBI) 2015 conference. This competition aimed to compare a range of different kinds of models in their ability to explain a large range of measurable in vivo DW human brain data. Specifically, we assessed the ability of models to predict the DW signal accurately for new diffusion gradients and b values. We did not evaluate the accuracy of estimated model parameters, as a ground truth is hard to obtain. We used the Connectome scanner at the Massachusetts General Hospital, using gradient strengths of up to 300 mT/m and a broad set of diffusion times. We focused on assessing the DW This work builds on the statistical modelling of the apparent diffusion coefficient 72 and tackles the modelling of axon fibre dispersion in single15,73and multiple fibre bundle cases. The method estimates empirically (rather than imposes) the distribution of tissue properties (axon radius, parallel diffusion, etc.), as well as the orientational distribution of the bundles. The general framework is as follows:• estimation of mean principal diffusion directions (PDD) per axon bundle;• selection of a dense set of orientationally focused basis directions that capture the discrete non-parametric fibre dispersion;• design of a dictionary of intra/extracellular synthetic DW signals, which are precomputed along the basis directions (see the DBF method in Ramirez-Manzanares et al. 74 );• computation of the size compartments per diffusion atom of the dictionary (model fitting).", 'arxivid': '1604.07287', 'author': ['Uran Ferizi [email protected] \nCentre for Medical Image Computing\nDepartment of Computer Science\nUniversity College London\nUK\n\nDepartment of Radiology\nNew York University School of Medicine\nUSA\n\nDepartment of Neuroinflammation\nInstitute of Neurology\nUniversity College London\nUK\n', "Benoit Scherrer \nComputational Radiology Laboratory\nBoston Children's Hosp\nHarvard University\nUSA\n", 'Torben Schneider \nDepartment of Neuroinflammation\nInstitute of Neurology\nUniversity College London\nUK\n\nPhilips Healthcare\nGuildfordSurreyUK\n', 'Mohammad Alipoor \nChalmers University of Technology\nGothenburgSweden\n', 'Odin Eufracio \nCentro de Investigacion en Matematicas AC\nGuanajuatoMexico\n', 'Rutger H J Fick \nAthena Project-Team\nINRIA Sophia Antipolis -Méditerranée\nFrance\n', 'Rachid Deriche \nAthena Project-Team\nINRIA Sophia Antipolis -Méditerranée\nFrance\n', 'Markus Nilsson \nLund University Bioimaging Center\nLund University\nSweden\n', 'Ana K Loya-Olivas \nCentro de Investigacion en Matematicas AC\nGuanajuatoMexico\n', 'Mariano Rivera \nCentro de Investigacion en Matematicas AC\nGuanajuatoMexico\n', 'Dirk H J Poot \nErasmus Medical Center\nDelft University of Technology\nthe Netherlands\n', 'Alonso Ramirez-Manzanares \nCentro de Investigacion en Matematicas AC\nGuanajuatoMexico\n', 'Jose L Marroquin \nCentro de Investigacion en Matematicas AC\nGuanajuatoMexico\n', 'Ariel Rokem \neScience Institute\nUniversity of Washington\nUSA\n\nCenter for Cognitive and Neurobiological Imaging\nStanford University\nUSA\n', 'Christian Pötter \nCenter for Cognitive and Neurobiological Imaging\nStanford University\nUSA\n', 'Robert F Dougherty \nCenter for Cognitive and Neurobiological Imaging\nStanford University\nUSA\n', 'Ken Sakaie \nImaging Institute\nThe Cleveland Clinic\nClevelandUSA\n', 'Claudia Wheeler-Kingshott \nDepartment of Neuroinflammation\nInstitute of Neurology\nUniversity College London\nUK\n', "Simon K Warfield \nComputational Radiology Laboratory\nBoston Children's Hosp\nHarvard University\nUSA\n", 'Thomas Witzel \nA.A. Martinos Center for Biomedical Imaging\nMGH\nHarvard University\nUSA\n\nCenter for Biomedical Imaging\nDepartment of Radiology\nCorrespondence Uran Ferizi\nPhD\n\nNew York University Langone Medical Center\n660 First Avenue, 4th floor10016New YorkNYUSA\n', 'Lawrence L Wald \nA.A. Martinos Center for Biomedical Imaging\nMGH\nHarvard University\nUSA\n\nCenter for Biomedical Imaging\nDepartment of Radiology\nCorrespondence Uran Ferizi\nPhD\n\nNew York University Langone Medical Center\n660 First Avenue, 4th floor10016New YorkNYUSA\n', 'José G Raya \nDepartment of Radiology\nNew York University School of Medicine\nUSA\n', 'Daniel C Alexander \nCentre for Medical Image Computing\nDepartment of Computer Science\nUniversity College London\nUK\n'], 'authoraffiliation': ['Centre for Medical Image Computing\nDepartment of Computer Science\nUniversity College London\nUK', 'Department of Radiology\nNew York University School of Medicine\nUSA', 'Department of Neuroinflammation\nInstitute of Neurology\nUniversity College London\nUK', "Computational Radiology Laboratory\nBoston Children's Hosp\nHarvard University\nUSA", 'Department of Neuroinflammation\nInstitute of Neurology\nUniversity College London\nUK', 'Philips Healthcare\nGuildfordSurreyUK', 'Chalmers University of Technology\nGothenburgSweden', 'Centro de Investigacion en Matematicas AC\nGuanajuatoMexico', 'Athena Project-Team\nINRIA Sophia Antipolis -Méditerranée\nFrance', 'Athena Project-Team\nINRIA Sophia Antipolis -Méditerranée\nFrance', 'Lund University Bioimaging Center\nLund University\nSweden', 'Centro de Investigacion en Matematicas AC\nGuanajuatoMexico', 'Centro de Investigacion en Matematicas AC\nGuanajuatoMexico', 'Erasmus Medical Center\nDelft University of Technology\nthe Netherlands', 'Centro de Investigacion en Matematicas AC\nGuanajuatoMexico', 'Centro de Investigacion en Matematicas AC\nGuanajuatoMexico', 'eScience Institute\nUniversity of Washington\nUSA', 'Center for Cognitive and Neurobiological Imaging\nStanford University\nUSA', 'Center for Cognitive and Neurobiological Imaging\nStanford University\nUSA', 'Center for Cognitive and Neurobiological Imaging\nStanford University\nUSA', 'Imaging Institute\nThe Cleveland Clinic\nClevelandUSA', 'Department of Neuroinflammation\nInstitute of Neurology\nUniversity College London\nUK', "Computational Radiology Laboratory\nBoston Children's Hosp\nHarvard University\nUSA", 'A.A. Martinos Center for Biomedical Imaging\nMGH\nHarvard University\nUSA', 'Center for Biomedical Imaging\nDepartment of Radiology\nCorrespondence Uran Ferizi\nPhD', 'New York University Langone Medical Center\n660 First Avenue, 4th floor10016New YorkNYUSA', 'A.A. Martinos Center for Biomedical Imaging\nMGH\nHarvard University\nUSA', 'Center for Biomedical Imaging\nDepartment of Radiology\nCorrespondence Uran Ferizi\nPhD', 'New York University Langone Medical Center\n660 First Avenue, 4th floor10016New YorkNYUSA', 'Department of Radiology\nNew York University School of Medicine\nUSA', 'Centre for Medical Image Computing\nDepartment of Computer Science\nUniversity College London\nUK'], 'corpusid': 215778497, 'doi': '10.1002/nbm.3734', 'github_urls': [], 'n_tokens_mistral': 30712, 'n_tokens_neox': 25950, 'n_words': 15927, 'pdfsha': 'db55b0b1b516e3064a6f658d2dc09138ab42da2c', 'pdfurls': None, 'title': ['Diffusion MRI microstructure models with in vivo human brain Connectome data: results from a multi-group comparison TE |G| b Nr (ms) (ms) (mT/m) (s/mm 2 ) Nr (ms) (ms) (mT/m) (s/mm 2 )', 'Diffusion MRI microstructure models with in vivo human brain Connectome data: results from a multi-group comparison TE |G| b Nr (ms) (ms) (mT/m) (s/mm 2 ) Nr (ms) (ms) (mT/m) (s/mm 2 )'], 'venue': []}
arxiv
A Silicon Nitride Microring Based High-Speed, Tuning-Efficient, Electro-Refractive Modulator Venkata Sai Department of ECE Department of ECE University of Kentucky Lexington KentuckyUSA Praneeth Karempudi Department of ECE Department of ECE University of Kentucky Lexington KentuckyUSA Ishan G Thakkar [email protected] Department of ECE University of Kentucky Lexington KentuckyUSA Jeffrey Todd Hastings [email protected] University of Kentucky Lexington KentuckyUSA A Silicon Nitride Microring Based High-Speed, Tuning-Efficient, Electro-Refractive Modulator Index Terms-Silicon nitridemodulationrefractive indexfree-carriersextinction ratio The use of the Silicon-on-Insulator (SOI) platform has been prominent for realizing CMOS-compatible, highperformance photonic integrated circuits (PICs). But in recent years, the silicon-nitride-on-silicon-dioxide (SiN-on-SiO2) platform has garnered increasing interest as an alternative to the SOI platform for realizing high-performance PICs. This is because of its several beneficial properties over the SOI platform, such as low optical losses, high thermo-optic stability, broader wavelength transparency range, and high tolerance to fabrication-process variations. However, SiN-on-SiO2 based active devices such as modulators are scarce and lack in desired performance, due to the absence of free-carrier based activity in the SiN material and the complexity of integrating other active materials with SiN-on-SiO2 platform. This shortcoming hinders the SiN-on-SiO2 platform for realizing active PICs. To address this shortcoming, we demonstrate a SiN-on-SiO2 microring resonator (MRR) based active modulator in this article. Our designed MRR modulator employs an Indium-Tin-Oxide (ITO)-SiN-ITO thin-film stack, in which the ITO thin films act as the upper and lower claddings of the SiN MRR. The ITO-SiN-ITO thin-film stack leverages the free-carrier assisted, high-amplitude refractive index change in the ITO films to effect a large electro-refractive optical modulation in the device. Based on the electrostatic, transient, and finite difference time domain (FDTD) simulations, conducted using photonics foundry-validated tools, we show that our modulator achieves 280 pm/V resonance modulation efficiency, 67.8 GHz 3-dB modulation bandwidth, ∼19 nm free-spectral range (FSR), ∼0.23 dB insertion loss, and 10.31 dB extinction ratio for optical on-off-keying (OOK) modulation at 30 Gb/s. I. INTRODUCTION Driven by the rise of CMOS-compatible processes for fabricating photonic devices, photonic integrated circuits (PICs) are inexorably moving from the domain of long-distance communications to chip-to-chip and even on-chip applications. It is common for the PICs to incorporate optical modulators, to enable efficient manipulation of optical signals, which is a necessity for the operation of active PICs. Recent advances in the CMOS-compatible silicon-on-insulator (SOI) photonic platform has fundamentally improved the applicability of SOI PICs [1], [2], [3]. But in the last few years, the siliconnitride-on-silicon-dioxide (SiN-on-SiO 2 ) platform has gained tremendous attention for realizing PICs. This is because the SiN-on-SiO 2 platform has several advantages over the SOI platform. Compared to silicon (Si), the SiN material has a much broader wavelength transparency range (500nm-3700nm), lower refractive index and smaller thermo-optic coefficient [4]. The lower refractive index of SiN means that SiN offers smaller index contrast with SiO 2 compared to Si. This in turn makes the SiN-on-SiO 2 based monomode passive devices (e.g., waveguides, microring resonators (MRRs)) less susceptible to (i) propagation losses due to the decreased sensitivity to edge roughness [5], and (ii) aberrations in the realized device dimensions caused due to fabrication-process variations [4]. In addition, the smaller thermo-optic coefficient of SiN makes it possible to design nearly athermal photonic devices using SiN [6]. Moreover, SiN devices and circuits exhibit increased efficiency of nonlinear parametric processes compared to Si [7]. Despite these favorable properties of the SiN-on-SiO 2 platform, SiN-on-SiO 2 based active devices such as modulators (e.g., [8]- [12]) are scarce and lack in modulation bandwidth, modulation efficiency and free spectral range (FSR) [9]. This is because of the lack of the free-carriers based activity in the SiN material and the general difficulty of incorporating other active materials with the SiN-on-SiO 2 platform. This in turn limits the use of the SiN-on-SiO 2 platform for realizing only passive PICs. To overcome this shortcoming, there is impetus to heterogeneously integrate active photonic materials and devices with SiN-on-SiO 2 passive devices [13]. When such efforts of integrating electro-optically active materials with the SiN-on-SiO 2 platform come to fruition, it will be possible to design extremely high-performance and energyefficient SiN-on-SiO 2 based active and passive PICs. Different from such prior efforts, in this article, we demonstrate for the first time the use of the high-amplitude electrorefractive activity of Indium-Tin-Oxide (ITO) thin films to realize a SiN-on-SiO 2 based optical on-off-keying (OOK) modulator. We show, based on the electrostatic, transient, and finite difference time domain (FDTD) simulations conducted using the photonics foundry-validated tools from Lumerical/Ansys, that our modulator achieves 280 pm/V resonance modulation efficiency, 67.8 GHz 3-dB modulation bandwidth, ∼19 nm free-spectral range (FSR), ∼0.23 dB insertion loss, and 10.31 dB extinction ratio for optical OOK modulation at 30 Gb/s. Based on the obtained simulation results, we advocate that our modulator can achieve better performance compared to the existing SiN modulators and several state-of-the-art Si and Lithium Niobate (LN) modulators from prior work. II. RELATED WORK AND MOTIVATION A plethora of Si, Lithium Niobate (LN) and SiN based integrated optical modulator designs have been formulated in prior work. But among these modulator designs, MRR based modulators have gained widespread attention due to their high wavelength selectivity, compact size, and compatibility for cascaded dense wavelength division multiplexing (DWDM). Here, we briefly review some relevant Si, LN and SiN MRR modulators from prior work. A. Silicon (Si) Based Modulators Over the last two decades, Si has emerged as the prominent material for fabricating PICs, mainly because the cost effectiveness of reusing the established CMOS manufacturing infrastructure promotes the use of Si for building complex PICs. Si material also exhibits high thermo-optic and electro-optic (free-carriers-induced) sensitivity, which enables the realization of optical modulators directly in Si substrate without requiring any auxiliary active materials. Si optical modulators based on free-carriers-induced plasma dispersion and absorption effects have become particularly more popular because of their low-power and high-speed operation. Although several designs of Si-based modulators have been reported, the most commonly adopted designs employ microring resonators (MRRs) (e.g., [14]- [23]). The recent work [18] has also demonstrated the use of an electrically active stack of Si-SiO 2 -ITO layers to substantially increase the electrooptic modulation efficiency of a Si MRR based modulator. In this design, the light-guiding core layer (i.e., Si) critically contributes to the electro-optic activity in the modulator. In contrast, our SiN-on-SiO 2 modulator presented in this paper employs for the first time an ITO-SiN-ITO thin-film stack in which the ITO thin films act as the active upper and lower claddings of the SiN MRR based core of the modulator. B. Lithium Niobate (LN) Based Modulators Lithium Niobate (LN) has recently emerged as the promising material for designing high-performance electro-optic modulators because of its wide bandgap and large secondorder electro-optic coefficient. Several LN modulators have been demonstrated so far in the literature (e.g., [24]- [28]). For instance, in [26] and [27], thin-film LN-based electro-optic modulators have been demonstrated. Similarly, in [24], [25] and [28], a hybrid Si-LN platform based MRR modulators have been presented. These LN modulators demonstrated in prior works, however, lack in modulation efficiency compared to the Si and SiN modulators from prior works. C. Silicon Nitride (SiN) Based Modulators Recently, silicon nitride (SiN) based PICs have gained tremendous attention due to their favorable properties compared to the traditional Si based PICs. As a result, several SiNon-SiO 2 modulators have been demonstrated (e.g., [8]- [12]). In [8], a graphene integrated electro-optic SiN MRR modulator has been reported. In [11], a hybrid SiN-LN platform based racetrack resonator modulator has been presented. Similarly, SiN modulators based on lead zirconate titanate and zinc oxide/zinc sulphide as active materials are demonstrated in [9] and [12]. In [10], a SiN modulator that achieves tuning via photo-elastic effect has been demonstrated. Compared to these modulator designs from prior work, we present a different, ITO-based electro-refractive SiN-on-SiO 2 modulator that achieves relatively better modulation bandwidth, modulation efficiency, and FSR. III. DESIGN OF OUR SIN-ON-SIO 2 MODULATOR In this section, firstly we describe the structure and operating principle of our modulator design. Then, we discuss the characterization results for our modulator that we have obtained through photonics foundry-validated simulations. We also compare our modulator with several Si, LN and SiN based MRR modulators from prior work, in terms of modulation bandwidth, modulation efficiency, and FSR. A. Structure and Operating Principle Fig. 1(a) and Fig. 1(b), respectively, show the top-view and cross-sectional schematics of our SiN-on-SiO 2 MRR modulator. The active region in the upper and lower claddings of the modulator consists of indium tin oxide (ITO) thin films with silicon nitride material (SiN) in between (creating an ITO-SiN-ITO thin-film stack). From Fig. 1(b), we have a 300 nm thick SiN-based MRR waveguide between two 10 nm thick ITO films. Upon applying voltage across the ITO-SiN-ITO stack (through the Au pads shown in Fig. 1(a)), free carriers accumulate in the ITO films at the ITO-SiN interfaces for up to 5 nm depth in the ITO films [1], making these accumulation regions in the ITO films high-carrier-density active regions. This is due to the free-carriers-assisted, largeamplitude modulation in the permittivity and refractive index of the ITO material previously reported in [1]. We evaluate this free-carriers based index modulation in the ITO films using the Drude-Lorentz model from [29]. It can be inferred from the Drude-Lorentz model that as the carrier concentration in the ITO accumulation regions increases, the refractive index of the ITO films decreases. Our modulator design from Fig. 1 leverages this electro-refractive phenomenon in ITO. The free-carriers-induced decrease in the refractive index of the ITO thin films decreases the effective refractive index of the SiN-on-SiO 2 modulator, causing a blue shift in its resonance wavelength that in turn causes a transmission modulation in the MRR modulator. The electro-refractive activity of our SiNon-SiO 2 MRR modulator is confined only in the ITO-based claddings. This is different from the Si-SiO 2 -ITO capacitor based MRR modulator from [18], which has the electrorefractive activity in both its Si-based MRR core and SiO 2 -ITO based cladding. B. Simulations based Characterization We performed electrostatic simulations of our ITO-SiN-ITO thin-film stack based SiN-on-SiO 2 modulator in the Table I and Fig. 2 respectively) at various applied voltages for the operation around 1.615 µm wavelength (L-band). From Fig. 2, our modulator achieves ∼4.5 nm resonance shift upon applying 17 V across the thinfilm stack, which renders the resonance tuning (modulation) efficiency of ∼280 pm/V. This is crucially significant as our MRR modulator has relatively very low overlap between the optical mode and free-carrier perturbation (only ∼10% of the guided optical mode overlaps with the ITO-based claddings) compared to the state-of-the-art ITO-based modulators (e.g., [18]). Further, from the simulated spectra in Fig. 2, we evaluate the FSR of our modulator to be ∼19 nm. Further, based on our device simulations using the Lumerical MODE tool, we evaluated the insertion loss and loaded Q-factor of our modulator to be ∼0.23 dB and ∼2300 respectively. We also evaluated the capacitance density of the ITO thin-films covering the MRR rim (using the Lumerical CHARGE tool) to be ∼0.18 fF/µm 2 for the 300 nm thick SiN layer, yielding the modulation bandwidth (3-dB RC bandwidth) of ∼67.8 GHz for the modulator. We also modeled our modulator in Lumerical INTERCONNECT, to simulate optical eye diagrams for the modulator at 30 Gb/s and 55 Gb/s operating bitrates (Fig. 3). As evident (Fig. 3(b)), our modulator can achieve 10.31 dB extinction ratio for OOK modulation at 30 Gb/s bitrate. Fig. 4 shows a comparison of our SiN-on-SiO 2 modulator with the best performing Si (ten; [14]- [23]), LN (five; [24]- [28]) and SiN (five; [8]- [12]) MRR modulators from prior work, in terms of three key attributes, namely modulation efficiency, FSR, and modulation bandwidth. As evident from Fig. 4, our modulator achieves better performance compared to the exisiting SiN modulators and the state-of-the-art Si and LN modulators from prior works, which in turn promotes its use in DWDM-based high-performance PICs. Since our SiN-on-SiO 2 modulator achieves modulation bandwidth of ∼67.8 GHz, it can be easily operated at the bitrate of >15 Gb/s to enable ultra-high-speed (potentially beyond Tb/s) DWDM-based PICs while ensuring minimal power-penalty from crosstalk [32] and self-heating [33]. In addition, our SiN-on-SiO 2 modulator also achieves a modulation efficiency of ∼280 pm/V. This in turn can enable dynamic operation of our modulator with energyefficiency of <100 fJ/bit [34]. Unfortunately, our modulator achieves relatively low loaded Q-factor of 2300. Nevertheless, we anticipate that the Q-factor can be increased to 5000-8000 by marginally trading the modulation bandwidth for better loss characteristics of the MRR cavity. Thus, future work should include an exhaustive search of design parameters, including the coupling gap, MRR waveguide width, MRR waveguide height, MRR radius, and the thicknesses of the ITO films, to minimize the coupling, bending and absorption losses in the MRR cavity without notably compromising the modulation efficiency. Having the loaded Q-factor of our modulator in the range of 5000-8000, while already having a greater than 12 nm FSR (Fig. 4), will enable balancing of the crosstalk penalty and modulation speed in our modulator, for high-performance DWDM based PICs [32]. Moreover, although ITO is not available in the CMOS process flow, it can be deposited at relatively low temperatures (less than 300°C) on top of the back-end-of-line (BEOL) metal layers of CMOS chips, independent of the CMOS FEOL process. This makes our SiN-on-SiO 2 modulator an excellent choice for implementing optical interconnect PICs on silicon interposers, to enable ultra-high-bandwidth inter-chiplet communication in emerging multi-chiplet systems [35]. N (cm −3 ) Re (η IT O ) Im (η IT O ) Re (η ef f ) Im (η ef f ) V ∆λr (pm)1×10 C. Comparison and Discussion In summary, we advocate that our SiN-on-SiO 2 modulator can achieve better performance compared to the SiN, Si and LN based MRR modulators from prior work. The obtained results corroborate our modulator's potential to consequently enable DWDM-based SiN-on-SiO 2 PICs that will offer highly scalable and energy-efficient solutions to a wide range of mature and emerging applications, including datacenter transceivers [36], high-performance computing [37], signal processing [38], optical computing [39], and artificial intelligence [40]. IV. CONCLUSION In recent years, the SiN-on-SiO 2 platform has attained tremendous attention for realizing PICs because it has several advantageous properties over the conventional SOI platform. Despite these advantages, the SiN-on-SiO 2 platform lacks high-performance active devices such as modulators. To address this drawback, we have demonstrated an ITO based SiNon-SiO 2 MRR modulator, which consists of ITO thin films as the active upper and lower claddings of the SiN MRR core. These ITO-based active claddings of our modulator leverage the free-carrier assisted, high-amplitude refractive index change in them to effect a large electro-refractive optical modulation in the device. To evaluate the performance of our SiNon-SiO 2 MRR modulator, we performed electrostatic, transient and finite difference time domain (FDTD) simulations using the foundry-validated Ansys/Lumerical tools. Based on these simulations, our modulator achieves superior performance with ∼280 pm/V modulation efficiency, 67.8 GHz 3-dB modulation bandwidth with ∼19nm FSR, ∼0.23 dB insertion loss and 10.31 dB extinction ratio for OOK modulation at 30 Gb/s. This excellent performance of our SiN-on-SiO 2 MRR modulator demonstrates its potential to enhance the performance and energy-efficiency of SiN-on-SiO 2 based PICs of the future. Fig. 1 . 1(a) Top view, (b) Cross-sectional view (along AA') of our SiN-on-SiO 2 MRR modulator. Fig. 2 . 2Transmission spectra of our modulator. Fig. 3 . 3Optical eye diagrams for (a) 30 Gb/s and (b) 55 Gb/s OOK inputs to our modulator. Fig. 4 . 4Modulation bandwidth, modulation efficiency and FSR (shown as the size of the bubbles and red data labels) of various Si, LN (LiNbO 3 ) and SiN MRR modulators from prior work, compared with our SiN-on-SiO 2 MRR modulator. TABLE I IFREE-CARRIER CONCENTRATION (N), REAL INDEX (RE(η IT O )), AND IMAGINARY INDEX (IM(η IT O )) FOR THE ITO ACCUMULATION LAYER IN OUR MODULATOR. THE REAL AND IMAGINARY EFFECTIVE INDEX (RE(η ef f ), IM(η ef f )), OPERATING VOLTAGE (V), AND INDUCED RESONANCE SHIFT (∆λr ) FOR OUR MODULATOR. Design methodologies for silicon photonic integrated circuits. L Chrostowski, Smart Photonic and Optoelectronic Integrated Circuits XVI. SPIE8989L. Chrostowski et al., "Design methodologies for silicon photonic integrated circuits," in Smart Photonic and Optoelectronic Integrated Circuits XVI, vol. 8989. SPIE, 2014, pp. 83-97. Programmable silicon photonic integrated circuits. W Bogaerts, IEEE IPC, 2020. W. Bogaerts et al., "Programmable silicon photonic integrated circuits," in IEEE IPC, 2020, pp. 1-2. Linear programmable nanophotonic processors. N C Harris, Optica. 512N. C. Harris et al., "Linear programmable nanophotonic processors," Optica, vol. 5, no. 12, pp. 1623-1631, 2018. A versatile silicon-silicon nitride photonics platform for enhanced functionalities and applications. Q Wilmart, Applied Sciences. 92255Q. Wilmart et al., "A versatile silicon-silicon nitride photonics platform for enhanced functionalities and applications," Applied Sciences, vol. 9, no. 2, p. 255, 2019. Ultra-low-loss high-aspect-ratio si 3 n 4 waveguides. J F Bauters, Optics express. 194J. F. Bauters et al., "Ultra-low-loss high-aspect-ratio si 3 n 4 waveg- uides," Optics express, vol. 19, no. 4, pp. 3163-3174, 2011. Silicon nitride o-band (de) multiplexers with low thermal sensitivity. G Gao, Optics Express. 2511G. Gao et al., "Silicon nitride o-band (de) multiplexers with low thermal sensitivity," Optics Express, vol. 25, no. 11, pp. 12 260-12 267, 2017. Integrated nonlinear optics in silicon nitride waveguides and resonators. J Levy, J. Levy, "Integrated nonlinear optics in silicon nitride waveguides and resonators," 2011. Graphene electro-optic modulator with 30 ghz bandwidth. C T Phare, Nature Photonics. 98C. T. Phare et al., "Graphene electro-optic modulator with 30 ghz bandwidth," Nature Photonics, vol. 9, no. 8, pp. 511-514, 2015. Nanophotonic pockels modulators on a silicon nitride platform. K Alexander, Nature communications. 91K. Alexander et al., "Nanophotonic pockels modulators on a silicon nitride platform," Nature communications, vol. 9, no. 1, pp. 1-6, 2018. Piezoelectrically tuned silicon nitride ring resonator. W Jin, Optics Express. 263W. Jin et al., "Piezoelectrically tuned silicon nitride ring resonator," Optics Express, vol. 26, no. 3, pp. 3174-3187, 2018. High-performance racetrack resonator in silicon nitride-thin film lithium niobate hybrid platform. A N R Ahmed, pp. 30 741-30 751Optics Express. 2721A. N. R. Ahmed et al., "High-performance racetrack resonator in silicon nitride-thin film lithium niobate hybrid platform," Optics Express, vol. 27, no. 21, pp. 30 741-30 751, 2019. Integrated silicon nitride electro-optic modulators with atomic layer deposited overlays. A Hermans, Optics letters. 445A. Hermans et al., "Integrated silicon nitride electro-optic modulators with atomic layer deposited overlays," Optics letters, vol. 44, no. 5, pp. 1112-1115, 2019. Enabling vcsel-on-silicon nitride photonic integrated circuits with micro-transfer-printing. J Goaeyverts, Optica. 812J. Goaeyverts et al., "Enabling vcsel-on-silicon nitride photonic inte- grated circuits with micro-transfer-printing," Optica, vol. 8, no. 12, pp. 1573-1580, 2021. Micrometre-scale silicon electro-optic modulator. X Qianfan, nature. 4357040X. Qianfan et al., "Micrometre-scale silicon electro-optic modulator," nature, vol. 435, no. 7040, pp. 325-327, 2005. 25 gbit/s silicon microring modulator based on misalignment-tolerant interleaved pn junctions. X Xiao, Optics express. 203X. Xiao et al., "25 gbit/s silicon microring modulator based on misalignment-tolerant interleaved pn junctions," Optics express, vol. 20, no. 3, pp. 2507-2515, 2012. Monolithic integration of o-band photonic transceivers in a "zero-change" 32nm soi cmos. S Moazeni, IEEE. S. Moazeni et al., "Monolithic integration of o-band photonic transceivers in a "zero-change" 32nm soi cmos," in IEEE IEDM), 2017, pp. 24-3. Monolithic silicon-photonic platforms in state-ofthe-art cmos soi processes. V Stojanovic, Optics express. 2610V. Stojanovic et al., "Monolithic silicon-photonic platforms in state-of- the-art cmos soi processes," Optics express, vol. 26, no. 10, pp. 13 106- 13 121, 2018. Silicon microring modulator with transparent conductive oxide gate. E Li, IEEE OI. E. Li et al., "Silicon microring modulator with transparent conductive oxide gate," in IEEE OI, 2019, pp. 1-2. Low-voltage, low-loss, multi-gb/s silicon microring modulator based on a mos capacitor. V Campenhout, OSA OFC. Optical Society of AmericaV. Campenhout et al., "Low-voltage, low-loss, multi-gb/s silicon micro- ring modulator based on a mos capacitor," in OSA OFC. Optical Society of America, 2012, pp. OM2E-4. 56gb/s ring modulator on a 300mm silicon photonics platform. M Pantouvaki, IEEE ECOC. M. Pantouvaki et al., "56gb/s ring modulator on a 300mm silicon photonics platform," in IEEE ECOC, 2015, pp. 1-3. A 128 gb/s pam4 silicon microring modulator with integrated thermo-optic resonance tuning. J Sun, IEEE JLT. 371J. Sun et al., "A 128 gb/s pam4 silicon microring modulator with integrated thermo-optic resonance tuning," IEEE JLT, vol. 37, no. 1, pp. 110-115, 2018. Low-voltage 60gb/s nrz and 100gb/s pam4 o-band silicon ring modulator. Y Ban, IEEE OI. Y. Ban et al., "Low-voltage 60gb/s nrz and 100gb/s pam4 o-band silicon ring modulator," in IEEE OI, 2019, pp. 1-2. A 112 gb/s pam4 silicon photonics transmitter with microring modulator and cmos driver. H Li, IEEE JLT. 381H. Li et al., "A 112 gb/s pam4 silicon photonics transmitter with microring modulator and cmos driver," IEEE JLT, vol. 38, no. 1, pp. 131-138, 2020. Hybrid silicon and lithium niobate electro-optical ring modulator. L Chen, Optica. 12L. Chen et al., "Hybrid silicon and lithium niobate electro-optical ring modulator," Optica, vol. 1, no. 2, pp. 112-118, 2014. Highly linear ring modulator from hybrid silicon and lithium niobate. L Chen, Optics Express. 2310L.Chen et al., "Highly linear ring modulator from hybrid silicon and lithium niobate," Optics Express, vol. 23, no. 10, pp. 13 255-13 264, 2015. Nanophotonic lithium niobate electro-optic modulators. C Wang, Optics express. 262C. Wang et al., "Nanophotonic lithium niobate electro-optic modulators," Optics express, vol. 26, no. 2, pp. 1547-1555, 2018. Monolithic lithium niobate photonic circuits for kerr frequency comb generation and modulation. C Wang, 10C.Wang et al., "Monolithic lithium niobate photonic circuits for kerr frequency comb generation and modulation," vol. 10, no. 1, pp. 1-6, 2019. Hybrid si-linbo 3 microring electro-optically tunable resonators for active photonic devices. Y S Lee, Optics letters. 367Y. S. Lee et al., "Hybrid si-linbo 3 microring electro-optically tunable resonators for active photonic devices," Optics letters, vol. 36, no. 7, pp. 1119-1121, 2011. Indium-tin-oxide for high-performance electro-optic modulation. Z Ma, Nanophotonics. 42Z. Ma et al., "Indium-tin-oxide for high-performance electro-optic modulation," Nanophotonics, vol. 4, no. 2, pp. 198-213, 2015. . L Inc, L. Inc. [Online]. Available: http://www-.lumerical.com/products Mid-infrared optical properties of thin films of aluminum oxide, titanium dioxide, silicon dioxide, aluminum nitride, and silicon nitride. J Kischkat, Applied optics. 5128J. kischkat et al., "Mid-infrared optical properties of thin films of aluminum oxide, titanium dioxide, silicon dioxide, aluminum nitride, and silicon nitride," Applied optics, vol. 51, no. 28, pp. 6789-6798, 2012. Crosstalk penalty in microring-based silicon photonic interconnect systems. M Bahadori, IEEE JLT. 3417M. Bahadori et al., "Crosstalk penalty in microring-based silicon pho- tonic interconnect systems," IEEE JLT, vol. 34, no. 17, pp. 4043-4052, 2016. 15 gb/s transmission with wide-fsr carrier injection ring modulator for tb/s optical links. M A Seyedi, OSA CLEO: Science and Innovations. M. A. Seyedi et al., "15 gb/s transmission with wide-fsr carrier injection ring modulator for tb/s optical links," in OSA CLEO: Science and Innovations, 2016, pp. SF2F-7. Theoretical analysis of energy efficiency and bandwidth limit of silicon photonic modulators. E Li, IEEE JLT. 3723E. Li et al., "Theoretical analysis of energy efficiency and bandwidth limit of silicon photonic modulators," IEEE JLT, vol. 37, no. 23, pp. 5801-5813, 2019. Popstar: a robust modular optical noc architecture for chiplet-based 3d integrated systems. Y Thonnart, IEEE/ACM DATE. Y. Thonnart at al., "Popstar: a robust modular optical noc architecture for chiplet-based 3d integrated systems," in IEEE/ACM DATE, 2020, pp. 1456-1461. A monolithic 25-gb/s transceiver with photonic ring modulators and ge detectors in a 130-nm cmos soi process. J F Buckwalter, IEEE Journal of Solid-State Circuits. 476J. F. Buckwalter et al., "A monolithic 25-gb/s transceiver with photonic ring modulators and ge detectors in a 130-nm cmos soi process," IEEE Journal of Solid-State Circuits, vol. 47, no. 6, pp. 1309-1322, 2012. Silicon photonics for exascale systems. S Rumley, IEEE JLT. 333S. Rumley et al., "Silicon photonics for exascale systems," IEEE JLT, vol. 33, no. 3, pp. 547-562, 2015. Photonic adc: overcoming the bottleneck of electronic jitter. A Khilo, Optics Express. 204A. Khilo et al., "Photonic adc: overcoming the bottleneck of electronic jitter," Optics Express, vol. 20, no. 4, pp. 4454-4469, 2012. Design exploration and scalability analysis of a cmos-integrated, polymorphic, nanophotonic arithmetic-logic unit. V S P Karempudi, ACM SenSys. New York, NY, USAV. S. P. Karempudi et al., "Design exploration and scalability analysis of a cmos-integrated, polymorphic, nanophotonic arithmetic-logic unit," in ACM SenSys, New York, NY, USA, 2021, p. 628-634. Pixel: Photonic neural network accelerator. K Shiflett, IEEE HPCA, 2020. K. Shiflett et al., "Pixel: Photonic neural network accelerator," in IEEE HPCA, 2020, pp. 474-487.
{'fraction_non_alphanumeric': 0.06121145452609914, 'fraction_numerical': 0.03420744473226642, 'mean_word_length': 4.557483731019523, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 1, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'The use of the Silicon-on-Insulator (SOI) platform has been prominent for realizing CMOS-compatible, highperformance photonic integrated circuits (PICs). But in recent years, the silicon-nitride-on-silicon-dioxide (SiN-on-SiO2) platform has garnered increasing interest as an alternative to the SOI platform for realizing high-performance PICs. This is because of its several beneficial properties over the SOI platform, such as low optical losses, high thermo-optic stability, broader wavelength transparency range, and high tolerance to fabrication-process variations. However, SiN-on-SiO2 based active devices such as modulators are scarce and lack in desired performance, due to the absence of free-carrier based activity in the SiN material and the complexity of integrating other active materials with SiN-on-SiO2 platform. This shortcoming hinders the SiN-on-SiO2 platform for realizing active PICs. To address this shortcoming, we demonstrate a SiN-on-SiO2 microring resonator (MRR) based active modulator in this article. Our designed MRR modulator employs an Indium-Tin-Oxide (ITO)-SiN-ITO thin-film stack, in which the ITO thin films act as the upper and lower claddings of the SiN MRR. The ITO-SiN-ITO thin-film stack leverages the free-carrier assisted, high-amplitude refractive index change in the ITO films to effect a large electro-refractive optical modulation in the device. Based on the electrostatic, transient, and finite difference time domain (FDTD) simulations, conducted using photonics foundry-validated tools, we show that our modulator achieves 280 pm/V resonance modulation efficiency, 67.8 GHz 3-dB modulation bandwidth, ∼19 nm free-spectral range (FSR), ∼0.23 dB insertion loss, and 10.31 dB extinction ratio for optical on-off-keying (OOK) modulation at 30 Gb/s.', 'arxivid': '2212.06326', 'author': ['Venkata Sai \nDepartment of ECE\nDepartment of ECE\nUniversity of Kentucky Lexington\nKentuckyUSA\n', 'Praneeth Karempudi \nDepartment of ECE\nDepartment of ECE\nUniversity of Kentucky Lexington\nKentuckyUSA\n', 'Ishan G Thakkar [email protected] \nDepartment of ECE\nUniversity of Kentucky Lexington\nKentuckyUSA\n', 'Jeffrey Todd Hastings [email protected] \nUniversity of Kentucky Lexington\nKentuckyUSA\n'], 'authoraffiliation': ['Department of ECE\nDepartment of ECE\nUniversity of Kentucky Lexington\nKentuckyUSA', 'Department of ECE\nDepartment of ECE\nUniversity of Kentucky Lexington\nKentuckyUSA', 'Department of ECE\nUniversity of Kentucky Lexington\nKentuckyUSA', 'University of Kentucky Lexington\nKentuckyUSA'], 'corpusid': 254591466, 'doi': '10.1109/ises54909.2022.00069', 'github_urls': [], 'n_tokens_mistral': 9109, 'n_tokens_neox': 7436, 'n_words': 4066, 'pdfsha': '2ec561b700f940f445e29f827d4d28149e9d0e97', 'pdfurls': ['https://export.arxiv.org/pdf/2212.06326v1.pdf'], 'title': ['A Silicon Nitride Microring Based High-Speed, Tuning-Efficient, Electro-Refractive Modulator', 'A Silicon Nitride Microring Based High-Speed, Tuning-Efficient, Electro-Refractive Modulator'], 'venue': []}
arxiv
New bounds for covering codes of radius 3 and codimension 3t + 1 * Alexander A Davydov [email protected] Department of Mathematics and Computer Science Perugia University 06123PerugiaItaly Stefano Marcugini [email protected] Department of Mathematics and Computer Science Perugia University 06123PerugiaItaly Fernanda Pambianco [email protected] Department of Mathematics and Computer Science Perugia University 06123PerugiaItaly New bounds for covering codes of radius 3 and codimension 3t + 1 * * The research of S. Marcugini and F. Pambianco was supported in part by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA -INDAM) (Contract No. U-UFMBAZ-2019-000160, 11.02.2019) and by University of Perugia (Project No. 98751: Strutture Geometriche, Combinatoria e loro Applicazioni, Base Research Fund 2017-2019).Covering codethe length functionsaturating setelliptic quadricpro- jective space Mathematics Subject Classification (2010) 94B0551E2151E22 The smallest possible length of a q-ary linear code of covering radius R and codimension (redundancy) r is called the length function and is denoted by q (r, R). In this work, for q an arbitrary prime power, we obtain the following new constructive upper bounds on q (3t + 1, 3):18 · q (r−3)/3 · 3 ln q, r = 3t + 1, t ≥ 1, q large enough.For t = 1, we use a one-to-one correspondence between codes of covering radius 3 and codimension 4, and 2-saturating sets in the projective space PG(3, q). A new construction providing sets of small size is proposed. The codes, obtained by geometrical methods, are taken as the starting ones in the lift-constructions (so-called "q m -concatenating constructions") to obtain infinite families of codes with radius 3 and growing codimension r = 3t + 1, t ≥ 1. The new bounds are essentially better than the known ones. Introduction Let F q be the Galois field with q elements. Let F * q = F q \ 0. Let F n q be the n-dimensional vector space over F q . A linear code in F n q with codimension (redundancy) r, minimum distance d, and covering radius R is said to be an [n, n − r, d] q R code; d is omitted when not relevant. All words in F r q can be obtained as a linear combination of at most R columns of a parity check matrix of an [n, n − r, d] q R code. Also, for this code, the space F n q is covered by spheres of radius R centered at the codewords. For an introduction to coding theory, see [28,32]. The minimum length n such that an [n, n−r] q R code exists is called the length function and is denoted by q (r, R). If covering radius and codimension are fixed then the covering problem for codes is that of finding codes with small length. Codes investigated from the point of view of the covering problem are called covering codes. Studying covering codes is a classical combinatorial problem. Covering codes are connected with many areas of theory and practice, see e.g. [8,Section 1.2], [17,Introduction], [7]. For an introduction to coverings of Hamming spaces over finite fields and covering codes, see [6,8,12,31,35]. This paper is devoted to the upper bound on the length function q (3t + 1, 3), t ≥ 1. Let PG(N, q) be the N -dimensional projective space over the Galois field F q . To obtain bounds on q (4, 3) we use 2-saturating sets in PG (3, q). A point set S ⊂ PG(3, q) is 2-saturating if any point of PG (3, q) belongs to a plane generated by three non-collinear points of S. Let s q (3, 2) be the smallest size of a 2-saturating set in PG (3, q). If the positions of a column of a parity check matrix of an [n, n − 4] q 3 code are treated as homogeneous coordinates of a point in PG (3, q) then the matrix defines a 2-saturating n-set in PG (3, q), and vice versa. So, there is a one-to-one correspondence between [n, n − 4] q 3 codes and 2-saturating n-sets in PG (3, q). Therefore, q (4, 3) = s q (3,2). For an introduction to geometries over finite fields and their connections with coding theory, see [12,22,23,[25][26][27]30]. Throughout the paper, c is a constant independent of q but it is possible that c is dependent on r and R. In [3,15], [20,Proposition 4.2.1], see also the references therein, the following lower bound is considered: q (r, R) ≥ cq (r−R)/R , R and r fixed. (1.1) In [12], the bound (1.1) is given in another (asymptotic) form. Let t, s be integer. Let q be a prime power. In the literature, the bound (1.1) is achieved in the following cases: [12,20,21,24]; R = sR , r = tR + s, q = (q ) R [12,14]; r = tR, q is an arbitrary prime power [11,12,14,18,19]. r = tR, q = (q ) R In the general case, for arbitrary r, R, q, in particular when r = tR and q is an arbitrary prime power, the problem of achieving the bound (1.1) is open. For r = tR + 1, in [1-5, 15, 16, 33] (for R = 2, 3) and [17] (for R ≥ 3) upper bounds of the following form are obtained: q (tR + 1, R) ≤ cq (r−R)/R · R ln q, q is an arbitrary prime power. (1.2) In the bounds (1.2), the "price" of the non-restrict structure of q is the factor R √ ln q. In this paper, see Sections 3.4 and 3.5, for q an arbitrary prime power, we obtain the following new constructive upper bounds on q (3t + 1, 3) of the form (1.2): • q (r, 3) 3 √ k · q (r−3)/3 · 3 ln q, r = 3t + 1, t ≥ 1, q ≥ W(k) , 18 < k ≤ 20.339, W(k) is a decreasing function of k; • q (r, 3) 3 √ 18 · q (r−3)/3 · 3 ln q, r = 3t + 1, t ≥ 1, q large enough. We consider the case t = 1 in projective geometry language. We propose a step-bystep algorithm obtaining a 2-saturating n-set in PG(3, q) that is a subset of an elliptic quadric and corresponds to an [n, n − 4, 4] q 3 code, see Section 2. Estimates of the size of the obtained set give the bounds on q (4, 3), see Section 3. For t ≥ 2, we use the lift-constructions for covering codes. These constructions are variants of the so-called "q m -concatenating constructions" proposed in [10] and developed in [11][12][13][16][17][18][19], see also the references therein and [6], [8,Section 5.4]. The q m -concatenating constructions obtain infinite families of covering codes with growing codimension using a starting code with a small one. We take the [n, n − 4, 4] q 3 codes with t = 1 as the starting ones for the q m -concatenating constructions and obtain infinite families of covering codes with growing codimension r = 3t + 1, t ≥ 1. These families provide the constructive upper bounds on q (3t + 1, 3), t ≥ 1. The new bounds are essentially better than the best known ones of [17]; for details see the figures and the table in Section 3 and the figure and the relations in Section 5. In particular, in the region 14983 ≤ q < 5 · 10 6 , the ratio of values of the known and new upper bounds lies in the region 2.167 . . . 1.96 and the asymptotic ratio is ≈ 1.89. The 2-saturating n-set in PG(3, q), obtained in Section 2 of this work, corresponds to an [n, n − 4, 4] q 3 code with r = 4 and d = 4, whereas the known bound for r = 4 is provided by a code with d = 3. Note also that, for small q in the region 7577 < q ≤ 7949, we obtain by computer search new small 2-saturating sets in PG(3, q), see Section 3.6. The paper is organized as follows. In Section 2, a construction of 2-saturating sets in PG(3, q) is proposed. In Section 3, estimates of sizes of 2-saturating sets in PG(3, q), obtained by the proposed construction, and new bounds on q (4, 3) are considered. In Section 4, with the help of the q m -concatenating lift-constructions the new bounds on q (3t + 1, 3), t ≥ 1, are obtained. Finally, in Section 5, the best known upper bounds of [17] are given and are compared with the new ones. A construction of 2-saturating sets in PG(3, q) We say that a point P of PG(3, q) is covered by a point set K ⊂ PG(3, q) if P lies in a plane through 3 non-collinear points of K or on a line through 2 points of K. In these cases, we say also that the set K covers the point P . Obviously, all points of K are covered by K. Let Q ⊂ PG(3, q) be an elliptic quadric [25]; then #Q = q 2 + 1. Let B u ∈ PG(3, q) be a point of Q. So, Q = {B 1 , B 2 , . . . , B q 2 +1 }. We construct a 2-saturating set as a subset of the elliptic quadric by a step-by-step iterative process which adds a new point to the current set in every step. Let w > 0 be a fixed integer. Consider the (w + 1)-st step of the process. This step starts from a w-subset K w = {B 1 , B 2 , . . . , B w } ⊂ Q obtained in the previous w steps. Let B w+1 ∈ Q \ K w be the point that will be added to the subset in the (w + 1)-st step, i.e. K w+1 = K w ∪ {B w+1 }. Let U w and U w+1 be the subset of points of P G(3, q) that are not covered by K w and K w ∪ {B w+1 }, respectively. Clearly, U w+1 ⊆ U w . To obtain the next subset K w+1 , we can add to K w any of q 2 + 1 − w points of Q \ K w . Thus, there exist q 2 + 1 − w distinct points that can be taken as B w+1 . Let ∆(H) be the number of the new covered points of P G(3, q) in the (w + 1)-st step after adding a point H ∈ Q \ K w to K w . Let N (H) be the set of new points of P G(3, q) covered by K w ∪ {H} with H ∈ Q \ K w . Obviously, #N (H) = ∆(H). Introduce the point multiset N ∪ w+1 such that N ∪ w+1 H∈Q\Kw N (H), #N ∪ w+1 = H∈Q\Kw ∆(H). Let P ∈ U w be a point that is not covered by K w . Let S w (P ) be the number of the inclusions of P to the multiset N ∪ w+1 when one sequentially adds all points H of Q \ K w to K w to obtain variants K w+1 = K w ∪ {H}. Let S min w min P ∈Uw S w (P ). Then #N ∪ w+1 = P ∈Uw S w (P ) ≥ S min w · #U w . (2.1) Let ∆ aver w+1 be the average number of new covered points in the (w+1)-st step calculated over all #Q \ K w = q 2 + 1 − w possible points H. By (2.1), ∆ aver w+1 = #N ∪ w+1 q 2 + 1 − w ≥ S min w · #U w q 2 + 1 − w . (2.2) Obviously, there exists a point H * ∈ Q \ K w providing the inequality ∆(H * ) ≥ ∆ aver w+1 . We take this point H * as B w+1 . Then, by (2.2), ∆(B w+1 ) = #U w − #U w+1 ≥ ∆ aver w+1 ≥ S min w · #U w q 2 + 1 − w ≥ S min w · #U w q 2 + 1 − w ; (2.3) #U w+1 ≤ #U w − S min w · #U w q 2 + 1 − w ≤ #U w 1 − S min w q 2 + 1 − w . (2.4) The iterative process ends when #U w+1 ≤ 1. Then one should add one point of Q to K w to obtain a 2-saturating set. Theorem 2.1. The 2-saturating n-set obtained by the construction above corresponds to an [n, n − 4, 4] q 3 code with minimum distance d = 4. Proof. All the points of the 2-saturating n-set belong to the elliptic quadric Q. 3 Estimates of sizes of 2-saturating sets in PG (3, q) and new bounds on q (4, 3) Let θ N,q = (q N +1 − 1)/(q − 1) be the number of points in the projective space PG(N, q). Obviously, any three points of Q cover some plane; therefore #U 3 = θ 3,q − θ 2,q = q 3 . (3.1) We consider our new bounds in the form (1.2). For this, note that if r = 4, R = 3, then q (r−R)/R · R √ ln q = 3 √ q ln q. Estimates of S min w Many important properties of the elliptic quadric are given in [25,Section 16]. In particular, a plane containing at least two points of Q intersects Q in a (q + 1)-arc. Through every point of Q we have one tangent plane and q(q + 1) secant planes. We have no planes external to Q. Lemma 3.1. Let A 1 and A 2 be two (q + 1)-arcs that are intersections of Q and planes π 1 and π 2 , respectively. Then #(A 1 ∩ A 2 ) = 2, 1, 0 if π 1 = π 2 and the intersection line π 1 ∩ π 2 is a bisecant, a tangent, and an external line to Q, respectively. Proof. It easy to see that A 1 ∩ A 2 = π 1 ∩ π 2 ∩ Q and the lemma follows immediately. Lemma 3.2. Let a point P of PG(3, q) be not covered by K w . It is possible P ∈ Q and P / ∈ Q as well. Then S min w ≥ w 2 q − w 2 if 2 w 2 − 1 ≤ q; (3.2) S min w ≥ q 2 − 1 4 if 2 w 2 − 1 > q and q is odd; (3.3) S min w ≥ q 2 4 if 2 w 2 − 1 > q and q is even. (3.4) Proof. Let π(T, Q, R) be the plane through three distinct points T, Q, R of P G(3, q). Let π w = {π(B i , B j , B k )|1 ≤ i < j < k ≤ w} be the multiset of w 3 planes through three distinct points of K w . For 1 ≤ i < j ≤ w, we have π(B i , B j , P ) / ∈ π w otherwise the point P would be covered by K w . We consider the (q + 1)-arc A i,j (P ) π(B i , B j , P ) ∩ Q; then A i,j (P ) ∩ K w = {B i , B j }. Every point A i,j (P )\{B i , B j } provides the inclusion of P into N ∪ w+1 when one sequentially adds all points H of Q \ K w to K w to obtain variants K w+1 = K w ∪ H. If P ∈ Q then one of these points is P . So, for a fixed pair (i, j), we have q − 1 inclusions of P into N ∪ w+1 . To estimate S w (P ), we should consider all the w 2 pairs (i, j) and take into account all the intersections points of the arcs A i,j (P ) for distinct (i, j) with each other. Let Π w (P ) {π(B i , B j , P ) | 1 ≤ i < j ≤ w}. If two planes from Π w (P ) coincide with each other then we have a plane containing P and three or four points of K w ; it implies that P is covered by K w , contradiction. So, all the planes of Π w (P ) are pairwise distinct. The set Π w (P ) defines the w 2 -set A w (P ) of the (q − 1)-arcs such that A w (P ) {(π(B i , B j , P ) ∩ Q) \ {B i , B j } | 1 ≤ i < j ≤ w}. By Lemma 3.1, two arcs from A w (P ) intersect each other in at most two points. To obtain a lower bound of S w (P ) we can assume that any two arcs of A w (P ) intersect each other in two points. Moreover, for P / ∈ Q, we can assume that all intersection points in all pair of the arcs are distinct. This is the worst case. But, for P ∈ Q, one intersection point is the same for all pairs, it is P . Therefore, the lower bound for P / ∈ Q is less than the one for P ∈ Q and we can consider only P / ∈ Q. Let P / ∈ Q. We take n arcs from A w (P ). We assume that every arc intersects all n − 1 other arcs in two points and all the intersection points are distinct; it is the worst case for the value of S w (P ). As (q − 1) − 2(n − 1) must be ≥ 0, the considered case is possible if 2n−1 ≤ q. In all n arcs, the total number of the intersection points is 2 n 2 = n(n−1). The total number of distinct points in the union of n the arcs is n(q − 1) − n(n − 1) = n(q − n). If 2 w 2 − 1 ≤ q we put n = w 2 that proves (3.2). Let 2 w 2 − 1 > q. Let q be odd. We put n = (q + 1)/2 that implies (q − 1) − 2(n − 1) = 0, n(q − n) = (q 2 − 1)/4. So, (3.3) is proved. If q is even, we put n = q/2 that gives (q − 1) − 2(n − 1) = 1, n(q − n) = q 2 /4, and proves (3.4). Implicit Bound A By above, in particular by (2.4) and (3.1), the implicit Bound A can be given as follows: #U 3 = q 3 , #U w+1 = #U w − S min w · #U w q 2 + 1 − w , #U w A q +1 ≤ 1, n A 4,q w A q + 1.(3.5) We put w = 3, #U 3 = q 3 , and then, sequentially increasing w by 1, we calculate #U w+1 , as it is written in (3.5), until for some w we obtain #U w+1 ≤ 1; we denote this w as w A q ; also, let n A 4,q w A q + 1. We have obtained the implicit upper Bound A s(2, 3) = q (4, 3) ≤ n A 4,q . (3.6) For illustration and comparison, the Bound A in the form n A 4,q / 3 √ q ln q is shown by the second curve in Figure 1, in the region 7951 ≤ q < 10 5 , and the bottom curve in Figure 2, where 10 5 < q < 5 · 10 6 . The curves are obtained using S w from Lemma 3.2. By Figure 1, for 7951 ≤ q < 10 5 , the Bound A takes the values 3.38 n A 4,q / 3 √ q ln q 2.79. By Figure 2, for 10 5 < q < 5 · 10 6 , the Bound A takes the values 2.79 n A 4,q / 3 √ q ln q 2.63. Other curves in the figures will be explained below. Implicit Bound B By (2.4) and (3.1), #U w+1 ≤ #U w 1 − S min w q 2 + 1 − w ≤ q 3 w j=3 1 − S min j q 2 + 1 − w . (3.7) Similarly to Section 3.2, we put w = 3, #U 3 = q 3 ; then, sequentially increasing w by 1, we calculate #U w+1 by (3.7), until for some w we obtain #U w+1 ≤ 1. Let q 0 be a value such that for q ≥ q 0 in all steps of the considered process we have 2 j 2 − 1 ≤ q and use the variant of S min j as in (3.2). Then, for q ≥ q 0 we have #U w+1 ≤ #U 3 w j=3 1 − j 2 q − j 2 q 2 + 1 − j = q 3 f q (w), (3.8) f q (w) w j=3 1 − j 2 q − j 2 q 2 + 1 − j . (3.9) Lemma 3.3. Let q ≥ q 0 . To provide #U w+1 ≤ q 3 f q (w) ≤ 1 it is sufficient to take w satisfying the inequality (w − 1) 3 − 0.3w 5 q ≥ 18q ln q. (3.10) Proof. By (3.9) and the inequality 1 − x < exp(−x), we have Table 1) vs the known bound (5.1)-(5.2) (the top, solid curve), 10 5 < q < 5 · 10 6 f q (w) = w j=3 1 − j(j − 1) (2q − j(j − 1)) 4(q 2 + 1 − j) < w j=3 exp − j(j − 1)(2q − j(j − 1)) 4(q 2 + 1 − j)< w j=3 exp − j(j − 1)(2q − j(j − 1)) 4q 2 = exp − w j=3 j(j − 1)(2q − j(j − 1)) 4q 2 . After simple transformations, using the results of [29, Section 1.2.3.1], we obtain w j=3 j(j − 1) = −2 + w j=1 j(j − 1) = w 3 − w − 6 3 > (w − 1) 3 3 ; w j=3 j 2 (j − 1) 2 = −4 + w j=1 j 2 (j − 1) 2 = w 5 5 − w 3 3 + 2w 15 − 4 < w 5 5 ; f q (w) < exp − w 3 − w − 6 6q + 3w 5 − 5w 3 + 2w − 60 60q 2 < exp − (w − 1) 3 6q + w 5 20q 2 . Taking the logarithm of both the parts of the inequality q 3 f q (w) ≤ 1, we obtain 3 ln q − (w − 1) 3 6q + w 5 20q 2 ≤ 0, that implies the assertion. Let w B q be the smallest integer satisfying the inequality (3.10) under the condition 2 w 2 − 1 ≤ q. Let n B 4,q w B q + 1, where "+1" takes into account that #U w+1 ≤ 1. We have obtained the implicit upper Bound B: s q (2, 3) = q (4, 3) ≤ n B 4,q . (3.11) Now we will estimate the value of q 0 . If w 2 − w − 1 ≤ q, then 2 w 2 − 1 ≤ q. So, we can consider w ≤ √ q. Let δ(q) be the difference between the left and right parts of the inequality (3.10) under condition that w = √ q. In other words, δ(q) ( √ q − 1) 3 − 0.3q √ q − 18q ln q. Considering the corresponding derivatives, it can be shown that for q ≥ 88274 we have δ(q) > 0. For simplicity of presentation, we can put q 0 = 10 5 . For illustration and comparison, the Bound B in the form n B 4,q / 3 √ q ln q is shown by the second, dashed curve in Figure 2, where q 0 = 10 5 < q < 5 · 10 6 . By Figure 2, for 10 5 < q < 5 · 10 6 , the Bound B takes the values 2.964 n B 4,q / 3 √ q ln q 2.7. Explicit Bound C We will find the solution of the inequality (3.10) in the form w = 3 √ kq ln q , where k > 0 is independent of q. For the convenience of research we write w = 3 √ kq ln q + 1. The inequality (3.10) takes the form (k − 18)q ln q − 0.3( 3 √ kq ln q + 1) 5 q ≥ 0 (3.12) that implies k > 18, 3 √ k > 3 √ 18 ≈ 2. 6207. Let > 0 be an arbitrary constant independent of q. Let V be a value such that 0.302( 3 (18 + )q ln q) 5 ≥ 0.3( 3 (18 + )q ln q + 1) 5 if q > V. (3.13) Considering the corresponding derivatives, it can be shown that V = 1516750. So, we can consider the inequality (k − 18)q ln q − 0.302( 3 √ kq ln q ) 5 q ≥ 0, q > V = 1516750, k > 18. (3.14) If (3.14) holds, then (3.12) holds also. We denote F (k, q) k − 18 0.302 3 1 k 5 − ln 2 q q , q > V, k > 18. (3.15) The inequality (3.14) is equivalent to F (k, q) ≥ 0. The derivative of F (k, q) with respect to q is d dq F (k, q) = ln q(ln q − 2) q 2 > 0, q > V. (3.16) So, F (k, q) is an increasing function of q for a fixed k and q > V. Therefore, for q > V, the equation F (k, q) = 0 with a fixed k has only one solution with respect to q. We denote this solution W(k). As W(k) can be non-integer, below we use W(k) . By above, W(k) is the smallest integer q satisfying (3.14) for the fixed k. Thus, for q ≥ W(k) ≥ W(k), the inequality (3.14) holds under the condition W(k) > V = 1516750. By (3.15), the equation F (k, q) = 0 can be written in the form ln 2 q q = k − 18 0.302 3 1 k 5 , k > 18. (3.17) Note that this equation is connected with Lambert W function, see e.g. [9]. By above we have the theorem. > V = 1516750. Let w C q (k) 3 √ kq ln q + 1, n C 4,q (k) w C q (k) + 1 = 3 √ kq ln q + 2. Then s q (2, 3) = q (4, 3) ≤ n C 4,q (k) = 3 kq ln q + 2 for q ≥ W(k) . (3.18) We call n C 4,q (k) explicit Bound C. Example 3.5. Some values of W(k) are given in Table 1. Also, in the 3-rd column of the table the values of n C 4,q (k)/ 3 √ q ln q are written such that n C 4,q (k) 3 √ q ln q = 3 √ k + 2 3 √ q ln q , q = W(k) . (3.19) They are useful, in particular, for comparison with the known results, see Section 5, the 4-th and 5-th columns of Table 1, and Figure 2, where the values of n C 4,q (k)/ 3 √ q ln q with k = 20.339, 20, 19.7 are presented. The derivative of F (k, q) with respect to k is d dk F (k, q) = (k − 18) 2 0.302 3 k 5 90 k − 2 > 0, 18 < k ≤ 20.339. (3.20) By (3.20), F (k, q) is an increasing function of k for a fixed q. Let F (k , q ) = 0 and let k > k . Then F (k , q ) > 0. As, by (3.16), F (k, q) is an increasing function of q for a fixed k, there exists q < q such that F (k , q ) = 0. So, W(k) is a decreasing function of k and, therefore, W (k) is a non-increasing function of k. As W(k) should be > V, by Table 1 Asymptotic Bound D The inequality (3.14) can be written in the form q ln q   (k − 18) − 0.302k 3 k 2 ln 2 q q   ≥ 0, q > V, 18 < k ≤ 20.339. (3.22) We have lim q→∞ 0.302k 3 k 2 ln 2 q q = 0, 18 < k ≤ 20.339. Therefore, for q large enough, the inequality (3.22) holds; together with it, the inequalities (3.14) and (3.12) hold also. Thus (see the beginning of Section 3.4), for w = 3 √ kq ln q + 1, 18 < k ≤ 20.339, the inequality (3.10) holds if q is large enough. We take k = 18+ε where ε > 0 is an arbitrary small constant independent of q. Let w D q 3 (18 + ε)q ln q + 1. Then s q (2, 3) = q (4, 3) ≤ n D 4,q w D q + 1 = 3 (18 + ε)q ln q + Bound E for relatively small q In [4,[15][16][17]19], see also the references therein, small 2-saturating sets in PG(3, q) for the region 13 ≤ q ≤ 7577 are obtained by computer search using the so-called "algorithms with the fixed order of points (FOP)" and "randomized greedy algorithms". These algorithms are described in detail in [4,15]. In this paper, we continue the computer search and obtain new small 2-saturating sets in PG(3, q) for the region 7577 < q ≤ 7949. We denote by s q (3, 2) the size of the smallest known 2-saturating sets in PG (3, q). The sets obtained in [4,[15][16][17]19] and in this paper (one set of [34] is used also) provide the following theorem. Theorem 3.6. In the projective space PG(3, q), for the smallest size s q (3, 2) of a 2saturating set the following upper bound holds: For illustration and comparison, the Bound E in the form n E 4,q / 3 √ q ln q is shown by the bottom curve in Figure 1, in the region 13 ≤ q < 7949. 4 Codes with growing codimension r = 3t + 1, t ≥ 1 s q (3, 2) ≤ s q (3, 2) ≤ n E 4, For upper bounds on the length function q (r, 3), r = 3t + 1 ≥ 7, an important tool is given by the inductive lifting constructions of [16,19] which provide Proposition 4.1. These constructions are variants of q m -concatenating constructions [10][11][12][13][16][17][18][19]. We denote ∆(r, q) 3 q (r−7)/3 + 2 q (r−10)/3 + δ r,13 , δ i,j is the Kronecker symbol. [19,Theorem 14] Let an [n 0 , n 0 − 4] q 3 code with n 0 < q exist. Then there is an infinite family of [n, n − r] q 3 codes with parameters n = n 0 q (r−4)/3 + ∆(r, q), r = 3t + 1 ≥ 4, t ≥ 1. (4.1) Corollary 4.2. Let n 0 = n • 4,q ∈ {n A 4,q , n B 4,q , n D 4,q , n E 4,q } or n 0 = n C 4,q (k), where n A 4,q , n B 4,q , n D 4,q , n E 4,q ,and n C 4,q (k) are given in Section 3. Let the region of q for n • 4,q and n C 4,q (k) be as in Section 3. Then, for the same region of q, there is an infinite family of [n • r,q , n • r,q − r] q 3 or [n C r,q (k), n C r,q (k) − r] q 3 codes with r = 3t + 1 ≥ 4, t ≥ 1, and length n • r,q = n • 4,q q (r−4)/3 + ∆(r, q), n • 4,q ∈ {n A 4,q , n B 4,q , n E 4,q }; (4.2) n C r,q (k) = 3 √ k · q (r−3)/3 · 3 ln q + 2q (r−4)/3 + ∆(r, q), 18 < k ≤ 20.339, q ≥ W(k) ; (4.3) n D r,q (k) = 3 √ 18 + ε · q (r−3)/3 · 3 ln q + 2q (r−4)/3 + ∆(r, q), q large enough. Proof. By Section 3, for all n A 4,q , n B 4,q , n C 4,q (k), n D 4,q , n E 4,q we have n 0 = n • 4,q < q and n 0 = n C 4,q (k) < q. By (3.21), (3.24), (4.3), (4.4), for r = 3t + 1 ≥ 4, t ≥ 2, we have lim q→∞ n C r,q (k) q (r−3)/3 · 3 √ ln q = lim q→∞ n C 4,q (k) q (4−3)/3 · 3 √ ln q = 3 √ k, 18 < k ≤ 20.339, k fixed; lim q→∞, ε→0 n D r,q q (r−3)/3 · 3 √ ln q = lim q→∞, ε→0 n D 4,q q (4−3)/3 · 3 √ ln q = 3 √ 18 ≈ 2.6207. (4.5) Comparison with the known results As far as the authors know, the best known bounds on q (3t + 1, 3) are given in [17] where for R = 3, r = 3t + 1, t ≥ 1, the following results are obtained: q (4, 3) ≤ Ω λ,3 (q) 3 q ln q + 6, q (r, 3) ≤ (Ω λ,3 (q) 3 q ln q + 6)q r−4 3 + 3θ t−1,q , (5.1) Ω λ,3 (q) λ + 36 β 2 λ,3 (q) 2 − 1 q − Υ λ,3 (q) , Υ λ,3 (q) λ 2 2 3 ln 2 q q , β λ,3 (q) λ − 2 3 √ q ln q . Here λ > 0 is a positive constant independent of q, its value can be assigned arbitrarily. where "knw" notes the known results. For illustration and comparison, the known bound n knw 4,q in the form n knw 4,q / 3 √ q ln q is shown by the top curve in Figure 1, for 14983 ≤ q < 10 5 , and Figure 2, for 10 5 < q < 5 · 10 6 . By Figure 1, for 14983 ≤ q < 10 5 , the known bound takes values 6.80 n knw 4,q / 3 √ q ln q 5.75. By Figure 2, for 10 5 < q < 5 · 10 6 , the known bound takes values 5.75 n knw 4,q / 3 √ q ln q 5.2. The ratio n knw 4,q /n A 4,q of the known upper bound n knw 4,q (5.1)-(5.2) on the length function q (4, 3) and the new one n A 4,q in the region 14983 ≤ q < 5 · 10 6 is shown in Figure 3, where the ratio lies in ≈ 2.167 . . . ≈ 1.96. The ratio n knw 4,q /n C 4,q (k) of the known bound and the new one n C 4,q (k) in the region 1517567 < q is shown in Table 1 where it lies in ≈ 1.918 . . . ≈ 1.89. By Table 1 Figure 3: The ratio n knw 4,q /n A 4,q of the known [17] upper bound n knw 4,q (5.1)-(5.2) on the length function q (4, 3) and the new one n A 4,q (3.5)-(3.6), 14983 ≤ q < 5 · 10 6 Figure 1 :Figure 2 : 12Upper bounds on the length function q (4, 3) divided by 3 √ q ln q: implicit Bound A (3.5)-(3.6) for 7951 ≤ q < 10 5 (the second curve), computer Bound E (3.25) for 13 ≤ q ≤ 7949 (the bottom curve) vs the known bound (5.1)-(5.2) for 14983 ≤ q < 10 5 (the top curve) Upper bounds on the length function q (4, 3) divided by3 √ q ln q: implicit Bound A (3.5)-(3.6) (the bottom, solid curve), implicit Bound B (3.10)-(3.11) (the second, dashed curve), and explicit Bound C (points 1, 2, and 3 correspond to n C 4,q (k)/ 3 √ q ln q with k = 20.339, 20, and 19.7, by Theorem 3. 4 . 4(explicit Bound C) Let W(k) be the solution with respect to q of the equation(3.17). Let k > 18 be such that W(k) Table 1 : 1Values of W(k) for 18.0001 ≤ k ≤ 20.340 and values of n C 4,q (k)/ 3 √ q ln q, n knw 4,q / 3 √ q ln q (the known bound), n knw 4,q /n C 4,q (k) for q = W(k) , 18.0001 ≤ k ≤ 20.339. (V = 1516750) k W(k) n C 4,q (k) 3 √ q ln q n knw 4,q 3 √ q ln q n knw 4,q n C 4,q (k) 20.340 1515738 ≈ 1.516 · 10 6 < V 20.339 1517567 ≈ 1.518 · 10 6 > V 2.7368 5.2500 1.9183 20.335 1524915 ≈ 1.525 · 10 6 > V 2.7367 5.2495 1.9182 20 2374364 ≈ 2.374 · 10 6 > V 2.7205 5.2087 1.9146 19.7 3820987 ≈ 3.821 · 10 6 > V 2.7059 5.1716 1.9112 19 19178705 ≈ 1.918 · 10 7 > V 2.6713 5.0828 1.9027 18.5 171670620 ≈ 1.717 · 10 8 > V 2.6461 5.0180 1.8963 18.1 30640000001 ≈ 3.064 · 10 10 > V 2.6258 4.9659 1.8912 18.05 294427001643 ≈ 2.944 · 10 11 > V 2.6233 4.9593 1.8905 18.01 52060446118120 ≈ 5.206 · 10 13 > V 2.6212 4.9542 1.8900 18.001 ≈ 7.880 · 10 16 > V 2.6208 4.9530 1.8899 18.0001 ≈ 1.109 · 10 20 > V 2.6207 4.9529 1.8899 qc E 4 3 q ln q, c E 4 =      2.61 if 13 ≤ q ≤ 4373 2.65 if 4373 < q ≤ 7723 2.69 if 7723 < q ≤ 7949 . (3.25) The bounds (5.1) hold if q > y where y is a solution of the equation Υ λ,3 (y) = 1 under the condition y > e 2 [17, (3.7), Remark 6.4].In [17, Section 6,Table 1], it is shown that λ = 3 √ 36 minimizes Ω λ,3 ; for this λ, the bounds (5.1) hold if q ≥ 14983. We denote 36,3 (q) 3 q ln q + 6, n knw r,qn knw 4,q Ω 3 √ n knw 4,q · q r−4 3 + 3θ t−1,q , r = 3t + 1, t ≥ 1, (5.2) By Figures 1-3,Table 1, and (5.4), one sees that the new bounds are essentially better than the known ones., (4.5), (5.1), for asymptotic estimates we have lim q→∞ n knw 4,q 3 √ q ln q = lim q→∞ n knw r,q q (r−3)/3 · 3 √ ln q = 3 2 3 √ 36 ≈ 4.953; (5.3) lim q→∞, ε→0 n knw 4,q n D 4,q ≈ 1.5 3 √ 2 ≈ 1.8899. (5.4) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10 6 1.95 2 2.05 2.1 2.15 New upper bounds on the smallest size of a saturating set in a projective plane. D Bartoli, A A Davydov, M Giulietti, S Marcugini, F Pambianco, 10.1109/RED.2016.77793202016 XV Int. Symp. Problems of Redundancy in Inform. and Control Systems (REDUNDANCY). St. Petersburg, RussiaIEEED. Bartoli, A.A. Davydov, M. Giulietti, S. Marcugini and F. Pambianco, New upper bounds on the smallest size of a saturating set in a projective plane, in 2016 XV Int. Symp. Problems of Redundancy in Inform. and Control Sys- tems (REDUNDANCY), IEEE, St. Petersburg, Russia, Sept. 2016, 18-22. https : //doi.org/10.1109/RED.2016.7779320 D Bartoli, A A Davydov, M Giulietti, S Marcugini, F Pambianco, New bounds for linear codes of covering radius 2, in Coding Theory and Applications (Proc. 5th. D. Bartoli, A. A. Davydov, M. Giulietti, S. Marcugini and F. Pambianco, New bounds for linear codes of covering radius 2, in Coding Theory and Applications (Proc. 5th . 10.1007/978-3-319-66278-7_1Lecture Notes in Computer Science. Int. Castle Meeting, ICMCTA 2017) (eds.Á. I. Barbero, V. Skachek, Ø. Ytrehus10495SpringerInt. Castle Meeting, ICMCTA 2017) (eds.Á. I. Barbero, V. Skachek, Ø. Ytrehus), Vihula, Estonia, Aug. 2017, Lecture Notes in Computer Science, Springer, Cham, 10495, 1-10 (2017) https : //doi.org/10.1007/978 − 3 − 319 − 66278 − 7 1 New bounds for linear codes of covering radii 2 and 3. D Bartoli, A A Davydov, M Giulietti, S Marcugini, F Pambianco, 10.1007/s12095-018-0335-0Crypt. Commun. 115D. Bartoli, A.A. Davydov, M. Giulietti, S. Marcugini and F. Pambianco, New bounds for linear codes of covering radii 2 and 3. Crypt. Commun. 11(5), 903-920 (2019) https : //doi.org/10.1007/s12095 − 018 − 0335 − 0 D Bartoli, A A Davydov, S Marcugini, F Pambianco, arXiv:1712.07078[csIT](2020Tables, bounds and graphics of short linear codes with covering radius 3 and codimension 4 and 5. D. Bartoli, A.A. Davydov, S. Marcugini and F. Pambianco, Tables, bounds and graphics of short linear codes with covering radius 3 and codimension 4 and 5, arXiv:1712.07078 [cs IT] (2020) https : //arxiv.org/abs/1712.07078 On defining sets for projective planes. E Boros, T Szőnyi, K Tichler, 10.1016/j.disc.2004.12.015Discrete Math. 303E. Boros, T. Szőnyi and K. Tichler, On defining sets for projective planes, Discrete Math., 303, 17-31 (2005) https : //doi.org/10.1016/j.disc.2004.12.015 Covering radius. R A Brualdi, S Litsyn, V Pless, Handbook of coding theory. V.S. Pless, W.C. HuffmanAmsterdamElsevier1R.A. Brualdi, S. Litsyn and V. Pless, Covering radius. In: V.S. Pless, W.C. Huffman (eds.) Handbook of coding theory, vol. 1, pp. 755-826. Elsevier, Amsterdam (1998) H Chen, L Qu, C Li, S Lyu, L Xu, arXiv:2208.01138[csIT](2020Generalized Singleton type upper bounds. H. Chen, L. Qu, C. Li, S. Lyu and L. Xu, Generalized Singleton type upper bounds, arXiv:2208.01138 [cs IT] (2020) https : //arxiv.org/abs/2208.01138 Covering codes. G Cohen, I Honkala, S Litsyn, A Lobstein, Elsevier54North-Holland Mathematical Library; Amsterdam, The NetherlandsG. Cohen, I. Honkala, S. Litsyn and A. Lobstein, Covering codes, North-Holland Mathematical Library, vol. 54. Elsevier, Amsterdam, The Netherlands (1997) R M Corless, G H Gonnet, D E G Hare, D J Jeffrey, D E Knuth, 10.1007/BF02124750On the Lambert W function (PostScript). 5R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey and D.E. Knuth, On the Lambert W function (PostScript), Adv. Computat. Math., 5(1), 329-359 (1996) https : //doi.org/10.1007/BF02124750 Construction of linear covering codes. A A Davydov, Problems Inform. Transmiss. 26A.A. Davydov, Construction of linear covering codes, Problems Inform. Transmiss. 26, 317-331 (1990) https : //www.researchgate.net/publication/268619798 Constructions and families of covering codes and saturated sets of points in projective geometry. A A Davydov, 10.1109/18.476339IEEE Trans. Inform. Theory. 6A.A. Davydov, Constructions and families of covering codes and saturated sets of points in projective geometry, IEEE Trans. Inform. Theory, 41(6), 2071-2080 (1995) https : //doi.org/10.1109/18.476339 Linear nonbinary covering codes and saturating sets in projective spaces. A A Davydov, M Giulietti, S Marcugini, F Pambianco, 10.3934/amc.2011.5.119Adv. Math. Commun. 51A.A. Davydov, M. Giulietti, S. Marcugini and F. Pambianco, Linear nonbinary cov- ering codes and saturating sets in projective spaces, Adv. Math. Commun. 5(1), 119-147 (2011) https : //doi.org/10.3934/amc.2011.5.119 Linear codes with covering radius 2, 3, and saturating sets in projective geometry. A A Davydov, S Marcugini, F Pambianco, 10.1109/TIT.2004.825503IEEE Trans. Inform. Theory. 503A.A. Davydov, S. Marcugini and F. Pambianco, Linear codes with covering radius 2, 3, and saturating sets in projective geometry, IEEE Trans. Inform. Theory, 50(3), 537-541 (2004) https : //doi.org/10.1109/TIT.2004.825503 New covering codes of radius R, codimension tR and tR + R 2 , and saturating sets in projective spaces. A A Davydov, S Marcugini, F Pambianco, 10.1007/s10623-019-00649-2Des. Codes Cryptogr. 87A.A. Davydov, S. Marcugini and F. Pambianco, New covering codes of radius R, codimension tR and tR + R 2 , and saturating sets in projective spaces, Des. Codes Cryptogr., 87, 2771-2792 (2019) https : //doi.org/10.1007/s10623 − 019 − 00649 − 2 New bounds for linear codes of covering radius 3 and 2-saturating sets in projective spaces. A A Davydov, S Marcugini, F Pambianco, 10.1109/REDUNDANCY48165.2019.9003348Proc. 2019 XVI Int. Symp. Problems Redundancy Inform. Control Systems (REDUN-DANCY). 2019 XVI Int. Symp. Problems Redundancy Inform. Control Systems (REDUN-DANCY)Moscow, RussiaA.A. Davydov, S. Marcugini and F. Pambianco, New bounds for linear codes of covering radius 3 and 2-saturating sets in projective spaces, in Proc. 2019 XVI Int. Symp. Problems Redundancy Inform. Control Systems (REDUN- DANCY), Moscow, Russia, Oct. 2019, IEEE Xplore, 47-52 (2020) https : //doi.org/10.1109/REDUNDANCY48165.2019.9003348 Bounds for complete arcs in PG(3,q) and covering codes of radius 3, codimension 4, under a certain probabilistic conjecture, Computational Science and Its Applications -ICCSA 2020. A A Davydov, S Marcugini, F Pambianco, https:/link.springer.com/chapter/10.1007/978-3-030-58799-4_8Lecture Notes in Computer Science. 2020SpringerA.A. Davydov, S. Marcugini and F. Pambianco, Bounds for complete arcs in PG(3,q) and covering codes of radius 3, codimension 4, under a certain probabilis- tic conjecture, Computational Science and Its Applications -ICCSA 2020. ICCSA 2020. Lecture Notes in Computer Science, vol 12249, pp 107-122. Springer, Cham. https : //link.springer.com/chapter/10.1007/978 − 3 − 030 − 58799 − 4 8 Upper bounds on the length function for covering codes with covering radius R and codimension tR + 1. A A Davydov, S Marcugini, F Pambianco, 10.3934/amc.2021074Adv. Math. Commun. 171A.A. Davydov, S. Marcugini and F. Pambianco, Upper bounds on the length func- tion for covering codes with covering radius R and codimension tR + 1, Adv. Math. Commun., 17(1), 98-118 (2023) https : //doi.org/10.3934/amc.2021074 Linear codes with covering radius R = 2, 3 and codimension tR. A A Davydov, P R J Östergård, 10.1109/18.904551IEEE Trans. Inform. Theory. 471A.A. Davydov and P.R.J.Östergård, Linear codes with covering radius R = 2, 3 and codimension tR, IEEE Trans. Inform. Theory, 47(1), 416-421 (2001) https : //doi.org/10.1109/18.904551 A A Davydov, P R J Östergård, 10.1007/s10623-009-9322-yLinear codes with covering radius 3, Designs, Codes and Cryptography. 54A.A. Davydov and P.R.J.Östergård, Linear codes with covering radius 3, Designs, Codes and Cryptography, 54(3), 253-271 (2010) https : //doi.org/10.1007/s10623 − 009 − 9322 − y Constructing saturating sets in projective spaces using subgeometries. L Denaux, 10.1007/s10623-021-00951-yDes. Codes Cryptogr. 909L. Denaux, Constructing saturating sets in projective spaces using subgeometries, Des. Codes Cryptogr., 90(9), 2113-2144 (2022) https : //doi.org/10.1007/s10623 − 021 − 00951 − y Higgledy-piggledy sets in projective spaces of small dimension. L Denaux, 10.37236/10736#P3.29Electron. J. Combin. 293L. Denaux, Higgledy-piggledy sets in projective spaces of small dimension, Electron. J. Combin. 29(3), #P3.29 (2022) https : //doi.org/10.37236/10736 T Etzion, L Storme, 10.1007/s10623-015-0156-5Galois geometries and coding theory. 78T. Etzion and L. Storme, Galois geometries and coding theory, Des. Codes Cryptogr., 78, 311-350 (2016) https : //doi.org/10.1007/s10623 − 015 − 0156 − 5 The geometry of covering codes: small complete caps and saturating sets in Galois spaces. M Giulietti, 10.1017/CBO9781139506748.003Surveys in Combinatorics. S. R. Blackburn, R. Holloway and M. WildonCambridgeCambridge University Press409M. Giulietti, The geometry of covering codes: small complete caps and sat- urating sets in Galois spaces, in Surveys in Combinatorics 2013, (eds. S. R. Blackburn, R. Holloway and M. Wildon) London Math. Soc., Lecture Note Series, 409, Cambridge University Press, Cambridge (2013), 51-90. https : //doi.org/10.1017/CBO9781139506748.003 Short minimal codes and covering codes via strong blocking sets in projective spaces. T Héger, Z Nagy, 10.1109/TIT.2021.3123730IEEE Trans. Inform. Theory. 682T. Héger and Z. Nagy, Short minimal codes and covering codes via strong blocking sets in projective spaces, IEEE Trans. Inform. Theory, 68(2), 881-890 (2022) https : //doi.org/10.1109/TIT.2021.3123730 J W P Hirschfeld, Finite Projective Spaces of Three Dimensions. OxfordOxford University PressJ.W.P. Hirschfeld, Finite Projective Spaces of Three Dimensions, Oxford University Press, Oxford, 1985. J W P Hirschfeld, Projective Geometries Over Finite Fields. Oxford mathematical monographs. OxfordClarendon Press2nd edn.J.W.P. Hirschfeld, Projective Geometries Over Finite Fields. Oxford mathematical monographs, Clarendon Press, Oxford, 2nd edn. (1998) The packing problem in statistics, coding theory and finite projective spaces: Update. J W P Hirschfeld, L Storme, 10.1007/978-1-4613-0283-4_13Finite Geometries (Proc. 4th Isle of Thorns Conf. A. Blokhuis, J.W.P. Hirschfeld, D. Jungnickel, J.A. ThasKluwer, Dordrecht3J.W.P. Hirschfeld and L. Storme, The packing problem in statistics, coding theory and finite projective spaces: Update 2001, in (eds. A. Blokhuis, J.W.P. Hirschfeld, D. Jungnickel, J.A. Thas), Finite Geometries (Proc. 4th Isle of Thorns Conf., July 16-21, 2000), Develop. Math., 3, Kluwer, Dordrecht, 2001, 201-246. https : //doi.org/10.1007/978 − 1 − 4613 − 0283 − 4 13 W C Huffman, V S Pless, Fundamentals of Error-Correcting Codes. Cambridge University PressW.C. Huffman and V.S. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, 2003 A Jeffrey, H H Dai, Handbook of Mathematical Formulas and Integrals, 4-th edition. AmsterdamAcademic PressA. Jeffrey and H.H. Dai, Handbook of Mathematical Formulas and Integrals, 4-th edition, Elsevier, Academic Press, Amsterdam, 2008. Galois geometry and coding theory. I Landjev, L Storme, Current Research Topics in Galois geometry. J. De Beule, L. StormeNew YorkI. Landjev and L. Storme, Galois geometry and coding theory, in Current Research Topics in Galois geometry, (eds. J. De Beule, L. Storme), Chapter 8, NOVA Aca- demic, New York, 2011, 187-214. Covering radius, an online bibliography. A Lobstein, A. Lobstein, Covering radius, an online bibliography (2019), https : //www.lri.fr/˜lobstein/bib − a − jour.pdf F J Macwilliams, N J A Sloane, The Theory of Error-Correcting Codes. Amsterdam, The NetherlandsNorth-Hollandthird ed.F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes, third ed., North-Holland, Amsterdam, The Netherlands, 1981 Saturating sets in projective planes and hypergraph covers. Z L Nagy, 10.1016/j.disc.2018.01.011Discrete Math. 3414Z.L. Nagy, Saturating sets in projective planes and hypergraph covers. Discrete Math. 341(4), 1078-1083 (2018) https : //doi.org/10.1016/j.disc.2018.01.011 7)-invariant 42-arcs in 3-dimensional projective spaces, Designs, Codes and Cryptography. A Sonnino, 10.1007/s10623-012-9778-zTransitive P SL. 722A. Sonnino, Transitive P SL(2, 7)-invariant 42-arcs in 3-dimensional projective spaces, Designs, Codes and Cryptography, 72(2), 455-463 (2014). https : //doi.org/10.1007/s10623 − 012 − 9778 − z Covering Codes. R Struik, 10.6100/IR425174The NetherlandsEindhoven University of TechnologyPh.D thesisR. Struik, Covering Codes, Ph.D thesis, Eindhoven University of Technology, The Netherlands, 1994. https : //doi.org/10.6100/IR425174
{'fraction_non_alphanumeric': 0.10802188903339795, 'fraction_numerical': 0.0895928934259282, 'mean_word_length': 3.0232994372593542, 'pattern_counts': {'":': 0, '<': 52, '<?xml version=': 0, '>': 43, 'https://': 0, 'lorem ipsum': 0, 'www.': 2, 'xml': 0}, 'pii_count': 4, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 31, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'The smallest possible length of a q-ary linear code of covering radius R and codimension (redundancy) r is called the length function and is denoted by q (r, R). In this work, for q an arbitrary prime power, we obtain the following new constructive upper bounds on q (3t + 1, 3):18 · q (r−3)/3 · 3 ln q, r = 3t + 1, t ≥ 1, q large enough.For t = 1, we use a one-to-one correspondence between codes of covering radius 3 and codimension 4, and 2-saturating sets in the projective space PG(3, q). A new construction providing sets of small size is proposed. The codes, obtained by geometrical methods, are taken as the starting ones in the lift-constructions (so-called "q m -concatenating constructions") to obtain infinite families of codes with radius 3 and growing codimension r = 3t + 1, t ≥ 1. The new bounds are essentially better than the known ones.', 'arxivid': '2305.11955', 'author': ['Alexander A Davydov [email protected] \nDepartment of Mathematics and Computer Science\nPerugia University\n06123PerugiaItaly\n', 'Stefano Marcugini [email protected] \nDepartment of Mathematics and Computer Science\nPerugia University\n06123PerugiaItaly\n', 'Fernanda Pambianco [email protected] \nDepartment of Mathematics and Computer Science\nPerugia University\n06123PerugiaItaly\n'], 'authoraffiliation': ['Department of Mathematics and Computer Science\nPerugia University\n06123PerugiaItaly', 'Department of Mathematics and Computer Science\nPerugia University\n06123PerugiaItaly', 'Department of Mathematics and Computer Science\nPerugia University\n06123PerugiaItaly'], 'corpusid': 258833455, 'doi': '10.48550/arxiv.2305.11955', 'github_urls': [], 'n_tokens_mistral': 17897, 'n_tokens_neox': 14702, 'n_words': 7910, 'pdfsha': 'bfe98f91a1d0c55d9fe71bc6888fe2c66e4cff03', 'pdfurls': ['https://export.arxiv.org/pdf/2305.11955v1.pdf'], 'title': ['New bounds for covering codes of radius 3 and codimension 3t + 1 *', 'New bounds for covering codes of radius 3 and codimension 3t + 1 *'], 'venue': []}
arxiv
EIGENVALUE ESTIMATES ON QUATERNION-KÄHLER MANIFOLDS 13 May 2021 Xiaolong Li Kui Wang EIGENVALUE ESTIMATES ON QUATERNION-KÄHLER MANIFOLDS 13 May 2021 We prove lower bound for the first closed or Neumann nonzero eigenvalue of the Laplacian on a compact quaternion-Kähler manifold in terms of dimension, diameter, and scalar curvature lower bound. It is derived as large time implication of the modulus of continuity estimates for solutions of the heat equation. We also establish lower bound for the first Dirichlet eigenvalue in terms of geometric data, via a Laplace comparison theorem for the distance to the boundary function.Theorem 1.1. Let (M n , g) be a compact Riemannian manifold (possibly with a smooth convex boundary) with diameter D and Ric ≥ (n − 1)κ for κ ∈ R. Let µ 1 be the first nonzero eigenvalue of the Laplacian on M (with Neumann boundary condition if ∂M = ∅). Then µ 1 ≥λ 1 (n, κ, D), 2020 Mathematics Subject Classification. 35P15, 53C26. Introduction Let (M n , g) be a compact n-dimensional Riemannian manifold (possibly with a smooth nonempty boundary). Denote by ∆ the Laplace-Beltrami operator associated to the metric g. It is well-known that the spectrum of ∆ (the Neumann boundary condition is imposed if ∂M is non-empty) consists of pure points spectrum that can be arranged in the order 0 = µ 0 < µ 1 ≤ µ 2 ≤ · · · → ∞. The study of the first nonzero eigenvalue µ 1 is an important issue in both mathematics and physics. In particular, the problem of establishing lower bounds for µ 1 in terms of geometric data of the manifold received considerable attention in the past few decades and a number of results have been obtained by various authors (see for example [Cha84] [SY94][BQ00] [LL10]). For instance, a classical result of Lichnerowicz states that µ 1 ≥ nκ if M is a closed ndimensional Riemannian manifold with Ric ≥ (n−1)κ > 0. This follows easily by integrating the Bochner formula. The rigidity was observed by Obata [Oba62], who showed that the equality occurs if and only if M is isometric to the round sphere of radius 1/ √ κ. In the nonnegative Ricci curvature case, by refining the gradient estimates of Li [Li79] and Li and Yau [LY80], Zhong and Yang [ZY84] proved the sharp lower bound µ 1 ≥ π 2 D 2 , where D denotes the diameter of M . Moreover, it was proved by Hang and Wang [HW07] that the equality happens if and only if M is a circle of radius D/π or the interval [−D/2, D/2]. Both Lichnerowicz and Zhong-Yang's results are special cases of the following theorem, which provides sharp lower bound for µ 1 depending on dimension, Ricci curvature lower bound, and diameter. whereλ 1 (n, κ, D) is the first nonzero Neumann eigenvalue of the one-dimensional eigenvalue problem ϕ ′′ − (n − 1)T κ ϕ ′ = −λϕ on the interval [−D/2, D/2], and T κ is defined in (1.1). Theorem 1.1 was first proved independently by Kröger [Krö92] using gradient estimate method and by Chen and Wang [CW94] using stochastic methods. The above explicit statement appeared first in the work of Bakry and Qian [BQ00], who also extended Theorem 1.1 to the the setting of smooth metric measure spaces using gradient estimates. In 2013, Andrews and Clutterbuck [AC13] gave a simple proof using modulus of continuity estimates (see also [ZW17] for an elliptic argument based on [AC13] and [Ni13]). The sharpness can be seen by constructing a sequence of Riemannian manifolds with Ric ≥ (n − 1)κ, which geometrically collapse to the interval [−D/2, D/2] (see for example [AC13]). However, there are very few results specifically for Kähler manifolds. Lichnerowicz [Lic58] showed that if M is closed Kähler manifold with Ric ≥ (n − 1)κ > 0, then µ 1 ≥ 2(n − 1)κ. Notice that this is a remarkable improvement of his well-known result in the Riemannian case. Lichnerowicz's proof makes use of a complex version Bochner formula (see also [Bal06,Theorem 6.14]). A different proof using harmonic maps was given by Urakawa [Ura87]. It is only until recently that the authors of the present paper established the following analogue of Theorem 1.1 for Kähler manifold in [LW21a]. Theorem 1.2. Let (M m , g, J) be a compact Kähler manifold of complex dimension m and diameter D, whose holomorphic sectional curvature is bounded from below by 4κ 1 and orthogonal Ricci curvature is bounded from below by 2(m − 1)κ 2 for some κ 1 , κ 2 ∈ R. Let µ 1 be the first nonzero eigenvalue of the Laplacian on M (with Neumann boundary condition if M has a strictly convex boundary). Then µ 1 ≥μ 1 (m, κ 1 , κ 2 , D), whereμ 1 (m, κ 1 , κ 2 , D) is the first Neumann eigenvalue of the one-dimensional eigenvalue problem ϕ ′′ − (2(m − 1)T κ2 + T 4κ1 ) ϕ ′ = −λϕ on [−D/2, D/2], and T κ is defined in (1.1). Theorem 1.2 provides the first diameter-depending lower bound for µ 1 for Kähler manifolds. Its proof uses the modulus of continuity approach of Andrews and Clutterbuck [AC13]. The key idea in taking the Kählarity into consideration is that the Ricci curvature can be decomposed as the sum of holomorphic sectional curvature and orthogonal Ricci curvature. The notion of orthogonal Ricci curvature was introduced recently by Ni and Zheng [NZ18] in the study of comparison theorems on Kähler manifolds (see also [NZ19] and [NWZ18] for more results on orthogonal Ricci curvature). It turns out that holomorphic section curvature and orthogonal Ricci curvature are more suitable conditions for various comparison theorems on Kähler manifolds. On one hand, they reflect more on the Kähler structure and lead to sharper results than the Ricci curvature. On the other hand, they are weaker than the well-studied bisectional curvature lower bound, under which comparions theorems for Kähler manifold were obtained by Li and Wang [LW05] and Tam and Yu [TY12]. See also [Liu14] for some comparison theorem for Kähler manifolds with Ricci curvature bounded from blow. Now let's turn to the quaternion-Kähler situation. When M is a closed quaternion-Kähler manifold of quaternionic dimension m ≥ 2 (i.e., the real dimension is 4m), it was shown by Alekseevsky and Marchiafava in [AM95] via a Bochner-type formula for 1-forms that µ 1 ≥ 8(m+1)κ, provided the scalar curvature is bound from below by 16m(m+2)κ > 0. Moreover, they showed that the equality characterize the quaternionic projective space. Given the results in Theorem 1.1 and Theorem 1.2, it is natural to ask whether one can prove lower bounds for µ 1 that depends on the diameter for a quaterion-Kähler manifold. The purpose of this paper is to establish analogous lower bounds for the first nonzero eigenvalue of the Laplacian on a compact quaternion-Kähler manifold. The main theorem states Theorem 1.3. Let (M m , g, I, J, K) be a compact quaternion-Kähler manifold (possibly with a smooth strictly convex boundary) of quaternionic dimension m ≥ 2 and diameter D. Suppose that the scalar curvature of M is bounded from below by 16m(m + 2)κ for some κ ∈ R. Let µ 1 be the first nonzero eigenvalue of the Laplacian on M (with Neumann boundary condition if M has a smooth strictly convex boundary). Then µ 1 ≥μ 1 (m, κ, D) whereμ 1 (m, κ, D) is the first nonzero Neumann eigenvalue of the one-dimensional eigenvalue problem ϕ ′′ − (4(m − 1)T κ + 3T 4κ ) ϕ ′ = −λϕ on [−D/2, D/2], and T κ is defined in (1.1). To the best of our knowledge, Theorem 1.3 provides the first diameter-dependent lower bound for µ 1 on a quaternion-Kähler manifold. It also seems to be the first lower bound for the Neumann boundary condition in the quaternion-Kähler setting. Moreover, the monotonicty ofμ 1 (m, κ, D) in D implies that when the diameter is small, the lower bound provided in Theorem 1.3 is better than any diameter-independent lower bound. The lower bound in Theorem 1.3 will be derived as large time implication of the modulus of continuity estimates for solutions of the heat equation. This is consistent with the Riemannian case in [AC13] and the Kähler case in [LW21a]. Recall that the modulus of continuity ω of a continuous function u defined on a metric space (X, d) is defined by It does not seem possible to expressμ 1 (m, κ, D) explicitly in terms of elementary functions, so we provide some explicit lower bounds below, which give a sort of interpolation between the Zhong-Yang and Licherowicz lower bounds. It is similar to the one obtained by Shi and Zhang in [SZ07] for the Riemannian case, which says forλ 1 (n, κ, D) as in Theorem 1.1, it holds that for κ ≥ 0, ω(s) := sup u(y) − u(x) 2 : d(x, y) = 2s .λ 1 (n, κ, D) ≥ sup s∈(0,1) 4s(1 − s) π 2 D 2 + s(n − 1)κ . A slight modification of their proof shows that in the Kähler setting, it holds that Proposition 1.1. Letμ 1 (m, κ 1 , κ 2 , D) be as in Theorem 1.2. If κ 1 , κ 2 ≥ 0, then µ 1 (m, κ 1 , κ 2 , D) ≥ sup s∈(0,1) 4s(1 − s) π 2 D 2 + 2s(m − 1)κ 2 + 4sκ 1 . We are grateful to Dr. Shoo Seto who pointed Proposition 1.1 to us. Similarly, in the quaternion-Kähler case, we have Proposition 1.2. Letμ 1 (m, κ, D) be as in Theorem 1.3. If κ ≥ 0, then µ 1 (m, κ, D) ≥ sup s∈(0,1) 4s(1 − s) π 2 D 2 + 4s(m + 2)κ . In Section 5, we also establish lower bound for the first Dirichlet eigenvalue in terms of geometric data (see Theorem 5.4). This is done via a Laplace comparison theorem for the distance to the boundary function on quaternion-Kähler manifold. Other sections are organized as follows. In Section 2, we review some basic properties of quaternion-Kähler manifolds. In Section 3, we derive the modulus of continuity estimates for solutions of quasilinear parabolic equations on a quaternion-Kähler manifold. As an application, we prove Theorem 1.3. The explicit lower bounds in Proposition 1.1 and 1.2 are proved in Section 4. Throughout the paper, we use the following notations. The function T κ is defined for κ ∈ R by (1.1) T κ (t) =      √ κ tan ( √ κt), κ > 0, 0, κ = 0, − √ −κ tanh ( √ −κt), κ < 0. The function c κ is defined for κ ∈ R by (1.2) c κ (t) =      cos √ κt if κ > 0, 1 if κ = 0, cosh √ −κt if κ < 0. Quaternion-Kähler Manifolds In this section, we recall some basic properties about quaternion-Kähler manifolds that will be needed in the sequel. These are proved by Berger [Ber66] and Ishibara [Ish74] (see also [Bes08]). We shall follow the presentation of [KLZ08] here. (2) If φ ∈ Γ(V ), then ∇ X φ ∈ Γ(V ) for all X ∈ T M . It is worth noting that then the tensors I, J, K may not be globally defined on M . For instance, the canonical quaternionic projective space admits no almost complex structure for topological reasons. However, the space spanned by I, J, K may always be defined globally according to the definition. A well known fact is that a 4m-dimensional Riemannian manifold is quaternion-Kähler if and only if its restricted holonomy group is contained in Sp(n)Sp(1). Since the 4-dimensional Riemannian manifolds with holonomy Sp(1)Sp(1) are simply the oriented Riemanian manifolds, we shall only consider the case m ≥ 2. In this paper, we are mostly concerned about the curvature properties of quaternion-Kähler manifolds. The Riemannian curvature tensor of (M, g) is defined by (2) The orthogonal Ricci curvature of M is defined as R(X, Y, Z, W ) = ∇ Y ∇ X Z − ∇ X ∇ Y Z + ∇ [X,Y ] Z, W .Ric ⊥ (X, X) := Ric(X, X) − Q(X) First of all, all quaternion-Kähler manifolds of quaternionic dimension m ≥ 2 are Einstein (see for instance [KLZ08, Theorem 1.2]), namely there exists a constant κ such that Ric = 4(m + 2)κ. Moreover, we have the following proposition. (2) M has constant orthogonal Ricci curvature, i.e., Ric ⊥ (X, X) = 4(m − 1)κ|X| 2 . Proof. For (1), see Theorem 1.3 in [KLZ08]. (2) follows directly from the definition Ric ⊥ (X, X) = Ric(X, X) − Q(X). We end this section with the following useful lemma. Lemma 2.1. Let (M m , g) be a quaternion-Kähler manifold of quaternionic dimension m ≥ 2 with Ric = 4(m + 2)κ for κ ∈ R. Let γ : [a, b] → M be a geodesic with unit speed and X I (t),X J (t),X K (t) are parallel vector fields along γ such that X I (a) = Iγ ′ (a), X J (a) = Jγ ′ (a), X K (a) = Kγ ′ (a). Then R(γ ′ , X I (t), γ ′ , X I (t)) + R(γ ′ , X J (t), γ ′ , X J (t)) + R(γ ′ , X K (t), γ ′ , X K (t)) = 12κ for all t ∈ [a, b]. Proof. See Lemma 1.5 in [KLZ08]. Modulus of Continuity Estimates In this section, we prove the modulus of continuity estimates for solution of a class of quasilinear isotropic parabolic equations on a compact quaternion-Kähler manifold. As in the Riemannian case in [AC13] or the Kähler case in [LW21a], we consider the following isotropic quasilinear equations: (3.1) ∂u ∂t = Q[u] := α(|∇u|) ∇ i u∇ j u |∇u| 2 + β(|∇u|) δ ij − ∇ i u∇ j u |∇u| 2 . Here α and β are smooth positive functions. Some important examples of (3.1) are the heat equation (with α = β = 1), the p-Laplacian heat flows (with α = (p − 1)|∇u| p−2 and β = |∇u| p−2 ) and the graphical mean curvature flow (with α = 1/(1 + |∇u| 2 ) and β = 1). In the quaternion-Kähler case, the associated one-dimensional operator F is given by (3.2) F ϕ := α(ϕ ′ )ϕ ′′ + (4(m − 1)T κ + 3T 4κ ) β(ϕ ′ )ϕ ′ , where the function T κ is the function defined in (1.1). The main result of this section is the following modulus of continuity estimates on a compact quaternion-Kähler manifold, in terms of initial oscillation, elapsed time, and scalar curvature lower bound. Theorem 3.1. Let (M m , g, I, J, K) be a compact quaternion-Kähler manifold with diameter D whose scalar curvature is bounded from below by 16m(m + 2)κ for κ ∈ R. Let u : M × [0, T ) → R be a solution of (3.1) (with Neumann boundary condition if M has a strictly convex boundary). Then the modulus of continuity ω : [0, D/2] × [0, T ) → R of u is viscosity subsolution of the one-dimensional equation (3.3) ω t = F ω, where the operator F is defined in (3.2). Proof. The proof proceeds as in the Kähler case in [LW21a] but is slightly more involved. By the definition of viscosity solutions (see [CIL92]), we need to show that for every smooth function ϕ that touches ω from above at (s 0 , t 0 ) ∈ (0, D/2) × (0, T ) in the sense that ϕ(s, t) ≥ ω(s, t) near (s 0 , t 0 ), ϕ(s 0 , t 0 ) = ω(s 0 , t 0 ), it holds that (3.4) ϕ t ≤ Lϕ, at the point (s 0 , t 0 ). It follows from the definition of ω that for such a function ϕ, we have (3.5) u (γ(b), t) − u (γ(a), t) − 2ϕ L[γ] 2 , t ≤ 0 for any t ≤ t 0 close to t 0 and any smooth path γ : |(∇ s γ r ) ⊥ | 2 − R(γ s , γ r , γ s , γ r ) ds + g(T, ∇ r γ r )| s0 −s0 , where T is the unit tangent vector to γ 0 . It will be convenient to work in the (quaternionic) Fermi coordinates along γ 0 chosen as follows. Choose an orthonormal basis {e i } 4m i=1 for T x0 M with e 1 = γ ′ 0 (−s 0 ), e 2 = Iγ ′ 0 (−s 0 ), e 3 = Jγ ′ 0 (−s 0 ), e 4 = Kγ ′ 0 (−s 0 ), and parallel transport it along γ 0 to produce an orthonormal basis {e i (s)} 4m i=1 for T γ0(s) M with e 1 (s) = γ ′ 0 (s) for each s ∈ [−s 0 , s 0 ]. Notice that the vector fields Iγ ′ 0 (t), Jγ ′ 0 (t) and Kγ ′ 0 (t) may not be parallel along γ 0 . The first variation consideration gives (3.6) Q[u](y 0 , t 0 ) − Q[u](x 0 , t 0 ) − 2ϕ t ≥ 0. and (3.7) ∇u(x 0 , t 0 ) = −ϕ ′ e 1 (−s 0 ), and ∇u(y 0 , t 0 ) = ϕ ′ e 1 (s 0 ). The variation γ(r, s) = γ 0 s + r 2s−1 L yields (3.8) u 11 (y 0 , t 0 ) − u 11 (x 0 , t 0 ) − 2ϕ ′′ ≤ 0. For i = 2, 3, 4, the variations γ(r, s) = exp γ0(s) (rη(s)e i (s)) produce (3.9) u ii (y 0 , t 0 ) − u ii (x 0 , t 0 ) − ϕ ′ s0 −s0 (η ′ ) 2 − η 2 R(e 1 , e i , e 1 , e i ) ds ≤ 0, and for 5 ≤ j ≤ 4m, the variations γ(r, s) = exp γ0(s) (rζ(s)e j (s)) yield (3.10) u jj (y 0 , t 0 ) − u jj (x 0 , t 0 ) − ϕ ′ s0 −s0 (ζ ′ ) 2 − ζ 2 R(e 1 , e j , e 1 , e j ) ds ≤ 0. We derive, by choosing η(s) = c4κ(s) c4κ(s0) and summing (3.9) for i = 2, 3, 4, that 4 i=2 (u ii (y) − u ii (x)) ≤ ϕ ′ s0 −s0 3(η ′ ) 2 − η 2 4 i=2 R(e 1 , e i , e 1 , e i ) ds ≤ 3ϕ ′ ηη ′ | s0 −s0 − s0 −s0 η 2 4 i=2 R(e 1 , e i , e 1 , e i ) − 12κ ds = −6T 4κ ϕ ′ (3.11) where we have used Choosing ζ(s) = cκ(s) cκ(s0) and summing (3.10) for 5 ≤ j ≤ 4m yield 4m j=5 (u jj (y) − u jj (x)) ≤ ϕ ′ s0 −s0   4(m − 1)(η ′ ) 2 − ζ 2   4m j=5 R(e 1 , e j , e 1 , e j )     ds ≤ 4(m − 1)ϕ ′ ζζ ′ | s0 −s0 − s0 −s0 ζ 2   4m j=5 R(e 1 , e j , e 1 , e j ) − 4(m − 1)κ   ds = −8T 4κ ϕ ′ . Combining (3.11) and (3.12) together, we have Q[u](y 0 , t 0 ) − Q[u](x 0 , t 0 ) = α(ϕ ′ ) (u 11 (y 0 , t 0 ) − u 11 (x 0 , t 0 )) + β(ϕ ′ ) 4m i=2 (u ii (y 0 , t 0 ) − u ii (x 0 , t 0 )) ≤ 2α(ϕ ′ )ϕ ′′ − 2β(ϕ ′ )ϕ ′ (4(m − 1)T κ + 3T 4κ ) = 2F ϕ. (3.12) It follows from (3.6) and (3.12) that ϕ t ≤ F ϕ. This completes the proof. In case M has a strictly convex boundary, the same argument as in [LW21a] rule out the possibility that either x 0 ∈ ∂M or y 0 ∈ ∂M . So the above argument remains valid. The following corollary is immediate. (1) ϕ t ≥ F ϕ; (2) ϕ ′ ≥ 0 on [0, D/2] × [0, T ); (3) |u(y, 0) − u(x, 0)| ≤ 2ϕ d(x,y) 2 , 0 Then |u(y, t) − u(x, t)| ≤ 2ϕ d(x, y) 2 , t for all x, y ∈ M and t ∈ [0, T ). As in the Riemannian or Kähler case, the modulus of continuity estimate implies lower bounds for the first nonzero eigenvalue of the Laplacian on a quaternion-Kähler manifold. We restate Theorem 1.3 here. whereμ 1 (m, κ, D) is the first nonzero Neumann eigenvalue of the one-dimensional eigenvalue problem ϕ ′′ − (4(m − 1)T κ + 3T 4κ ) ϕ ′ = −λϕ on [−D/2, D/2], and T κ is defined in (1.1). Proof of Theorem 3.3. The proof is a slight modification of the proof in the Kähler case in [LW21a], so we leave the details to interested reader. Explicit Lower Bounds Let a, b be positive integers and κ 1 , κ 2 be nonnegative real numbers. Consider the onedimensional eigenvalue problem (4.1) ϕ ′′ − [2a(m − 1) √ κ 2 tan( √ κ 2 t) + 2b √ κ 1 tan(2 √ κ 1 )] ϕ ′ + λϕ = 0 with Neumann boundary condition ϕ ′ (−D/2) = ϕ ′ (D/2) = 0. Proposition 4.1. Let λ 1 be the first nonzero Neumann eigenvalue of (4.1). Then λ 1 ≥ sup s∈(0,1) 4s(1 − s) π 2 D 2 + s(2a(m − 1)κ 2 + 4bκ 1 ) . Proof. Let ϕ be the eigenfunction of (4.1) associated to the first nonzero Neumann eigenvalue λ 1 . It's easy to observe that the function y = ϕ ′ satisfies the ODE y ′′ − [2a(m − 1) √ κ 2 tan( √ κ 2 t) + 2b √ κ 1 tan(2 √ κ 1 )] y ′ − 2a(m − 1)κ 1 sec 2 ( √ κ 2 t) + 4bκ 1 sec 2 (2 √ κ 1 ) y + λy = 0. with Dirichlet boundary condition y(−D/2) = y(D/2) = 0. Multiplying by y γ−1 for some γ > 1 and integrating both sides, we have D 2 − D 2 y ′′ y γ−1 dt + λ D 2 − D 2 y γ dt (4.2) = D 2 − D 2 2a(m − 1) √ κ 2 tan( √ κ 2 t) + 2b k 1 tan(2 √ κ 1 t) y ′ y γ−1 dt + D 2 − D 2 2a(m − 1)κ 2 sec 2 ( √ κ 2 t) + 4bκ 1 sec 2 (2 √ κ 1 ) y γ dt By integration by parts, we get √ κ 2 D 2 − D 2 tan( √ κ 2 t)y γ−1 y ′ dt (4.3) = √ κ 2 γ D 2 − D 2 tan( √ κ 2 t)(y γ ) ′ dt = − κ 2 γ D 2 − D 2 sec 2 ( √ κ 2 t)y γ dt. Similarly, we have that (4.4) 2 √ κ 1 D 2 − D 2 tan(2 k 1 t)y ′ y γ−1 dt = − 4κ 1 γ D 2 − D 2 sec 2 (2 k 1 t)y γ dt On the other hand, integration by parts implies D 2 − D 2 y γ−1 y ′′ dt = −(γ − 1) D 2 − D 2 y γ−2 y ′ y ′ dt = − 4(γ − 1) γ 2 D 2 − D 2 (y γ 2 ) ′ 2 dt. By Wirtinger's inequality, we get that (4.5) D 2 − D 2 y γ−1 y ′′ dt ≤ − 4(γ − 1) γ 2 π 2 D 2 D 2 − D 2 y γ dt Plugging (4.3), (4.4), and (4.5) into (4.2), we obtain − 4(γ − 1) γ 2 π 2 D 2 D 2 − D 2 y γ dt + λ D 2 − D 2 y γ dt ≥ 1 − 1 γ D 2 − D 2 2a(m − 1)κ 2 sec 2 ( √ κ 2 t) + 4bκ 1 sec 2 (2 √ κ 1 t) y γ dt ≥ 1 − 1 γ (2a(m − 1)κ 2 + 4bκ 1 ) D 2 − D 2 y γ dt where we have used sec 2 (x) ≥ 1 for all x in the last line. It then follows that we must have λ ≥ 4(γ − 1) γ 2 π 2 D 2 + 1 − 1 γ (2a(m − 1)κ 2 + 4bκ 1 ) Letting s = 1 − 1 γ , we have λ ≥ 4s(1 − s) π 2 D 2 + s(2a(m − 1)κ 2 + 4bκ 1 ). Since γ > 1 is arbitrary, we get λ ≥ sup s∈(0,1) 4s(1 − s) π 2 D 2 + s(2a(m − 1)κ 2 + 4bκ 1 ) . Proof of Proposition 1.1 and 1.2. Proposition 1.1 is a special case of Proposition 4.1 with a = b = 1 and Proposition 1.2 is a special case of Proposition 4.1 with a = 2 and b = 3. First Dirichlet Eigenvalue Throughout this section, d(x, ∂M ) denotes the distance function to ∂M given by d(x, ∂M ) = inf{d(x, y) : y ∈ ∂M } and R denotes the inradius of M given by R = sup{d(x, ∂M ) : x ∈ M }. For convenience, denote by C κ,Λ (t) the unique solution of the initial value problem (5.1) φ ′′ + κφ = 0, φ(0) = 1, φ ′ (0) = −Λ, and define T κ,Λ for κ, Λ ∈ R by (5.2) T κ,Λ (t) := − C ′ κ,Λ (t) C κ,Λ (t) . In the Riemannian setting, Li and Yau [LY80] and Kause [Kas84] proved the following well-known result. Theorem 5.1. Let (M n , g) be a compact Riemannian manifold with smooth boundary ∂M = ∅. Suppose that the Ricci curvature of M is bounded from below by (n − 1)κ and the mean curvature of ∂M is bounded from below by (n − 1)Λ for some κ, Λ ∈ R. Let λ 1 be the first Dirichlet eigenvalue of the Laplacian on M . Then λ 1 ≥λ 1 (n, κ, Λ, R), whereλ 1 (n, κ, Λ, R) is the first eigenvalue of the one-dimensional eigenvalue problem (5.3) ϕ ′′ − (n − 1)T κ,Λ ϕ ′ = −λϕ, ϕ(0) = 0, ϕ ′ (R) = 0. In [LW21a], the authors of the present paper obtained an analogous theorem in the Kähler setting. Theorem 5.2. Let (M m , g, J) be a compact Kähler manifold with smooth nonempty boundary ∂M . Suppose that the holomorphic sectional curvature is bounded from below by 4κ 1 and the orthogonal Ricci curvature is bounded from below by 2(m−1)κ 2 for some κ 1 , κ 2 ∈ R, and the second fundamental form on ∂M is bounded from below by Λ ∈ R. Let λ 1 be the first Dirichlet eigenvalue of the Laplacian on M . Then λ 1 ≥λ 1 (m, κ 1 , κ 2 , Λ, R) whereλ 1 (m, κ 1 , κ 2 , Λ, R) is the first eigenvalue of the one-dimensional eigenvalue problem (5.4) ϕ ′′ − (2(m − 1)T κ2,Λ + T 4κ1,Λ ) ϕ ′ = −λϕ, ϕ(0) = 0, ϕ ′ (R) = 0. Remark 5.3. Diameter-independent lower bounds for λ 1 were obtained by Guedj, Kolev and Yeganefar [GKY13]. It has been generalized via a p-Reilly formula to the first Dirichlet eigenvalue of the p-Laplacian by Blacker and Seto [BS19] when p ≥ 2. Here we prove the following quaternion-Kähler version. Theorem 5.4. Let (M m , g, I, J, K) be a compact quaternion-Kähler manifold with smooth nonempty boundary ∂M . Suppose that the scalar curvature is bounded from below by 16m(m+ 2)κ for some κ ∈ R, and the second fundamental form on ∂M is bounded from below by Λ ∈ R. Let λ 1 be the first Dirichlet eigenvalue of the Laplacian on M . Then λ 1 ≥λ 1 (m, κ, Λ, R) whereλ 1 (m, κ, Λ, R) is the first eigenvalue of the one-dimensional eigenvalue problem (5.5) ϕ ′′ − (4(m − 1)T κ,Λ + 3T 4κ,Λ ) ϕ ′ = −λϕ, ϕ(0) = 0, ϕ ′ (R) = 0. Remark 5.5. When the boundary is convex, namely Λ = 0, it is easily seen that we havē λ 1 (m, κ, 0, R) =μ 1 (m, κ, R). Thus we can obtain the same explicit lower bounds as in Proposition 1.2 forλ 1 (m, κ, 0, R). Remark 5.6. With the help of a generalized Barta's inequality for the p-Laplacian (see [LW20, Theorem 3.1]), the same argument here indeed yields such lower bounds for the first Dirichlet eigenvalue of the p-Laplacian for all 1 < p < ∞. The proof of Theorem 5.4 relies on a comparison theorem for the second derivatives of d(x, ∂M ). Theorem 5.7. Let (M m , g, I, J, K) be a compact quaternion-Kähler manifold with smooth nonempty boundary ∂M . Suppose that the scalar curvature of M is bounded from below by 16m(m + 2)κ for some κ ∈ R, and the second fundamental form on ∂M is bounded from below by Λ ∈ R. Let ϕ : [0, R] → R + be a smooth function with ϕ ′ ≥ 0. Then the function v(x) = ϕ (d(x, ∂M )) is a viscosity supersolution of Q[v] = [α(ϕ ′ )ϕ ′′ − β(ϕ ′ )ϕ ′ (4(m − 1)T κ,Λ + 3T 4κ,Λ )]| d(x,∂M) , on M , where Q is the operator defined in (3.1). Proof of Theorem 5.7. By approximation, it suffices to consider the case ϕ ′ > 0 on [0, R]. By definition of viscosity solutions (see [CIL92]), it suffices to prove that for any smooth function ψ touching v from below at x 0 ∈ M , i.e., ψ(x) ≤ v(x) on M, ψ(x 0 ) = v(x 0 ), it holds that Q[ψ](x 0 ) ≤ [α(ϕ ′ )ϕ ′′ − β(ϕ ′ )ϕ ′ (4(m − 1)T κ,Λ + 3T 4κ,Λ )]| d(x0,∂M) . Since the function d(x, ∂M ) may not be smooth at x 0 , so we need to replace it by a smooth functiond(x) defined in a neighborhood U (x 0 ) of x 0 satisfyingd(x) ≥ d(x, ∂M ) for x ∈ U (x 0 ) andd(x 0 ) = d(x 0 , ∂Mi } 4m i=1 for T x0 M with e 1 = γ ′ 0 (0), e 2 = Iγ ′ 0 (0), e 3 = Jγ ′ 0 (0), e 4 = Kγ ′ 0 (0), and parallel transport it along γ 0 to produce an orthonormal basis {e i (s)} 4m i=1 for T γ0(s) M with e 1 (s) = γ ′ 0 (s) for each s ∈ [0, s 0 ]. For any vector X ∈ exp −1 x0 U (x 0 ) , let X(s), s ∈ [0, s 0 ] be the vector field obtained by parallel translating X along γ, and decompose it as X(s) = aX 1 (s) + bγ ′ (s) + cX 2 (s), where a, b and c are constants along γ with a 2 + b 2 + c 2 = |X| 2 , and X 1 (s) and X 2 (s) are unit components in span{e 5 (s), · · · , e 4m (s)} and span{e 2 (s), e 3 (s), e 4 (s)}. Define W (s) = a η(s)X 1 (s) + b 1 − s s 0 γ ′ (s) + c ζ(s)X 2 (s), where η, ζ : [0, s 0 ] → R + are two C 2 functions to be chosen later. Next we define the 4m-parameter family of curves γ X : [0, s 0 ] → M such that (1) γ 0 = γ; (2) γ X (0) = exp x0 (W (0)) and γ X (s 0 ) ∈ ∂M ; (3) W (s) is induced by the one-parameter family of curves ε → γ εX (s) for ε ∈ (−ε 0 , ε 0 ) and s ∈ [0, s 0 ]; (4) γ X depends smoothly on X. Finally letd(x) be the length of the curve γ X where x = exp x0 (X) ∈ U (x 0 ). Then we havē d(x) ≥ d(x, ∂M ) on U (x 0 ),d(x 0 ) = d(x 0 , ∂M ). Recall the first and second variation formulas: (5.6) ∇d = −e 1 (0), and ∇ 2d (X, X) = −A aη(s 0 )X 1 (s 0 ) + cζ(s 0 )X 2 (s 0 ), aη(s 0 )X 1 (s 0 ) + cζ(s 0 )X 2 (s 0 ) + s0 0 a 2 (η ′ ) 2 + c 2 (ζ ′ ) 2 − R(aηX 1 + cζX 2 , γ ′ , aηX 1 + cζX 2 , γ ′ ) ds where A denotes the second fundamental form of ∂M at y 0 . Here and below the derivatives ofd are all evaluated at x 0 . Then we have (5.7) ∇ 2d (e 1 (0), e 1 (0)) = 0. Since the function ψ(x)− ϕ (d(x, ∂M )) attains its maximum at x 0 and ϕ ′ > 0, it follows that the function ψ(x) − ϕ(d(x)) attains a local maximum at x 0 . The first and second derivative tests yield ∇ψ(x 0 ) = −ϕ ′ e 1 (0), ψ 11 (x 0 ) ≤ ϕ ′′ , and ψ ii (x 0 ) ≤ ϕ ′ ∇ 2d (e i (0), e i (0)) for 2 ≤ i ≤ 4m, where we used (5.6) and (5.7). Here and below the derivatives of ϕ are all evaluated at s 0 = d(x 0 , ∂M ). Thus we have Q[ψ](x 0 ) = α(ϕ ′ )ψ 11 + β(ϕ ′ ) 4m i=2 ψ ii (5.11) ≤ α(ϕ ′ )ϕ ′′ + β(ϕ ′ )ϕ ′ 4m i=2 ∇ 2d (e i (0), e i (0)) ≤ α(ϕ ′ )ϕ ′′ − β(ϕ ′ )ϕ ′ 4(m − 1)T κ,Λ + 3T 4κ,Λ . The proof is complete. By choose ϕ(s) = s, we have Corollary 5.8. Let (M m , g, I, J, K) be a compact quaternion-Kähler manifold with smooth nonempty boundary ∂M . Suppose that the scalar curvature of M is bounded from below by 16m(m + 2)κ for some κ ∈ R, and the second fundamental form on ∂M is bounded from below by Λ ∈ R. Then ∆d(x, ∂M ) ≤ −4(m − 1)T κ,Λ (d(x, ∂M )) − 3T 4κ,Λ (d(x, ∂M )) in the viscosity sense. Proof of Theorem 5.4. The proof is a slight modification of the Kähler case presented in [LW21a, Section 7], so we omit the details. Definition 2.1. A quaternion-Kähler manifold (M m , g) of quaternionic dimension m (the real dimension is 4m) is a Riemannian manifold with a rank three vector bundle V ⊂ End(T M ) satisfying (1) In any coordinate neighborhood U of M , there exists a local basis {I, J, K} of V such that I 2 = J 2 = K 2 = 1, IJ = −JI = K, JK = −KJ = I, KI = −IK = J, and for all X, Y ∈ T M X, Y = IX, IY = JX, JY = KX, KY Definition 2 . 2 . 22Let (M, g) be a quaternion-Kähler manifold. ( 1 ) 1The quaternionic sectional curvature of M is defined as Q(X) := R(X, IX, X, IX) + R(X, JX, X, JX) + R(X, KX, X, KX) |X| 2 . Proposition 2. 1 . 1Let (M m , g) be a quaternion-Kähler manifold of quaternionic dimension m ≥ 2 with Ric = 4(m + 2)κ for κ ∈ R. Then ( 1 ) 1M has constant quaternionic sectional curvature, i.e., Q(X) = 12κ|X| 2 . [a, b] → M with length close to 2s 0 . Moreover, since M is compact, there exist points x 0 and y 0 in M (assume for a moment that ∂M = ∅), with d(x 0 , y 0 ) = 2s 0 such that the equality in (3.5) holds for γ 0 : [−s 0 , s 0 ] → M , a length-minimizing unit speed geodesic connecting x 0 and y 0 . The key idea is to derive useful inequalities from the first and second tests along smooth family of variations of the curve γ 0 . For this purpose, we need to recall the first and second variation formulas of arc length. If γ : (r, s) → M is a smooth variation of γ 0 (s) 1 , e i , e 1 , e i ) = Ric(e 1 , e 1 ) − 4 i=2 R(e 1 , e i , e 1 , e i ) ≥ 4(m − 1)κ. Corollary 3 . 2 . 32Let M and u be the same as in Theorem 3.1. Suppose ϕ : [0, D/2]×[0, T ) → R satisfies Theorem 3 . 3 . 33Let (M m , g, I, J, K) be a compact quaternion-Kähler manifold (possibly with a smooth strictly convex boundary) of quaternionic dimension m and diameter D. Suppose that the scalar curvature of M is bounded from below by 16m(m + 2)κ for some κ ∈ R. Let µ 1 be the first nonzero eigenvalue of the Laplacian on M (with Neumann boundary condition if ∂M = ∅). Then µ 1 ≥μ 1 (m, κ, D) d(x 0 0, y 0 ) = d(x 0 , ∂M ) := s 0 . Let γ : [0, s 0 ] → M be the unit speed length-minimizing geodesic with γ(0) = x 0 and γ(s 0 ) = y 0 . Choose an orthonormal basis {e For 2 R(e i , γ ′ , e i , γ ′ 2≤ i ≤ 4, we obtain by choosing ζ(s) = C 4κ,Λ (s 0 − s)/C 4κ,Λ (s 0 ) that∇ 2d (e i (0), e i (0)) (5.8) = −ζ 2 (s 0 )A(e i (s 0 ), e i (s 0 )) + s0 0 (ζ ′ ) 2 − ζ 2 R(e i , γ ′ , e i , γ ′ ) ′ ) 2 − ζ 2 R(e i , γ ′ , e i , γ ′ ) ds Summing over 2 ≤ i ζ ′ ) 2 − 12κζ 2 ds = −3T 4κ,Λ (s 0 ). For 5 ≤ i ≤ 4m, we have ∇ 2d (e i (0), e i (0)) = −η 2 (s 0 )A(e i (s 0 ), e i (s 0 )) + s0 0 (η ′ ) 2 − η 2 R(e i , γ ′ , e i , γ ′ ) ds.Summing over 5 ≤ i ≤ 4m and choosing η(s) = C κ,Λ (s 0 − s)/C κ,Λ (s 0 ) gives 4m i=5 ∇ 2d (e i (0), e i (′ ) 2 − κη 2 ds = −4(m − 1)T κ,Λ (s 0 ). ). The construction is standard (see e.g. [Wu79, pp. 73-74] or [AX19, pp. 1187]), which we state below for reader's convenience.Since M is compact, there exists y 0 ∈ ∂M such that Lipschitz bounds for solutions of quasilinear parabolic equations in one space variable. Ben Andrews, Julie Clutterbuck, J. Differential Equations. 24611Ben Andrews and Julie Clutterbuck. Lipschitz bounds for solutions of quasilinear parabolic equa- tions in one space variable. J. Differential Equations, 246(11):4268-4283, 2009. Time-interior gradient estimates for quasilinear parabolic equations. Ben Andrews, Julie Clutterbuck, Indiana Univ. Math. J. 581Ben Andrews and Julie Clutterbuck. Time-interior gradient estimates for quasilinear parabolic equations. Indiana Univ. Math. J., 58(1):351-380, 2009. Proof of the fundamental gap conjecture. Ben Andrews, Julie Clutterbuck, J. Amer. Math. Soc. 243Ben Andrews and Julie Clutterbuck. Proof of the fundamental gap conjecture. J. Amer. Math. Soc., 24(3):899-916, 2011. Sharp modulus of continuity for parabolic equations on manifolds and lower bounds for the first eigenvalue. Ben Andrews, Julie Clutterbuck, Anal. PDE. 65Ben Andrews and Julie Clutterbuck. Sharp modulus of continuity for parabolic equations on manifolds and lower bounds for the first eigenvalue. Anal. PDE, 6(5):1013-1024, 2013. Transformations of a quaternionic Kähler manifold. V Dmitry, Stefano Alekseevsky, Marchiafava, C. R. Acad. Sci. Paris Sér. I Math. 3206Dmitry V. Alekseevsky and Stefano Marchiafava. Transformations of a quaternionic Kähler man- ifold. C. R. Acad. Sci. Paris Sér. I Math., 320(6):703-708, 1995. Eigenvalue comparison on Bakry-Emery manifolds. Ben Andrews, Lei Ni, Comm. Partial Differential Equations. 3711Ben Andrews and Lei Ni. Eigenvalue comparison on Bakry-Emery manifolds. Comm. Partial Differential Equations, 37(11):2081-2092, 2012. Moduli of continuity, isoperimetric profiles, and multi-point estimates in geometric heat equations. Ben Andrews, Surveys in differential geometry 2014. Regularity and evolution of nonlinear equations. Somerville, MAInt. Press19Ben Andrews. Moduli of continuity, isoperimetric profiles, and multi-point estimates in geometric heat equations. In Surveys in differential geometry 2014. Regularity and evolution of nonlinear equations, volume 19 of Surv. Differ. Geom., pages 1-47. Int. Press, Somerville, MA, 2015. Gradient estimates via two-point functions for elliptic equations on manifolds. Ben Andrews, Changwei Xiong, Adv. Math. 349Ben Andrews and Changwei Xiong. Gradient estimates via two-point functions for elliptic equa- tions on manifolds. Adv. Math., 349:1151-1197, 2019. Werner Ballmann, Lectures on Kähler manifolds. ESI Lectures in Mathematics and Physics. EMS), ZürichEuropean Mathematical SocietyWerner Ballmann. Lectures on Kähler manifolds. ESI Lectures in Mathematics and Physics. European Mathematical Society (EMS), Zürich, 2006. Remarques sur les groupes d'holonomie des variétés riemanniennes. Marcel Berger, C. R. Acad. Sci. Paris Sér. A-B. 262Marcel Berger. Remarques sur les groupes d'holonomie des variétés riemanniennes. C. R. Acad. Sci. Paris Sér. A-B, 262:A1316-A1318, 1966. Arthur L Besse, Einstein manifolds. Classics in Mathematics. BerlinSpringer-VerlagReprint of the 1987 editionArthur L. Besse. Einstein manifolds. Classics in Mathematics. Springer-Verlag, Berlin, 2008. Reprint of the 1987 edition. Some new results on eigenvectors via dimension, diameter, and Ricci curvature. Dominique Bakry, Zhongmin Qian, Adv. Math. 1551Dominique Bakry and Zhongmin Qian. Some new results on eigenvectors via dimension, diameter, and Ricci curvature. Adv. Math., 155(1):98-153, 2000. First eigenvalue of the p-Laplacian on Kähler manifolds. Casey Blacker, Shoo Seto, Proc. Amer. Math. Soc. 1475Casey Blacker and Shoo Seto. First eigenvalue of the p-Laplacian on Kähler manifolds. Proc. Amer. Math. Soc., 147(5):2197-2206, 2019. Eigenvalues in Riemannian geometry. Isaac Chavel, Pure and Applied Mathematics. 115Academic Press, IncIncluding a chapter. With an appendix by Jozef DodziukIsaac Chavel. Eigenvalues in Riemannian geometry, volume 115 of Pure and Applied Mathemat- ics. Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol, With an appendix by Jozef Dodziuk. User's guide to viscosity solutions of second order partial differential equations. G Michael, Hitoshi Crandall, Pierre-Louis Ishii, Lions, Bull. Amer. Math. Soc. (N.S.). 271Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions. User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.), 27(1):1-67, 1992. Application of coupling method to the first eigenvalue on manifold. Feng Yu Mu Fa Chen, Wang, Sci. China Ser. A. 371Mu Fa Chen and Feng Yu Wang. Application of coupling method to the first eigenvalue on manifold. Sci. China Ser. A, 37(1):1-14, 1994. Fundamental gap estimate for convex domains on sphere -the case n = 2. Xianzhe Dai, Shoo Seto, Guofang Wei, arXiv:1803.01115Comm. Anal. Geom. to appearXianzhe Dai, Shoo Seto, and Guofang Wei. Fundamental gap estimate for convex domains on sphere -the case n = 2. Comm. Anal. Geom., to appear, arXiv:1803.01115, 2018. A Lichnerowicz estimate for the first eigenvalue of convex domains in Kähler manifolds. Vincent Guedj, Boris Kolev, Nader Yeganefar, Anal. PDE. 65Vincent Guedj, Boris Kolev, and Nader Yeganefar. A Lichnerowicz estimate for the first eigenvalue of convex domains in Kähler manifolds. Anal. PDE, 6(5):1001-1012, 2013. A remark on Zhong-Yang's eigenvalue estimate. Fengbo Hang, Xiaodong Wang, Art. ID rnm064. 9Fengbo Hang and Xiaodong Wang. A remark on Zhong-Yang's eigenvalue estimate. Int. Math. Res. Not. IMRN, (18):Art. ID rnm064, 9, 2007. Fundamental gap of convex domains in the spheres. Chenxu He, Guofang Wei, Qi S Zhang, Amer. J. Math. 1424Chenxu He, Guofang Wei, and Qi S. Zhang. Fundamental gap of convex domains in the spheres. Amer. J. Math., 142(4):1161-1192, 2020. . Shigeru Ishihara. Quaternion Kählerian manifolds. J. Differential Geometry. 9Shigeru Ishihara. Quaternion Kählerian manifolds. J. Differential Geometry, 9:483-500, 1974. On a lower bound for the first eigenvalue of the Laplace operator on a Riemannian manifold. Atsushi Kasue, Ann. Sci.École Norm. Sup. 174Atsushi Kasue. On a lower bound for the first eigenvalue of the Laplace operator on a Riemannian manifold. Ann. Sci.École Norm. Sup. (4), 17(1):31-44, 1984. Spectrum of the Laplacian on quaternionic Kähler manifolds. Shengli Kong, Peter Li, Detang Zhou, J. Differential Geom. 782Shengli Kong, Peter Li, and Detang Zhou. Spectrum of the Laplacian on quaternionic Kähler manifolds. J. Differential Geom., 78(2):295-332, 2008. On the spectral gap for compact manifolds. Pawel Kröger, J. Differential Geom. 362Pawel Kröger. On the spectral gap for compact manifolds. J. Differential Geom., 36(2):315-330, 1992. A lower bound for the first eigenvalue of the Laplacian on a compact manifold. Peter Li, Indiana Univ. Math. J. 286Peter Li. A lower bound for the first eigenvalue of the Laplacian on a compact manifold. Indiana Univ. Math. J., 28(6):1013-1019, 1979. Moduli of continuity for viscosity solutions. Xiaolong Li, Proc. Amer. Math. Soc. 1444Xiaolong Li. Moduli of continuity for viscosity solutions. Proc. Amer. Math. Soc., 144(4):1717- 1724, 2016. Modulus of continuity estimates for fully nonlinear parabolic equations. Xiaolong Li, arXiv:2006.16631Xiaolong Li. Modulus of continuity estimates for fully nonlinear parabolic equations. arXiv:2006.16631, 2020. . Géométrie des groupes de transformations. André Lichnerowicz, Travaux et Recherches Mathématiques, III. Dunod. ParisAndré Lichnerowicz. Géométrie des groupes de transformations. Travaux et Recherches Mathématiques, III. Dunod, Paris, 1958. Kähler manifolds with Ricci curvature lower bound. Gang Liu, Asian J. Math. 181Gang Liu. Kähler manifolds with Ricci curvature lower bound. Asian J. Math., 18(1):69-99, 2014. Bounds of eigenvalues on Riemannian manifolds. Jun Ling, Zhiqin Lu, Trends in partial differential equations. Somerville, MAInt. Press10Jun Ling and Zhiqin Lu. Bounds of eigenvalues on Riemannian manifolds. In Trends in partial dif- ferential equations, volume 10 of Adv. Lect. Math. (ALM), pages 241-264. Int. Press, Somerville, MA, 2010. On a class of quasi-linear operators on smooth metric measure spaces. Xiaolong Li, Yucheng Tu, Kui Wang, arXiv:2009.10418Xiaolong Li, Yucheng Tu, and Kui Wang. On a class of quasi-linear operators on smooth metric measure spaces. arXiv:2009.10418, 2020. Comparison theorem for Kähler manifolds and positivity of spectrum. Peter Li, Jiaping Wang, J. Differential Geom. 691Peter Li and Jiaping Wang. Comparison theorem for Kähler manifolds and positivity of spectrum. J. Differential Geom., 69(1):43-74, 2005. Moduli of continuity for viscosity solutions on manifolds. Xiaolong Li, Kui Wang, J. Geom. Anal. 271Xiaolong Li and Kui Wang. Moduli of continuity for viscosity solutions on manifolds. J. Geom. Anal., 27(1):557-576, 2017. Sharp lower bound for the first eigenvalue of the weighted p-Laplacian I. Xiaolong Li, Kui Wang, J. Geom. Anal. to appear arxiv:1910.02295Xiaolong Li and Kui Wang. Sharp lower bound for the first eigenvalue of the weighted p-Laplacian I. J. Geom. Anal, to appear arxiv:1910.02295,, 2019. Sharp lower bound for the first eigenvalue of the weighted p-Laplacian II. Xiaolong Li, Kui Wang, arXiv:1911.04596Math. Res. Lett. to appearXiaolong Li and Kui Wang. Sharp lower bound for the first eigenvalue of the weighted p-Laplacian II. Math. Res. Lett, to appear, arXiv:1911.04596, 2019. First Robin eigenvalue of the p-Laplacian on Riemannian manifolds. Xiaolong Li, Kui Wang, arXiv:2002.06472Math. Z. to appearXiaolong Li and Kui Wang. First Robin eigenvalue of the p-Laplacian on Riemannian manifolds. Math. Z., to appear, arXiv:2002.06472, 2020. Lower bounds for the first eigenvalue of the Laplacian on Kähler manifolds. Xiaolong Li, Kui Wang, arXiv:2010.12792Trans. Amer. Math. Soc. to appearXiaolong Li and Kui Wang. Lower bounds for the first eigenvalue of the Laplacian on Kähler manifolds. Trans. Amer. Math. Soc., to appear, arXiv:2010.12792, 2021. Sharp lower bound for the first eigenvalue of the weighted p-Laplacian I. Xiaolong Li, Kui Wang, arxiv:1910.02295J. Geom. Anal, to appear. Xiaolong Li and Kui Wang. Sharp lower bound for the first eigenvalue of the weighted p-Laplacian I. J. Geom. Anal, to appear, arxiv:1910.02295, 2021. Estimates of eigenvalues of a compact Riemannian manifold. Peter Li, Shing Tung Yau, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii. Honolulu, Hawaii; Providence, R.I.Amer. Math. SocProc. Sympos. Pure Math., XXXVIPeter Li and Shing Tung Yau. Estimates of eigenvalues of a compact Riemannian manifold. In Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), Proc. Sympos. Pure Math., XXXVI, pages 205-239. Amer. Math. Soc., Providence, R.I., 1980. Estimates on the modulus of expansion for vector fields solving nonlinear equations. Lei Ni, J. Math. Pures Appl. 999Lei Ni. Estimates on the modulus of expansion for vector fields solving nonlinear equations. J. Math. Pures Appl. (9), 99(1):1-16, 2013. Manifolds with positive orthogonal ricci curvature. Lei Ni, Qingsong Wang, Fangyang Zheng, arXiv:1806.10233v2Amer. J. Math. to appearLei Ni, Qingsong Wang, and Fangyang Zheng. Manifolds with positive orthogonal ricci curvature. Amer. J. Math., to appear, arXiv:1806.10233v2, 2018. Comparison and vanishing theorems for Kähler manifolds. Lei Ni, Fangyang Zheng, Art. 151Calc. Var. Partial Differential Equations. 57631Lei Ni and Fangyang Zheng. Comparison and vanishing theorems for Kähler manifolds. Calc. Var. Partial Differential Equations, 57(6):Art. 151, 31, 2018. On orthogonal Ricci curvature. Lei Ni, Fangyang Zheng, Advances in complex geometry. Providence, RIAmer. Math. Soc735Lei Ni and Fangyang Zheng. On orthogonal Ricci curvature. In Advances in complex geometry, volume 735 of Contemp. Math., pages 203-215. Amer. Math. Soc., Providence, RI, 2019. Certain conditions for a Riemannian manifold to be isometric with a sphere. Morio Obata, J. Math. Soc. Japan. 14Morio Obata. Certain conditions for a Riemannian manifold to be isometric with a sphere. J. Math. Soc. Japan, 14:333-340, 1962. Sharp fundamental gap estimate on convex domains of sphere. Shoo Seto, Lili Wang, Guofang Wei, J. Differential Geom. 1122Shoo Seto, Lili Wang, and Guofang Wei. Sharp fundamental gap estimate on convex domains of sphere. J. Differential Geom., 112(2):347-389, 2019. R Schoen, S.-T Yau, Lectures on differential geometry. Conference Proceedings and Lecture Notes in Geometry and Topology. Cambridge, MAI. International PressR. Schoen and S.-T. Yau. Lectures on differential geometry. Conference Proceedings and Lecture Notes in Geometry and Topology, I. International Press, Cambridge, MA, 1994. Lower bounds for the first eigenvalue on compact manifolds. Min Yu, Hui Chun Shi, Zhang, Chinese Ann. Math. Ser. A. 286Yu Min Shi and Hui Chun Zhang. Lower bounds for the first eigenvalue on compact manifolds. Chinese Ann. Math. Ser. A, 28(6):863-866, 2007. Some comparison theorems for Kähler manifolds. Luen-Fai Tam, Chengjie Yu, Manuscripta Math. 1373-4Luen-Fai Tam and Chengjie Yu. Some comparison theorems for Kähler manifolds. Manuscripta Math., 137(3-4):483-495, 2012. Stability of harmonic maps and eigenvalues of the Laplacian. Hajime Urakawa, Trans. Amer. Math. Soc. 3012Hajime Urakawa. Stability of harmonic maps and eigenvalues of the Laplacian. Trans. Amer. Math. Soc., 301(2):557-589, 1987. An elementary method in the study of nonnegative curvature. H Wu, Acta Math. 1421-2H. Wu. An elementary method in the study of nonnegative curvature. Acta Math., 142(1-2):57-78, 1979. An alternative proof of lower bounds for the first eigenvalue on manifolds. Yuntao Zhang, Kui Wang, Math. Nachr. 29016Yuntao Zhang and Kui Wang. An alternative proof of lower bounds for the first eigenvalue on manifolds. Math. Nachr., 290(16):2708-2713, 2017. On the estimate of the first eigenvalue of a compact Riemannian manifold. Qing Jia, Hong Cang Zhong, Yang, Sci. Sinica Ser. A. 2712Jia Qing Zhong and Hong Cang Yang. On the estimate of the first eigenvalue of a compact Riemannian manifold. Sci. Sinica Ser. A, 27(12):1265-1273, 1984.
{'fraction_non_alphanumeric': 0.09679398332235123, 'fraction_numerical': 0.04998750312421894, 'mean_word_length': 3.4078117175763647, 'pattern_counts': {'":': 0, '<': 5, '<?xml version=': 0, '>': 9, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 53, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'We prove lower bound for the first closed or Neumann nonzero eigenvalue of the Laplacian on a compact quaternion-Kähler manifold in terms of dimension, diameter, and scalar curvature lower bound. It is derived as large time implication of the modulus of continuity estimates for solutions of the heat equation. We also establish lower bound for the first Dirichlet eigenvalue in terms of geometric data, via a Laplace comparison theorem for the distance to the boundary function.Theorem 1.1. Let (M n , g) be a compact Riemannian manifold (possibly with a smooth convex boundary) with diameter D and Ric ≥ (n − 1)κ for κ ∈ R. Let µ 1 be the first nonzero eigenvalue of the Laplacian on M (with Neumann boundary condition if ∂M = ∅). Then µ 1 ≥λ 1 (n, κ, D), 2020 Mathematics Subject Classification. 35P15, 53C26.', 'arxivid': '2105.06303', 'author': ['Xiaolong Li ', 'Kui Wang '], 'authoraffiliation': [], 'corpusid': 234482984, 'doi': '10.1007/s12220-022-01141-5', 'github_urls': [], 'n_tokens_mistral': 18004, 'n_tokens_neox': 14663, 'n_words': 8209, 'pdfsha': 'd7106a6ec169f1026cddf755ccee4b9c94b063c1', 'pdfurls': ['https://arxiv.org/pdf/2105.06303v1.pdf'], 'title': ['EIGENVALUE ESTIMATES ON QUATERNION-KÄHLER MANIFOLDS', 'EIGENVALUE ESTIMATES ON QUATERNION-KÄHLER MANIFOLDS'], 'venue': []}
arxiv
DESIGNING A 3D-AWARE STYLENERF ENCODER FOR FACE EDITING 19 Feb 2023 Songlin Yang [email protected] School of Artificial Intelligence University of Chinese Academy of Sciences BeijingChina Center for Research on Intelligent Perception and Computing NLPR BeijingCASIAChina Wei Wang Center for Research on Intelligent Perception and Computing NLPR BeijingCASIAChina Bo Peng Center for Research on Intelligent Perception and Computing NLPR BeijingCASIAChina Jing Dong [email protected] Center for Research on Intelligent Perception and Computing NLPR BeijingCASIAChina DESIGNING A 3D-AWARE STYLENERF ENCODER FOR FACE EDITING 19 Feb 2023Index Terms-Neural Radiance Field (NeRF)GAN Inversion3D Consistent Face Manipulation GAN inversion has been exploited in many face manipulation tasks, but 2D GANs often fail to generate multi-view 3D consistent images. The encoders designed for 2D GANs are not able to provide sufficient 3D information for the inversion and editing. Therefore, 3Daware GAN inversion is proposed to increase the 3D editing capability of GANs. However, the 3D-aware GAN inversion remains underexplored. To tackle this problem, we propose a 3D-aware (3Da) encoder for GAN inversion and face editing based on the powerful StyleNeRF model. Our proposed 3Da encoder combines a parametric 3D face model with a learnable detail representation model to generate geometry, texture and view direction codes. For more flexible face manipulation, we then design a dual-branch StyleFlow module to transfer the StyleNeRF codes with disentangled geometry and texture flows. Extensive experiments demonstrate that we realize 3D consistent face manipulation in both facial attribute editing and texture transfer. Furthermore, for video editing, we make the sequence of frame codes share a common canonical manifold, which improves the temporal consistency of the edited attributes. INTRODUCTION Face editing via GAN (Generative Adversarial Network) inversion [1] enables users to flexibly edit a wide range of facial attributes in real face images. Existing methods [2,3,4,5] first invert face images into the latent space of 2D GANs such as StyleGAN [6], then manipulate the style codes, and finally feed the edited codes into the pre-trained generator to obtain the edited face images. However, 2D GANs lack the knowledge of the underlying 3D structure of the faces, and their 3D consistency in multi-view generation is limited, as shown in Fig. 1. In order to increase the 3D consistency of the generators in the GAN-inversion-based manipulation pipeline, one intuitive idea is replacing the 2D GANs with 3D-aware GANs [11,12,13,14,15]. However, the vanilla encoders designed for 2D GAN inversion fail to provide sufficient 3D information for the 3D-aware GAN inversion. Furthermore, the SOTA 2D encoders like e4e [16] bring much variety in the inversion stage, which degrades the video consistency in video editing. Therefore, to obtain better 3D consistency in multiview facial attribute editing, we propose a 3D-aware (3Da) StyleN-eRF [13] encoder which encodes geometry and texture separately to have more flexible manipulation capability. Our proposed 3Da encoder combines a parametric 3D face model with a learnable detail representation model to generate the * Corresponding author. Fig. 1. The comparisons of the multi-view editing effects, using StyleRig [7], InterfaceGAN [8], GANSpace [9], StyleFlow [10] and ours. Our method not only achieves better results in novel views, but also preserves the multi-view 3D consistency of the edited results. geometry, texture and view direction codes. By introducing the 3D face model, we can enhance the stability of the generated faces. The 3D-aware inversion codes are then fed into a well trained dualbranch StyleFlow [10] module which makes for the flexible face manipulation. We realize the 3D consistent face manipulation in both facial attribute editing and texture transfer. Moreover, we extend our pipeline to video editing. We make the video frames share a common canonical representation manifold, which improves the temporal consistency of the edited attributes. The main contributions of this work are as follows: We propose the first 3D-aware (3Da) StyleNeRF encoder for the face editing. Our 3Da encoder is able to encode geometry, texture and view direction information separately, achieving multi-view generation and facial attribute editing simultaneously. By introducing the parametric 3D face model, we are able to enhance the stability of the generated faces, which aligns the facial details with the morphable model adaptively. METHOD 3D-Aware StyleNeRF Inversion for Face Embedding The StyleNeRF [13] is adopted as our pre-trained generator. It has two inputs for conditioning the style and camera view respectively. Its NeRF-based [17] architecture performs volume rendering only to produce a low-resolution feature map, and progressively applies up- : to-be-trained Fig. 2. The framework of our 3D-aware (3Da) encoder. Note that when embedding the video frames, the shape β and albedo α should be the same for the same face among different frames, i.e., frame-irrelevant. Therefore, in the video setting, we first extract these two coefficients for all the frames. Then we use the averaged β and α for encoding all the frames. Pre-process Post-process Editing ! Editing Step 1: Inverse with original attribute " Step 2: Forward with target attribute " # Fig. 3. The portrait manipulation pipeline with our 3Da StyleN-eRF encoder for the image and video setting. Our 3D-aware GAN inversion module realizes the disentanglement of canonical and morphable modulations, as well as the separate editing of geometry and texture. Our attribute editing module extends the StyleFlow [10] to two branches for texture and geometry, respectively. Note that our 3Da encoder is able to disentangle the frame-irrelevant information when encoding the video frame code sequence, which is beneficial to increase temporal consistency of the video editing. " # = {"$, "%, " & # , … , "' } " " = {"$, "%, "&, … , "' } … … … … CNF CNF CNF CNF CNF CNF CNF sampling in 2D to obtain high-resolution images. Our first motivation is that, two branches of geometry and texture should be adopted to match the 3D-aware architecture. The second motivation is the disentanglement of canonical and morphable information. So we adopt the parametric 3D face model DECA [18] as 3D prior and use the ResNet-based [19] encoder to provide the detail information. This has two benefits: (1) For training, the sampled style codes with their corresponding synthetic images are used to train the encoder for encoding the images into the GAN space. Our proposed methods can accelerate the convergence and avoid overfitting to the synthetic data. (2) For the video setting, this can guide every code in the code sequence of video frames to share a canonical representation manifold of the target face, preserving temporal consistency of the inversion and editing. Encoder. As shown in Fig. 2 of geometry g, texture t and camera direction c. Then, geometry g and texture t are input into fully-connected mapping networks Mgeo and Mtex to obtain the morphable codes of geometry w morph geo and texture w morph tex . Camera direction c is input into Mcam to get the view direction codes d. Specifically, geometry g is concatenated by shape β and expression ψ and displacement δ. Texture t is concatenated by albedo α, light l. The camera direction c is concatenated by camera C and pose θ. Discriminator. To encourage the style codes to lie within the distribution of the latent style code space of StyleNeRF, denoted as W , a discriminator Dw is used to discriminate between real samples from the W space and the learned latent space of our 3Da encoder. This discriminator is important because it is able to not only accelerate convergence, but also avoid the mode collapse (see Fig. 9). Source geometry Target texture Fig. 6. Texture transfer. The images in the first row provide the source geometry, and the images in the first column provide the target texture. Our 3Da encoder is able to transfer the target texture to the source geometry. To optimize our encoder and discriminator in an adversarial manner, we use the non-saturating GAN loss function [20] to train these networks as follows: L D,E adv = − w∼W [logDw(w)]− x∼p X [log(1−Dw(Ew(x)))],(1)L E rec = Lsim + λ1L style + λ2Lview,(2) where Lsim, L style and Lview are as follows: Lsim = x − G(Ew(x), E d (x)) 2 + vgg(x, G(Ew(x), E d (x))),(3)L style = wgeo − w GT geo 1 + wtex − w GT tex 1,(4)Lview = d − d GT 1,(5) The target image x is the style-mixing image with the groundtruth geometry style code w GT geo , texture style code w GT tex and view direction code d GT . The G is the fixed pre-trained generator. The vgg denotes perceptual loss [21]. We set λ1 and λ2 as 0.5 and 5. Dual-Branch StyleFlow for Face Editing We adopt StyleFlow [10] as the attribute editing method. However, the original StyleFlow only has a single branch of Continuous Normalizing Flow (CNF) blocks, failing to fully utilize the advantages of our 3Da encoder. Therefore, as shown in the Fig. 3, we train two branches of Continuous Normalizing Flows {φs}s=geo,tex. Note that φgeo and φtex are used to obtain geometry style code wgeo and texture style code wtex respectively, for controllable editing. We denote v as the variable of the given StyleNeRF space, while t is the time variable. We suppose that wgeo and wtex are mapped from a latent variable z in a normal distribution. We use {φs}s=geo,tex to conduct the inversion inference as follows: where v(t0) = v(t1) + t 0 t 1 φs(v(t), t, a)dt,(6)v(t0) is the z. For s = geo, the v(t1) is wgeo, while v(t1) is wtex if s = tex. Note that a is the original attribute vector. Then, we modify a according to the given editing instruction, to obtain the edited attribute vector a ′ . After that, we perform a forward inference to produce the edited style code w ′ geo = v(t1) or w ′ tex = v(t1), conditioned on a ′ as follows: v(t1) = v(t0) + t 1 t 0 φs(v(t), t, a ′ )dt,(7) The above is the inference process, and the training details of CNF blocks can be found in this work [10]. EXPERIMENTS Implementation Details Network Architectures. ResNet [19] is used as the backbone for encoding networks Egeo and Etex, to extract the feature vectors, corresponding to the input dimensions of StyleNeRF [13]. Mgeo and Mtex are fully-connected networks with 5 layers, while Mcam has 3 layers. The LeakyReLU is selected as the activation function. We conduct all the experiments on one NVIDIA RTX 3090. We conducted some preliminary prototyping using the MindSpore framework during our implementation. Our encoder requires 4 days, while the editing module requires 2 days. Training Data and Annotation. We randomly sample and save 10,000 groups of style-mixing style codes w, view direction codes d and their corresponding StyleNeRF-generated images as training data. Moreover, the attribute vectors of these generated images are annotated using Microsoft Face API [22], which every dimension of the vector represents an attribute. These attribute vectors and their corresponding style codes are used for training our dual-branch StyleFlow-based attribute editing module. Baseline and Compared Methods. The baseline of our experiments is the StyleGAN [6] with e4e [16] encoder that is wildly used in this field. Our differences are as follows: (1) Inputs: StyleNeRF has input of style codes and view direction codes, while StyleGAN has Fig. 8. Comparison of the temporal consistency of video editing. The 'StyleGAN + e4e' method generates same changing degree of Glasses leads to Black rimmed glasses in the previous frames, while Sun glasses in the later frames. Our 3Da with StyleNeRF is able to maintain the temporal consistency of the edited attributes. [26] as the facial attribute prediction model to output the attribute vectors of face images. Method PSNR ↑ SSIM ↑ VIF ↑ FVD ↓ StyleGAN + Face Image Inversion and Editing Attribute Editing. As shown in Fig. 4, we select Age, Glasses, Beards and Hair as the examples. As shown in Tab. 1, we evaluate the identity consistency scores of edited face images compared with their original images in the FFHQ [27] dataset. The generation quality is quantitatively evaluated in Tab. 2. Multi-Attribute Editing and Multi-View Generation. As shown in Fig. 1, our method has good 3D consistency among different views of edited images. Furthermore, as shown in Fig. 5, our method can handle the multi-view generation and simultaneously edit multiple attributes. Texture Transfer. As shown in Fig. 6, we can realize texture transfer among different real images by combing the geometry style code of one image with the texture style code of another and then inputting the style-mixing codes to the StyleNeRF. This illustrates the good geometry-texture disentanglement ability of our method. Table 3. Temporal attribute inconsistency scores of video editing. Portrait Video Manipulation As shown in Fig. 3, our video manipulation pipeline is composed of three main stages, inspired by the STIT [5] method. First, we use DECA [18] to extract the frame-irrelevant information (shape β and albedo α), encode the cropped face images and smooth the style code sequence over a window of two frames by weighted sum rules. Then, the cropped images and style code sequence are used to finetune the StyleNeRF generator. Note that the style code sequence is fixed in this fine-tuning process. And lastly, the style code sequence is input to our dual-branch StyleFlow to obtain an edited style code sequence conditioned on the required attribute vector, and the finetuned generator is used to obtain the edited frame sequence. Video Inversion. As shown in Fig. 7, we evaluate different methods under the same setting as STIT [5]. The StyleNeRF with our 3Da encoder can achieve better results, in aspects of reconstruction and temporal consistency of identity similarity. Video Editing. As shown in Fig. 8, our method has more consistent video editing effects. As shown in Tab. 3, we quantitatively measure the Mean Absolute Error of the attribute vectors between the first frame and the following frames on 50 testing videos. Each of them has 100 frames with different attribute edited. Our 3Da encoder embeds the frame sequence more stably, as shown by the lower temporal attribute inconsistency. Fig. 9 (b) shows that only using DECA coefficients is insufficient. As shown in Fig. 9 (c) and Fig. 9 (d), 3Da without the discriminator leads to the mode collapse, and using real data in training degrades the image quality. Using the real images without ground-truth style codes for training only has the Lsim loss, which is less effective than the embedding supervision from L style and Lview. Ablation Study CONCLUSION In this paper, we propose a 3D-aware (3Da) StyleNeRF encoder to encode geometry, texture and view direction of the real face images. Extensive experiments qualitatively and quantitatively demonstrate that, we are able to realize high-quality multi-view generation and facial attribute editing. Moreover, we extend our method to the portrait video manipulation, achieving better temporal consistency over the 2D-GAN-based editing methods. ) : shape * : expression + : camera , : pose -: albedo . : light ? : displacement g : geometry $ : texture / : camera direction @ : view direction code ( !"# 1#ABC / ( $"% 1#ABC : morphable code D !"# / D $"% : detail code !"# / $"% : encoding network & !"# / & /01 / & $"% : mapping network ' ( : discriminator 2 Fig. 4 . 4The edited face images of different attributes. Zoom in the digital version for better view. Fig. 5 . 5The multi-view generation of multi-attribute editing. Fig. 7 . 7The inversion of video frame sequences. The StyleNeRF with 3Da (fifth row) achieves better ID preservation results than others. The StyleGAN with e4e (second row) generates some inversion frames with low ID similarity (marked yellow). The e4e fails to keep the consistency of StyleNeRF, as shown in the third row. , for the geometry style code, wgeo = w morph geo + ∆geo. For the texture style code, wtex = w morph tex + ∆tex. The addition operation combines the 3D information and content details. The CNN-based encoding networks Egeo and Etex are used to extract the detail codes ∆geo and ∆tex respectively. For the morphable codes w morph geo , w morph tex and view direction code d, DECA[18] is used to extract the semantic feature vectorsInversion Age Glasses Beards/Hair Original StyleNeRF + 3Da StyleGAN + e4e StyleNeRF + 3Da StyleGAN + e4e Loss Function. For formulation, we denote Ew(x) = {w i } 1≤i≤N as style codes and E d (x) = d as view direction codes, where N is the number of style modulation layers (N = 21 for StyleNeRF). The Ew and E d represent the networks of our 3Da encoder. Note that each code in {w i } 1≤i≤7 is the same and referred to as wgeo, while each code in {w i } 8≤i≤21 is the same and referred to as wtex.Method Age ↑ Glasses ↑ Beards ↑ Hair ↑ StyleGAN + e4e 0.637 0.653 0.651 0.694 StyleNeRF + 3Da 0.794 0.791 0.803 0.811 Table 1. The identity consistency scores of edited face images. Fig. 9. Ablation study of inversion results. (a) shows 3Da encoder with discriminator and without real training data. (b) is mapping DECA coefficients to style codes. (c) is 3Da encoder without the discriminator. (d) is 3Da encoder with real data and the discriminator. Note that they are trained with the same epochs.a ours c w/o dis d with real data b only DECA Input image Method Age ↓ Glasses ↓ Beards ↓ Hair ↓ StyleGAN + e4e 0.498 0.533 0.502 0.497 StyleNeRF + 3Da 0.454 0.223 0.385 0.441 ACKNOWLEDGMENTS Gan inversion: A survey. Weihao Xia, Yulun Zhang, Yujiu Yang, Jing-Hao Xue, Bolei Zhou, Ming-Hsuan Yang, IEEE Transactions on Pattern Analysis and Machine Intelligence. Weihao Xia, Yulun Zhang, Yujiu Yang, Jing-Hao Xue, Bolei Zhou, and Ming-Hsuan Yang, "Gan inversion: A survey," IEEE Transactions on Pattern Analysis and Machine Intelli- gence, 2022. Talk-to-edit: Fine-grained facial editing via dialog. Yuming Jiang, Ziqi Huang, Xingang Pan, Chen Change Loy, Ziwei Liu, Proceedings of the IEEE/CVF International Conference on Computer Vision. the IEEE/CVF International Conference on Computer VisionYuming Jiang, Ziqi Huang, Xingang Pan, Chen Change Loy, and Ziwei Liu, "Talk-to-edit: Fine-grained facial editing via dialog," in Proceedings of the IEEE/CVF International Con- ference on Computer Vision, 2021, pp. 13799-13808. Styleheat: One-shot high-resolution editable talking face generation via pretrained stylegan. Fei Yin, Yong Zhang, Xiaodong Cun, Mingdeng Cao, Yanbo Fan, Xuan Wang, Qingyan Bai, Baoyuan Wu, Jue Wang, Yujiu Yang, arXiv:2203.04036arXiv preprintFei Yin, Yong Zhang, Xiaodong Cun, Mingdeng Cao, Yanbo Fan, Xuan Wang, Qingyan Bai, Baoyuan Wu, Jue Wang, and Yujiu Yang, "Styleheat: One-shot high-resolution editable talking face generation via pretrained stylegan," arXiv preprint arXiv:2203.04036, 2022. Video2stylegan: Disentangling local and global variations in a video. Rameen Abdal, Peihao Zhu, J Niloy, Peter Mitra, Wonka, arXiv:2205.13996arXiv preprintRameen Abdal, Peihao Zhu, Niloy J Mitra, and Peter Wonka, "Video2stylegan: Disentangling local and global variations in a video," arXiv preprint arXiv:2205.13996, 2022. Rotem Tzaban, Ron Mokady, Rinon Gal, H Amit, Daniel Bermano, Cohen-Or, arXiv:2201.08361Stitch it in time: Gan-based facial editing of real videos. arXiv preprintRotem Tzaban, Ron Mokady, Rinon Gal, Amit H Bermano, and Daniel Cohen-Or, "Stitch it in time: Gan-based facial edit- ing of real videos," arXiv preprint arXiv:2201.08361, 2022. Analyzing and improving the image quality of stylegan. Tero Karras, Samuli Laine, Miika Aittala, Janne Hellsten, Jaakko Lehtinen, Timo Aila, CVPR. Tero Karras, Samuli Laine, Miika Aittala, Janne Hellsten, Jaakko Lehtinen, and Timo Aila, "Analyzing and improving the image quality of stylegan," in CVPR, 2020, pp. 8110-8119. Stylerig: Rigging stylegan for 3d control over portrait images. Ayush Tewari, Mohamed Elgharib, Gaurav Bharaj, Florian Bernard, Hans-Peter Seidel, Patrick Pérez, Michael Zollhofer, Christian Theobalt, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionAyush Tewari, Mohamed Elgharib, Gaurav Bharaj, Florian Bernard, Hans-Peter Seidel, Patrick Pérez, Michael Zollhofer, and Christian Theobalt, "Stylerig: Rigging stylegan for 3d control over portrait images," in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2020, pp. 6142-6151. Interfacegan: Interpreting the disentangled face representation learned by gans. Yujun Shen, Ceyuan Yang, Xiaoou Tang, Bolei Zhou, IEEE transactions. 2020Yujun Shen, Ceyuan Yang, Xiaoou Tang, and Bolei Zhou, "In- terfacegan: Interpreting the disentangled face representation learned by gans," IEEE transactions on pattern analysis and machine intelligence, 2020. Ganspace: Discovering interpretable gan controls. Erik Härkönen, Aaron Hertzmann, Jaakko Lehtinen, Sylvain Paris, Advances in Neural Information Processing Systems. 33Erik Härkönen, Aaron Hertzmann, Jaakko Lehtinen, and Syl- vain Paris, "Ganspace: Discovering interpretable gan con- trols," Advances in Neural Information Processing Systems, vol. 33, pp. 9841-9850, 2020. Styleflow: Attribute-conditioned exploration of stylegangenerated images using conditional continuous normalizing flows. Rameen Abdal, Peihao Zhu, J Niloy, Peter Mitra, Wonka, ACM Transactions on Graphics (ToG). 403Rameen Abdal, Peihao Zhu, Niloy J Mitra, and Peter Wonka, "Styleflow: Attribute-conditioned exploration of stylegan- generated images using conditional continuous normalizing flows," ACM Transactions on Graphics (ToG), vol. 40, no. 3, pp. 1-21, 2021. Graf: Generative radiance fields for 3d-aware image synthesis. Katja Schwarz, Yiyi Liao, Michael Niemeyer, Andreas Geiger, Advances in Neural Information Processing Systems. 33Katja Schwarz, Yiyi Liao, Michael Niemeyer, and Andreas Geiger, "Graf: Generative radiance fields for 3d-aware im- age synthesis," Advances in Neural Information Processing Systems, vol. 33, pp. 20154-20166, 2020. Giraffe: Representing scenes as compositional generative neural feature fields. Michael Niemeyer, Andreas Geiger, CVPR. Michael Niemeyer and Andreas Geiger, "Giraffe: Represent- ing scenes as compositional generative neural feature fields," in CVPR, 2021, pp. 11453-11464. Stylenerf: A style-based 3d-aware generator for highresolution image synthesis. Jiatao Gu, Lingjie Liu, Peng Wang, Christian Theobalt, 2022Jiatao Gu, Lingjie Liu, Peng Wang, and Christian Theobalt, "Stylenerf: A style-based 3d-aware generator for high- resolution image synthesis," ICLR, 2022. Efficient geometry-aware 3d generative adversarial networks. Connor Z Eric R Chan, Lin, A Matthew, Koki Chan, Boxiao Nagano, Pan, Orazio Shalini De Mello, Leonidas J Gallo, Jonathan Guibas, Sameh Tremblay, Khamis, CVPREric R Chan, Connor Z Lin, Matthew A Chan, Koki Nagano, Boxiao Pan, Shalini De Mello, Orazio Gallo, Leonidas J Guibas, Jonathan Tremblay, Sameh Khamis, et al., "Effi- cient geometry-aware 3d generative adversarial networks," in CVPR, 2022, pp. 16123-16133. 3d-aware image synthesis via learning structural and textural representations. Yinghao Xu, Sida Peng, Ceyuan Yang, Yujun Shen, Bolei Zhou, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionYinghao Xu, Sida Peng, Ceyuan Yang, Yujun Shen, and Bolei Zhou, "3d-aware image synthesis via learning struc- tural and textural representations," in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recog- nition, 2022, pp. 18430-18439. Designing an encoder for stylegan image manipulation. Omer Tov, Yuval Alaluf, Yotam Nitzan, Or Patashnik, Daniel Cohen-Or, ACM Transactions on Graphics (TOG). 404Omer Tov, Yuval Alaluf, Yotam Nitzan, Or Patashnik, and Daniel Cohen-Or, "Designing an encoder for stylegan image manipulation," ACM Transactions on Graphics (TOG), vol. 40, no. 4, pp. 1-14, 2021. Nerf: Representing scenes as neural radiance fields for view synthesis. Ben Mildenhall, P Pratul, Matthew Srinivasan, Jonathan T Tancik, Ravi Barron, Ren Ramamoorthi, Ng, ECCV. SpringerBen Mildenhall, Pratul P Srinivasan, Matthew Tancik, Jonathan T Barron, Ravi Ramamoorthi, and Ren Ng, "Nerf: Representing scenes as neural radiance fields for view synthe- sis," in ECCV. Springer, 2020, pp. 405-421. Learning an animatable detailed 3d face model from in-thewild images. Yao Feng, Haiwen Feng, J Michael, Timo Black, Bolkart, ACM Transactions on Graphics (ToG). 404Yao Feng, Haiwen Feng, Michael J Black, and Timo Bolkart, "Learning an animatable detailed 3d face model from in-the- wild images," ACM Transactions on Graphics (ToG), vol. 40, no. 4, pp. 1-13, 2021. Deep residual learning for image recognition. Kaiming He, Xiangyu Zhang, Shaoqing Ren, Jian Sun, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionKaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun, "Deep residual learning for image recognition," in Proceed- ings of the IEEE conference on computer vision and pattern recognition, 2016, pp. 770-778. Generative adversarial nets. Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, Yoshua Bengio, Advances in neural information processing systems. 27Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio, "Generative adversarial nets," Advances in neural information processing systems, vol. 27, 2014. Perceptual losses for real-time style transfer and super-resolution. Justin Johnson, Alexandre Alahi, Li Fei-Fei, SpringerJustin Johnson, Alexandre Alahi, and Li Fei-Fei, "Percep- tual losses for real-time style transfer and super-resolution," in European conference on computer vision. Springer, 2016, pp. 694-711. . Azure, MicrosoftAzure, "https://azure.microsoft.com/en-in/services/cognitive- services/face/," Microsoft, 2020. Towards accurate generative models of video: A new metric & challenges. Thomas Unterthiner, Karol Sjoerd Van Steenkiste, Raphael Kurach, Marcin Marinier, Sylvain Michalski, Gelly, arXiv:1812.01717arXiv preprintThomas Unterthiner, Sjoerd van Steenkiste, Karol Kurach, Raphael Marinier, Marcin Michalski, and Sylvain Gelly, "To- wards accurate generative models of video: A new metric & challenges," arXiv preprint arXiv:1812.01717, 2018. The unreasonable effectiveness of deep features as a perceptual metric. Richard Zhang, Phillip Isola, Alexei A Efros, Eli Shechtman, Oliver Wang, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionRichard Zhang, Phillip Isola, Alexei A Efros, Eli Shechtman, and Oliver Wang, "The unreasonable effectiveness of deep fea- tures as a perceptual metric," in Proceedings of the IEEE con- ference on computer vision and pattern recognition, 2018, pp. 586-595. Arcface: Additive angular margin loss for deep face recognition. Jiankang Deng, Jia Guo, Niannan Xue, Stefanos Zafeiriou, CVPR. Jiankang Deng, Jia Guo, Niannan Xue, and Stefanos Zafeiriou, "Arcface: Additive angular margin loss for deep face recogni- tion," in CVPR, 2019, pp. 4690-4699. Large-scale celebfaces attributes (celeba) dataset. Ziwei Liu, Ping Luo, Xiaogang Wang, Xiaoou Tang, Retrieved August. 1511Ziwei Liu, Ping Luo, Xiaogang Wang, and Xiaoou Tang, "Large-scale celebfaces attributes (celeba) dataset," Retrieved August, vol. 15, no. 2018, pp. 11, 2018. A style-based generator architecture for generative adversarial networks. Tero Karras, Samuli Laine, Timo Aila, CVPR. Tero Karras, Samuli Laine, and Timo Aila, "A style-based generator architecture for generative adversarial networks," in CVPR, 2019, pp. 4401-4410.
{'fraction_non_alphanumeric': 0.05255531208029971, 'fraction_numerical': 0.024351452604792535, 'mean_word_length': 4.689724512366781, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 1, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'GAN inversion has been exploited in many face manipulation tasks, but 2D GANs often fail to generate multi-view 3D consistent images. The encoders designed for 2D GANs are not able to provide sufficient 3D information for the inversion and editing. Therefore, 3Daware GAN inversion is proposed to increase the 3D editing capability of GANs. However, the 3D-aware GAN inversion remains underexplored. To tackle this problem, we propose a 3D-aware (3Da) encoder for GAN inversion and face editing based on the powerful StyleNeRF model. Our proposed 3Da encoder combines a parametric 3D face model with a learnable detail representation model to generate geometry, texture and view direction codes. For more flexible face manipulation, we then design a dual-branch StyleFlow module to transfer the StyleNeRF codes with disentangled geometry and texture flows. Extensive experiments demonstrate that we realize 3D consistent face manipulation in both facial attribute editing and texture transfer. Furthermore, for video editing, we make the sequence of frame codes share a common canonical manifold, which improves the temporal consistency of the edited attributes.', 'arxivid': '2302.09467', 'author': ['Songlin Yang [email protected] \nSchool of Artificial Intelligence\nUniversity of Chinese Academy of Sciences\nBeijingChina\n\nCenter for Research on Intelligent Perception and Computing\nNLPR\nBeijingCASIAChina\n', 'Wei Wang \nCenter for Research on Intelligent Perception and Computing\nNLPR\nBeijingCASIAChina\n', 'Bo Peng \nCenter for Research on Intelligent Perception and Computing\nNLPR\nBeijingCASIAChina\n', 'Jing Dong [email protected] \nCenter for Research on Intelligent Perception and Computing\nNLPR\nBeijingCASIAChina\n', 'Songlin Yang [email protected] \nSchool of Artificial Intelligence\nUniversity of Chinese Academy of Sciences\nBeijingChina\n\nCenter for Research on Intelligent Perception and Computing\nNLPR\nBeijingCASIAChina\n', 'Wei Wang \nCenter for Research on Intelligent Perception and Computing\nNLPR\nBeijingCASIAChina\n', 'Bo Peng \nCenter for Research on Intelligent Perception and Computing\nNLPR\nBeijingCASIAChina\n', 'Jing Dong [email protected] \nCenter for Research on Intelligent Perception and Computing\nNLPR\nBeijingCASIAChina\n'], 'authoraffiliation': ['School of Artificial Intelligence\nUniversity of Chinese Academy of Sciences\nBeijingChina', 'Center for Research on Intelligent Perception and Computing\nNLPR\nBeijingCASIAChina', 'Center for Research on Intelligent Perception and Computing\nNLPR\nBeijingCASIAChina', 'Center for Research on Intelligent Perception and Computing\nNLPR\nBeijingCASIAChina', 'Center for Research on Intelligent Perception and Computing\nNLPR\nBeijingCASIAChina', 'School of Artificial Intelligence\nUniversity of Chinese Academy of Sciences\nBeijingChina', 'Center for Research on Intelligent Perception and Computing\nNLPR\nBeijingCASIAChina', 'Center for Research on Intelligent Perception and Computing\nNLPR\nBeijingCASIAChina', 'Center for Research on Intelligent Perception and Computing\nNLPR\nBeijingCASIAChina', 'Center for Research on Intelligent Perception and Computing\nNLPR\nBeijingCASIAChina'], 'corpusid': 257038059, 'doi': '10.1109/icassp49357.2023.10094932', 'github_urls': [], 'n_tokens_mistral': 8518, 'n_tokens_neox': 7472, 'n_words': 4191, 'pdfsha': '41fbd58779db0947b27051b8014b8a9d76baa9fa', 'pdfurls': ['https://export.arxiv.org/pdf/2302.09467v1.pdf'], 'title': ['DESIGNING A 3D-AWARE STYLENERF ENCODER FOR FACE EDITING', 'DESIGNING A 3D-AWARE STYLENERF ENCODER FOR FACE EDITING', 'DESIGNING A 3D-AWARE STYLENERF ENCODER FOR FACE EDITING', 'DESIGNING A 3D-AWARE STYLENERF ENCODER FOR FACE EDITING'], 'venue': []}
arxiv
THE EISENSTEIN CYCLES AND MANIN-DRINFELD PROPERTIES 14 Apr 2022 Debargha Banerjee Loïc Merel THE EISENSTEIN CYCLES AND MANIN-DRINFELD PROPERTIES 14 Apr 2022arXiv:2204.06379v2 [math.NT] Let Γ be a subgroup of finite index of SL 2 (Z). We give an analytic criterion for a cuspidal divisor to be torsion in the Jacobian J Γ of the corresponding modular curve X Γ . By Belyi's theorem, such a criterion would apply to any curve over a number field. Our main tool is the explicit description, in terms of modular symbols, of what we call Eisenstein cycles. The latter are representations of relative homology classes over which integration of any holomorphic differential forms vanishes. Our approach relies in an essential way on the specific case Γ ⊂ Γ(2), where we can consider convenient generalized Jacobians instead of J Γ .The Eisenstein classes are the real part of certain homology classes with complex coefficients. The imaginary part of those classes are related to the scattering constants attached to Eisenstein series. Finally, we illustrate our theory by considering Fermat curves. application in mind: the divisor D is torsion in the Jacobian J Γ of X Γ if and only if E D ∈ H 1 (X Γ , ∂ Γ ; Q).It is known since Manin and Drinfeld that D is torsion whenΓ is a congruence subgroup [9, 16]. Scholl [23], Murty-Ramakrishnan [19] and recently Burrin [5] have given criteria for D being torsion in J Γ , without appealing to E D . Our approach to this question is different from those authors and is a continuation of [4] and [17] (where we limited ourselves to congruence subgroups).We impose a more specific setting (as we will see, without losing generality), in connection with the theory of Belyi maps and dessins d'enfants, see section 2. We assume in this article that Γ is contained in Γ(2) (the principal congruence subgroup of level 2) and that −Id ∈ Γ. The corresponding modular INTRODUCTION Let Γ be a subroup of finite index of SL 2 (Z), acting on the upper half-plane H. Consider the modular curve Y Γ = Γ\H. Let ∂ Γ = Γ\P 1 (Q) be the set of cusps and X Γ be the compact modular curve obtained from Y Γ = Γ\H by adding the cusps. We have a long exact sequence of relative homology: 0 → H 1 (X Γ ; Z) → H 1 (X Γ , ∂ Γ ; Z) δ − → Z[∂ Γ ] → Z → 0, where the map H 1 (X Γ , ∂ Γ ; Z) δ − → Z[∂ Γ ] is obtained from the boundary map H 1 (X Γ , ∂ Γ ; Z) → H 0 (∂ Γ ; Z). Let H 0 (X Γ , Ω 1 ) denote the complex vector space of global sections of the sheaf of holomorphic differentials on the Riemann surface X Γ . Let D = j∈∂Γ m j [j] be a divisor of degree 0 supported on ∂ Γ . We call Eisenstein classes corresponding to D to be the unique element E D ∈ H 1 (X Γ , ∂ Γ ; R) such that ED ω = 0 for all ω ∈ H 0 (X Γ , Ω 1 ) and δ(E D ) = D. We will determine this class in this article, with the following S D (x) = 1 2πi lim s→1 + j∈∂Γ m j ∞ r=1 σ∈Γj \Γ/Γj e 2πir(x+ d c ) |c| 2s , where the entries c and d are such that σ −1 j σσ j = ( ⋆ ⋆ c d ) and c is required to be > 0. The internal sum over σ is sometimes called a Kloosterman zeta function [13,Chapter 9,p. 121], [23, p. 14]. A priori, such Dirichlet series converge absolutely for ℜ(s) > 1. Their values at 1 exist since D is of degree 0. Let D + ∈ Div 0 (∂ + Γ ) (resp. D − ∈ Div 0 (∂ − Γ )). Define F D + and F D + : Γ(2) → C by F D + (h) = S D + (h(−1)) − S D + (h(1)) and F D − (h) = S D − (h(∞)) − S D − (h(0)). We will see that those Γ-invariant quantities can be understood as periods of certain Eisenstein series. Set E D + = g∈Γ\Γ (2) F D + (g){g0, g∞} + and E D − = g∈Γ\Γ (2) F D − (g){g(1), g(−1)} − in H 1 (X Γ − ∂ − Γ , ∂ + Γ ; C) and H 1 (X Γ − ∂ + Γ , ∂ − Γ ; C) respectively. We decompose them into their real and imaginary parts: E D + = R D + + 1 2πi I D + and E D − = R D − + 1 2πi I D − . Theorem 1. For all differential forms of the third kind ω on X Γ with poles in ∂ − Γ (resp. ∂ + Γ ), one has R D + ω = 0 (resp. R D − ω = 0) and the boundary of R D + (resp. R D − ) is −D + (resp. −D − ). Moreover E D + (resp. E D − ) has boundary D + (resp. D − ). It follows that the images of R D + and R D − in H 1 (X Γ , ∂ Γ ; R) are Eisenstein classes. They are the classes of what we call Eisenstein cycles. We will explain in section 10 how to derive all Eisenstein classes in H 1 (X Γ , ∂ Γ ; R) from such images. We show in section 7 that I D − and I D + are deduced from the scattering constants, obtained from specializing scattering matrices of Eisenstein series at s = 1. We have a canonical map λ + : H 0 (∂ − Γ , R) → H 1 (X Γ − ∂ − Γ , ∂ + Γ , R) (resp. λ − : H 0 (∂ + Γ , R) → H 1 (X Γ − ∂ + Γ , ∂ − Γ , R)) dual to the boundary map. Its image contains I D + (resp. I D − ), see section 8. Let J − Γ (resp. J + Γ ) be the generalized Jacobian of X Γ with respect to the set ∂ − Γ (resp. ∂ + Γ ). Theorem 2. The class of D + (resp. D − ) in J − Γ (resp. J + Γ ) is torsion if and only if for every g ∈ Γ\Γ(2), one has F D + (g) ∈ Q (resp. F D − (g) ∈ Q). The class of D + (resp. D − ) in J Γ is torsion if and only if one has g∈Γ\Γ(2) F D + (g)[g] ∈ Q[Γ\Γ (2) ] + λ + (R[∂ − Γ ]) (resp. g∈Γ\Γ(2) F D − (g)[g] ∈ Q[Γ\Γ(2)] + λ − (R[∂ + Γ ]). The first criterion is similar to the following statement: e 2πiz is torsion in C × if and only if the real part of z is rational, and the imaginary part of z is 0. Note that the first statement is substantial even if X Γ has genus 0. The second criterion can be derived from the first. The maps λ + and λ − are computed explicitly by proposition 5. The principal weakness of our approach resides in the fact that the functions F D + and F D − are not easily computable for two reasons : (i) the convergence of Dirichlet series is notoriously difficult to understand, and (ii) we have no convenient way to pass from a combinatorial description of P Γ(2) = Γ(2)/{±1} as a subgroup of the free group on two generators P Γ(2) to a description of the elements of Γ as 2 × 2 matrices. But it is expected that the Eisenstein classes are of a transcendental nature in general, and we hope that our method cast a new light on this nature. We computed the functions F + D and F − D when Γ is congruence subgroup [2,3,4]. Here is a classical instance of a non-congruence subgroup where the computation of these formulas is possible. Let N be an integer > 0. Let Φ N be the kernel of the composed maps Γ(2) → Γ(2) ab → Γ(2) ab /N , where the first map is the abelianization. It is a non-congruence subgroup, when N > 8. The corresponding modular curve X ΦN is the N -th Fermat curve (i.e. it admits the model X N + Y N = 1, the cusps correspond to the points such that X N = 0, 1 or ∞). Rohrlich has shown that every cuspidal divisor of degree 0 is torsion in J ΦN [22]. As an application of our theory, we describe the cuspidal subgroups of J + ΦN and J − ΦN ; as expected they are finite. THE COMBINATORICS OF BELYI MAPS Let X be a smooth projective curve over C. A Belyi map is a morphism f : X → P 1 (C) unramified outside 0, 1, ∞ [25]. A dessin d'enfant [12] is a connected graph, composed of a finite set of vertices V and a finite set of edges E, with the additional structures : • a map from V to the pair {0, ∞} (the graph is bicolored) • the extremities of a given edge have different colors • the set of of edges adjacent to a given vertice is endowed with a cyclic ordering (i.e. a transitive action of Z). The dessin d'enfant associated to f is the bicolored graph G whose vertices of color 0 (resp. ∞) constitute f −1 (0) (resp. f −1 (∞)). The edges of the graph are the components the inverse images by f of the arc form ∞ to 0 in P 1 (C) (we depart here from the usual convention, where 0 and 1 are used instead of 0 and ∞). Consider the function on the upper half-plane [6, p. 86 and p. 105]: λ(z) = 16e 2πiz n≥1 1 + e 2nπiz 1 + e (2n−1)πiz 8 . It identifies the modular curve X(2) to the projective line. We resist the tentation to use 1/λ; instead we follow here the customary convention so that λ(∞) = 0, λ(0) = ∞, λ(1) = λ(−1) = 1. Thus the arc from ∞ to 0 on the projective line is the image by λ of the arc from 0 to ∞ in the upper half-plane. Since Γ(2)/{−1, 1} identifes to the fundamental group of the affine modular curve Y (2), and the upper half-plane is the universal covering of Y (2), the morphism f can be regarded as the morphism X Γ → X(2), for an appropriate subgroup Γ ⊂ Γ(2) of finite index, and containing −Id. The group Γ(2)/{−1, 1} is freely generated by {A, B} where A is the image of 1 2 0 1 and B is the image of 1 0 2 1 . The dessin d'enfant attached to X Γ is the graph whose vertices are the cusps of X Γ above 0 (of colour ∞) and ∞ (of colour 0). The edges are the translates by Γ(2) of the image in X Γ of the arc from 0 to ∞ (in the upper half-plane). The action of Z on the edges attached to a vertice of color 0 (resp. ∞) follows from the action of A (resp. B −1 ). Conversely, Γ can be recovered as follows from the dessin. It acts on the edges of the dessin: consider an edge e attached to a pair of vertices v e (0) and v e (∞) (coloured 0 and ∞ respectively), the image of e by A (resp. B) is the successor (resp. predecessor) of e along the vertice v e (0) (resp. v e (∞)). The stabilizers of all edges constitute Γ. Thus a dessin d'enfant encodes the situation considered in this article. Question: Is there a natural interpretation of the Kloosterman zeta function purely in terms of the geometry of the dessin, without involving the entries of matrices in Γ(2)? PRELIMINARIES AND PREVIOUS WORK Let X be a compact, connected, non-empty Riemann surface. Let S be a finite subset of X. Let D be a divisor of degree 0 supported on S. Recall that a differential of the third kind on X is a meromorphic differential whose poles are simple and whose residues are integers. Let ω be a differential of the third kind on X, with divisor of poles equal to D. It exists by the Riemann Roch theorem. There exists a unique holomorphic differential form ω ′ ∈ Ω 1 (X) such that, for every c ∈ H 1 (X, S; Z), one has Re( c ω) = Re( c ω ′ ). Therefore there exists a unique differential of the third kind ω D of residue divisor D such that Re( c ω D ) = 0 for every c ∈ H 1 (X, S; Z). This is the canonical differential of the third kind associated to D. The notion extends obviously to the situation where D is a divisor of degree 0 with real coefficients. Let f D be a multivalued function on X −S such that df D = ω D . Since the periods of ω D are imaginary, the real part g D of f D is single-valued and harmonic. We turn now to the specific situation of modular curves. Let Γ be a subgroup of finite index of the modular group. The cases of noncongruence subgroups has been studied since Atkin and Swinnerton-Dyer [1]. Suppose X is the modular curve X Γ associated to Γ, and S is the set ∂ Γ of cusps. The pullbacks to the upper half-plane of differential of the third kind care of the form 2πif (z) dz, where f is a holomorphic modular form of weight 2 for Γ. The pullback of the canonical differential of the third kind associated to D is of the form 2πiG D (z) dz, where G D is by definition the Eisenstein series associated to D. The Eisenstein series G D has been determined explicitly see section 6. We find convenient to follow Scholl's account, which in turn follows Hecke, Selberg, Kubota etc. Let j ∈ ∂ Γ and choose σ j ∈ SL 2 (Z) such that σ j ∞ ∈ j and denote by Γ j the stabilizer of σ j ∞ in Γ. For s ∈ C, with Re(s) > 1 and z = x + iy ∈ H, set E j (z, s) = σ∈Γj \Γ Im(σ −1 j σ(z)) s = σ∈Γj \Γ y s |cz + d| s , where (c, d) is the lower row of σ −1 j σ. As a function of s, E j (z, s) admits a meromorphic continuation to C, with a simple pole at s = 1. Its study involves the corresponding Kloosterman zeta function at the pair of cusps (j, k), which has been introduced by Selberg, even though the terminology seems due in print to Goldfeld and Sarnak [10], see also the monographies of Kubota [ φ jk,r (s) := σ∈Γj \Γ/Γ k e 2πird c |c| 2s , where the entries c and d are such that σ −1 j σσ k = ( ⋆ ⋆ c d ) and c is required to be > 0. The series is absolutely convergent if Re(s) > 1, and can be extended to a meromorphic function on the complex plane, which is due to Selberg [24], see for instance [13,Chapter 9,p. 122]. It has a pole at s = 1 when r = 0, and extends holomorphically to the neighbourhood of s = 1, when r = 0. The notation φ jk,r comes from Kubota's book [14], and has been retained by Goldstein and Scholl [11,23], and has been modified by Iwaniec in [13]. The expansion of E j at σ k (∞) is given by the formula : (3.1) E j (σ k (z), s) = r∈Z a jk,r (y, s)e 2πirx . where, for r ≥ 1, (3.2) a jk,r (y, s) = φ jk,r (s)F r (y, s), and (3. 3) a jk,0 (y, s) = δ j,k y s + φ jk,0 (s)y 1−s π 1/2 Γ(s − 1/2) Γ(s) here δ j,k is the Kronecker symbol, and F r (y, s) is built out of Bessel functions, and does not concern us, except for the value at s = 1. Note that Scholl's formula for a jk,0 (y, s) seems to contain a misprint: it involves π s instead of π 1/2 , contra Kubota and Iwaniec. The coefficients φ jk,0 (s)π 1/2 Γ(s − 1/2)/Γ(s) are the entries of the scattering matrix at the pair (j, k). Such expressions have all a simple pole at s = 1, whose residue does not depend on j and k. More precisely, the following limit exists (3.4) C j,k = lim s→1 (φ jk,0 (s)π 1/2 Γ(s − 1/2) Γ(s) − 2 π|Γ(2)/Γ|(s − 1) ) = lim s→1 (πφ jk,0 (s) − 2 π|Γ(2)/Γ|(s − 1) ). It is the scattering constant, see [15], at the pair (i, j). As a remark on the terminology, we note that the term seems to be of more recent use than the calculations that justifies its existence. It does not appear in [13], for instance. The constants could be, and have been [20], normalized differently, for instance par using ζ(s) instead of 1/(s − 1). Since we will only be interested in differences between those constants, this doesn't matter to us. Following Scholl [23, p. 15], define G j (z, s) = 2i ∂ ∂z E j (z, s). Furthermore, write G j (z) = lim s→1 G j (z, s). The Kloosterman zeta function is involved in the qexpansion of G D by the formula [23] (based on [14]) G j (σ k (z))J(σ j , z) −2 = δ j,k − πC y − 4π 2 ∞ r=1 rφ jk,r (1)q r where z = x+ iy, q = e 2πiz , J(γ, z) = cz + d when γ = ( ⋆ ⋆ c d ), C is a constant independent of j, and δ j,k is the Kronecker symbol. Note that if we write G j|σj (z) = G j (σ j (z))J(σ j , z) −2 , one has G j|σj = G σ −1 j Γj = G σ −1 j Γσj ∞ . Write D = j∈∂Γ m j [j] . The limit lim s→1 E j (z, s) makes no sense, because of the pole at s = 1. But E D (z) = lim s→1 j∈∂Γ m j E j (z, s) exists, as the residue at s = 1 disappears. The modular form G D is given by G D (z) = m j=1 m j G j (z). It is holomorphic since D is of degree 0. Write G D (z) = ∞ r=0 a r e 2πirz . Scholl, using Waldschmidt's work in transcendental number theory [27], established the following criterion. Theorem 3 (Scholl,[23]). The divisor D is torsion in J Γ if and only if, for every integer r ≥ 1, the coefficient (3.5) a r = −4π 2 r j∈∂Γ m j φ jΓ∞,r (1) is an algebraic number. Note that the differential form G D (z) dz is determined by finitely many (in terms of the genus of X Γ ) coefficients a r . Therefore the criterion can be reduced to verifying the algebricity of finitely many numbers. The coefficients a r have been computed slightly more explicitly by K. Murty and Ramakrishnan, in terms of what they call generalized Ramanujan sums [19]. In [15], Kühn expressed the Néron-Tate pairing of two divisors of degree zero supported on ∂ Γ as a rational linear combination of (1) logarithms of integers and (2) products of π by scattering constants, at least when X Γ is defined over Q. He derived as a consequence a formula for the scattering constants when all such divisors are torsion in J Γ . MIXED HOMOLOGY GROUPS We retain the notations of the introduction. We follow [17] and study the mixed homology groups H 1 (X Γ − ∂ − Γ , ∂ + Γ ; Z) and H 1 (X Γ − ∂ + Γ , ∂ − Γ ; Z). The intersection pairing provides a perfect bilinear pairing • : H 1 (X Γ − ∂ + Γ , ∂ − Γ ; Z) × H 1 (X Γ − ∂ − Γ , ∂ + Γ ; Z) → Z. For g ∈ Γ(2), set ξ + (Γg) = {g0, g∞} + and ξ − (Γg) = {g1, g(−1)} − . By linearity, we extend the maps ξ + and ξ − to Z[Γ\Γ(2)] → H 1 (X Γ − ∂ − Γ , ∂ + Γ ; Z) and Z[Γ\Γ(2)] → H 1 (X Γ − ∂ + Γ , ∂ − Γ ; Z) respectively. Theorem 4 ( [17] ). The map ξ + and ξ − thus obtained are group isomorphisms. Furthermore, for g, h ∈ Γ(2) the intersection pairing ξ + (g) • ξ − (h) is equal to 1 if Γg = Γh and to 0 otherwise. Consider the maps: H 1 (X Γ − ∂ − Γ , ∂ + Γ ; Z) → H 1 (X Γ , ∂ + Γ ; Z) and H 1 (X Γ − ∂ + Γ , ∂ − Γ ; Z) → H 1 (X Γ , ∂ − Γ ; Z) . The kernel of those maps are the image of the maps λ + : Z[∂ − Γ ] ≃ H 0 (∂ − Γ , Z) → H 1 (X Γ − ∂ + Γ , ∂ − Γ ; Z) and λ − : Z[∂ + Γ ] ≃ H 0 (∂ + Γ ; Z) → H 1 (X Γ − ∂ − Γ , ∂ + Γ ; Z) respectively. To be precise, λ − (x) (resp. λ + (x) ) is homologous to a counterclockwise loop around the cusp x. The width of the cusp is the ramification index of π 0 at that cusp. Recall that A (resp. B −1 , resp. BA −1 ) is the generator of the stabiliser of ∞ (resp. 0, resp. 1) in P Γ(2) such that, for z in the upper half-plane, the image in Y (2) of a path from z to Az (resp. B −1 z, resp. BA −1 z) is a counterclockwise loop around Γ(2)∞ (resp. Γ(2)0, resp. Γ(2)1). Proposition 5. Let g ∈ Γ(2) and denote by w ∞ the width of the cusp j = Γg∞. One has λ − (Γg∞) = − w∞−1 k=0 ξ − (gA k ) = h∈Γ\Γ(2),h∞=j ξ − (h). Denote by w 0 the width of the cusp Γg0. One has λ − (Γg0) = w0−1 k=0 ξ − (gB k ) = h∈Γ\Γ(2),h0=j ξ − (h). Denote by w 1 the width of the cusp Γg1. One has λ + (Γg1) = w1−1 k=0 ξ + (g(AB −1 ) k B) − ξ + (g(AB −1 ) k ) = h∈Γ\Γ(2),h(−1)=j ξ − (h) − h∈Γ\Γ(2),h1=j ξ − (h). Proof. Note that A∞ = ∞, B0 = 0, A(−1) = B(−1) = 1. The class of a loop around the cusp Γg∞ is given by {g(−1), (gAg −1 ) w∞ g(−1)} − = {g(−1), gA w∞ (−1)} − = {g(−1), gA w∞−1 1} − = {g(−1), gA w∞−1 (−1)} − + {gA w∞−1 (−1), gA w∞−1 1} − = {g(−1), gA w∞−1 (−1)} − + {gA w∞−1 (−1), gA w∞−1 1} − = {g(−1), gA w∞−1 (−1)} − − ξ − (gA w∞−1 ). Which gives the first formula by iteration. The second formula is proved by the same method (replace A by B −1 ). The third formula is obtained similarly. The class of a loop around the cusp Γg1 is given by {g∞, (gBA −1 g −1 ) w1 g∞} + = {g∞, g(BA −1 ) w1−1 B∞} + = {g∞, g(BA −1 ) w1−1 B0} + + {g(BA −1 ) w1−1 B0, g(BA −1 ) w1−1 B∞} + = {g∞, g(BA −1 ) w1−1 0} + + ξ + (g(BA −1 ) w1−1 B) = {g∞, g(BA −1 ) w1−1 ∞} + + {g(BA −1 ) w1−1 ∞, g(BA −1 ) w1−1 0} + +ξ + (g(AB −1 ) w1−1 B) = {g∞, g(BA −1 ) w1−1 ∞} + − ξ + (g(BA −1 ) w1−1 ) + ξ + (g(AB −1 ) w1−1 B), which leads to the third formula, by iterating again. The boundary maps H 1 (X Γ − ∂ − Γ , ∂ + Γ ; Z) → Z[∂ + Γ ] (resp. H 1 (X Γ − ∂ + Γ , ∂ − Γ ; Z) → Z[∂ − Γ ]) associates to ξ − (g) (resp. ξ + (g)) the divisor (Γg1) − (Γg(−1)) (resp. (Γg∞) − (Γg0)). They are dual to the maps λ − and λ + respectively. There is a notion of Eisenstein class in H 1 (X Γ − ∂ − Γ , ∂ + Γ ; R) and H 1 (X Γ − ∂ + Γ , ∂ − Γ ; R). Let D + (resp. D − ) be a divisor of degree 0 supported on ∂ + Γ (resp. ∂ − Γ ). The corresponding Eisenstein class belongs to H 1 (X Γ − ∂ − Γ , ∂ + Γ ; R) (resp. H 1 (X Γ − ∂ + Γ , ∂ − Γ ; R) ). It is the unique element c of boundary D + (resp. D − ) such that ℜ( c ω) = 0 for all ω differential form of the third kind whose poles are supported on ∂ − Γ (resp. ∂ + Γ ) and whose residues are real. INNER PRODUCT FORMULA Let M 2 (Γ) be the space of holomorphic modular forms of weight 2 for Γ. The following formula is akin to the formula [18, p. 21, Theorem 2]. Theorem 6. Let f + and f − be elements M 2 (Γ) such that the poles of f + (z)dz and f − (z)dz belong to ∂ + Γ and ∂ − Γ respectively. Thus the Petersson inner product < f + , f − > is well defined. We have the formula: < f + , f − >= 1 12i[Γ(2) : Γ] g∈Γ\Γ(2) g(−1) g(1) f + (z) dz g∞ g0 f − (z)dz. Proof. We prove first a more abstract statement which seems a basic statement in the theory of Riemann surfaces. However we did not know any reference for this. Proposition 7. Let X be a connected, compact, non-empty Riemann surface endowed with a quadrangulation. In such a situation, the set of vertices of the quadrangulation is partitioned in two subsets E + and E − such that no two vertices of E + (resp. E − ) are connected by an edge. Let Q be the set of faces. For every q ∈ Q, we fix an oriented path δ + q (resp. δ − q ) whose extremities are the vertices of q in E + (resp. in E − ). We impose furthermore that the intersection product is given by δ + q • δ − q = 1 (here, our convention is that δ + q and δ − q cross counterclockwise). Let ω + and ω − be meromorphic differential forms on X whose poles are at most simple and reside in E + and E − respectively. Then one has X ω + ∧ω − = q∈Q δ − q ω + δ + qω − , where q runs through the faces of the quadrangulation. Proof. Note that the formula makes sense. Set V = E − . Let Q be the set of faces of the quadrangulation. We can suppose that for every q ∈ Q, the paths δ + q and δ − q are supported on a set that divides q in two connected components. Denote by V q the subset of vertices in V that are adjacent to q. For v ∈ V , denote by τ (q, v) the connected component containing v of q deprived of the support of δ + q . Denote by Q v the set of faces of the quadrangulation which are adjacent to v. One has a disjoint union (up to a negligible subset) X = ∪ v∈V ∪ q∈Qv τ (q, v). We turn now to the computation. One has X ω + ∧ω − = v∈V q∈Qv τ (v,q) ω + ∧ω − = v∈V ∪q∈Q v τ (v,q) ω + ∧ω − . The boundary of ∪ q∈Qv τ (v, q) is q∈Qv α(q, v)δ + q , where α(q, v) = 1 (resp. −1) if δ + q goes left (resp. right) from v's point of view. For v ∈ V , and z ∈ ∪ q∈Qv τ (v, q) denote by F v (z) = z v ω + , which is well defined since ∪ q∈Qv τ (v, q) is simply connected. One has dF v ω − = ω + ∧ ω − . By Stokes' theorem, one has X ω + ∧ω − = v∈V q∈Qv α(q, v) δ + q F v (z)ω − . Note that q∈Qv α(q, v)F v = δ − q ω + , because of the condition δ + q • δ − q = 1. Thus we get the desired formula. The theorem follows. Indeed, the translates by Γ of the fundamental domain D 0 of X(2) given by the hyperbolic quadrangle with vertices 0, 1, −1 and ∞ provide a quadrangulation of X Γ . The faces are given by the ΓgD 0 , for g ∈ Γ\Γ(2). The δ + ΓgD0 and δ − ΓgD0 can be chosen as the image of the geodesic paths from g0 to g∞ and g1 to g(−1) respectively. The Petersson inner product [8, p. 182]: < f + , f − >= 1 [SL 2 (Z) : Γ] DΓ f + (z)f − (z)dxdy = 1 12i[Γ(2) : Γ] XΓ ω f+ ∧ ω f− , where D Γ is a fundamental domain for Γ in the upper half-plane. The last term can be expressed using proposition 7. Let D + and D − be divisors of degree 0 supported on ∂ + Γ and ∂ − Γ respectively. Proposition 8. With the notations of the preceeding theorem, suppose furthermore that f + (z) dz (resp. f − (z) dz) is the pullback of a canonical differential form of the third kind. One has < f + , f − >= 0. Proof. Call ω + the differential form whose pullback is f + (z) dz. The proposition is true whenever ω + is holomorphic on X Γ . Thus we can suppose that ω + is a canonical differential form of the third kind. It can be written as ∂h, where h is a harmonic function on X Γ . Hence XΓ ∂h ∧ω = 0, for all meromorphic differential forms ω whose poles are concentrated in ∂ − Γ . PERIODS OF EISENSTEIN SERIES We use the setup of section 3. Let m = 2π|Γ/Γ(2)| be the hyperbolic volume of the modular curve X Γ (it will only play a transitory role below). Set q = e 2iπz . Consider the function on the upper half-plane: log(η Γ,j (z)) = − mz 4i − πm[ ∞ r=1 φ jjr (1)q r ]. It is connected to the Eisenstein series E j (z, s) via a version of the Kronecker limit theorem established by Goldstein [11, Theorem 3-1; 3-3] (6.1) lim s→1 [ 1 2π E j (z, s) − 1 2πm(s − 1) ] = β j 2 − 1 πm log(2) − 1 πm log | √ yη Γ,j (z) 2 |, where z = x + iy and β j is a complex number independent of z (the scattering constants C j,j up to a scalar multiple independent of j). Goldstein has explored the properties of the function η Γ,j , which is a modular form of weight 1/2 for the group σ j Γσ −1 j , and whose periods he sees as analogues of Dedekind sums [11]. The multivalued function on the upper half-plane associated to the divisor D is then given by log(η Γ,D (z)) = j m j log(η Γ,j )(z). Note that E D (z) := 2 m log |η Γ,D (z) 2 |. Proposition 9. We have an equality of differential forms 2πiG D (z) dz = 4π m dη Γ,D . Proof. We compare the q-expansions. One has, taking into account that D is of degree 0, dη Γ,D (z) = 2πi j m j d log(η Γ,j )(z) = 2πi j −m j πm[ ∞ r=1 rφ jjr (1)q r dz]. Similarly, constant terms disappear in G D . So we just need to check that the q-expansions corresponding to each cusp agree. When j = Γ∞, it is obvious by examining the q-expansions. The other terms agree as well. Indeed, one has log(η Γ,j )(σ j z) = log(η σ −1 j Γσj ,Γ∞(z) ). On the other side we use the identity G j|σj = G σ −1 j Γσj ∞ , and we are left with the case of the cusp σ −1 j Γσ j ∞. Proposition 9 can be reformulated as G D (z) dz = 2 mi dη Γ,D (z) = 2πi j∈∂Γ m j d( ∞ r=1 φ jj,r (1)q r ). Recall that S D (x) = 1 2πi j∈∂Γ m j ∞ r=1 φ jj,r (1)e 2riπx . Equipped with these formulas, we relate the periods of Eisenstein series to the functions defined in the introduction. Proposition 10. One has, for g ∈ Γ(2), g(−1) g(1) G D + (z) dz = F D + (g) and g(∞) g(0) G D − (z) dz = F D − (g). Proof. We apply Proposition 9 and its reformulation Proposition 6.2 to D = D + . Thus we have g(−1) g(1) G D + (z) dz = 2πi g(−1) g(1) j∈∂Γ m j ∞ r=1 φ jj,r (1)q r dz = S D + (g(−1)) − S D + (g(1)). The other statement is proved similarly. The integrals occuring in Proposition 10 play a key role in our work. It is tempting to ask whether they have a p-adic counterpart, using for instance Coleman's integration. A priori, the notion of canonical differential of the third kind does not make sense without making choice in the p-adic world. However, Pierre Colmez suggests that it still does using the notion of Wintenberger splitting [7]. SCATTERING CONSTANTS We provide another expression for the functions I D + and I D − of the introduction, in terms of scattering constants. Recall that, for j, k ∈ ∂ Γ , one has C j,k = lim s→1 π( σ∈Γj \Γ/Γ k 1 |c| 2s ) − 2 π|Γ(2)/Γ|(s − 1) where the entries c and d are such that σ −1 j σσ k = ( ⋆ ⋆ c d ) and c is required to be > 0. Proposition 11. Let g ∈ Γ(2). One has I D + = π g∈Γ\Γ(2) j∈∂Γ m j (C j,Γg(−1) − C j,Γg(1) ){g0, g∞} + and I D − = π g∈Γ\Γ(2) j∈∂Γ m j (C j,Γg(∞) − C j,Γg(0) ){g(1), g(−1)} − . Proof. We prove the first equality. Let g ∈ Γ. We use the formula g(−1) g(1) G D + (z) dz = F D + (g). Thus we get F D + (g) = 2 mi [log(η Γ,D )] g(−1) g (1) . Recall that I D + is defined as the real part of 2πiE D + = 2πi g∈Γ\Γ(2) F D + (g){g0, g∞} + . We get ℜ(2πiF D + (g)) = ℜ( 4π m [log(η Γ,D )] g(−1) g(1) ) = π[E D ] g(−1) g (1) . We use the formulas 3.1 and 3.3 for the development 3.1 of E j (z, s) at the cusps g(1) and g(−1). Note that , when m j = 0, the cusp j is distinct from both cusps Γg(1) and Γg(−1), since Γ(2)j = Γ(2)0 or Γ(2)∞. The constant terms given by equation 3.3 are φ jΓg(1),0 (1)π 1/2 Γ(1/2)/Γ(1) = πφ jΓg(1),0 (1) and πφ jΓg(−1),0 (1) respectively. We get ℜ(2πif D + (g)) = j∈∂Γ π 2 m j (φ jΓg(−1),0 (1) − φ jΓg(1),0 (1)) = π j∈∂Γ m j (C j,Γg(−1) − C j,Γg(1) ) (because of the cancellation of the poles, the middle term makes sense). The formula follows. The second formula is proved similarly. It is interesting to compare to the work of Kühn [15]. For instance, when D + (resp. D − ) is torsion, and the Belyi map is defined over Q, Kühn shows that e 2π j∈∂ Γ mj (C j,Γg(−1) −C j,Γg(1) ) (resp. e 2π j∈∂ Γ mj (Cj,Γg∞−Cj,Γg0) ) is a rational number for every g ∈ Γ. More precisely, still in the case where the Belyi map is defined over Q, Kühn proves that the Néron-Tate pairing of two divisors D = j m j [j] and D ′ = j ′ m ′ j ′ , both of degree 0, satisfies (7.1) [D, D ′ ] NT ∈ log(Q × + ) + 2π j,j ′ ∈∂Γ m j m ′ j ′ C j,j ′ . A similar formula holds even if the Belyi map is not defined over Q; but one needs to consider the conjugates of the Belyi map. The second term in fomula 7.1 can be expressed as follows, when D = D + and D ′ = D − , 2π j,j ′ ∈∂Γ m j m ′ j ′ C j,j ′ = −2I D + • R D − . Indeed, we just have to note that the boundary of R D − is −D − (that will be proved in section 8). Note that Néron-Tate pairing considered by Kühn is obtained by summing contribution coming from all places of Q and is relative to the jacobian J Γ , whereas our intersection product, of complex analytic nature, takes place relatively to the 1-motives we are considering. One wonders whether an adjustment of the Néron-Tate pairing for the mixed situation would not provide a clearer connection between Kühn's formula and ours. PROOF OF THEOREM 1 Let D a divisor supported on ∂ − Γ (resp. ∂ + Γ ). Let c D be a cycle on X Γ −∂ + Γ (resp. X Γ −∂ − Γ ) of boundary D. Then the real number ℑ( 1 2iπ cD ω D + ) (resp. ℑ( 1 2iπ cD ω D − )) depends only on D. Let j, j 0 in ∂ − Γ (resp ∂ + Γ ). When D = [j] − [j 0 ], we abuse notations and write ℑ( 1 2iπ j j0 ω D − ) (resp. ℑ( 1 2iπ j j0 ω D + )) instead. Since j∈∂ − Γ λ + (j) = 0 (resp. j∈∂ + Γ λ − (j) = 0), the class j∈∂ − Γ ℑ( 1 2iπ j j0 ω D + )λ + (j) in the real vector space H 1 (X Γ − ∂ + Γ , ∂ − Γ ; R) (resp. j∈∂ + Γ ℑ( 1 2iπ j j0 ω D − )λ + (j) in H 1 (X Γ − ∂ − Γ , ∂ + Γ ; R)) is indepen- dent of the choice of j 0 . In other words, j∈∂ − Γ ℜ( j j0 ω D + )λ + (j) (resp. j∈∂ + Γ ℜ( j j0 ω D − )λ − (j) ) makes sense and is independent of j 0 . Recall that I D + (resp. I D − ) is defined as the real part of 2πiE D + (resp. 2πiE D − ). Proposition 12. One has I D + = j∈∂ − Γ ℜ( j j0 ω D + )λ + (j) and I D − = − j∈∂ + Γ ℜ( j j0 ω D − )λ − (j). In particular, those classes belong to the real vector spaces spanned by the images of λ + and λ − respectively. Proof. By proposition 10, one has, by using proposition 5 I D + = g∈Γ\Γ(2) ℜ( g(−1) g(1) ω D + )ξ + (g) = g∈Γ\Γ(2) ℜ( g(−1) j0 ω D + − g(1) j0 )ξ + (g) = g∈Γ\Γ(2) ℜ( g(1) j0 ω D + )(ξ + (gB) − ξ + (g)) = j∈∂ − Γ g∈Γ\Γ(2),Γg(1)=j ℜ( g(1) j0 ω D + )(ξ + (gB) − ξ + (g)) = j∈∂ − Γ ℜ( j j0 ω D + )λ + (j). Similarly, one has I D − = g∈Γ\Γ(2) ℜ( g∞ g0 ω D − )ξ − (g) = g∈Γ\Γ(2) ℜ( g∞ j0 ω D − − g0 j0 ω D − )ξ − (g) = j∈∂ + Γ g∈Γ\Γ(2),Γg∞=j)ℜ( g∞ j 0 ω D − )ξ − (g)− g∈Γ\Γ(2),Γg0=j ℜ g0 j0 ω D − )ξ − (g) = − j∈∂ − Γ ,Γ(2)j=Γ(2)∞ ℜ( j j0 ω D + )λ − (j) − j∈∂ − Γ ,Γ(2)j=Γ(2)0 ℜ( j j0 ω D + )λ − (j) = − j∈∂ + Γ ℜ( j j0 ω D − )λ − (j). Corollary 13. One has I D + • I D − = 0. Proof. Indeed, one has λ + (j) • λ − (j ′ ), for j ∈ ∂ − Γ = 0, j ′ ∈ ∂ + Γ (small loops do not intersect). The following consequence is certainly well-known. However it follows nicely from our proposition. Corollary 14. One has I D + = 0 (resp. I D − = 0) if and only if, for all j, j ′ ∈ ∂ − Γ (resp. j, j ′ ∈ ∂ − Γ ) one has ℜ( j j ′ ω D + ) = 0 (resp. ℜ( j j ′ ω D − ) = 0). Proof. Indeed, the kernel of the map λ + : Z[∂ − Γ ] → H 1 (X Γ − ∂ − Γ , ∂ + Γ ; Z) is spanned by j∈∂ − Γ [j]. It follows that all the coefficients in the expression of I D + need to be equal for I D + to vanish. The other statement is proved similarly. Proof. We prove theorem 1. By proposition 8 and theorem 5, one has E D + ω = 0. We write ω = ω D − + ω 0 , where ω 0 is holomorphic on X Γ and ω D − is the canonical differential form of the third kind associated to D − , where D − is the divisor of ω. Since ω 0 has no pole on X Γ , proposition 12 implies that I D + ω 0 = 0, and therefore R D + ω 0 = 0. Thus one gets E D + ω = E D + ω D − = 0. One has in C, with the notations of the introduction, and using the formula for the intersection products and proposition 8, E D + • E D − = 0. Since I D + • I D − = 0, if follows that we have the following identities in R R D + • R D − = 0, and R D + • I D − + I D + • R D − = 0. Thus we get ℜ( R D + ω) = ℜ( R D + ω D − ) = ℜ(R D + • R D − + 1 2πi R D + • I D − ) = R D + • R D − = 0. It remains to prove that the boundary of E D + (resp. E D − ) is −D + (resp. −D − ). Indeed, δ(E D + ) = g∈Γ\Γ(2) g(−1) g(1) G D + (z) dz((Γg∞) − (Γg0)) = j∈∂ + Γ g∈Γ\Γ(2),Γg∞=j g(−1) g(1) G D + (z) dz(j) − j∈∂ + Γ g∈Γ\Γ(2),Γg0=j g(−1) g(1) G D + (z) dz(j) = j∈∂ + Γ g∈Γ\Γ(2),Γg∞=j 1 2πi ξ − (g) ω D + (j) − j∈∂ + Γ g∈Γ\Γ(2),Γg0=j 1 2πi ξ − (g) ω D + (j). Since, by proposition 5, for every j ∈ ∂ + Γ , one has 1 2πi λ + (j) ω D + = − g∈Γ\Γ(2),Γg∞=j 1 2πi ξ − (g) ω D + and, similarly, 1 2πi λ + (j) ω D + = j∈∂ + Γ g∈Γ\Γ(2),Γg0=j 1 2πi ξ − (g) ω D + , one gets δ(E D + ) = − j∈∂ + Γ Res j (ω D + ) = −D + . The boundary of E D − is computed similarly. MANIN-DRINFELD PROPERTIES AND GENERALIZED JACOBIANS Let D be an effective divisor on X Γ supported on S. Recall that the generalized Jacobian of X Γ with respect to D admits a complex analytic description as follows. Denote by Ω(−D) the space of meromorphic differential forms on X Γ whose divisor is ≥ −D. Denote by E = Hom C (Ω(−D), C). Denote by L the image of the map H 1 (X Γ − S; Z) → E which associates to c the map ω → c ω. The complex Lie group formed by the complex points of the generalized Jacobian of X Γ with respect to D is isomorphic to E/L. Suppose D = s∈S [s] , then the group of complex point of the generalized Jacobian can be identified to a subgroup of H 1 (X Γ −S; C)/H 1 (X Γ −S; Z). This subgroup is (H 1 (X Γ −S; R)+iλ(H 0 (S; R)))/H 1 (X Γ − S; Z), where λ is the canonical map H 0 (S; R) → H 1 (X Γ − S; R). When D ′ is a divisor of degree 0 of X Γ supported outside D, its image in the generalized Jacobian is the class of the map ω → c ′ D ω, where c D ′ ∈ H 1 (X Γ − S, D ′ ; Z) has boundary D ′ . In the case of interest to us S = ∂ − Γ (resp. ∂ + ) and D ′ = D + (resp. D − ) is supported on ∂ + Γ (resp. ∂ − Γ ). We define the cuspidal subgroup of J − Γ (resp. J + Γ ) as the subgroup spanned by the divisors of degree 0 supported on ∂ + Γ (resp. ∂ − Γ ). To simplify notation, we see H 1 (X Γ − ∂ − Γ ; Z) (resp. H 1 (X Γ − ∂ + Γ ; Z)) as a subgroup of H 1 (X Γ − ∂ − Γ , ∂ + Γ ; Z) (resp. H 1 (X Γ − ∂ + Γ , ∂ − Γ ; Z)). Proposition 15. The converse of the map c → (ω → c ω) provides the following group isomorphisms : J − Γ (C) ≃ (H 1 (X Γ − ∂ − Γ ; R) + iλ + (H 0 (∂ − Γ ; R)))/H 1 (X Γ − ∂ − Γ ; Z) and J + Γ (C) ≃ (H 1 (X Γ − ∂ + Γ ; R) + iλ − (H 0 (∂ + Γ ; R)))/H 1 (X Γ − ∂ + Γ ; Z). The groups on the right-hand sides identify canonically to subgroups of (H 1 (X Γ − ∂ − Γ , ∂ + Γ ; R) + iλ + (H 0 (∂ − Γ ; R)))/H 1 (X Γ − ∂ − Γ , ∂ + Γ ; Z) and (H 1 (X Γ − ∂ + Γ , ∂ − Γ ; R) + iλ − (H 0 (∂ + Γ , R)))/H 1 (X Γ − ∂ + Γ , ∂ − Γ ; Z) respectively. In the latter groups, the class of D + (resp. D − ) in J − Γ (C) (resp. J + Γ (C)) is the class of −E D + (resp. −E D − ). Proof. The first statement follows from what we have just recalled on generalized Jacobians. We prove now the second statement regarding D + . Let c ∈ H 1 (X Γ − ∂ − Γ , ∂ + Γ ; Z) such that the boundary of c is D + . By Abel's theorem, the image of D + in J − Γ (C) is the map ω → c ω. Let ω be a meromorphic differential form of the third kind with divisor supported on ∂ − Γ . One has E D + ω = 0. Thus c ω = c−E D + ω. The boundary of c − E D + ∈ H 1 (X Γ − ∂ + Γ , ∂ − Γ ; R) + iλ − (H 0 (∂ + Γ ; R)) is 0. The desired result follows. The assertion concerning D − is proved similarly. Corollary 16. The cuspidal subgroup of the generalized Jacobian J − Γ (resp. J + Γ ) is isomorphic to the subgroup of H 1 (X Γ − ∂ − Γ , ∂ + Γ ; C)/H 1 (X Γ − ∂ − Γ , ∂ + Γ ; Z) (resp. H 1 (X Γ − ∂ + Γ , ∂ − Γ ; C)/H 1 (X Γ − ∂ + Γ , ∂ − Γ ; Z) ) generated by the classes E D + (resp. E D − ) when D + runs through the divisors of degree 0 supported on ∂ + Γ (resp. ∂ − Γ ). We now turn to theorem 2. The first part follows directly from the following proposition. Corollary 17. The divisor D + (resp. D − ) is torsion in J − Γ (resp. J + Γ ) if and only if one has R D + ∈ H 1 (X Γ − ∂ − Γ , ∂ + Γ ; Q) (resp. R D − ∈ H 1 (X Γ − ∂ + Γ , ∂ − Γ ; Q)) and I D + = 0 (resp. I D − = 0). Proof. It follows from proposition generalizedjacobian that D + is torsion if and only if E D + ∈ H 1 (X Γ − ∂ − Γ , ∂ + Γ ; Q) which translates immediately into R D + ∈ H 1 (X Γ − ∂ − Γ , ∂ + Γ ; Q) and I D + = 0. The alternate statement is proved similarly. The second part of theorem 2 can be deduced from the first part. Indeed, it follows from the fact that the image of λ + (resp. λ − ) is the kernel of the map H 1 (X Γ − ∂ − Γ , ∂ + Γ ; Z) → H 1 (X Γ , ∂ + Γ ; Z) (resp. H 1 (X Γ − ∂ + Γ , ∂ − Γ ; Z) → H 1 (X Γ , ∂ − Γ ; Z)). PASSAGE TO ANY MODULAR CURVE Our purpose in this section is to derive from Theorem 1 all Eisenstein classes in H 1 (X Γ , ∂ Γ ; R), where Γ is still a subgroup of Γ(2). In fact we will need to apply Theorem 1 for several conjugates of Γ in the full modular group. Recall that Manin [16] has defined map ξ : Z[Γ\SL 2 (Z)] → H 1 (X Γ , ∂ Γ ; Z), which to [Γg] associates the class {g0, g∞} of the image in X Γ of the path from g0 to g∞ in the upper half-plane. Such a map is surjective and its kernel is fully described by Manin. Denote by U = 0 1 −1 1 and S = 0 1 −1 0 . The element U permutes cyclicly 0, 1 and ∞. Denote by ∂ 0 , ∂ 1 and ∂ ∞ the cusps of X Γ above the cusps Γ(2)0, Γ(2)1 and Γ(2)∞ respectively. Let D be a divisor of degree 0 supported on ∂ Γ . Write D = −E 0 + E ∞ where E 0 and E ∞ are divisors of degree 0 supported on ∂ ∞ ∪ ∂ 1 and ∂ 0 ∪ ∂ 1 respectively. This decomposition is well-defined up to a divisor of degree 0 supported on ∂ 1 . When D is supported on ∂ + Γ (resp. ∂ 0 Γ ), we extend the function F D to a map still denoted F D : SL 2 (Z) → C by F (g) = 0 whenever g / ∈ Γ. Proposition 18. The element E ′ D = g∈Γ\SL2(Z) (F U −1 E∞ (U −1 g) − F UE0 (U g))ξ(g) belongs to H 1 (X Γ , ∂(X Γ ); R) and is the Eisenstein class of boundary D. Proof. Note first that U E 0 and U −1 E ∞ are divisors of degree 0 supported on ∂ + UΓU −1 ⊂ X UΓU −1 and ∂ + U −1 ΓU ⊂ X U −1 ΓU respectively. Consider the corresponding Eisenstein elements E UE0 and E U −1 E∞ and their respective images E ′ UE0 and E ′ U −1 E∞ in H 1 (X UΓU −1 , ∂ UΓU −1 , C) and H 1 (X U −1 ΓU , ∂ U −1 ΓU , C). Those images have in fact real coefficients, see section 1. Then U −1 E ′ UE0 and U E U −1 E ′ ∞ are Eisenstein elements in H 1 (X Γ , ∂ Γ ; R) of boundary E 0 and E ∞ respectively. For g ∈ Γ(2), the image of ξ + (g) via the composition of maps H 1 (X Γ − ∂ − Γ , ∂ + Γ , Z) → H 1 (X Γ , ∂ + Γ , Z) → H 1 (X Γ , ∂ Γ , Z) is ξ(g). Thus, one has E ′ U −1 E∞ = g∈U −1 ΓU\Γ(2) F U −1 E∞ (g)ξ(g) and E ′ UE0 = g∈UΓU −1 \Γ(2) F UE0 (g)ξ(g). Those Eisenstein elements have boundary U −1 E ∞ and U E 0 respectively. Thus U E U −1 E∞ − U −1 E UE0 is an Eisenstein elements of boundary E ∞ − E 0 = D. It is precisely E ′ D , as can be seen by changing variable in the sums. When Γ ′ is a subgroup of SL 2 (Z) containing Γ, consider the degeneracy map π : X Γ → X Γ ′ . It defines a map π * : H 1 (X Γ , ∂(X Γ ); R) → H 1 (X Γ ′ , ∂(X Γ ); R), which is compatible with the boundary maps and respects Eisenstein elements. Thus proposition 18 provides a formula for all Eisenstein elements for modular curves associated to any subgroup of finite index of SL 2 (Z). The formula is unsatisfactory insofar as the decomposition D = E ∞ − E 0 is not unique. Note a dual statement using ξ − instead of ξ + is missing. We make no use of ξ − in proposition 18. Corollary 19. The class of the divisor D is torsion in J Γ if and only if there exists F ′ , F ′′ : Z[Γ\SL 2 (Z)] → R which are right S-invariant and right U -invariant respectively such that for all g ∈ SL 2 (Z) (F U −1 E∞ (U −1 g) − F UE0 (U g)) − F ′ (g) + F ′′ (g) ∈ Q. Proof. Indeed, the class of the divisor D is torsion if and only if the Eisenstein element E D belongs to H 1 (X Γ , ∂(X Γ ); Q). The kernel of Manin's map ξ is spanned by the sum of elements invariant by U and elements invariant by S [16]. The corollary follows. FERMAT CURVES Let N be an integer > 0. The N -th Fermat curve F N is given by the projective equation: X N + Y N = Z N . Fermat curves and their points at infinity (cusps) are studied extensively by Rohrlich [21], [22], Vélu [26] and Posingies [20]. We use the notations of the latter author. In particular, we have the Belyi map β N : F N → P 1 given by (X : Y : Z) → (X N : Z N ). The map β N satisfy the following properties: • It is of degree N 2 . • It is ramified only above the points 0, 1, ∞. • The corresponding ramification points are given by a j = (0 : ζ j : 1), b j = (ζ j : 0 : 1), c j = (ǫζ j : 1 : 0), for j ∈ Z/N Z. Here, ζ = e 2πi/N is the primitive N th root of unity and ǫ = e πi/N . Each of the above points has ramification index N over P 1 . Since the λ function identifies P 1 − {0, 1, ∞} to Y (2). A covering of P 1 − {0, 1, ∞} can be understood as a covering of Y (2), i.e. a modular curve. Consider the group morphism P Γ(2) → (Z/N Z) 2 which sends A to (1, 0) and sends B to (0, 1). An explicit formula for this morphism in terms of the entries of the matrix (and not in terms of its decomposition as product of generators) is due to Murty and Ramakrishnan [19]. Denote by Φ N its kernel, which is a subgroup of index N 2 of P Γ(2). Denote by X Φ(N ) the corresponding modular curve. A system of representatives for the cosets Φ N \Γ(2) is given by A a B b with a, b ∈ {0, 1...(N − 1)}. The dessin d'enfant of the Fermat curve enjoys a special combinatorial property: its edges are in a one-to-one correspondence with the pairs of opposite color vertices. Proposition 20. The map Φ N \Γ(2) → Φ N \Γ(2)0 × Φ N \Γ(2)∞ which to Φ N g associates (Φ N g0, Φ N g∞) is bijective. Proof. All sets involved have cardinality N 2 . It is sufficient to show that the map is injective. For the first statement, let g, g ′ ∈ Γ(2) such that (Φ N g0, Φ N g∞) = (Φ N g ′ 0, Φ N g ′ ∞). There exists a, b ∈ Z such that Φ N g = Φ N g ′ A a and Φ N g = Φ N g ′ B b . Thus one has Φ N gA −a = Φ N gB −b . Hence a and b belong to N Z. Thus Φ N g = Φ N g ′ .Φ N A a 0 (resp. Φ N B b ∞) is such that the successor of {Φ N A a 0, Φ N B b ∞} is {Φ N A a+1 0, Φ N B b ∞} (resp. {Φ N A a 0, Φ N B b−1 ∞}). The N -th roots x := N √ λ, y := N √ 1 − λ define modular units for Φ N . We recover thus the familiar model of the Fermat curve. The cusps a j , b j and c j have been introduced above. The divisors of the following modular functions are given by: div(x − ζ j ) = N b j − j c j , div(y − ζ j ) = N a j − j c j , div(x − ǫξ j y) = N c j − j c j . It follows that the cuspidal subgroup of J ΦN is annihilated by N . Suppose from now that N is an odd integer. Rohrlich [22] has determined the structure of the cuspidal group of J ΦN (we use Vélu's alternative proof and description [26] ). Theorem 21. (Rohrlich,[22], [26]) The group of principal divisors of X Φ(N ) is spanned by by N [∂ ΦN ] 0 together with the following set (the Rohrlich relations) { N −1 i=0 [a i ] − [P ], N −1 i=0 [b i ] − [P ], N −1 i=0 [c i ] − [P ], N −1 i=0 i([a i ] − [b i ]), N −1 i=0 i([a i ] − [c i ]), N −1 i=0 i 2 ([a i ] + [b i ] + [c i ] − 3[P ])}, where P is any cusp of X ΦN . Consequently the cuspidal subgroup of J ΦN is isomorphic to (Z/N Z) 3N −7 . We turn now to the analogue of this result for the generalized Jacobian. We start with the Eisenstein cycles. Proof. Set f + j,k = y−ζ j x−ǫζ k y . It is a modular unit of divisor N (a j − c k ). One has ω (aj )−(c k ) = d(log f j,k ) = dy y − ζ j − dy x − ǫζ k y . Recall that 2πiG (aj )−(c k ) dz is the pullback on the upper half-plane of ω (aj )−(c k ) . Thus one has G (aj )−(c k ) (z) dz{g0, g∞} + . The second statement is proved similarly, using the function f − j,k = (x − ζ j )/(x − ζ k ), a modular unite of divisor N ((b j ) − (b k )). Thus which to (b j ) associates g∈ΦN \Γ(2),g1=bj [g] − g∈ΦN \Γ(2),g(−1)=bj [g]. The kernel of θ ′ is spanned by the element j (b j ). Thus θ − is injective. Remark 24. We did not use the sums coming from the Kloosterman zeta function to determine the Eisenstein classes for the Fermat curves. A priori, it seems difficult to understand a sum over the elements of Φ N . Indeed, how to characterize the elements of Φ N in terms of the entries of the corresponding matrices? Murty and Ramakrishnan [19] provide an answer in terms of Dedekind sums, but it is unclear how this would be useful. Thus the corresponding dessin d'enfant admits as 0-vertices the cusps Φ N A a 0, for a ∈ {0, 1, ..., N − 1}, as ∞-vertices the cusps Φ N B b ∞, for b ∈ {0, 1, ..., N − 1}, as edges the pairs {Φ N A a 0, Φ N B b ∞} for a, b ∈ {0, 1, ..., N −1}. The cyclic ordering (i.e. action of Z) on the edges attached to Theorem 22 . 22Let j, k ∈ Z/N Z, one has E (aj )−(c k ) F (aj )−(c k ) ( g0,aj − δ g∞,c k ]where δ is the Kronecker symbol. The result follows from the formulaE (aj )−(c k ) F (bj − δ g(−1),b k − δ g(1),bj + δ g(−1),b k ].An evident consequence of theorem is that the imaginary parts of E (aj )−(c k ) and E (bj )−(b k ) vanish. This is in fact a consequence of proposition 11 and of theorem 8.1. of , where Posingies determines the scattering constants for the Fermat curves.Corollary 23. The kernel of the mapZ[∂ + ΦN ] 0 → J − ΦN is generated by N Z[∂ + ΦN ] 0 ∪{ N −1 j=0 [a j ]− [c j ]} Therefore the cuspidal subgroup of J − ΦN is isomorphic to (Z/N Z) 2N −2 . The kernel of the map Z[∂ − ΦN ] 0 → J + ΦN is generated by 2N Z[∂ − ΦN ] 0 . Therefore the cuspidal subgroup of J + ΦN is isomorphic to (Z/2N Z) N −1 .Proof. We use corollary 16 to understand the cuspidal subgroup of J − ΦN as a subgroup ofH 1 (X ΦN − ∂ − ΦN , ∂ + ΦN ; Q)/H 1 (X ΦN − ∂ − ΦN , ∂ + ΦN ; Z). Since the denominator of E (aj )−(c k ) is N , by theorem 22, the cuspidal subgroup is contained in 1 N H 1 (X ΦN − ∂ − ΦN , ∂ + ΦN ; Z)/H 1 (X ΦN − ∂ − ΦN , ∂ + ΦN ; Z). Recall that the group H 1 (X ΦN − ∂ − ΦN , ∂ + ΦN ; Z)is freely generated by the {g0, g∞} + when g runs through Φ N \Γ(2) by theorem 4. Thus we are left to determine the kernel of the map :θ + : (Z/N Z)[∂ + ΦN ] 0 → (Z/N Z)[Φ N \Γ(2)] which to (a j ) − (c k ) associates g∈ΦN \Γ(2),g0=aj [g] − g∈ΦN \Γ(2),g∞=c k [g].We use now proposition 20.It follows that the kernel of θ + is spanned byN −1 j=0 [a j ] − [c j ].The other case can be treated in a similar manner. The group H 1 (X ΦN − ∂ + ΦN , ∂ − ΦN ; Z) is freely generated by the {g0, g∞} − when g runs through Φ N \Γ(2). Therefore we have to examine the mapθ − : (Z/2N Z)[∂ + ΦN ] 0 → (Z/2N Z)[Φ N \Γ(2)] which to (b j ) − (b k ) associates g∈ΦN \Γ(2),g1=bj [g] − g∈ΦN \Γ(2),g1=b k [g] − g∈ΦN \Γ(2),g(−1)=bj [g] + g∈ΦN \Γ(2),g(−1)=b k [g]. It is induced by the map θ ′ : (Z/2N Z)[∂ + ΦN ] → (Z/2N Z)[Φ N \Γ(2)] 1.1. Acknowledgements. It is a pleasure to acknowledge several e-mail communication, advice and remark of Professors Rohrlich and Professor Kumar Murty. The first author was partially supported by the SERB grant MTR/2017/000357 and CRG/2020/000223. 14, Chapter 2, p. 16] and Iwaniec [13, Chapter 9, p. 121]. For r ∈ Z, it is defined as A O L Atkin, H P F Swinnerton-Dyer, Modular forms on noncongruence subgroups, in Combinatorics (Proc. Sympos. Pure Math. Los Angeles, CalifXIXUniv. CaliforniaA. O. L. Atkin and H. P. F. Swinnerton-Dyer, Modular forms on noncongruence subgroups, in Combinatorics (Proc. Sympos. Pure Math., Vol. XIX, Univ. California, Los Angeles, Calif., 1968), 1-25 (1971). A note on the Eisenstein elements of prime square level. D Banerjee, Proc. Amer. Math. Soc. 14211D. Banerjee, A note on the Eisenstein elements of prime square level, Proc. Amer. Math. Soc. 142 (2014), no. 11, 3675-3686. Eisenstein elements inside the modular symbols for level product of two distinct odd primes. D Banerjee, S Krishnamoorthy, Pacific Journal of Mathematics. 2D. Banerjee and S. Krishnamoorthy, Eisenstein elements inside the modular symbols for level product of two distinct odd primes, Pacific Journal of Mathematics. (2016), no. 2, 257-285. The Eisenstein cycles as modular symbols. D Banerjee, L Merel, London Mathematical SocietyD. Banerjee and L. Merel, The Eisenstein cycles as modular symbols, J. London Mathematical Society (2018) . The Manin-Drinfeld theorem and the rationality of Rademacher symbols. C Burrin, Journal de Théorie des Nombres de. C. Burrin, The Manin-Drinfeld theorem and the rationality of Rademacher symbols, Journal de Théorie des Nombres de Bordeaux (2021) . K Chandrasekharan, 3-540-15295-4Elliptic functions. BerlinSpringer-Verlag281of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical SciencesK. Chandrasekharan, Elliptic functions, Vol. 281 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin (1985), ISBN 3-540-15295-4. P Colmez, Intégration sur les variétés p-adiques. 155P. Colmez, Intégration sur les variétés p-adiques, Astérisque (1998), no. 248, viii+155. A first course in modular forms. F Diamond, J Shurman, ISBN 0-387-23229-XGraduate Texts in Mathematics. 228Springer-VerlagF. Diamond and J. Shurman, A first course in modular forms, Vol. 228 of Graduate Texts in Mathematics, Springer-Verlag, New York (2005), ISBN 0-387-23229-X. Two theorems on modular curves. V G Drinfeld, Funkcional. Anal. i Prilov zen. 72V. G. Drinfeld, Two theorems on modular curves, Funkcional. Anal. i Prilov zen. 7 (1973), no. 2, 83-84. Sums of Kloosterman sums. D Goldfeld, P Sarnak, Invent. Math. 712D. Goldfeld and P. Sarnak, Sums of Kloosterman sums, Invent. Math. 71 (1983), no. 2, 243-250. Dedekind sums for a Fuchsian group. I, Nagoya Math. L J Goldstein, J. 50L. J. Goldstein, Dedekind sums for a Fuchsian group. I, Nagoya Math. J. 50 (1973) 21-47. Esquisse d'un programme. A Grothendieck, Geometric Galois actions. CambridgeCambridge Univ. Press242A. Grothendieck, Esquisse d'un programme, in Geometric Galois actions, Vol. 242 of London Math. Soc. Lecture Note Ser., 5-48, Cambridge Univ. Press, Cambridge (1997). H Iwaniec, ISBN 0-8218-3160-7Spectral methods of automorphic forms. Providence, RI; MadridAmerican Mathematical Society53Graduate Studies in Mathematics. second editionH. Iwaniec, Spectral methods of automorphic forms, Vol. 53 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI; Revista Matemática Iberoamericana, Madrid, second edition (2002), ISBN 0-8218-3160-7. Elementary theory of Eisenstein series. T Kubota, Kodansha Ltd. John Wiley & SonsT. Kubota, Elementary theory of Eisenstein series, Kodansha Ltd., Tokyo; Halsted Press [John Wiley & Sons], New York- London-Sydney (1973). Néron-Tate heights on algebraic curves and subgroups of the modular group. U Kühn, Manuscripta Math. 1164U. Kühn, Néron-Tate heights on algebraic curves and subgroups of the modular group, Manuscripta Math. 116 (2005), no. 4, 401- 419. Parabolic points and zeta functions of modular curves. J I Manin, Izv. Akad. Nauk SSSR Ser. Mat. 36J. I. Manin, Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972) 19-66. L'accouplement de Weil entre le sous-groupe de Shimura et le sous-groupe cuspidal de J 0 (p). L Merel, J. Reine Angew. Math. 477L. Merel, L'accouplement de Weil entre le sous-groupe de Shimura et le sous-groupe cuspidal de J 0 (p), J. Reine Angew. Math. 477 (1996) 71-115. Symboles de Manin et valeurs de fonctions L, in Algebra, arithmetic, and geometry: in honor of Yu. Boston, MABirkhäuser Boston, IncII---, Symboles de Manin et valeurs de fonctions L, in Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, Vol. 270 of Progr. Math., 283-309, Birkhäuser Boston, Inc., Boston, MA (2009). The Manin-Drinfeld theorem and Ramanujan sums. V K Murty, D Ramakrishnan, Proc. Indian Acad. Sci. Math. Sci. 971-3V. K. Murty and D. Ramakrishnan, The Manin-Drinfeld theorem and Ramanujan sums, Proc. Indian Acad. Sci. Math. Sci. 97 (1987), no. 1-3, 251-262 (1988). Belyi pairs and scattering constants. A E Posingies, BerlinHumboldt-University BerlinPh.D. thesisA. E. Posingies, Belyi pairs and scattering constants, Ph.D. thesis, Humboldt-University Berlin, Berlin (2010). Modular functions and the Fermat's curves, ProQuest LLC. D E Rohrlich, Thesis (Ph.D.)-Yale University. Ann Arbor, MID. E. Rohrlich, Modular functions and the Fermat's curves, ProQuest LLC, Ann Arbor, MI (1976). Thesis (Ph.D.)-Yale Uni- versity. Points at infinity on the Fermat curves. 39---, Points at infinity on the Fermat curves, Invent. Math. 39 (1977), no. 2, 95-127. Fourier coefficients of Eisenstein series on noncongruence subgroups. A J Scholl, Math. Proc. Cambridge Philos. Soc. 991A. J. Scholl, Fourier coefficients of Eisenstein series on noncongruence subgroups, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 1, 11-17. On the estimation of Fourier coefficients of modular forms. A Selberg, Proc. Sympos. SymposAmer. Math. Soc., Providence, R.IVIIIA. Selberg, On the estimation of Fourier coefficients of modular forms, in Proc. Sympos. Pure Math., Vol. VIII, 1-15, Amer. Math. Soc., Providence, R.I. (1965). J.-P Serre, 3-528-28968-6Lectures on the Mordell-Weil theorem. Brown and SerreBraunschweigthird editionJ.-P. Serre, Lectures on the Mordell-Weil theorem, Aspects of Mathematics, Friedr. Vieweg & Sohn, Braunschweig, third edi- tion (1997), ISBN 3-528-28968-6. Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt, With a foreword by Brown and Serre. J Vélu, FrenchSéminaire Delange-Pisot-Poitou, 20e année: 1978/1979. Théorie des nombres. Paris11Le groupe cuspidal des courbes de FermatJ. Vélu, Le groupe cuspidal des courbes de Fermat, in Séminaire Delange-Pisot-Poitou, 20e année: 1978/1979. Théorie des nom- bres, Fasc. 2 (French), Exp. No. 28, 11, Secrétariat Math., Paris (1980). M Waldschmidt, Nombres transcendants et groupes algébriques. Astérisque, Société mathématique de France69M. Waldschmidt, Nombres transcendants et groupes algébriques, Vol. 69-70 of Astérisque, Société mathématique de France (1979). . INDIAN INSTITUTE OF SCIENCE EDUCATION AND RESEARCH. INDIAN INSTITUTE OF SCIENCE EDUCATION AND RESEARCH, PUNE, INDIA . Université Paris, Cnrs Sorbonne Université, F Imj-Prg, PARIS, FRANCEUNIVERSITÉ PARIS CITÉ AND SORBONNE UNIVERSITÉ, CNRS, IMJ-PRG, F-75013 PARIS, FRANCE
{'fraction_non_alphanumeric': 0.1121803127874885, 'fraction_numerical': 0.029898804047838085, 'mean_word_length': 3.0093685452935968, 'pattern_counts': {'":': 0, '<': 4, '<?xml version=': 0, '>': 13, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 90, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': "Let Γ be a subgroup of finite index of SL 2 (Z). We give an analytic criterion for a cuspidal divisor to be torsion in the Jacobian J Γ of the corresponding modular curve X Γ . By Belyi's theorem, such a criterion would apply to any curve over a number field. Our main tool is the explicit description, in terms of modular symbols, of what we call Eisenstein cycles. The latter are representations of relative homology classes over which integration of any holomorphic differential forms vanishes. Our approach relies in an essential way on the specific case Γ ⊂ Γ(2), where we can consider convenient generalized Jacobians instead of J Γ .The Eisenstein classes are the real part of certain homology classes with complex coefficients. The imaginary part of those classes are related to the scattering constants attached to Eisenstein series. Finally, we illustrate our theory by considering Fermat curves. application in mind: the divisor D is torsion in the Jacobian J Γ of X Γ if and only if E D ∈ H 1 (X Γ , ∂ Γ ; Q).It is known since Manin and Drinfeld that D is torsion whenΓ is a congruence subgroup [9, 16]. Scholl [23], Murty-Ramakrishnan [19] and recently Burrin [5] have given criteria for D being torsion in J Γ , without appealing to E D . Our approach to this question is different from those authors and is a continuation of [4] and [17] (where we limited ourselves to congruence subgroups).We impose a more specific setting (as we will see, without losing generality), in connection with the theory of Belyi maps and dessins d'enfants, see section 2. We assume in this article that Γ is contained in Γ(2) (the principal congruence subgroup of level 2) and that −Id ∈ Γ. The corresponding modular", 'arxivid': '2204.06379', 'author': ['Debargha Banerjee ', 'Loïc Merel '], 'authoraffiliation': [], 'corpusid': 248157359, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 21899, 'n_tokens_neox': 20388, 'n_words': 11181, 'pdfsha': 'b3a8cf1baa0cd914a9211ee7d4b1b469c518b34e', 'pdfurls': ['https://arxiv.org/pdf/2204.06379v2.pdf'], 'title': ['THE EISENSTEIN CYCLES AND MANIN-DRINFELD PROPERTIES', 'THE EISENSTEIN CYCLES AND MANIN-DRINFELD PROPERTIES'], 'venue': []}
arxiv
A Fermionic bi-Doublet Effective Field Theory for Dark Matter A Fermionic bi-Doublet EFT for DM 31 August -23 September, 2016 31 Mar 2017 D Karamitros University of Ioannina Greece D Karamitros University of Ioannina Greece A Fermionic bi-Doublet Effective Field Theory for Dark Matter A Fermionic bi-Doublet EFT for DM 31 August -23 September, 2016 31 Mar 2017Corfu Summer Institute 2016 "School and Workshops on Elementary Particle Physics and Gravity" Corfu, Greece * Speaker. c Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). We study an effective field theory which includes the Standard Model extended by a Dark Sector consisting of two fermionic SU(2) L -doublets. A Z 2 parity guarantees that, after electroweak symmetry breaking, the lightest neutral particle is stable, acting as a WIMP. The dark sector interacts with the Higgs and gauge bosons through renormalizable and non-renormalizable d = 5 operators. We find that a WIMP with a mass around the electroweak scale, i.e. accessible at the LHC, is consistent with collider and astrophysical data only when non-trivial magnetic dipole interactions with the gauge bosons exist. Introduction There is a number of evidence suggesting that the mass content of the universe is dominated by Dark Matter (DM). From CMB measurements, the DM seems to account for about 25% of the total energy of the universe [1]. One of the most promising DM candidates is the so-called Weakly Interacting Massive Particle (WIMP). The WIMP is assumed to be a massive electrically neutral stable particle interacting weakly with the Standard Model (SM). Under these assumptions, the mass of the WIMP seems to lie naturally at the electroweak scale, due to the so-called WIMPmiracle [2]. This makes the WIMP accessible at LHC as well as direct detection experiments. Here we present the work of [3], where the SM is extended by a pair of fermionic SU(2) Ldoublets, which constitutes the Dark Sector (DS). Assuming that the SM and the DS do not mix, due to a Z 2 parity, the complete set of d = 5 non-renormalizable operators is introduced. We show that after electroweak (EW) symmetry breaking there is a stable neutral particle, which can act as a WIMP. The EFT at hand generalises the discussion on the bi-doublet DM scenario. Among the models incorporating a pair of fermion doublets in their low-energy spectrum, one finds the higgsino DM case [4], some simplified models such as the doublet-triplet DM [5], non-supersymmetric SO(10) GUTs [6] and its left-right symmetric subgroup [7]. Performing a phenomenological analysis, we show that a viable WIMP with a mass close to the EW scale, i.e. suitable for LHC searches, acquires sizeable magnetic dipole moments with the gauge bosons. The EFT content In the SM we add a pair of fermion SU(2) L -doublets, D 1,2 , with opposite hypercharges, Y (D 1 ) = −Y (D 2 ) = −1. We impose a Z 2 parity which separates the DS from the SM and ensures that the lightest neutral particle is stable and thus a WIMP candidate. Apart from the renormalizable interactions, we also introduce the complete set of d = 5 non-renormalizable operators, which are responsible for the dipole interactions between the DS fermions and the SM gauge bosons, the Yukawa interactions and the mass splitting between the components of the doublets. The Yukawa interactions Since there are no renormalizable interactions between the Higgs boson and D 1,2 , they appear at the d = 5 level. The d = 5 Yukawa along with the mass terms are 1 −L mass+Yukawa ⊃ y 1 2 Λ (H T εD 1 ) ( H T εD 1 ) + y 2 2Λ (H † D 2 ) (H † D 2 ) (2.1) + y 12 Λ (H T εD 1 ) (H † D 2 ) + ξ 12 Λ ( D T 1 εD 2 )(H † H) + M D D T 1 εD 2 + H.c., where Λ is the cut-off of the EFT and ε is the SU(2) L anti-symmetric tensor (in the fundamental representation). Also, for simplicity we assume that the parameters are real numbers, while the mass parameter M D can be redefined to be positive. Finally, as it can be seen from eq.(2.1), there are four independent operators with their respective Wilson coefficients y 1,2,12 and ξ 12 . The dipole interactions Apart from the operators of eq. 2.1, there are also interactions with the gauge bosons at d = 5 level. These are L dipoles ⊃ d γ Λ D T 1 σ µν εD 2 B µν + d W Λ D T 1 σ µν ε τ D 2 · W µν + i e γ Λ D T 1 σ µν εD 2 B µν + i e W Λ D T 1 σ µν ε τ D 2 · W µν + H.c. , (2.2) B µν ( W µν ) the U(1) Y (SU(2) L )H = −H 0 * H + H − H 0 ,(2. The charge conjugation symmetry In addition to the custodial symmetry, there is also a charge conjugation (c.c.) symmetry, which is a symmetry of the entire set of d = 5 operators. For y 1 = y 2 = y (and ∀y 12 ), the interactions 2.1 and 2.2 are invariant under exchanging D 1 → D 2 according to 2 C −1 D 2a C = ε ba D 1b . (2.7) This symmetry, basically, exchanges the columns of the matrix 2.5. Finally, we should point out that in our phenomenological analysis we are going to study two Benchmark scenarios in the the c.c. symmetric limit i.e. y 1 = y 2 . These two case are: (a) y 12 = −y, (b) y 12 = 0. (2.8) The first one is the SU(2) R symmetric limit, while the second violates the custodial symmetry, but is employed since it gives us a distinct mass spectrum. 3. The physical states Mass spectrum After EW symmetry breaking, the Higgs field is shifted by its vacuum expectation value (vev), v, resulting to mixing between the components of D 1,2 . After rotating to the mass basis, physical states are χ 0 1 = 1 √ 2 (D 0 1 + D 0 2 ) , χ 0 2 = − i √ 2 (D 0 1 − D 0 2 ) , χ + = i D + 2 , χ − = i D − 1 . (3.1) For these particles the mass terms become L ⊃ −m χ ± χ − χ + − 1 2 2 ∑ i=1 m χ 0 i χ 0 i χ 0 i + H.c. , (3.2) where the masses are m χ ± = M D + ξ 12 ω , m χ 0 1 = m χ ± + ω (y − y 12 ) , ω ≡ v 2 Λ , (3.3) m χ 0 2 = m χ ± − ω (y + y 12 ) . As stated in the previous section, we consider the two Benchmark scenarios shown in 2.8. These produce two distinct hierarchies for the masses m χ ± , m χ 0 1,2 . The hierarchies are shown in Fig. 1. We note that for y < 0 the lightest particle is always χ 0 1 , while for y > 0, the lightest particle becomes χ 0 2 without changing the phenomenology. Therefore, for the following analysis we are going to restrict y to be negative, which makes χ 0 1 our WIMP candidate. The masses of the particles for the two cases under study are: (a) y = −y 12 < 0. In this case, the heavy fermion is degenerate with the charged one, where the various masses are given by m χ 0 2 = m χ ± , m χ 0 1 = m χ ± − 2ω|y|. (b) y < 0, y 12 = 0. There is no degeneracy between the particles and the various masses are m χ 0 2 = m χ ± + 2ω|y|, m χ 0 1 = m χ ± − 2ω|y|. Interactions We calculate the Yukawa interactions for the rotated fields. The Lagrangian describing the 3-point DS-Higgs interactions 3 is L dim=5 χ χ h ⊃ −Y hχ − χ + h χ − χ + − 1 2 Y hχ 0 i χ 0 j h χ 0 i χ 0 j , (3.4) with Y hχ − χ + = √ 2 ξ 12 ω v , Y hχ 0 1 χ 0 1 = √ 2 ω v (ξ 12 + y − y 12 ), Y hχ 0 2 χ 0 2 = √ 2 ω v (ξ 12 − y − y 12 ), Y hχ 0 1 χ 0 2 = 0. (3.5) Interestingly, due to the c.c. symmetry, the interaction of the WIMP (χ 0 1 ) with the Higgs follows Y hχ 0 1 χ 0 1 ∼ ξ 12 + y − y 12 . Thus current Direct Detection experimental constraints (discussed later) can be avoided easily without the need for parameter fine tuning. Since D 1,2 are charged under SU(2) L ×U(1) Y , there are renormalizable interactions between them and the corresponding gauge bosons. The neutral ones are given by L dim=4 neutral ⊃ −(+e) (χ + ) †σ µ χ + A µ − (−e) (χ − ) †σ µ χ − A µ + g c W O L (χ + ) †σ µ χ + Z µ − g c W O R (χ − ) †σ µ χ − Z µ + g c W O L i j (χ 0 i ) †σ µ χ 0 j Z µ ,(3.6) where with s W = sinθ W and θ W being the weak mixing angle. Notably, the interaction χ 1 0 χ 1 0 Z vanishes due to the charge conjugation symmetry. O L = O R = − 1 2 (1 − 2s 2 W ) and O L = − i 2 0 1 −1 0 ,(3. Furthermore, the non-renormalizable operators of eq. 2.2 also contribute to the neutral interactions, where the 3-point ones are: L dim=5 neutral 3−point ⊃ − ω v 2 (d γ s W + d W c W ) O L i j χ 0 i σ µν χ 0 j F µν Z − ω v 2 (d γ s W − d W c W ) χ − σ µν χ + F µν Z + ω v 2 (d γ c W − d W s W ) O L i j χ 0 i σ µν χ 0 j F µν γ + (3.8) ω v 2 (d γ c W + d W s W ) χ − σ µν χ + F µν γ + H.c., where F γ and F Z are the field strength tensors of the photon and Z, respectively. Interestingly, the dipole operators which arise at d = 5 level, generate interactions between the neutral dark particles and the photon proportional to C γ ≡ d W s W − d γ c W . There are also other interactions between W ± and the Dark Sector. The renormalizable ones are: L dim=4 charged 3−point ⊃ g O L i (χ 0 i ) †σ µ χ + W − µ − g O R i (χ − ) †σ µ χ 0 i W − µ + g O L * i (χ + ) †σ µ χ 0 i W + µ − g O R * i (χ 0 i ) †σ µ χ − W + µ , (3.9) with O L i = 1 2 i −1 , O R i = 1 2 i −1 . (3.10) From eq. 2.2, the 3-point interactions between W ± and the dark fermions become L dim=5 charged 3−point ⊃ −2 ω v 2 d W O R * i χ − σ µν χ 0 i F µν W + + 2 ω v 2 d W O L i χ + σ µν χ 0 i F µν W − + H.c. (3.11) Finally, we should point out that, due to an alignment of couplings in eqs. 3.6 and 3.9 with the those in eqs. 3.8 and 3.11, a "natural" cancellation of the d=4 and d = 5 contributions in the annihilation cross-section χ 0 1 χ 0 1 → VV (with V being W and Z) can be achieved. This, as we shall see later, will be important in obtaining the observed relic abundance for WIMP masses at the electroweak scale. "Earth constraints" In this section we study constraints, from WIMP(χ 0 1 )-nucleon scattering experiments, searches for heavy charged fermions at LEP and from the LHC data for the Higgs boson decay to two photons. We collectively refer to these as "Earth constraints". Nucleon-WIMP direct detection bounds For the Spin-Independent cross-section the current limit set by the LUX Collaboration [9,10] is σ SI ∼ {1 − 3.5}10 −45 cm 2 , for m DM ∼ {100 − 500}GeV. This translates to |Y hχ 0 1 χ 0 1 | {0.04, 0.06} . (4.1) LEP bounds We next examine constrains for heavy charged fermions from LEP. From Fig. 1 we observe the next-to-lightest particle is the charged dark fermion, χ ± , with mass m χ ± = M D + ξ 12 ω that is assumed to be positive. The bound on m χ ± from such experiments is [11] m χ ± 100 GeV, which in terms of ξ 12 , ω and M D becomes ξ 12 100−M D ω . Bound from h → γγ measurements From the interactions 2.1 and 3.8, the ratio R h→γγ ≡ Γ(h→γγ) Γ(h→γγ) (SM) is given by [5] R h→γγ = 1 + 1 A SM √ 2Y hχ − χ + v m χ ± A 1/2 (τ) 2 ,(4.2) where A SM −6.5 for m h = 125 GeV, τ ≡ m 2 h /4m 2 χ ± and A 1/2 is the well known function given in Ref. [12]. The ratio R is currently under experimental scrutiny at LHC. The current value is R h→γγ = 1.15 +0.28 −0.25 [13]. From eq. 3.5 we expect that ξ 12 would be restricted to small values from the loop induced h → γγ bound. This would also result to a lower bound on M D at ∼ 100 GeV. Combined "Earth constraints" A numerical example of the combination of the "Earth constraints" is shown in Fig. 2. Generally, combining all the aforementioned bounds, results to a lower allowed value for the doublet mass parameter (M D ) at ∼ 90 GeV. Furthermore, ξ 12 is restricted to (relatively) small values. Also the allowed Yukawa couplings follow the relation ξ 12 ≈ −(2)y ± 0.16, for y 12 = 0 (y 12 = −y). This relation, then, also restricts y to small values. Cosmological and astrophysical constraints Having examined the constraints imposed from earth-based experiments, we now can the calculate the relic abundance of the lightest particle (χ 0 1 ) of this model and delineate the parameter space in which Ωh 2 the observed one. After that, various others astrophysical constraints are going to be considered. The role of the dipoles Before moving on, we should remind that the dipole operators (2.2) are essential in our study. This is because d W acts as a regulator that minimizes the total annihilation cross-section as the desired EW WIMP mass tends to amplify it 4 . Thus this minimization is vital for obtaining cosmologically acceptable relic abundance for m χ 0 1 at the electroweak scale. The behaviour of the annihilation cross sections of Fig. 3 shows that there are two minima for the channels χ 0 1 χ 0 1 → ZZ (ZZ−channel) W + W − (WW −channel) and γZ (γZ−channel) and one minimum for γγ (γγ−channel). The first minimum of the annihilations to ZZ and W + W − coincides with d γ c W d W s W (C γ 0) which gives small cross-sections for the γγ− and γZ− channels. On the other hand, the second minimum of the ZZ− and WW − channels is in a region where the annihilation to γγ and γZ blows up. Furthermore, for negative d W , there are no minima and every cross section becomes quite large. Therefore, if the minimization of the cross section is indeed needed, we expect d W to be bounded to non-vanishing positive values close to C γ 0. Relic abundance constraints Since the role of the dipole operators is pointed out, we can calculate the relic abundance and set further constraints on the parameter space. In Fig. 4 we show the d γ − d W plane of the parameter From Fig. 5 we observe that M D vastly affects the allowed values for d W for which we obtain the observed relic abundance. This is mainly due to the dependence of the minimum of the total annihilation cross section on M D . Additionally, as M D increases, d W moves to lower values, since for larger WIMP masses the minimization of the cross section is less needed (we can obtain the desired relic abundance at the renormalizable level). Also the allowed values of M D − y are shown in Fig. 6. Finally, the Yukawa couplings and the mass parameter M D showed above, fix the masses and their differences, as expected from eq. 3.3. These masses are shown in Fig. 7 for y 12 = −y (similar region holds also for y 12 = 0). We observe that the WIMP can be at the EW scale and its mass, as expected, is dominated by M D . Constraints from Gamma-ray monochromatic spectrum In this paragraph, we calculate the cross-sections for processes that could give gamma-ray lines from the Galactic Center (GC). As input, we use the parameter space that evade all the other, previously examined, restrictions and use the results from Fermi-LAT [14] to set additional bounds to the parameters of this model. ∼ 5 × 10 −28 for E γ ∼ 500 GeV. For the photon production channel χ 0 1 χ 0 1 → γZ, we need to rescale this bound by a factor of two. This process also results to different value for the photon energy given by E γ = m χ 0 1 (1 − m 2 Z /4m 2 χ 0 1 ). The values of the relevant cross sections in the allowed region of the parameter space are shown in Fig. 8. Applying the bounds discussed above, the parameter space remains virtually unaffected, apart from the d W − d γ plane shown in figure 4. The values of d W − d γ which respect the bounds discussed here are shown in Fig. 9 along with C γ (contours). We observe that the allowed values of C γ are concentrated around zero, which forces d W and d γ to have the same sign. Thus the latter is restricted to positive values, while accepted range of d W remains as in Fig. 4. LHC searches Having found the viable area in the parameter space, in which the observed DM relic abundance is obtained while avoiding all the other experimental and observational constraints, we move on to discuss possible observational effects at the LHC. In Fig. 10 we show the mono-jet channel cross section, which seems to be the most promising one, at least for this model. The current bound [15] on pp → χ 0 1 (χ 0 2 → χ 0 1 + νν) + jet at center of mass energyŝ = 8 TeV is σ / E T + jet 6.1 f b. It is apparent that this bound is easily evaded in the allowed parameter space. For LHC (RunII) at √ŝ = 13 TeV, the mono-jet channel can provide us with a relatively large number of events. From Fig. 10b, we observe that the production of a jet accompanied with missing E T , can reach cross sections up to ∼ 2.5 f b. Therefore the number of events that can, in principle, be observed is around 250 (750) for LHC expected luminosity reach of 100 (300) f b −1 . Conclusions We have extended the SM particle spectrum by a fermionic pair of doublets, D 1,2 , with opposite hypercharges. In addition, we have assumed a discrete Z 2 -symmetry that distinguishes this Dark Sector from the SM fields. At the renormalizable level there are a neutral, and a charged Dirac, fermion. After EW symmetry breaking, due the presence of d = 5 operators, the neutral Dirac fermion splits to two Majorana states. The lightest of them (χ 0 1 ), which is the WIMP candidate and a heavier neutral state, χ 0 2 . Moreover, the d = 5 operators include magnetic dipole operators which are, in principle, generated by a UV-complete model, at the TeV scale. The question we ask here is whether the WIMP, with a mass around the EW scale, is compatible to the various experimental and observational data. In order to reduce fine tuning and further simplify the parameter space, in section 2, we adopted two scenarios based on a charge conjugation symmetric limit. Then, in section 3 we showed the mass spectrum of the physical states. In section 4, we performed an analysis based a) on scattering WIMP-nucleus scattering experiments, b) LEP searches for heavy charged fermions, as well as c) on LHC searches for the decay h → γγ. We found the collective bounds showed in Fig. 2. In section 5 we calculated Ωh 2 for our WIMP candidate. In the presence of non-vanishing d = 5 dipole interactions, the WIMP annihilation cross sections acquire minima, which allow for the WIMP mass to be as low as 200 GeV. Following this we also considered constraints based of monochromatic gamma-ray spectrum observations from the Galactic Center. These set the final restrictions on the parameter space, which confined the photon dipole coupling (C γ ) to be ∼ ±0.1. Since our main goal was to be able to produce a WIMP at the electroweak scale, in order to be accessible at the LHC, in section 6 we estimated the cross section for producing χ 0 1 in association with a jet (monojet) with center of mass energy √ŝ = 8, 13 TeV. Although current bounds are weak, we found that the monojet channel, can produce few hundred of events at √ŝ = 13 TeV and with m χ 0 1 200 − 350 GeV (see Fig. 10). SM Higgs sector is invariant under SU(2) L × SU(2) R with the transformation rule H → U L H U R . It turns out that in the EFT at hand the Yukawa sector exhibits the same symmetry, when y 1 = y 2 = −y 12 . as D → U L DU R . Then the equation 2.1 obtains the form: −L Yuk ⊃ y Λ Tr(H † D) 2 + M D detD + H.c., (2.6) which is clearly invariant under SU(2) L × SU(2) R . Figure 1 : 1The mass hierarchies for the two Benchmark scenarios under study. 4-point interactions are not shown for simplicity. Figure 2 : 2The allowed values of y-ξ 12 , in order to satisfy the earth constraints, for Λ = 1 TeV and M D = 300 GeV. Figure 3 : 3The dependence of the cross section for the various annihilation channels on d W for vanishing relative velocity, M D = 400GeV, Λ = 1TeV, y = −y 12 = − ξ 12 2 = −0.8 and d γ = 0. Figure 4 : 4The plane d W − d γ of the parameter space that gives the observable relic abundance, for Λ = 1 TeV, y 12 = −y and allowing the other parameters to vary. Similar region holds for y 12 = 0.space that is compatible with the observed DM relic density, varying all the other parameters, while keeping Λ = 1 TeV and M D 500GeV. The parameter d W (d γ ) is bounded to be (mostly) positive in order to explain the DM relic abundance for a WIMP mass at electroweak scale, as expected from the minimization of the annihilation cross section discussed in the previous paragraph. Figure 5 : 5The acceptable values on the plane M D − d W for the two cases (a) y 12 = −y and (b) y 12 = 0, for Λ = 1 TeV. Again we allow for the other parameters to vary. Figure 6 : 6The acceptable values on the plane M D − y for the two cases (a) y 12 = −y and (b) y 12 = 0, for Λ = 1 TeV. The other free parameters are allowed to vary. Figure 7 : 7The allowed mass region for the case y 12 = −y. A similar region is also allowed for y 12 = 0. Figure 8 : 8Fro observations of gamma-ray lines from the GC show that the annihilation cross-section for χ 0 1 χ 0 1 → γγ cannot be above ∼ 10 −28 cm 3 s −1 for photon energy (E γ = m χ The allowed, region of the parameter space, in terms of the photon energy and the coupling d W s W − d γ c W . The contours show the values of the cross-section of the channels χ 0 1 χ 0 1 → γγ (a) and γZ (b) in cm 3 s −1 for y 12 = −y. Again y 12 = 0 results in an almost identical plot. Figure 9 : 9Allowed regions on the M D −C γ plane, consistent with "Earth" constraints, the observed relic abundance and the bounds from gamma-ray monochromatic spectrum. Almost identical regions are allowed for y 12 = 0. The contour lines show the values of the χ 0 1 χ 0 2 -photon coupling C γ . Figure 10 : 10The cross-section for the mono-jet channel with (a) √ŝ = 8TeV and (b) √ŝ = 13TeV. The areas are obtained from randomly selected values from the ones satisfying all the constraints discussed in the previous sections. gauge boson, d γ and d W are real numbers. Additionally, since we are not concerned about CP violation, e γ = e W = 0.2.3 Symmetries of the Dark sector The custodial symmetry It is known [8] that, the SM Higgs sector is invariant under a global SU(2) R (custodial) sym- metry. defining The spinor and gauge indices are suppressed for simplicity. In general the Higgs field is transformed as H → H † , but in the Kibble parametrization H remains unaffected. Generally the cross section scales as M −2 D at the renormalizable level. P A R Ade, Planck CollaborationarXiv:1502.01589Planck 2015 results. XIII. Cosmological parameters. 594Planck Collaboration, P. A. R. Ade et al., Planck 2015 results. XIII. Cosmological parameters, Astron. Astrophys. 594 (2016) A13, [arXiv:1502.01589]. Particle Dark Matter. D Hooper, arXiv:0901.4090Proceedings of Theoretical Advanced Study Institute in Elementary Particle Physics on The dawn of the LHC era (TASI. Theoretical Advanced Study Institute in Elementary Particle Physics on The dawn of the LHC era (TASIBoulder, USAD. Hooper, Particle Dark Matter, in Proceedings of Theoretical Advanced Study Institute in Elementary Particle Physics on The dawn of the LHC era (TASI 2008): Boulder, USA, June 2-27, 2008, pp. 709-764, 2010. arXiv:0901.4090. Effective Theory for Electroweak Doublet Dark Matter. A Dedes, D Karamitros, V C Spanos, arXiv:1607.05040Phys. Rev. 949A. Dedes, D. Karamitros, and V. C. Spanos, Effective Theory for Electroweak Doublet Dark Matter, Phys. Rev. D94 (2016), no. 9 095008, [arXiv:1607.05040]. Light neutralino as dark matter. L Roszkowski, Phys. Lett. 262L. Roszkowski, Light neutralino as dark matter, Phys. Lett. B262 (1991) 59-67. Doublet-Triplet Fermionic Dark Matter. A Dedes, D Karamitros, arXiv:1403.7744Phys. Rev. 8911A. Dedes and D. Karamitros, Doublet-Triplet Fermionic Dark Matter, Phys. Rev. D89 (2014), no. 11 115002, [arXiv:1403.7744]. Weakly-Interacting Massive Particles in Non-supersymmetric SO(10) Grand Unified Models. N Nagata, K A Olive, J Zheng, arXiv:1509.00809JHEP. 19310N. Nagata, K. A. Olive, and J. Zheng, Weakly-Interacting Massive Particles in Non-supersymmetric SO(10) Grand Unified Models, JHEP 10 (2015) 193, [arXiv:1509.00809]. C Garcia-Cely, J Heeck, arXiv:1512.03332Phenomenology of left-right symmetric dark matter. JCAP1603,021(2016)C. Garcia-Cely and J. Heeck, Phenomenology of left-right symmetric dark matter, arXiv:1512.03332. [JCAP1603,021(2016)]. Isospin Breaking in Technicolor Models. P Sikivie, L Susskind, M B Voloshin, V I Zakharov, Nucl. Phys. 173P. Sikivie, L. Susskind, M. B. Voloshin, and V. I. Zakharov, Isospin Breaking in Technicolor Models, Nucl. Phys. B173 (1980) 189-207. Improved Limits on Scattering of Weakly Interacting Massive Particles from Reanalysis of 2013 LUX Data. D S Akerib, LUX CollaborationarXiv:1512.03506Phys. Rev. Lett. 11616LUX Collaboration, D. S. Akerib et al., Improved Limits on Scattering of Weakly Interacting Massive Particles from Reanalysis of 2013 LUX Data, Phys. Rev. Lett. 116 (2016), no. 16 161301, [arXiv:1512.03506]. First results from the LUX dark matter experiment at the Sanford Underground Research Facility. D S Akerib, LUX CollaborationarXiv:1310.8214Phys. Rev. Lett. 11291303LUX Collaboration, D. S. Akerib et al., First results from the LUX dark matter experiment at the Sanford Underground Research Facility, Phys. Rev. Lett. 112 (2014) 091303, [arXiv:1310.8214]. Search for heavy neutral and charged leptons in e + e − annihilation at LEP. P Achard, L3 Collaborationhep-ex/0107015Phys. Lett. 517L3 Collaboration, P. Achard et al., Search for heavy neutral and charged leptons in e + e − annihilation at LEP, Phys. Lett. B517 (2001) 75-85, [hep-ex/0107015]. The Anatomy of electro-weak symmetry breaking. I: The Higgs boson in the standard model. A Djouadi, hep-ph/0503172Phys.Rept. 457A. Djouadi, The Anatomy of electro-weak symmetry breaking. I: The Higgs boson in the standard model, Phys.Rept. 457 (2008) 1-216, [hep-ph/0503172]. Combined Measurement of the Higgs Boson Mass in pp Collisions at √ s = 7 and 8 TeV with the ATLAS and CMS Experiments. G Aad, CMS CollaborationarXiv:1503.07589Phys. Rev. Lett. 114ATLASATLAS, CMS Collaboration, G. Aad et al., Combined Measurement of the Higgs Boson Mass in pp Collisions at √ s = 7 and 8 TeV with the ATLAS and CMS Experiments, Phys. Rev. Lett. 114 (2015) 191803, [arXiv:1503.07589]. Updated search for spectral lines from Galactic dark matter interactions with pass 8 data from the Fermi Large Area Telescope. M Ackermann, Fermi-LAT CollaborationarXiv:1506.00013Phys. Rev. 9112Fermi-LAT Collaboration, M. Ackermann et al., Updated search for spectral lines from Galactic dark matter interactions with pass 8 data from the Fermi Large Area Telescope, Phys. Rev. D91 (2015), no. 12 122002, [arXiv:1506.00013]. Search for dark matter, extra dimensions, and unparticles in monojet events in proton-proton collisions at √ s = 8 TeV. V Khachatryan, CMS CollaborationarXiv:1408.3583Eur. Phys. J. 755CMS Collaboration, V. Khachatryan et al., Search for dark matter, extra dimensions, and unparticles in monojet events in proton-proton collisions at √ s = 8 TeV, Eur. Phys. J. C75 (2015), no. 5 235, [arXiv:1408.3583].
{'fraction_non_alphanumeric': 0.061041021379466646, 'fraction_numerical': 0.04517639948628843, 'mean_word_length': 3.544284243048404, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 50, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'We study an effective field theory which includes the Standard Model extended by a Dark Sector consisting of two fermionic SU(2) L -doublets. A Z 2 parity guarantees that, after electroweak symmetry breaking, the lightest neutral particle is stable, acting as a WIMP. The dark sector interacts with the Higgs and gauge bosons through renormalizable and non-renormalizable d = 5 operators. We find that a WIMP with a mass around the electroweak scale, i.e. accessible at the LHC, is consistent with collider and astrophysical data only when non-trivial magnetic dipole interactions with the gauge bosons exist.', 'arxivid': '1703.10916', 'author': ['D Karamitros \nUniversity of Ioannina\nGreece\n', 'D Karamitros \nUniversity of Ioannina\nGreece\n'], 'authoraffiliation': ['University of Ioannina\nGreece', 'University of Ioannina\nGreece'], 'corpusid': 56307485, 'doi': '10.22323/1.292.0057', 'github_urls': [], 'n_tokens_mistral': 9277, 'n_tokens_neox': 7551, 'n_words': 4940, 'pdfsha': '77aa2a70a26313aedcabb09f2cb68ad86d81f4f2', 'pdfurls': ['https://arxiv.org/pdf/1703.10916v1.pdf'], 'title': ['A Fermionic bi-Doublet Effective Field Theory for Dark Matter A Fermionic bi-Doublet EFT for DM', 'A Fermionic bi-Doublet Effective Field Theory for Dark Matter A Fermionic bi-Doublet EFT for DM'], 'venue': []}
arxiv
VARIATIONAL AUTOENCODERS WITH DECREMENTAL INFORMATION BOTTLENECK FOR DISENTANGLEMENT Jiantao Wu University of Surrey Shentong Mo Carnegie Mellon University Muhammad Awais University of Surrey Sara Atito University of Surrey Xingshen Zhang University of Jinan Lin Wang University of Jinan Xiang Yang Mingyi Tech VARIATIONAL AUTOENCODERS WITH DECREMENTAL INFORMATION BOTTLENECK FOR DISENTANGLEMENT Preprint One major challenge of disentanglement learning with variational autoencoders is the trade-off between disentanglement and reconstruction fidelity. Previous incremental methods with only on latent space cannot optimize these two targets simultaneously, so they expand the Information Bottleneck while training to optimize from disentanglement to reconstruction. However, a large bottleneck will lose the constraint of disentanglement, causing the information diffusion problem. To tackle this issue, we present a novel decremental variational autoencoder with disentanglement-invariant transformations to optimize multiple objectives in different layers , termed DeVAE, for balancing disentanglement and reconstruction fidelity by decreasing the information bottleneck of diverse latent spaces gradually. Benefiting from the multiple latent spaces, DeVAE allows simultaneous optimization of multiple objectives to optimize reconstruction while keeping the constraint of disentanglement, avoiding information diffusion. DeVAE is also compatible with large models with high-dimension latent space. Experimental results on dSprites and Shapes3D that DeVAE achieves a good balance between disentanglement and reconstruction. DeVAE shows high tolerant of hyperparameters and on high-dimensional latent spaces. * INTRODUCTION Unsupervised learning (Wu et al., 2018;Chen et al., 2020a;b;Grill et al., 2020;He et al., 2020;Chen et al., 2020c;Mo et al., 2021;Chen & He, 2021;Mo et al., 2022;2023a) for sensing the properties of objects is crucial to reduce the gap between humans and machines intelligence. Inline with human intelligence disentanglement learning (Bengio et al., 2013) is considered to be a promising direction to obtain explanatory representations from observations to understand and reason objects without any supervision. In the recent years, various approaches Chen et al., 2018;Kim & Mnih, 2018; have been proposed to successfully extract basic properties of objects, such as position, color, orientation, and scale (Burgess & Kim, 2018;. The commonly-used methods are based on variational autoencoder (VAE) (Kingma & Welling, 2014). In particular, β-VAE introduced an extra parameter β on the Kullback-Leibler (KL) divergence to promote disentanglement. However, there is a trade-off between disentanglement and reconstruction fidelity on β-VAE, which is a problem to be solved in the following works. One common direction for dealing with the trade-off is to penalize the Total Correlation (TC) between latent variables, avoiding reducing the mutual information, such as FactorVAE (Kim & Mnih, 2018), β-TCVAE (Chen et al., 2018), and DIPVAE (Kumar et al., 2018). As pointed out in , TC-based VAEs have a strong prior assumption that the factors are statistically independent. Beyond that, when it comes to high-dimension latent space, the estimation of TC becomes inaccurate due to the curse of dimensionality, as our experiments observed in Section 3.2. The realistic problems usually have numerous factors, therefore it would need a large model with high latent space to extract representations. For example, the popular deep model ResNet50 (He et al., 2016) has 2048 dimensional feature space. However, the current TC estimations are not scaled to high dimensional problems, causing the low performance of BC-based methods in practice. In this work, instead of calculating TC, we leverage the information bottleneck (IB) (Tishby et al., 1999; to promote disentanglement. In the meanwhile, previous information bottleneck (IB)-based methods Shao et al., 2022;Wu et al., 2022) have tried to solve the obstacle of trade-off between disentanglement and reconstruction fidelity. A narrow IB enforces the model to find efficient codes for representing the data, which encourages disentanglement. Therefore, they first set a high pressure with a narrow IB and then expand the IB gradually to promote disentanglement to reconstruction fidelity , termed incremental methods. For example, DynamicVAE (Shao et al., 2022) initiated β with a large value at the beginning of training for disentanglement and stably increase the KL divergence for reconstruction by a non-linear PI controller (Åström & Hägglund, 2006). However, they lost the constraint of disentanglement when expanding the IB, which causes the information diffusion problem (Wu et al., 2022). In this work, to avoid information diffusion, we aim to optimize reconstruction while keeping the constraint of disentanglement. Different from IB-Incremental based approaches listed above, our key motivation is to optimize disentanglement and reconstruction simultaneously. revious methods only have one latent space and are unable to optimize disentanglement and reconstruction at the same time, which causes them to have to change the target from disentanglement to reconstruction during training. Instead, our work proposes a novel multi-layer framework with its own latent spaces and objectives in each layer, allowing optimizing multiple targets at a time. In this way, the first layer is a vanilla VAE to rebuild high-quality images, and the subsequent layers will distill some important variables by narrow IBs to promote disentanglement. To inherit disentanglement from the subsequent layers, we introduce disentanglement-invariant transformations to connect the layers one by one. These extra layers can be seen as regularizations for disentanglement to constrain the representation. To achieve this, we propose a simple yet effective VAE framework composed of multiple continuous latent sub-spaces with a novel IB-Decremental strategy and disentanglement-invariant transform operators, which we call DeVAE. Specifically, we decrease the information bottleneck of each latent space layer by layer, where we constrain the first space for informativeness to recover the input image, and other disentangled spaces for learning factors of the image by narrow IBs. Furthermore, we introduce the disentanglement-invariant transform operator to ensure simultaneous optimization of disentanglement across continuous latent sub-spaces, which avoids the information diffusion. Our decremental model avoids ID by keeping the constraints of disentanglement while optimizing reconstruction. We also conducted comprehensive comparisons with popular methods quantitatively and qualitatively. Experimental results demonstrate that DeVAE is robust in hyperparameters and the size of latent spaces. Our contributions can be summarized as follows: • We introduce several latent spaces sharing disentanglement by disentanglement-invariant transformations. • We propose a novel diagram for disentanglement learning by decreasing IB, termed decremental VAE (DeVAE). Our decremental model can handle large-scale problems and show robustness on several datasets. METHODOLOGY PRELIMINARIES Problem Setup & Notations. Disentanglement learning aims to learn the factors of variation which raises the change of observations. Given a set of samples x ∈ X , they can be uniquely described by a set of ground-truth factors c ∈ C. Generally, the generation process g(·) is invisible x = g(c). We say a representation for factor c i is disentangled if it is invariant for the samples with c j . We use variational inference to learn the disentangled representation for a given problem. p(z|x) denotes the probability of z = f (x), p(x|z) denotes the probability of x = g(z). The representation function is a conditional Bayesian network of the form q φ (z|x) to estimate p(z|x). The generative model is another network of the form p θ (x|z)p(z). φ, θ are trainable parameters. The solid lines denote the information flow of the encoding process. The dash lines denote the decoding process which randomly selects one layer's representation and concatenates the corresponding embedding vector. v i denotes a layer embedding. τ i denotes a disentanglement-invariant transformation. µ i , σ i denote the parameters of latent variables z i . N denotes a random noise. Each layer has a pressure β i to control the capacity of IB. Revisit VAE & β-VAE. The VAE framework (Kingma & Welling, 2014) computes the representation function by introducing q φ (z|x) and optimizing the variational lower bound (ELBO). β-VAE introduces the hyperparameter β to control the IB: L(θ, φ) = E q φ (z|x) [log p θ (x|z)] − βD KL (q φ (x|z)||p(z)). (1) Consider using β-VAE to learn a representation of the data; the representation will be disentangled but lose information when β is large . We can set a large β to learn a disentangled representation and a small β to learn an informative representation. However, previous disentanglement methods Chen et al., 2018; are limited in low-dimension latent space and poorly deal with the trade-off between disentanglement and reconstruction. Current state-of-the-art approach (Shao et al., 2022) with an annealing manner from high pressure to low pressure will loosen the constraint of disentanglement when reducing the pressure. To address this issue, we propose a novel decremental variational autoencoder with hierarchical latent spaces, namely DeVAE, to optimize disentanglement and reconstruction fidelity simultaneously, which can handle high-dimensional latent spaces, as shown in Figure 1. Our DeVAE applies a hierarchical structure with a decremental information bottleneck and disentanglement-invariant transformation to produce latent variables layer by layer. The decoder part randomly selects one layer's latents concatenating an embedding vector to generate images. HIERARCHICAL LATENT SPACES WITH DECREMENTAL INFORMATION BOTTLENECK In order to retain the disentanglement constraint while optimizing the reconstruction fidelity, we introduce a Hierarchical Latent Space (HiS) with K layers and assign a pressure β i for the ith layer Z i to promote disentanglement. The first layer aims to rebuild the dataset and uses the ELBO as objective. The subsequent layers will promote disentanglement by reducing the IB. Therefore, the objective of the ith layer is L i (θ, φ) = E q φ (zi|x) [log p θ (x|z i , v i )] − β i D KL (q φ (z i |x)||p(z i )),(2) where v i ∈ R 1×D denotes the learnable layer embedding for the ith layer, p θ (x|z i , v i ) is the decoder network shared with all layers, q φ (z i |x) is the encoder network dependent on previous ones, β ∈ R K is a set of coefficients to penalize the IB, particularly β 0 = 1. The parameters of each layer are parameterized as a bottom-up process: q(z i |x) = N (µ i (x), σ i (x) 2 ), q(z 0 |x) = N (µ 0 (x), σ 0 (x) 2 ), µ i (x), σ i (x) = τ i (µ i−1 (x), σ i−1 (x)), i > 0 (3) where the first layer q φ (z 0 |x) is a conditional Bayesian network, τ i denotes a transformation to modify the poster distribution of the previous layer to fit the layer objective L i (θ, φ). According to information theory, information can only decrease while processing, therefore we gradually decrease the IB in the sequential layers, i.e., β i+1 > β i . In this way, the last layer with a narrow IB can promote disentanglement only, and the reconstruction fidelity will become better and better from the bottom to the top. DISENTANGLEMENT-INVARIANT TRANSFORMATION Though we create multiple latent spaces, these objectives only encourage the local representations to be disentangled or informative. We need a mechanism to connect these objectives for balancing disentanglement and reconstruction in one layer. In order to make sure disentanglement across all latent layers, we propose a disentanglement-invariant transformation (DiT) denoted as τ . Theorem 1 w · z is disentangled if z is disentangled, w is a diagonal matrix. Proof in Appendix A.2. According to Theorem 1, we can scale the latent space to keep disentanglement. Scaling the posterior z i violates the generation process which wants the marginal distribution q(z) = q φ (z|x)p(x) to be close to a standard normal distribution. Besides, most downstream tasks use the mean representation instead of sampled representation. Therefore, we only need the mean representations disentanglement-invariant. The disentanglement-invariant transformation scales the parameters of the ith layer: τ w 1 ,w 2 (µ, σ) = h(w 1 )µ, h(w 2 )σ,(4) where w 1 , w 2 are learnable diagonal matrices belonging to the ith layer, h(w) = e w is the power function to make sure the scaling values greater than 0. Therefore, we get the parameters of ith latent variables µ i = h( i−1 j=0 w 1 j )µ 0 , σ i = h( i−1 j=0 w 2 j )σ 0 , i > 0.(5) and the ith KL divergence D KLi = 1 2 (1 + 2 i−1 j=0 w 2 j + 2 log(σ 0 ) − h(2 i−1 j=0 w 2 j )σ 2 0 − h(2 i−1 j=0 w 1 j )µ 2 0 ).(6) OPTIMIZATION ALGORITHM In this section, we combine the above components and introduce the optimization algorithm for the multiple objectives. We use a random process to optimize one layer's objective from K latent spaces: L(θ, φ) = E p(zi) [L i (θ, φ)],(7) where p(z i ) denotes the probability of optimizing the ith latent space z i , which is defined as: p(z i ) =    1 K , for s = 1 (1 − s) s i 1 − s i+1 , for s > 1(8) where s denotes the power annealing of scale hyper-parameter for each p(z i ). In experiments, we empirically find that s = 1 achieves better performance, as observed in Section 3.3. Note that we do idx = np.random.randint(K) not aggregate the objectives into a loss, instead, DeVAE only rebuilds the images and optimizes the objective of one layer in one mini-batch. In our model, q φ (z|x) and decoder p θ (x|z) are modelled by two neural networks, a K-size sequence 'betas' denotes the penalties on the KL divergences of corresponding layers, w 1 , w 2 stores the learnable parameters for transforming latent spaces. First, we randomly sample a mini-batch and choose a target layer to optimize. Then use the algorithm introduced in Section 2.3 to obtain the representation of the target layer and reconstruct the corresponding images. Instead of using K separated decoders to rebuild images, we apply a shared decoder with layer embeddings to reduce the parameter size. The only extra computational cost comes from w 1 , w 2 and layer embeddings. Therefore, the parameter size and overhead are similar to the vanilla VAE. The PyTorch-like algorithm is shown in Algorithm 1. EXPERIMENTS EXPERIMENTAL SETUP Datasets. We evaluate our method on two widely-used datasets (dSprites, Shapes3D). dSprites has 737,280 binary 64 × 64 x 1 images generated from five factors: shape (5), orientation (40), scale (6), position X (32), and position Y (32). Shapes3D (Burgess & Kim, 2018) has 480,000 RGB 64 × 64 × 3 images of 3D shapes generated from six factors: floor color (10), wall color (10), object color (10), object size (8), object shape (4), and azimuth (15). Evaluation Metrics. We apply the following metrics to evaluate the performance of disentanglement and reconstruction. MIG (Chen et al., 2018): the mutual information gap between two variables with the highest and the second-highest mutual information. FactorVAE metric (Kim & Mnih, 2018): the error rate of the classifier, which predicts the latent variable with the lowest variance. DCI Dis.: abbreviation for DCI Disentanglement (Eastwood & Williams, 2018), a matrix of relative importance by regression. Recon.: abbreviation for Reconstruction Error, a measure of the distance between images; we use Mean Squared Error for RGB images and Binary Cross Entropy for binary images. Implementation. We use a convolutional neural network as the encoder and a deconvolutional neural network as the decoder. Detailed architecture can be found in Appendix A.1. The activation function is ReLU. The optimizer is Adam (Kingma & Ba, 2015) with a learning rate of 1e-4, β 1 = 0.9, β 2 = 0.999. The batch size is 256, which accelerates the training process. All experiments train 300, 000 iterations by default. For the hyperparameters, we set β = 12 for β-TCVAE, β = 6 for β-VAE, and K i = 0.001, K p = 0.01 for DynamicVAE. According to the information freezing points (IFP) (Wu et al., 2022), beta=40 can filter factors orientation and shape, beta=10 can only filter factor orientation, so we set {β i } = [1, 10, 40], s = 1 for DeVAE. Disentanglement & Reconstruction. We conducted experiments on dSprites and Shapes3D to compare the above methods. Each trail was repeated 10 times with different random seeds. We draw the distributions of three disentanglement scores and reconstruction errors in Figure 2. The visualization of sampling from the best models is shown in Appendix A.5. Experimental results reveal that DeVAE achieves an average improvement of 8% comparing to β-TCVAE and 47% to β-VAE on dSprites for disentanglement. DeVAE has a lower average reconstruction error on two datasets by 2% for β-TCVAE and by 30% for β-VAE. Though the improvement is not significant, β-TCVAE and β-VAE are not robust to one hyperparameter setting. Though DynamicVAE achieves the best overall results, it still suffers from ID problems and is incapable of dealing with highdimensional space, see Figure 5 and Appendix A.7. Qualitative Visualization. We also conducted a qualitative analysis to assess disentanglement. We show the selected latent traversals whose KL divergence is larger than 0.5 in Figure 3. We can see that DeVAE disentangles position X and position Y perfectly. Shape, scale, and orientation are hard to be disentangled. We show the latent traversals of the best models with the highest MIG in Appendix A.4. Preventing Information Diffusion. Information diffusion is a phenomenon of disentangling that one factor's information diffuses into other latent variables while training, causing the disentanglement scores to fluctuate during training (Wu et al., 2022). We hypothesize that losing the constraint of disentanglement is the reason for ID. To prove it, we monitored the changes in mutual information during training. From Figure 4, we see that DynamicVAE has a significant trend of losing information on iteration 3e5. It means that the learned structure of representation was destroyed when expanding the IB. In contrast, DeVAE shows a relatively steady trend of increasing information for consistent regularizing. DeVAE overcomes the drawback of traditional information bottleneck-based methods by keeping the constraint of disentanglement. Scaling to High-dimensional Latent Space. Most disentanglement methods evaluate their performance on simple scenes with only one object and few factors. It is a challenge to extend these methods to complex scenes. However, whether these methods adapt to a large latent space to fit more factors is questionable. In particular, the dimension of latent space affects the estimation accuracy of MI for the TC-based methods. To study the effect of high-dimensional latent space on estimating TC, we first generate samples from a D-dimensional multi-variable normal distribution x which is divided into two groups x 1 and x 2 with D/2 dimensions. The variables in a group are independent Cov(x m i , x m j ) = 0, i = j; the variables between groups are correlative Cov(x m i , x n i ) = ρ, m = n; each variable is a standard normal distribution. So, the TC of x is TC(x) = − D 4 log(1 − ρ 2 ).(9) We trained a discriminator for 2000 iterations to estimate the TC introduced in FactorVAE ( Kim & Mnih, 2018). We compared the estimated TC and the real TC over dimensions and ρ. Each trail was repeated 10 times, and we report the average results as shown in Table 1. One can see that increasing ρ or dimension diminishes the accuracy of estimation, and the estimators always have low errors when the dimension is 10. However, the estimation becomes extremely inaccurate when Recon. Figure 5: Distributions of MIG scores and reconstruction errors for low-dimensional space (blue) and high-dimensional space (green). The points in the bottom right have a better balance of disentanglement and reconstruction. the dimension raises to 1000, which means such estimation will fail to penalize the TC for large models. MIG We further conduct experiments on dSprites to validate the above conclusion. The experimental settings are the same except for increasing the dimension of latent space to 1024. From Figure 5, we can see that β-TCVAE and DynamicVAE have significant performance decay. Higher dimensional space increases the complexity of calculating the TC and leads to significant estimation errors and also increases the chance of the ID problem for DynamicVAE see in Appendix A.7. β-VAE and DeVAE show robustness in high-dimensional latent spaces, which is necessary to train a large model on large data. EXPERIMENTAL ANALYSIS In this section, we performed ablation studies on the benefit of the proposed Hierarchical Latent Spaces (HiS) and Disentanglement-invariant Transformation (DiT). We also conducted extensive experiments to explore the effect of β and s on disentanglement and reconstruction performance. Hierarchical Latent Spaces & Disentanglement-invariant Transformation. To demonstrate the effectiveness of the introduced Hierarchy Latent Spaces (HiS) and Disentanglement-invariant Transformation (DiT), we performed ablation experiments on the following situations: 1) The model has one single latent space; 2) The model applies a parallel structure instead of the hierarchy that latent spaces are independent; 3) We replace DiT with Linear Transformation (τ i (z i ) = wz i ), w is an arbitrary matrix; 4) The proposed model DeVAE. We report the MIG and Recon. for each layer in Table 2. MS and HiS can optimize multiple objectives for these layers separately. DiT enforces all layers to share disentanglement. In this way, the first layer aims to optimize the ELBO, and the subsequent layers optimize disentanglement jointly by DiT which works like a constraint of disentanglement. Therefore, the key of DeVAE is to connect the multiple latent spaces by DiT to form a hierarchical structure with a decremental IB. Effect of β. More latent layers mean more chance to explore disentanglement solutions but need more time to converge. Though Wu. etc. (Wu et al., 2022) proposes the Annealing Test to determine the value of β, it requires labels to learn the information freezing point (IFP). Choosing a suitable β for each layer is difficult without knowing the information of factors. Fortunately, DeVAE is insensitive to the choice of β, which means we can create redundant latent layers to cover all suitable βs. In Table 3, we compared tree cases: redundant betas ([1,10,20,40,80]), just betas ([1,10,40]), insufficient betas ([1,10]). Redundant betas slightly diminish the performance of disentanglement and reconstruction. It is an advantage to increase the parameter size and training iterations without rebooting Effect of Scale s. Increasing s will add the weights of higher beta, encouraging disentanglement more than reconstruction fidelity. It is a crucial hyperparameter to balance the objectives of latent layers. Note that our model equals the vanilla VAE when s = 0. In Table 4, we compared the effects of choosing s and reported the mean±std scores of MIG (Chen et al., 2018) and reconstruction. For most cases, s = 1 is a good choice. LIMITATION Since our model creates several diverse latent spaces, it is a challenge to optimize multiple objectives. Though there are numerous combinations for setting pressures and weighting these objectives, we only search a limited range of hyper-parameters. Even so, DeVAE shows compatible performance on the benchmarks. Though we validated that our model is adequate for high-dimensional space, we did not test it on real problems. It is challenging to train a disentanglement model on large-scale problems, such as ImageNet. RELATED WORK Disentanglement Learning. Disentanglement learning aims at learning generative factors existing in the dataset, that is, disentangled representation learning. Though the definition of disentanglement is still an open topic (Kumar et al., 2018;Do & Tran, 2020;Abdi et al., 2019;Duan et al., 2020;Mo et al., 2023b), it is widely accepted that the redundancy between latent variables diminishes disentanglement. Penalizing the Total Correlation (TC) (Watanabe, 1960) is an important direction in disentanglement learning, and many SOTA methods are based on it (Chen et al., 2018;Kim & Mnih, 2018;Esmaeili et al., 2019;Kumar et al., 2018;Wei et al., 2021). PM algorithm promotes factorial codes but only works for binary codes (Schmidhuber, 1992); Though ICA (Comon, 1994) and PCA (Wold et al., 1987) ensure independence theoretically, they extract linear representations. Until recently, deep learning has made it workable. FactorVAE (Kim & Mnih, 2018) applies an adversarial training method to approximate and penalize the TC term. β-TCVAE (Chen et al., 2018) decomposed the KL term into three parts: mutual information (MI), total correlation (TC), and dimensional-wise KL (DWKL). However, these methods rely on the estimation of TC, which is extremely hard for high-dimensional spaces. Information Bottleneck. Information bottleneck theory (Tishby et al., 1999;Shannon, 1948) plays a vital role in interpreting neural networks. Some methods encourage disentanglement by increasing the information bottleneck while training (Jeong & Song, 2019;Shao et al., 2022;Dupont, 2018;Wu et al., 2022). These methods vary in the way of expanding the IB. Cascade-VAE (Jeong & Song, 2019) sequentially relieves one latent variable at one stage to increase the IB. DynamicVAE (Shao et al., 2022) designs a non-linear PI controller for manipulating β to control IB steadily increasing. DEFT (Wu et al., 2022) applies a multi-stage training strategy with separated encoders to extract one factor at one stage according to its information freezing point (IFP). However, the above incremental models, increasing the IB while training, suffer from the information diffusion (ID) problem (Wu et al., 2022) that the disentangled representation may diffuse the learned information into other variables. This work presents a novel framework with a decremental information bottleneck to solve the ID problem. Hierarchical Latent Variables. Normalizing Flow (Rezende & Mohamed, 2015;Kingma et al., 2016) also uses hierarchical latent layers to generate an arbitrary distribution. Unlike Normalizing Flow, each layer aims to encourage disentanglement or reconstruction. Besides, Normalizing Flow gradually increases the complexity of the output distribution after entering a new layer. In contrast, our model reduces the complexity layer by layer. LadderVAE (Sønderby et al., 2016) also applies hierarchical latent variables in the encoder, but it using a symmetry structure decodes these latent variables in hierarchy. Therefore, the information among the i-th layer will increase comparing to the last layer. CONCLUSION We propose a novel framework with a decremental information bottleneck for disentanglement. Hierarchical latent spaces with disentanglement-invariant transformation are the key to overcoming the problem of losing disentanglement constraint while expanding the information bottleneck. The decremental method is compatible with high-dimensional problems and reduces the information diffusion problem. In practice, the typical disentanglement methods have to refine suitable hyperparameters for a dataset without labels by trail and error. In contrast, DeVAE is tolerant to redundant layers such that we can set a large parameter set to fit kinds of datasets. Broader Impact Unlike previous works that spread the conflict of the trade-off over time, our work demonstrates a novel direction to solve the trade-off by spreading the conflict in spaces. Our work provides insights on balancing disentanglement and reconstruction. The details of architectures are listed in Table 5. A.2 DISENTANGLEMENT-INVARIANT REPRESENTATIONS In this section, we prove the proposed disentanglement-invariant transformation. Consider that we have a new representation by multiplying a diagonal matrix: z = wz, w. We can calculate the Covariance between any two latent variables: Cov(w i z i , w j z j ) = E[(w i z i − E[w i z i ])(w j z j − E[w j z j ])] = w i w j (E[z j ] − E[z i ]E[z j ]) = w i w j Cov(z i , z j ),(10) where the subscript denotes the index of latent variables. Note that we use a different notion in this section to simplify the formula. Then we can get the correlation coefficient by ρ(w i z i , w j z j ) = Cov(w i z i , w j z j ) Var[w i z i ] Var[w j z j ] = ρ(z i , z j ).(11) Therefore, the correlation matrix will not change by multiplying a diagonal matrix w, w = 0. We could create a disentanglement-invariant representation by multiplying a diagonal matrix. A.3 ESTIMATION OF I(z j ; c i ) Given an inference network q(z|x), we use the Markov chain Monte Carlo (MCMC) method to get samples from q(z) by the formula q(z) = q(z|x)p(x). We use 10, 000 points to estimate q(z). Then, we discretize these latent variables by a histogram with 20 bins. After discretizing one latent variable, we call a discrete mutual information estimation algorithm to calculate I(w j z j ; c i ) by a 2D histogram. A.4 LATENT TRAVERSALS We compare DeVAE to others with latent traversals on Shapes3D and dSprites. Each column denotes the generated images by traversing one latent variable from -2 to 2. We also interpret the extracted factor at the bottom. From Figure 6 and Figure 7, we can see that DeVAE has a lower entanglement level. Note that only DeVAE disentangles object size isolated on Shapes3D. Figure 6: Latent traversal on Shapes3D. "back." denotes background color, "floor" denotes floor color, "obj." denotes object, and "wall" denotes wall color. A.5 RANDOM SAMPLING We show the visualization of random sampling from the best models with the highest MIG trained on dSprites and Shapes3D in Figure 8 9 10. A.6 DECREMENTAL DIAGRAM Figure 11 shows how the mutual information between factors and latent variables decreases over layers on dSprites. One can see the mutual information decreases along the layers, and information of shape, scale, and orientation is totally disappeared in layer2. The last layer is more likely to disentangle them, and that property will be preserved and passed to the first layer for a constraint of disentanglement. A.7 HIGH-DIMENSIONAL LATENT SAPCE DeVAE has significant advantages for handling high-dimensional latent spaces. Though Dynamic-VAE outperforms low-dimensional latent spaces, there is a gap in the high-dimensional latent spaces. We trained DynamicVAE and DeVAE with 1024-dimensional latent spaces on dSprites to investigate what causes the difference. (Cao et al., 2022) found that existing disentanglement metrics fail to make meaningful measurements for high-dimensional representation models, therefore we apply the proposed metric by them in this experiment. Active variables denote the latent variables containing information, and those containing no information will collapse into one point, so the active variables will have large variances. From Figure 12, DynamicVAE has more active variables and performs worse than DeVAE. DynamicVAE expanding the IB smoothly could have a good performance on low-dimensional spaces, but the increment of dimensions raises the chance of leaking information to others. As a result, the number of active variables increases quickly when expanding the IB, see iteration 20000 to 100000. A.8 COMPARISON WITH FACTORVAE AND CASCADEVAE FactorVAE (Kim & Mnih, 2018) and CascadeVAE (Jeong & Song, 2019) are two relevant methods for disentanglement. We compare six disentanglement methods on dSprites in Table 6. Each tail repeats 5 times to get the mean and std scores. Similar to β-TCVAE, FactorVAE can not consistently outperform one hyperparameter on two datasets. Though CascadeVAE has good reconstruction fidelity, it cannot disentangle all factors properly. Figure 1 : 1Illustration of our Decremental Variational Autoencoder (DeVAE). -like implementation of DeVAE loss. Figure 2 : 2Box plots of quantitative benchmarks MIG, FactorVAE, Disentanglement, and reconstruction error on dSprites and Shapes3D.3.2 COMPARISON TO PRIOR WORKTo demonstrate the effectiveness of the proposed DeVAE, we compare it to previous all types of baselines: 1) β-VAE: the popular method for disentanglement and also the baseline model for DeVAE when there is only one latent layer; 2) β-TCVAE(Chen et al., 2018): the TC-based method with a good balance of simplicity and effectiveness; 3) Dynamic-VAE(Shao et al., 2022): the latest method with incremental information bottleneck. Figure 3 :Figure 4 : 34Latent traversal of DeVAE with the best MIG score on dSprites. Each block shows the generated images of traversing the latent variable from −2 to 2. Each group shows the traversal images sampling from 3 different random noise. The title above each group denotes the index of latent variable. Comparison results of information diffusion. Each colored curve denotes the learned information that belongs to one factor over training iterations. Figure 7 : 7Latent traversal on dSprites. Figure 9 :Figure 10 :Figure 11 :Figure 12 : 9101112Generated images of β-TCVAE and DeVAE from sampling random noise on Shapes3D. Generated images of DynamicVAE and β-VAE from sampling random noise on The mutual information between factors and latent variables over layers on dSprites. We select five informative variables, and the title denotes the index of the latent variable. The rows denote the factors, shape, scale, orientation, posX, and posY respectively. The column denotes the layer where the latent variable (title) is. Comparison of active variables and MED scores on 1024-dimensional models on dSprites. DynamicVAE is unstable while expanding the information bottleneck. Table 1 : 1The estimated MI of FactorVAE and the real MI on high dimensional spaces. The cases having large error are bold. ρ denotes the correlation of two random variables.Estimated TC TC Error Dim ρ 10 0.3 0.23 0.24 0.03 0.6 1.08 1.12 0.03 0.9 4.14 4.15 0.00 100 0.3 2.17 2.36 0.08 0.6 10.27 11.16 0.08 0.9 23.39 41.52 0.44 1000 0.3 8.62 23.58 0.63 0.6 17.40 111.57 0.84 0.9 22.47 415.18 0.95 Table 2 : 2Ablation Study on Multiple Spaces (MS), Hierarchical Structure (HiS) and Disentanglement-invariant Transformation (DiT). The reconstruction fidelity becomes better and better from the bottom layer to the top layer, but the disentanglement gets worse. DiT can prevent disentanglement to degenerate.MS HiS DiT MIG Recon. Layer0 Layer1 Layer2 Layer0 Layer1 Layer2 0.19 - - 47.49 - - 0.24 0.32 0.35 22.21 40.79 62.40 0.24 0.29 0.30 38.82 45.48 63.78 0.35 0.35 0.35 43.29 75.11 175.99 Table 3 : 3Exploration study of betas on disentangle- ment (MIG) and reconstruction (Recon.). l1, l2, l3 de- note [1,10,20,40,80], [1,10,40], [1,10] respectively. s is fixed to 1. Dataset MIG Recon. No. betas dSprites l1 0.30±0.03 79.65±16.06 l2 0.35±0.02 51.99±26.99 l3 0.16±0.11 38.19±02.35 Shapes3D l1 0.54±0.06 65.01±25.37 l2 0.57±0.01 43.24±11.41 l3 0.55±0.04 39.31±06.96 Table 4 : 4Comparison of scale. We report the mean±std of MIG and reconstruction for 5 trails on dSprites and Shapes3D. The sequence of betas is fixed to [1,10,40]. MIG Recon. scale 0.3 0.21±0.14 16.01±01.14 0.5 0.29±0.09 22.40±01.78 1.0 0.35±0.02 51.99±26.99 0.3 0.55±0.02 24.43±01.37 0.5 0.57±0.02 28.48±03.31 1.0 0.57±0.01 43.24±11.41 Table 5 : 5The architecture details. "FC." denotes fully connected layer, "conv." denotes convolutional layer, "deconv" denotes transposed convolution layer. c is the dimension of color channel.Encoder Decoder 4 × 4 conv. 32 stride 2 FC.256 4 × 4 conv. 32 stride 2 FC. 4 × 4 × 64 4 × 4 conv. 64 stride 2 4 × 4 deconv. 64 stride 2 4 × 4 conv. 64 stride 2 4 × 4 deconv. 32 stride 2 FC. 256 4 × 4 deconv. 32 stride 2 FC. 10 4 × 4 deconv. c stride 2 A APPENDIX A.1 ARCHITECTURE Table 6 : 6Comparison of reconstruction error (Recon.), MIG score, and ELBO for six disentanglement methods. 55±0.84 0.34±0.04 -46.05±2.24 CascadeVAE 12.04±1.23 0.20±0.07 -32.14 ± 1.29 Dynamic 19.81±1.19 0.35±0.01 -37.83±1.17 beta-TCVAE 73.04±3.41 0.29±0.10 -82.29±3.48±2.28 0.38±0.28 -38.08±1.87 CascadeVAE 14.84±1.98 0.46±0.11 -32.54±2.10 Dynamic 29.70±4.15 0.54±0.04 -47.68±4.28 beta-TCVAE 44.53±5.69 0.49±0.11 -60.01±6.29 beta-VAE 34.95±2.34 0.42±0.18 -49.09±2.72 DeVAE 46.80±13.97 0.52±0.10 -74.73±31.66dataset model Recon. MIG ELBO dSprites FactorVAE 21.71 beta-VAE 48.75±2.84 0.17±0.05 -61.17±3.13 DeVAE 36.02±20.02 0.32±0.11 -51±22.26 Shapes3D FactorVAE 18. def loss_fn(x,encoder,decoder,W,embeddings,betas,K): A preliminary study of disentanglement with insights on the inadequacy of metrics. H Amir, Purang Abdi, Sidney Abolmaesumi, Fels, arXiv:1911.11791arXiv preprintAmir H Abdi, Purang Abolmaesumi, and Sidney Fels. A preliminary study of disentanglement with insights on the inadequacy of metrics. arXiv preprint arXiv:1911.11791, 2019. 10 Pid control. IEEE Control Systems Magazine, 1066. Karl Jåström, Tore Hägglund, Karl JÅström and Tore Hägglund. Pid control. IEEE Control Systems Magazine, 1066, 2006. 2 Representation learning: A review and new perspectives. Yoshua Bengio, Aaron Courville, Pascal Vincent, IEEE Transactions on Pattern Analysis and Machine Intelligence. 358Yoshua Bengio, Aaron Courville, and Pascal Vincent. Representation learning: A review and new perspectives. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(8):1798-1828, 2013. 1 . Chris Burgess, Hyunjik Kim, 153d shapes datasetChris Burgess and Hyunjik Kim. 3d shapes dataset. https://github.com/deepmind/3dshapes-dataset/, 2018. 1, 5 Understanding disentangling in β-vae. Christopher P Burgess, Irina Higgins, Arka Pal, Loic Matthey, Nick Watters, Guillaume Desjardins, Alexander Lerchner, International Conference on Machine Learning (ICML). 10Christopher P. Burgess, Irina Higgins, Arka Pal, Loic Matthey, Nick Watters, Guillaume Desjardins, and Alexander Lerchner. Understanding disentangling in β-vae. In International Conference on Machine Learning (ICML), 2018. 1, 2, 3, 10 An empirical study on disentanglement of negative-free contrastive learning. Jinkun Cao, Ruiqian Nai, Qing Yang, Jialei Huang, Yang Gao, Advances in Neural Information Processing Systems. Alice H. Oh, Alekh Agarwal, Danielle Belgrave, and Kyunghyun ChoJinkun Cao, Ruiqian Nai, Qing Yang, Jialei Huang, and Yang Gao. An empirical study on disentan- glement of negative-free contrastive learning. In Alice H. Oh, Alekh Agarwal, Danielle Belgrave, and Kyunghyun Cho (eds.), Advances in Neural Information Processing Systems, 2022. URL https://openreview.net/forum?id=fJguu0okUY1. 16 Isolating sources of disentanglement in variational autoencoders. Xuechen Tian Qi Chen, Roger B Li, David Grosse, Duvenaud, Neural Information Processing Systems (NeurIPS). 910Tian Qi Chen, Xuechen Li, Roger B. Grosse, and David Duvenaud. Isolating sources of disentan- glement in variational autoencoders. In Neural Information Processing Systems (NeurIPS), 2018. 1, 3, 5, 6, 9, 10 A simple framework for contrastive learning of visual representations. Ting Chen, Simon Kornblith, Mohammad Norouzi, Geoffrey Hinton, 2020a. 1Proceedings of International Conference on Machine Learning (ICML). International Conference on Machine Learning (ICML)Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton. A simple framework for contrastive learning of visual representations. In Proceedings of International Conference on Machine Learning (ICML), 2020a. 1 Big selfsupervised models are strong semi-supervised learners. Ting Chen, Simon Kornblith, Kevin Swersky, Mohammad Norouzi, Geoffrey Hinton, 2020b. 1Proceedings of Advances in Neural Information Processing Systems (NeurIPS). Advances in Neural Information Processing Systems (NeurIPS)Ting Chen, Simon Kornblith, Kevin Swersky, Mohammad Norouzi, and Geoffrey Hinton. Big self- supervised models are strong semi-supervised learners. In Proceedings of Advances in Neural Information Processing Systems (NeurIPS), 2020b. 1 Infogan: Interpretable representation learning by information maximizing generative adversarial nets. Xi Chen, Yan Duan, Rein Houthooft, John Schulman, Ilya Sutskever, Pieter Abbeel, Neural Information Processing Systems (NeurIPS). Xi Chen, Yan Duan, Rein Houthooft, John Schulman, Ilya Sutskever, and Pieter Abbeel. Infogan: Interpretable representation learning by information maximizing generative adversarial nets. In Neural Information Processing Systems (NeurIPS), pp. 2172-2180, 2016. 1 Exploring simple siamese representation learning. Xinlei Chen, Kaiming He, IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). 2021Xinlei Chen and Kaiming He. Exploring simple siamese representation learning. In IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2021. 1 Xinlei Chen, Haoqi Fan, arXiv:2003.04297Ross Girshick, and Kaiming He. Improved baselines with momentum contrastive learning. arXiv preprintXinlei Chen, Haoqi Fan, Ross Girshick, and Kaiming He. Improved baselines with momentum contrastive learning. arXiv preprint arXiv:2003.04297, 2020c. 1 Independent component analysis, a new concept? Signal processing. Pierre Comon, 36Pierre Comon. Independent component analysis, a new concept? Signal processing, 36(3):287-314, 1994. 10 On the transfer of disentangled representations in realistic settings. Andrea Dittadi, Frederik Träuble, Francesco Locatello, Manuel Wüthrich, Vaibhav Agrawal, Ole Winther, Stefan Bauer, Bernhard Schölkopf, arXiv:2010.14407arXiv preprintAndrea Dittadi, Frederik Träuble, Francesco Locatello, Manuel Wüthrich, Vaibhav Agrawal, Ole Winther, Stefan Bauer, and Bernhard Schölkopf. On the transfer of disentangled representations in realistic settings. arXiv preprint arXiv:2010.14407, 2020. 1 Theory and evaluation metrics for learning disentangled representations. Kien Do, Truyen Tran, 2020. 10International Conference on Learning Representations (ICLR. Kien Do and Truyen Tran. Theory and evaluation metrics for learning disentangled representations. In International Conference on Learning Representations (ICLR), 2020. 10 Unsupervised model selection for variational disentangled representation learning. Sunny Duan, Loic Matthey, Andre Saraiva, Nick Watters, Christopher Burgess, Alexander Lerchner, Irina Higgins, 2020. 10International Conference on Learning Representations (ICLR. Sunny Duan, Loic Matthey, Andre Saraiva, Nick Watters, Christopher Burgess, Alexander Lerch- ner, and Irina Higgins. Unsupervised model selection for variational disentangled representation learning. In International Conference on Learning Representations (ICLR), 2020. 10 Learning disentangled joint continuous and discrete representations. Emilien Dupont, Neural Information Processing Systems (NeurIPS). Emilien Dupont. Learning disentangled joint continuous and discrete representations. In Neural Information Processing Systems (NeurIPS), pp. 708-718, 2018. 10 A framework for the quantitative evaluation of disentangled representations. Cian Eastwood, K I Christopher, Williams, International Conference on Learning Representations (ICLR). Cian Eastwood and Christopher K. I. Williams. A framework for the quantitative evaluation of disentangled representations. In International Conference on Learning Representations (ICLR), 2018. 5 Structured disentangled representations. Babak Esmaeili, Hao Wu, Sarthak Jain, Alican Bozkurt, Narayanaswamy Siddharth, Brooks Paige, H Dana, Jennifer Brooks, Jan-Willem Dy, Meent, PMLRThe 22nd International Conference on Artificial Intelligence and Statistics. Babak Esmaeili, Hao Wu, Sarthak Jain, Alican Bozkurt, Narayanaswamy Siddharth, Brooks Paige, Dana H Brooks, Jennifer Dy, and Jan-Willem Meent. Structured disentangled representations. In The 22nd International Conference on Artificial Intelligence and Statistics, pp. 2525-2534. PMLR, 2019. 10 Bootstrap your own latent -a new approach to self-supervised learning. Jean-Bastien Grill, Florian Strub, Florent Altché, Corentin Tallec, Pierre Richemond, Elena Buchatskaya, Carl Doersch, Bernardo Avila Pires, Zhaohan Guo, Mohammad Gheshlaghi Azar, Bilal Piot, Remi Koray Kavukcuoglu, Michal Munos, Valko, Proceedings of Advances in Neural Information Processing Systems (NeurIPS). Advances in Neural Information Processing Systems (NeurIPS)2020Jean-Bastien Grill, Florian Strub, Florent Altché, Corentin Tallec, Pierre Richemond, Elena Buchatskaya, Carl Doersch, Bernardo Avila Pires, Zhaohan Guo, Mohammad Gheshlaghi Azar, Bilal Piot, koray kavukcuoglu, Remi Munos, and Michal Valko. Bootstrap your own latent -a new approach to self-supervised learning. In Proceedings of Advances in Neural Information Processing Systems (NeurIPS), 2020. 1 Deep residual learning for image recognition. Kaiming He, Xiangyu Zhang, Shaoqing Ren, Jian Sun, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionKaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recog- nition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770-778, 2016. 2 Momentum contrast for unsupervised visual representation learning. Kaiming He, Haoqi Fan, Yuxin Wu, Saining Xie, Ross Girshick, Proceedings of IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)Kaiming He, Haoqi Fan, Yuxin Wu, Saining Xie, and Ross Girshick. Momentum contrast for un- supervised visual representation learning. In Proceedings of IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 9729-9738, 2020. 1 Shakir Mohamed, and Alexander Lerchner. beta-vae: Learning basic visual concepts with a constrained variational framework. Irina Higgins, Loïc Matthey, Arka Pal, Christopher Burgess, Xavier Glorot, Matthew Botvinick, International Conference on Learning Representations (ICLR. 36Irina Higgins, Loïc Matthey, Arka Pal, Christopher Burgess, Xavier Glorot, Matthew Botvinick, Shakir Mohamed, and Alexander Lerchner. beta-vae: Learning basic visual concepts with a constrained variational framework. In International Conference on Learning Representations (ICLR), 2017. 1, 3, 6 Learning discrete and continuous factors of data via alternating disentanglement. Yeonwoo Jeong, Hyun Oh Song, International Conference on Machine Learning (ICML). 1021Yeonwoo Jeong and Hyun Oh Song. Learning discrete and continuous factors of data via alternating disentanglement. In International Conference on Machine Learning (ICML), 2019. 10, 21 Disentangling by factorising. Hyunjik Kim, Andriy Mnih, International Conference on Machine Learning (ICML). 21Hyunjik Kim and Andriy Mnih. Disentangling by factorising. In International Conference on Machine Learning (ICML), 2018. 1, 5, 7, 10, 21 Adam: A method for stochastic optimization. P Diederik, Jimmy Kingma, Ba, 3rd International Conference on Learning Representations (ICLR). Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In 3rd Interna- tional Conference on Learning Representations (ICLR), 2015. 5 Auto-encoding variational bayes. P Diederik, Max Kingma, Welling, International Conference on Learning Representations (ICLR). 13Diederik P. Kingma and Max Welling. Auto-encoding variational bayes. In International Conference on Learning Representations (ICLR), 2014. 1, 3 Improved variational inference with inverse autoregressive flow. P Durk, Tim Kingma, Rafal Salimans, Xi Jozefowicz, Ilya Chen, Max Sutskever, Welling, Advances in neural information processing systems. 29Durk P Kingma, Tim Salimans, Rafal Jozefowicz, Xi Chen, Ilya Sutskever, and Max Welling. Im- proved variational inference with inverse autoregressive flow. Advances in neural information processing systems, 29, 2016. 10 Variational inference of disentangled latent concepts from unlabeled observations. Abhishek Kumar, Prasanna Sattigeri, Avinash Balakrishnan, International Conference on Learning Representations (ICLR). 10Abhishek Kumar, Prasanna Sattigeri, and Avinash Balakrishnan. Variational inference of disen- tangled latent concepts from unlabeled observations. In International Conference on Learning Representations (ICLR), 2018. 1, 10 Demis Hassabis, and Alexander Lerchner. dsprites: Disentanglement testing sprites dataset. Loic Matthey, Irina Higgins, 15Loic Matthey, Irina Higgins, Demis Hassabis, and Alexander Lerchner. dsprites: Disentanglement testing sprites dataset. https://github.com/deepmind/dsprites-dataset/, 2017. 1, 5 Siamese prototypical contrastive learning. Shentong Mo, Zhun Sun, Chao Li, BMVC. Shentong Mo, Zhun Sun, and Chao Li. Siamese prototypical contrastive learning. In BMVC, 2021. 1 Rethinking prototypical contrastive learning through alignment, uniformity and correlation. Shentong Mo, Zhun Sun, Chao Li, BMVC. 2022Shentong Mo, Zhun Sun, and Chao Li. Rethinking prototypical contrastive learning through align- ment, uniformity and correlation. In BMVC, 2022. 1 Multi-level contrastive learning for self-supervised vision transformers. Shentong Mo, Zhun Sun, Chao Li, 2023 IEEE/CVF Winter Conference on Applications of Computer Vision (WACV). Shentong Mo, Zhun Sun, and Chao Li. Multi-level contrastive learning for self-supervised vision transformers. In 2023 IEEE/CVF Winter Conference on Applications of Computer Vision (WACV), pp. 2777-2786, 2023a. 1 Representation disentanglement in generative models with contrastive learning. Shentong Mo, Zhun Sun, Chao Li, 2023 IEEE/CVF Winter Conference on Applications of Computer Vision (WACV). Shentong Mo, Zhun Sun, and Chao Li. Representation disentanglement in generative models with contrastive learning. In 2023 IEEE/CVF Winter Conference on Applications of Computer Vision (WACV), pp. 1531-1540, 2023b. 10 Variational inference with normalizing flows. Danilo Rezende, Shakir Mohamed, PMLRInternational conference on machine learning. Danilo Rezende and Shakir Mohamed. Variational inference with normalizing flows. In Interna- tional conference on machine learning, pp. 1530-1538. PMLR, 2015. 10 Learning factorial codes by predictability minimization. Jürgen Schmidhuber, 08997667. 10Neural Computation. 46Jürgen Schmidhuber. Learning factorial codes by predictability minimization. Neural Computation, 4(6):863-879, 1992. ISSN 08997667. 10 A mathematical theory of communication. Claude Elwood Shannon, Bell system technical journal. 273Claude Elwood Shannon. A mathematical theory of communication. Bell system technical journal, 27(3):379-423, 1948. 10 Rethinking controllable variational autoencoders. Huajie Shao, Yifei Yang, Haohong Lin, Longzhong Lin, Yizhuo Chen, Qinmin Yang, Han Zhao, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern Recognition610Huajie Shao, Yifei Yang, Haohong Lin, Longzhong Lin, Yizhuo Chen, Qinmin Yang, and Han Zhao. Rethinking controllable variational autoencoders. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 19250-19259, 2022. 2, 3, 6, 10 Ladder variational autoencoders. Tapani Casper Kaae Sønderby, Lars Raiko, Maaløe, Ole Søren Kaae Sønderby, Winther, NIPS. Casper Kaae Sønderby, Tapani Raiko, Lars Maaløe, Søren Kaae Sønderby, and Ole Winther. Ladder variational autoencoders. In NIPS, 2016. 10 The information bottleneck method. Naftali Tishby, Fernando C Pereira, William Bialek, Proceedings of the 37-th Annual Allerton Conference on Communication, Control and Computing. the 37-th Annual Allerton Conference on Communication, Control and Computing210Naftali Tishby, Fernando C. Pereira, and William Bialek. The information bottleneck method. In In Proceedings of the 37-th Annual Allerton Conference on Communication, Control and Comput- ing, 1999. 2, 10 Frederik Träuble, Elliot Creager, Niki Kilbertus, Francesco Locatello, Andrea Dittadi, Anirudh Goyal, Bernhard Schölkopf, Stefan Bauer, arXiv:2006.07886On disentangled representations learned from correlated data. arXiv preprintFrederik Träuble, Elliot Creager, Niki Kilbertus, Francesco Locatello, Andrea Dittadi, Anirudh Goyal, Bernhard Schölkopf, and Stefan Bauer. On disentangled representations learned from correlated data. arXiv preprint arXiv:2006.07886, 2020. 1 Information theoretical analysis of multivariate correlation. Satosi Watanabe, IBM Journal of research and development. 410Satosi Watanabe. Information theoretical analysis of multivariate correlation. IBM Journal of re- search and development, 4:66-82, 1960. 10 Orthogonal jacobian regularization for unsupervised disentanglement in image generation. Yuxiang Wei, Yupeng Shi, Xiao Liu, Zhilong Ji, Yuan Gao, Zhongqin Wu, Wangmeng Zuo, Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV). the IEEE/CVF International Conference on Computer Vision (ICCV)Yuxiang Wei, Yupeng Shi, Xiao Liu, Zhilong Ji, Yuan Gao, Zhongqin Wu, and Wangmeng Zuo. Orthogonal jacobian regularization for unsupervised disentanglement in image generation. In Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), pp. 6721- 6730, October 2021. 10 Principal component analysis. Chemometrics and intelligent laboratory systems. Svante Wold, Kim Esbensen, Paul Geladi, 2Svante Wold, Kim Esbensen, and Paul Geladi. Principal component analysis. Chemometrics and intelligent laboratory systems, 2(1-3):37-52, 1987. 10 DEFT: distilling entangled factors by preventing information diffusion. Jiantao Wu, Lin Wang, Bo Yang, Fanqi Li, Chunxiuzi Liu, Jin Zhou, 10.1007/s10994-022-06134-7s10994-022-06134-7. 2Mach. Learn. 111610Jiantao Wu, Lin Wang, Bo Yang, Fanqi Li, Chunxiuzi Liu, and Jin Zhou. DEFT: dis- tilling entangled factors by preventing information diffusion. Mach. Learn., 111(6):2275- 2295, 2022. doi: 10.1007/s10994-022-06134-7. URL https://doi.org/10.1007/ s10994-022-06134-7. 2, 5, 6, 9, 10 Unsupervised feature learning via nonparametric instance discrimination. Zhirong Wu, Yuanjun Xiong, Stella X Yu, Dahua Lin, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR). the IEEE Conference on Computer Vision and Pattern Recognition (CVPR)Zhirong Wu, Yuanjun Xiong, Stella X. Yu, and Dahua Lin. Unsupervised feature learning via non- parametric instance discrimination. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2018. 1 Layer0 Layer1 Layer2 DeVAE Figure 8: Generated images of β-TCVAE and DeVAE from sampling random noise on dSprites. Layer0 Layer1 Layer2 DeVAE Figure 8: Generated images of β-TCVAE and DeVAE from sampling random noise on dSprites.
{'fraction_non_alphanumeric': 0.05017338534893801, 'fraction_numerical': 0.03720560612628233, 'mean_word_length': 4.7971940110983144, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 4, 'https://': 4, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 6, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'One major challenge of disentanglement learning with variational autoencoders is the trade-off between disentanglement and reconstruction fidelity. Previous incremental methods with only on latent space cannot optimize these two targets simultaneously, so they expand the Information Bottleneck while training to optimize from disentanglement to reconstruction. However, a large bottleneck will lose the constraint of disentanglement, causing the information diffusion problem. To tackle this issue, we present a novel decremental variational autoencoder with disentanglement-invariant transformations to optimize multiple objectives in different layers , termed DeVAE, for balancing disentanglement and reconstruction fidelity by decreasing the information bottleneck of diverse latent spaces gradually. Benefiting from the multiple latent spaces, DeVAE allows simultaneous optimization of multiple objectives to optimize reconstruction while keeping the constraint of disentanglement, avoiding information diffusion. DeVAE is also compatible with large models with high-dimension latent space. Experimental results on dSprites and Shapes3D that DeVAE achieves a good balance between disentanglement and reconstruction. DeVAE shows high tolerant of hyperparameters and on high-dimensional latent spaces. *', 'arxivid': '2303.12959', 'author': ['Jiantao Wu \nUniversity of Surrey\n\n', 'Shentong Mo \nCarnegie Mellon University\n\n', 'Muhammad Awais \nUniversity of Surrey\n\n', 'Sara Atito \nUniversity of Surrey\n\n', 'Xingshen Zhang \nUniversity of Jinan\n\n', 'Lin Wang \nUniversity of Jinan\n\n', 'Xiang Yang \nMingyi Tech\n\n'], 'authoraffiliation': ['University of Surrey\n', 'Carnegie Mellon University\n', 'University of Surrey\n', 'University of Surrey\n', 'University of Jinan\n', 'University of Jinan\n', 'Mingyi Tech\n'], 'corpusid': 257687624, 'doi': '10.48550/arxiv.2303.12959', 'github_urls': ['https://github.com/deepmind/3dshapes-dataset/,', 'https://github.com/deepmind/dsprites-dataset/,'], 'n_tokens_mistral': 16977, 'n_tokens_neox': 14325, 'n_words': 7966, 'pdfsha': '57c415b831e74e95deed67a66fc8941278278a3e', 'pdfurls': ['https://export.arxiv.org/pdf/2303.12959v1.pdf'], 'title': ['VARIATIONAL AUTOENCODERS WITH DECREMENTAL INFORMATION BOTTLENECK FOR DISENTANGLEMENT', 'VARIATIONAL AUTOENCODERS WITH DECREMENTAL INFORMATION BOTTLENECK FOR DISENTANGLEMENT'], 'venue': []}
arxiv
Dynamic Benchmarking of Masked Language Models on Temporal Concept Drift with Multiple Views Katerina Margatina [email protected] AWS AI Labs University of Sheffield Shuai Wang AWS AI Labs University of Sheffield Yogarshi Vyas AWS AI Labs University of Sheffield † Neha AWS AI Labs University of Sheffield Anna John AWS AI Labs University of Sheffield Yassine Benajiba AWS AI Labs University of Sheffield Miguel Ballesteros AWS AI Labs University of Sheffield Dynamic Benchmarking of Masked Language Models on Temporal Concept Drift with Multiple Views Temporal concept drift refers to the problem of data changing over time. In NLP, that would entail that language (e.g. new expressions, meaning shifts) and factual knowledge (e.g. new concepts, updated facts) evolve over time. Focusing on the latter, we benchmark 11 pretrained masked language models (MLMs) on a series of tests designed to evaluate the effect of temporal concept drift, as it is crucial that widely used language models remain upto-date with the ever-evolving factual updates of the real world. Specifically, we provide a holistic framework that (1) dynamically creates temporal test sets of any time granularity (e.g. month, quarter, year) of factual data from Wikidata, (2) constructs fine-grained splits of tests (e.g. updated, new, unchanged facts) to ensure comprehensive analysis, and (3) evaluates MLMs in three distinct ways (singletoken probing, multi-token generation, MLM scoring). In contrast to prior work, our framework aims to unveil how robust an MLM is over time and thus to provide a signal in case it has become outdated, by leveraging multiple views of evaluation. Introduction In the real world, what people talk about and how they tend to speak and write changes constantly over time. In Natural Language Processing (NLP), this entails a challenging shift of the textual data distribution that is commonly referred to as temporal concept drift. Prior work has identified that pretrained language models (PLMs) tend to become outdated soon after new topics and concepts are emerging (Lazaridou et al., 2021;Dhingra et al., 2022;Agarwal and Nenkova, 2022;Luu et al., 2022), limiting their capability to be robust to newly generated data. We consider the desiderata of language models' robustness to temporal drift to be twofold. First, * Work done during an internship at AWS AI Labs. LMs should be well adapted to the dynamic use of language, from the linguistic perspective. Language changes over time, pronunciations evolve, new words and expressions are borrowed or invented, the meaning of old words drifts, and morphology develops or decays (Blank, 1999;Traugott and Dasher, 2001;Kulkarni et al., 2015). Second, LMs should be aware of the ever-changing reality of the world, from a factual perspective. Models' factual knowledge should be up-to-date with new facts and concepts (e.g. to be of use continuously. In this work, we focus on the latter; the temporal robustness of LMs to facts that change over time. In an ideal scenario, we would like to know exactly when the factual knowledge of a model is "expired" so that we could adapt it to the new (or updated) set of facts. In reality, this is a challenging task. A large body of work has focused on the part of (continually) adapting an "outdated" model to the new data distribution (Guu et al., 2020;Yogatama et al., 2021;Sun et al., 2020;Biesialska et al., 2020;Jang et al., 2022b;Jin et al., 2022;Chakrabarty et al., 2022). This line of work is parallel to ours, as we focus on the crucial step before adaptation, the evaluation of the model on temporal concept drift: How can we know if a language model is outdated or not? Let us consider the case where we desire a language model to be up-to-date with the Prime Minister of the United Kingdom ( Figure 1). 1 A plausible way to evaluate this is to use the LAMA-probe paradigm (Petroni et al., 2019) and query the LM as a knowledge base (KB). This would mean that we could form the query as "The surname of the Prime Minister of the United Kingdom is <mask>.", give it as an input to a (masked) LM and inspect the output token distribution for the <mask> token. Figure 1 shows the top prediction for a series of ROBERTA models. 2 We first observe that the most widely used ROBERTA base and large models are both outdated in terms of factual knowledge, as they predict the names of PMs that served from 2010 until 2019. Next, while the last three models (2020-2022) answer correctly, the 2019 model answers the (correct) first name of the PM (Boris), not the surname (Johnson) which is asked for. This is a handy illustration of the many challenges in evaluating MLMs for temporal robustness in the LMs-as-KBs framework. First, this 2019 model would be considered to have made a mistake (as the prediction is different than the gold label and the metric is accuracy), even though the factual knowledge was correct (the name of the PM of the UK). Second, notice that we designed the query to ask for the surname (instead of the name of the PM), as this results in a single mask. The LAMA-probe and related frameworks do not handle multi-token queries for MLMs (e.g., Boris Johnson). Finally, we mark with a ? the answers of the first two ROBERTA models, because even though their answers are out-of-date for our current evaluation (October 2022), their answers could have been correct in an evaluation setting in the time of the training data (2019). This illustrates the obscurity of the temporal window in which the model is expected to be correct, if the model is not trained with a temporally-aware design (Lazaridou et al., 2021;Dhingra et al., 2022;Loureiro et al., 2022;Jang et al., 2022a). In this work, we aim to address such limitations and provide a holistic framework for dynamic benchmarking of masked language models on temporal concept drift, with a focus on facts that change over time. Following the propositions of Kiela et al. (2021) and Søgaard et al. (2021) that advocate for a focus on dynamic (i.e., test sets should not become saturated) and targeted (i.e., use of multiple, independent test sets for realistic performance estimates) benchmarking respectively, and building on prior work (Jiang et al., 2020b;Dhingra et al., 2022;Jang et al., 2022a), we create a large opensource test set that can be dynamically updated over time, containing temporal fine-grained subsets of examples that can be used to query masked language models and evaluate their factual knowledge over time. Contributions (1) We release DYNAMICTEM-PLAMA, an improved version of the static TEM-PLAMA (Dhingra et al., 2022) test set consisting of Wikidata relations, that is used to evaluate temporal robustness of MLMs. We provide data and code to dynamically keep DYNAMICTEMPLAMA up-to-date over time. 3 (2) We propose a novel evaluation framework to first create temporal splits of test sets of any granularity (month, quarter, year) and then to further create fine-grained splits of facts that are unchanged, updated, new or deleted, aiming to improve comprehensiveness ( §3.1). (3) We introduce three distinct evaluation views with multiple metrics ( §3.3) to ensure comprehensive results and provide analysis of benchmarking a large set open-source temporal ROBERTA models ( §3.2). Related Work Temporal Concept Drift Evaluation of the robustness of language models on temporal concept drift has seen a rising interest in the recent years. Previous work has focused on methods to continually adapt models over time (Hombaiah et al., 2021;Rosin et al., 2022;Lazaridou et al., 2022). Another area of research is evaluation of temporal robustness which has been explored both in the upstream LM pretraining task (Jiang et al., 2020b;Lazaridou et al., 2021;Dhingra et al., 2022;Jang et al., 2022a;Loureiro et al., 2022) and in downstream tasks such as sentiment analysis (Lukes and Søgaard, 2018;Agarwal and Nenkova, 2022), named entity recognition (Rijhwani and Preotiuc-Pietro, 2020;Onoe et al., 2022), question answering (Mavromatis et al., 2021;Liška et al., 2022), and rumor detection (Mu et al., 2023). It has also been studied for model explanations (Zhao et al., 2022) and for text classifi-cation in legal, biomedical (Chalkidis and Søgaard, 2022), and social media (Röttger and Pierrehumbert, 2021) domains. Luu et al. (2022) explore the setting of temporal misalignment (i.e., training and test data drawn from different periods of time) for both upstream and downstream tasks and find that temporal adaptation should not be seen as a substitute for finding temporally aligned labeled data for fine-tuning. The closest work to ours is TEMPLAMA (Dhingra et al., 2022). However, we differ across four axes: (i) TEMPLAMA is static, while we provide code to dynamically download facts in a finegrained fashion from any periods of time (not only yearly), (ii) we evaluate the same models over time focusing on the evaluation of robustness over time, we do not explore the best adaptation technique to address the problem, (iii) we do not fine-tune the models to adapt them to the domain/format of the test data, and (iv) we address benchmarking of masked LMs (not auto-regressive) including more evaluation techniques. Finally, similar to our motivation, Jang et al. (2022a) recently explored lifelong adaptation and evaluation of temporal concept drift in LMs and introduced TEMPORALWIKI for continual adaptation and TWIKI-PROBES for evaluation. The major difference is that the authors focus on providing corpora to adapt an LM over time, while in our paper we focus on evaluating temporal robustness of LMs. DYNAMICTEM-PLAMA is a holistic evaluation framework, while "TWIKI-PROBES are not natural sentences; they are factual phrases synthetically generated from a naive concatenation of Subject, Relation, and Object". Language Models as Knowledge Bases The cloze-style LM evaluation framework for factual knowledge, LAMA Petroni et al. (2019), follows the setting depicted in Figure 1. A knowledge base relation is transformed into natural language text with a manually created template and then passed as an input to an LM. The framework is based on treating the output distribution for the mask token as the retrieved answers to the query (AlKhamissi et al., 2022). The LAMA probe has since been extensively used to evaluate factual knowledge in LMs (Petroni et al., 2020;Talmor et al., 2020;Sung et al., 2021;Dhingra et al., 2022;Fierro and Søgaard, 2022), while other works have been exploring its limitations and ways to improve it (Kassner and Schütze, 2020;Haviv et al., 2021;Elazar et al., 2021;Zhong et al., 2021;Qin and Eisner, 2021). A particular challenge in our experimental setting, is the text compatibility between the model (i.e., its pre-training data) and the format of test examples, named as "language mismatch" by Talmor et al. (2020). Dhingra et al. (2022) opts to fine-tune the model under evaluation with part of the test set to adapt it to the format of the task. We argue that this process suffers from many caveats; it is inefficient and impractical to fine-tune a model whose capabilities are under evaluation, it risks optimization stability and overfitting issues due to the small training dataset, and enforces extra biases and errors, especially in the case of temporal robustness evaluation. Dynamic Benchmarking of Temporal Concept Drift In this section we describe in detail the steps to (re)create DYNAMICTEMPLAMA, our dynamically updated test set with facts from Wikidata ( §3.1). We then present the open-source temporal ROBERTA models (TIMELMS) (Loureiro et al., 2022) that we use for benchmarking ( §3.2). Finally, we introduce the evaluation framework under which we investigate how well the TimeLMs perform in terms of temporal robustness ( §3.3). The research question that we try to address with our work is: How can we measure temporal drift robustness of PLMs with an evaluation framework that is: unsupervised (no labeled downstream data), efficient (quality test set of facts-no need to run inference on a large corpus to compute perplexity for every token), dynamic (test set easily generated per request-can be used to dynamically evaluate new concepts over time), general (option to create test sets of any time granularity), and comprehensive (battery of targeted test sets that evaluate different LM capabilities and multiple views of evaluation). DYNAMIC-TEMPLAMA We base our implementation on the TEM-PLAMA (Dhingra et al., 2022) code, while we make several changes in terms of accessibility (i.e. option to dynamically update the test set), flexibility (i.e. option to adjust the granularity of the temporal splits) and comprehensiveness (i.e. finegrained splits and multiple evaluation views). We provide a high-level overview of the process to create DYNAMICTEMPLAMA in Figure 2. Data Collection We start the process by selecting a set of relations collected from the Wikidata KB ( Figure 2a). 4 Specifically, we use the 9 relations used in the TEMPLAMA dataset, followed by 7 more that we also decided to collect. We collect all relations from Wikidata in the span of 2019 − 2022. We then manually craft a cloze style query, i.e template, for each relation. Table 1 shows a few examples of relations and templates, along with dataset statistics. 5 We explain the data collection process in detail in Appendix A.1. Temporal Splits In this stage, we have a very large collection of facts for which we have temporal information (i.e., that the fact is true) in the time range we investigate (2019 − 2022). In the TEM-PLAMA dataset, the facts are divided yearly. However, we would ideally like to benchmark temporal models of any time granularity. Specifically, since we benchmark temporal models that are trained quarterly ( §3.2), a yearly split would not be useful to evaluate temporal concept drift of the four models trained on each quarter of a year. Consequently, we divide the large set of collected facts per quarter (Figure 2b), while adding the functionality to our implementation to split the facts in any time granularity (monthly, quarterly, yearly). to measure adaptation (i.e. how well a model adapts to new information/facts). Finally, we can measure overall temporal robustness by evaluating a temporal model from timestep t on D UPDATED t+1 and D NEW t+1 in timesteps for t ∈ [t + 1, t + 2, ...). We believe that this framework is particularly useful for insightful evaluation of methods that aim to adapt language models over time (Guu et al., 2020;Yogatama et al., 2021;Sun et al., 2020;Biesialska et al., 2020;Jang et al., 2022b;Jin et al., 2022;Chakrabarty et al., 2022). Fine-grained Splits Temporal Models In contrast with prior work that uses private, inhouse models for temporal robustness evaluation that are not accessible by the community (Lazaridou et al., 2021;Dhingra et al., 2022), we instead benchmark a series of open-source temporal models. Despite our aim for transparency, energy efficiency (Strubell et al., 2019) and reproducibility, we also believe that the dynamic nature of the task at hand requires accessibility to past, present and future models, to ensure that the findings of evaluation studies in temporal concept drift are meaningful, trustworthy and serve their purpose in evaluating models in a ever-evolving world. Under this assumption, we believe that studies on temporal robustness should ideally build on each other, so that we can have a holistic view as to how these models truly evolve over time. To this end, we use the Diachronic Language Models (TIMELMS) (Loureiro et al., 2022) that are publicly available in the HuggingFace hub (Wolf et al., 2019). 6 TIMELMS are ROBERTAmodels (Liu et al., 2019) trained quarterly on Twitter data. All models are initialised from the original roberta-base model checkpoint and are later trained using data from the previous quarters and the new temporal data from the new time period. For instance, the first model (2019-Q4) was trained with data sampled from Twitter until December 2019, while the second model (2020-Q1) was trained on the concatena-6 https://huggingface.co/cardiffnlp tion of all the data used to train 2019-Q4 and temporally-aligned data sampled from the first quarter of 2020. There are 11 TIMELMS in total, from 2019-Q4 until 2022-Q2. Finally, we would like to draw attention to two specific points. First, all TIMELMS are trained using the same ROBERTA (base) tokenizer and thus have the same vocabulary. This is crucial when evaluating models in a Cloze-style format, like the LAMA-probe, in order to evaluate fair comparison among the models. Second, Loureiro et al. (2022) aim to continue training and releasing TIMELMS every quarter, which is a very important and promising initiative to help with the dynamic evaluation of LMs in temporal concept drift in the future. Temporal Concept Drift Evaluation Single-token probing Our first evaluation type is single-token probing, which was introduced in the seminal LAMA-probe work of Petroni et al. (2019). The idea is simple and follows the fillin-the-blank format. Specifically, we convert each relation using its template to natural language text (see Figure 2(a)) replacing the <object> with the mask token (i.e., <mask> for ROBERTA). Then, as shown in Figure 1, we give the prompt as an input to the MLM and obtain a probability distribution over the vocabulary for the <mask> token. We use the metrics from Petroni et al. (2019), that are Accuracy, Mean Reciprocal Rank (MRR) and Precision at k (P@k). 7 Note that a crucial limitation of this approach is that it considers only facts with single-token objects. This results in trimming down the test sets by 95%, while limiting the actual value of the test (as most facts and concepts contain multiple words). Multi-token generation We aim to address this limitation and include multi-token objects to our evaluation framework. It is important to note that we are benchmarking masked language models instead of autoregressive left-to-right language models like Dhingra et al. (2022). This is crucial because the latter, decoder-based family of models, can be used off-the-shelf to generate multiple tokens. In contrast, MLMs are trained with 15% of their inputs masked and optimized to predict only the masked tokens. We therefore use the formulation introduced by Wang and Cho (2019), that is essentially a decoding-based strategy for MLMs based on Gibbs sampling. Specifically, we consider the setting that we do not know a priori the correct number of masks for each label. Instead, we enumerate from a single mask up to M masks, i.e., m = 1, ..., M . Following Jiang et al. (2020a), we choose M = 5, as all our facts are in the English language. When m > 1, we add m consecutive masks to the input and we pass the input to the model m times, when each time we sequentially sample each mask from left to right. At each iteration we replace the mask with the corresponding token prediction of the previous iteration. This way, we can extend the LAMA probe to include multitoken labels in our test set. The setting is entirely different than the single-token approach, as here we have m predictions from the model with an increasing number of tokens, while the correct label can consist of any number of tokens in the range of 1, ..., M . Another difference here is the evaluation metrics. Because we converted the task to text generation, we borrow generation metrics such as ROUGE (Lin, 2004), while also including standard metrics like F 1 -macro. Finally, we also include as a metric BERT-score (Zhang* et al., 2020) as an additional informative metric from the perspective of contextual semantics. In effect, we evaluate factual knowledge over time of MLMs, where facts include multiple correct answers and each answer consists of multiple tokens. We consider a prediction correct if the model correctly predicts any of the acceptable answers. MLM scoring Finally, as a third lens of evaluation we use the MLM scoring framework of Salazar et al. (2020). Contrary to the previous approaches, MLM scoring aims to measure the probability of the correct answer (i.e., of the masks), instead of generating the most probable answer. More specifically, we evaluate MLMs out of the box via their pseudo-log-likelihood scores (PLLs), which are computed by masking tokens one by one. PLLs have been widely used to measure the equivalence of perplexity (of autoregressive language models) for MLMs in unlabelled data (Lazaridou et al., 2021). Still, computing PLLs for large corpora is a very costly process in terms of time and resources (Loureiro et al., 2022). Instead, we propose to combine the LAMA and MLM scoring frameworks to create an efficient and targeted evaluation framework for temporal factual knowledge. Dataset Analysis We consider different subsets of the DYNAM-ICTEMPLAMA test sets for the three different evaluation settings ( §3.3). For the multi-token and MLM scoring settings, we keep the full dataset, for single-token we first tokenize the labels and keep only the test examples that have at least one label with a single token. This results in a very aggressive filtering of the dataset. Specifically, each quarterly temporal split consists of 8500 test examples on average for the multi-token setting, but for the single-token this results in only 450 examples, marking a loss of 95% of the data. 8 Additionally, the distribution of the fine-grained splits is of great interest, as it will shape the interpretation of the results and the general challenges of the evaluation framework. D UPDATED and D UNCHANGED (i.e., the splits of the most interest) constitute around 96% and 0.3%, respectively, of the total examples for the single-token evaluation, and 95% and 1.8% for the multi-token. This is arguably a very skewed distribution, showing the importance of our work in diving the temporal splits into further fine-grained splits. This is essential, because we would have different expectations for a model trained on timestep t while tested on data from both t and t − 1; for unchanged facts it would be desirable to keep equal performance in both sets (i.e., knowledge preservation §4.2), while for updated facts we would like to see improved performance in timestep t (i.e., adaptation §4.3). Results Temporal robustness We first evaluate temporal robustness of the 11 TIMELMS, defined as the overall performance over time ( §3.1). Figure 3 shows the average performance in all temporal and fine-grained splits in the time range from 2019-Q4 to 2022-Q2 for two types of evaluation, single-token probing and multitoken generation. For the former evaluation type, (Fig. 3a), all models perform similarly for all metrics. However, when we evaluate multi-token generation the models gradually improve over time. (Fig. 3b). This difference shows the importance of considering multiple views and evaluations for the same LM capability (i.e., temporal robustness). We attribute the similar single-token performance to the fact that these temporal datasets con- tain almost exclusively unchanged facts ( §3.4). It is therefore a positive outcome to observe that TIMELMS can preserve acquired knowledge ( §4.2). The findings for overall multi-token evaluation corroborate the intuition that more recent models, that are trained with temporal data of the entire range, should perform better than "past" (e.g. 2020) models that have not seen "future" data (e.g. 2022) during training. We also provide the overall results with MLM scoring in Table 2. We also observe that the last model performs best across all temporal splits, showing the effectiveness of adaptation with more recent unlabelled data ( §3.2). Even though we observe that this pattern holds for most temporal splits (i.e., scores improving for each column ↓), the 2020-Q4 and 2021-Q1 TIMELMS produce worse PLL scores than their previous or later versions. This is more evident in the overall density plot in Figure 5. This finding entails that either the distribution shift in these quarters was a lot stronger than the other temporal periods, or the training of these particular models was not as successful as it would have been expected. Knowledge preservation We use the D UNCHANGED split to evaluate the capability of MLMs to preserve knowledge over time. Figure 6 shows that for both single and multi-token evaluation all TIMELMS demonstrate similar performance over time, showing strong knowledge preserving skills. Surprisingly, different metrics show different patterns among the models for a single split. While in general we should not compare the performance of the single model over time (as the test sets are different), the comparision is valid in this case because the splits contain unchanged facts, and hence most temporal test sets are almost identical. All plots are shown in Figure 7 in the Appendix. Adaptation to emerging & evolving concepts Finally, we use the D NEW and D UPDATED splits for evaluation of emerging and evolving concepts, respectively. Here to ensure fair comparison, we evaluate the TIMELMS for a specific time window; for each model trained on timestep t, we keep the test sets from t − 1, t and t + 1. We observe in Figure 4 that in these cases the results vary among the models. There is not a very clear pattern as before, so case-by-case examination would be required. Still, a common pattern for the UPDATED split is that the middle set tends to have the highest performance (∧ shape). This means that models manage to effectively adapt to the updated facts of that timestep (t), but on the next timestep (t + 1) they underpeform as they are unaware of the factual changes, thus requiring adaptation. We provide all plots in the Appendix, including the DELETED split, which is more difficult to interpret intuitively (i.e., why are some facts deleted from Wikidata after a certain point?). Table 3 provides some examples from the DY-NAMICTEMPLAMA test set that can help us further interpret our results and inspect existing challenges. We first observe that all examples have multi-token labels (i.e., objects from the Subject-relation-object format) and are in ef-fect discarded in the single-token evaluation setup, making the inclusion of multiple views essential for this task. Qualitative Analysis More specifically, in 1, we observe that one label (United States women's national soccer team) has more than M = 5 tokens. It is therefore excluded even from the multi-token the test set, leaving MLM scoring to be the only method that could evaluate it. Interestingly, we manually tested the 2021-Q4 temporal model and found that it produces 1.6 and 307.3 average PLL scores for the two options respectively, making the disregarded label a far more confident prediction. In the second and third example, we observe how the correct answer for the query changes over time, making the granularity of the evaluation (i.e., yearly, quarterly, monthly) an important factor in the correct assessment of the model's temporal factual knowledge. For instance, for the example 3, we can carefully inspect how the predictions of the models change for facts that change over time (Table 4). However, even though PLL scores can follow intuitive temporal patterns (i.e., the PLL value can increase or decrease according to the point in time that the fact has changed), comparison between scores is not always helpful (i.e., word frequency can obscure factual knowledge) leaving room for improving the LAMA formulation. Conclusion & Future Work We addressed MLMs' robustness on temporal concept drift and introduced DYNAMICTEMPLAMA: a dataset for dynamic benchmarking of factual knowledge in temporal, fine-grained splits, from 2019-Q4 to 2022-Q2 that contain facts over time. We release our codebase to dynamically update the current test set over time and the option to extend it with custom (i) templates, (ii) relations from Wikidata, (iii) any period of time (years) and (iv) granularity of time (month/quarter/year). We include multiple views of evaluation, showing that it is essential in order to properly interpret the results of our benchmarking study of 11 temporal ROBERTA models. We consider experimentation with improving MLM decoding and addressing "domain mismatch" as open areas of research for future work. Our code can be found at https: //github.com/amazon-science/temporal-robustness. Limitations Lower bound estimate A very common issue with the LAMA probe evaluation framework (Petroni et al., 2019) is that it constitutes a lower bound estimate for its performance on factual knowledge retrieval. Specifically, if a model performs well, one can infer that it has the tested reasoning skill. However, failure does not entail that the reasoning skill is missing, as it is possible that there is a problem with the lexical-syntactic construction we picked (Talmor et al., 2020). Any given prompt only provides a lower bound estimate of the knowledge contained in an LM (Jiang et al., 2020b). Domain mismatch Despite the advantages of zero-shot evaluation, performance of a model might be adversely affected by mismatches between the language the pre-trained LM was trained on and the language of the examples in our tasks (Jiang et al., 2020b). It is quite possible that a fact that the LM does know cannot be retrieved due to the prompts not being effective queries for the fact (Jiang et al., 2020b). Prior work proposes to fine-tune the model with a small set of examples taken from the test set (and removed of course) in order to address the incompatibility problem or 'language mismatch' (Talmor et al., 2020;Dhingra et al., 2022). We argue that this process suffers for multiple limitations, such as that it not practical for a fast evaluation of the capabilities of a PLM at hand and it faces optimization stability issues due to the small training dataset, inter alia. The major limitation, however, is that such fine-tuning enforces extra biases and errors, especially in the case of temporal robustness evaluation. MLM decoding (multi-token labels) In this work we tried to address the problem of decoding from masked language models, by incorporating two distinct approaches to the evaluation framework; multi-token generation with MLMs (Wang and Cho, 2019) and MLM scoring (Salazar et al., 2020). Still, we observe that both methods provide results that are hard to interpret ( §5), leaving the problems of (i) decoding or generating multiple tokens from MLMs and (ii) evaluation of factual knowledge in LMs as open areas of research. Manual Templates For LAMA-style probing (Petroni et al., 2019), prior work creates the templates manually. This is a limitation both in terms of scale (i.e., generalization to many different kinds of inputs) and consistency (i.e., how do models perform with minimal changes to their inputs?). LMs do not reason in an abstract manner and are contextdependent (Talmor et al., 2020). It is therefore essential to address this problem and include functionalities to incorporate a set of diverse templates for each evaluation setup. English Twitter MLMs Finally, our dataset, DY-NAMICTEMPLAMA, following prior work (Dhingra et al., 2022), collects and evaluates facts from the Wikidata in the English language alone, and benchmarks RoBERTa language models trained in English Twitter data. We understand that this is a limitation and further data collection and experimentation in more languages would be strongly encouraged. TEMPORAL SPLIT UNCHANGED UPDATED DELETED NEW TOTAL %UNCHANGED %UPDATED %LOST A Appendix A.1 Data Collection for DYNAMICTEMPLAMA Following Dhingra et al. (2022), we identify all facts in the Wikidata snapshot, which have either a start or an end date after 2010 and whose subjects and objects are both entities with Wikipedia pages.1 Among these 482K facts, we identify subject and relation pairs which have multiple objects at different times and select 16 relations with the most such subjects. Then, for these relations we manually write template cloze queries (i.e., templates) and populate them with the 1000 most frequent subjects per relation. For each subject and each relation we gather all the objects with their associated time interval and construct a separate query for each year in that interval. When intervals for the object entities overlap, we add all of them to the list of correct answers. The query and the corresponding year form the input texts and the temporal information t, while the object entity is the target that we want to predict (i.e., gold label). In contrast to Dhingra et al. (2022), we do extra temporal divisions. Specifically, we get each yearly split and divide it further in quarterly splits ( §3.1, Figure 2b), following the same algorithm. A.2 Full Results We provide the full results with all metrics for the UNCHANGED split in Figure 7, and the UPDATED, NEW and DELETED splits for multi-token generation in Figure 9. Figure 1 : 1Querying pretrained MLMs on their knowledge about the Prime Minister of the United Kingdom. Figure 3 : 3Overall performance over time (2019−2022) for both single and multi-token evaluation. X-axis corresponds to the TIMELMS and the Y -axis to different metrics depending on the type of the evaluation. Figure 4 : 4Multi-token evaluation for evolving and emerging facts. EXAMPLE INPUT GROUND TRUTH LABELS #TOKENS #ANSWERS SPLIT Figure 5 : 5Overall PLL distributions for TIMELMS. Figure 6 : 6Single and multi-token evaluation for the UN- For a given time range, from timestep t to t + 1 (e.g. 2019-Q1→2019-Q2), we further create comprehensive test sets that contain examples with unchanged, updated, new or deleted facts, denoted by D UNCHANGEDFigure 2c). We create these splits to be able to measure different capabilities of the MLM in terms of robustness to temporal concept drift. The motivation for this stems from limitations of prior work(Dhingra et al., 2022) to shed light into what kind of data each temporalFigure 2:The process for creating DYNAMICTEM-PLAMA. We first collect data from Wikidata (a), we then divide it to quarterly temporal splits (b) and finally we create more targeted fine-grained sets (c). test set contains. For instance, we pose questions like How many facts were updated from timestep t → t + 1? How many facts remained unchanged? What was the change? The object or the subject? Are there new facts in timestep t + 1 that were not present before? We argue that it is essential to distinguish between these sub-tests, so that each split can target specific capabilities of the LM. First, we can use D UNCHANGED t+1 to evaluate knowledge preservation (i.e. how well a model can preserve knowledge over time). Second, we can use D UPDATED t+1 , D NEW t+1 and D DELETEDt+1 , D UPDATED t+1 , D NEW t+1 and D DELETED t+1 respectively (t+1 Table 2 : 2MLM scoring (median pseudo-log-likelihood scores) averaged for each temporal split. Table 3 : 3Qualitative analysis of certain examples in DYNAMICTEMPLAMA. Table 4 : 4PLL scores for Example 2 fromTable 3. Table 5 : 5Total number of examples for each temporal and fine-grained split in DYNAMICTEMPLAMA. We show both the single-token and the multi-token datasets (up to M = 5 tokens). Cell scheme to be read single | multi. %UNCHANGED and %UPDATED show the percentage of the total examples that are part of the UNCHANGED and UPDATED set respectively. %LOST shows the percentage of examples we lose when we filter out the dataset for the single-token evaluation setting. The time of writing of this paper is September 2022. 2 Except for the ROBERTA base and large models, we also show the predictions of models trained with Twitter data until 2019, 2020, 2021, and 2022, respectively(Loureiro et al., 2022). https://github.com/amazon-science/temporal-robustness All possible relations from Wikidata can be found here https://www.wikidata.org/wiki/Wikidata:List_of_properties. 5 Details on all relations and templates of DYNAMICTEM-PLAMA can be found in Tables 6 & 7 in the Appendix A.1. (a) Data collection (b) Temporal Splits (c) Fine-grained Splits P@k= 1, if the gold label is in the top-k predictions of the model, therefore P@1 corresponds to Accuracy. Table 5in the Appendix shows all the statistics in detail. AcknowledgementsWe would like to thank the anonymous reviewers and the AWS AI team for their valuable feedback on the paper. We also thank Robert Vacareanu and Karthikeyan K for their help with running experiments.we keep the test sets from t − 1, t and t + 1. Temporal effects on pre-trained models for language processing tasks. Oshin Agarwal, Ani Nenkova, Transactions of the Association for Computational Linguistics. 10Oshin Agarwal and Ani Nenkova. 2022. Temporal ef- fects on pre-trained models for language processing tasks. Transactions of the Association for Computa- tional Linguistics, 10:904-921. . Millicent Badr Alkhamissi, Asli Li, Celikyilmaz, 10.48550/ARXIV.2204.06031Mona Diaband Marjan Ghazvininejad. 2022. A review on language models as knowledge basesBadr AlKhamissi, Millicent Li, Asli Celikyilmaz, Mona Diab, and Marjan Ghazvininejad. 2022. A re- view on language models as knowledge bases. Continual lifelong learning in natural language processing: A survey. Magdalena Biesialska, Katarzyna Biesialska, Marta R Costa-Jussà, 10.18653/v1/2020.coling-main.574Proceedings of the 28th International Conference on Computational Linguistics. the 28th International Conference on Computational LinguisticsBarcelona, SpainInternational Committee on Computational LinguisticsMagdalena Biesialska, Katarzyna Biesialska, and Marta R. Costa-jussà. 2020. Continual lifelong learning in natural language processing: A survey. In Proceedings of the 28th International Conference on Computational Linguistics, pages 6523-6541, Barcelona, Spain (Online). International Committee on Computational Linguistics. Why do new meanings occur? a cognitive typology of the motivations for lexical semantic change. Andreas Blank, Andreas Blank. 1999. Why do new meanings occur? a cognitive typology of the motivations for lexical semantic change. 2022. Fine-tuned language models can be continual learners. Tuhin Chakrabarty, Thomas Scialom, Smaranda Muresan, Challenges & Perspectives in Creating Large Language Models. Tuhin Chakrabarty, Thomas Scialom, and Smaranda Muresan. 2022. Fine-tuned language models can be continual learners. In Challenges & Perspectives in Creating Large Language Models. Improved multi-label classification under temporal concept drift: Rethinking group-robust algorithms in a labelwise setting. Ilias Chalkidis, Anders Søgaard, Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics. the 60th Annual Meeting of the Association for Computational LinguisticsDublin, IrelandIlias Chalkidis and Anders Søgaard. 2022. Improved multi-label classification under temporal concept drift: Rethinking group-robust algorithms in a label- wise setting. In Proceedings of the 60th Annual Meeting of the Association for Computational Lin- guistics, Dublin, Ireland. Time-aware language models as temporal knowledge bases. Bhuwan Dhingra, Jeremy R Cole, Julian Martin Eisenschlos, Daniel Gillick, Jacob Eisenstein, William W Cohen, 10.1162/tacl_a_00459Transactions of the Association for Computational Linguistics. 10Bhuwan Dhingra, Jeremy R. Cole, Julian Martin Eisenschlos, Daniel Gillick, Jacob Eisenstein, and William W. Cohen. 2022. Time-aware language models as temporal knowledge bases. Transactions of the Association for Computational Linguistics, 10:257-273. Measuring and improving consistency in pretrained language models. Yanai Elazar, Nora Kassner, Shauli Ravfogel, Abhilasha Ravichander, Eduard Hovy, Hinrich Schütze, Yoav Goldberg, 10.1162/tacl_a_00410Transactions of the Association for Computational Linguistics. 9Yanai Elazar, Nora Kassner, Shauli Ravfogel, Abhi- lasha Ravichander, Eduard Hovy, Hinrich Schütze, and Yoav Goldberg. 2021. Measuring and im- proving consistency in pretrained language models. Transactions of the Association for Computational Linguistics, 9:1012-1031. Factual consistency of multilingual pretrained language models. Constanza Fierro, Anders Søgaard, 10.18653/v1/2022.findings-acl.240Findings of the Association for Computational Linguistics: ACL 2022. Dublin, IrelandAssociation for Computational LinguisticsConstanza Fierro and Anders Søgaard. 2022. Fac- tual consistency of multilingual pretrained language models. In Findings of the Association for Compu- tational Linguistics: ACL 2022, pages 3046-3052, Dublin, Ireland. Association for Computational Lin- guistics. Realm: Retrievalaugmented language model pre-training. Kelvin Guu, Kenton Lee, Zora Tung, Panupong Pasupat, Ming-Wei Chang, Proceedings of the 37th International Conference on Machine Learning. the 37th International Conference on Machine LearningICML'20. JMLR.orgKelvin Guu, Kenton Lee, Zora Tung, Panupong Pasu- pat, and Ming-Wei Chang. 2020. Realm: Retrieval- augmented language model pre-training. In Pro- ceedings of the 37th International Conference on Machine Learning, ICML'20. JMLR.org. BERTese: Learning to speak to BERT. Adi Haviv, Jonathan Berant, Amir Globerson, 10.18653/v1/2021.eacl-main.316Proceedings of the 16th Conference of the European Chapter of the Association for Computational Linguistics: Main Volume. the 16th Conference of the European Chapter of the Association for Computational Linguistics: Main VolumeOnline. Association for Computational LinguisticsAdi Haviv, Jonathan Berant, and Amir Globerson. 2021. BERTese: Learning to speak to BERT. In Proceedings of the 16th Conference of the European Chapter of the Association for Computational Lin- guistics: Main Volume, pages 3618-3623, Online. Association for Computational Linguistics. Dynamic language models for continuously evolving content. Tao Spurthi Amba Hombaiah, Mingyang Chen, Michael Zhang, Marc-Alexander Bendersky, Najork, Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining. the 27th ACM SIGKDD Conference on Knowledge Discovery & Data MiningSpurthi Amba Hombaiah, Tao Chen, Mingyang Zhang, Michael Bendersky, and Marc-Alexander Najork. 2021. Dynamic language models for continuously evolving content. Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining. Temporalwiki: A lifelong benchmark for training and evaluating ever-evolving language models. Joel Jang, Seonghyeon Ye, Changho Lee, Sohee Yang, Joongbo Shin, Janghoon Han, Gyeonghun Kim, Minjoon Seo, 10.48550/ARXIV.2204.14211Joel Jang, Seonghyeon Ye, Changho Lee, Sohee Yang, Joongbo Shin, Janghoon Han, Gyeonghun Kim, and Minjoon Seo. 2022a. Temporalwiki: A lifelong benchmark for training and evaluating ever-evolving language models. Towards continual knowledge learning of language models. Joel Jang, Seonghyeon Ye, Sohee Yang, Joongbo Shin, Janghoon Han, Kim Gyeonghun, Stanley Jungkyu Choi, Minjoon Seo, International Conference on Learning Representations. Joel Jang, Seonghyeon Ye, Sohee Yang, Joongbo Shin, Janghoon Han, Gyeonghun KIM, Stanley Jungkyu Choi, and Minjoon Seo. 2022b. Towards continual knowledge learning of language models. In Interna- tional Conference on Learning Representations. X-FACTR: Multilingual factual knowledge retrieval from pretrained language models. Zhengbao Jiang, Antonios Anastasopoulos, Jun Araki, Haibo Ding, Graham Neubig, 10.18653/v1/2020.emnlp-main.479Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP). the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP)Zhengbao Jiang, Antonios Anastasopoulos, Jun Araki, Haibo Ding, and Graham Neubig. 2020a. X- FACTR: Multilingual factual knowledge retrieval from pretrained language models. In Proceedings of the 2020 Conference on Empirical Methods in Nat- ural Language Processing (EMNLP), pages 5943- 5959, Online. Association for Computational Lin- guistics. How can we know what language models know?. Zhengbao Jiang, Frank F Xu, Jun Araki, Graham Neubig, 10.1162/tacl_a_00324Transactions of the Association for Computational Linguistics. 8Zhengbao Jiang, Frank F. Xu, Jun Araki, and Graham Neubig. 2020b. How can we know what language models know? Transactions of the Association for Computational Linguistics, 8:423-438. Lifelong pretraining: Continually adapting language models to emerging corpora. Xisen Jin, Dejiao Zhang, Henghui Zhu, Wei Xiao, Shang-Wen Li, Xiaokai Wei, Andrew Arnold, Xiang Ren, 10.18653/v1/2022.bigscience-1.1Proceedings of BigScience Episode #5 -Workshop on Challenges & Perspectives in Creating Large Language Models. BigScience Episode #5 -Workshop on Challenges & Perspectives in Creating Large Language Modelsvirtual+Dublin. Association for Computational LinguisticsXisen Jin, Dejiao Zhang, Henghui Zhu, Wei Xiao, Shang-Wen Li, Xiaokai Wei, Andrew Arnold, and Xiang Ren. 2022. Lifelong pretraining: Continually adapting language models to emerging corpora. In Proceedings of BigScience Episode #5 -Workshop on Challenges & Perspectives in Creating Large Language Models, pages 1-16, virtual+Dublin. As- sociation for Computational Linguistics. Multilingual LAMA: Investigating knowledge in multilingual pretrained language models. Nora Kassner, Philipp Dufter, Hinrich Schütze, 10.18653/v1/2021.eacl-main.284Proceedings of the Conference of the European Chapter of the Association for Computational Linguistics. the Conference of the European Chapter of the Association for Computational LinguisticsNora Kassner, Philipp Dufter, and Hinrich Schütze. 2021. Multilingual LAMA: Investigating knowl- edge in multilingual pretrained language models. In Proceedings of the Conference of the European Chapter of the Association for Computational Lin- guistics. Negated and misprimed probes for pretrained language models: Birds can talk, but cannot fly. Nora Kassner, Hinrich Schütze, 10.18653/v1/2020.acl-main.698Proceedings of the Annual Meeting of the Association for Computational Linguistics. the Annual Meeting of the Association for Computational LinguisticsNora Kassner and Hinrich Schütze. 2020. Negated and misprimed probes for pretrained language models: Birds can talk, but cannot fly. In Proceedings of the Annual Meeting of the Association for Compu- tational Linguistics. Dynabench: Rethinking benchmarking in NLP. Douwe Kiela, Max Bartolo, Yixin Nie, Divyansh Kaushik, Atticus Geiger, Zhengxuan Wu, Bertie Vidgen, Grusha Prasad, Amanpreet Singh, Pratik Ringshia, Zhiyi Ma, Tristan Thrush, Sebastian Riedel, Zeerak Waseem, Pontus Stenetorp, Robin Jia, Mohit Bansal, Christopher Potts, Adina Williams, 10.18653/v1/2021.naacl-main.324Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies. the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language TechnologiesOnlineAssociation for Computational LinguisticsDouwe Kiela, Max Bartolo, Yixin Nie, Divyansh Kaushik, Atticus Geiger, Zhengxuan Wu, Bertie Vid- gen, Grusha Prasad, Amanpreet Singh, Pratik Ring- shia, Zhiyi Ma, Tristan Thrush, Sebastian Riedel, Zeerak Waseem, Pontus Stenetorp, Robin Jia, Mo- hit Bansal, Christopher Potts, and Adina Williams. 2021. Dynabench: Rethinking benchmarking in NLP. In Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Tech- nologies, pages 4110-4124, Online. Association for Computational Linguistics. Statistically significant detection of linguistic change. Vivek Kulkarni, Rami Al-Rfou, Bryan Perozzi, Steven Skiena, 10.1145/2736277.2741627Proceedings of the 24th International Conference on World Wide Web, WWW '15. the 24th International Conference on World Wide Web, WWW '15Republic and Canton of Geneva. Wide Web Conferences Steering CommitteeVivek Kulkarni, Rami Al-Rfou, Bryan Perozzi, and Steven Skiena. 2015. Statistically significant de- tection of linguistic change. In Proceedings of the 24th International Conference on World Wide Web, WWW '15, page 625-635, Republic and Canton of Geneva, CHE. International World Wide Web Con- ferences Steering Committee. Internetaugmented language models through few-shot prompting for open-domain question answering. Angeliki Lazaridou, Elena Gribovskaya, Wojciech Stokowiec, Nikolai Grigorev, 10.48550/ARXIV.2203.05115Angeliki Lazaridou, Elena Gribovskaya, Wojciech Stokowiec, and Nikolai Grigorev. 2022. Internet- augmented language models through few-shot prompting for open-domain question answering. Kris Cao, Susannah Young, and Phil Blunsom. 2021. Mind the gap: Assessing temporal generalization in neural language models. Angeliki Lazaridou, Adhiguna Kuncoro, Elena Gribovskaya, Devang Agrawal, Adam Liska, Tayfun Terzi, Mai Gimenez, Cyprien De Masson D&apos;autume, Tomáš Kočiský, Sebastian Ruder, Dani Yogatama, Advances in Neural Information Processing Systems. Angeliki Lazaridou, Adhiguna Kuncoro, Elena Gri- bovskaya, Devang Agrawal, Adam Liska, Tayfun Terzi, Mai Gimenez, Cyprien de Masson d'Autume, Tomáš Kočiský, Sebastian Ruder, Dani Yogatama, Kris Cao, Susannah Young, and Phil Blunsom. 2021. Mind the gap: Assessing temporal generalization in neural language models. In Advances in Neural In- formation Processing Systems. ROUGE: A package for automatic evaluation of summaries. Chin-Yew Lin, Text Summarization Branches Out. Barcelona, SpainAssociation for Computational LinguisticsChin-Yew Lin. 2004. ROUGE: A package for auto- matic evaluation of summaries. In Text Summariza- tion Branches Out, pages 74-81, Barcelona, Spain. Association for Computational Linguistics. . Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, Veselin Stoyanov, 10.48550/ARXIV.1907.11692Roberta: A robustly optimized bert pretraining approachYinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Man- dar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. 2019. Roberta: A robustly optimized bert pretraining ap- proach. Devang Agrawal, Cyprien de Masson d'Autume, Tim Scholtes, Manzil Zaheer. Adam Liška, Tomáš Kočiský, Elena Gribovskaya, Tayfun Terzi, Eren Sezener, 10.48550/ARXIV.2205.11388Blunsom, and Angeliki Lazaridou. 2022. Streamingqa: A benchmark for adaptation to new knowledge over time in question answering models. Susannah Young, Ellen Gilsenan-McMahon, Sophia Austin, PhilAdam Liška, Tomáš Kočiský, Elena Gribovskaya, Tay- fun Terzi, Eren Sezener, Devang Agrawal, Cyprien de Masson d'Autume, Tim Scholtes, Manzil Za- heer, Susannah Young, Ellen Gilsenan-McMahon, Sophia Austin, Phil Blunsom, and Angeliki Lazari- dou. 2022. Streamingqa: A benchmark for adapta- tion to new knowledge over time in question answer- ing models. TimeLMs: Diachronic language models from Twitter. Daniel Loureiro, Francesco Barbieri, Leonardo Neves, Luis Espinosa Anke, Jose Camacho-Collados, 10.18653/v1/2022.acl-demo.25Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics: System Demonstrations. the 60th Annual Meeting of the Association for Computational Linguistics: System DemonstrationsDublin, IrelandAssociation for Computational LinguisticsDaniel Loureiro, Francesco Barbieri, Leonardo Neves, Luis Espinosa Anke, and Jose Camacho-collados. 2022. TimeLMs: Diachronic language models from Twitter. In Proceedings of the 60th Annual Meet- ing of the Association for Computational Linguistics: System Demonstrations, pages 251-260, Dublin, Ire- land. Association for Computational Linguistics. Sentiment analysis under temporal shift. Jan Lukes, Anders Søgaard, 10.18653/v1/W18-6210Proceedings of the Workshop on Computational Approaches to Subjectivity, Sentiment and Social Media Analysis. the Workshop on Computational Approaches to Subjectivity, Sentiment and Social Media AnalysisJan Lukes and Anders Søgaard. 2018. Sentiment analy- sis under temporal shift. In Proceedings of the Work- shop on Computational Approaches to Subjectivity, Sentiment and Social Media Analysis, pages 65-71. Time waits for no one! analysis and challenges of temporal misalignment. Kelvin Luu, Daniel Khashabi, Suchin Gururangan, Karishma Mandyam, Noah A Smith, 10.18653/v1/2022.naacl-main.435Proceedings of the 2022 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies. the 2022 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language TechnologiesSeattle, United StatesAssociation for Computational LinguisticsKelvin Luu, Daniel Khashabi, Suchin Gururangan, Kar- ishma Mandyam, and Noah A. Smith. 2022. Time waits for no one! analysis and challenges of tem- poral misalignment. In Proceedings of the 2022 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pages 5944-5958, Seattle, United States. Association for Computational Lin- guistics. Costas Mavromatis, Lakkur Prasanna, Subramanyam, N Vassilis, Soji Ioannidis, Phillip R Adeshina, Howard, 10.48550/ARXIV.2112.05785Tetiana Grinberg, Nagib Hakim, and George Karypis. 2021. Tempoqr: Temporal question reasoning over knowledge graphs. Costas Mavromatis, Prasanna Lakkur Subramanyam, Vassilis N. Ioannidis, Soji Adeshina, Phillip R. Howard, Tetiana Grinberg, Nagib Hakim, and George Karypis. 2021. Tempoqr: Temporal ques- tion reasoning over knowledge graphs. It's about time: Rethinking evaluation on rumor detection benchmarks using chronological splits. Yida Mu, Kalina Bontcheva, Nikolaos Aletras, Findings of the Conference of the European Chapter. Association for Computational LinguisticsYida Mu, Kalina Bontcheva, and Nikolaos Aletras. 2023. It's about time: Rethinking evaluation on rumor detection benchmarks using chronological splits. In Findings of the Conference of the Euro- pean Chapter of the Association for Computational Linguistics. Entity cloze by date: What LMs know about unseen entities. Yasumasa Onoe, Michael Zhang, Eunsol Choi, Greg Durrett, 10.18653/v1/2022.findings-naacl.52Findings of the Association for Computational Linguistics: NAACL 2022. Seattle, United StatesAssociation for Computational LinguisticsYasumasa Onoe, Michael Zhang, Eunsol Choi, and Greg Durrett. 2022. Entity cloze by date: What LMs know about unseen entities. In Findings of the Association for Computational Linguistics: NAACL 2022, pages 693-702, Seattle, United States. Associ- ation for Computational Linguistics. . Fabio Petroni, S H Patrick, Aleksandra Lewis, Tim Piktus, Yuxiang Rocktäschel, Alexander H Wu, Miller, and Sebastian Riedel. 2020. How context affects language models' factual predictions. CoRR, abs/2005.04611Fabio Petroni, Patrick S. H. Lewis, Aleksandra Pik- tus, Tim Rocktäschel, Yuxiang Wu, Alexander H. Miller, and Sebastian Riedel. 2020. How context affects language models' factual predictions. CoRR, abs/2005.04611. Language models as knowledge bases?. Fabio Petroni, Tim Rocktäschel, Sebastian Riedel, Patrick Lewis, Anton Bakhtin, Yuxiang Wu, Alexander Miller, 10.18653/v1/D19-1250Proceedings of the Conference on Empirical Methods in Natural Language Processing and the International Joint Conference on Natural Language Processing. the Conference on Empirical Methods in Natural Language Processing and the International Joint Conference on Natural Language ProcessingFabio Petroni, Tim Rocktäschel, Sebastian Riedel, Patrick Lewis, Anton Bakhtin, Yuxiang Wu, and Alexander Miller. 2019. Language models as knowl- edge bases? In Proceedings of the Conference on Empirical Methods in Natural Language Processing and the International Joint Conference on Natural Language Processing. Learning how to ask: Querying LMs with mixtures of soft prompts. Guanghui Qin, Jason Eisner, 10.18653/v1/2021.naacl-main.410Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies. the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language TechnologiesOnline. Association for Computational LinguisticsGuanghui Qin and Jason Eisner. 2021. Learning how to ask: Querying LMs with mixtures of soft prompts. In Proceedings of the 2021 Conference of the North American Chapter of the Association for Computa- tional Linguistics: Human Language Technologies, pages 5203-5212, Online. Association for Compu- tational Linguistics. Temporally-informed analysis of named entity recognition. Shruti Rijhwani, Daniel Preotiuc-Pietro, 10.18653/v1/2020.acl-main.680Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics. the 58th Annual Meeting of the Association for Computational LinguisticsOnline. Association for Computational LinguisticsShruti Rijhwani and Daniel Preotiuc-Pietro. 2020. Temporally-informed analysis of named entity recognition. In Proceedings of the 58th Annual Meeting of the Association for Computational Lin- guistics, pages 7605-7617, Online. Association for Computational Linguistics. Time masking for temporal language models. Guy D Rosin, Ido Guy, Kira Radinsky, Proceedings of the Fifteenth ACM International Conference on Web Search and Data Mining. Guy D. Rosin, Ido Guy, and Kira Radinsky. 2022. Time masking for temporal language models. Pro- ceedings of the Fifteenth ACM International Confer- ence on Web Search and Data Mining. Temporal adaptation of BERT and performance on downstream document classification: Insights from social media. Paul Röttger, Janet Pierrehumbert, 10.18653/v1/2021.findings-emnlp.206Findings of the Association for Computational Linguistics: EMNLP 2021. Punta Cana, Dominican RepublicAssociation for Computational LinguisticsPaul Röttger and Janet Pierrehumbert. 2021. Tempo- ral adaptation of BERT and performance on down- stream document classification: Insights from social media. In Findings of the Association for Computa- tional Linguistics: EMNLP 2021, pages 2400-2412, Punta Cana, Dominican Republic. Association for Computational Linguistics. Masked language model scoring. Julian Salazar, Davis Liang, Toan Q Nguyen, Katrin Kirchhoff, 10.18653/v1/2020.acl-main.240Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics. the 58th Annual Meeting of the Association for Computational LinguisticsOnline. Association for Computational LinguisticsJulian Salazar, Davis Liang, Toan Q. Nguyen, and Ka- trin Kirchhoff. 2020. Masked language model scor- ing. In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, pages 2699-2712, Online. Association for Compu- tational Linguistics. We need to talk about random splits. Anders Søgaard, Sebastian Ebert, Jasmijn Bastings, Katja Filippova, 10.18653/v1/2021.eacl-main.156Proceedings of the 16th Conference of the European Chapter of the Association for Computational Linguistics: Main Volume. the 16th Conference of the European Chapter of the Association for Computational Linguistics: Main VolumeOnline. Association for Computational LinguisticsAnders Søgaard, Sebastian Ebert, Jasmijn Bastings, and Katja Filippova. 2021. We need to talk about random splits. In Proceedings of the 16th Confer- ence of the European Chapter of the Association for Computational Linguistics: Main Volume, pages 1823-1832, Online. Association for Computational Linguistics. Energy and policy considerations for deep learning in NLP. Emma Strubell, Ananya Ganesh, Andrew Mccallum, abs/1906.02243CoRREmma Strubell, Ananya Ganesh, and Andrew McCal- lum. 2019. Energy and policy considerations for deep learning in NLP. CoRR, abs/1906.02243. {LAMAL}: {LA}nguage modeling is all you need for lifelong language learning. Fan-Keng Sun, Cheng-Hao Ho, Hung-Yi Lee, International Conference on Learning Representations. Fan-Keng Sun, Cheng-Hao Ho, and Hung-Yi Lee. 2020. {LAMAL}: {LA}nguage modeling is all you need for lifelong language learning. In International Conference on Learning Representations. Can language models be biomedical knowledge bases?. Mujeen Sung, Jinhyuk Lee, Sean Yi, Minji Jeon, Sungdong Kim, Jaewoo Kang, 10.18653/v1/2021.emnlp-main.388Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing. the 2021 Conference on Empirical Methods in Natural Language ProcessingOnline and Punta Cana, Dominican RepublicAssociation for Computational LinguisticsMujeen Sung, Jinhyuk Lee, Sean Yi, Minji Jeon, Sung- dong Kim, and Jaewoo Kang. 2021. Can language models be biomedical knowledge bases? In Pro- ceedings of the 2021 Conference on Empirical Meth- ods in Natural Language Processing, pages 4723- 4734, Online and Punta Cana, Dominican Republic. Association for Computational Linguistics. 2020. oLMpics-on what language model pre-training captures. Alon Talmor, Yanai Elazar, Yoav Goldberg, Jonathan Berant, 10.1162/tacl_a_00342Transactions of the Association for Computational Linguistics. Alon Talmor, Yanai Elazar, Yoav Goldberg, and Jonathan Berant. 2020. oLMpics-on what language model pre-training captures. Transactions of the As- sociation for Computational Linguistics. Elizabeth Closs Traugott, Richard B Dasher, 10.1017/CBO9780511486500Regularity in Semantic Change. Cambridge Studies in Linguistics. Cambridge University PressElizabeth Closs Traugott and Richard B. Dasher. 2001. Regularity in Semantic Change. Cambridge Studies in Linguistics. Cambridge University Press. BERT has a mouth, and it must speak: BERT as a Markov random field language model. Alex Wang, Kyunghyun Cho, 10.18653/v1/W19-2304Proceedings of the Workshop on Methods for Optimizing and Evaluating Neural Language Generation. the Workshop on Methods for Optimizing and Evaluating Neural Language GenerationMinneapolis, MinnesotaAssociation for Computational LinguisticsAlex Wang and Kyunghyun Cho. 2019. BERT has a mouth, and it must speak: BERT as a Markov random field language model. In Proceedings of the Workshop on Methods for Optimizing and Eval- uating Neural Language Generation, pages 30-36, Minneapolis, Minnesota. Association for Computa- tional Linguistics. Huggingface's transformers: State-of-the-art natural language processing. Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, Rémi Louf, abs/1910.03771CoRRMorgan Funtowicz, and Jamie BrewThomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pier- ric Cistac, Tim Rault, Rémi Louf, Morgan Funtow- icz, and Jamie Brew. 2019. Huggingface's trans- formers: State-of-the-art natural language process- ing. CoRR, abs/1910.03771. Adaptive semiparametric language models. Dani Yogatama, Lingpeng Cyprien De Masson D&apos;autume, Kong, 10.1162/tacl_a_00371Transactions of the Association for Computational Linguistics. 9Dani Yogatama, Cyprien de Masson d'Autume, and Lingpeng Kong. 2021. Adaptive semiparametric language models. Transactions of the Association for Computational Linguistics, 9:362-373. Bertscore: Evaluating text generation with bert. Tianyi Zhang, * , Varsha Kishore, * , Felix Wu, * , Kilian Q Weinberger, Yoav Artzi, International Conference on Learning Representations. Tianyi Zhang*, Varsha Kishore*, Felix Wu*, Kilian Q. Weinberger, and Yoav Artzi. 2020. Bertscore: Eval- uating text generation with bert. In International Conference on Learning Representations. On the impact of temporal concept drift on model explanations. Zhixue Zhao, George Chrysostomou, Kalina Bontcheva, Nikolaos Aletras, 10.48550/ARXIV.2210.09197Findings of the Conference on Empirical Methods in Natural Language Processing. Zhixue Zhao, George Chrysostomou, Kalina Bontcheva, and Nikolaos Aletras. 2022. On the impact of temporal concept drift on model expla- nations. In Findings of the Conference on Empirical Methods in Natural Language Processing. Factual probing is [MASK]: Learning vs. learning to recall. Zexuan Zhong, Dan Friedman, Danqi Chen, 10.18653/v1/2021.naacl-main.398Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies. the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language TechnologiesOnline. Association for Computational LinguisticsZexuan Zhong, Dan Friedman, and Danqi Chen. 2021. Factual probing is [MASK]: Learning vs. learning to recall. In Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Tech- nologies, pages 5017-5033, Online. Association for Computational Linguistics.
{'fraction_non_alphanumeric': 0.044518385014711094, 'fraction_numerical': 0.029345163824575998, 'mean_word_length': 4.953595011005135, 'pattern_counts': {'":': 0, '<': 5, '<?xml version=': 0, '>': 6, 'https://': 3, 'lorem ipsum': 0, 'www.': 1, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 4, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'Temporal concept drift refers to the problem of data changing over time. In NLP, that would entail that language (e.g. new expressions, meaning shifts) and factual knowledge (e.g. new concepts, updated facts) evolve over time. Focusing on the latter, we benchmark 11 pretrained masked language models (MLMs) on a series of tests designed to evaluate the effect of temporal concept drift, as it is crucial that widely used language models remain upto-date with the ever-evolving factual updates of the real world. Specifically, we provide a holistic framework that (1) dynamically creates temporal test sets of any time granularity (e.g. month, quarter, year) of factual data from Wikidata, (2) constructs fine-grained splits of tests (e.g. updated, new, unchanged facts) to ensure comprehensive analysis, and (3) evaluates MLMs in three distinct ways (singletoken probing, multi-token generation, MLM scoring). In contrast to prior work, our framework aims to unveil how robust an MLM is over time and thus to provide a signal in case it has become outdated, by leveraging multiple views of evaluation.', 'arxivid': '2302.12297', 'author': ['Katerina Margatina [email protected] \nAWS AI Labs\nUniversity of Sheffield\n\n', 'Shuai Wang \nAWS AI Labs\nUniversity of Sheffield\n\n', 'Yogarshi Vyas \nAWS AI Labs\nUniversity of Sheffield\n\n', '† Neha \nAWS AI Labs\nUniversity of Sheffield\n\n', 'Anna John \nAWS AI Labs\nUniversity of Sheffield\n\n', 'Yassine Benajiba \nAWS AI Labs\nUniversity of Sheffield\n\n', 'Miguel Ballesteros \nAWS AI Labs\nUniversity of Sheffield\n\n'], 'authoraffiliation': ['AWS AI Labs\nUniversity of Sheffield\n', 'AWS AI Labs\nUniversity of Sheffield\n', 'AWS AI Labs\nUniversity of Sheffield\n', 'AWS AI Labs\nUniversity of Sheffield\n', 'AWS AI Labs\nUniversity of Sheffield\n', 'AWS AI Labs\nUniversity of Sheffield\n', 'AWS AI Labs\nUniversity of Sheffield\n'], 'corpusid': 257206033, 'doi': '10.48550/arxiv.2302.12297', 'github_urls': ['https://github.com/amazon-science/temporal-robustness'], 'n_tokens_mistral': 18572, 'n_tokens_neox': 16068, 'n_words': 9382, 'pdfsha': 'aad8b1ca56aef4a7512789102ce0cc3fc8b064e4', 'pdfurls': ['https://export.arxiv.org/pdf/2302.12297v1.pdf'], 'title': ['Dynamic Benchmarking of Masked Language Models on Temporal Concept Drift with Multiple Views', 'Dynamic Benchmarking of Masked Language Models on Temporal Concept Drift with Multiple Views'], 'venue': []}
arxiv
Molecular sunscreen: water protects pyrrole from radiation damage Short title: "Water protects molecules from radiation damage" (Dated: 2020-10-02) Melby Johny Center for Free-Electron Laser Science Deutsches Elektronen-Synchrotron DESY Notkestrasse 8522607HamburgGermany Constant A Schouder Department of Physics Aarhus University Langelandsgade 1408000Aarhus CDenmark Ahmed Al-Refaie Center for Free-Electron Laser Science Deutsches Elektronen-Synchrotron DESY Notkestrasse 8522607HamburgGermany Lanhai He Center for Free-Electron Laser Science Deutsches Elektronen-Synchrotron DESY Notkestrasse 8522607HamburgGermany Joss Wiese Center for Free-Electron Laser Science Deutsches Elektronen-Synchrotron DESY Notkestrasse 8522607HamburgGermany Department of Chemistry Universität Hamburg Martin-Luther-King-Platz 620146HamburgGermany Center for Ultrafast Imaging Universität Hamburg Luruper Chaussee 14922761HamburgGermany Henrik Stapelfeldt Department of Chemistry Aarhus University Langelandsgade 1408000Aarhus CDenmark Sebastian Trippel Center for Free-Electron Laser Science Deutsches Elektronen-Synchrotron DESY Notkestrasse 8522607HamburgGermany Center for Ultrafast Imaging Universität Hamburg Luruper Chaussee 14922761HamburgGermany Jochen Küpper Center for Free-Electron Laser Science Deutsches Elektronen-Synchrotron DESY Notkestrasse 8522607HamburgGermany Center for Ultrafast Imaging Universität Hamburg Luruper Chaussee 14922761HamburgGermany Department of Physics Universität Hamburg Luruper Chaussee 14922761HamburgGermany Molecular sunscreen: water protects pyrrole from radiation damage Short title: "Water protects molecules from radiation damage" (Dated: 2020-10-02) Radiation-induced damage of biological matter is an ubiquitous problem in nature. The influence of the hydration environment is widely discussed, but its exact role remains elusive. We present the experimental observation of a hydrogen-bonded water molecule acting as a radiation protection agent for ionized pyrrole, a prototypical aromatic biomolecule. Pure samples of pyrrole and pyrrole(H2O) were outer-valence ionized and the subsequent damage and relaxation processes were studied. Bare pyrrole fragmented through the breaking of the C-C or N-C covalent bonds. However, for pyrrole(H2O), we observed a strong protection of the pyrrole ring through the dissociative release of neutral water or by transferring an electron or proton across the hydrogen bond. Furthermore, for pyrrole(H2O) a smaller probability for double ionization was observed. Overall, a single water molecule strongly reduces the fragmentation probability and thus the persistent radiation damage of ionized pyrrole.One sentence summary: "A single water molecule strongly reduces the fragmentation probability and thus the persistent radiation damage of an ionized prototypical biomolecular chromophore." arXiv:2010.00453v1 [physics.chem-ph] 1 Oct 2020 INTRODUCTION The damage of biological matter upon the interaction with UV radiation [1] or ionizing radiation [2], such as x-rays [2,3], γ-rays [4], and α- [5], or other fast charged particles [3,6,7] is a major environmental impact on living organisms [1,2]. For instance, inner-shell-, innervalence-, or outer-valence-ionized states can relax in various pathways that form cationic species, which can result in molecular fragmentation [3,8,9]. In addition, recent studies show that one highly relevant mechanism of DNA strand breaks is via autoionization or excitation caused by low-energy secondary electrons [3,6,7,10]. Regarding the radiation damage of molecules, ionization and excitation are similar: Vacancies are created in the occupied molecular orbitals in both cases, which can lead to corresponding bond breaking. In the case of ionization, the electron is directly transferred to the continuum, leaving the molecule, while excitation may result in the population of dissociative excited states [11,12]. Typical sources for single ionization of biological matter in aqueous environments are deep UV ionization or the interaction with radicals, slow electrons, or ions [2,3,12,13]. While deep UV radiation is efficiently blocked by the earth's atmosphere [14] it is omnipresent in outer space [15]; harder radiation penetrates the atmosphere. Larger molecular assemblies such as clusters, droplets, and even large molecules like proteins in their natural solvation environment are known to allow for addi-tional relaxation pathways due to intermolecular interactions [8,[16][17][18][19][20][21][22][23][24]. These pathways may lead to a protection of the molecule especially if the biomolecule is directly affected by the radiation [1]. On the other hand, secondary species originated from the ionization of surrounding solvent molecules can open up new pathways that lead to biomolecular destruction. Hydrogen-bonded solute-solvent complexes allow for quantitative investigations of these effects [25][26][27][28]. One of the important electronic relaxation channels of such dimers after x-ray ionization, electron-impact ionization, or α-particle irradiation was ascribed to intermolecular Coulombic decay (ICD) [8,9,[25][26][27]29]. ICD results in the formation of mainly charge-separated di-cationic complexes which undergo fragmentation via Coulomb repulsion. A competing ultrafast relaxation channel of hydrogen-bonded complexes after inner-shell ionization, which may protect biomolecules, is intermolecular electron-or proton-transfer-mediated charge separation [22,24,27]. This was observed following x-ray ionization of the water dimer [28] and liquid water [30,31]. A key ingredient for the indirect destruction pathways of biological matter is the radiolysis of water [13,32,33], which is probable because ∼3/4 of the volume of the cell comprises of aqueous environment [3,13]. In this case, reactive cations, radicals, anions, or aqueous electrons are produced inside the water environment, which can trigger biomolecular fragmentation [2,[32][33][34][35][36]. In this context, it is still under discussion whether the hydration environment inhibits or enhances radiation-induced biological FIG. 1. Schematic representation of the ionization scheme and radiation-protection mechanism in pyrrole(H2O). Single ionization of the cluster is followed by its dissociation, with the leaving water molecule allowing the aromatic ring to stay intact without further fragmentation. damage [4,13,[36][37][38][39][40][41][42]. Pyrrole, a heterocyclic aromatic molecule, is a UVabsorbing chromophore, e. g., in hemes and chlorophylls [43]. Pyrrole is also a subunit of indole, 3methylindole, and tryptophan, which are of great relevance as the principal UV-absorbers of proteins [44,45]. The photophysical and photochemical properties of indole and pyrrole are sensitive to the hydration environment [46][47][48][49][50]: upon UV absorption these chromophores indirectly populate an excited 1 πσ * state, which is repulsive along the N-H-stretching coordinate [44][45][46]. This triggers an ultrafast internal-conversion process to the ground state, essential for the photostability of proteins. The pyrrole(H 2 O) cluster has a well-defined structure with a hydrogen bond from the N-H site to the water [51]. This reflects the strongest interaction between pyrrole and surrounding H 2 O in aqueous solution. H-elimination dynamics from the N-H site of pyrrole, mediated by the electronic excitation of the 1 πσ * state [44,[52][53][54] or by vibrationally mediated photodissociation [55], was studied by time-resolved photoion and photoelectron spectroscopy. Theoretical calculations for electronically excited pyrrole(H 2 O) clusters predicted electron transfer across the hydrogen bond without photodissociation of the pyrrole moiety [48,56]. Direct ionization of pyrrole [57] and pyrrole(H 2 O) [58] led to molecular fragmentation. Here, we experimentally investigated the damage incurred in singly and doubly ionized pyrrole molecules and the effect of solvation by comparing the fragmentation pathways of bare pyrrole and microsolvated pyrrole(H 2 O) heterodimers using pure samples of either species. This scheme enables the systematic investigation of the role of water solvation on the photophysics of pyrrole and hence for hydrated biomolecules in general. A schematic representation of our ionization strategy is represented in Figure 1. We mimicked radiation damage through outer valence ionization. Pyrrole and pyrrole(H 2 O) were site-specifically ionized, see Meth- ods, through the removal of electrons from the HOMO or HOMO−1 orbitals, which are localized on the aromatic ring. This was achieved by strong-field ionization using 800 nm laser pulses with a peak intensity of ∼1 × 10 14 W/cm 2 and a pulse duration of 30 fs. Valence ionization of bare pyrrole resulted in extensive fragmentation. On the other hand, for singly ionized pyrrole(H 2 O) we mainly observed breaking of the hydrogen bond and a water molecule leaving an intact pyrrole ring. In this case, breaking of the actual biomolecule is strongly suppressed. RESULTS Our comparison of the fragmentation dynamics of bare and microsolvated pyrrole built on the production of very pure molecular beams of pyrrole and pyrrole(H 2 O), respectively; see Methods for details. The molecular beam had, in the case of pyrrole, a purity of ∼95 %, with a contamination of ∼5 % given by pyrrole dimer. The purity of the pyrrole(H 2 O) beam was ∼99 % with the major contamination by water dimer [59]. Figure 2 shows the time-of-flight mass spectrum (TOF-MS) and corresponding velocity-map images (VMIs) of all ions resulting from strong-field ionization of pyrrole. All data were recorded simultaneously using a Timepix3 camera [60,61]. Rings in the VMIs occur for two-body Coulomb explosion, i. e., a charge-repulsion-driven fast breakup into two positively charged fragments. These fragmentation channels obey momentum conservation in the recoil frame. Low kinetic energy (KE) features correspond to charged intact parent ions or ions created from dissociative single ionization. The data shows no signatures of three-body breakup beyond hydrogen atom/proton loss. The most prominent feature in the TOF-MS is the narrow peak at m/q = 67 u/e, assigned to pyrrole + , on top of a broader pedestal. The peak corresponds to the sharp central dot in the corresponding VMI. The pedestal correlates with the rings in the VMI which are assigned to pyrrole + from Coulomb explosion of the pyrrole dimer. The TOF-MS peak at m/q = 33.5 u/e and the central dot in the corresponding VMI are assigned to the doubly ionized pyrrole. Its pedestal in the TOF-MS and the corresponding rings in the VMI are attributed to pyrrole 2+ from Coulomb explosion of multiply-ionized pyrrole dimer. In both cases 95 % of the signal strengths are in the central peaks, i. e., originating from the monomer. The signals in the mass-to-charge regions m/q = 24 . . . 30 u/e and m/q = 35 . . . 44 u/e correspond to fragments from the breakup of the pyrrole ring. Some possible ionic products are labeled in Figure 2, in line with the mass peak assignment after electron impact-and photoionization of pyrrole [57,62]. A clear assignment of the various individual peaks observed in the two mass regions to specific fragments is not possible due to overlapping signals and ambiguities in the construction of specific mass to charge ratios out of possible feasible fragments. The situation is compounded by the fact that hydrogen loss off fragments was present. Proton loss after double ionization can be ruled out due to the lack of correlations between the detected protons and the fragments observed in the two mass regions. The small proton peak in the spectrum is therefore attributed to the initial charge states > 2. Intense central peaks in the VMIs correspond to fragments from dissociative ionization of singly charged pyrrole. The rings in these VMIs, contributing ∼30 % of the total ion count, correspond to fragments from Coulomb explosion of doubly charged pyrrole. The Coulomb explosion signals of the two mass regions are correlated, as confirmed by kinetic-energy selected coincidence maps and momentum conservation. Figure 3 shows the TOF-MS and VMIs of all ions resulting from strong-field ionization of purified pyrrole(H 2 O). Again all data were recorded simultaneously using the time-stamping detector. All VMIs exhibit a central low-KE part due to single ionization as well as sharp or diffuse higher KE signals. The peak at m/q = 85 u/e, with a central dot in the VMI, corresponds to the pyrrole(H 2 O) + parent ion. The strongest peak in the TOF-MS is again at m/q = 67 u/e, the pyrrole-monomer cation, which resulted from the dissociation of the hydrogen bond in singly ionized pyrrole(H 2 O). This is confirmed by its broader KE distribution in the VMI owing to recoil from the momentum conservation with the leaving neutral water molecule. The peaks at m/q = 66 u/e and m/q = 19 u/e correspond to C 4 H 4 N + and H 3 O + , respectively. Both fragments exhibit sharp rings in their VMIs, with corre- As for pyrrole, fragments within the regions m/q = 24 . . . 30 u/e and m/q = 35 . . . 44 u/e were detected due to the breakup of the aromatic ring. However, they show broad structureless distributions in the VMIs which are not correlated with each other. The high KE ions in these regions originate from pyrrole(H 2 O) 2+ after three-body fragmentation processes. These channels always involve H 3 O + as a second ionic partner, as well as a third neutral aromatic fragment. The small peak at m/q = 36 u/e is attributed to a singly ionized water dimer ((H 2 O) 2 ). DISCUSSION Overall, single and double valence ionization of pyrrole led to a significant breakup of the aromatic ring. Single ionization caused fragmentation of the pyrrole moiety through various dissociative pathways which resulted in low kinetic energy ions. Double ionization caused fragmentation of the aromatic ring driven by Coulomb explosion. Surprisingly, the scenario is very different for pyrrole(H 2 O). Despite of the localized ionization with the removal of electrons from pyrrole's same π orbitals, different relaxation pathways emerged through the hydrogenbonded water molecule. For example, in the case of singly ionized pyrrole(H 2 O), we observed the dissociation of the hydrogen bond, i. e., the loss of neutral water, which protected the pyrrole ring from fragmentation. Furthermore, after double ionization, additional Coulomb explosion channels appeared for pyrrole(H 2 O), such as the H 3 O + channel with its counter ion C 4 H 4 N + . Both fragments showed sharp rings in the corresponding VMIs, thus leaving the pyrrole ring intact with only a proton lost. Additionally, two body Coulomb explosion channels of pyrrole(H 2 O) in the regions m/q = 24 . . . 30 u/e and m/q = 35 . . . 44 u/e are absent and the uncorrelated signal is much weaker than the strong signals observed for the pyrrole monomer. In total, the fragmentation pathways were strongly influenced, i. e., pyrrole fragmentation was strongly reduced, by a single water molecule attached to pyrrole. In order to unravel the influence of the microsolvation on the radiation induced damage and to quantify the degree of protection for ionized pyrrole in the presence of a single water molecule, we normalized the ion yields for the pyrrole and pyrrole(H 2 O) species with respect to each other. We compared the laser intensity-dependent shape of the low-intensity single-ionization ion yield for both species as described in the Methods. Furthermore, the single, double, and higher-order ionization channels were separated to provide a direct comparison of the fragmentation yields for both species as a function of the initial charge state. In addition, the fragmentation channels were classified into ring-fragmenting and ringprotecting channels. The momentum maps for the specific mass-to-charge regions were taken into account in order to separate single, double, and higher-order ionization channels. Almost all VMIs shown in Figure 2 and Figure 3 had contributions from ions with low as well as with high kinetic energy. We attribute the low-KE ions to single ionization and the high-KE ions to double or higher-order ionization, respectively. Gating on the momenta with p < 30 u · km/s resulted in a normalized mass spectrum (NORMS) for single-ionization channels whereas gating on the momenta with p > 30 u · km/s resulted in a NORMS for the higherionization channels. The resulting normalized gated timeof-flight mass spectra for pyrrole and pyrrole(H 2 O) are shown in Figure 4 for m/q = 0 . . . 60 u/e, which contains all ring-breaking fragments. The upper panel corresponds to the single-ionization channels (z = +1), whereas the lower panel corresponds to double (z = +2) or higher (z > +2) ionization-channels. Both species showed similar fragmentation products, especially in the mass-to-charge regions m/q = 24 . . . The contributions from triple ionization were statistically estimated to less than 5 % and are thus negligible, see Methods. Furthermore, the NORMS peaks in the region m/q = 1 . . . 14 u/e originate from charge states with z > +2, confirmed by a covariance analysis. In the case of higher-order ionization, a direct comparison of the ion yield of pyrrole and pyrrole(H 2 O) from the NORMS is not feasible due to the complex fragmentation processes and overlapping fragmentation channels. To estimate the extent of fragmentation protection after single-and double ionization of pyrrole in a microsolvated environment the observed fragmentation channels were classified into ring-breakup and ring-protection channels. Based on this, the ring-fragmentation probability is defined as P = N b /( N b + N i ) with the number of fragments where the ring is broken (intact) specified by N b (N i ). First, we considered ring-breakup and ring-protection channels following single ionization of pyrrole and pyrrole(H 2 O). For pyrrole, the parent ion was considered as an intact channel. nant ring-protection channel is the dissociation of the hydrogen bond, i. e., the loss of neutral water. Furthermore, the dissociative single ionization processes resulting in low KE ions of H 2 O + , H 3 O + , and C 4 H 4 N + , prevent the aromatic ring from fragmentation. All other low KE ions in the mass-region m/q = 15 . . . 60 u/e are considered as ionic products that originated from the fragmentation of the aromatic ring. The ring-fragmentation probabilities for pyrrole and pyrrole(H 2 O) after single ionization were then determined by counting the ions in the momentummap images with a cut on the low KE part in the specific mass-to-charge regions. The projection of ions from higher charge states (z = +2) into the center of the momentummap images is estimated statistically, see Methods, and this contribution was subtracted. We estimated the ringfragmentation probability individually for pyrrole and pyrrole(H 2 O) to 48 % and 7 %, respectively, see also Figure 5. Overall, the ring-fragmentation probability of pyrrole following single ionization is reduced by a factor of ∼7 in pyrrole(H 2 O) compared to pyrrole. We then classified the channels following double ionization of pyrrole and pyrrole(H 2 O). For pyrrole, the only ring-protecting fragmentation channel was the formation of the doubly charged parent ion, C 4 H 5 N 2+ , marked by the asterisks in the upper panel of Figure 4. All other channels broke up the aromatic ring. The dominant ring protection channel for the pyrrole(H 2 O) 2+ was the intermolecular proton transfer from the N-H site of the pyrrole moiety to the water moiety, producing H 3 O + and C 4 H 4 N + . A second channel was the electron transfer process across the hydrogen bond, which leads to the formation of C 4 H 5 N + and H 2 O + . All other double ionization channels correspond to the breaking of the aromatic ring. The ring-fragmentation probability for pyrrole and pyrrole(H 2 O) after double ionization were determined by counting ions in the high KE part of the momentum-map images while taking into account that two ions might have been produced after double ionization, leading to two hits in the corresponding momentum-maps for a single fragmentation event. The finite detection efficiency of 0.5 for each ion was taken into account. This led to a similar ring-fragmentation probability after double ionization for pyrrole(H 2 O), P = 0.80 ± 0.04, as for pyrrole, P = 0.79 ± 0.04. However, we observed a significant reduction in the total absolute ion yield of the microsolvated system for double ionization. This was indicated qualitatively through the comparison of the ion signals in the NORMS in Figure 4. To quantify this reduction we counted ions in the normalized momentum maps of the specific channels originating from double ionization of pyrrole and pyrrole(H 2 O). The direct comparison revealed a reduction of the total ion yield of pyrrole(H 2 O) by a factor of 2.6 ± 0.2. Thus, a single water molecule attached to the aromatic molecule pyrrole reduces its probability of being doubly ionized, with a subsequent high probability of ring breaking, 2.6fold. CONCLUSION We demonstrated that a single water molecule strongly protected the pyrrole ring from fragmentation by ionization. For single ionization, microsolvation led to a strongly reduced fragmentation of the aromatic ring. Moreover, the solvation also significantly reduced the probability of double ionization and the corresponding probabilities for ring breaking. These quantitative studies were enabled by our ability to provide pure samples of the bare molecule and the microsolvated complex, as well as by using a novel time-stamping VMI detector. Singly ionized pyrrole underwent radiation-induced damage through the breaking of, typically two, C-C or N-C bonds. Outstandingly, the dissociation mechanisms after single ionization of pyrrole(H 2 O), through breaking of the intermolecular bond or by transferring an electron or proton across this hydrogen bond, strongly reduces the ring breaking probability by a remarkable factor of 7. After double ionization, similar ring-fragmentation probabilities were observed for pyrrole and pyrrole(H 2 O). For the microsolvated system, intermolecular proton-and electron transfer processes occurring across the hydrogen bond increase the redistribution of charges, initially created in the pyrrole ring, to the water molecule. Nevertheless, we observed a reduction of 2.6 in the double-ionization probability for pyrrole(H 2 O) compared to pyrrole, demonstrating a reduced double-ionization cross section and the corresponding protection of the microsolvated system. Our experiments employed strong-field ionization for the ionization, typically removing electrons from the HOMO or HOMO-1 orbitals of pyrrole, which are also localized on the pyrrole moiety in the microsolvated systems. This mimics the ionization by neighboring molecules in aqueous systems, such as cells, as well as electronic excitation through UV radiation. Thus, our results provide a test case of how an aqueous microsolvation environment can strongly reduce the radiation damage of biological molecules induced by UV radiation as well as by secondary effects of ionizing radiation where single outer-valence ionization is the scenario. Furthermore, the doubly ionized systems in our experiment resemble to a large extent the fate of a molecule after Auger decay processes subsequent to core-shell ionization [8,9]. Biomolecules and proteins in nature are actively solvated by the surrounding water molecules, which allow for efficient charge redistribution to the solvent environment through electron-and proton transfer pathways quantified here. In aqueous solution, the loss of the attached, neutral, ionized, or protonated, water could easily be repaired by the many solvent molecules around. Our analysis of the protection in pyrrole(H 2 O) provides a quantitative analysis of radiation protection and serves as the basis for further investigations of the existing uncertainties [4,13,[36][37][38][39][40][41][42] regarding the role of the solvent environment on the radiation damage of biomolecules. MATERIALS AND METHODS Experimental Setup Details of the experimental setup were described elsewhere [63]. A pulsed valve was operated at 100 Hz to supersonically expand a few millibars of pyrrole (Sigma Aldrich, > 98 %) and traces of water in ∼90 bar of helium into vacuum. The resulting molecular beam contained atomic helium, individual pyrrole and water molecules, and various aggregates thereof. The electric deflector was used to create pure samples of pyrrole(H 2 O) [58,64]. Pyrrole and pyrrole(H 2 O) were strong-field ionized by 800 nm laser pulses with linear polarization, a duration of ∼30 fs, focused to ∅ ≈ 35 µm (full width at half maximum intensity) with a peak intensity of ∼1 × 10 14 W/cm 2 . The ions generated were extracted perpendicular to the molecular beam and laser propagation directions using a velocitymap-imaging spectrometer (VMIS). All ions were detected using a position-and time-sensitive detector consisting of a micro-channel plate (MCP) in combination with a fast phosphor screen (P46). A visible-light-sensitive Timepix3 detector [60,65] in an event-driven mode recorded all signals, which were stored and centroided using our homebuilt pymepix software [61]. Normalization of the mass spectra The normalization of the TOF-MS was necessary due to the different densities of the two species, pyrrole and pyrrole(H 2 O), in the molecular beam. Due to the very similar first ionization energies (E i ) of pyrrole and pyrrole(H 2 O) the ionization probabilities for both species are also very similar. The calculated (HF/MP2-aug-cc-pVTZ using GAMESS-US) first vertical E i of pyrrole and pyrrole(H 2 O) are 8.59 eV and 8.15 eV, respectively. To quantify the relative ionization probability experimentally [66,67] the ion yield as a function of the laser peak intensity was measured for the single ionization channel for pyrrole, pyrrole(H 2 O), and water, see Figure 6. Here, the ion yield from each pure species is normalized to its value for the highest laser intensity. In the low-intensity region, 1-8 × 10 13 W/cm 2 , for pyrrole only the parent ion was observed. A saturation intensity for the parent ion signal of ∼6 × 10 13 W/cm 2 was obtained from the measured low-intensity ion-yield curve. For pyrrole(H 2 O) we summed the signals for parent ion and pyrrole + [58], which yielded a saturation intensity for single ionization of ∼4.5 × 10 13 W/cm 2 . Based on the similar saturation intensities and the very comparable intensity dependence of the ionization yields, Figure 6, we normalized the TOF-MS of pyrrole and pyrrole(H 2 O) using a normalization factor of 1.94 ± 0.1. For the ionization of water, we obtained a very different intensity dependence and saturation intensity, which was The molecular orbitals calculated (GAMESS-US, HF/MP2-aug-cc-pVTZ) for the geometry-optimized ground state structure of pyrrole(H 2 O) are shown in Figure 7. The electron densities of HOMO to HOMO-3 are localized on the aromatic ring. The highest-energy bound molecular orbital with significant density on the water moiety is HOMO-4. This orbital has an energy that is 6.5 eV lower than the HOMO. Therefore, under the applied laser intensities ionization from this orbital can be neglected [69] and localized ionization of pyrrole(H 2 O) at the pyrrole moiety can be safely assumed. Triple-ionization contributions To estimate the contributions of individual ions over a given mass-to-charge range from the measured 2Dprojection of the 3D momenta, we made specific cuts in these experimental momentum maps in order to isolate contributions from single-, double-, and triple-ionization processes. The measured momentum map for m/q = 35 . . . 45 u/e is shown with Figure 8 as an example. Circles indicate corresponding cuts in the 2D-projection of the 3D-momentum sphere formed from each ionization process: The white circle with a radius of p r = 60 u · km/s represents the edge of the momentum for dissociative single ionization, the green circle with p r = 140 u · km/s corresponds to the maximum momentum of ions from Coulomb explosion following the double ionization, and the red circle with p r = √ 2 * 140 ≈ 200 u · km/s is the maximum momentum of ions from triple ionization, assuming a two-body fragmentation into a singly-charged and a doubly-charged ion. The total ion count corresponding to the disks defined by these specific radii and the signal in the two outer rings are provided in Table I. Areas of the specific disks and rings are also given. Ion counts inside the outer ring, 140 < p r < 200, correspond to triple ionization without contribution from single and double ionization. However, the middle ring, 60 < p r < 140, represents the double ionization channels and it has contributions from triple ionization; similarly, the innermost disk, 0 < p r < 60, representing single ionization has contributions from ions originating in double and triple ionization. The corrected total number of ions from single, double, and triple ionization are provided in Table II assuming a flat ion distribution in the inner part of the corresponding rings and disks. The relative contribution of the ion yield from the triple ionization process to the total ion yield in the given mass-to-charge region is < 5 %, i. e., negligible. radius pr ion counts in disk ion counts in ring area of disk/π area of ring/ π 60 167522 167522 3600 3600 140 225860 58338 19600 16000 200 229120 3260 40000 20400 FIG. 2 . 2Time-of-flight mass spectrum and corresponding velocity-map images of the ions generated by strong-field ionization of pyrrole. The structure of pyrrole is given on the right of the lower panel. The colormap and velocity scale holds for all velocity map images. FIG. 3 . 3Time-of-flight mass spectrum of pyrrole(H2O) with the velocity-map images of the ions, which were recorded simultaneously after strong-field ionization. Also shown are the structure of the dimer, sum formulas of all fragments, and the velocity ranges and colormap for all velocity-map images. lated ions that obey momentum conservation, demonstrating a two-body Coulomb explosion break-up channel of pyrrole(H 2 O) including a proton transfer from pyrrole to water. A weaker H 2 O + channel, ∼3/4 of which is correlated to pyrrole(H 2 O) whereas the remaining signal can be attributed to the water dimer, shows the direct breakup of pyrrole(H 2 O) 2+ across the hydrogen bond. FIG. 4 . 430 u/e and m/q = 35 . . . 44 u/e, which are the channels arising Comparison of the normalized TOF-MS of pyrrole (red) and pyrrole(H2O) (blue). The number of charges created on the system after strong-field ionization is denoted by z. The upper panel corresponds to an initial charge state z = +1. The peak marked with the asterisk corresponds to intact C4H5N 2+ from double ionization of pyrrole. The lower section is for the charge states z > +1. Signals for m/q = 2 . . . 14 u/e in the lower section originated from initial charge states with z > +2.from the breaking of the C-C and N-C covalent bonds of the pyrrole ring. However, they vastly differed in the specific ion yield and we observed a significant reduction of fragments arising from ring-opening in pyrrole(H 2 O) as compared to pyrrole for both cases of single and multiple ionization. For both, pyrrole and pyrrole(H 2 O), the contributions in the NORMS for z ≥ +2 in the region m/q = 15 . . . 60 u/e were dominated by double ionization. FIG. 5 . 5For single ionization of pyrrole(H 2 O), in addition to the parent ion, the domi-Schematic representation of the ring protection and breaking probabilities for pyrrole and pyrrole(H2O) after single ionization, showing the very strongly reduced fragmentation probability of pyrrole in the cluster. For simplicity, only one of the observed fragments is shown for each channel. The width of the arrows (purple) and the numbers written above it represent the total ring-protection and breaking probabilities for pyrrole + and pyrrole(H2O) + . FIG. 8 . 8The momentum map for all ions detected within a mass-to-charge region m/q = 35 . . . 45 u/e is shown. Marked circles with specific radii in the momenta map represent edges for single, double, and triple ionization, respectively. FIG. 7. Molecular orbital picture of HOMO, HOMO-1, and HOMO-4 for the geometry optimized ground state structure of the pyrrole(H2O). determined as ∼1.4×10 14 W/cm 2 . This is consistent with the larger E i of 12.62 eV [68] and further demonstrates the similarities in the ionization cross sections of pyrrole and pyrrole(H 2 O).HOMO HOMO-1 HOMO-4 Highest occupied molecular orbitals of pyrrole(H2O) TABLE I . ITotal ion counts as well as the areas of disks and rings within specific radii chosen in the momenta map. 167522 − 3.486 · 3600 − 0.1598 · 3600 = 154397 154397/3600 = 42.89 +2 (double) 58338 · (19600/16000) − 0.1598 · 19600 = 68332 68332/19600 = 3.486 TABLE II. Number of ion counts per area of each disk, and the estimated number of ions formed after the single, double, and triple ionization, respectively.charge state ion counts ion counts/area +1 (single) +3 (triple) 3260 · (40000/20400) = 6392 6392/40000 = 0.1598 AcknowledgmentsWe thank Jolijn Onvlee, Jonathan Correa, and David Pennicard for helpful discussions.Competing interestsThe authors declare that they have no competing interests.Data and materials availabilityAll data needed to interpret the conclusions in the paper are provided in the paper or the Methods section. Additional data related to this paper can be requested from the authors. . * Email, [email protected]* Email: [email protected]; Ultrafast excited-state dynamics in nucleic acids. C E Crespo-Hernández, B Cohen, P M Hare, B Kohler, 10.1021/cr0206770Chem. Rev. 1041977C. E. Crespo-Hernández, B. Cohen, P. M. Hare, and B. Kohler, Ultrafast excited-state dynamics in nucleic acids, Chem. Rev. 104, 1977 (2004). Biomolecular Action of Ionizing Radiation. S Lehnert, 10.1201/9781420011920Taylor & Francis, LondonS. Lehnert, Biomolecular Action of Ionizing Radiation (Taylor & Francis, London, 2007). Biomolecular damage induced by ionizing radiation: The direct and indirect effects of low-energy electrons on DNA. E Alizadeh, T M Orlando, L Sanche, 10.1146/annurev-physchem-040513-103605Annu. Rev. Phys. Chem. 66379E. Alizadeh, T. M. Orlando, and L. Sanche, Biomolecular damage induced by ionizing radiation: The direct and indirect effects of low-energy electrons on DNA, Annu. Rev. Phys. Chem. 66, 379 (2015). Dependence of the yield of strand breaks induced by γ-rays in DNA on the physical conditions of exposure: Water content and temperature. T Ito, S C Baker, C D Stickley, J G Peak, M J Peak, 10.1080/09553009314550391Int. J. Radiat. Biol. 63289T. Ito, S. C. Baker, C. D. Stickley, J. G. Peak, and M. J. Peak, Dependence of the yield of strand breaks induced by γ-rays in DNA on the physical conditions of exposure: Water content and temperature, Int. J. Radiat. Biol. 63, 289 (1993). . H.-K Kim, J Titze, M Schöffler, F Trinter, M Waitz, J Voigtsberger, H Sann, M Meckel, C Stuck, U Lenz, M Odenweller, N Neumann, S Schössler, K Ullmann-Pfleger, B Ulrich, R C Fraga, N Petridis, D Metz, A Jung, R Grisenti, A Czasch, O Jagutzki, L Schmidt, T Jahnke, H Schmidt-Böcking, R Dörner, 10.1073/pnas.1104382108PNAS. 10811821Enhanced production of low energy electrons by alpha particle impactH.-K. Kim, J. Titze, M. Schöffler, F. Trinter, M. Waitz, J. Voigtsberger, H. Sann, M. Meckel, C. Stuck, U. Lenz, M. Odenweller, N. Neumann, S. Schössler, K. Ullmann- Pfleger, B. Ulrich, R. C. Fraga, N. Petridis, D. Metz, A. Jung, R. Grisenti, A. Czasch, O. Jagutzki, L. Schmidt, T. Jahnke, H. Schmidt-Böcking, and R. Dörner, Enhanced production of low energy electrons by alpha particle im- pact, PNAS 108, 11821 (2011). Resonant formation of DNA strand breaks by low-energy (3 to 20 ev) electrons. B Boudaïffa, P Cloutier, D Hunting, M A Hues, L Sanche, 10.1126/science.287.5458.1658Science. 2871658B. Boudaïffa, P. Cloutier, D. Hunting, M. A. Hues, and L. Sanche, Resonant formation of DNA strand breaks by low-energy (3 to 20 ev) electrons, Science 287, 1658 (2000). Unexpected electrons. T D Märk, P Scheier, 10.1038/nphys1526Nat. Phys. 682T. D. Märk and P. Scheier, Unexpected electrons, Nat. Phys. 6, 82 (2010). Interatomic and intermolecular Coulombic decay: the coming of age story. T Jahnke, 10.1088/0953-4075/48/8/082001J. Phys. B. 4882001T. Jahnke, Interatomic and intermolecular Coulombic decay: the coming of age story, J. Phys. B 48, 082001 (2015). Giant intermolecular decay and fragmentation of clusters. L S Cederbaum, J Zobeley, F Tarantelli, 10.1103/PhysRevLett.79.4778Phys. Rev. Lett. 794778L. S. Cederbaum, J. Zobeley, and F. Tarantelli, Giant intermolecular decay and fragmentation of clusters, Phys. Rev. Lett. 79, 4778 (1997). DNA strand breaks induced by 0-4 ev electrons: The role of shape resonances. F Martin, P D Burrow, Z Cai, D Cloutier, Pand Hunting, L Sanche, 10.1103/PhysRevLett.93.068101Phys. Rev. Lett. 9368101F. Martin, P. D. Burrow, Z. Cai, D. Cloutier, Pand Hunt- ing, and L. Sanche, DNA strand breaks induced by 0-4 ev electrons: The role of shape resonances, Phys. Rev. Lett. 93, 068101 (2004). Charge and energy transfer dynamics in molecular systems. V May, O Kühn, Wiley-VCHV. May and O. Kühn, Charge and energy transfer dynam- ics in molecular systems (Wiley-VCH, 2004). Vacuum Ultraviolet Photoionization and Photodissociation of Molecules and Clusters. C Y Ng, 10.1142/1244World Scientific Publishing Co. Pte. LtdC. Y. Ng, Vacuum Ultraviolet Photoionization and Pho- todissociation of Molecules and Clusters (World Scientific Publishing Co. Pte. Ltd, 1991). . B C Garrett, D A Dixon, D M Camaioni, D M Chipman, M A Johnson, C D Jonah, G A Kimmel, J H Miller, T N Rescigno, P J Rossky, S S Xantheas, S D Colson, A H Laufer, D Ray, P F Barbara, D M Bartels, K H Becker, K H Bowen, S E Bradforth, I Carmichael, J V Coe, L R Corrales, J P Cowin, M Dupuis, K B , J. A. Franz, M. S. Gutowski, K. D. Jordan, B. D. Kay, J. A. LaVerne, S. V. Lymar, T. E. Madey, C. W. McCurdy, D. Meisel, S. Mukamel, A. R. Nilsson, T. M. Orlando, N. G. Petrik, S. M. Pimblott, J. R. Rustad, G. K. Schenter, S. J. Singer, A. Tokmakoff, L.-SB. C. Garrett, D. A. Dixon, D. M. Camaioni, D. M. Chipman, M. A. Johnson, C. D. Jonah, G. A. Kimmel, J. H. Miller, T. N. Rescigno, P. J. Rossky, S. S. Xantheas, S. D. Colson, A. H. Laufer, D. Ray, P. F. Barbara, D. M. Bartels, K. H. Becker, K. H. Bowen, S. E. Bradforth, I. Carmichael, J. V. Coe, L. R. Corrales, J. P. Cowin, M. Dupuis, K. B. Eisenthal, J. A. Franz, M. S. Gutowski, K. D. Jordan, B. D. Kay, J. A. LaVerne, S. V. Lymar, T. E. Madey, C. W. McCurdy, D. Meisel, S. Mukamel, A. R. Nilsson, T. M. Orlando, N. G. Petrik, S. M. Pimblott, J. R. Rustad, G. K. Schenter, S. J. Singer, A. Tokmakoff, L.-S. Role of water in electron-initiated processes and radical chemistry: Issues and scientific advances. T S Wang, Zwier, 10.1021/cr030453xChem. Rev. 105355Wang, , and T. S. Zwier, Role of water in electron-initiated processes and radical chemistry: Issues and scientific advances, Chem. Rev. 105, 355 (2005). Effects of clouds and stratospheric ozone depletion on ultraviolet radiation trends. D Lubin, E H Jensen, 10.1038/377710a0Nature. 377710D. Lubin and E. H. Jensen, Effects of clouds and strato- spheric ozone depletion on ultraviolet radiation trends, Nature 377, 710 (1995). M A Barstow, J B Holberg, 10.1017/CBO9780511536144Extreme Ultraviolet Astronomy. CambridgeCambridge University PressM. A. Barstow and J. B. Holberg, Extreme Ultraviolet As- tronomy (Cambridge University Press, Cambridge, 2003). Efficient deactivation of a model base pair via excited-state hydrogen transfer. T Schultz, E Samoylova, W Radloff, I V Hertel, A L Sobolewski, W Domcke, 10.1126/science.1104038Science. 3061765T. Schultz, E. Samoylova, W. Radloff, I. V. Hertel, A. L. Sobolewski, and W. Domcke, Efficient deactivation of a model base pair via excited-state hydrogen transfer., Science 306, 1765 (2004). Mapping GFP structure evolution during proton transfer with femtosecond raman spectroscopy. C Fang, R R Frontiera, R Tran, R A Mathies, 10.1038/nature08527Nature. 462200C. Fang, R. R. Frontiera, R. Tran, and R. A. Mathies, Mapping GFP structure evolution during proton transfer with femtosecond raman spectroscopy, Nature 462, 200 (2009). The nature of the hydrated excess proton in water. D Marx, M E Tuckerman, J Hutter, M Parrinello, 10.1038/17579Nature. 397601D. Marx, M. E. Tuckerman, J. Hutter, and M. Parrinello, The nature of the hydrated excess proton in water, Nature 397, 601 (1999). Ionization of dimethyluracil dimers leads to facile proton transfer in the absence of hydrogen bonds. A Golan, K B Bravaya, R Kudirka, O Kostko, S R Leone, A I Krylov, M Ahmed, 10.1038/nchem.1298Nat. Chem. 4323A. Golan, K. B. Bravaya, R. Kudirka, O. Kostko, S. R. Leone, A. I. Krylov, and M. Ahmed, Ionization of dimethy- luracil dimers leads to facile proton transfer in the absence of hydrogen bonds, Nat. Chem. 4, 323 (2012). Hydrogen bonds in excited state proton transfer. D A Horke, H M Watts, A D Smith, E Jager, E Springate, O Alexander, C Cacho, R T Chapman, R S Minns, 10.1103/PhysRevLett.117.163002Phys. Rev. Lett. 117163002D. A. Horke, H. M. Watts, A. D. Smith, E. Jager, E. Springate, O. Alexander, C. Cacho, R. T. Chapman, and R. S. Minns, Hydrogen bonds in excited state proton transfer, Phys. Rev. Lett. 117, 163002 (2016). Ultrafast relaxation of photoexcited superfluid He nanodroplets. M Mudrich, A C Laforge, A Ciavardini, P O&apos;keeffe, C Callegari, M Coreno, A Demidovich, M Devetta, M D Fraia, M Drabbels, P Finetti, O Gessner, C Grazioli, A Hernando, D M Neumark, Y Ovcharenko, P Piseri, O Plekan, K C Prince, R Richter, M P Ziemkiewicz, T Möller, J Eloranta, M Pi, M Barranco, F Stienkemeier, 10.1038/s41467-019-13681-6Nat. Commun. 11112M. Mudrich, A. C. LaForge, A. Ciavardini, P. O'Keeffe, C. Callegari, M. Coreno, A. Demidovich, M. Devetta, M. D. Fraia, M. Drabbels, P. Finetti, O. Gessner, C. Grazi- oli, A. Hernando, D. M. Neumark, Y. Ovcharenko, P. Piseri, O. Plekan, K. C. Prince, R. Richter, M. P. Ziemkiewicz, T. Möller, J. Eloranta, M. Pi, M. Barranco, and F. Stienkemeier, Ultrafast relaxation of photoex- cited superfluid He nanodroplets, Nat. Commun. 11, 112 (2020). Relaxation processes in aqueous systems upon x-ray ionization: Entanglement of electronic and nuclear dynamics. P Slavíček, N V Kryzhevoi, E F Aziz, B Winter, 10.1021/acs.jpclett.5b02665The Journal of Physical Chemistry Letters. 7234P. Slavíček, N. V. Kryzhevoi, E. F. Aziz, and B. Winter, Relaxation processes in aqueous systems upon x-ray ion- ization: Entanglement of electronic and nuclear dynamics, The Journal of Physical Chemistry Letters 7, 234 (2016). Relaxation mechanisms of UV-photoexcited DNA and RNA nucleobases. M Barbatti, A J A Aquino, J J Szymczak, D Nachtigallová, P Hobza, H Lischka, 10.1073/pnas.1014982107PNAS. 10721453M. Barbatti, A. J. A. Aquino, J. J. Szymczak, D. Nachti- gallová, P. Hobza, and H. Lischka, Relaxation mechanisms of UV-photoexcited DNA and RNA nucleobases, PNAS 107, 21453 (2010). Electronic decay in weakly bound heteroclusters: Energy transfer versus electron transfer. J Zobeley, R Santra, L S Cederbaum, 10.1063/1.1395555J. Chem. Phys. 1155076J. Zobeley, R. Santra, and L. S. Cederbaum, Electronic decay in weakly bound heteroclusters: Energy transfer versus electron transfer, J. Chem. Phys. 115, 5076 (2001). Experimental evidence for ultrafast intermolecular relaxation processes in hydrated biomolecules. X Ren, E Wang, A D Skitnevskaya, A B Trofimov, G Kirill, A Dorn, 10.1038/s41567-018-0214-9Nat. Phys. 791745X. Ren, E. Wang, A. D. Skitnevskaya, A. B. Trofimov, G. Kirill, and A. Dorn, Experimental evidence for ul- trafast intermolecular relaxation processes in hydrated biomolecules, Nat. Phys. 79, 1745 (2018). Ultrafast energy transfer between water molecules. T Jahnke, H Sann, T Havermeier, K Kreidi, C Stuck, M Meckel, M Schöffler, N Neumann, R Wallauer, S Voss, A Czasch, O Jagutzki, A Malakzadeh, F Afaneh, T Weber, H Schmidt-Böcking, R Dörner, 10.1038/nphys1498Nat. Phys. 6139T. Jahnke, H. Sann, T. Havermeier, K. Kreidi, C. Stuck, M. Meckel, M. Schöffler, N. Neumann, R. Wallauer, S. Voss, A. Czasch, O. Jagutzki, A. Malakzadeh, F. Afaneh, T. Weber, H. Schmidt-Böcking, and R. Dörner, Ultrafast energy transfer between water molecules, Nat. Phys. 6, 139 (2010). Damaging intermolecular energy and proton transfer processes in alpha-particleirradiated hydrogen-bonded systems. S Xu, D Guo, X Ma, X Zhu, W Feng, S Yan, D Zhao, Y Gao, S Zhang, X Ren, Y Zhao, Z Xu, A Dorn, L S Cederbaum, N V Kryzhevoi, 10.1002/anie.201808898Angew. Chem. Int. Ed. 5717023S. Xu, D. Guo, X. Ma, X. Zhu, W. Feng, S. Yan, D. Zhao, Y. Gao, S. Zhang, X. Ren, Y. Zhao, Z. Xu, A. Dorn, L. S. Cederbaum, and N. V. Kryzhevoi, Damaging intermolecu- lar energy and proton transfer processes in alpha-particle- irradiated hydrogen-bonded systems, Angew. Chem. Int. Ed. 57, 17023 (2018). Competition between proton transfer and intermolecular Coulombic decay in water. C Richter, D Hollas, C M Saak, M Förstel, T Miteva, M Mucke, O Björneholm, N Sisourat, P Slavicek, U Hergenhahn, 10.1038/s41467-018-07501-6Nat. Commun. 94988C. Richter, D. Hollas, C. M. Saak, M. Förstel, T. Miteva, M. Mucke, O. Björneholm, N. Sisourat, P. Slavicek, and U. Hergenhahn, Competition between proton transfer and intermolecular Coulombic decay in water, Nat. Commun. 9, 4988 (2018). Hergenhahn, A hitherto unrecognized source of low-energy electrons in water. M Mucke, M Braune, S Barth, M Förstel, T Lischke, V Ulrich, T Arion, U Becker, A Bradshaw, U , 10.1038/nphys1500Nat. Phys. 6143M. Mucke, M. Braune, S. Barth, M. Förstel, T. Lischke, V. Ulrich, T. Arion, U. Becker, A. Bradshaw, and U. Her- genhahn, A hitherto unrecognized source of low-energy electrons in water, Nat. Phys. 6, 143 (2010). On the nature and origin of dicationic, charge-separated species formed in liquid water on X-ray irradiation. S Thürmer, M Ončák, N Ottosson, R Seidel, U Hergenhahn, S E Bradforth, P Slavíček, B Winter, 10.1038/nchem.1680Nat. Chem. 5590S. Thürmer, M. Ončák, N. Ottosson, R. Seidel, U. Her- genhahn, S. E. Bradforth, P. Slavíček, and B. Winter, On the nature and origin of dicationic, charge-separated species formed in liquid water on X-ray irradiation, Nat. Chem. 5, 590 (2013). Proton-transfer mediated enhancement of nonlocal electronic relaxation processes in X-ray irradiated liquid water. P Slavíček, B Winter, L S Cederbaum, N V Kryzhevoi, 10.1021/ja5117588J. Am. Chem. Soc. 13618170P. Slavíček, B. Winter, L. S. Cederbaum, and N. V. Kryzhevoi, Proton-transfer mediated enhancement of non- local electronic relaxation processes in X-ray irradiated liquid water, J. Am. Chem. Soc. 136, 18170 (2014). . Z.-H Loh, G Doumy, C Arnold, L Kjellsson, S H Southworth, A Haddad, Y Kumagai, M.-F Tu, P J Ho, A M March, R D Schaller, M S Bin Mohd Yusof, T Debnath, M Simon, R Welsch, L Inhester, K Khalili, K Nanda, A I Krylov, S Moeller, G Coslovich, J Koralek, M P Minitti, W F Schlotter, J.-E Rubensson, R Santra, L Young, 10.1126/science.aaz4740Science. 367179Observation of the fastest chemical processes in the radiolysis of waterZ.-H. Loh, G. Doumy, C. Arnold, L. Kjellsson, S. H. Southworth, A. Al Haddad, Y. Kumagai, M.-F. Tu, P. J. Ho, A. M. March, R. D. Schaller, M. S. Bin Mohd Yusof, T. Debnath, M. Simon, R. Welsch, L. Inhester, K. Khalili, K. Nanda, A. I. Krylov, S. Moeller, G. Coslovich, J. Ko- ralek, M. P. Minitti, W. F. Schlotter, J.-E. Rubensson, R. Santra, and L. Young, Observation of the fastest chem- ical processes in the radiolysis of water, Science 367, 179 (2020). Realtime observation of water radiolysis and hydrated electron formation induced by extreme-ultraviolet pulses. V Svoboda, R Michiels, A C Laforge, J Med, F Stienkemeier, P Slavíček, H J Wörner, 10.1126/sciadv.aaz0385Science Advances. 6385V. Svoboda, R. Michiels, A. C. LaForge, J. Med, F. Stienkemeier, P. Slavíček, and H. J. Wörner, Real- time observation of water radiolysis and hydrated electron formation induced by extreme-ultraviolet pulses, Science Advances 6, eaaz0385 (2020). A sting in the tail of electron tracks. B D Michael, P O&apos;neill, 10.1126/science.287.5458.1603Science. 1871603B. D. Michael and P. O'Neill, A sting in the tail of electron tracks, Science 187, 1603 (2000). Bond breaks of nucleotides by dissociative electron transfer of nonequilibrium prehydrated electrons: A new molecular mechanism for reductive DNA damage. C.-R Wang, J Nguyen, Q.-B Lu, 10.1021/ja902675gJ. Am. Chem. Soc. 13111320C.-R. Wang, J. Nguyen, and Q.-B. Lu, Bond breaks of nucleotides by dissociative electron transfer of nonequilib- rium prehydrated electrons: A new molecular mechanism for reductive DNA damage, J. Am. Chem. Soc. 131, 11320 (2009). DNA strand breaks induced by near-zero-electronvolt electron attachment to pyrimidine nucleotides. X Bao, J Wang, J Gu, J Leszczynski, 10.1073/pnas.0510406103PNAS. 1035658X. Bao, J. Wang, J. Gu, and J. Leszczynski, DNA strand breaks induced by near-zero-electronvolt electron attach- ment to pyrimidine nucleotides, PNAS 103, 5658 (2006). Reactivity of prehydrated electrons toward nucleobases and nucleotides in aqueous solution. J Ma, F Wang, S A Denisov, A Adhikary, M Mostafavi, 10.1126/sciadv.1701669Science Advances. 31J. Ma, F. Wang, S. A. Denisov, A. Adhikary, and M. Mostafavi, Reactivity of prehydrated electrons toward nucleobases and nucleotides in aqueous solution, Science Advances 3, 1 (2017). Observation of dissociative quasi-free electron attachment to nucleoside via excited anion radical in solution. J Ma, A Kumar, Y Muroya, S Yamashita, T Sakurai, S A Denisov, M D Sevilla, A Adhikary, S Seki, M Mostafavi, 10.1038/s41467-018-08005-zNat. Commun. 61J. Ma, A. Kumar, Y. Muroya, S. Yamashita, T. Sakurai, S. A. Denisov, M. D. Sevilla, A. Adhikary, S. Seki, and M. Mostafavi, Observation of dissociative quasi-free elec- tron attachment to nucleoside via excited anion radical in solution, Nat. Commun. 6, 1 (2019). Interactions of electrons with bare and hydrated biomolecules: From nucleic acid bases to DNA segments. J Gu, J Leszczynski, H F Schaefer, 10.1021/cr3000219Chem. Rev. 1125603J. Gu, J. Leszczynski, and H. F. Schaefer, Interactions of electrons with bare and hydrated biomolecules: From nucleic acid bases to DNA segments, Chem. Rev. 112, 5603 (2012). Electroninduced hydrogen loss in uracil in a water cluster environment. M Smyth, J Kohanoff, I I Fabrikant, 10.1063/1.4874841J. Chem. Phys. 140184313M. Smyth, J. Kohanoff, and I. I. Fabrikant, Electron- induced hydrogen loss in uracil in a water cluster environ- ment, J. Chem. Phys. 140, 184313 (2014). Microhydration prevents fragmentation of uracil and thymine by low-energy electrons. J Kočišek, A Pysanenko, M Fárník, J Fedor, 10.1021/acs.jpclett.6b01601J. Phys. Chem. Lett. 273401J. Kočišek, A. Pysanenko, M. Fárník, and J. Fedor, Micro- hydration prevents fragmentation of uracil and thymine by low-energy electrons, J. Phys. Chem. Lett. 27, 3401 (2016). Water acting as a catalyst for electron-driven molecular break-up of tetrahydrofuran. E Wang, X Ren, W Baek, H Rabus, T Pfeifer, A Dorn, 10.1038/s41467-020-15958-7Nat. Commun. 112194E. Wang, X. Ren, W. Baek, H. Rabus, T. Pfeifer, and A. Dorn, Water acting as a catalyst for electron-driven molecular break-up of tetrahydrofuran, Nat. Commun. 11, 2194 (2020). Structural basis for the photoconversion of a phytochrome to the activated Pfr form. A T Ulijasz, G Cornilescu, C C Cornilescu, J Zhang, M Rivera, J L Markley, R D Vierstra, 10.1038/nature08671Nature. 463250A. T. Ulijasz, G. Cornilescu, C. C. Cornilescu, J. Zhang, M. Rivera, J. L. Markley, and R. D. Vierstra, Structural basis for the photoconversion of a phytochrome to the activated Pfr form, Nature 463, 250 (2010). The role of πσ * excited states in the photodissociation of heteroaromatic molecules. M Ashfold, B Cronin, A L Devine, R Dixon, M G D Nix, 10.1126/science.1125436Science. 3121637M. Ashfold, B. Cronin, A. L. Devine, R. Dixon, and M. G. D. Nix, The role of πσ * excited states in the photodissociation of heteroaromatic molecules, Science 312, 1637 (2006). Excited-state hydrogen detachment and hydrogen transfer driven by repulsive 1 πσ * states: A new paradigm for nonradiative decay in aromatic biomolecules. A L Sobolewski, W Domcke, C Dedonder-Lardeux, C Jouvet, 10.1039/B110941NPhys. Chem. Chem. Phys. 41093A. L. Sobolewski, W. Domcke, C. Dedonder-Lardeux, and C. Jouvet, Excited-state hydrogen detachment and hydrogen transfer driven by repulsive 1 πσ * states: A new paradigm for nonradiative decay in aromatic biomolecules, Phys. Chem. Chem. Phys. 4, 1093 (2002). Conical intersections induced by repulsive 1 πσ * states in planar organic molecules: malonaldehyde, pyrrole and chlorobenzene as photochemical model systems. A L Sobolewski, W Domcke, 10.1016/S0301-0104(00)00203-2Chem. Phys. 259181A. L. Sobolewski and W. Domcke, Conical intersections in- duced by repulsive 1 πσ * states in planar organic molecules: malonaldehyde, pyrrole and chlorobenzene as photochem- ical model systems, Chem. Phys. 259, 181 (2000). Ultrafast photoinduced processes in indole-water clusters. H Lippert, V Stert, L Hesse, C P Schulz, I V Hertel, W Radloff, 10.1016/S0009-2614(03)00921-7Chem. Phys. Lett. 37640H. Lippert, V. Stert, L. Hesse, C. P. Schulz, I. V. Her- tel, and W. Radloff, Ultrafast photoinduced processes in indole-water clusters, Chem. Phys. Lett. 376, 40 (2003). Photoejection of electrons from pyrrole into an aqueous environment: ab initio results on pyrrole-water clusters. A L Sobolewski, W Domcke, 10.1016/S0009-2614(00)00404-8Chem. Phys. Lett. 321479A. L. Sobolewski and W. Domcke, Photoejection of elec- trons from pyrrole into an aqueous environment: ab initio results on pyrrole-water clusters, Chem. Phys. Lett. 321, 479 (2000). Indole-H2O in the gas phase. Structures, barriers to internal motion, and S1 ← S0 transition moment orientation. Solvent reorganization in the electronically excited state. T M Korter, D W Pratt, J Küpper, 10.1021/jp982456xJ. Phys. Chem. A. 1027211T. M. Korter, D. W. Pratt, and J. Küpper, Indole-H2O in the gas phase. Structures, barriers to internal motion, and S1 ← S0 transition moment orientation. Solvent re- organization in the electronically excited state, J. Phys. Chem. A 102, 7211 (1998). Biological water at the protein surface: Dynamical solvation probed directly with femtosecond resolution. S K Pal, J Peon, A H Zewail, 10.1073/pnas.042697899PNAS. 991763S. K. Pal, J. Peon, and A. H. Zewail, Biological water at the protein surface: Dynamical solvation probed directly with femtosecond resolution, PNAS 99, 1763 (2002). Microwave spectrum and structure of a hydrogen-bonded pyrrole-water complex. M J Tubergen, A M Andrews, R L Kuczkowski, 10.1021/j100131a011J. Phys. Chem. 977451M. J. Tubergen, A. M. Andrews, and R. L. Kuczkowski, Microwave spectrum and structure of a hydrogen-bonded pyrrole-water complex, J. Phys. Chem. 97, 7451 (1993). Probing ultrafast dynamics in photoexcited pyrrole: timescales for 1 πσ * mediated H-atom elimination. G M Roberts, C A Williams, H Yu, A S Chatterley, J D Young, S Ullrich, V G Stavros, 10.1039/c2fd20140bFaraday Disc. 16395G. M. Roberts, C. A. Williams, H. Yu, A. S. Chatterley, J. D. Young, S. Ullrich, and V. G. Stavros, Probing ul- trafast dynamics in photoexcited pyrrole: timescales for 1 πσ * mediated H-atom elimination, Faraday Disc. 163, 95 (2013). Non-radiative relaxation dynamics of pyrrole following excitation in the range 249.5-200 nm. O M Kirkby, M A Parkes, S P Neville, G A Worth, H H Fielding, 10.1016/j.cplett.2017.04.035Chem. Phys. Lett. 683179O. M. Kirkby, M. A. Parkes, S. P. Neville, G. A. Worth, and H. H. Fielding, Non-radiative relaxation dynamics of pyrrole following excitation in the range 249.5-200 nm, Chem. Phys. Lett. 683, 179 (2017). Femtosecond time-resolved hydrogen-atom elimination from photoexcited pyrrole molecules. H Lippert, H.-H Ritze, I V Hertel, W Radloff, 10.1002/cphc.200400079Chem. Phys. Lett. 51423H. Lippert, H.-H. Ritze, I. V. Hertel, and W. W Radloff, Femtosecond time-resolved hydrogen-atom elimination from photoexcited pyrrole molecules, Chem. Phys. Lett. 5, 1423 (2004). Vibrationally mediated photodissociation dynamics of pyrrole. K Grygoryeva, J Rakovský, I Vinklárek, O Votava, M Fárník, V Poterya, 10.1063/1.5091974AIP Adv. 9351511K. Grygoryeva, J. Rakovský, I. Vinklárek, O. Votava, M. Fárník, , and V. Poterya, Vibrationally mediated pho- todissociation dynamics of pyrrole, AIP Adv. 9, 0351511 (2019). Excited state dynamics in pyrrole-water clusters: First principles simulation. I Frank, K Damianos, 10.1016/j.chemphys.2007.08.029Chem. Phys. 343347I. Frank and K. Damianos, Excited state dynamics in pyrrole-water clusters: First principles simulation, Chem. Phys. 343, 347 (2008). Photodissociation and photoionization of pyrrole following the multiphoton excitation at 243 and 364.7 nm. A J Van Den Brom, M Kapelios, T N Kitsopoulos, N H Nahler, B Cronin, M N R Ashfold, 10.1039/B415766DPhys. Chem. Chem. Phys. 75704A. J. van den Brom, M. Kapelios, T. N. Kitsopoulos, N. H. Nahler, B. Cronin, and M. N. R. Ashfold, Pho- todissociation and photoionization of pyrrole following the multiphoton excitation at 243 and 364.7 nm, Phys. Chem. Chem. Phys. 7, 5704 (2005). Spatial separation of pyrrole and pyrrolewater clusters. M Johny, J Onvlee, T Kierspel, H Bieker, S Trippel, J Küpper, 10.1016/j.cplett.2019.01.052arXiv:1901.05267Chem. Phys. Lett. 721M. Johny, J. Onvlee, T. Kierspel, H. Bieker, S. Trippel, and J. Küpper, Spatial separation of pyrrole and pyrrole- water clusters, Chem. Phys. Lett. 721, 149-152 (2019), arXiv:1901.05267 [physics]. Pure molecular beam of water dimer. H Bieker, J Onvlee, M Johny, L He, T Kierspel, S Trippel, D A Horke, J Küpper, 10.1021/acs.jpca.9b06460arXiv:1904.08716J. Phys. Chem. A. 1237486H. Bieker, J. Onvlee, M. Johny, L. He, T. Kierspel, S. Trip- pel, D. A. Horke, and J. Küpper, Pure molecular beam of water dimer, J. Phys. Chem. A 123, 7486 (2019), arXiv:1904.08716 [physics]. Coincidence velocity map imaging using Tpx3Cam, a time stamping optical camera with 1.5 ns timing resolution. A Zhao, M Van Beuzekom, B Bouwens, D Byelov, I Chakaberia, C Cheng, E Maddox, A Nomerotski, P Svihra, J Visser, V Vrba, T Weinacht, 10.1063/1.4996888arXiv:1707.06253Rev. Sci. Instrum. 88113104A. Zhao, M. van Beuzekom, B. Bouwens, D. Byelov, I. Chakaberia, C. Cheng, E. Maddox, A. Nomerotski, P. Svihra, J. Visser, V. Vrba, and T. Weinacht, Coin- cidence velocity map imaging using Tpx3Cam, a time stamping optical camera with 1.5 ns timing resolution, Rev. Sci. Instrum. 88, 113104 (2017), arXiv:1707.06253 [physics]. PymePix: A python library for SPIDR readout of Timepix3. A Al-Refaie, M Johny, J Correa, D Pennicard, P Svihra, A Nomerotski, S Trippel, J Küpper, 10.1088/1748-0221/14/10/P10003arXiv:1905.07999J. Instrum. 141010003A. Al-Refaie, M. Johny, J. Correa, D. Pennicard, P. Svihra, A. Nomerotski, S. Trippel, and J. Küpper, PymePix: A python library for SPIDR readout of Timepix3, J. Instrum. 14 (10), P10003, arXiv:1905.07999 [physics]. Fragmentation dynamics of size-selected pyrrole clusters prepared by electron impact ionization: Forming a solvated dimer ion core. V Profant, V Poterya, M Fárník, P Slavíček, U Buck, 10.1021/jp0751561J. Phys. Chem. A. 11112477V. Profant, V. Poterya, M. Fárník, P. Slavíček, and U. Buck, Fragmentation dynamics of size-selected pyrrole clusters prepared by electron impact ionization: Forming a solvated dimer ion core, J. Phys. Chem. A 111, 12477 (2007). Strongly aligned and oriented molecular samples at a kHz repetition rate. S Trippel, T Mullins, N L M Müller, J S Kienitz, K Długołęcki, J Küpper, 10.1080/00268976.2013.780334arXiv:1301.1826Mol. Phys. 1111738S. Trippel, T. Mullins, N. L. M. Müller, J. S. Kienitz, K. Długołęcki, and J. Küpper, Strongly aligned and ori- ented molecular samples at a kHz repetition rate, Mol. Phys. 111, 1738 (2013), arXiv:1301.1826 [physics]. Küpper, Spatially-controlled complex molecules and their applications. Y.-P Chang, D A Horke, S Trippel, J , 10.1080/0144235X.2015.1077838arXiv:1505.05632Int. Rev. Phys. Chem. 34557Y.-P. Chang, D. A. Horke, S. Trippel, and J. Küp- per, Spatially-controlled complex molecules and their applications, Int. Rev. Phys. Chem. 34, 557 (2015), arXiv:1505.05632 [physics]. First demonstration of 3D optical readout of a TPC using a single photon sensitive Timepix3 based camera. A Roberts, P Svihra, A Al-Refaie, H Graafsma, J Küpper, K Majumdar, K Mavrokoridis, A Nomerotski, D Pennicard, B Philippou, S Trippel, C Touramanis, J Vann, 10.1088/1748-0221/14/06/p06001arXiv:1810.09955J. Instrum. 14066001A. Roberts, P. Svihra, A. Al-Refaie, H. Graafsma, J. Küp- per, K. Majumdar, K. Mavrokoridis, A. Nomerotski, D. Pennicard, B. Philippou, S. Trippel, C. Touramanis, and J. Vann, First demonstration of 3D optical readout of a TPC using a single photon sensitive Timepix3 based camera, J. Instrum. 14 (06), P06001, arXiv:1810.09955 [physics]. Intense-field laser ionization rates in atoms and molecules. S Hankin, D Villeneuve, P Corkum, D Rayner, 10.1103/PhysRevA.64.013405Phys. Rev. A. 6413405S. Hankin, D. Villeneuve, P. Corkum, and D. Rayner, Intense-field laser ionization rates in atoms and molecules, Phys. Rev. A 64, 013405 (2001). Strong-field photoelectron momentum imaging of OCS at finely resolved incident intensities. J Wiese, J.-F Olivieri, A Trabattoni, S Trippel, J Küpper, 10.1088/1367-2630/ab34e8arXiv:1904.07519New J. Phys. 2183011J. Wiese, J.-F. Olivieri, A. Trabattoni, S. Trippel, and J. Küpper, Strong-field photoelectron momentum imaging of OCS at finely resolved incident intensities, New J. Phys. 21, 083011 (2019), arXiv:1904.07519 [physics]. High-resolution photoionization spectrum of water molecules in a supersonic beam. R H Page, Y R Shen, Y T Lee, 10.1063/1.454058J. Chem. Phys. 885362R. H. Page, Y. R. Shen, and Y. T. Lee, High-resolution photoionization spectrum of water molecules in a super- sonic beam, J. Chem. Phys. 88, 5362 (1988). Keldysh theory of strong field ionization: history, applications, difficulties and perspectives. S V Popruzhenko, 10.1088/0953-4075/47/20/204001J. Phys. B. 47204001S. V. Popruzhenko, Keldysh theory of strong field ioniza- tion: history, applications, difficulties and perspectives, J. Phys. B 47, 204001 (2014).
{'fraction_non_alphanumeric': 0.06673769240554636, 'fraction_numerical': 0.05104312428444218, 'mean_word_length': 4.295915789473685, 'pattern_counts': {'":': 0, '<': 8, '<?xml version=': 0, '>': 7, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'Radiation-induced damage of biological matter is an ubiquitous problem in nature. The influence of the hydration environment is widely discussed, but its exact role remains elusive. We present the experimental observation of a hydrogen-bonded water molecule acting as a radiation protection agent for ionized pyrrole, a prototypical aromatic biomolecule. Pure samples of pyrrole and pyrrole(H2O) were outer-valence ionized and the subsequent damage and relaxation processes were studied. Bare pyrrole fragmented through the breaking of the C-C or N-C covalent bonds. However, for pyrrole(H2O), we observed a strong protection of the pyrrole ring through the dissociative release of neutral water or by transferring an electron or proton across the hydrogen bond. Furthermore, for pyrrole(H2O) a smaller probability for double ionization was observed. Overall, a single water molecule strongly reduces the fragmentation probability and thus the persistent radiation damage of ionized pyrrole.One sentence summary: "A single water molecule strongly reduces the fragmentation probability and thus the persistent radiation damage of an ionized prototypical biomolecular chromophore." arXiv:2010.00453v1 [physics.chem-ph] 1 Oct 2020', 'arxivid': '2010.00453', 'author': ['Melby Johny \nCenter for Free-Electron Laser Science\nDeutsches Elektronen-Synchrotron DESY\nNotkestrasse 8522607HamburgGermany\n', 'Constant A Schouder \nDepartment of Physics\nAarhus University\nLangelandsgade 1408000Aarhus CDenmark\n', 'Ahmed Al-Refaie \nCenter for Free-Electron Laser Science\nDeutsches Elektronen-Synchrotron DESY\nNotkestrasse 8522607HamburgGermany\n', 'Lanhai He \nCenter for Free-Electron Laser Science\nDeutsches Elektronen-Synchrotron DESY\nNotkestrasse 8522607HamburgGermany\n', 'Joss Wiese \nCenter for Free-Electron Laser Science\nDeutsches Elektronen-Synchrotron DESY\nNotkestrasse 8522607HamburgGermany\n\nDepartment of Chemistry\nUniversität Hamburg\nMartin-Luther-King-Platz 620146HamburgGermany\n\nCenter for Ultrafast Imaging\nUniversität Hamburg\nLuruper Chaussee 14922761HamburgGermany\n', 'Henrik Stapelfeldt \nDepartment of Chemistry\nAarhus University\nLangelandsgade 1408000Aarhus CDenmark\n', 'Sebastian Trippel \nCenter for Free-Electron Laser Science\nDeutsches Elektronen-Synchrotron DESY\nNotkestrasse 8522607HamburgGermany\n\nCenter for Ultrafast Imaging\nUniversität Hamburg\nLuruper Chaussee 14922761HamburgGermany\n', 'Jochen Küpper \nCenter for Free-Electron Laser Science\nDeutsches Elektronen-Synchrotron DESY\nNotkestrasse 8522607HamburgGermany\n\nCenter for Ultrafast Imaging\nUniversität Hamburg\nLuruper Chaussee 14922761HamburgGermany\n\nDepartment of Physics\nUniversität Hamburg\nLuruper Chaussee 14922761HamburgGermany\n'], 'authoraffiliation': ['Center for Free-Electron Laser Science\nDeutsches Elektronen-Synchrotron DESY\nNotkestrasse 8522607HamburgGermany', 'Department of Physics\nAarhus University\nLangelandsgade 1408000Aarhus CDenmark', 'Center for Free-Electron Laser Science\nDeutsches Elektronen-Synchrotron DESY\nNotkestrasse 8522607HamburgGermany', 'Center for Free-Electron Laser Science\nDeutsches Elektronen-Synchrotron DESY\nNotkestrasse 8522607HamburgGermany', 'Center for Free-Electron Laser Science\nDeutsches Elektronen-Synchrotron DESY\nNotkestrasse 8522607HamburgGermany', 'Department of Chemistry\nUniversität Hamburg\nMartin-Luther-King-Platz 620146HamburgGermany', 'Center for Ultrafast Imaging\nUniversität Hamburg\nLuruper Chaussee 14922761HamburgGermany', 'Department of Chemistry\nAarhus University\nLangelandsgade 1408000Aarhus CDenmark', 'Center for Free-Electron Laser Science\nDeutsches Elektronen-Synchrotron DESY\nNotkestrasse 8522607HamburgGermany', 'Center for Ultrafast Imaging\nUniversität Hamburg\nLuruper Chaussee 14922761HamburgGermany', 'Center for Free-Electron Laser Science\nDeutsches Elektronen-Synchrotron DESY\nNotkestrasse 8522607HamburgGermany', 'Center for Ultrafast Imaging\nUniversität Hamburg\nLuruper Chaussee 14922761HamburgGermany', 'Department of Physics\nUniversität Hamburg\nLuruper Chaussee 14922761HamburgGermany'], 'corpusid': 222090187, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 21882, 'n_tokens_neox': 17849, 'n_words': 9435, 'pdfsha': '2685dd5c4b8dce71a3bbadf3d3131a1104bb61a2', 'pdfurls': ['https://arxiv.org/pdf/2010.00453v1.pdf'], 'title': ['Molecular sunscreen: water protects pyrrole from radiation damage Short title: "Water protects molecules from radiation damage"', 'Molecular sunscreen: water protects pyrrole from radiation damage Short title: "Water protects molecules from radiation damage"'], 'venue': []}
arxiv
arXiv:quant-ph/0001061v1 18 Jan 2000 Causality and probabilistic interpretation of quantum mechanics * D A Slavnov [email protected] Department of Physics Moscow State University 119899MoscowRussia arXiv:quant-ph/0001061v1 18 Jan 2000 Causality and probabilistic interpretation of quantum mechanics * * Submitted to "Theoretical and Mathematical Physics" It is shown that the probabilistic treatment of quantum mechanics can be coordinated with causality of all physical processes. The physical interpretation of quantummechanical phenomena such as process of measurement and collapse of quantum state is given.The purpose of the present paper is the substantiation of a thesis that the probabilistic interpretation of quantum mechanics is quite compatible with the supposition about unique causality of all physical processes. The consideration is performed within the framework of the binary model (see papers [1]) of quantum mechanics in which it is supposed that there are separate material carriers of corpuscular and wave properties in quantum objects.Following the basic idea of the papers [1] we shall consider that any quantum object has dynamical and phase degrees of freedom. Carriers of the dynamical degrees of freedom are local. Further they will refer to as nucleuses of the quantum object (to not confuse to atomic nucleuses). Carriers of the phase degrees of freedom are fields, which further will refer to as an information fields or a shell of a quantum object. These fields are spread in the space.An elementary quantum object consists of one nucleus and a shell (information field) which are coherent each other. It is possible to assume, though it is not necessary, that nucleuses exist in pulsatory (in the time and the space) regime like some splashes of the information field. In this case in microscopic scale a nucleus will not have continuous trajectory in the space -time but in macroscopic scale the trajectory will exist.Action onto a quantum object can be dynamical and phase. The dynamical action is accompanied by transmission of dynamical quantities. It acts onto the dynamical degrees of freedom, i.e. onto the nucleuses. Respectively, the nucleuses are responsible for corpuscular properties of the quantum object and they contain the information about observables quantities in the latent shape. The phase action is not accompanied by transmission of dynamical quantities (or by very small transmission) and it acts on the phase degrees of freedom. A classical measuring device reacts to the dynamical action of the quantum object. Therefore the device reacts to an elementary quantum object as onto one aggregate. But the concrete result of this reaction is defined by structure of the shell of the quantum object. This structure depends on state of the information field. Further we shall name it physical state of the quantum object and we shall designate by symbol ϕ. Now we try to formalize these physical ideas about the quantum object. In order to take into account the latency of the information about observables quantities we postulate, that the observablesB are Hermite elements of noncommutative involute algebra (*-algebras) A. Let's consider that to each physical state ϕ of a quantum object univalently there corresponds a functional on the algebra A. This functional we shall designate by the same symbol ϕ. The term "the physical state" will denote structure of the information field, and the functional, corresponding to this structure. Thus, ifB ∈ A andB * =B then ϕ(B) = B is a real number (value of the observableB) which is obtained in the concrete measurement. Let {Q} is some maximal set of mutually commuting Hermite elements of the algebra A, i.e. {Q} ⊂ A and ifQ i ,Q j ∈ {Q} then [Q i ,Q j ] = 0; ifQ i ∈ {Q},Q j ∈ A and [Q i ,Q j ] = 0 thenQ j ∈ {Q}; ifQ i ∈ {Q},Q j / ∈ {Q j } then [Q i ,Q j ] = 0. The functional ϕ maps the set {Q} in a set of real numbers We shall take one fundamental supposition touching the physical state: each physical state is unique, i.e. there are no two identical states in the world and the physical states never repeat. It is possible to consider the physical state is determined by all previous history of the concrete physical object and this history for each object is individual. In particular, in two different experiments we necessarily deal with two different physical states. Different physical states ϕ i , ϕ j correspond to different functionals ϕ i ( ), ϕ j ( ), i.e. always there will be such observableB, that ϕ i (B) = ϕ j (B). From this supposition follows, that each physical state ϕ i can be exhibited in form of the functional ϕ i ( ) no more, than in one experiment. For each observable we shall introduce concept "an actual state". It is a physical state, in which the observable was measured or will be measured. A set of such states we shall denote by [ϕ]Â. A set {Q} of equivalent states, actual for an observableÂ, we shall designate {ϕ} Q . Following the standard quantum mechanics, we shall consider that only mutually commuting observables can be measured in one experiment. The functional ϕ is not linear. However we shall require, that on Hermite elements of algebra A a functional ϕ( ), corresponding to actual states, would satisfy to the following postulates: 1) ϕ(λÎ) = λ,Î is unity of algebra A, λ is real number; (1) 2) ϕ( +B) = ϕ(Â) + ϕ(B), ϕ(ÂB) = ϕ(Â)ϕ(B), if [Â,B] = 0; 3) sup ϕ ϕ( *  ) > 0, if = 0; ϕ( *  ) = 0, if = 0; 4) for each set {Q} and each ∈ A there is lim n→∞ 1 n n i=1 ϕ i (Â) ≡ Ψ Q (Â), where {ϕ 1 , . . . , ϕ n } is a random sample of the set{ϕ} Q ; 5) for everyÂ,B ∈ A Ψ Q ( +B) = Ψ(Â) + Ψ(B). We shall extend the functionals ϕ( ) onto anti-Hermitian elements of the algebra A with the help of the equality ϕ(iÂ) = iϕ(Â). The functional Ψ Q ( ), appearing in the fourth postulate, has a meaning of a functional ϕ( ) which is averaged over all {Q}-equivalent actual states. Symbolically it can be represented in the form of Monte-Carlo integral over the actual states: Ψ Q (Â) = ϕ∈{ϕ} Q dµ(ϕ) ϕ(Â). We shall note that ϕ∈{ϕ} Q dµ(ϕ) = 1.(2) Let us connect the functional Ψ Q ( ) with each quantum state Ψ Q . The fourth postulate assumes that this functional does not depend on a concrete random sample. Further the term "a quantum state Ψ Q " we shall use both for the set {ϕ} Q of physical states and for the corresponding functional Ψ Q ( ) (the quantum average). Let *  =B ∈ {Q}. If ϕ ∈ {ϕ} Q , then ϕ( *  ) = B, where B ∈ {Q}. Therefore Ψ Q ( *  ) = B = ϕ( *  ) ϕ∈{ϕ} Q . From here  2 ≡ sup Q Ψ Q ( *  ) = sup ϕ∈[ϕ] ϕ( *  ) > 0, if = 0. As Ψ Q ( * B ) is a linear (in * B ) positive semidefinite functional then the Cauchy-Bunkyakovsky-Schwarz inequality is valid (see for example [2]) Ψ Q ( * B )Ψ Q (B *  ) ≤ Ψ Q ( *  )Ψ Q (B * B ). Therefore for Ψ Q ( *  ) the postulates for square of seminorm of the element are fulfilled. Respectively it is possible to accept  for the norm ofÂ. The algebra A becomes C * -algebra at such definition of the norm . According to the Gelfand-Naumark theorem (see [2]) every C * -algebra can be realized as algebra of linear operators in some Hilbert space. Thus, the proposed here construction of quantum mechanics permit the standard realization. As against usual scheme of quantum mechanics in the proposed construction one additional element is present. It is the physical state ϕ and the corresponding nonlinear functional ϕ( ). The functional ϕ( ) describes results of individual measurement in a concrete experiment, and the functional Ψ Q ( ) describes average value of the observable in a series of experiments, performed in identical conditions from the point of view of the observer, i.e. at the same quantum state. The fact of existence of the functionals ϕ( ) and Ψ Q ( ), satisfying to the enumerated postulates, is proved by all complex of quantum experiments. The standard quantum mechanics is busy in problems describing by the functionals Ψ Q ( ). However now single quantum phenomena take a great meaning. For example, such phenomena underlie an operation of quantum computer. Therefore it is desirable to supplement the formalism of the standard quantum mechanics by positions, which would allow considering the single quantum phenomena. On the other hand, it is extremely desirable, that such expanded formalism did not give rise to deductions, which would not agree with deductions of the standard quantum mechanics. Namely for sufficing this condition the postulate of uniqueness of each physical state is accepted. It follows this postulate that the physical state can not be univalently fixed. Really, to fix the functional ϕ( ), we should know its value ϕ(B) for all independent observableŝ B. Physically it is not realizable. In one experiment we can find ϕ(B i ) only for mutually commuting observablesB i , and in different experiments we necessarily deal with different functionals ϕ( ). The most hard fixing of the functional ϕ( ), which can physically be realized, consists in reference it to some set {ϕ} Q , i.e. to a certain quantum state. For this purpose it is enough to perform measurements of mutually commuting observables. It is possible to be restricted only by independent measurements. In principle it can be done in one experiment. Thus we can not have the complete information about a physical state of a concrete physical object in essence. The maximal observable and controllable information on the physical object is concentrated in the quantum state. For example, let a spin-free particle decays into two particles A and B with spins 1/2 which scatter at large distance. Let's measure a projection of spin onto the axis z for the particle A. Let result will be S z (A). Then using the conservation law, we can state that for the particle B with absolute probability the projection of spin onto the axis z is equal S z (B) = −S z (A). It denotes that the quantum state of the particle B corresponds to such value of the projection of spin onto axis z. However for the particle A we could measure the projection of spin onto axis x. Let result would be S x (A). Then we could state that the particle B is in the quantum state, which corresponds to the value S x (B) = −S x (A) of the observableŜ x (B). As any physical operations with the particle B are not fulfilled, the physical state in both cases will be same ϕ ∈ {ϕ} −Sz(A) ∩ {ϕ} −Sx(A) . The different quantum states of the particle B are related only to our subjective choice of the device for measurement of the physical state of the particle A. This example is the description of the experiment proposed by Bohm [3] for demonstrating of the Einstein-Podolsky-Rosen paradox [4]. In proposed here treatment any paradox is absent. The physical state ϕ particles B, which is an objective reality, does not depend on our manipulations with the remote particle A. Any transmission of an action on distance is absent. The described experiment is an example of indirect measurement at which the information about state of the quantum object is obtained without physical action onto it. Usually interpretation of the indirect experiment arouses the greatest difficulties, first of all, bound with concept of collapse of quantum state (reduction of wave function). In adduced example by our wish we "channelize" the particle B into the quantum state {ϕ} −Sz(A) , or into the quantum state {ϕ} −Sx(A) , physically not acting onto the particle B. Such collapse can be named subjective (or passive) in contrast to objective (active) collapse, which is related to the actual physical action onto the quantum object. About the objective collapse we shall talk later. Here we shall note that the subjective collapse is related to physical impossibility for us to receive the complete objective information about the quantum object (the complete information about the physical state). We should be content by the partial information (quantum state). It depends on our desire what concrete part we prefer to be satisfied by. It is possible also to use this experiment for demonstrating that, strictly speaking, it would be possible to receive larger the information about state of quantum object, than ascertaining of its membership to this or that quantum state. In the experiment we can measure the projection of spin onto the axis z for the particle A and the projection onto the axis x for the particle B. In this case we can establish that the physical state ϕ of the particles B belongs to intersection of the corresponding quantum states ϕ ∈ {ϕ} −Sz(A) ∩ {ϕ} Sx(B) .(3) But this information has specific character. It refer to the past, more precisely, to the restricted interval in the past from the moment of decay of the spin-free particle to the moment of measurement of the projection of spin of the particle B. In this moment the information described by the equation (3) will be garbled and will be useless for the further monitoring (or control) of the particle B. Now we shall discuss dynamics and temporal evolution of quantum object. As was already spoken, the action onto quantum object can be dynamical and phase. By hypothesis elementary quantum object is nonlocal due to the shell. The different parts of the quantum object can undergo different exterior action and lose mutual coherence. In turn, it should result in disintegration of the quantum object, since all constituent parts of the elementary quantum object must be coherent by hypothesis. As the elementary quantum objects are rather stable structures, there should be a cause, which hinders loss of the coherence. Let's assume that such cause is a strong phase interaction within the elementary quantum object. It recovers coherence. Let's assume also that at the microscopic level the direct phase action onto quantum object is much feebler than dynamical action. At the direct phase action the exterior objects act onto phase degrees of freedom of quantum object directly. However indirect phase action is possible. It is carried out as follows. The exterior objects dynamically act onto dynamical degrees of freedom (onto nucleus) of the quantum object. Further this action is transmitted to phase degrees of freedom through strong interior phase interaction. It brings to reorganization of the shell of the quantum object, i.e. to modification of its physical state. Such phase action is not weak as against dynamical. Neglecting the direct phase action, we shall consider that the evolution of a physical state of quantum object is determined by dynamical action, and it is controlled by a dynamical equation of motion, which is coordinated with usual quantum-mechanical equation of motion. Namely we shall assume that the physical state ϕ evolves in the time so, that the functional, corresponding to this state, ϕ(Â) ( ∈ A) varies as follows: ϕ 0 (Â) → ϕ t (Â) ≡ ϕ 0 (Â(t)),(4) where dÂ(t) d t = ī h Ĥ ,Â(t) ,Â(0) =Â.(5) HereĤ is a usual Hamiltonian (considered as an element of the algebra A) of the quantum object. The equations (4) and (5) quite unequivocally describe temporal evolution of the physical state. Therefore when only dynamical action is accounted (it corresponds to that that von Neumann [5] names as action of the second type) physical processes are strictly determinable. Other matter, that with the help of observations we can determine the initial value ϕ 0 (Â) of the functional (physical state) only to within its membership to some quantum state {ϕ} Q . Therefore majority of our predictions about the further observed dates for the considered quantum object can be only probabilistic. Now we will turn to consideration of direct phase action. In the previous reasoning we considered that they can be neglected. A situation however is possible when it cannot be done. This situation is realized when the very large number of exterior objects act the quantum object. As the nucleus is local, it feels action of small number of the exterior objects if the long-range action is absent. As opposed to this, the shell having nonlocal structure feels action of the large number of the exterior objects. The mass character of the action can cancel weakness of the separate action. It happens only in that case when the separate weak actions do not cancel each other. It is possible to assume that exactly such situation is realized at action of a classical measuring device onto a quantum object. I.e. distinctive feature of a measuring device is that its separate microscopic elements exerts synchronous direct phase action onto the quantum object. The typical classical measuring device comprises analyzer and detector. The analyzer is a classical device with one inlet and several exits. In the analyzer due to of direct phase action the united shell of quantum object decomposes onto several coherent constituents. Symbolically we shall figure it so: ϕ → ϕ 1 ⊕ ϕ 2 ⊕ . . . ⊕ ϕ i ⊕ . . . . I.e. the united structure (the physical state ϕ) decomposes onto the direct (coherent) sum of constituents ϕ i . Inside the analyzer everyone evolves somehow , but all constituents preserve a mutual coherence. Therefore, if in the further the constituents will incorporate they will be able to interfere among themselves. The constituent ϕ i quits the analyzer through i-th exit. Everyone (i-th) exit corresponds to a particular value A i of certain observable (or of several mutually commuting observables). Thus, the analyzer is a point of branching of the initial physical state ϕ. The part of the shell, falling into the i-th branch, belongs to a set [ϕ] A i , which contains all information fields ϕ ′ such, that the corresponding functionals satisfy to equality ϕ ′ (Â) = A i . If only the dynamical action onto the quantum object is taken into account point of branching of the shell is a point of bifurcation for motion of the nucleus. In this point the interaction of the nucleus with the information field plays a role of "random" force, which guides the nucleus along one of the branch. Let's consider that the nucleus is retracted into the branch, for which ϕ ∈ [ϕ] A i ,(6) where ϕ is information field of the quantum object bef ore points of branching. Such motion through the analyzer is admissible for the shell not changing structure. Let's consider that the nucleus should be in a resonance with neighbouring part of the shell. Then such motion is admissible for nucleus for which the resonant condition does not vary. Actually the structure of the shell varies at the analyzer. Therefore equation (6) is necessary to consider, as the requirement of an invariance of the resonant condition for the nucleus at the point of bifurcation. The equation (6) is certain condition of continuity for the motion of the nucleus. This equation guarantees that at the point of bifurcation the evolution of the quantum object is uniquely determinated by its physical state ϕ. The observable evolution has probabilistic character. Firstly, it is not controlled by a dynamical equation of motion (the bifurcation point). Secondly, the physical state ϕ before the bifurcation point can be fixed only to within membership ϕ to certain quantum state {ϕ} Q . Due to the equation (6) the probability W i of falling into the i-th branch for the nucleus is determined by the equality W i = ϕ∈{ϕ} Q ∩[ϕ] A i dµ(ϕ).(7) Now we shall discuss the detector. It is a classical object, which has strong dynamical and phase interaction with quantum object. The detector is in a macroscopically unstable state. As a result of dynamical action of nucleus of the quantum object it goes out equilibrium. A catastrophic process, which makes macroscopically observable result, develops in it. The phase action of quantum object onto the detector is proportioned onto large number of microscopical constituents of the detector and does not give macroscopically observable effect. Thus, the detector macroscopically reacts only to nucleus of the quantum object, i.e. it reacts to the quantum object as onto one aggregate. If the detector is located at the i-th branch then it works with probability W i (formula (7)). The nucleus of quantum object falls into the i-th branch with such probability. If the detector has worked, it denotes ϕ ∈ [ϕ] A i , i.e. ϕ(Â) = A i . Thus, on the one hand, value of the functional ϕ( ) really characterizes result of individual measurement. On the other hand, using formula (7) for average value < A > of the observable we can receive < A >= i W i A i = i ϕ∈{ϕ} Q ∩[ϕ] A i dµ(ϕ) ϕ(Â) = ϕ∈{ϕ} Q dµ(ϕ) ϕ(Â) = Ψ Q (Â). It agrees with deductions of the standard quantum mechanics and with property (2) of the functional Ψ Q ( ). The inverse action of the detector onto the quantum object can go along two scenarios. The first scenario is realized when the nucleus is at the branch where there is the detector. The detector dynamically acts onto the nucleus and strongly (directly and indirectly) onto that part of the shell of the quantum object, which has fallen into the i-th branch. Due to this action these parts of the quantum object lose coherence with those parts of the shell, which have fallen into other branches. As a result, firstly, they lose possibility to interfere with the part of the shell, which has passed through the i-th branch. Secondly, only this part of the shell remains in the structure of the quantum object, as only it does not lose coherence with the nucleus due to strong interior phase interaction. Thus, there is sharp reorganization of the shell of the quantum object (of its physical state). In standard quantum mechanics this phenomenon is treated as a collapse of the quantum state. Here this phenomenon can be named objective or active collapse. By the exterior displays the objective collapse is quite similar to the earlier described passive collapse, but the physical essences of these phenomena are completely different. It is necessary to note, that the modification of the physical state and, as the consequence, its quantum state happens due to actual modification of the part of the quantum object, which is at the detector. But not as a result of vanishing (of reduction) of those parts, which are not at the detector. Nothing happens with them. Nevertheless, they cease to be constituents of the quantum object. In this case the action of the detector onto the quantum object is dynamical and phase. Either first type, or the second type of interaction can predominate. If the overwhelming contribution gives the dynamical action then this contribution can be described by the dynamical equations (4), (5). If overwhelming or the essential contribution gives direct phase action then this contribution can not be described by the dynamical equations. By terminology of von Neumann it is interaction of the first type. However in this case (as opposed to the von Neumann's opinion ) the physical evolution of the quantum object is uniquely determined by structure of that part of the shell, which has hit the detector. Other matter, that we have only the information, which there is in the equation ϕ i ∈ {ϕ} Q ∩ {ϕ} A i for this part ϕ i of the shell. Therefore we can do only probabilistic predictions. Most probably, if the direct phase action plays main role then the modified part ϕ ′ i of the shell will belong to the set {ϕ} A i , but it will cease to belong to the set {ϕ} Q . In favour of such supposition speaks experiment, as a particular quantum state is practically prepared usually so. Let's consider now second scenario, when the nucleus falls into that branch, in which the detector is absent. For simplicity of reasoning we shall consider that the analyzer has only two branches. The detector is located in the second branch, and the nucleus falls into the first branch. In this case the detector does not work (negative experiment). However the standard quantum mechanics states that there is a collapse of a quantum state also. Let's look, how it can be justified within the framework of considered here model. In this case at the analyzer the field ϕ decomposes onto coherent constituents ϕ 1 and ϕ 2 : ϕ → ϕ 1 ⊕ ϕ 2 . The field ϕ 2 falls into operative zone of the detector. There this part of the shell undergoes strong direct action of the detector. As a result of this action ϕ 2 loses a coherence with the nucleus and ϕ 1 . Therefore ϕ 2 ceases to be a part of the shell of the quantum object. Now the quantum object will have physical state ϕ 1 . There is a modification of the physical state of the quantum object. This modification is not controlled by the dynamical equations (objective collapse of a quantum state). In this case we quite definitely can state, that ϕ 1 ∈ [ϕ] A 1 . Let's note that both at the first and second scenario, on the one hand, as a result of action of the detector there is a (objective) collapse of the quantum state of the quantum object, on the other hand, a long-range action of the detector is absent. The proposed scheme of quantum mechanics gives obvious and almost classical explanation of the most fundamental quantum phenomena. The scheme is free from any paradoxes. However there is a problem, maybe this scheme one of variants of scheme with hidden parameters. In a certain measure it is so, but the reasonings, over which schemes with the hidden parameters are rejected, is not correct in this case. The famous proof of von Neumann [5] about impossibility of the hidden parameters in quantum mechanics essentially founds on linearity of quantum mechanics. One of main elements, functional ϕ( ), is not linear in the scheme, proposed here. Therefore proof of von Neumann does not concern the present case. Other, not less famous, argument against schemes with the hidden parameters is Bell inequality [6]. We shall reproduce a typical deduction of this inequality. Let a quantum object Q (particle with spin 0 in the elementary variant of experiment) decays into two objects A and B (particles with spins 1/2). The objects A and B scatter on large distance and hit detectors D(A) and D(B), respectively, in which the measurements are independent. The object A has a set of observables a (double projection of spin onto the direction a). The observables corresponding to different values of an index a, are not simultaneously measurable. Each of observables can take two values ±1. In a concrete experiment the device D(A) measures an observable a with a particular index a. For the object B everything is similar. Let's assume that a quantum object Q has a hidden parameter λ. In each individual event the parameter λ has a particular value. The distribution of events according to the parameter λ is characterized by a measure µ(λ) with usual properties µ(λ) ≥ 0, dµ(λ) = 1. All magnitudes, connected with individual event, depend on the parameter λ. In particular, the values of observables a andB b , obtained in a concrete experiment, are functions A a , B b of the parameter λ. For individual event the correlation of observables a andB b is characterized by the magnitude A a (λ)B b (λ). The average value of the magnitude is referred to as correlation function E(a, b): E(a, b) = dµ(λ) A a (λ) B b (λ). Giving various values to the indexes a and b and taking into account that A a (λ) = ±1, B b (λ) = ±1,(8) we shall obtain the following inequality |E(a, b) − E(a, b ′ )| + |E(a ′ , b) + E(a ′ , b ′ )| ≤ (9) ≤ dµ(λ) [|A a (λ)| |B b (λ) − B b ′ (λ)| + |A a ′ (λ)| |B b (λ) + B b ′ (λ)|] = = dµ(λ) [|B b (λ) − B b ′ (λ)| + |B b (λ) + B b ′ (λ)|]. In the right-hand side of the formula (9), due to equalities (8), one of the expressions |B b (λ) − B b ′ (λ)|, |B b (λ) + B b ′ (λ)|(10) is equal to zero, and another is equal to two for each value of λ. From here we obtain the Bell inequality |E(a, b) − E(a, b ′ )| + |E(a ′ , b) + E(a ′ , b ′ )| ≤ 2. The correlation function E(a, b) is easily calculated within the framework of the standard quantum mechanics. In particular, when A and B are particles with spin 1/2 E(a, b) = − cos θ ab , θ ab is an angle between a and b. It is easy to verify that there are directions a, b, a ′ , b ′ , for which formulas (11) and (12) contradict each other. It would seem it is possible to repeat this derivation in the proposed here model, having made replacements of type A(λ) → ϕ(Â), B(λ) → ϕ(B), dµ(λ) . . . → ϕ∈{ϕ}ÂB Q dµ(ϕ)ϕ(. . .). However this opinion is erroneous. For derivation of the Bell inequality it is essential, that in the left-hand side of the inequality (9) it is possible to represent all terms in the form of united integral over one parameter λ. It is not valid for the quantum average substituting this integral, as it is necessary to integrate over actual states in it . The elements, appearing in different correlation functions, aBb , a ′B b , aBb ′ , a ′B b ′ , do not commute among themselves. Therefore sets of actual states, corresponding to these operators, do not intersect. In derivation of the inequality (11) we tacitly supposed that expression (10) exist for each λ. However there is no physical state ϕ, which would be by actual state both for the observablê B b and for the observableB b ′ . In summary it is necessary especially to note, that the present paper is not at all attempt to formulate a new rival theory to quantum mechanics. The proposed scheme does not contradict any statement of the standard quantum mechanics. Maybe some theses gain slightly other physical interpretation. All deductions of the standard quantum mechanics are valid in the described scheme. At the same time there are additional elements (for example, functional ϕ( ), equality (1)) in this scheme. They allow to include individual events in the domain of its application. Strictly speaking, the formalism of the standard quantum mechanics assumes that the classical relations are reproduced only for average values of quantum observables. Meantime the practice shows that such relations are reproduced in each individual experiment. Of course, it concerns only those observables, what can be measured in one experiment. In the proposed approach this fact is consequence of the equalities (1). For the different functionals ϕ i , ϕ j the sets {ϕ i (Q)}, {ϕ j (Q)} can differ and can coincide. If for allQ ∈ {Q} is valid ϕ i (Q) = ϕ j (Q) = Q then we shall term the physical states ϕ i and ϕ j as {Q}-equivalent. Let's denote by {ϕ} Q set of all {Q}-equivalent physical states. Let's term the set {ϕ} Q as a quantum state and we shall designate Ψ Q . At the same time, the quantum states have some subjective element. From the standard quantum mechanics it is well known that any quantum state can be represented in form of superposition of quantum states which are fixed by one maximal set {Q} of mutually commuting observables. In the proposed here construction it corresponds to the fact that one physical state can belong to the different quantum states {ϕ} Q and {ϕ} R . I.e. ϕ ∈ {ϕ} Q ∩ {ϕ} Q , where the states {ϕ} Q are classified by values of the set {Q} of mutually commuting observablesQ, and {ϕ} R are classified by values of observablesR ∈ {R}. The observablesQ andR do not commute among themselves. Then depending on what set ({Q} or {R}) we shall choose for classification the physical state ϕ will be referred either to the quantum state {ϕ} Q or to the quantum state {ϕ} R . . D A Slavnov// Theor, Moscow University. Phys. Bull. 10630Math. Phys.. N 1, 24; N 2, 13; N 3, 12; N 4D.A. Slavnov// Theor. Math. Phys., 106, (1996), 220; 110, (1997), 235; Moscow Uni- versity. Phys. Bull. (1996), N 1, 24; N 2, 13; N 3, 12; N 4, 30. G G Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory. Wiley-Interscience, a Division of John Wiley & Sons, INC. New YorkG.G. Emch. Algebraic Methods in Statistical Mechanics and Quantum Field Theory. Wiley-Interscience, a Division of John Wiley & Sons, INC. New York (1972). D Bohm, Quantum Theory. Englewood Cliffs, N.Y.Prentice-HallD. Bohm. Quantum Theory. Prentice-Hall, Englewood Cliffs, N.Y. (1951). . A Einstein, B Podolsky, N Rosen //, Phys. Rev. 47777A. Einstein, B. Podolsky, and N. Rosen //Phys. Rev., 47, (1935), 777. Mathematical Foundation of Quantum Mechanics. J Neumann, Prentice-HallNew YorkJ. von Neumann. Mathematical Foundation of Quantum Mechanics. Prentice-Hall, New York (1952) . J S Bell, Physics. 1195J.S. Bell//Physics (Long Island City, N.Y.)., 1, (1965), 195.
{'fraction_non_alphanumeric': 0.04772622167411746, 'fraction_numerical': 0.0066903036842916605, 'mean_word_length': 4.239218219996769, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 5, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'It is shown that the probabilistic treatment of quantum mechanics can be coordinated with causality of all physical processes. The physical interpretation of quantummechanical phenomena such as process of measurement and collapse of quantum state is given.The purpose of the present paper is the substantiation of a thesis that the probabilistic interpretation of quantum mechanics is quite compatible with the supposition about unique causality of all physical processes. The consideration is performed within the framework of the binary model (see papers [1]) of quantum mechanics in which it is supposed that there are separate material carriers of corpuscular and wave properties in quantum objects.Following the basic idea of the papers [1] we shall consider that any quantum object has dynamical and phase degrees of freedom. Carriers of the dynamical degrees of freedom are local. Further they will refer to as nucleuses of the quantum object (to not confuse to atomic nucleuses). Carriers of the phase degrees of freedom are fields, which further will refer to as an information fields or a shell of a quantum object. These fields are spread in the space.An elementary quantum object consists of one nucleus and a shell (information field) which are coherent each other. It is possible to assume, though it is not necessary, that nucleuses exist in pulsatory (in the time and the space) regime like some splashes of the information field. In this case in microscopic scale a nucleus will not have continuous trajectory in the space -time but in macroscopic scale the trajectory will exist.Action onto a quantum object can be dynamical and phase. The dynamical action is accompanied by transmission of dynamical quantities. It acts onto the dynamical degrees of freedom, i.e. onto the nucleuses. Respectively, the nucleuses are responsible for corpuscular properties of the quantum object and they contain the information about observables quantities in the latent shape. The phase action is not accompanied by transmission of dynamical quantities (or by very small transmission) and it acts on the phase degrees of freedom. A classical measuring device reacts to the dynamical action of the quantum object. Therefore the device reacts to an elementary quantum object as onto one aggregate. But the concrete result of this reaction is defined by structure of the shell of the quantum object. This structure depends on state of the information field. Further we shall name it physical state of the quantum object and we shall designate by symbol ϕ.', 'arxivid': 'quant-ph/0001061', 'author': ['D A Slavnov [email protected] \nDepartment of Physics\nMoscow State University\n119899MoscowRussia\n'], 'authoraffiliation': ['Department of Physics\nMoscow State University\n119899MoscowRussia'], 'corpusid': 117344152, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 8458, 'n_tokens_neox': 7663, 'n_words': 5494, 'pdfsha': 'da6e38c578f2a39767ed462d2e6945c77b8484e2', 'pdfurls': ['https://export.arxiv.org/pdf/quant-ph/0001061v1.pdf'], 'title': ['arXiv:quant-ph/0001061v1 18 Jan 2000 Causality and probabilistic interpretation of quantum mechanics *', 'arXiv:quant-ph/0001061v1 18 Jan 2000 Causality and probabilistic interpretation of quantum mechanics *'], 'venue': []}
arxiv
The Spin-Orbit Torque from a Magnetic Heterostructure with High-Entropy Alloy Tian-Yue Chen Department of Materials Science and Engineering National Taiwan University 10617TaipeiTaiwan Tsao-Chi Chuang Department of Physics National Taiwan University 10617TaipeiTaiwan Ssu-Yen Huang Hung-Wei Yen Department of Materials Science and Engineering National Taiwan University 10617TaipeiTaiwan Department of Physics National Taiwan University 10617TaipeiTaiwan Chi-Feng Pai [email protected] Department of Materials Science and Engineering National Taiwan University 10617TaipeiTaiwan The Spin-Orbit Torque from a Magnetic Heterostructure with High-Entropy Alloy 1 High-entropy alloy (HEA) is a family of metallic materials with nearly equal partitions of five or more metals, which might possess mechanical and transport properties that are different from conventional binary or tertiary alloys. In this work, we demonstrate current-induced spin-orbit torque (SOT) magnetization switching in a Ta-Nb-Hf-Zr-Ti HEA-based magnetic heterostructure with perpendicular magnetic anisotropy (PMA). The maximum damping-like SOT efficiency from this particular HEA-based magnetic heterostructure is further determined to be HEA 0.033 DL   by hysteresis loop shift measurements, while that for the Ta control sample is Ta 0.04 DL   . Our results indicate that HEA-based magnetic heterostructures can serve as a new group of potential candidates for SOT device applications.  In recent years, high-entropy alloys (HEAs) have attracted considerable attention from the materials science community, since HEAs offer a new route of metallurgy research beyond traditional binary and/or tertiary alloy systems. HEAs typically consist of five or more principle elements with equimolar or nearly-equimolar ratios and can be viewed as composites at atomic scale [1,2], therefore the high mixing entropy. The mechanical properties of HEAs have been widely-studied, while their physical properties, especially spin-transport properties, have yet to be reported. Meanwhile, in the field of spintronics or spin-orbitronics, researchers have identified that the spin Hall effects (SHEs) [3][4][5] are strong in pure heavy transition metals (TM) such as Pt [6][7][8], β-Ta [9], and β-W [10] as well as in metallic alloys such as Cu-Bi [11], Cu-Ir [12], Cu-Pb [13], Au-W [14], Au-Cu [15], Pt-Hf (Al) [16], and even in antiferromagnetic alloying systems [17,18]. In order to extend the horizon of spin-Hall material exploration to beyond binary alloying systems, it would be very interesting and important to study the SHE in HEAs. In this work, we demonstrate the SHE-induced spin-orbit torque (SOT) magnetization switching from an HEA/Ta/CoFeB/Hf/MgO magnetic heterostructure, with the HEA being nominally-equimolar Ta-Nb-Hf-Zr-Ti. By using SOT-assisted hysteresis loop shift measurements [19] [27]. Although the present work focuses on room-temperature transport properties of an HEA-based system with a different molar composition, the rich physics in high-entropy alloys make this particular HEA-based system attractive. We prepared our heterostructure samples by DC (for metallic materials) and RF (for MgO) magnetron sputtering in a high vacuum system with a base pressure 8 3 10 Torr   . The working 4 Ar pressure and the DC sputtering power for metallic materials deposition are 3 mTorr and 30 W, respectively. The condition for RF sputtering of MgO is 10 mTorr Ar with 50W forward power. The growth rates of each material were further characterized by X-ray reflectivity (XRR) and atomic force microscopy (AFM). To check the quality of our sputter-deposited Ta-Nb-Hf-Zr-Ti HEA film, we first performed cross-sectional high resolution transmission electron microscopy (HR-TEM, FEI Tecnai G2 F20) and in-situ X-ray energy dispersive spectrometry (EDS) on a 57 nm-thick HEA film, which was deposited on a Si/SiO2 substrate. The TEM sample was prepared by a lift-out technique with Helios NanoLab 600i focus ion beam. As shown in Fig. 1(a), the HR-TEM image indicates that the sputtered HEA film is amorphous. The electrical resistivity of the HEA film is measured to be around HEA 138 μΩ-cm   , which is comparable to the reported values of other amorphous films that give rise to large spin Hall effects [26]. The in-situ EDS result ( Fig. 1(b)) further indicates that the sputtered film composition to be fairly uniform across the whole thickness range, with its averaged atomic ratio being Ta24.9Nb18.7Hf17.7Zr18.3Ti20.4. For simplicity, we will refer to the sputtered Ta24.9Nb18.7Hf17.7Zr18.3Ti20.4 (nominal composition: Ta20Nb20Hf20Zr20Ti20) film as "HEA" for the rest of the paper. We utilized HEA-based PMA devices to investigate the SOT therein. (nominal thickness). We further performed annealing for all samples at 300°C for 1 hour to promote PMA in the same high vacuum chamber after thin film depositions. The purpose of depositing Ta (0.5 nm) and/or Hf (0.5 nm) insertion layers on the opposite sides of CoFeB in these films is to facilitate the enhancement of PMA in these magnetic heterostructures [20,28]. Both insertions layers should have minimal effects on the spin-Hall transport properties of the whole structure, since the Ta layer thickness is ultra-thin and the Hf layer will be mostly oxidized by the MgO on-top [28]. The magnetic properties of both series of samples were characterized by vibrating sample magnetometer (VSM). As shown in Fig. 2 (a) t t t  ) for both series of samples are shown in Fig. 2 CoFeB in between Ta and Hf dusting layers will give rise to similar s M , dead t , and eff K , independent of the adopted buffer layer (either HEA or Ta). This feature allows us to study spintransport properties that vary with the buffer layer or the spin-Hall source material, while keeping the magnetic layer or the spin current sink material (CoFeB) unchanged. As schematically shown in Fig. 3 (a) switching data indicate that the switching is a domain nucleation/propagation process, which is similar to the switching curves in several previous reports [30,31]. The current-induced switching symmetry is also consistent with that from Ta-based [9] and W-based heterostructures [22,30], which indicates a negative spin Hall ratio for this particular HEA. To further quantify SOT efficiencies of these HEA-based magnetic heterostructures, 7 especially the damping-like (Slonczewski-like) torque contribution, we performed AH voltage hysteresis loop shift measurements [19] on a series of HEA-based devices with various HEA thicknesses. The nominal CoFeB thickness in these heterostructures was fixed at 1.4 nm, since this particular thickness provides maximum eff K . In Fig. 4 (a) Fig. 4 (c). When [35]. Therefore, we believe that our observation here cannot be simply explained by adopting Vegard's mixing rule. This deviation of the transport property from an ideal mixing scenario is consistent with the discovery that the electrical degree of freedom does not follow a "cocktail effect" of the constituent elements in an HEA [27], though the mixture is mostly random and in an amorphous solid solution fashion. In conclusion, we show that Ta-Nb-Hf-Zr-Ti HEA-based magnetic heterostructures can possess PMA as well as SOT efficiencies that are comparable to those of Ta-based heterostructures, though the concentration of the major spin Hall or spin-orbit coupling sources, such as Ta and/or Hf, have been diluted down to less than 50% (at %). We demonstrate currentinduced SOT switching in these HEA-based devices and further characterize the maximum damping-like SOT efficiency to be HEA 0.033 DL   . Our discovery suggests that by working on randomly-mixing alloys beyond binary systems, it is possible to explore electronic degree of freedom that does not follow Vegard's mixing rule in other similar HEA-based magnetic heterostructures, thereby improving the SOT efficiency for possible next-generation SOT memory device applications. For comparison, magnetic heterostructures with PMA were deposited on HEA and Ta buffer layers, as two series of samples: (I) HEA series: || HEA(3.5) /Ta(0.5)/Co20Fe60B20(tCoFeB)/Hf(0.5)/MgO(2)/Ta( , we patterned samples into micron-sized Hall-bar devices with lateral dimensions of 5 μm by 60 μm for anomalous Hall (AH) voltage measurements. A representative AH loop was obtained from an HEA( HEA t )/Ta(0.5)/CoFeB(1.4)/Hf(0.5)/MgO(2)/Ta(2) sample with HEA 5 nm t  , as shown in Fig. 3 (b), which suggests PMA indeed exists on the HEA buffer layer with out-of-plane coercive field 10 Oe c H  . To demonstrate the efficacy of SOT from these HEA-based magnetic heterostructures, we performed current-induced SOT switching measurements. The currentinduced switching data from this representative HEA 5 nm t  Hall-bar device with opposite inplane bias fields ( 100 Oe x H  ) are shown in Fig. 3 (c) and (d). The steps in current-induced J SOT plus Dzyaloshinskii-Moriya interaction (SOT+DMI) scenario[19,32,33], the in-plane bias fieldx H will re-align Néel domain walls in the magnetic heterostructure while the DC current DC I flowing through the buffer layer (HEA in this case) will further generate SOT acting upon the re-aligned walls, thereby resulting in an out-of-plane effective field eff H that shifts the whole hysteresis loop. As shown in Fig. 4 (b), the linear trend between eff H and DC I then can be obtained to further calculate "current-induced effective field per current density" is the charge current density flowing in the buffer layer. Typically,  can be considered as the figure of merit for evaluating the SOT efficacy from different magnetic heterostructures. We summarize the in-plane field x H dependence of  in , we find that the maximum damping-like SOT efficiency from our HEA-based magnetic heterostructures to be deposited from an equimolar Ta20Nb20Hf20Zr20Ti20 (in at. %) sputter target. We picked this particular composition of HEA for several reasons. First of all, from the perspective of gainingHEA 0.033 DL   , which is comparable to that from the Ta control samples Ta 0.04 DL   . The HEA thin film (buffer layer) that we employed in our magnetic heterostructure was 3 interfacial perpendicular magnetic anisotropy (PMA) in a TM/CoFeB/MgO heterostructure, both Ta and Hf [20-23] are widely-used TM buffer layers for achieving this goal. Moreover, Zr has been shown to be a good boron sink to facilitate the crystallization of CoFe from amorphous CoFeB hence its templating with respect to the MgO (100)-orientation [24]. Recently, it has also been demonstrated that PMA can be obtained from both Zr/CoFeB/MgO and Nb/CoFeB/MgO systems after suitable heat treatments [25]. Secondly, a pure Ta in its highly resistive phase has been shown to be a strong spin Hall source [9,26], which can be utilized to efficiently generate SOTs. We would like to observe the change in terms of SOT efficiency if the Ta component has been diluted in a solid solution fashion. Lastly, though studies on the electron transport properties of HEAs are still rare, it has been reported that the Ta34Nb33Hf8Zr14Ti11 HEA is a BCS type superconductor with a transition temperature 7.3K c T      . The results from two control samples, Ta-based heterostructures for comparison. It can be seen that the HEA-based magnetic heterostructures possess damping-like torque efficiencies that are about 50% to 80% of their Ta-based counterparts, though the composition of the major spin Hall material, Ta, is only 24.9% (at %) in the deposited HEA films. Only if we take the maximum possible Hf contribution (17.7%, with the largest based on Vegard's mixing rule for solid solution. However, other reports also suggest that Hf has a limited spin Hall effect ( Hf 0The magnitude of sat  is smaller yet comparable to reported values for pure Ta-based magnetic heterostructures [19,33]. The negative sign of sat  8 also confirms a negative spin Hall ratio (or spin Hall angle) of HEA, similar to that of Ta [9] and W [10] systems. Based on the theory of SHE-induced SOT, sat  of a magnetic heterostructure with PMA can be related to its damping-like torque efficiency DL  by    eff sat 0 FM / 2 / 2 DL s e M t      [34], where eff FM t is the effective thickness of ferromagnetic layer. Therefore, by using VSM- determined 3 1490emu/cm s M  ( 6 1.49 10 A/m  in SI units) and eff CoFeB 0.69 nm t  , we can estimate DL  of these samples. We summarize the magnitude of damping-like SOT efficiencies DL  of HEA-based magnetic heterostructures with different HEA thicknesses ( HEA t ) in Fig. 4 (d). The estimated DL  for the HEA-based samples increases with respect to the buffer layer thickness, with a maximum of HEA 0.033 0.004 DL   at HEA 5 nm t  . This thickness dependence of HEA DL  can be well-fitted to a simple spin-diffusion model [6],       HEA HEA HEA HEA 1 sech DL DL s t t t           , with spin diffusion length HEA 2.5nm s   and   HEA 0.04 DL t with PMA (Ta( Ta 3.5 nm and 4 nm t  )/CoFeB(1.4)/Hf(0.5)/MgO(2)/Ta(2)), are also plot in Fig. 4 (d) 9 reported Hf 0.11 DL   [26]) into account, the damping-like torque efficiency will reach HEA 0.03 DL   DL   ) [22], especially when it's thin ( 2 nm  ) . J W Yeh, S K Chen, S J Lin, J Y Gan, T S Chin, T T Shun, C H Tsau, S , J. W. Yeh, S. K. Chen, S. J. Lin, J. Y. Gan, T. S. Chin, T. T. Shun, C. H. Tsau, and S. Y. . Chang , Advanced Engineering Materials. 6299Chang, Advanced Engineering Materials 6, 299 (2004). . J W Yeh, Annales De Chimie-Science Des Materiaux. 31633J. W. Yeh, Annales De Chimie-Science Des Materiaux 31, 633 (2006). . M I Dyakonov, V I Perel, Physics Letters A. 35459M. I. Dyakonov and V. I. Perel, Physics Letters A 35, 459 (1971). . J E Hirsch, Physical Review Letters. 831834J. E. Hirsch, Physical Review Letters 83, 1834 (1999). . S F Zhang, Physical Review Letters. 85393S. F. Zhang, Physical Review Letters 85, 393 (2000). . L Q Liu, T Moriyama, D C Ralph, R A Buhrman, Physical Review Letters. 10636601L. Q. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, Physical Review Letters 106, 036601 (2011). . I M Miron, Nature. 476189I. M. Miron et al., Nature 476, 189 (2011). . L Q Liu, O J Lee, T J Gudmundsen, D C Ralph, R A Buhrman, Physical Review Letters. 10996602L. Q. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and R. A. Buhrman, Physical Review Letters 109, 096602 (2012). . L Q Liu, C.-F Pai, Y Li, H W Tseng, D C Ralph, R A Buhrman, Science. 336555L. Q. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Science 336, 555 (2012). . C.-F Pai, L Q Liu, Y Li, H W Tseng, D C Ralph, R A Buhrman, Applied Physics Letters. 101122404C.-F. Pai, L. Q. Liu, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Applied Physics Letters 101, 122404 (2012). . Y Niimi, Y Kawanishi, D H Wei, C Deranlot, H X Yang, M Chshiev, T Valet, A Fert, Y Otani, Physical Review Letters. 109156602Y. Niimi, Y. Kawanishi, D. H. Wei, C. Deranlot, H. X. Yang, M. Chshiev, T. Valet, A. Fert, and Y. Otani, Physical Review Letters 109, 156602 (2012). . Y Niimi, M Morota, D H Wei, C Deranlot, M Basletic, A Hamzic, A Fert, Y Otani, Physical Review Letters. 106126601Y. Niimi, M. Morota, D. H. Wei, C. Deranlot, M. Basletic, A. Hamzic, A. Fert, and Y. Otani, Physical Review Letters 106, 126601 (2011). . Y Niimi, H Suzuki, Y Kawanishi, Y Omori, T Valet, A Fert, Y Otani, Physical Review B. 8954401Y. Niimi, H. Suzuki, Y. Kawanishi, Y. Omori, T. Valet, A. Fert, and Y. Otani, Physical Review B 89, 054401 (2014). . P Laczkowski, Applied Physics Letters. 10411P. Laczkowski et al., Applied Physics Letters 104, 4870835 (2014). 11 . L K Zou, S H Wang, Y Zhang, J R Sun, J W Cai, S S Kang, Physical Review B. 9314422L. K. Zou, S. H. Wang, Y. Zhang, J. R. Sun, J. W. Cai, and S. S. Kang, Physical Review B 93, 014422 (2016). . M.-H Nguyen, M Zhao, D C Ralph, R A Buhrman, Applied Physics Letters. 108242407M.-H. Nguyen, M. Zhao, D. C. Ralph, and R. A. Buhrman, Applied Physics Letters 108, 242407 (2016). . S Fukami, C L Zhang, S Duttagupta, A Kurenkov, H Ohno, Nature Materials. 15535S. Fukami, C. L. Zhang, S. DuttaGupta, A. Kurenkov, and H. Ohno, Nature Materials 15, 535 (2016). . Y X Ou, S J Shi, D C Ralph, R A Buhrman, Physical Review B. 93220405Y. X. Ou, S. J. Shi, D. C. Ralph, and R. A. Buhrman, Physical Review B 93, 220405 (2016). . C F Pai, M Mann, A J Tan, G S D Beach, Physical Review B. 93144409C. F. Pai, M. Mann, A. J. Tan, and G. S. D. Beach, Physical Review B 93, 144409 (2016). . S Ikeda, Nature Materials. 9721S. Ikeda et al., Nature Materials 9, 721 (2010). . T Liu, J W Cai, L Sun, AIP Advances. 232151T. Liu, J. W. Cai, and L. Sun, AIP Advances 2, 032151 (2012). . C.-F Pai, M H Nguyen, C Belvin, L H Vilela-Leao, D C Ralph, R A Buhrman, Applied Physics Letters. 10482407C.-F. Pai, M. H. Nguyen, C. Belvin, L. H. Vilela-Leao, D. C. Ralph, and R. A. Buhrman, Applied Physics Letters 104, 082407 (2014). . Y W Oh, K D Lee, J R Jeong, B G Park, Journal of Applied Physics. 115Y. W. Oh, K. D. Lee, J. R. Jeong, and B. G. Park, Journal of Applied Physics 115, 17c724 (2014). . A T Hindmarch, V Harnchana, A S Walton, A P Brown, R M D Brydson, C H Marrows, Applied Physics Express. 413002A. T. Hindmarch, V. Harnchana, A. S. Walton, A. P. Brown, R. M. D. Brydson, and C. H. Marrows, Applied Physics Express 4, 013002 (2011). . P J Chen, Y L Iunin, S F Cheng, R D Shull, IEEE Transactions on Magnetics. 521P. J. Chen, Y. L. Iunin, S. F. Cheng, and R. D. Shull, IEEE Transactions on Magnetics 52, 1 (2016). . J Liu, T Ohkubo, S Mitani, K Hono, M Hayashi, Applied Physics Letters. 107232408J. Liu, T. Ohkubo, S. Mitani, K. Hono, and M. Hayashi, Applied Physics Letters 107, 232408 (2015). . P Kozelj, S Vrtnik, A Jelen, S Jazbec, Z Jaglicic, S Maiti, M Feuerbacher, W Steurer, J Dolinsek, Physical Review Letters. 113107001P. Kozelj, S. Vrtnik, A. Jelen, S. Jazbec, Z. Jaglicic, S. Maiti, M. Feuerbacher, W. Steurer, and J. Dolinsek, Physical Review Letters 113, 107001 (2014). . Y Ou, D C Ralph, R A Buhrman, Applied Physics Letters. 110192403Y. Ou, D. C. Ralph, and R. A. Buhrman, Applied Physics Letters 110, 192403 (2017). . M T Johnson, P J H Bloemen, F J A , J J Devries, Reports on Progress in Physics. 591409M. T. Johnson, P. J. H. Bloemen, F. J. A. denBroeder, and J. J. deVries, Reports on Progress in Physics 59, 1409 (1996). . Q Hao, G Xiao, Physical Review Applied. 334009Q. Hao and G. Xiao, Physical Review Applied 3, 034009 (2015). . S Cho, S H C Baek, K D Lee, Y Jo, B G Park, Scientific Reports. 514668S. Cho, S. H. C. Baek, K. D. Lee, Y. Jo, and B. G. Park, Scientific Reports 5, 14668 (2015). . O J Lee, L Q Liu, C F Pai, Y Li, H W Tseng, P G Gowtham, J P Park, D C Ralph, R A Buhrman, Physical Review B. 8924418O. J. Lee, L. Q. Liu, C. F. Pai, Y. Li, H. W. Tseng, P. G. Gowtham, J. P. Park, D. C. Ralph, and R. A. Buhrman, Physical Review B 89, 024418 (2014). . S Emori, E Martinez, K J Lee, H W Lee, U Bauer, S M Ahn, P Agrawal, D C Bono, G S D Beach, Physical Review B. 90184427S. Emori, E. Martinez, K. J. Lee, H. W. Lee, U. Bauer, S. M. Ahn, P. Agrawal, D. C. Bono, and G. S. D. Beach, Physical Review B 90, 184427 (2014). . A Thiaville, S Rohart, E Jue, V Cros, A Fert, Europhys. Lett. 10057002A. Thiaville, S. Rohart, E. Jue, V. Cros, and A. Fert, Europhys. Lett. 100, 57002 (2012). . R Ramaswamy, X P Qiu, T Dutta, S D Pollard, H Yang, Applied Physics Letters. 108202406R. Ramaswamy, X. P. Qiu, T. Dutta, S. D. Pollard, and H. Yang, Applied Physics Letters 108, 202406 (2016).
{'fraction_non_alphanumeric': 0.07634934233734818, 'fraction_numerical': 0.048782946127097715, 'mean_word_length': 3.5839685839685838, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'High-entropy alloy (HEA) is a family of metallic materials with nearly equal partitions of five or more metals, which might possess mechanical and transport properties that are different from conventional binary or tertiary alloys. In this work, we demonstrate current-induced spin-orbit torque (SOT) magnetization switching in a Ta-Nb-Hf-Zr-Ti HEA-based magnetic heterostructure with perpendicular magnetic anisotropy (PMA). The maximum damping-like SOT efficiency from this particular HEA-based magnetic heterostructure is further determined to be HEA 0.033 DL \uf07a \uf0bb by hysteresis loop shift measurements, while that for the Ta control sample is Ta 0.04 DL \uf07a \uf0bb . Our results indicate that HEA-based magnetic heterostructures can serve as a new group of potential candidates for SOT device applications. \uf02a', 'arxivid': '1705.07248', 'author': ['Tian-Yue Chen \nDepartment of Materials Science and Engineering\nNational Taiwan University\n10617TaipeiTaiwan\n', 'Tsao-Chi Chuang \nDepartment of Physics\nNational Taiwan University\n10617TaipeiTaiwan\n', 'Ssu-Yen Huang ', 'Hung-Wei Yen \nDepartment of Materials Science and Engineering\nNational Taiwan University\n10617TaipeiTaiwan\n\nDepartment of Physics\nNational Taiwan University\n10617TaipeiTaiwan\n', 'Chi-Feng Pai [email protected] \nDepartment of Materials Science and Engineering\nNational Taiwan University\n10617TaipeiTaiwan\n'], 'authoraffiliation': ['Department of Materials Science and Engineering\nNational Taiwan University\n10617TaipeiTaiwan', 'Department of Physics\nNational Taiwan University\n10617TaipeiTaiwan', 'Department of Materials Science and Engineering\nNational Taiwan University\n10617TaipeiTaiwan', 'Department of Physics\nNational Taiwan University\n10617TaipeiTaiwan', 'Department of Materials Science and Engineering\nNational Taiwan University\n10617TaipeiTaiwan'], 'corpusid': 118924782, 'doi': '10.1103/physrevapplied.8.044005', 'github_urls': [], 'n_tokens_mistral': 7645, 'n_tokens_neox': 6397, 'n_words': 3277, 'pdfsha': '2086ce89eac21d877fa0be832c13747d66e30cb6', 'pdfurls': ['https://export.arxiv.org/pdf/1705.07248v1.pdf'], 'title': ['The Spin-Orbit Torque from a Magnetic Heterostructure with High-Entropy Alloy', 'The Spin-Orbit Torque from a Magnetic Heterostructure with High-Entropy Alloy'], 'venue': []}
arxiv
DotDFS: A Grid-based high-throughput file transfer system Alireza Poshtkohi Department of Electrical Engineering Shahed University 331911865TehranIran M B Ghaznavi-Ghoushchi Department of Electrical Engineering Shahed University 331911865TehranIran DotDFS: A Grid-based high-throughput file transfer system Data GridsFile Transfer ProtocolsHigh Throughput File TransferGrid ComputingGrid SecurityModeling Parallel TCP Throughput DotGrid platform is a Grid infrastructure integrated with a set of open and standard protocols recently implemented on the top of Microsoft .NET in Windows and MONO .NET in UNIX/Linux. DotGrid infrastructure along with its proposed protocols provides a right and solid approach to targeting other platforms, e.g., the native C/C++ runtime. In this paper, we propose a new file transfer protocol called DotDFS as a high-throughput distributed file transfer component for DotGrid. DotDFS introduces some open binary protocols for efficient file transfers on current Grid infrastructures. DotDFS protocol also provides mechanisms for multiple file streams to gain high-throughput file transfer similar to GridFTP protocol, but by proposing and implementing a new parallel TCP connection-oriented paradigm. In our LAN tests, we have achieved better results than Globus GridFTP implementation particularly in multiple TCP streams and directory tree transfers. Our LAN experiences in memory-to-memory tests show that DotDFS accesses to the 94% bottleneck bandwidth while GridFTP is accessing 91%. In LAN disk-to-disk tests, comparing DotDFS protocol with GridFTP protocol unveils a set of interesting and technical problems in GridFTP for both the nature of the protocol and its implementation by Globus. In the WAN experimental studies, we propose a new idea for analytical modeling of file transfer protocols like DotDFS inspired by sampling, experimentation and mathematical interpolation approaches. The cross-platform and open standard-based features of DotDFS provide a substantial framework for unifying data access and resource sharing in real heterogeneous Grid environments.A. Poshtkohi, M.B. Ghaznavi-Ghoushchi transfer, partial file transfer, automatic negotiation of TCP buffer/window sizes, and support for reliable and restartable data transfer.[4,5,6] describe GridFTP protocol and its full features. The authors in [7] also addressed the implementation problems of GridFTP for .NET and Windows platforms. Some GridFTP problems formerly studied by researchers are long-time GridFTP setup due to any application setup, connection and authentication, firewalls, and network address translation[8,9,10,11].We have developed DotGrid platform[12,13,14]which enables the creation of Grid infrastructure using Microsoft . NET [15] in Windows and MONO . NET [16] in UNIX and Linux environments. It provides Grid services and toolkits for fast developing Grid applications.The main goal of DotGrid project[12,13,14]is to develop a cross-platform framework for sharing computational resources between heterogeneous DotGrid nodes. To build a cross-platform desktop Grid infrastructure in heterogeneous platforms, DotGrid makes use of high-performance implementation of the native .NET Framework in Windows and MONO .NET project in UNIX/Linux family of operating systems, respectively. Resource sharing and bulk data transfer are two major architectural needs that we have investigated during the design process of DotGrid.Currently, we are implementing a new cross-platform Grid/Cloud infrastructure, which extends and utilizes the open and standard protocols suggested in DotGrid platform for native runtime, i.e., C/C++ stack. This new platform is applying the ECMA international open standards [33,34] to provide a much more standardized native implementation of DotGrid for scientific and enterprise Grid/Cloud communities in the most recently operating systems including UNIX, Linux, and Windows.Recently, WAN-based file systems like Lustre[18], and Gfarm [20] are maturated. Due to our knowledge, it seems that DotDFS is more like to GridFTP than the above mentioned file systems. For example, DotDFS may be used, as well as GridFTP, in Gfarm as the underlying file transfer protocol. DotDFS has many different aspects ranging from structure to architecture with GridFTP protocol. Most of them are declared during this paper.In this paper, we introduce a Grid-based high-throughput file transfer system, called DotDFS. This can be used as a component of a computational Grid [12,13,14]or data Grid [17] environments.The rest of the paper is organized as follows. In section two, we present DotDFS protocol. Section 3 infers highperformance server design patterns for the DotDFS protocol. Sections four and six describe DotDFS implementation and DotSec GSI. Section five focuses on comparison of DotDFS protocol with GridFTP protocol. The LAN and WAN experimental studies are described in section 7. We conclude in section 8.DotDFS ProtocolThe current version of DotDFS protocol is a binary protocol model like TCP/IP, DNS and SOCKS protocols. This platform-independent feature allows DotDFS protocol to be re-implemented in other platforms and gains more interoperability. This approach results in more performance and throughput in the cost of implementation complexity.[7] reports interoperability problems of the Globus GridFTP server due to the MPI used in Globus core. The two major problems in the current .NET GridFTP implementation are: no support of the authentication in data channels and no interoperability of the stripped data transfers with the Globus GridFTP. The .NET GridFTP extensively used from Windows services and native Win32 APIs. This causes it could not be ported to UNIX/Linux platforms via MONO .NET.DotDFS is a high-throughput file transfer protocol which meets the requirements to set it as a background for applications in the areas of distributed, cluster, grid and cloud computing. These requirements are discussed in this section and have been considered in our current DotDFS implementation.DotDFS heavily depends upon a set of abstractions that are used to hide infrastructure dependencies. DotDFS APIs can be used for implementing applications in high-throughput data transfers similar to GridFTP. Other applications that can be implemented relied upon on DotDFS are: data intensive Grid applications, Grid-based resource sharing, booting operating systems from high-speed networks, peer-to-peer file systems and distributed databases. Introduction Recently, integrated use of system resources are utilized by Grid and Cloud platforms. Grid [1,2,3] infrastructures provide the ability to share, select and aggregate distributed resources as computers, storage systems or other devices in an integrated way. Effective end-to-end transmission of data demands a system approach in which file systems, computers, network interfaces, and network protocols are managed in an integrated fashion to meet performance and robustness goals [4]. Secure and high-speed data transfers are vital and profound in Grid environments. GridFTP protocol is recently used as the de-facto standard for bulk data transfers in various Grid projects in high-bandwidth wide-area networks all over the world [4,5,6]. Globus [4] has developed GridFTP protocol for UNIX-class style operating systems [4] , and a team at the University of Virginia implemented GridFTP for Windows-based operating systems via Microsoft .NET Platform [7]. Globus has fully implemented the GridFTP protocol with the following major features: thirdparty control of data transfer, authentication, data integrity, data confidentiality, striped data transfer, parallel data In DotDFS protocol, upon establishment of a connection to a DotDFS server, each client negotiates its favorite session with a set of flexible parameters. The topological architecture of DotDFS protocol is depicted in Fig. 1. The following paragraphs present the details of the protocol. The default port of DotDFS server is 2799. When a client starts to negotiate with the server, it sends a one-byte binary header to notify the server from its requested service type. In the current version of DotGrid, values of 0, 1, and 2 are defined for specifying the request modes of DotDfsMode, DotGridRemoteProcessMode and DotGridThreadMode service, respectively. DotDFS protocol requires that all established connections to the server must be authenticated and authorized after service selection through DotSec GSI and DotSec TSI. This follows with X-Channels or TSI X-Channels selection between client and server. DotSec protocol is explained in section 6 with more details. The current version of DotDFS supports three servicing modes from the viewpoint of the connected clients to a DotDFS server including DFSM, FTSM, and PathM. Distributed File System Mode (DFSM) This requested mode supports the access to the files and data sharing mechanisms which are used in conventional distributed file systems. Moreover, this mode can be used for stripped and third-party data transfers. One good example of this mode is a situation with one or more transport streams between m network endpoints sending side and n network endpoints on the receiving side. In [12,13], we developed a distributed cryptographic engine to encrypt/decrypt and transfer petabytes scale files by using data striping approaches. The major of DFSM origination is inspired by the unique features of FileStream class and System.IO namespace declarations for file/directory manipulations stated in Microsoft .NET Framework 1 and later versions. FileStream features in file IO streaming is the main reason for selecting it for DFSM mode of DotDFS protocol. Due to the wide range of methods and arguments shown in Fig. 1 (like Seek(), Read(), and Flush()), DotDFS protocol not only supports POSIX file semantics, but also it can be used in other scope of applications like remote file streaming and remote named pipes for inter-process communications (IPC). Recently, we are investigating some approaches to fully supporting this feature relied upon DotDFS protocol for unifying IPC's via in UNIX/Linux and Windows operating systems on Grids. Access to NFS, Microsoft DFS (this will be discussed with PathM mode in section 2.3) and other pluggable file systems are also the applications of DFSM mode. Here is an example for the binary model feature of the DotDFS protocol. Let's have a DotDFS session with a Windows platform client and a Linux platform server. The DFSM mode for copying a 100MB remote file to local storage is examined. The C++ pseudo code of this scenario is shown in Fig. 2. The required binary header for remote->Read() method of Fig. 2 is illustrated in Fig. 3. DotDFS server replies to the Read() request of client with the value of zero in Method block. This value is used for declaring the response of Read() request. The length of transmitted buffer is stored in RW-Length field. RW-Mode specifies the value of RW-Length in turn. We have applied a different approach in this header and in spite of all other binary protocols with RW-Length default value of 4-byes, we set it variable and controlled with the actual length of the read buffer from the storage system. For example, if the submitted buffer size is less than 2 16 and greater than 2 8 , then RW-Mode is filled with value of 2. This technique in buffer space strength reduction is widely used during the design of our proposed DotDFS protocol. File Transfer System Mode (FTSM) In this mode, DotDFS protocol not only supports the Grid computing demands investigated by GridFTP developer teams [4,5,6] but also includes procedures and methods for high-throughput and high-performance in dedicated client-server architectures based on DotDFS protocol. FTSM mode results in high throughput and performance file transfer in DotDFS protocol. The following paragraphs describe the major features of FTSM. DotDFS protocol defines a set of specifications for increasing file transfer throughput through establishing multiple TCP connections in parallel to accelerate start-up in the TCP slow rate, and negotiating the TCP Socket Window buffer size between a DotDFS server and client before starting the DotDFS session according to the bandwidth-delay product of a network. This case meets when X-Channels are changed to be TCP/IP channels as shown in Fig. 1. It must be noticed that in DotDFS protocol, eXtensible channels are an abstraction concept for lowlevel transmission control protocols between endpoints. The TCP/IP stack is the main default protocol used in DotGridSocket layer in which all network I/O requests are processed in the specifications and implementation of the DotDFS protocol. Due to the WAN-based TCP overheads, the SCTP protocol [32] will be considered as a replacement for TCP protocol in our future research development. Fig. 4 shows the client-server DotDFS protocol communications in FTSM mode. All the shown communications are binary forms. Initially client sends a FTSM service selection header to DotDFS server and requests the service type from DotDFS server. Then, if there is an available service, the server sends an available DotDFS service header, and puts the client in authentication stage and negotiation enforcement for sending DotSec TSI parameters. Upon completion of session selection mode (stage 5 in Fig. 4), client sends FTSM mode request. If the supported mode notification header is submitted from the server, the first stream connection of client is now established and a new DotDFS session is created in server side. In this step, the client sends file transfer session parameters including the number of parallel streams, TCP Window buffer size, Transfer Type (Upload/Download), and a unique GUID. In the next step, server is in waiting mode until all the specified parallel streams by stream1 are established. The connection schemas of the remaining streams are similar to the first stream. When all of the parallel streams are established, due to the Transfer Type (Upload/Download), the bulk data transfer of stage 10 is started. It is necessary to note that in both upload and download sessions, the stream connections are initiated from client side to server side. Furthermore, depending on the data flow the following operations are done in the client-server sides: reading from local storage and sending to the remote server (in upload scenario initiated by the client), and receiving from the remote server and writing to local storage (in download scenario initiated by the client). In the DotDFS protocol, server does not connect to the client, and it seems this is a good feature in our proposed protocol. This results in omitting concepts like control channel used in GridFTP. This feature also solves some problems of GridFTP protocol addressed in [9,10,11]. These problems in more detail are discussed in section 2.4.4. In Fig. 5, our suggested and implemented binary protocol headers during data transfer channels in stage 10 of Fig. 4 are shown. In Fig. 5, the SeekValue is the value of file offset for transferred file block and ReadValue is the transferred data length. These two blocks have variable length specified by L 1 and L 2 respectively. If both of L 1 and L 2 are zero, this is a notification of successful transfer of all file blocks. The connection side receives the zero value, closes the established streams and releases the used OS resources. FTSM mode supports the feature of reusable transfer channels simply by adding the required headers shown in Fig. 5. These headers must be included during the DotDFS session initiation on the connection establishment. PathM Mode PathM mode is designed to support basic features like creation/deletion of remote files/directories and related features that are depicted in Fig. 1. Fig. 6 shows the client-server communication architecture of DotDFS protocol in PathM mode. Steps 1-7 in Fig. 4 are also used in Fig. 6. In the PathM mode, DotDFS server operates like a RPC server, but all the requested methods of client are previously defined as binary in the client-server negotiation protocol. The PathM mode methods for client-server communication are based on the mechanisms shown in Fig. 1 and 6. PathM mode also supports a unique feature for upload/download of huge small size files in the client side. This is implemented via Upload/Download SmallFile methods. The reusable channels support in PathM gains the RPC server-like functionality to our proposed protocol. In PathM mode each client may ask consecutive commands to execute from the server many times during one open DotDFS session. In section 4.2, a sample scenario of this functionality is presented for DotDFS Directory Tree Transfers. Client Server conn ecte d strea m More DotDFS Protocol Features The DotDFS protocol includes many other features and specifications. In this section, we briefly state a few other features available at the current proposed protocol. Striped Data Transfers In FTSM mode, the TSI/X-Channels (shown in Fig. 7 and described in sections 2.2 and 4.1) may actually consist of several TCP streams from multiple hosts identified by GUID's that cause the realization of the striped data transfer support in DotDFS protocol. Data may be striped or interleaved across multiple servers, as in a parallel file system. As well as, stripped transfers is fully supported in DFSM mode [12,13]. Partial File Transfers Partial file transfer helps Grid processes and applications tend to transfer only fragments of a file. This mechanism is supported by pre-defined DotDFS headers. Stateless Architecture DotDFS protocol introduces a natural stateless architecture in DFSM operating mode. This means that DotDFS servers do not keep track of DotDFS client requests according to opened files, file positions and etc. This leads DotDFS clients to be able to fail and resume without disturbing our system as whole and allows fast development of Grid-based parallel cluster file systems, in future. The Firewalls' Issue For intermediate solving diverse problems with the firewalls stated in [10,11], all DotGrid services are running on default listening port 2799 or client API calls are enforced to be established on this port. On each machine, one process called DotGrid Listener listens to this default port for Grid task requests. After receiving a service request to this process, based on the sent binary header request from client, the selected request is relayed to an appropriate service runtime manager like DotGridThread, DotDFS and DotGridRemoteProcess stacks. The remote service selection is done by DotGrid binary request header protocol. Then, as stated in the sections 2.1, 2.2 and 2.3, all upload and download requests in DotDFS protocol must be originated from client to the server. High-Performance Server Design Patterns for DotDFS Protocol Constructing highly concurrent systems is inherently sophisticated and difficult. This issue is more obvious in communication protocols where the protocol specification can greatly influence on the implementation and development methods of the protocol. While threads are the most commonly used tools to specify concurrency but large resource consumption and scalability limitations in many systems relied upon threading models have made researchers focus on event-driven techniques. Building a system merely based on threading models or event-driven methods is the main issue to make the system complex and difficult to evolve from different design aspects. A very good way for designing a concurrent system is to establish a level of balance between these two models. Eventdriven techniques are advantageous to achieve high concurrency; but when systems architects take the problem of developing real time systems into consideration, threads bring out significant importance to exploit multicoremultiprocessor parallelism and deal with blocking I/O mechanisms appropriately. Since file transfer protocols like FTP and NFS are more disk I/O-bound, almost no research work has been conducted to suggest a concurrent file transfer protocol that simultaneously employs threaded and event-driven models in the protocol level. Also, the specification and implementation of legacy file transfer protocols nearly remain intact in the level of the processed model, multithreaded model or threaded pool models. Due to our knowledge DotDFS is the first concurrent file transfer protocol that, from this viewpoint, presents a new computing paradigm in the field of data transmission protocols. In the remainder of this section we examine the benefits and disadvantages of two these models and will finally introduce the hybrid server architecture for DotDFS protocol discussed in section 2. Threaded-Based Concurrency Pattern (TBCP) A thread of execution is the smallest processing unit that can be scheduled by an operating system. Operating systems usually allow processes to switch between different threads through thread scheduling primitives such as preemptive and cooperative scheduling; this operation is called the term context switch. A primary overhead exposed by threads is O(n) in the number of threads. While the number of threads is increasing, the system experiences overheads, including scheduling and context switching, memory pressure due to thread footprints, cache and TLB misses, and contention for shared resources such as locks. In highly threaded systems, the instruction cache tends to take many misses as the thread's control passes through many unrelated code modules to process the task. When a context switch explicitly takes place by the OS preemption or cooperatively by the requested thread, other threads will invariably flush the waiting thread's state out of the on-chip cache. Event-Driven Concurrency Pattern (EDCP) As stated in the previous section, threads exhibit many scalability limitations in developing highly concurrent systems. Event-driven methods are used to overcome problems of threads. In EDCP model, a server consists of a certain number of threads that loop frequently and process events with different types from a queue. In this way, each request is described as a set of finite state machines (FSMs). Events are usually generated by operating system kernel and accessible through system calls in forms of network and disk I/O readiness, completion notification, and timers. As seen in TBCP model; the number of context switches decreases due to multiplexing requests in a definite amount of threads; avoiding synchronization mechanisms not only increase the performance but reducing the number of consumed threads strongly boosts instruction and data locality. Hybrid DotDFS Concurrency Pattern (HCP) As shown in two EDCP and TBCP models both have advantages and disadvantages. In this section, we propose a hybrid concurrency pattern for DotDFS protocol, which directly dictates the protocol implementation and specification. The main idea of this methodology is the use of an approach to balance between EDCP and TBCP models in favor of the HCP model. In this section, to show the efficiency of our proposed approach we only examine the structure of HCP model for DotDFS FTSM upload mode in the server as the wildly-used case in Grids. Because DotDFS is a disk I/O bound protocol, to detach the protocol from non-standard asynchronous disk I/O interfaces for maintaining the protocol portability and universality features we have made use of threads to provide a nonblocking disk I/O architecture. Each thread manages one DotDFS session. To use the benefits of event-driven techniques, multiple TCP streams are processed through event dispatching methods for each thread of transfer session. Therefore, in the proposed architecture, threads are used to execute parallelism on multiprocessors in order to eliminate blocking disk I/O drawbacks, and event-driven techniques are used to exploit high-throughput data transfer through nonblocking network I/O interfaces. Considering the diversity and heterogeneity of the system functions that allow asynchronous network I/O to be used, our current focus on the event-driven component of HCP architecture has been given to the system function select(). Fig. 7 illustrates a typical timing diagram of the event-driven DotDFS server architecture per each FTSM upload session in which n parallel connections are processed in a thread of execution. In this diagram, the infinite event dispatcher loop is the main part of the event-driven system that continuously checks the read-readiness of sockets in the black-border loop. After selecting m sockets by the event dispatcher loop, the finite I/O loop in an iteration with the size of m writes the packets received from the protocol subsystem through the system function recv() into the hard disk by calling the system function write(). The top side of the timing diagram shows mode switches (mode transitions), and the bottom side shows copy operations, between user and kernel space. As this diagram tells us for every move from step 0 to step 6, there to the number of 2 + 4 * mode switches occur. Mode switches in operating systems are very similar to a simple function call in user space in which the return operation is done to the user space after jumping to the kernel. On most architectures, the cost of these mode transitions consists of saving all (or some, or none) of the registers to the stack, pushing the function arguments to the stack (or putting them in registers), incrementing the stack pointer and jumping to the beginning of the new code. Hence the HCP model only slightly suffers from mode switch overheads, this negative cost is notably little and can be completely ignored against the overheads exposed by the TBCP model. It is necessary to note that the overheads of mode switches cannot be removed in any way because user space applications (such as GridFTP server) must call system functions in having access to the hardware functionalities provided by proper kernel interfaces. While our concern in the initial phase of DotDFS implementation was to maintain its portability running on the most existing platforms through POSIX-compliant system function standards like write(), select() and recv(), this approach exposes extra overheads due to additional copies occurred between hardware, user space and kernel memory buffers (this issue is also exactly concerned with GridFTP server). These overheads can be eliminated by using zero-copy mechanisms, considering these techniques are beyond the scope of this paper. As the bottom side of Fig. 7 shows, 4 * copies happen in steps 3 to 6 for each iteration of the inner loop highlighted as the green border, including CPU and DMA copies. We have applied a novel method in the design of HCP model. As earlier stated, we have used select() to maintain more DotDFS portability. But in practice we cannot implement a full DotDFS server core with one thread and one loop containing the select() function; because this method not only reduces the server parallelism, but also passing a large set of socket descriptors to select() significantly decreases the scalability of the developed server in high-traffic environments due to frequent copies during the transition from user space to the kernel and vice versa. Since using five parallel streams is normally common in Grid, for example, to upload large files; thus it is more reasonable that these network streams are processed in a separate thread through multiplexing network I/O techniques. Using different event-driven models with more capabilities, which ultimately make DotDFS protocol to have more sophisticated architectures, will be one of our straightforward future research works. Our Implementation of the Proposed DotDFS Protocol This section presents an implementation survey of the main units of the DotDFS protocol proposed in section 2. Architecture of DotDFS Implementation in FTSM Upload Mode In this section, we describe how a DotDFS client in DFSM mode can transfer large file sizes to a DotDFS server by using parallel streams to achieve a high transfer rate. Fig. 8 shows the architecture of client-server DotDFS implementation in FTSM upload mode. 1. When a client wants to connect to a DotDFS server, the client calls the UploadClient API (the client has to declare some parameters such as the number of parallel streams, file size, TCP Windows size and the underlying DotSec TSI parameters when accessing to this client API). The UploadClient API creates a master thread in which management of n parallel connections is flowed. At the beginning of the session, this thread generates a GUID that will be used to distinguish all the counterpart parallel connections for a DotDFS transfer session in the server side. All streams ( ) are established by this thread to the remote DotDFS server. We have implemented a Local File Read Queue (LFRQ) in DotDFS APIs to decrease side effects like extra local storage activity, opening large file handles for large parallel streams and random file access in using extensive Seek() file system calls. LFRQ sequentially reads file blocks from the file system and put them into a queue. A locking mechanism is used when the stream j issues a request to access a file block from the buffer. While a read request is being received to the queue, all other file blocks in LFRQ are locked to provide a coherency model until the request of file block read completed. The throughput of file block read from LFRQ is directly depended on available external bandwidth and network latency in which the DotDFS session is being executed via n parallel streams. With regard to this approach, there are always enough file blocks provided by LFRQ for the requesting streams. After the complete establishment of a DotDFS session, the master thread managing the DotDFS session investigates writability of sockets, which constitute the endpoint streams, by successive calling the system socket select() function. In this time, the master thread performs requesting the needed file blocks to the number of selected sockets by the system select() function from LFRQ and sends them over the writable streams. All opened file and socket handles will be released and returned to the host operating system at the end of each DotDFS session (it means that all file blocks had been sent to DotDFS server) by the manager thread. 2. The following operations are performed at DotDFS server in each DotDFS session. A thread listening on default port 2799 receives the requests from the network adaptor. When a request is received, this thread examines the sent parameters by the stream j, and extracts the GUID parameter which the stream contains. The listener thread investigates whether the GUID exists in the session pool's hash table or not. If there is not any GUID related to the received GUID, then it represents the starting of a new DotDFS session. In this state, if n (the number of parallel streams) is greater than zero, then the listener thread creates a secondary thread, named as Manager Thread (MT), and invokes Waiter Loop Module (WLM) method, which contains in the class's member method that is executed by MT (also parameters like n, remote file name, local file name and TCP Window size are passed to this module). Also, MT puts the received GUID into the session pool. In next requests that are received by the listener thread, if these requests' GUID is existed in the session pool then CurrentReceivedStreams variable related to MT is increased with value of 1 by the listener thread. Since the beginning of the execution of WLM by the listener thread, WLM is continuously checking the value of CurrentReceivedStreams variable and comparing it with the passed value of n. When CurrentReceivedStreams equals to n, WLM terminates its loop iteration and runs Parallel IO Executor (PIOE) module, while deleting references to the counterpart connections related to the current session specified by its GUID. PIOE is the main core of the DotDFS server. POIE in an infinite loop, similar to the mentioned client side model, investigates the readability of sockets, which constitute the endpoint streams, by successive calling system socket select() function. POIE selects the readable sockets, receives file blocks from these sockets and writes them into the storage system. Because networks like WAN usually have high latency behavior, we cannot consider any certain coherency from the aspect of file block offset for receiving the sent file packets from client to server side. When file blocks are conveyed based on DotDFS protocol, a set of information is sent to the destination server encapsulated in the headers of DotDFS protocol by DotDFS client, such as the block offset and block count. Hence in regard to the random receiving of these file blocks, POIE must call Seek() file system function in a repetitive sequence. To decrease this overhead, we have implemented a queue module similar to the LFRQ in the DotDFS client side. It attempts to queue file blocks, that are located in an offset interval, and write the whole queue capacity into the storage system. The overall effect of this technique is that the deteriorated coherency can somewhat be increased, and consequently, decreasing the Seek() system calls. In this way, we make use of buffering the new received file blocks in a typical 1MB buffer for coherency preservation and decreasing random file access through Seek() file system calls with writing the total buffer to the storage system. At the end of each DotDFS server side session, all consumed operating system resources will be released like opened native file handles and threads by session thread manager. In the above mentioned paragraphs, only a total of two threads are created in client and server sides that are responsible to manage a shared DotDFS session. This fact implicitly explains that the optimized kernel operating system methods in the proposal and implementation of the DotDFS protocol have been considered. DotDFS Directory Tree Transfers There are many situations where moving a directory tree is desired. For example, a grid user may wants to migrate from his/her whole home directory to another machine for load balancing or capacity-limit reasons. The DotDFS client and server architectures are extended to gain the directory tree transfer using the FTSM and PathM modes. The new implemented APIs are then included in DotGrid Platform SDK. The overall view of operations is illustrated in Fig. 9. In Fig. 9, the upload scenario of a local directory from client to a DotDFS server is depicted. When the client calls DirectoryUploadClient API, the following tasks are done and managed via the Manager Worker (MW): 1. First, MW establishes the PathM 1 route to the server. PathM 1 is the operating agent of recursive creation of the remote directory tree. Moreover, MW runs the Recursive Local Tree Directory Traversal Module (RLTDTM). Not only RLTDTM is the responsible agent for remote directory tree creation, but also it queues the found files into Small File Queue (SFQ) and Large File Queue (LFQ) modules based on small or large file sizes in each state of its directory traversal. 2. After completion of the local directory traversal and creation of the remote directory tree, MW creates m-1 connections (m is the number of parallel connections in PathM mode) to DotDFS server. Files queued in SFQ are transferred between two endpoints through these established channels. Also, MW simultaneously calls DotDFS UploadClient APIs stated in section 3.1 to transfer large LFQ files by using n parallel streams. As seen in this scenario, m+n transfer channels are established between two endpoints in parallel. If the LFQ and SFQ are not empty and LFQ contains minimum m queued files, then m+1 files are being transferred between a DotDFS client and server in parallel. In this approach, PathM and FTSM executor threads are in competition and do transmit SFQ and LFQ fetched files. There is no default compression algorithm in our implemented APIs. But these APIs have interfaces that a Grid/Cloud user may extend his/her own APIs with compression algorithms like GZIP, LZ77, BZip2, etc enabled. The above example illustrates the usefulness of the layered model of implementation in DotDFS protocol, and features like multi-level parallelism and reusable channels mentioned in sections 2.2 and 2.3, for gaining highthroughput and high-performance directory tree transfers. Comparison of DotDFS Protocol with GridFTP Protocol In this section, we compare DotDFS protocol with FTP and GridFTP protocols. Some GridFTP weaknesses and structural differences between DotDFS and GridFTP are discussed, so that the reader can understand more reasons (in addition to those considered in sections 2, 3 and 4) why a new concurrent file transfer protocol is proposed. The aim of this paper is not to decline the GridFTP protocol; rather we will show when a new protocol is designed from the ground-up for the next generation of distributed computing paradigms how it can lead to different and better results from a completely new and deep perspective. However, Globus in [] expresses the following phrase for selecting and extending the FTP protocol: "We chose the FTP protocol because it is the most commonly used protocol for bulk data transfers on the Internet and of the existing candidates from which to start (http, DPSS, HPSS, SRB, etc.) ftp comes closest to meet the needs of Grid applications." []. Table 1 shows a comprehensive comparison of different file transfer mechanisms including DotDFS, FTP, and GridFTP. Operating System Recourse Consumption Low High High DotDFS Protocol vs. GridFTP Protocol in Upload Mode In this section, we concentrate on comparing DotDFS and GridFTP protocols. Since the complete comparison of these two protocols is out of the number of pages of this paper, we consider just a specific taxonomy to explain. In this scenario, we assume that a client wants to upload a large file to a server via n parallel connections. DotDFS and GridFTP protocols use FTSM and passive X PUT modes, respectively. The server-side CFSMs of these two modes are illustrated in Fig. 10 and 11. These CFSMs are concurrently representing the protocol implementations and largely the protocol specifications as a collection of FSMs. The red boundaries mean those parts are executing in parallel. A suitable model for describing communication protocols and concurrent systems are communicating finite state machines (CFSMs). In the CFSM model, a protocol is defined as a collection of processes (i.e., the protocol entities) which exchange messages over error-free simplex channels. Obviously, the CFSM model virtually has become the de-facto standard to specify, verify and test communication protocols in the telecommunication industry. DotDFS CFSM, to a large extent, is representing the descriptions discussed in sections xx and yy. After authenticating the client through DotSec GSI, choosing the FTSM mode by the client, and receiving the FTSM parameters; the server checks whether the session has already been created using its GUID by the client or not. If the session has already been existed and the number of sockets in the hash table is not corresponding (equal) to the value of n received from the client, the server adds the new client stream to the hash table in state 8. In step 7, the server concurrently checks, so that if the number of client streams is equal to the value of n then moves the CFSM flow to state 9. If an error occurs during states 1 to 8, the next state will be 15. States 9 through 14 are the main DotDFS server core where file blocks are received from the client only in a loop in one thread of execution, through network I/O event dispatching mechanisms, and the server writes them to the hard disk in state 12. As the CFSM of the GridFTP server core shows (Fig. 11, inferred from GridFTP drafts and Globus published papers), it consists of + 1 processes that contain the control channel process and data channel processes. GridFTP protocol in this scenario cannot use event dispatching mechanisms to open only one process; because, firstly the control channel and data channels are two separate modules that must be executed in parallel, and secondly the different types of the Descriptor field in the X mode header (such as EOR, EODC, EOD, etc.) do not permit the server to uniformly manage n sockets in a loop with event dispatching mechanisms used. Additionally, in the source code level, Globus Toolkit makes extensive use of the UNIX fork()/exec() system functions to implement GridFTP client and server programs, and IPC methods to implement synchronization and communication between the control channel process and n data channel processes. Section zzz will argue the overheads of GridFTP implementation that are due to using these two functions. In this CFSM, the GridFTP server remains waiting for the client to choose PUT or GET mode after accepting the client connection and authenticating the client according to the RFC 2228. In addition to the parameters of the PUT command, the server in state 4 receives the number of parallel connections via the OPTS command. In state 5, the server creates n processes and remains waiting on ports p 1 , p 2 ,…, p n to accept ongoing client connections which will constitute the data channels. In state 6, GridFTP server receives the client requests as FTP commands in an infinite loop over the control channel. This allows the client to be aware of the transfer status (such as performance monitoring commands during X mode sessions that are sent and received by the client over the control channel) and to control the server-side GridFTP session over data channels. The server in state 7 accepts data channel connections and authenticates them through GSS-API mechanisms. At this stage, a separate file handle is opened and a memory buffer with the length of block size is allocated to perform the receive operation from socket and the write operation to the hard disk, for each data channel. The above description reveals other facts during each server-side GridFTP session with the passive X PUT mode in which n file handles, 2 + 1 socket handles (n socket handles to listen on n ports, n socket handle for accepted connections belonging to data channels, and a socket handle for control channel) and a discrete chain of buffers with size of * are assigned. As it can be seen, the GridFTP implementation suffers from these overheads due to the inherent protocol nature. In contrast to GridFTP, DotDFS only requires assigning one memory buffer with the size of block size, opening a file handle and n accepted socket handles. Lightweight DotSec's Grid Security Infrastructure Model Virtual organizations share different and wide range of resources distributed in Grids. This urges the need for more security issues in developing Grid Computing [21]. In DotSec protocol, we propose a model of new lightweight grid security infrastructure. It has the security structure like Globus GSI [22] and SSL (as well as TLS [23]), and also provides some other security requirements for DotGrid services. Recently, the Globus GSI has being played a key role in the security of Grid Computing. Mutual authentication and delegation, and single sign-on are major services of Globus GSI. Based on Public Key Infrastructure (PKI), Globus GSI clients must manage and process long-term credentials. PKI-based GSI is somehow a heavyweight approach. This is mainly due to public key certificates and proxy certificates [24]. This section is about the overall view of our new lightweight GSI solution of DotSec. DotSec is a lightweight GSI but not Globus GSI. It is designed as a unified security model in DotGrid platform. It uses extensively from all protocols and methods described here. The layered DotSec model is shown in Fig. 12. As seen in Fig. 12, the DotSec layer not only supports Grid-based authentication/authorization mechanisms, but also includes TSI core, a critical service of data transfer security like SSL and TLS. DotSec also reduces the heavyweight interface of the most common protocols used in Globus GSI and SSL. It suggests services like change between TSI X-Channel state to Normal X-Channel and vice versa at each point of the transfer session among endpoints. The private and public RSA keys import/export as Authenticode X.509 v.3 certification was added to gain more flexibility in DotSec protocol. The main DotSec core has a 4-layers structure and all the communications between services, clients, and DotGrid APIs are conducted through DotGridSocket API into DotSec sub-layers. DotSec One-Way Verification Protocol In the proposed DotSec protocol, the sharing of a private shared key is done for twofold purposes: the first is for using it in encryption tasks of all data transfers between two endpoints over TSI layer, the second is to verify and certify the clients connected to the server. In Fig. 13, a DotSec session between a client and server based on our proposed DotSec protocol is illustrated. As shown in Fig. 13, the client waits to receive RSAPublicHeader from the server just after service selection of stage 3. The RSAPublicHeader structure is depicted in Fig. 14.a. It contains both of server public key and RSA encryption modulus. Just after RSAPublicHeader receives, client manages the following tasks: first a private key and an initial vector (IV) are generated relied upon a symmetric cryptographic (SC) algorithm, second the hash of key+IV is calculated based on a hashing algorithm. And finally by using the public-key and modulus of RSAPublicHeader, all the data are encrypted and ready to submit to the server. The final data structure is shown in Fig. 14.b (the value of key+IV is used for data encryption between two endpoints by TSI). In the next stage, server manages the following tasks: first it decrypts the received data using its own RSA private key, second calculates its hash, and finally compares the calculated hash with the hash in arrived SharedKeyHeader. The client is certified and verified only if these two values meet. The certification/verification stage initiates the TSI between client and server. This in turn allows to use client submitted shared key for encryption if there is a need for secure data transfer in DotSec protocol or if there is a request for whole encryption in the DotSec sessions. DotSec Authentication and Authorization After client verification, client submits the username, password and all the other requested service parameters (e.g., remote path in DotDFS PathM mode) using TSI as encrypted with shared key. In the server side, based on the submitted data and resource access control, the user is authorized if it has been validated and the communication session has already legally been started. DotGrid permissions service is described in [12,13] and presents the way in which a client is allowed or not to use the provided Grid resources during DotGrid and DotSec sessions. DotSec Cryptography Specification Protocol (DotSec CSP) DotSec CSP is used for selection and negotiation of Symmetric Cryptographic (SC) and hashing algorithms used for shared key generation and encrypting the data sent over X-Channels. As shown in Fig. 14.b, the crypto block in SharedKeyHeader is indicating the type of used SC algorithm in the client side toward the server side and the server must validate and acknowledge the demanded SC type based on DotGrid service provider's administrative rules. In the current version of DotSec, the following algorithms are supported by default: SC algorithms including Rijndael proposed by NIST, Triple DES (using one key and IV for encryption), and hashing algorithms of SHA-1 and MD5. DotSec Transmission Security Interface (DotSec TSI) TSI is responsible for data encryption and transfer using shared key between two endpoints of the connections. TSI transfers the encrypted data encapsulated as SecuredDataHeader blocks (illustrated in Fig. 14.c) which contain important information about the encrypted data. Data integrity is assured through TSI in the following manner. As shown in Fig. 14.c, the hash block is the hash value of unencrypted pure data. It is used for verification of transferred encrypted data block. Further, this is used to assure whether the data in the DotSec session is intentionally tampered or not. If the values of mode1 and mode2 in SecuredDataHeader are zero filled and submitted, then the receiver changes the working mode from DotSec TSI X-Channels to normal X-Channels shown in Fig. 13. This is a new feature in DotSec TSI protocol. It means that in this new mode, the transferred data is not encrypted and directly placed on X-Channels layer from DotGridSocket interface. We called this new mode in TSI as semi-secure. As illustrated in Fig. 13, from step 9, user or DotGrid APIs' developer in use, extending or designing of new services for integration with DotGrid must set to enable/disable the mode of TSI in each step of data transfer explicitly with the structure of mutual transfer flow shown in Fig. 13. Experimental Studies Single Stream Performance in LAN In the first experiment, the evaluation of DotDFS versus GridFTP in a single stream over LAN is done. Fig. 15 shows the performance and throughput for transferring files with size of ranging from 200MB to 4000MB. While our current implementation is suffering from the overheads of MONO .NET implementation and .NET Platform Invoke calls (more meaningfully due to the virtual machine nature of the .NET Framework), the proposed DotDFS protocol has reasonable performance over GridFTP. An interesting observation is the reduction of throughput for files larger than 1GB. The measured throughput in this state is rather much less than the speeds of the local read and write over the storage system. This phenomenon is discussed with more detail in section 5.2. Harnessing Parallelism in LAN In the next step, we experiment and evaluate the effect of parallel TCP connections to increase the throughput. Fig. 16 shows the results for large set of streams. Fig. 13 shows three distinct evaluation modes: memory-to-memory (/dev/zero to /dev/null), disk-to-disk and Iperf test. In disk-to-disk tests a file with 1GB size is transferred between a client and a server. In the memory-to-memory tests, DotDFS and GridFTP reached the 94% and 91% of the bottleneck bandwidth, respectively. As seen in Fig. 16, with increasing the number of parallel streams from 50 to 500, the memory-to-memory throughput is rapidly reduced in Globus, while is nearly constant in DotDFS. This phenomenon is more discussed in this section. In our belief, the DotDFS results in section 5.1, disk-to-disk and memory-to-memory for parallel streams are not manifesting the real performance of our proposed DotDFS protocol. We think this is due to the overheads raised from using .NET Platform Invoke technology of .NET Framework in our current DotDFS implementation. .NET Platform Invoke is a technology for calling native functions relied on metadata to locate exported functions and marshal their arguments at run time [25]. The .NET Platform Invoke technology is built right into the CLR runtime to enable managed programs (means the CLI runtime) to invoke ordinary dynamically linked unmanaged code. When working with unmanaged code, whether it is a native system function or native libraries in C++, there is a type system gap that must be bridged. For instance, a string to .NET Framework is not the same thing as a string in C++. Marshalling performs the transformations to the bits such that data instances can be used on both sides of the runtimes (managed runtime against unmanaged runtime). This operation may be a simple bit-for-bit copy from the managed to unmanaged runtime and vice versa, but just as well might involve a complete reorganization of the contents of a data structure as the copy occurs. This translation adds extra overheads. Hence, the marshaling mechanism can be very expensive; it can add tens of native instructions per argument for even simple native function calls. Calling native socket (through current DotGridSocket interface), file system and threading APIs by .NET Platform Invoke expose some overheads and shortcomings on our current DotDFS client and server implementations. To unveil this deduction, we also implemented two other applications for data transfer between client and server in a memory-to-memory scenario with native standard C code and the .NET Framework based on C# language. Our new experiments show that the socket system calls in .NET add at least an overhead of 5% to our system. This means that if DotDFS is implemented in native code by C or C++, it may reach about 99% of the performance and throughput of Iperf standard in the memory-to-memory scenario as well as the maximum performance in speed for read from the file in the sender and write to the file in the receiver in the disk-to-disk scenario (again, this means that if the current DotDFS implementation is ported to naive code rather than Microsoft .NET, one can access performance of benchmarks like Bonnie file system benchmark [26] with the notion of a highthroughput file transfer system). As seen in Fig. 16, the disk-to-disk performance of DotDFS is better than Globus. By increasing the number of parallel streams (i.e. 500), the Globus throughput is reduced in disk-to-disk and memory-to-memory tests while DotDFS throughput remains remarkably high. The throughputs shown in Fig. 13 promote that our DotDFS protocol is more suitable than Globus GridFTP for high-throughput data transfer. Globus GridFTP server widely uses from Linux forks instead of kernel threads like POSIX threads for multitasking server architecture support. This is specially used in parallel TCP streams implementation proposed by GridFTP protocol. Threads are lightweight processes while forks are heavyweight processes. This explains that using UNIX forks in server-side results in higher overload and cost over using threads. This, in turn, causes more unstable server applications in dynamic, high traffic and actual server environments. The second overhead factor of GridFTP is the context switching. A context switch is the switching of the CPU from one process or thread to another in operating system or hardware level. Context switching is generally computationally intensive. GridFTP protocol supports parallel TCP connections by the command PORT and other GridFTP protocol extensions. Fig. 17 shows an example of GridFTP passive file upload scenario in X mode. In this example, the GridFTP server replies on PORT command and go to listening state to accept new data channels for establishing parallel TCP streams. According to this scenario, in GridFTP protocol, it is mandatory to use multi-threading or forks for parallel data channels establishment on the GridFTP server. Furthermore, the number of client-side requested threads and or forks are equal to the number of parallel data channels plus the one thread or fork for the data control channel. Client This action results in more context switching (i.e. in large parallel streams or large clients' connections) in Globus GridFTP server. On the other hand, context switches for threads are faster than Linux forks, while Globus GridFTP extensively makes use of forks. As previously stated in sections 2.2 and 3.1, in DotDFS protocol, for example, in an upload from client to the server, there is only one openen thread in the server-side, and data and session managements are all proceeded by this thread using consecutive socket select() calls in parallel TCP channels. This approach is somehow like the Single-Process Event-Driven (SPED) model [27]. This may describe clearly why the DotDFS throughput in disk-to-disk and memory-to-memory remains fixed with parallel streams increased while reduced in Globus GridFTP. This observation leads to an interesting real world solution. In practical situations, for example with 500 concurrently connected clients and 5 parallel streams (which is widely used in WAN), then DotDFS server needs 500 threads and GridFTP server needs 2500 forks. This practically leads to more stable servers developed based on DotDFS protocol implementation. To show the effect of more GridFTP protocol overheads, the percent of used physical memory in both client and servers of DotDFS and GridFTP servers are measured with the increase in parallel streams. The results are shown in Fig. 18. As seen in Fig. 18, the rate of rise in Globus is much more than DotDFS by increasing the number of parallel streams. The main responsible factor for this increase in Globus is the more opened forks and used memory buffers for GridFTP transfer sessions when parallel streams are increased. In our experiments, we also observed an interesting phenomenon. In all of our experiments for LAN, we observed that with the increase of file sizes more than 1GB, the performance and overall throughput are reduced with a factor and reached to a fixed value which is below the real measured speed of read and write on the hard disks used for client and server machines. This shows saturation on throughput. We call this speed as saturation speed. Saturation speed is achieved earlier with file sizes like 4GB and with increased parallel streams. We think that the main reasons for this observation are: first is data copies via reads and writes between user space and kernel mode for file descriptors and sockets, and second is the scheduling and context switching overheads due to the event handling and notification via socket select() calls in DotDFS or Linux forks handling in Globus GridFTP. Transfer Large Collection of Small Files In another experiment, we evaluate the performance of directory tree movement by DotDFS and Globus GridFTP. The directory to transfer contained 1000 files in 100 sub-directories. File sizes range from 1KB to around 512KB. The total data volume size in the directory tree is about 80MB. The achieved results are depicted in Fig. 19. For Globus case in GUC, we used the -pp option that provides GridFTP Pipelining which improves the transfer performance of lots of small file [28]. As stated in section 3.2, DotDFS for directory transfers uses the full parallelism features in our proposed protocol and the efficient paralleled client side implementation. These are the main reasons that with increasing the number of streams, the whole performance in the directory tree movement is improved significantly in the DotDFS case rather than the GridFTP case. WAN Experiments In this section, a new idea for description and modeling of data transfer protocols in TCP/IP stack based on WAN's is presented. Due to our limited access to various topologies of MAN and WAN networks, the detailed development of this model is in our next tasks. There have been a large number of studies in the area of buffer optimization and parallel stream optimization. However, a comprehensive combination of tuned buffer size and parallel streams can even give more effective results than the single applications of these two methods. Unfortunately, there are not any practical approach to counterpoising the buffer size and the number of parallel streams to achieve the optimal network throughput. In WAN, there are various facts including distance, TCP window size, packet loss due to the TCP congestion, and random network packet losses that mainly affect overall throughput of a single TCP flow. Furthermore, the lack of tuned TCP options also drastically reduces the throughput and wastes the useful bandwidth. Many researchers have investigated to increase the effective WAN throughput (WAN goodput) especially in networks with high bandwidth-delay products and random losses. In [29], the authors present a simple stochastic model of steady-state TCP behavior in its congestion-avoidance mode. This model states that the well-known result for a single TCP stream throughput is formulated in equation 1 of Table 1. In this equation, MSS is the maximum segment size, RTT is the round trip time, C is a constant term that reflects details of the TCP acknowledgment algorithm (C=1.22 if every segment is acknowledged and C=0.87 if at least every other segment is acknowledged), and p is the packet loss ratio which is the number of retransmitted packets divided by the total number of packets transmitted. Based on this formula the mean TCP Window size is W. With the formulation of this model, [30] proposes a similar equation to model WAN-related multi-stream TCP goodput and the authors investigate the verification of their formula. Based on the results in [30], then we can model Table 1. In this equation, n is the number of parallel TCP streams for an end-to-end transfer session and BWagg is the aggregate TCP throughput. Using the approach and results of [30], if we assume that all the values of RTTi, Pi and MSSi are fixed, then for n parallel streams, we obtain the equation 3 of Table 1. In relationship 3, W is the representation of the TCP Window size. As seen from this relationship, the aggregate throughput rises with the increase in the values of n and W. This equation is valid for a specified interval. Increase or decrease in one of the n or W value affects the extensive variations in another parameter, which have been studied by many researchers to achieve a high throughput transfer rate. One of the chief reasons on this phenomenon may be the direct relation between TCP Window size and the packet loss formulated in equation 3 and the empirical relationship between packet loss and n value. For example, the excessive increase in the n parameter can be conducted to the TCP congestion phenomenon. This side effect increases packet loss due to dropping the TCP data segments exposed by direct interferences of gateways and routers in a typical network. Packet loss, however, may be due to other factors, such as intermittent hardware faults or physical layer distortions. In this section, we briefly express our experiences, related to measuring the throughputs of current DotDFS protocol implementation and Iperf tool, to validate the inferences of this model, to propose a new general abstraction to analytical model file transport protocols, and to optimally select the values of n and W in a typical WAN network with the testbed topology stated in section 5 (the main goal of this section is to model the behavior of DotDFS protocol and Iperf tool). Since the most important effective parameters to achieve a high throughput rate in TCP-based WAN networks are n and W formulated in equation 3 of Table 1, we assume that the throughput is generally a function of two variables, n and W values (the equation 3 of Table 1 theoretically shows this fact). Therefore, we derive the equation 4 of Table 1. As illustrated in equation 4, the main goal is to find a compact geometrical equation of a curve or surface which relates the variations of z coordinate according to the variations of x and y coordinates. When the implicit equation representing the f function can be found, the exterma or saddle points of the f function of two variables can be obtained, and therefore, we can find a set of points that maximize the z value corresponding to the variations of x and y coordinates. Ultimately, these obtained values are to represent the optimum values of n and W, which will contribute to a high throughput transfer rate for being used in Grid-based file transport protocols. Diagrams 20 and 21 were obtained from nearly 500 collected raw data sets inspired by sampling, experimentation and mathematical interpolation approaches. Examining these diagrams reveals interesting results. From the most key inferences of this model are followed. Fig. 17 and 18 illustrate which the equation z=f(x,y) is expressing an open surface with a set of exterma points. This set presents the points that maximize z value along x and y variables meaning the pairs (x,y)=(n,W). These pairs are the same points to achieve a high throughput transfer rate in our TCP-based WAN model. Equally, from these figures we can observe the validity of equations 1 to 3 for the total throughput of single and multiple (parallel) TCP streams. For example, the Fig. 20 shows to achieve a high WAN goodput and using the optimum network utilization the number of TCP streams must be set to 45 to 55 for a large TCP Window size such as 2MB for Iperf. Fig. 20 and 21 implicitly verify the same approaches proposed by Globus for accessing an optimal WAN goodput [31]. The final goal of key subjects covered in this section is to propose a new general model to optimal select n and W values and to develop a tool enabling the Grid developers to automatically measure these values without worrying about the network dynamic and the used Grid-based file transfer protocol like DotDFS or GridFTP soon. Conclusion and Future Works In this paper, we introduced the DotDFS protocol and expressed the characteristics that one can achieve to a Grid-based high-throughput file transfer system by using the DFSM, FTSM and PathM modes in the proposed and implemented protocol. We believe to achieve this goal, the procedures stated in this paper must be considered. Equally, we pointed out some weaknesses in the nature of the proposed GridFTP protocol and its implementation by the Globus team. The performance of the both .NET-based DotDFS and native Globus GridFTP were compared in different scenarios to validate the well-designed approaches used in the proposal and implementation of DotDFS protocol. Our LAN experiences in memory-to-memory tests showed that DotDFS accessed to the 94% bottleneck bandwidth while GridFTP was accessing 91%. In LAN disk-to-disk tests, comparison between DotDFS and GridFTP protocol unveiled a set of interesting and technical problems in GridFTP for both the nature of that protocol and its implementation by Globus. We plan to do the following prioritized research works upon the DotDFS protocol based on results and experiences obtained and stated in this paper for the very near future: 1. The main future plan is to port the current .NET-based DotDFS implementation into the native runtime by using the standard C++ language in a cross-platform manner to be run-able in all operating systems. It will integrate the Win32/UNIX-compliant-POSIX APIs and utilize the prominent modern hardware advancements over the networks and storage systems. To probably prevent the objected-oriented overheads of C++ stack and to achieve the best optimized throughput, some critical cores of the new implemented DotDFS protocol will be written in pure C language. 2. Because of the verified overheads exposed by the widely-used TCP protocol, particularly in WAN networks upon file transport protocols, the other future work will be to provide a SCTP [32] stack driver for the underlying DotGridSocket interface to intercept SCTP packets instead of the TCP packets. 3. Another plan will be to expand the ideas stated in section 5.4 to analytical model the DotDFS protocol. We expect to provide an API-based network tool to optimally select the best network parameters such as the number of parallel streams and TCP Window buffer size in TCP-based transfer channels in pursuit of a highthroughput file transfer system. 4. And final, as stated during this paper, due to our knowledge we discovered an interesting phenomenon called the saturation speed. We plan to investigate in more detail what factors affect on throughput of large file transfers, especially for parallel streams and to find a solution for fixing the main problem. Fig. 1 . 1Topological DotDFS protocol representation. Fig. 2 . 2DFSM Mode pseudo code sample.Fig. 3. Read/Write method header for DFSM mode. Fig. 4 . 4Client-server DotDFS protocol communications in FTSM mode, n is the number of parallel streams. Fig. 6 . 6Client-server DotDFS protocol communications in PathM mode. Fig. 7 . 7A typical timing diagram showing the event-driven DotDFS server architecture per each FTSM upload mode's thread of execution. Fig. 8 . 8Architecture of client-server DotDFS implementation in FTSM upload mode. Fig. 9 . 9Client DotDFS architecture for directory tree transfers. Fig. 10 . 10DotDFS server communicating finite state machine in FTSM upload mode Fig. 11 . 11GridFTP server communicating finite state machine in passive X PUT mode. Fig. 12 . 12Layered DotSec grid security infrastructure model. Fig. 13 . 13Client-server DotSec layer protocol communications. Fig. 14 . 14DotSec protocol headers. Our current implementation of the DotDFS protocol includes all the features and specifications mentioned in this paper. For test and evaluation of our proposed protocol, some sort of experiments are prepared and tested. The results of our experiments show the effectiveness of the proposed protocol. Our experiments are done in three categories: LAN, WAN and directory tree transfer experiments. Our LAN platform was a network with 1Gb/s and 0.05 ms RTT. The host machines of both client and server had Intel Dual Core 3 GHZ CPU, 1GB RAM, 360GB RAID hard disk. The target operating systems were CERN Scientific Linux 4 for x86_64 with 1G byte swap space. In LAN experiments, the TCP buffer size was set to 4MB. The directory tree transfer experiments are described in section 5.3. The WAN experiments are done between two machines with Windows Server 2003 placed apart on Miami, FL and Los Angeles, CA. The target platform was a network with 20Mbit/s and 76 ms RTT measured by Iperf tool. Current version of DotDFS protocol is implemented on the top of DotGrid platform based upon Microsoft .NET Framework 1.1. We fully ported the developed toolset on Linux via MONO .NET Framework. This gains our tools to be portable on UNIX and Solaris clones too. In our experiments, Windows machines were with Microsoft .NET Framework 3.5 installed and Linux Machines were with MONO .NET Framework 1.9.1 installed. In our LAN experiments, the developed DotDFS protocol is compared with the Globus GridFTP protocol. We used Globus GridFTP of GT4.2.1 and GT4.2.1's globus-url-copy (GUC) GridFTP client utility [4]. The current version of the implemented DotDFS client enriches with a set of APIs for Grid developers. A new command-line DotDFS client utility with the most features of Globus GUC is also developed. In all experiments, the TCP channels' authentications in DotDFS are based on DotSec and while in GridFTP are based on GSI (GSS-API Grid Security Infrastructure) with Kerberos. All data points are the means of 10 runs. Fig. 15 . 15Single-stream throughput on LAN. Fig. 16 . 16Parallel throughput on LAN. Fig. 17 . 17GridFTP passive file upload scenario in X mode. Fig. 18 . 18The percentage of total client-server memory usage. Fig. 19 . 19Comparison of directory tree transfer performance. multi-stream TCP bandwidth with the relationship 2 in Fig. 20and 21 show the three dimensional diagrams of the measured throughputs for DotDFS protocol and Iperf tool stated earlier in this section. Fig. 20 . 203D interpolated WAN Iperf throughput Fig. 21 . 213D interpolated WAN DotDFS disk-to-disk throughput for a 1GB file transfer. Fig. 5. Download/Upload binary header for FTSM mode.Read Mode Seek Mode Seek Value Read Value Data 4 bits 4 bits L2 bytes L1 bytes ReadValue bytes L1 L2 TABLE 1 COMPARISON 1OF DIFFERENT WIDELY-USED FILE TRANSFER PROTOCOLSFeature DotDFS FTP GridFTP Creation year 2010 1971 2003 Protocol standards DotDFS v.1 (2010), xDFS v.2 (2010) 20 RFCs GFD-R-P.020 (2003), GFD.47 (2005) Low-Level Transmission Protocol Multi-protocol support via DotGridSocket Interface TCP/IP only Multi-protocol support via Globus XIO Platforms Cross-platform Cross-platform UNIX/Linux POSIX-compliant I/O standard support Fully No Partially Local-area file access support (e.g. NFS) Fully No No WAN improvements (TCP window size/parallelism) Strong/Strong No/weak Strong/Moderate Native event-driven protocol- level architecture Fully No No Statefull/stateless architecture Fully/fully Fully/No Fully/No NAT compliance Strong Weak Weak Firewall compliance Strong Weak Weak Internationalization Strong Moderate No Extensibility Strong Weak Moderate Protocol message interchange exchange format Binary ASCII ASCII Security extensions DotSec 7 RFCs Globus GSI, GSS-API, and FTP RFCs Scalability/Large concurrent requests support Strong Weak Weak File/pipe-based inter-process communication support Strong No No Large size file support Strong Weak Strong Protocol-level zero-copy extensions Yes No No Distributed file systems semantics Yes No No Recursive Directory tree transfer support Strong Weak Moderate TABLE 2 WAN 2FORMULAS In this table, c p is equal to W and M SS R T T is equal to k. We expect to provide and popularize a universal DotDFS framework for a variety of runtimes including .NET, Java, and C/C++ native code, which will unify the most available computing systems (desktop, server, notebook, netbook and mobile platforms) for highly secure, high-performance and high-throughput file transfers utilizing our protocol. The Anatomy of the Grid Enabling Scalable Virtual Organizations. I Foster, C Kesselman, S Tuecke, International J. Supercomputer Applications. 153I. Foster, C. Kesselman, and S. Tuecke, The Anatomy of the Grid Enabling Scalable Virtual Organizations, International J. Supercomputer Applications, 15(3), 2001. The Physiology of the Grid: An Open Grid Services Architecture for Distributed Systems Integration. I Foster, C Kesselman, J Nick, S Tuecke, Open Grid Service Infrastructure WG. Global Grid ForumI. Foster, C. Kesselman, J. Nick, and S. Tuecke, The Physiology of the Grid: An Open Grid Services Architecture for Distributed Systems Integration, Open Grid Service Infrastructure WG, Global Grid Forum, June 22, 2002. Globus Toolkit Version 4: Software for Service-Oriented Systems. I Foster, IFIP International Conference on Network and Parallel Computing. Springer-Verlag3779I. Foster, Globus Toolkit Version 4: Software for Service-Oriented Systems, IFIP International Conference on Network and Parallel Computing, Springer-Verlag LNCS 3779, pp 2-13, 2005. W Allcock, J Bresnahan, R Kettimuthu, M Link, C Dumitrescu, I Raicu, I Foster, Proceedings of Super Computing 2005 (SC05). Super Computing 2005 (SC05)W. Allcock, J. Bresnahan, R. Kettimuthu, M. Link, C. Dumitrescu, I. Raicu and I. Foster, The Globus Striped GridFTP Framework and Server, in: Proceedings of Super Computing 2005 (SC05), November 2005. W Allcock, GridFTP: Protocol extensions to ftp for the Grid. W. Allcock, GridFTP: Protocol extensions to ftp for the Grid, http://www.ggf.org/documents/GFD.20.pdf, 2010. B Allcock, I Mandrichenko, T Perelmutov, GridFTP v2 protocol description. B. Allcock, I. Mandrichenko, and T. Perelmutov, GridFTP v2 protocol description, http://www.ggf.org/documents/GFD.47.pdf, 2010. Toward Seamless Grid Data Access: Design and Implementation of GridFTP on .NET. J Feng, L Cui, G Wasson, M Humphrey, Proceedings of the 2005 Grid Workshop. the 2005 Grid WorkshopSeattle, USAJ. Feng, L. Cui, G. Wasson, and M. Humphrey, Toward Seamless Grid Data Access: Design and Implementation of GridFTP on .NET, in: Proceedings of the 2005 Grid Workshop, Seattle, USA, Nov 13-14, 2005. A Comparison of GSIFTP and RFIO on a WAN. R Kalmady, B Tierney, Proceedings of CHEP'01. CHEP'01Beijing, ChinaR. Kalmady, and B. Tierney, A Comparison of GSIFTP and RFIO on a WAN, in: Proceedings of CHEP'01, Beijing, China, Sept 3-7, 2001. A parallel I/O mechanism for distributed systems. T Baer, P Wyckoff, Proceedings of the 2004 IEEE International Conference on Cluster Computing. the 2004 IEEE International Conference on Cluster ComputingCalifornia, USAT. Baer, and P. Wyckoff, A parallel I/O mechanism for distributed systems, in: Proceedings of the 2004 IEEE International Conference on Cluster Computing, California, USA, Sept 20-23, 2004. . R Niederberger, W Allcock, L Gommans, E Grünter, T Metsch, I Monga, G L Volpato, C Grimm, Firewall Issues Overview. R. Niederberger, W. Allcock, L. Gommans, E. Grünter, T. Metsch, I. Monga, G. L. Volpato, and C. Grimm, Firewall Issues Overview, http://www.ggf.org/documents/GFD.83.pdf, 2010. T Metsch, L Gommans, E Grünter, R Niederberger, A Smet, G L Volpato, Requirements on operating Grids in Firewalled Environments. T. Metsch, L. Gommans, E. Grünter, R. Niederberger, A. de Smet, and G. L. Volpato, Requirements on operating Grids in Firewalled Environments, http://www.ggf.org/documents/GFD.142.pdf, 2010. DotGrid: a .NET-based cross-platform software for desktop grids. A Poshtkohi, A H Abutalebi, S Hessabi, International Journal of Web and Grid Services. 33A. Poshtkohi, A.H. Abutalebi, and S. Hessabi, DotGrid: a .NET-based cross-platform software for desktop grids, International Journal of Web and Grid Services 2007, Vol. 3, No.3 pp. 313 -332. DotGrid: A .NET-based Infrastructure for Global Grid Computing. A Poshtkuhi, A Abutalebi, L Ayough, S Hessabi, Proceedings of the 6th IEEE International Symposium on Cluster Computing and the Grid. the 6th IEEE International Symposium on Cluster Computing and the GridSingaporeA. Poshtkuhi, A. Abutalebi, L. Ayough, and S. Hessabi, DotGrid: A .NET-based Infrastructure for Global Grid Computing, in: Proceedings of the 6th IEEE International Symposium on Cluster Computing and the Grid (CCGrid2006), 16-19 May 2006, Singapore. DotGrid: A .NET-based Cross-Platform Grid Computing Infrastructure. A Poshtkuhi, A Abutalebi, L Ayough, S Hessabi, Proceedings of the IEEE International Conference On Computing and Informatics 2006 (ICOCI06). the IEEE International Conference On Computing and Informatics 2006 (ICOCI06)MalaysiaA. Poshtkuhi, A. Abutalebi, L. Ayough, and S. Hessabi, DotGrid: A .NET-based Cross-Platform Grid Computing Infrastructure, in: Proceedings of the IEEE International Conference On Computing and Informatics 2006 (ICOCI06), Malaysia, June 6-8, 2006. Microsoft Corporation, .NET Framework Home. Microsoft Corporation, .NET Framework Home, http://msdn.microsoft.com/netframework/, 2010. MONO .NET Project Home Page. MONO .NET Project Home Page, http://www.mono-project.com/, 2010. The Open Grid Services Architecture, and data Grids, Grid Computing Making the Global Infrastructure a Reality. Z , Peter Kunszt, P. Leanne Guy, John WileyZ. Peter Kunszt and P. Leanne Guy, The Open Grid Services Architecture, and data Grids, Grid Computing Making the Global Infrastructure a Reality, John Wiley, 2003. . Lustre Cluster File System. Lustre Cluster File System, http://www.lustre.org/, 2010. General Parallel File System (GPFS). General Parallel File System (GPFS), http://www-1.ibm. com/servers/eserver/clusters/software/gpfs.html, 2010. . Gfarm File System. Gfarm File System, http://datafarm.apgrid.org/, 2010. Security for Grid Services. V Welch, F Siebenlist, I Foster, J Bresnahan, K Czajkowski, J Gawor, C Kesselman, S Meder, L Pearlman, S Tuecke, Twelfth International Symposium on High Performance Distributed Computing. IEEE PressV. Welch, F. Siebenlist, I. Foster, J. Bresnahan, K. Czajkowski, J. Gawor, C. Kesselman, S. Meder, L. Pearlman, and S. Tuecke, Security for Grid Services, Twelfth International Symposium on High Performance Distributed Computing, IEEE Press, June 2003. A security architecture for computational Grids. I Foster, C Kesselman, G Tsudik, S Tuecke, Proceedings of the 5th ACM Computer and Communications Security Conference (CCS '98). the 5th ACM Computer and Communications Security Conference (CCS '98)ACM PressI. Foster, C. Kesselman, G. Tsudik, and S. Tuecke, A security architecture for computational Grids, in: Proceedings of the 5th ACM Computer and Communications Security Conference (CCS '98), pages 83-92. ACM Press, November 1998. The TLS protocol version 1.0, IETF. T Dierks, C Allen, T. Dierks, and C. Allen, The TLS protocol version 1.0, IETF, 1999. A certificate-free grid security infrastructure supporting password-based user authentication. J Crampton, H W Lim, K G Paterson, G Price, Proceedings of the 6th Annual PKI R&D Workshop. the 6th Annual PKI R&D WorkshopNISTJ. Crampton, H. W. Lim, K. G. Paterson, and G. Price, A certificate-free grid security infrastructure supporting password-based user authentication, in: Proceedings of the 6th Annual PKI R&D Workshop 2007, NIST, 2007. Calling Win32 DLLs in C# with P/Invoke, MSDN Magazine. J Clark, The Microsoft Journal for Developers. J. Clark, Calling Win32 DLLs in C# with P/Invoke, MSDN Magazine, The Microsoft Journal for Developers, July, 2003, http://msdn.microsoft.com/msdnmag/issues/03/07/NET/default.aspx, 2010. Bonnie file system benchmark. Bonnie file system benchmark, http://www.textuality.com/bonnie, 2010. Flash: An efficient and portable web server. V S Pai, P Druschel, W Zwaenepoel, Proceedings of the USENIX 1999 Annual Technical Conference. the USENIX 1999 Annual Technical ConferenceV.S. Pai, P. Druschel, and W. Zwaenepoel, Flash: An efficient and portable web server, in: Proceedings of the USENIX 1999 Annual Technical Conference, June 1999. J Bresnahan, M Link, R Kettimuthu, D Fraser, I Foster, Gridftp Pipelining, Proceedings of the 2007 TeraGrid Conference. the 2007 TeraGrid ConferenceJ. Bresnahan, M. Link, R. Kettimuthu, D. Fraser and I. Foster, GridFTP Pipelining, in: Proceedings of the 2007 TeraGrid Conference, June, 2007. The Macroscopic Behavior of the TCP Congestion Avoidance Algorithm. M Mathis, J Semke, J Mahdavi, T Ott, Computer Communication Review. 273M. Mathis, J. Semke, J. Mahdavi, and T. Ott, The Macroscopic Behavior of the TCP Congestion Avoidance Algorithm, Computer Communication Review, volume 27, number3, July 1997. The End-to-End Performance Effects of Parallel TCP Sockets on a Lossy Wide-Area Network. T J Hacker, B D Athey, B Noble, Proceedings of the 16th International Parallel and Distributed Processing Symposium. the 16th International Parallel and Distributed Processing SymposiumFlorida, USAT.J. Hacker, B.D. Athey, and B. Noble, The End-to-End Performance Effects of Parallel TCP Sockets on a Lossy Wide-Area Network, in: Proceedings of the 16th International Parallel and Distributed Processing Symposium, Florida, USA, April 15-19, 2004. Applied Techniques for High Bandwidth Data Transfers Across Wide Area Networks. J Lee, D Gunter, B Tierney, B Allcock, J Bester, J Bresnahan, S Tuecke, Proceedings of International Conference on Computing in High Energy and Nuclear Physics. International Conference on Computing in High Energy and Nuclear PhysicsBeijing, ChinaJ. Lee, D. Gunter, B. Tierney, B, Allcock, J. Bester, J. Bresnahan, S. Tuecke, Applied Techniques for High Bandwidth Data Transfers Across Wide Area Networks, in: Proceedings of International Conference on Computing in High Energy and Nuclear Physics, Beijing, China, September 2001. An Introduction to the Stream Control Transmission Protocol (SCTP), IETF, RFC 3286. L Ong, J Yoakum, L. Ong, and J. Yoakum, An Introduction to the Stream Control Transmission Protocol (SCTP), IETF, RFC 3286, 2002. ECMA-334: C# Language Specification. ECMA-334: C# Language Specification, http://www.ecma-international.org/publications/standards/Ecma- 334.htm, 2010. Ecma-335, Common Language Infrastructure (CLI). ECMA-335: Common Language Infrastructure (CLI), http://www.ecma- international.org/publications/techreports/E-TR-084.htm, 2010.
{'fraction_non_alphanumeric': 0.030982893246706802, 'fraction_numerical': 0.011694581868033618, 'mean_word_length': 4.707307116614506, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 10, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 11, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'DotGrid platform is a Grid infrastructure integrated with a set of open and standard protocols recently implemented on the top of Microsoft .NET in Windows and MONO .NET in UNIX/Linux. DotGrid infrastructure along with its proposed protocols provides a right and solid approach to targeting other platforms, e.g., the native C/C++ runtime. In this paper, we propose a new file transfer protocol called DotDFS as a high-throughput distributed file transfer component for DotGrid. DotDFS introduces some open binary protocols for efficient file transfers on current Grid infrastructures. DotDFS protocol also provides mechanisms for multiple file streams to gain high-throughput file transfer similar to GridFTP protocol, but by proposing and implementing a new parallel TCP connection-oriented paradigm. In our LAN tests, we have achieved better results than Globus GridFTP implementation particularly in multiple TCP streams and directory tree transfers. Our LAN experiences in memory-to-memory tests show that DotDFS accesses to the 94% bottleneck bandwidth while GridFTP is accessing 91%. In LAN disk-to-disk tests, comparing DotDFS protocol with GridFTP protocol unveils a set of interesting and technical problems in GridFTP for both the nature of the protocol and its implementation by Globus. In the WAN experimental studies, we propose a new idea for analytical modeling of file transfer protocols like DotDFS inspired by sampling, experimentation and mathematical interpolation approaches. The cross-platform and open standard-based features of DotDFS provide a substantial framework for unifying data access and resource sharing in real heterogeneous Grid environments.A. Poshtkohi, M.B. Ghaznavi-Ghoushchi transfer, partial file transfer, automatic negotiation of TCP buffer/window sizes, and support for reliable and restartable data transfer.[4,5,6] describe GridFTP protocol and its full features. The authors in [7] also addressed the implementation problems of GridFTP for .NET and Windows platforms. Some GridFTP problems formerly studied by researchers are long-time GridFTP setup due to any application setup, connection and authentication, firewalls, and network address translation[8,9,10,11].We have developed DotGrid platform[12,13,14]which enables the creation of Grid infrastructure using Microsoft . NET [15] in Windows and MONO . NET [16] in UNIX and Linux environments. It provides Grid services and toolkits for fast developing Grid applications.The main goal of DotGrid project[12,13,14]is to develop a cross-platform framework for sharing computational resources between heterogeneous DotGrid nodes. To build a cross-platform desktop Grid infrastructure in heterogeneous platforms, DotGrid makes use of high-performance implementation of the native .NET Framework in Windows and MONO .NET project in UNIX/Linux family of operating systems, respectively. Resource sharing and bulk data transfer are two major architectural needs that we have investigated during the design process of DotGrid.Currently, we are implementing a new cross-platform Grid/Cloud infrastructure, which extends and utilizes the open and standard protocols suggested in DotGrid platform for native runtime, i.e., C/C++ stack. This new platform is applying the ECMA international open standards [33,34] to provide a much more standardized native implementation of DotGrid for scientific and enterprise Grid/Cloud communities in the most recently operating systems including UNIX, Linux, and Windows.Recently, WAN-based file systems like Lustre[18], and Gfarm [20] are maturated. Due to our knowledge, it seems that DotDFS is more like to GridFTP than the above mentioned file systems. For example, DotDFS may be used, as well as GridFTP, in Gfarm as the underlying file transfer protocol. DotDFS has many different aspects ranging from structure to architecture with GridFTP protocol. Most of them are declared during this paper.In this paper, we introduce a Grid-based high-throughput file transfer system, called DotDFS. This can be used as a component of a computational Grid [12,13,14]or data Grid [17] environments.The rest of the paper is organized as follows. In section two, we present DotDFS protocol. Section 3 infers highperformance server design patterns for the DotDFS protocol. Sections four and six describe DotDFS implementation and DotSec GSI. Section five focuses on comparison of DotDFS protocol with GridFTP protocol. The LAN and WAN experimental studies are described in section 7. We conclude in section 8.DotDFS ProtocolThe current version of DotDFS protocol is a binary protocol model like TCP/IP, DNS and SOCKS protocols. This platform-independent feature allows DotDFS protocol to be re-implemented in other platforms and gains more interoperability. This approach results in more performance and throughput in the cost of implementation complexity.[7] reports interoperability problems of the Globus GridFTP server due to the MPI used in Globus core. The two major problems in the current .NET GridFTP implementation are: no support of the authentication in data channels and no interoperability of the stripped data transfers with the Globus GridFTP. The .NET GridFTP extensively used from Windows services and native Win32 APIs. This causes it could not be ported to UNIX/Linux platforms via MONO .NET.DotDFS is a high-throughput file transfer protocol which meets the requirements to set it as a background for applications in the areas of distributed, cluster, grid and cloud computing. These requirements are discussed in this section and have been considered in our current DotDFS implementation.DotDFS heavily depends upon a set of abstractions that are used to hide infrastructure dependencies. DotDFS APIs can be used for implementing applications in high-throughput data transfers similar to GridFTP. Other applications that can be implemented relied upon on DotDFS are: data intensive Grid applications, Grid-based resource sharing, booting operating systems from high-speed networks, peer-to-peer file systems and distributed databases.', 'arxivid': '1703.03905', 'author': ['Alireza Poshtkohi \nDepartment of Electrical Engineering\nShahed University\n331911865TehranIran\n', 'M B Ghaznavi-Ghoushchi \nDepartment of Electrical Engineering\nShahed University\n331911865TehranIran\n'], 'authoraffiliation': ['Department of Electrical Engineering\nShahed University\n331911865TehranIran', 'Department of Electrical Engineering\nShahed University\n331911865TehranIran'], 'corpusid': 13055052, 'doi': '10.1016/j.parco.2010.12.003', 'github_urls': [], 'n_tokens_mistral': 20818, 'n_tokens_neox': 18749, 'n_words': 12922, 'pdfsha': '6bc28034702a72aea193d55892d81c22cc2fe71d', 'pdfurls': ['https://export.arxiv.org/pdf/1703.03905v2.pdf'], 'title': ['DotDFS: A Grid-based high-throughput file transfer system', 'DotDFS: A Grid-based high-throughput file transfer system'], 'venue': []}
arxiv
Offline Reinforcement Learning with Additional Covering Distributions Chenjie Mao [email protected] School of Computer Science and Technology Huazhong University of Science and Technology 430074WuhanChina Offline Reinforcement Learning with Additional Covering Distributions We study learning optimal policies from a logged dataset, i.e., offline RL, with function approximation. Despite the efforts devoted, existing algorithms with theoretic finite-sample guarantees typically assume exploratory data coverage or strong realizable function classes, which is hard to be satisfied in reality. While there are recent works that successfully tackle these strong assumptions, they either require the gap assumptions that only could be satisfied by part of MDPs or use the behavior regularization that makes the optimality of learned policy even intractable. To solve this challenge, we provide finite-sample guarantees for a simple algorithm based on marginalized importance sampling (MIS), showing that sample-efficient offline RL for general MDPs is possible with only a partial coverage dataset and weak realizable function classes given additional side information of a covering distribution. Furthermore, we demonstrate that the covering distribution trades off prior knowledge of the optimal trajectories against the coverage requirement of the dataset, revealing the effect of this inductive bias in the learning processes.Preprint. Under review. Introduction and related works In offline reinforcement learning (offline RL, also known as batch RL), the learner tries to find good policies with a pre-collected dataset. This data-driven paradigm eliminates the heavy burden of environmental interaction required in online learning, which could be dangerous or costly (e.g., in robotics (Kalashnikov et al., 2018;Sinha and Garg, 2021) and healthcare (Gottesman et al., 2018(Gottesman et al., , 2019Tang et al., 2022)), making offline RL a promising approach in real-world applications. In early theoretic studies of offline RL (e.g., Munos (2003Munos ( , 2005Munos ( , 2007; Ernst et al. (2005); Antos et al. (2007); Munos and Szepesvari (2008);Farahmand et al. (2010)), researchers analyzed the finite-sample behavior of algorithms under the assumptions such as exploratory datasets, realizable or Bellman-complete function classes. However, despite some error propagation bounds and sample complexity guarantees achieved in these works, the strong coverage assumption made on datasets and the non-monotonic assumptions made on function classes-which are always hard to be satisfied in reality-drive people to try to find sample-efficient offline RL algorithms under only weak assumptions about dataset and function classes (Chen and Jiang, 2019). From the dataset perspective, partial coverage, which means that only some specific (or even none) policies are covered by the dataset (Rashidinejad et al., 2021;Xie et al., 2021;Uehara and Sun, 2021;Song et al., 2022), is studied. To address the problem of insufficient information, most algorithms use behavior regularization (e.g., Laroche and Trichelair (2017); Kumar et al. (2019); Zhan et al. (2022)) or pessimism in the face of uncertainty (e.g., Liu et al. (2020); Jin et al. (2020); Rashidinejad et al. (2021); Xie et al. (2021); Uehara and Sun (2021); Cheng et al. (2022); Zhu et al. (2023)) to constrain the learned policies to be close to the behavior policy. Most of the algorithms in this setting (except some that we will discuss later) require function assumptions in some sense of completeness-Bellman-completeness or strict realization according to another function class (we attribute it as strong realization). From the function classes perspective, while the primary concern is Bellman-completeness assumption which is criticized for its non-monotonicity, some recent works (Zhan et al., 2022;Chen and Jiang, 2022;Ozdaglar et al., 2022) have noticed that the realizability according to another function class is also non-monotonic. These non-monotonic properties contradict the intuition in supervised learning that rich function classes perform better (or at least no worse). Typical examples of these assumptions are the "realizability of all candidate policies' value functions" (e.g., Jiang and Huang (2020); Zhu et al. (2023)) and the "realizability of all candidate policies' density ratio" (e.g., Xie and Jiang (2020)). These assumptions are equally strong as Bellman-completeness, and we classify them as "strong realizability" (Zhan et al. (2022); Ozdaglar et al. (2022) attribute it as "completeness-type") for clarification. We also classify assuming that the function class realizes specific elements as "weak realizability" correspondingly (Chen and Jiang (2022) attributes this as "realizability-type"). We argue that this taxonomy is justified also because Bellman-completeness can be converted to the realizability assumption between two function classes with the minimax algorithm (Chen and Jiang, 2019). This conversion aligns the behavior of Bellman-completeness with strong realizability assumptions. On the one hand, Bellman-completeness assumption is always made in the classical finite-sample analyses of offline RL (e.g., analysis of FQI (Ernst et al., 2005;Antos et al., 2007)) to ensure closed updates of value functions (Sutton and Barto, 2018;Wang et al., 2021). This assumption is notoriously hard to mitigate, and Foster et al. (2021) even suggests an information-theoretic lower bound stating that without Bellman-completeness, sample-efficient offline RL is impossible even with an exploratory dataset and a function class containing all candidate policies' value functions. Therefore, it is clear that additional assumptions are required to circumvent Bellman-completeness. On the other hand, as marginalized importance sampling (MIS, see, e.g., Liu et al. (2018); Uehara et al. (2019); Jiang and Huang (2020); Huang and Jiang (2022)) has shown its effect of eliminating Bellmancompleteness with only a partial coverage dataset by assuming the realizability of density ratios in off-policy evaluation (OPE), there are works try to adapt it to policy optimization. These adaptations retain the elimination of Bellman-completeness, but most come up with other drawbacks.Some works (e.g., Jiang and Huang (2020);Zhu et al. (2023)) use OPE as an intermediate evaluation step for policy optimization yet require the strong realizability assumption on the value function class. The others borrow the idea of discriminators from MIS. Lee et al. (2021);Zhan et al. (2022) take value functions as discriminators for the optimal density ratio, using MIS to approximate the linear programming approach of Markov Decision Processes (Manne, 1960;Puterman, 1994). Nachum et al. (2019); Chen and Jiang (2022); Uehara et al. (2023) take distribution density ratios as discriminators for optimal value function by replacing the Bellman equation in OPE with its optimality variant. While in most cases, theoretic finite-sample guarantees with these discriminators would require strong realizable function classes (e.g., Rashidinejad et al. (2022)), Zhan et al. (2022); Chen and Jiang (2022); Uehara et al. (2023) avoid this with additional gap assumptions or an alternative criterion of optimality-performance degradation w.r.t. the regularized optimal policy. To the best of our knowledge, they are the only works that achieve theoretic sample-efficient guarantees under only weak realizability and partial coverage assumptions. However, on the one hand, the gap (margin) assumption (Chen and Jiang, 2022;Uehara et al., 2023) causes that only some specific Markov decision processes (MDPs)-under which the optimal value functions have gaps-can be solved. On the other hand, sub-optimality compared with a regularized optimal policy (Zhan et al., 2022) could be meaningless in some cases, and the actual performance of the learned policy is intractable. As summarized above, the following question arises: Is sample-efficient offline RL possible with only partial coverage and weak realizability assumptions for general MDPs? We answer this question in the positive and propose an algorithm that achieves finite-sample guarantees under weak assumptions with the help of an additional covering distribution. We assume that the covering distribution covers all non-stationary near-optimal policies, and the dataset covers the trajectories induced by an optimal policy from it. The covering distribution is adaptive such that both "non-stationary" and "near-optimal" above would be alleviated as the gap of optimal value function increases. The covering distribution also gives a trade-off against the data coverage assumption: the more accurate it is, the fewer redundant trajectories are required to be covered by the dataset. Chen and Jiang (2022) optimal conc. w ⋆ ∈ W, and Q ⋆ ∈ Q assume gap (margin) Rashidinejad et al. (2022) optimal conc. w ⋆ ∈ W, V ⋆ ∈ V, u ⋆ w ∈ U ∀w and ζ ⋆ w ⋆ ,u ∈ Z ∀u strong realizability Zhu et al. (2023) optimal conc. w ⋆ ∈ W, and ∀π ∈ Π, Qπ ∈ Q strong realizability Uehara et al. (2023) optimal conc from d D w ⋆ ∈ W, and Q ⋆ ∈ Q assume gap (margin) Ours (VOPR) optimal conc. from dc w ⋆ ∈ W, β ⋆ ∈ B and Q ⋆ ∈ Q assume a covering dc Furthermore, we can directly use the data distribution as the covering distribution as done in Uehara et al. (2023), if the near-optimal variant of their data assumptions are also satisfied. For comparison, we summarize algorithms with partial coverage that do not need Bellmancompleteness and model realizability (which is even stronger (Chen and Jiang, 2019;Zhan et al., 2022)) in Table 1. Necessary transfers are made to get the sub-optimality bound. We have removed additional definitions of notations for simplicity and refer the interested reader to the original papers for more detail. In conclusion, our contributions are as follows: • (Section 3) We identify the novel mechanism of non-stationary near-optimal concentrability in policy optimization under weak assumptions. • (Section 4) We demonstrate the trade-off brought by additional covering distributions for the coverage requirement of the dataset. • (Section 4) We propose the first algorithm that achieves finite-sample guarantees for general MDPs under only weak realizability and partial coverage assumptions. Preliminaries This section introduces base concepts and notations in offline RL with function approximation and MIS. See Table 2 in Appendix A for a more complete list of definitions of notations. Markov Decision Processes (MDPs) We consider infinite-horizon discounted MDPs defined as (S, A, P, R, γ, µ 0 ), where S is the state space, A is the action space, P : S × A → ∆(S) is the transition probability, R : S × A → [0, R max ] is the deterministic reward function, γ ∈ (0, 1) is the discount factor that unravels the problem of infinite horizons, and µ 0 ∈ ∆(S) is the initial state distribution. With a policy π : S → ∆(A), we say that it induces a random trajectory {s 0 , a 0 , r 0 , s 1 , a 1 , r 1 , . . . , s i , a i , r i , s i+1 , . . . } if: s 0 ∼ µ 0 , a i ∼ π(·|s i ), r i = R(s i , a i ) and s i+1 ∼ P (·|s i , a i ). We define the expected return of a policy π as J π = E ∞ i=0 γ i r i | µ 0 , π . We also denote the value function of π as the expected return starting from some specific state s or state-action pair (s, a) as V π (s) = E ∞ i=0 γ i r i | s, π and Q π (s, a) = E ∞ i=0 γ i r i | (s, a), π . We denote the optimal policies that achieve the maximum return J ⋆ from µ 0 as Π ⋆ , and its member as π ⋆ . We say a policy is optimal almost everywhere if its state value function is maximized at every state and denote it as π ⋆ e (π ⋆ e is not always unique). We represent the value functions of π ⋆ e as V ⋆ and Q ⋆ . It worth noting that V ⋆ and Q ⋆ , the unique solutions of both Bellman optimality equation and the primal part of LP approach of MDPs (Puterman, 1994), are not the value functions of all optimal policies. For ease of discussion, we assume S, A, S × A are compact measurable spaces and, with abuse of notation, we use ν to denote the corresponding finite uniform measure on each space (e.g., Lebesgue measure). We use P π to denote the state-action transition operator for density d as P π d(s ′ , a ′ ) := S×A π(a ′ | s ′ )P (s ′ | s, a)d(s, a)dν(s, a). Offline policy learning with function approximation In the standard theoretical setup of offline RL, we are given with a dataset D consisting of N (s, a, r, s ′ ) tuples, which is collected with state distribution µ D and behavior policy π b such that s ∼ µ D , a ∼ π b (·|s), r = R(s, a), s ′ ∼ P (·|s, a). We use d D (s, a) := µ(s)π b (a | s) to denote the composed state-action distribution of the dataset. When the state space and action space become rather complex, function approximation is typically used. For this, we assume there are some function classes at hand that satisfy certain assumptions and have limited complexity (measured by cardinality, metric entropy and so forth). The function classes considered in this paper are state-action value function class Q ⊆ (S × A → R), state distribution ratio class W ⊆ (S × A → R), and policy ratio class B ⊆ (S × A → R). MIS with density discriminators and L 2 error bound One of the most popular ways to estimate the optimal value function is via the Bellman optimality equation: ∀s ∈ S, a ∈ A, Q ⋆ (s, a) = T ⋆ Q ⋆ (s, a)(1) where T ⋆ q(s, a) := R(s, a) + γE s ′ ∼P (·|s,a) [max q(s ′ , ·)] denotes the Bellman optimality operator. However, when we try to utilize the constraints from Eq. (1) (e.g., through the L 1 error ∥q−T ⋆ q∥ 1,d D ), the expectation in T ⋆ would introduce the infamous double-sampling issue (Baird, 1995), making the estimation intractable. To overcome this, privious works with MIS tried to take weight functions as discriminators and minimize a weighted sum of Eq. (1). In fact, even the L 1 error itself could be written as a weighted sum with some sign function (take 1 if q ≥ T ⋆ q and −1 otherwise (Ozdaglar et al., 2022)). Namely, we approximate Q ⋆ througĥ q = argmin q∈Q max w∈W E d D [w(s, a)(q(s, a) − T ⋆ q(s, a)].(2) Since the weight function class W is marginalized into the state-action space (instead of trajectories), this approach is called marginalized importance sampling ( 3 From Q ⋆ to optimal policy, the minimum requirement Uehara et al. (2023) shows that accurately estimating optimal value function Q ⋆ under d D is possible if d D covers the optimal trajectories starting from itself. This "self-covering" assumption could be relieved and generalized if we only require an accurate estimator under some state-action distribution d c such that d c ≪ d D (we use µ c and π c to denote the state distribution and policy decomposed from d c ). In fact, d c provides a trade-off for the coverage requirement of the dataset: the fewer state-action pairs on the support of d c , the weaker data coverage assumptions we will make. Nevertheless, how much trade-off can d c provide while preserving the desired result? In policy learning, our goal is to derive an optimal policyπ from the estimated Q ⋆ (denoted aŝ q). While there are methods (see Section 4.3 for a brief discussion) that induce policies fromq by exploiting pessimism or data regularization, one of the most straightforward ways is to take the actions covered by d c that achieve the maximumq in each state. This can be done with the help of policy ratio class B, viaβ = argmax β∈B ⟨µ c ,q(·, π β )⟩ and takeπ = πβ, where π β (a | s) = π b (a | s)β(s, a) (normalized if needed). With the optimal realizability of B and concentrability of π c , Eq. (3) is actually equivalent to ⟨µ c , Q ⋆ (·,π) − Q ⋆ (·, π ⋆ e )⟩ = 0,(4) which guides us to exploit the coverage provided by µ c . Recall that our goal is to use d c to trade off the coverage assumption of d D . Therefore, the question left, which forms the primary subject of this section, is With which µ c can we conclude thatπ is optimal from ⟨µ c , Q ⋆ (·,π) − Q ⋆ (·, π ⋆ e )⟩ = 0, and what is the minimum requirement of it? Since µ c and d c are to provide additional coverage, we also call them "covering distributions". The remainder of this section is organized as follows: we first show why single optimal concentrability of µ c is not enough in Section 3.1, and then we introduce the alternative "all optimal concentrability" in Section 3.2 and the adapted version of it in Section 3.3 to accommodate statistical errors. The above MDP is deterministic, and we initially start from state 1. We can take actions L (left) and R (right) in each state. In states 1 and 3, action L (R) will transfer us to its left (right) hand state, and taking actions in other states will only cause a self-loop. We can only obtain non-zero rewards by taking actions in states 2 and 4, with values 1 and 1 γ correspondingly. There are two trajectories that could lead to the optimal γ /(1−γ) return: {(1, R), 2, . . . } and {(1, L), (3, L), 4 . . . }. We take γ as the discount factor. The dilemma of single optimal contentrability Single optimal concentrability is standard (Liu et al., 2020;Xie et al., 2021;Cheng et al., 2022) when we try to mitigate exploratory data assumptions (e.g., all-policy concentrability). However, this framework suffers from a conundrum if only making weak realizability assumptions: we will know that the learned policy performs well only if we are informed with trajectories induced by it-rather than the ones induced by the covered policy. More specifically, as the optimality ofπ could be quantified as J ⋆ − Jπ, the performance gap, we can telescope it through the performance difference lemma. Lemma 1 (The performance difference lemma). We can decompose the performance gap as (1 − γ)(J π1 − J π2 ) = ⟨µ π1 , Q π2 (·, π 1 ) − Q π2 (·, π 2 )⟩. Thus, with Eq. (4), if we want J ⋆ − Jπ (i.e., J π ⋆ e − Jπ) to be equal to zero, it might be necessary to require µ c to cover µπ (µ c ≫ µπ) since the right part of the inner product in Eq. (4) is always non-positive. However, asπ is estimated and is even random when considering approximating it from a dataset, µ c ≫ µπ is usually achieved through all-policy concentrability-µ c ≫ µ π for all π in the hypothesis class. Single optimal concentrability fails to provide the desired result. For instance, consider the counterexample in Figure 1 which is adapted from Zhan et al. (2022); Chen and Jiang (2022). Suppose we finally get the following covering distribution and policy: µ c (s) = 1 /2 if s = 1 1 /2 if s = 2 andπ(s) =    L if s = 1 R if s = 3 Random elsewise. While µ c achieves single optimal concentrability andπ achieves the maximized value of Q ⋆ in each state on the support of µ c ,π is not an optimal policy since it would end up with 0 return. How gap assumptions avoid this While both Chen and Jiang (2022) and Uehara et al. (2023) consider single optimal concentrability and weak realizability assumptions (Uehara et al. (2023) also assumes additional structures of the dataset), the gap (margin) assumptions ensure that only taking π ⋆ asπ could achieve Eq. (4). Moreover, Chen and Jiang (2022) shows that with the gap assumption, we can even use a value-based algorithm to derive a near-optimal policy without accurately estimating Q ⋆ . All-optimal concentrability While single optimal concentrability suffers the hardness revealed before, there is still an alternative for the exploratory covering µ c , which is shown in the following lemma: Lemma 2 (From advantage to optimality). If µ c covers all distributions induced by non-stationary optimal policies (i.e., µ c ≫ µ π ⋆ non for any π ⋆ non ) and Eq. (4) holds, thenπ is optimal and Jπ = J ⋆ . Remark 1. Non-stationary policies are frequently employed in the analysis of offline RL (Munos, 2003(Munos, , 2005Scherrer and Lesner, 2012;Chen and Jiang, 2019;Liu et al., 2020). If we make the gap assumption, the "all non-stationary" requirement is discardable since the action in each state that could lead to the optimal return is unique. Remark 2. Wang et al. (2022) also utilizes the all-optimal concentrability assumption, but they consider the tabular setting and they require additionally gap assumptions to achieve the near-optimal guarantees. We now provide a short proof of Lemma 2, showing by induction thatπ i -the non-stationary policy that adoptsπ at the beginning 0-th to i-th (include the i-th) steps and then follows π ⋆ e -is optimal. Proof. We first rewrite the telescoping equation in the performance difference lemma as (1 − γ)(Jπ i − J ⋆ ) =⟨µπ i , Q ⋆ (·,π i ) − Q ⋆ (·, π ⋆ e )⟩ (5) =⟨µ 0:î πi , Q ⋆ (·,π) − Q ⋆ (·, π ⋆ e )⟩ + ⟨µ i+1:∞ πi , Q ⋆ (·, π ⋆ e ) − Q ⋆ (·, π ⋆ e )⟩ (6) =⟨µ 0:î πi , Q ⋆ (·,π) − Q ⋆ (·, π ⋆ e )⟩(7) where µ i:j π denotes the i-th to j-th steps (include the i-th and j-th) part of µ π . Thus, the optimality of π i only depends on the first 0-th to i-th steps, andπ i is optimal if this part is on the support of µ c . Now we inductively show that, for any natural number i, the initial 0-th to i-th steps part is covered: • The step-0 part of µπ (i.e., (1 − γ)µ 0 ) is on the support of µ c since there is some (non- stationary) optimal policy π ⋆ covered by it, µ c ≫ µ π ⋆ ≫ µ 0 . Therefore, ⟨µ 0:0 π0 , Q ⋆ (·,π) − Q ⋆ (·, π ⋆ e )⟩ = 0. From Eq. (7),π 0 is optimal. • We next show that ifπ i is optimal (which means that it's covered µ c ), then the first 0-th to (i + 1)-th steps part of µπ is covered by µ c , which means thatπ i+1 is optimal. This comes from the fact that the initial 0-th to (i + 1)-th steps part of the state distribution induced by a policy only depends on its previous 0-th to i-th decisions: µ c ≫ µπ i ≫ µ 0:i+1 πi = µ 0:i+1 π . From Eq. (7),π i+1 is optimal. Thus, for any ϵ > 0, there exists natural number i ≥ log γ ϵ Vmax such that J ⋆ − Jπ ≤ J ⋆ − J 0:î π ≤ J ⋆ − (Jπ i − γ i+1 V max ) ≤ γ i+1 V max ≤ ϵ, where J i:j π denotes the i-th to j-th steps part of the return. Therefore,π is optimal. Consequently, instead of the exploratory data assumption, all non-stationary optimal coverage is sufficient for policy optimization. Dealing with statistical error While Lemma 2 is adequate at the population level (i.e., with an infinite amount of data), covering all non-stationary optimal policies is not enough when considering the empirical setting (i.e., with finite samples) due to the introduced statistical error. This motivates us to adapt Lemma 2 with a more refined µ c . Assumption 1 (All near-optimal concentrability). We are given with a covering distribution d c such that its state distribution part µ c covers the distributions induced by any non-stationary ε c near-optimal policyπ: µπ µ c ∞ ≤ C c , ∀π ∈ Π ⋆ εc,non .(8) We call a policy π is ε near-optimal if J ⋆ − J π ≤ ε and denote Π ⋆ ε,non as the class of all non-stationary ε near-optimal policies. We also define 0 0 = 1 to suppress the extreme cases. With this refined µ c , we can now derive the optimality ofπ even with some statistical errors. Lemma 3 (From advantage to optimality, with statistical errors). If ⟨µ c , Q ⋆ (·,π) − Q ⋆ (·, π ⋆ )⟩ ≥ −ε , and Assumption 1 holds with ε c ≥ Ccε 1−γ ,π is Ccε 1−γ near-optimal. We defer the proof of this lemma to Appendix C.1. Remark 3 (The asymptotic property of ε c ). One of the most important properties of all near-optimal concentrability is that ε c depends on the statistical error, which decreases as the amount of data increases. Algorithm and analysis After discussing the minimum requirement of estimating Q ⋆ , this section will demonstrate how to fulfill it and accomplish the policy learning task. Our algorithm, which is based on the optimal value estimator from Uehara et al. (2023), follows the pseudocode in Algorithm 1. Algorithm 1: VOPR (Value-Based Offline RL with Policy Ratio) Input :Dataset D, value function class Q, distribution density ratio class W, policy ratio function class B, and covering distribution d c 1 Estimate the optimal value functionq aŝ q = argmin q∈Q max w∈WL (d c , q, w) (9) wherê L(d, q, w) := 0.5E d [q 2 (s, a)] + 1 N D (s,a,r,s ′ )∈D w(s, a) γ max q(s ′ , ·) + r − q(s, a)(10) 2 Derive the approximated optimal policy ratio: β = argmax β∈B E µc [q(s, π β )] Output :π = πβ We organized the rest of this section as follows: we first discuss the trade-off provided by the additional covering distribution d c and how to deduce d c in reality in Section 4.1; we then provide the finite-sample analysis of Algorithm 1 and its proof sketch in Section 4.2; we finally conclude this section by comparing our algorithms with the others in Section 4.3. We defer the main proofs in this section to Appendix D. Data assumptions and trade-off As investigated in recent works (Huang and Jiang, 2022;Uehara et al., 2023), value function estimation under a given distribution requires a dataset that contains trajectories rolled out from it. Thus, our data assumption is as follows. Assumption 2 (Partial concentrability from d c ). The shift from d D to the induced state-action distribution by π ⋆ e from d c is bounded: d dc,π ⋆ e d D ∞ ≤ C D .(11) It is clear that with Assumption 2, d c is also covered by d D . Proposition 4. If Assumption 2 holds, by definition of d dc,π ⋆ e , d c d D ∞ ≤ d dc,π ⋆ e /(1 − γ) d D ∞ ≤ C D 1 − γ . We now clarify the order of the learning process: we are first given with a dataset D with some good properties; then we try to find a d c from the support of the state-action distribution of D through some inductive bias (with necessary approximation); finally, we apply Algorithm 1 with D and d c . The choice of d c constructs a trade-off between the knowledge about optimal policy and the requirement of data coverage. On the one hand, the most casual choice of d c is d D (as in Uehara et al. (2023)), which means we have no prior knowledge about optimal policies. Employing d D as d c will not only requires the dataset to cover unnecessary suboptimal trajectories, but also makes the dataset non-monotonic (adding new data points to it would break this assumption). On the other hand, if we have perfect knowledge about optimal policies, Assumption 2 could be significantly alleviated. More concretely, if d c strictly consists of the state-action distribution of trajectories induced by near-optimal policies, our data assumption reduces to the per-step version of near-optimal concentrability. Lemma 5. If d c is a linear combination of the state-action distributions induced by non-stationary ε near-optimal policies Π ⋆ ε,non under a fixed probability measure λ: d c = Π ⋆ ε,non dπdλ(π).(12) And d D covers all admissible distributions of Π ⋆ ε,non : ∀π ∈ Π ⋆ ε,non , i ∈ N, dπ ,i d D ∞ ≤ C, where d π,i denotes the normalized distribution of the i-th step part of d π . The distribution shift from d D is bounded as d dc,π ⋆ e d D ∞ ≤ C. While the above case is impractical in reality, it reveals the power of this inductive bias: the more auxiliary information we obtain about optimal paths, the weaker coverage assumptions of the dataset are required. Finite-sample guarantee We now give the finite-sample guarantee of Algorithm 1, but before proceeding, we should state necessary function class assumptions. The first are the weak realizability assumptions: Assumption 3 (Realizability of W). There exists state-action distribution density ratio w ⋆ ∈ W such that w ⋆ • d D = (I − γP π ⋆ e ) −1 d c Q ⋆ . Assumption 4 (Realizability of B). There exists policy ratio β ⋆ ∈ B such that β ⋆ • π c = π ⋆ e and for all s ∈ S, A β(s, a)π c (s, a)dν(a) = 1. Assumption 5 (Realizability of Q). Q contains the optimal value function: Q ⋆ ∈ Q. On the other hand, we gather all the bounding assumptions here. Assumption 6 (Boundness of Q). For any q ∈ Q, we assume q ∈ (S × A → [0, V max ]). Assumption 7 (Boundness of B). For any β ∈ B, we assume β ∈ (S × A → [0, U B ]). Assumption 8 (Boundness of W). For any w ∈ W, we assume w ∈ (S × A → [0, U W ]). Remark 4 (Validity). The invertibility of I − γP π ⋆ e is shown by Lemma 10 in Appendix B.1. While Assumptions 3 and 8 actually subsumes Assumption 2, we make it explicit for clarity of explanation. Assumption 4 implicitly assumes that π c covers π ⋆ e , this can easily be done by directly choosing π b as π c . Remark 5. Although we include the normalization step in Assumption 4, this can also be achieved with some preprocessing steps. Remark 6. There is an overlap in the above assumptions: we can derive a policy ratio class B directly from W and Q. With these prerequisites in place, we can finally state our finite-sample guarantee. Theorem 1 (Sample complexity of learning a near-optimal policy). If Assumptions 1, 2, 3, 4, 5, 6, 7 and 8 hold with ε c ≥ 4CcU B √ εstat 1−γ where ε stat = U W V max 2 log(2|Q||W|/δ) N D , then with probability at least 1 − δ, the outputπ from Algorithm 1 is near-optimal: J ⋆ − Jπ ≤ 4C c U B √ ε stat 1 − γ . Proof sketch of Theorem 1 As we can obtain the near-optimality guarantee via Lemma 3, the remaining task is to approximate Eq. (4). This comes from the following two lemmas. Lemma 6 (L 2 error ofq under d c , adapted from theorem 2 in Uehara et al. (2023)). If Assumptions 2, 3, 5, 6 and 8 hold, with probability at least 1 − δ, the estimatedq from Algorithm 1 satisfies ∥q − Q ⋆ ∥ dc,2 ≤ 2 √ ε stat . Lemma 7 (From L 1 distance to Eq. (4)). If Assumptions 4 and 7 hold, ⟨Q ⋆ (·, π ⋆ e ) − Q ⋆ (·,π), µ c ⟩ ≤2U B ∥q − Q ⋆ ∥ dc,1 . Combine them, we have that with probability at least 1 − δ, ⟨Q ⋆ (·, π ⋆ e ) − Q ⋆ (·,π), µ c ⟩ ≤2U B ∥q − Q ⋆ ∥ dc,1 ≤ 2U B ∥q − Q ⋆ ∥ dc,2 ≤ 4U B √ ε stat . Comparison with related works We now provide a brief comparison of our method with some related algorithms. Algorithms with gap assumptions Chen and Jiang (2022) and Uehara et al. (2023) assume that there are (soft) gaps in the optimal value function, which is only satisfied by part of MDPs, whereas our goal is to deal with general problems. Moreover, while our algorithm is based on the optimal value estimator proposed by Uehara et al. (2023), we use the policy ratio to ensure a finite distribution shift and our near-optimality guarantee does not require the soft margin assumption. Besides, Uehara et al. (2023) use d D as d c , assuming that the dataset covers the optimal trajectories from itself. This assumption is non-monotonic and hard to be satisfied in reality. Instead, we propose using an additional covering distribution d c as an alternative, which can effectively utilize the prior knowledge about the optimal trajectories and trade off the dataset requirement. 2023)) are often closely related to approximate dynamic programming (ADP). They "pessimistically" estimate the given policies and update (or choose) policies "pessimistically" with the estimated value functions. However, the evaluation step used in these algorithms always requires the strong realization of all candidate policies' value functions, which our algorithm avoids. Algorithms with behavior regularization Limitations of our algorithm On the one hand, the additional covering distribution may be hard to access in some scenarios, leading back to using d D as d c . On the other hand, although mitigated with increasing dataset size, the assumption of covering all near-optimal policies is still stronger than the classic single-optimal concentrability. In addition, the "non-stationary" coverage requirement is also somewhat restrictive. Conclusion and further work This paper present VOPR, a new MIS-based algorithm for offline RL with function approximations. VOPR is inspired by the optimal value estimator proposed in Uehara et al. (2023), and it circumvents the soft margin assumption in the original paper with the near-optimal coverage assumption. While it still works if using the data distribution as the covering distribution, VOPR can trade off data assumptions with more refined choices. Compared with other algorithms considering partial coverage, VOPR does not make strong function class assumptions and works under general MDPs. Finally, despite the successes, a refined additional covering distribution may be difficult to obtain, and the near-optimal coverage assumption is still stronger than single optimal concentrability. We leave them for further investigation. Hanlin Zhu, Paria Rashidinejad, and Jiantao Jiao. Importance weighted actor-critic for optimal conservative offline reinforcement learning. ArXiv, abs/2301.12714, 2023. policy ratio function class β members of B V π state value function for policy π Q π state-action value function for policy π V ⋆ optimal state value function Q ⋆ optimal state-action value function ν uniform measure of A, S, or S × A, depending on the context D dataset used in the algorithm d D state-action distribution of dataset µ D state distribution of dataset π b behaviour policy d c the additional covering distribution µ c state distribution of the additional covering distribution π c policy of the additional covering distribution ⟨a, b⟩ inner product of a and b, usually as ab dν A Notations f 1 • f 2 (f 1 • f 2 )(s, a) = f 1 (s, a)f 2 (s, a), normalizing it if needed (e.g., density) µ × π (µ × π)(s, a) = µ(s)π(a | s) T ⋆ Bellman optimality operator, T ⋆ q(s, a) := R(s, a) + γE s ′ ∼P (·|s,a) [max q(s ′ , ·)] µ 0 initial state distribution µ i:j π the i-th to j-th steps part of µ π d 1 ≫ d 2 d 2 is absolutely continuous w.r.t. d 1 d π,i normalize i-th step part of state-action distribution induced by π d d,π state-action distribution induced by π from d π i policy take π in the previous 0-th to i-th (include the i-th) steps, and take π ⋆ e after this π β π β (a | s) = π c (a | s)β(s, a)/ A π c (a | s)β(s, a)dν(a) Π ⋆ ε,non the class of all non-stationary ε near-optimal policies P π state-action transition kernel with policy π O ⋆ conjucate operator of some operator O While π, µ, and d are mainly used to denote the Radon-Nikodym derivatives of the underlying probability measures w.r.t. ν, we sometimes also use them to represent the correspondent distribution measure with abuse of notation. B Helper Lemmas B.1 Properties of P π We first provide some properties of P π (for any policy π) as an operator on the L ∞ -space of S × A, and similar results should also hold for transition operators with policies defined on S. Note that the integrations of the absolute value of the functions considered in this subsection are always finite, which means that we can change the orders of integrations via Fubini's Theorem. As we will consider conjugate operators, we define the inner product as ⟨q, d⟩ = S×A q(s, a)d(s, a)dν(s, a). Lemma 8. P π is linear. Proof. Recall the definition of P π , P π d(s ′ , a ′ ) = S×A π(a ′ | s ′ )P (s ′ | s, a)d(s, a)dν(s, a) For any d 1 , d 2 ∈ L ∞ (S × A), P π (α 1 d 1 + α 2 d 2 )(s ′ , a ′ ) = S×A π(a ′ | s ′ )P (s ′ | s, a)(α 1 d 1 + α 2 d 2 )(s, a)dν(s, a) = S×A α 1 π(a ′ | s ′ )P (s ′ | s, a)d 1 (s, a)dν(s, a) + S×A α 2 π(a ′ | s ′ )P (s ′ | s, a)d 2 (s, a)dν(s, a) =α 1 S×A π(a ′ | s ′ )P (s ′ | s, a)d 1 (s, a)dν(s, a) + α 2 S×A π(a ′ | s ′ )P (s ′ | s, a)d 2 (s, a)dν(s, a) =α 1 P π d 1 (s ′ , a ′ ) + α 2 P π d 2 (s ′ , a ′ ) This compeletes the proof. Lemma 9. The adjoint operator of P π is P ⋆ π q(s, a) = S×A q(s ′ , a ′ )π(a ′ | s ′ )P (s ′ | s, a)dν(s ′ , a ′ ). Remark 7. Intuitively, we can see P π d(s ′ , a ′ ) as one-step forward of d, such that we start from (s, a) ∼ d, transit into s ′ ∼ P (· | s, a) and take a ′ ∼ π(· | s ′ ). Also, we can view P ⋆ π q(s, a) as one-step backward of q, such that we compute the value of (s, a) through the one step transferred state-action distribution with the help of q. Proof. Consider the inner products ⟨q, P π d⟩ and ⟨P ⋆ π q, d⟩, we should prove that these two are equal. By definition, (Fubini's Theorem) This completes the proof. ⟨q, P π d⟩ = S×A q(s ′ , a ′ ) S×A π(a ′ | s ′ )P (s ′ | s, a)d(s, a)dν(s, a) dν(s ′ , a ′ ) = S×A S×A q(s ′ , a ′ )π(a ′ | s ′ )P (s ′ | s, Lemma. ∥P ⋆ π ∥ ∞ = ∥P π ∥ ∞ ≤ 1 Remark 8. This upper bound should be intuitive since that P π can be seen as a probability transition kernel from S × A to itself. Proof. Fix any s ∈ S, a ∈ A, we define p(s ′ , a ′ ) = P (s ′ | s, a)π(a ′ | s ′ ), By Fubini's theorem, we have that ∥p∥ 1,ν = S×A |p|dν = S×A pdν = S A P (s ′ | s, a)π(a ′ | s ′ )dν(a ′ )dν(s ′ ) = S P (s ′ | s, a) A π(a ′ | s ′ )dν(a ′ ) dν(s ′ ) = S P (s ′ | s, a)dν(s ′ ) =1. For another function q on S × A such that ∥q∥ ∞,ν ≤ 1, we can use Hölder's inequality, which yields ∥pq∥ 1,ν ≤ ∥q∥ ∞,ν ∥p∥ 1,ν ≤ 1. Thus, for any s ∈ S, a ∈ A, and function q with ∥q∥ ∞,ν ≤ 1, P ⋆ π q(s, a) = S×A q(s ′ , a ′ )π(a ′ | s ′ )P (s ′ | s, a)dν(s ′ , a ′ ) = ∥pq∥ 1,ν ≤ 1. So, we have that ∥P π ∥ ∞ = ∥P ⋆ π ∥ ∞ = max ∥q∥∞≤1 ∥P ⋆ π q∥ ∞,ν ≤ max ∥q∥∞≤1 max s∈S,a∈A P ⋆ π q(s, a) ≤ 1. This completes the proof. Lemma 10. I − γP π is invertible and (I − γP π ) −1 = ∞ i=0 (γP π ) i . Proof. Since ∥P π ∥ ∞ ≤ 1, ∞ i=0 (γP π ) i converges. Take multiplication (I − γP π )[ ∞ i=0 (γP π ) i ] = ∞ i=0 (γP π ) i − ∞ i=1 (γP π ) i =(γP π ) 0 =I. This completes the proof. Proposition 11. By definition, d d, π = (1 − γ) ∞ i=0 (γP π ) i d = (1 − γ)(I − γP π ) −1 d. B.2 Other useful lemmas Lemma (Performance difference lemma). We can decompose the performance gap as (1 − γ)(J π1 − J π2 ) = ⟨µ π1 , Q π2 (·, π 1 ) − Q π2 (·, π 2 )⟩. Proof. By definition, ⟨µ π1 , Q π2 (·, π 1 ) − Q π2 (·, π 2 )⟩ =E s∼µπ 1 R(·, π 1 ) + γE a∼π1(·|s),s ′ ∼P (·|s,a) [Q π2 (s ′ , π 2 )] − Q π2 (·, π 2 )⟩ =E s∼µπ 1 R(·, π 1 ) + E s∼µπ 1 γE a∼π1(·|s),s ′ ∼P (·|s,a) [Q π2 (s ′ , π 2 )] − E s∼µπ 1 Q π2 (·, π 2 )⟩ =E s∼µπ 1 R(·, π 1 ) + γE s∼µπ 1 ,a∼π1(·|s),s ′ ∼P (·|s,a) [Q π2 (s ′ , π 2 )] − E s∼µπ 1 Q π2 (·, π 2 )⟩ =E s∼µπ 1 R(·, π 1 ) + E s∼µπ 1 [Q π2 (s, π 2 )] − (1 − γ)E s∼µ0 [Q π2 (s, π 2 )] − E s∼µπ 1 Q π2 (·, π 2 )⟩ =E s∼µπ 1 R(·, π 1 ) − (1 − γ)E s∼µ0 [Q π2 (s, π 2 )] =(1 − γ)(J π1 − J π2 ) The first equality comes from Bellman equation, and the fourth equality comes from the definition of µ π . This completes the proof. C Detailed proofs for Section 3 C.1 Proof of Lemma 3 Lemma (From advantage to optimality, restatement of Lemma 3). If ⟨µ c , Q ⋆ (·,π)−Q ⋆ (·, π ⋆ )⟩ ≥ −ε , and Assumption 1 holds with ε c ≥ Ccε 1−γ ,π is Ccε 1−γ near-optimal. Proof. We begin with using induction to prove thatπ i is Ccε 1−γ near-optimal for any i ∈ N: • We first show thatπ 0 is Ccε 1−γ near-optimal. From Assumption 1, we can use anyπ ∈ Π ⋆ εc,non to conclude that µ 0 µ c ∞ ≤ µπ/(1 − γ) µ c ∞ ≤ C c 1 − γ . Thus, we can the show optimality ofπ ⋆ 0 by the advantage: ⟨µπ 0 , Q ⋆ (·,π 0 ) − Q ⋆ (·, π ⋆ e )⟩ =⟨µ 0:0 π0 , Q ⋆ (·,π) − Q ⋆ (·, π ⋆ e )⟩ + ⟨µ 1:∞ π0 , Q ⋆ (·, π ⋆ e ) − Q ⋆ (·, π ⋆ e )⟩ =⟨µ 0:0 π ⋆ 0 , Q ⋆ (·,π) − Q ⋆ (·, π ⋆ e )⟩ =(1 − γ)⟨µ 0 , Q ⋆ (·,π) − Q ⋆ (·, π ⋆ e )⟩ ≥C c ⟨µ c , Q ⋆ (·,π) − Q ⋆ (·, π ⋆ e )⟩ (Q ⋆ (·,π) − Q ⋆ (·, π ⋆ e ) is non-positive) ≥ − C c ε. By performance difference lemma, (1 − γ)(Jπ 0 − J ⋆ ) =⟨µπ 0 , Q ⋆ (·,π 0 ) − Q ⋆ (·, π ⋆ e )⟩ ≥ − C c ε. • Next, we show that ifπ i is Ccε 1−γ near-optimal,π i+1 is Ccε 1−γ near-optimal. Since thatπ i is Ccε 1−γ optimal, the distribution shift from µ c to µπ i is bounded, which means, µ 0:i+1 π µ c ∞ = µ 0:i+1 πi µ c ∞ ≤ µπ i µ c ∞ ≤ C c . Then, we have ⟨µπ i+1 , Q ⋆ (·,π i+1 ) − Q ⋆ (·, π ⋆ e )⟩ =⟨µ 0:i+1 πi+1 , Q ⋆ (·,π) − Q ⋆ (·, π ⋆ e )⟩ + ⟨µ i+2:∞ πi+1 , Q ⋆ (·, π ⋆ e ) − Q ⋆ (·, π ⋆ e )⟩ =⟨µ 0:i+1 πi+1 , Q ⋆ (·,π) − Q ⋆ (·, π ⋆ e )⟩ =⟨µ 0:i+1 π , Q ⋆ (·,π) − Q ⋆ (·, π ⋆ e )⟩ ≥C c ⟨µ c , Q ⋆ (·,π) − Q ⋆ (·, π ⋆ e )⟩ (Q ⋆ (·,π) − Q ⋆ (·, π ⋆ e ) is non-positive) ≥ − C c ε. By performance difference lemma, (1 − γ)(Jπ i+1 − J ⋆ ) =⟨µπ i+1 , Q ⋆ (·,π i+1 ) − Q ⋆ (·, π ⋆ e )⟩. ≥ − C c ε Therefore,π i+1 is Ccε 1−γ near-optimal. Thus, for any ϵ > 0, there exists natural number i ≥ log γ ϵ Vmax such that J ⋆ − Jπ ≤ J ⋆ − J 0:î π ≤ J ⋆ − (Jπ i − γ i+1 V max ) ≤ C c ε 1 − γ + γ i+1 V max ≤ C c ε 1 − γ + ϵ, where J i:j π denotes the i-th to j-th steps part of the return. Therefore,π is Ccε 1−γ near-optimal. D Detailed proofs for Section 4 D.1 Proof of Lemma 5 Lemma (Restatement of Lemma 5). If d c is a linear combination of the state-action distributions induced by ε near-optimal non-stationary policies Π ⋆ ε,non under a fixed probability measure λ: d c = Π ⋆ ε,non dπdλ(π).(13) And d D covers all admissible distributions of Π ⋆ ε,non : ∀π ∈ Π ⋆ ε,non , i ∈ N, dπ ,i d D ∞ ≤ C. The distribution shift from d D is bounded as d dc,π ⋆ e d D ∞ ≤ C. Proof. Define the state-action distribution of policy π from s ∈ S, a ∈ A at step i as d s,a,π,i (s ′ , a ′ ) = P (s i = s ′ , a i = a ′ |s 0 = s, a 0 = a, s 1 ∼ P (· | s 0 , a 0 ), a 1 ∼ π(· | s 1 ) . . . s j ∼ P (· | s j−1 , a j−1 ), a j ∼ π(· | s j ) . . . ). Also, define the global version of it as d s,a,π (s ′ , a ′ ) = (1 − γ) ∞ i=0 d s,a,π,i (s ′ , a ′ ). We can rewrite d dc,π ⋆ e (s, a) as d dc,π ⋆ e (s, a) = S×A d s1,a1,π ⋆ e (s, a)d c (s 1 , a 1 )dν(s 1 , a 1 ) = S×A d s1,a1,π ⋆ e (s, a) Π dπ(s 1 , a 1 )dλ(π) dν(s 1 , a 1 ) = Π S×A d s1,a1,π ⋆ e (s, a)dπ(s 1 , a 1 )dν(s 1 , a 1 ) dλ(π) (Fubini's Theorem) = Π S×A (1 − γ) ∞ i=0 γ i d s1,a1,π ⋆ e (s, a)dπ ,i (s 1 , a 1 ) dν(s 1 , a 1 ) dλ(π) = Π (1 − γ) ∞ i=0 γ i S×A d s1,a1,π ⋆ e (s, a)dπ ,i (s 1 , a 1 )dν(s 1 , a 1 ) dλ(π) = Π (1 − γ) ∞ i=0 d i:∞ πi (s, a) dλ(π). The last equation comes from that γ i S×A d s1,a1,π ⋆ e (s, a)dπ ,i (s 1 , a 1 )dν(s 1 , a 1 ) =γ i S×A d s1,a1,π ⋆ e (s, a) S A d s2,a2,π,i (s 1 , a 1 )π(a 2 | s 2 )dν(a 2 ) µ 0 (s 2 )dν(s 2 ) dν(s 1 , a 1 ) = S A γ i S×A d s1,a1,π ⋆ e (s, a)d s2,a2,π,i (s 1 , a 1 )dν(s 1 , a 1 ) π(a 2 | s 2 )dν(a 2 ) µ 0 (s 2 )dν(s 2 ), (Fubini's Theorem) since γ i S×A d s1,a1,π ⋆ e (s, a)d s2,a2,π,i (s 1 , a 1 )dν(s 1 , a 1 ) =γ i S×A (1 − γ) ∞ k=0 γ k d s1,a1,π ⋆ e ,k (s, a) d s2,a2,π,i (s 1 , a 1 )dν(s 1 , a 1 ) =(1 − γ) ∞ k=0 γ k+i S×A d s1,a1 ,π ⋆ e ,k (s, a)d s2,a2,π,i (s 1 , a 1 )dν(s 1 , a 1 ) =(1 − γ) ∞ k=0 γ k+i d s2,a2,πi,k+i (s, a) =(1 − γ) ∞ k=i γ k d s2,a2,πi,k (s, a) =d i:∞ s2,a2,πi (s, a), we get γ i S×A d s1,a1,π ⋆ e (s, a)dπ ,i (s 1 , a 1 )dν(s 1 , a 1 ) = S A d i:∞ s2,a2,πi (s, a) π(a 2 | s 2 )dν(a 2 ) µ 0 (s 2 )dν(s 2 ) =d i:∞ πi (s, a). Finally, ∀s ∈ S, a ∈ A, d dc,π ⋆ e (s, a) d D (s, a) = Π (1 − γ) ∞ i=0 d i:∞ πi (s, a) d D (s, a) dλ(π) = Π (1 − γ) ∞ i=0 (1 − γ) ∞ j=i γ j dπ i,j (s, a) d D (s, a) dλ(π) = Π (1 − γ) ∞ i=0 (1 − γ) ∞ j=i γ j dπ i,j (s, a) d D (s, a) dλ(π) ≤ Π C(1 − γ) 2 ∞ i=0 ∞ j=i γ j dλ(π) (π ∈ Π ⋆ ε,non indicatesπ i ∈ Π ⋆ ε,non ) ≤ Π C(1 − γ) 2 ∞ i=0 γ i 1 − γ dλ(π) ≤ Π Cdλ(π) =C. This completes the proof. D.2 Proof of Lemma 6 Note that the lemmas and proofs of this subsection are mainly adapted from Uehara et al. (2023), similar statements could also be found in the original paper. However, since that we use d c to replace d D , we present them for clarity of explanation and to make our paper self-contained. We refer interested readers to the original paper for another detail. We first define the expected version of Eq. (10) as L(d, q, w) :=0.5E d [q 2 (s, a)] + E (s,a)∼d D w ,r=R(s,a),s ′ ∼P (·|s,a) γ max q(s ′ , ·) + r − q(s, a) =0.5E d [q 2 (s, a)] + E Dw γ max q(s ′ , ·) + r − q(s, a) where d D w = d D • w, and E Dw denotes taking expectation with respect to the reweighted data collecting process. Lemma 12 (Expectation). The expected value ofL(d, q, w) w.r.t. the data collecting process is L(d, q, w): E D [L(d, q, w)] = L(d, q, w). Proof. Since only the second term ofL is random, we additional definê L W (q, w) := 1 N D (s,a,r,s ′ )∈D E D w(s, a) γ max q(s ′ , ·) + r − q(s, a) . We can rearrange the expectation as follows, E D [L(d, q, w)] =E D 0.5E d [q 2 (s, a)] +L W (q, w)(14)=E D 0.5E d [q 2 (s, a)] + E D L W (q, w) (15) =0.5E d [q 2 (s, a)] + E D L W (q, w)(16) Then, by the i.i.d. assumption of samples and linear property of MIS, This compeletes the proof. E D [L(d, q, w)] =0.5E d [q 2 (s, a)] + E D 1 N D (s,a,r,s ′ )∈D w(s, a) γ max q(s ′ , ·) + r − q(s, a) =0.5E d [q 2 (s, a)] + 1 N D (s,a,r,s ′ )∈D E D w(s, a) γ max q(s ′ , ·) + r − q(s, a) =0.5E d [q 2 (s, a)] + E D w(s, a) γ max q(s ′ , ·) + r − q(s, a) =0.5E d [q 2 (s, a)] + E (s, Lemma 13 (Concentration). For any fixed d, with probability at least 1 − δ, for any q ∈ Q, w ∈ W, L(d, q, w) −L(d, q, w) ≤ ε stat . Proof. The statistical error only comes fromL W , as L(d, q, w) −L(d, q, w) = E D [L(d, q, w)] −L(d, q, w) (Lemma 12) = E D [L W (q, w)] −L W (q, w) . (Eq. (16)) Since each entry of L W is bounded: ∀q ∈ Q, w ∈ W, a ∈ A, s ′ ∈ S, w(s, a) γ max q(s ′ , ·) + r − q(s, a) ≤ U W V max , we can apply Hoeffding's inequality which yields that, for any q ∈ Q, w ∈ W, with probability at least 1 − δ/(|Q||W|), E D [L W (q, w)] −L W (q, w) ≤ U W V max 2 log(2|Q||W|/δ) N D . Finally, we can use union bound, rearranging terms to get that, for any fixed d, with probability at least 1 − δ, for any q ∈ Q, w ∈ W, L(d, q, w) −L(d, q, w) ≤ U W V max 2 log(2|Q||W|/δ) N D = ε stat This compeletes the proof. Lemma 14. If w is non-negative ν-a.e. (e.g., w ∈ W), for any q : S × A → [0, V max ], L(d, q, w) − L(d, Q ⋆ , w) ≥ 0.5⟨d, q 2 − (Q ⋆ ) 2 ⟩ + ⟨(γP π ⋆ e − I)d D w , q − Q ⋆ ⟩.(17) Proof. This result simply comes from the definition: L(d, q, w) − L(d, Q ⋆ , w) =0.5E d [q 2 (s) − (Q ⋆ ) 2 (s)] + E Dw [γ max q(s ′ , ·) + r − q(s, a)] − E Dw [γ max Q ⋆ (s ′ , ·) + r − Q ⋆ (s, a)] =0.5E d [q 2 (s) − (Q ⋆ ) 2 (s)] + E Dw [γ max q(s ′ , ·) + r − q(s, a)] − E Dw [γQ ⋆ (s ′ , π ⋆ e ) + r − Q ⋆ (s, a)] ≥0.5E d [q 2 (s) − (Q ⋆ ) 2 (s)] + E Dw [γq(s ′ , π ⋆ e ) + r − q(s, a)] − E Dw [γQ ⋆ (s ′ , π ⋆ e ) + r − Q ⋆ (s, a)] =0.5E d [q 2 (s) − (Q ⋆ ) 2 (s)] + E Dw [γ(q − Q ⋆ )(s ′ , π ⋆ e ) − (q − Q ⋆ )(s, a)] =0.5⟨d, q 2 − (Q ⋆ ) 2 ⟩ + ⟨d D w , (γP ⋆ π ⋆ e − I)(q − Q ⋆ )⟩ (Rewrite the expectation with inner products) =0.5⟨d, q 2 − (Q ⋆ ) 2 ⟩ + ⟨(γP π ⋆ e − I)d D w , q − Q ⋆ ⟩. This compeletes the proof. Lemma 15. If Assumption 5 holds, with probability at least 1−δ, for any w ∈ W and any state-action distribution d, we have L(d,q, w) − L(d, Q ⋆ , w) ≤ 2ε stat . Proof. We can decompose Eq. (18) whereŵ(q) = argmax w∈WL (d, q, w). For the terms above, we have that: • (2) and (3) are non-positive since the optimization process. • (1) and (4) could be bound by concentration. • For (5), as Bellman optimality equation holds, ∀s ∈ S, a ∈ A, E s ′ ∼P (·|s,a) γ max Q ⋆ (s ′ , ·) + R(s, a) − Q ⋆ (s, a) = 0. We have that Thus, we conclude that with probability at least 1 − δ, L(q, w) − L(Q ⋆ , w) ≤ L(q, w) −L(q, w) (1) +L(Q ⋆ ,ŵ(Q ⋆ )) − L(Q ⋆ ,ŵ(Q ⋆ )) (4) ≤|L(q, w) −L(q, w)| + |L(Q ⋆ ,ŵ(Q ⋆ )) − L(Q ⋆ ,ŵ(Q ⋆ ))| ≤2ε stat . ( Lemma 13) This compeletes the proof. With lemmas above, it's time to prove Lemma 6. Lemma (L 2 error ofq under d c , restatement Lemma 6). If Assumptions 2, 3, 5, 6 and 8 hold, with probability at least 1 − δ, the estimatedq from Algorithm 1 satisfies ∥q − Q ⋆ ∥ dc,2 ≤ 2 √ ε stat . Proof. By Assumption 3, d D w ⋆ = (I − γP π ⋆ ) −1 d c Q ⋆ , and from Lemma 14 we have L(d c ,q, w ⋆ ) − L(d c , Q ⋆ , w ⋆ ) ≥0.5⟨d c ,q 2 − (Q ⋆ ) 2 ⟩ − ⟨(I − γP π ⋆ )(I − γP π ⋆ ) −1 d c Q ⋆ , (q − Q ⋆ )⟩ =0.5⟨d c ,q 2 − (Q ⋆ ) 2 ⟩ − ⟨d c Q ⋆ , (q − Q ⋆ )⟩ =0.5⟨d c ,q 2 − (Q ⋆ ) 2 ⟩ − ⟨d c , Q ⋆ (q − Q ⋆ )⟩ =0.5⟨d c , (q − Q ⋆ ) 2 ⟩ =0.5∥q − Q ⋆ ∥ 2 dc,2 . Together with Lemma 15, with probability at least 1 − δ, 0.5∥q − Q ⋆ ∥ 2 dc,2 ≤ L(d c ,q, w ⋆ ) − L(d c , Q ⋆ , w ⋆ ) ≤ 2ε stat . Rearrange this and we can get ∥q − Q ⋆ ∥ dc,2 ≤ 2 √ ε stat This compeletes the proof. D.3 Proof of Lemma 7 Lemma (Restatement of Lemma 7). If Assumptions 4 and 7 hold, ⟨Q ⋆ (·, π ⋆ e ) − Q ⋆ (·,π), µ c ⟩ ≤2U B ∥q − Q ⋆ ∥ dc,1 . Proof. We can rearrange the above term as ⟨Q ⋆ (·, π ⋆ e ) − Q ⋆ (·,π), µ c ⟩ =⟨Q ⋆ (·, π ⋆ e ) −q(·, π ⋆ e ), µ c ⟩ + ⟨q(·, π ⋆ e ) −q(·,π), µ c ⟩ + ⟨q(·,π) − Q ⋆ (·,π), µ c ⟩ ≤⟨Q ⋆ (·, π ⋆ e ) −q(·, π ⋆ e ), µ c ⟩ + ⟨q(·,π) − Q ⋆ (·,π), µ c ⟩ (Assumption 4) ≤∥Q ⋆ (·, π ⋆ e ) −q(·, π ⋆ e )∥ µc,1 + ∥q(·,π) − Q ⋆ (·,π)∥ µc,1 =∥Q ⋆ −q∥ µc×π ⋆ e ,1 + ∥q − Q ⋆ ∥ µc×π,1 ≤2U B ∥Q ⋆ −q∥ dc,1 The distribution shift comes from the fact that µ × π 1 µ × π 2 ∞ = π 1 π 2 ∞ , and shifts from π c to π ⋆ e andπ are both bound by U B due to Assumptions 4 and 7. This completes the proof. D.4 Proof of Theorem 1 Theorem (Finite sample guarantee of Algorithm 1, restatement of Theorem 1). If Assumptions 1, 2, 3, 4, 5, 6, 7 and 8 hold with ε c ≥ 4CcU B √ εstat 1−γ , then with probability at least 1 − δ, the outputπ from Algorithm 1 is near-optimal: J ⋆ − Jπ ≤ 4C c U B √ ε stat 1 − γ . Proof. From Lemma 6, we have that with probability at least 1 − δ, ∥q − Q ⋆ ∥ dc,1 ≤ ∥q − Q ⋆ ∥ dc,2 ≤ 2 √ ε stat . Then apply Lemma 7 to bound the weighted advantage, ⟨Q ⋆ (·, π ⋆ e ) − Q ⋆ (·,π), µ c ⟩ ≤2U B ∥q − Q ⋆ ∥ dc,1 ≤ 4U B √ ε stat . Finally, according to Lemma 3,π is 4CcU B √ εstat 1−γ optimal. This completes the proof. MIS) (Liu et al., 2018). While theoretic guarantees in MIS under weak realizability and partial coverage assumptions are typically made for scalar values (e.g., the return (Uehara et al., 2019; Jiang and Huang, 2020)), recently, Zhan et al. (2022); Huang and Jiang(2022);Uehara et al. (2023) have gone beyond this and derived L 2 error guarantees for the estimators by using some strongly convex functions. Among them, the optimal value function estimator fromUehara et al. (2023) constructs the base of this work. Figure 1 : 1Figure 1: The above MDP is deterministic, and we initially start from state 1. We can take actions L (left) and R (right) in each state. In states 1 and 3, action L (R) will transfer us to its left (right) hand state, and taking actions in other states will only cause a self-loop. We can only obtain non-zero rewards by taking actions in states 2 and 4, with values 1 and 1 γ correspondingly. There are two trajectories that could lead to the optimal γ /(1−γ) return: {(1, R), 2, . . . } and {(1, L), (3, L), 4 . . . }. We take γ as the discount factor. Zhan et al. (2022) use behavior regularization to ensure that the learned policy is close to the dataset. Nevertheless, the regularization makes the optimality of the learned policy intractable.Algorithms with pessimism in the face of uncertainty These algorithms (e.g.,Jiang and Huang (2020); Liu et al. (2020); Xie et al. (2021); Cheng et al. (2022); Zhu et al. ( s ′ , a ′ )π(a ′ | s ′ )P (s ′ | s, a)dν(s ′ , a ′ ) dν(s, a) = S×A S×A d(s, a)q(s ′ , a ′ )π(a ′ | s ′ )P (s ′ | s, a)dν(s ′ , a ′ )dν(s, a) = S×A S×A q(s ′ , a ′ )π(a ′ | s ′ )P (s ′ | s, a)d(s, a)dν(s, a)dν(s ′ , a ′ ). a)∼d D ,r=R(s,a),s ′ ∼P (·|s,a) w(s, a) γ max q(s ′ , ·) + r − q(s, a) =0.5E d [q 2 (s, a)] + E (s,a)∼d D w(s, a) E s ′ ∼P (·|s,a) [γ max q(s ′ , ·)] + R(s, a) − q(s, a) =0.5E d [q 2 (s, a)] + E (s,a)∼d D w E s ′ ∼P (·|s,a) [γ max q(s ′ , ·)] + R(s, a) − q(s, a) =0.5E d [q 2 (s, a)] + E (s,a)∼d D w ,r=R(s,a),s ′ ∼P (·|s,a) γ max q(s ′ , ·) + r − q(s, a) =L(d, q, w). as follows,L(d,q, w) − L(d, Q ⋆ , w) = L(d,q, w) −L(d,q, w) d,q,ŵ) −L(d, Q ⋆ ,ŵ(Q ⋆ )) (3) +L(d, Q ⋆ ,ŵ(Q ⋆ )) − L(d, Q ⋆ ,ŵ(Q ⋆ )) (4) + L(d, Q ⋆ ,ŵ(Q ⋆ )) − L(d, Q ⋆ , w) ( 5 ) 5=L(d, Q ⋆ ,ŵ(Q ⋆ )) − L(d, Q ⋆ , w) =0.5E d [(Q ⋆ ) 2 (s, a)] + E Dŵ (Q ⋆ ) γ max q(s ′ , ·) + r − q(s, a) − 0.5E d [(Q ⋆ ) 2 (s, a)] + E Dw γ max Q ⋆ (s ′ , ·) + r − Q ⋆ (s, a) =E (s,a)∼d D w(Q ⋆ ) ,r=R(s,a),s ′ ∼P (·|s,a) γ max Q ⋆ (s ′ , ·) + r − Q ⋆ (s, a) − E (s,a)∼d D w ,r=R(s,a),s ′ ∼P (·|s,a) γ max Q ⋆ (s ′ , ·) + r − Q ⋆ (s, a) =E (s,a)∼d D w(Q ⋆ ) γE s ′ ∼P (·,s,a) [max Q ⋆ (s ′ , ·)] + R(s, a) − Q ⋆ (s, a) − E (s,a)∼d D w γE s ′ ∼P (·,s,a) [max Q ⋆ (s ′ , ·)] + R(s, a) − Q ⋆ (s, a) =0. Table 1 : 1Comparison of offline RL algorithm (conc. stand for concentrability)Algorithm Data assumptions Function assumptions Major drawbacks Jiang and Huang (2020) optimal conc. w ⋆ ∈ W, and ∀π ∈ Π, Qπ ∈ C(Q) strong realizability Zhan et al. (2022) optimal conc. w ⋆ α ∈ W, and v ⋆ α ∈ V compare with π ⋆ α Jongmin Lee, Wonseok Jeon, Byung-Jun Lee, Joëlle Pineau, and Kee-Eung Kim. Optidice: Offline policy optimization via stationary distribution correction estimation. In International Conference on Machine Learning, 2021. Qiang Liu, Lihong Li, Ziyang Tang, and Dengyong Zhou. Breaking the curse of horizon: Infinitehorizon off-policy estimation. In Neural Information Processing Systems, 2018. Ruosong Wang, Yifan Wu, Ruslan Salakhutdinov, and Sham M. Kakade. Instabilities of offline rl with pre-trained neural representation. In International Conference on Machine Learning, 2021. Xinqi Wang, Qiwen Cui, and Simon Shaolei Du. On gap-dependent bounds for offline reinforcement learning. ArXiv, abs/2206.00177, 2022.Yao Liu, Adith Swaminathan, Alekh Agarwal, and Emma Brunskill. Provably good batch reinforce- ment learning without great exploration. ArXiv, abs/2007.08202, 2020. A.S. Manne. Linear programming and sequential decisions. In Management Science, volume 6, page 259-267, 1960. Rémi Munos. Error bounds for approximate policy iteration. In International Conference on Machine Learning, 2003. Rémi Munos. Error bounds for approximate value iteration. In AAAI Conference on Artificial Intelligence, 2005. Tengyang Xie and Nan Jiang. Q* approximation schemes for batch reinforcement learning: A theoretical comparison. ArXiv, abs/2003.03924, 2020. Tengyang Xie, Ching-An Cheng, Nan Jiang, Paul Mineiro, and Alekh Agarwal. Bellman-consistent pessimism for offline reinforcement learning. In Neural Information Processing Systems, 2021. Wenhao Zhan, Baihe Huang, Audrey Huang, Nan Jiang, and Jason D. Lee. Offline reinforcement learning with realizability and single-policy concentrability. ArXiv, abs/2202.04634, 2022. Table 2 : 2NotationsS state space A action space Q state-action value function class W state-action distribution ratio function class B Fitted q-iteration in continuous action-space mdps. András Antos, Rémi Munos, Csaba Szepesvári, NIPS. András Antos, Rémi Munos, and Csaba Szepesvári. Fitted q-iteration in continuous action-space mdps. In NIPS, 2007. Residual algorithms: Reinforcement learning with function approximation. C Leemon, Baird, International Conference on Machine Learning. Leemon C. Baird. Residual algorithms: Reinforcement learning with function approximation. In International Conference on Machine Learning, 1995. Information-theoretic considerations in batch reinforcement learning. Jinglin Chen, Nan Jiang, abs/1905.00360ArXiv. Jinglin Chen and Nan Jiang. Information-theoretic considerations in batch reinforcement learning. ArXiv, abs/1905.00360, 2019. Offline reinforcement learning under value and density-ratio realizability: the power of gaps. Jinglin Chen, Nan Jiang, abs/2203.13935ArXiv. Jinglin Chen and Nan Jiang. Offline reinforcement learning under value and density-ratio realizability: the power of gaps. ArXiv, abs/2203.13935, 2022. Adversarially trained actor critic for offline reinforcement learning. Ching-An Cheng, Tengyang Xie, Nan Jiang, Alekh Agarwal, abs/2202.02446ArXiv. Ching-An Cheng, Tengyang Xie, Nan Jiang, and Alekh Agarwal. Adversarially trained actor critic for offline reinforcement learning. ArXiv, abs/2202.02446, 2022. Tree-based batch mode reinforcement learning. Damien Ernst, Pierre Geurts, Louis Wehenkel, J. Mach. Learn. Res. 6Damien Ernst, Pierre Geurts, and Louis Wehenkel. Tree-based batch mode reinforcement learning. J. Mach. Learn. Res., 6:503-556, 2005. Error propagation for approximate policy and value iteration. Rémi Amir Massoud Farahmand, Csaba Munos, Szepesvari, NIPS. Amir Massoud Farahmand, Rémi Munos, and Csaba Szepesvari. Error propagation for approximate policy and value iteration. In NIPS, 2010. Offline reinforcement learning: Fundamental barriers for value function approximation. Dylan J Foster, Akshay Krishnamurthy, David Simchi-Levi, Yunzong Xu, Annual Conference Computational Learning Theory. Dylan J. Foster, Akshay Krishnamurthy, David Simchi-Levi, and Yunzong Xu. Offline reinforce- ment learning: Fundamental barriers for value function approximation. In Annual Conference Computational Learning Theory, 2021. Evaluating reinforcement learning algorithms in observational health settings. Omer Gottesman, D Fredrik, Joshua Johansson, Jack Meier, Donghun Dent, Srivatsan Lee, Linying Srinivasan, Yi Zhang, David Ding, Xuefeng Wihl, Jiayu Peng, Isaac Yao, Lage, abs/1805.12298ArXiv. Christopher Mosch, Li wei H. Lehman, Matthieu Komorowski, A. Aldo Faisal, Leo Anthony Celi, David A. Sontag, and Finale Doshi-VelezOmer Gottesman, Fredrik D. Johansson, Joshua Meier, Jack Dent, Donghun Lee, Srivatsan Srinivasan, Linying Zhang, Yi Ding, David Wihl, Xuefeng Peng, Jiayu Yao, Isaac Lage, Christopher Mosch, Li wei H. Lehman, Matthieu Komorowski, A. Aldo Faisal, Leo Anthony Celi, David A. Sontag, and Finale Doshi-Velez. Evaluating reinforcement learning algorithms in observational health settings. ArXiv, abs/1805.12298, 2018. Guidelines for reinforcement learning in healthcare. Omer Gottesman, Fredrik Johansson, Matthieu Komorowski, Aldo Faisal, David Sontag, Finale Doshi-Velez, Leo Anthony Celi, Nature Medicine. 25Omer Gottesman, Fredrik Johansson, Matthieu Komorowski, Aldo Faisal, David Sontag, Finale Doshi-Velez, and Leo Anthony Celi. Guidelines for reinforcement learning in healthcare. Nature Medicine, 25:16 -18, 2019. Beyond the return: Off-policy function estimation under user-specified error-measuring distributions. Audrey Huang, Nan Jiang, abs/2210.15543ArXiv. Audrey Huang and Nan Jiang. Beyond the return: Off-policy function estimation under user-specified error-measuring distributions. ArXiv, abs/2210.15543, 2022. Minimax value interval for off-policy evaluation and policy optimization. Nan Jiang, Jiawei Huang, arXiv: LearningNan Jiang and Jiawei Huang. Minimax value interval for off-policy evaluation and policy optimization. arXiv: Learning, 2020. Is pessimism provably efficient for offline rl. Ying Jin, Zhuoran Yang, Zhaoran Wang, International Conference on Machine Learning. Ying Jin, Zhuoran Yang, and Zhaoran Wang. Is pessimism provably efficient for offline rl? In International Conference on Machine Learning, 2020. Qt-opt: Scalable deep reinforcement learning for vision-based robotic manipulation. Dmitry Kalashnikov, Alex Irpan, Peter Pastor, Julian Ibarz, Alexander Herzog, Eric Jang, Deirdre Quillen, Ethan Holly, Mrinal Kalakrishnan, Vincent Vanhoucke, Sergey Levine, abs/1806.10293ArXiv. Dmitry Kalashnikov, Alex Irpan, Peter Pastor, Julian Ibarz, Alexander Herzog, Eric Jang, Deirdre Quillen, Ethan Holly, Mrinal Kalakrishnan, Vincent Vanhoucke, and Sergey Levine. Qt-opt: Scal- able deep reinforcement learning for vision-based robotic manipulation. ArXiv, abs/1806.10293, 2018. Stabilizing off-policy q-learning via bootstrapping error reduction. Aviral Kumar, Justin Fu, G Tucker, Sergey Levine, Neural Information Processing Systems. Aviral Kumar, Justin Fu, G. Tucker, and Sergey Levine. Stabilizing off-policy q-learning via bootstrapping error reduction. In Neural Information Processing Systems, 2019. Safe policy improvement with baseline bootstrapping. Romain Laroche, Paul Trichelair, abs/1712.06924ArXiv. Romain Laroche and Paul Trichelair. Safe policy improvement with baseline bootstrapping. ArXiv, abs/1712.06924, 2017. Performance bounds in lp-norm for approximate value iteration. Rémi Munos, SIAM J. Control. Optim. 46Rémi Munos. Performance bounds in lp-norm for approximate value iteration. SIAM J. Control. Optim., 46:541-561, 2007. Finite-time bounds for fitted value iteration. Rémi Munos, Csaba Szepesvari, J. Mach. Learn. Res. 9Rémi Munos and Csaba Szepesvari. Finite-time bounds for fitted value iteration. J. Mach. Learn. Res., 9:815-857, 2008. Algaedice: Policy gradient from arbitrary experience. Ofir Nachum, Bo Dai, Ilya Kostrikov, Yinlam Chow, Lihong Li, Dale Schuurmans, abs/1912.02074ArXiv. Ofir Nachum, Bo Dai, Ilya Kostrikov, Yinlam Chow, Lihong Li, and Dale Schuurmans. Algaedice: Policy gradient from arbitrary experience. ArXiv, abs/1912.02074, 2019. Revisiting the linear-programming framework for offline rl with general function approximation. Asuman E Ozdaglar, Sarath Pattathil, Jiawei Zhang, K Zhang, abs/2212.13861ArXiv. Asuman E. Ozdaglar, Sarath Pattathil, Jiawei Zhang, and K. Zhang. Revisiting the linear-programming framework for offline rl with general function approximation. ArXiv, abs/2212.13861, 2022. Markov decision processes: Discrete stochastic dynamic programming. Martin L Puterman, Wiley Series in Probability and Statistics. Martin L. Puterman. Markov decision processes: Discrete stochastic dynamic programming. In Wiley Series in Probability and Statistics, 1994. Bridging offline reinforcement learning and imitation learning: A tale of pessimism. Paria Rashidinejad, Banghua Zhu, Cong Ma, Jiantao Jiao, Stuart J Russell, IEEE Transactions on Information Theory. 68Paria Rashidinejad, Banghua Zhu, Cong Ma, Jiantao Jiao, and Stuart J. Russell. Bridging offline reinforcement learning and imitation learning: A tale of pessimism. IEEE Transactions on Information Theory, 68:8156-8196, 2021. Optimal conservative offline rl with general function approximation via augmented lagrangian. Paria Rashidinejad, Hanlin Zhu, Kunhe Yang, Stuart J Russell, Jiantao Jiao, abs/2211.00716ArXiv. Paria Rashidinejad, Hanlin Zhu, Kunhe Yang, Stuart J. Russell, and Jiantao Jiao. Optimal conservative offline rl with general function approximation via augmented lagrangian. ArXiv, abs/2211.00716, 2022. On the use of non-stationary policies for stationary infinite-horizon markov decision processes. Bruno Scherrer, Boris Lesner, NIPS. Bruno Scherrer and Boris Lesner. On the use of non-stationary policies for stationary infinite-horizon markov decision processes. In NIPS, 2012. S4rl: Surprisingly simple self-supervision for offline reinforcement learning. Samarth Sinha, Animesh Garg, abs/2103.06326ArXiv. Samarth Sinha and Animesh Garg. S4rl: Surprisingly simple self-supervision for offline reinforcement learning. ArXiv, abs/2103.06326, 2021. Hybrid rl: Using both offline and online data can make rl efficient. Yuda Song, Yi Zhou, Ayush Sekhari, J Andrew Bagnell, Akshay Krishnamurthy, Wen Sun, abs/2210.06718ArXiv. Yuda Song, Yi Zhou, Ayush Sekhari, J. Andrew Bagnell, Akshay Krishnamurthy, and Wen Sun. Hybrid rl: Using both offline and online data can make rl efficient. ArXiv, abs/2210.06718, 2022. Reinforcement Learning: An Introduction. Richard S Sutton, Andrew G Barto, The MIT Presssecond editionRichard S. Sutton and Andrew G. Barto. Reinforcement Learning: An Introduction. The MIT Press, second edition, 2018. Leveraging factored action spaces for efficient offline reinforcement learning in healthcare. Shengpu Tang, Maggie Makar, Michael W Sjoding, Finale Doshi-Velez, Jenna Wiens, Shengpu Tang, Maggie Makar, Michael W. Sjoding, Finale Doshi-Velez, and Jenna Wiens. Leveraging factored action spaces for efficient offline reinforcement learning in healthcare. 2022. Pessimistic model-based offline reinforcement learning under partial coverage. Masatoshi Uehara, Wen Sun, International Conference on Learning Representations. Masatoshi Uehara and Wen Sun. Pessimistic model-based offline reinforcement learning under partial coverage. In International Conference on Learning Representations, 2021. Minimax weight and q-function learning for off-policy evaluation. Masatoshi Uehara, Jiawei Huang, Nan Jiang, International Conference on Machine Learning. Masatoshi Uehara, Jiawei Huang, and Nan Jiang. Minimax weight and q-function learning for off-policy evaluation. In International Conference on Machine Learning, 2019. Refined value-based offline rl under realizability and partial coverage. Masatoshi Uehara, Nathan Kallus, Jason D Lee, Wen Sun, abs/2302.02392ArXiv. Masatoshi Uehara, Nathan Kallus, Jason D. Lee, and Wen Sun. Refined value-based offline rl under realizability and partial coverage. ArXiv, abs/2302.02392, 2023.
{'fraction_non_alphanumeric': 0.1009580070951912, 'fraction_numerical': 0.02802131682998109, 'mean_word_length': 3.6297662976629765, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 41, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'We study learning optimal policies from a logged dataset, i.e., offline RL, with function approximation. Despite the efforts devoted, existing algorithms with theoretic finite-sample guarantees typically assume exploratory data coverage or strong realizable function classes, which is hard to be satisfied in reality. While there are recent works that successfully tackle these strong assumptions, they either require the gap assumptions that only could be satisfied by part of MDPs or use the behavior regularization that makes the optimality of learned policy even intractable. To solve this challenge, we provide finite-sample guarantees for a simple algorithm based on marginalized importance sampling (MIS), showing that sample-efficient offline RL for general MDPs is possible with only a partial coverage dataset and weak realizable function classes given additional side information of a covering distribution. Furthermore, we demonstrate that the covering distribution trades off prior knowledge of the optimal trajectories against the coverage requirement of the dataset, revealing the effect of this inductive bias in the learning processes.Preprint. Under review.', 'arxivid': '2305.12679', 'author': ['Chenjie Mao [email protected] \nSchool of Computer Science and Technology\nHuazhong University of Science and Technology\n430074WuhanChina\n'], 'authoraffiliation': ['School of Computer Science and Technology\nHuazhong University of Science and Technology\n430074WuhanChina'], 'corpusid': 258832646, 'doi': '10.48550/arxiv.2305.12679', 'github_urls': [], 'n_tokens_mistral': 24047, 'n_tokens_neox': 21678, 'n_words': 11820, 'pdfsha': '3f1302446ae40cceea8a804efc9fc2dba1d056d0', 'pdfurls': ['https://export.arxiv.org/pdf/2305.12679v2.pdf'], 'title': ['Offline Reinforcement Learning with Additional Covering Distributions', 'Offline Reinforcement Learning with Additional Covering Distributions'], 'venue': []}
arxiv
Quantitative atom counting of Zn and O atomsby atomic resolution off-axis and in-line holography U Bhat International Centre for Materials Science, Chemistry and Physics of Materials Unit Jawaharlal Nehru Centre for Advanced Scientific Research 560064BangaloreIndia R Datta International Centre for Materials Science, Chemistry and Physics of Materials Unit Jawaharlal Nehru Centre for Advanced Scientific Research 560064BangaloreIndia Quantitative atom counting of Zn and O atomsby atomic resolution off-axis and in-line holography 1 Quantitative atom counting of Zn and O atoms in zinc oxide(ZnO)epitaxial thin film by three different routes; reconstruction of phase from side and central band of atomic resolution off-axis and in-line electron holography are presented. It is found that the reconstructed phase from both side and central band and corresponding atom number for both Zn (Z = 30) and O (Z = 8) atom columns are in close agreementalong with the systematic increase in thickness for thinner sample area.However, complete disagreement is observed for thethicker sample area. On the other hand,the reconstructed phase obtained via in-line holography showsno systematic change with thickness.Phase detection limits and atomic model used to count the atoms are discussed.Corresponding author e-mail address: 2 I. Introduction Phase is the fundamental information obtained by high resolution transmission electron interferometerexperiment [1].Phase shift encodes information on the potential of atomic ensembles and detailed knowledge of charge distribution which may be used to deducethe atomic arrangement and properties ofmaterials [2][3][4].Two established approaches,i.e., off-axis and in-line electron holography can be usedto retrieve phase information experimentally at atomic and sub-atomic length scale. In-line holography is popularly known as HRTEM (high resolution transmission electron microscopy). The first approach,which has the origin in Gabor's proposal of holographyand subsequent development of off-axis geometry by Leith and Upatnieks [5,6]. Off-axis geometry eliminates the twin image problem associated with the Gabor's original idea of in-line holography [6]. Gabor's proposalwas based on using a reference optical wavefront to interfere with the object wave,e.g., an electron micrographto overcome the resolution limit imposed by the geometrical aberrations of the electron lens. Such a hologram contains all the information about the object and the imaging system. Practical off-axis electron holography technique makes use of an electrostatic bi-prism for the electron interference developed by Möllensted and Düeker [7]. The two side bands (SBs) of the off-axis hologram containpure phase information. The central band (CB) is equivalent to inline holography with mixed amplitude and phase signals. Off-axis holography is a routine techniquefor medium resolution imaging of electric and magnetic fields [8][9][10]. Only recently, atomic resolution offaxis holography has been possible with the development of a special holography microscopeequipped withdouble bi-prism set up [11][12][13][14]. Double bi-prism set up eliminates Fresnel fringes and Vignetting effect essential for good quality atomic resolution off-axis 3 hologram which usually has a small field of view [15,16]. Atomic resolution off-axis electron holography is a recent development where sub-atomic electron interference fringes encode phase information at that length scale. On the other hand,reconstruction of phase from in-line holography requires series of imagesto be recorded at different focus values. Various reconstruction schemes for object exit wave(OEW) function have been developed from the experimental image series [1,[17][18][19][20]. Development of both the experimental approaches to obtain phase information dates back to the BRITE EURAM program [21]. Comparisons of phase information by two different approaches have been performed by few groups both at medium and atomic scale resolutions.However, quantitative phase information obtained so far through off-axis and inline holography do not correspond to each other for the same sample area and depends on frequency range considered for the analysis[14, [22][23][24]. Quantitative imaging is a recent area of active research inatomic resolution microscopy community [12,[25][26][27][28][29][30][31][32]and understanding the accuracy on the experimental phase determination and its correlation with the property of materials is crucial for its success and contribution to material and microscopyscience as a whole. Both aberration corrected HRTEM and atomic resolution off-axis holography provide a unique opportunity to record phase information at the atomic and sub-atomic length scale. In the present report, we compare the atomic scale phase information quantitatively by three different methods; off-axis electron holography using both SB and CB and in line holography. It is found that the peak phase values and corresponding atom numbers for both heavy Zn (Z =30) 4 and light O (Z = 8) atoms are in close agreement between the SB and CB of off-axiselectron holography for thinner specimenarea with a systematic change in sample thickness. However, for thicker sample the agreement no longer holds. On the other hand, the phase information obtained via HRTEM method show a much lesser number of atoms than expected and does not change systematically with sample thickness.Phase detection limit in both the methods and atomic model used to count the atoms is discussed. II. Experimental details and data analysis A. Crystal growth The ZnO epitaxial thin films were grown on 'c' plane ZnO substrate under two different oxygen partial pressure ( 2 ) conditions using pulsed laser deposition (PLD) technique as described earlier [33,34]. Electron carrier concentrations can be controlled between 10 19 to 10 16 cm -3 with 2 10 -5 and 10 -2 Torr, respectively. Though the original aim was to compare the difference in point defect distribution leading to change in carrier concentrations in these two samples, however, due to technical limitations at this point of time we restrict ourselves only to analyze atom counting by twodifferent phase contrast routes with sample thickness. B. Off-axis electron holography method, instrumentation and data analysis The principle behind HRTEM and holography image acquisitionfor phase retrieval is shown schematically in Fig. 1 Fig. 1 (c)] [35]. The present data were acquired using aberration-corrected FEI TITAN 80-300 Berlin holography special TEM operated at 300kV in adouble bi-prism setup.Through focalimage series was acquired at a focus range of -10 to +10 nm with ∆ 1 nm. Third order spherical aberration coefficient ( )was set close to zero. It was already mentioned before that the aberration correction improved the phase detection limit by a factor of 4, i.e., 2π/20 to 2π/80 [11]. Through focal holography method provides extraction of phase through CB using standard algorithm used for HRTEM, in the present case combination ofPAM (Paraboloid method) and MAL(Maximum-likelihood). MAL corrects exit wave function iteratively,based on a least square formalism. Series of images improves the signal to noise ratio significantly in the phase detection from the SB reconstruction using the Berlin code [14].Earlier comparison of phase values based onthe medium resolution reported poor signal to noise ratio for a single image SB reconstruction [24]. The details of the principle behind the method can be found inref. [14]and is shown schematically in Fig. 2 (a). Example FFT of the atomic resolution hologram from ZnO is shown in Fig. 2 III. Results and discussion A. Phase detection limit Resolution is the most important parameter in high resolution transmission electron microscopy. In the presence of aberration, the point resolution is defined by the first zero crossing of the phase contrast transfer function (PCTF) on the frequency axis under optimum C s and defocus∆ [ Fig. S2].The information limit of a microscope is the maximum information which can be transferred and isdefinedby the last point of the PCTF functionjust above the noise level and usually damped by various incoherent aberrations. The information encoded between the point resolution and information limit is not directly interpretable. For example, in an 7 aberration corrected microscope one can obtain resolution better than 0.8Å, which is sufficient to resolve any chemical bonds in the crystalline material along high symmetry orientation. This reveals the structure of the material in terms of periodic arrangement of atoms. Similar to resolution, minimum detectable amplitude and the phase signal of an electron waveafter interacting with the specimenpotential is equally important to evaluatethe smallest gradient of electric and magnetic fields, distinguishing atoms between the columns and counting atoms along the columns. Below is the brief discussion on phase detection limit in both off-axis and in-line holography in the context of present data. In electron holography, following the procedure described by Lichte [36], the phase detection limit in a medium resolution hologram is given by = 2 2 0 ( )(1) Where,e is the charge of the electron, Vis the fringe visibility, ( ) is the signal transfer efficiency of the CCD camera and 0 is the current density during the exposure time over the area 2 . Lichte has shown that the phase detection limit improves with increasing electron dose . . −ln ( ) 2 ∈ ( ) × 4 3 (2) Where,snr is the signal to noise ratio,q max is the resolution in reciprocal space, is the degree of spatial coherence, , , are the hologram contrast arising due to inelastic scattering, instabilities and Modulation Transfer Function (MTF) of the CCD respectively. ( ) is the Detection Quantum Efficiency of CCD camera, B ax is the brightness of the electron source, electron wave number k and e is the charge of the electron [11]. In the present experimental hologram with fringe spacing (s) of 0.0469 nm, the phase detection limit is 0.00023rad for an area 2 ~ 100 nm 2 (512×512 pixels), V = 15%, and electron dose 16×10 6 nm -2 , which is the area of reconstruction in the present case.At the limit of resolution where the lateral resolution of wave should be selected as 4 times psf, i.e., for p =0.32 nm, phase detection limit is 0.007365 rad.With this phase detection limit counting of incremental atoms both for O (0.109 rad) and Zn (0.284rad) atoms is possible. The dependence of theoretical phase detection limit on V, electron dose and lateral resolution are given in supplementary[ Fig. S3]. It can be seen that the phase detection limit changes within the same order of the magnitude with some variation in V, p and N thus should not affect the atom counting both in the case of Zn and O.In this context, Lehman et al [11]reported phase detection limit 2π/80 for an aberration corrected holography microscope.Cooper and Voelkl improved the phase detection limit to 0.001 and 2π/1000 (0.00628) by long exposure and multiplicity of holograms along with bi-prism and sample drift correction, respectively [37,38]. However, none of the latter two cases above used 9 double bi-prism set up which eliminates Fresnel fringe and improves the phase detection limit significantly. On the other hand, in the context of HRTEM, phase detection limit has not been discussed. Experimentally, distinguishing between B and N atoms has been reported with peak phase values as 0.022 and 0.026 rad, respectivelywith a difference of 0.004rad between the two atoms [39] [ Fig. S4]. It is the shape and contrasts both responsible for the detection of atoms. The peak phase value on the atom position depends on atomicscattering and structure factors, microscope transfer function, and resolution. This will be reflected in the recorded image intensity as well. The changes in peak values for both phase and intensity can be calculated theoretically [see section II.B.]. However, there is another factor, i.e., the standard deviation in the vacuum phase value from reconstruction methoddetermines the experimental phase detection limit. Experimentally, it is the standard deviation of intensity and reconstructed phase in the vacuum will limit the interpretable phase change, i.e., typically 0.023 rad from the present result.In case of in holography,the number is better, i.e., 0.00488 rad (see also section III). B. Atomic potential model It is necessary to compare the results with the theoretical reference values to quantify the atom numbers from the reconstructed phase shift. This method involves modeling the atomic potentialas imaging electron directly interacts with it giving rise to what is called object exit wave (OEW) function.Moreover, the lens phase contrast transfer function (PCTF) and aperturediameter (k in Å -1 ) modify the phase of the OEW further on the way to the recording device. The size of the nucleus (1.6 to 15 fm) is extremely tiny compared to the size of the atoms 10 consisting of nucleus and surrounding electron clouds (0.1 to 0.5 nm). For a stationary atom, the Coulomb potential is∝ 1 , and there is a singularity at the center of the atom. The imaging electrons mostly see the nuclear potential, and the surrounding electrons shield the effect [40]. Inelastic events are negligible compared to elastic events(imaging electrons) for thin sample.Various theoretical atomic potential models are available in the literature [40][41][42]. In the present investigation, Hartree-Fock atomic model projected along the z-direction is considered which is given by , = , , +∞ −∞ = 4 2 0 3 =1 0 2 + 2 2 0 3 =1 exp (− 2 2 / )(3)with 2 = 2 + 2 Where, 0 is the Bohr radius, , , , are the parameterized coefficients. 0 ( )is the modified Bessel function of order zero [19]. The projected atomic potential of Zn and O atoms calculated by the above equation is given in supplementary [Fig. S5]. The potential function is asymptotic due to 1 dependence. Therefore,it is necessary to consider inner and outer bound of the potential while calculating the phase shift and image of the atom.The atomic scattering factor ( )(according to Moliere) and image of the atoms depends on the inner and outer cut off potential [ Fig. S6]. However, it is observed that there is a limit in both inner and outer cut off, beyond which the change in ( Fig.3.The two curves corresponding to peak phase values match well for fewer atoms in a column but deviates from linearity due to dynamical scattering for a higher number of atoms. The mean phase is found to be smaller (~ factor of 0.5) compared to the peak phase value, and this has implications on the atom number assignment by two different reference parameters and is discussed next. 13 C. Atomic resolution off-axis electron holography In this section the experimental phase information retrieved from both SB and CB off-axis hologram ofZnO film with varyingthickness is analyzed. ZnO films with two different thickness along <11-20> and <01-10> orientations are considered.The extinction lengths ( ) are108.6 and 142.4 nm for <11-20> and <01-10> Z.A., respectively. Fig.4&6are the amplitude and phase images corresponding to CB and SB obtained for the two different areas marked as P and Q. The IV. Conclusions In conclusion, atomic resolution reconstructed phase of Zn and O atoms in ZnO epitaxial thin film is compared between off-axis and in-line holography techniques. While holography method has an excellent match in atom numbers for both Zn and O atoms extracted from SB and CB for thin sample area,however, for thicker sample the atom numbers do not match. In case of in-line holographic reconstruction of HRTEM data, the atom number do not change systematically with increasing sample thickness, and a constant atom number of one is obtained throughout the reconstructed area. Acknowledgement R. Datta sincerely thanks Prof. Micheal Lehmann and Dr. Tore Niermann for hosting and assisting in the acquisition of atomic resolution off-axis holograms and data analysis. R. Datta also thanks DFG and ICMS for the funding.U. Bhat sincerely acknowledges JNCASR and ICMS for the financial support. Supplementary material Supplementary material contains information on example HRTEM images with and without digital aberration-correction, PCTF, phase detection limit in off-axis holography, atomic potential model and corresponding images and exit wave function. S 3. On the phase detection limit In off-axis holography, the phase detection limit has been discussed by Lichte,[Ref. 36] and is given by = 2 2 0 ( ) (S1) The above equation can be written as = 2 / 2 ( ) (S2) 1 0 2 N is the number of electrons/nm 2 , j 0 is the mean current density in the detector plane, 2 is the area of reconstruction, and is the exposure time. ( ) is the signal transfer function of the CCD camera. The three important parameters in the above equations are, N, p and V. The fundamental limit in phase detection is governed by the shot noise or stochastic impacts of single electrons, due to probabilistic nature of the electron wave. This is given by = 2 2 (S3) The fundamental phase detection limit improves with the increasing electron dose N. Lichte shown that for V=0.4, STE=0.8, and N=9000/nm 2 is 0.0314rad for 2 =1nm 2 . However, there is almost no changes in the phase detection limit by improving contrast up to 0.8. rather decreases with further increase in contrast. In our experimental holograms, acquired in Berlin, the average electron dose is 16*10 6 /nm 2 ,V=15%. Therefore, for a reconstructed area 2 =100nm 2 (512 X 512), = 0.00023 rad. Lehmann has modified the equation S3 to incorporate the effect of Cs and smallest area of reconstruction (w hol ≥ 4psf). Lehmann has shown improvement in the phase detection limit by factor of 4 using Cs corrected microscope. The minimum area of reconstruction for the present data is approximately (4 × 0.8) 2 =(0.32nm) 2 , where 0.8 Å is the point resolution of the microscope.Thus, the phase detection limit for the smallest area of reconstruction, correspondind to the present data is 0.007365rad. Theoretical model suggest, the change in peak (mean) phase 34 due to incremental change in Zn and O atoms in the atomic column are 0.284 (0.122)and 0.1098 (0.052) rad respectively. The mean phase has been calculated for an inner and outer cut off potential 10-50pm for Zn atom. Therefore, it is possible to count the incremental Zn and O atom in the atomic columns of the ZnO from the present atomic resolution holography data irrespective of area of reconstruction. In this context, Cooper and Voelkl improved the phase detection limit to 0.001 and 2π/1000 (0.00628) by long exposure and multiplicity of holograms along with bi-prism and sample drift correction, respectively [37,38]. However, none of the latter two cases above used double bi-prism set ups which eliminates Fresnel fringe and improves the Where, 0 is the Bohr radius, , , , are the parameterized coefficients. Then the mean phase shift in the absence of dynamical scattering is calculated by the equation, Φ , = , ,(S5) Where = is the interaction parameter with wavelength λ and accelerating voltage E [44]. The projected atomic potential integrated along z-direction can be calculated from the equation above and is given in the main text equation (3). The atomic potential is asymptotic and has a singularity at the center of the atom. Various resolution limiting factors such as diffraction limit, thermal vibration, aberration of the microscope result in measurable peak phase value in the phase image of the atom. The phase image of the atom can be approximated to a Gaussian function. Gaussian function is parameterized by the peak height and the full width half maxima (FWHM). Therefore, the reference phase shift can be considered either based on the peak value of the phase or the mean value of the phase. The mean value will depend on both the peak value and FWHM of the phase distribution function. The mean value of the phase can be calculated by integrating threedimensional atomic potential between the two limits and dividing with the volume. Figure S5. Projected potential of Zn and O atoms calculated using equation (3) Table S1 . Table of (b). The cut off frequency for CB and SB are 14 and 12 nm -1 , respectively. The cut off frequencies are chosen in such a way that it does not overlap with the neighboring band. C. In-line holography method and data analysis HRTEM data was acquired in a double aberration-corrected FEI TITAN 80-300 kV transmission electron microscope available at ICMS, JNCASR, Bangalore. An optimized phase contrast 6 transfer function (PCTF) with C s = -35 µm,f = 8 nm and a point to point resolution better than 0.8 Å at 300 kV was set for the experimentation. Image series were recorded under these conditions; = −35 and focus range -10 to 10 nm with ∆ 1 nm with the exposure time 1s. However, only 10 numbers of images are contemplated for the reconstruction. We did not observe any difference in the reconstructed phase image between 10 & 20 number of imagesconsidered for the reconstruction.The image series was reconstructed using the Gerchberg-Saxton scheme as implemented in MacTempas. Following are the parameters employed for the reconstruction; = −35 , acceleration voltage 300kV, area of reconstruction 1024×1024 (pixel), and objective aperture size g max =2 Å -1 ,neither any phase plate nor any filter is used. Strong central beam condition is considered.The phase image obtained was further corrected for the residual aberration using the digital aberration correction scheme available within the package. Example images before and after aberration correction are shown in Fig. S1 and the schematic of reconstruction steps are shown in Fig. 2 (e). Nnm - 2 2andincreasing lateral resolution of reconstructed wave.For atomic resolution holography as the width of hologram ( ) is related to the resolution ( ≥ 4 ), where psf is the point spread function of the electron microscope.The above equation can be modified for a ) or peak values of image of the atom do not change significantly. For the present report, the limits 11 selected are 0.01 and 1 Å, for calculating images of isolated Zn and O atoms using standard formula [See supplementary for more information].The image of single isolated atom based on the model potential above, equation(3)can be calculated directly using electron scattering amplitude as given by the following equation; the electron scattering factor in the Moliere approximation using the projected atomic potential.is the aberration function, = α max /λis the maximum spatial frequency in the objective aperture and 0 ( ) is the Bessel function of order zero.The effect of inner and outer bound of potential and the resolution of the microscope as set by the objective aperture diameter on the shape and peak values of single atomphase shift and image intensity are given in supplementary[Fig. S7and S8]. The real part of the wave transfer function,cos is neglected (i.e. set to zero) and imaginary part , which is the pure phase partis set to 1 tomimic Scherzer like transfer function for the atoms in a periodic lattice within weak phase object approximation. Similar to the scattering factor, the peak value of phase and intensity do not change significantly below an inner cut off of 0.01Å, and no significant change is observed with the outer bound. This is true in case of both Zn and O atoms. The peak phaseand intensity values changes with the size of the aperture[Fig. S9]. In the present case an aperture size of 2Å -1 is used.12 Two different theoretical phase values;peak and mean for a given atom which can be considered to interpret the reconstructed phase for counting the number of atoms. However, atoms are never stationary in the crystal and due to finite temperature atoms oscillate (0.0073and 0.0072 Å, for Zn and O atoms in ZnO at 293K[43]) about the mean position. Therefore,an incoming probe electron sees a blurred atom position. Aberration of the microscope will cause further blurring. However, the amplitude of thermal vibrationat room temperature and resulting blurring is smaller compared to the blurring due to aberration and is not considered in the present investigation. By numerical evaluation, one can find that the peak phase shift values have a coarse dependence of Z 0.6-0.7 and deviation can be observed due to valence electron filling with the atomic number[44]. On the other hand, mean phase shift value is sensitive to the inner and outer bound of potential [Table. S1]. Mean phase shift value does not change strongly with the inner cut off for less than 0.01 Å but changes significantly with outer cut off for less than 1 Å of the potential. But beyond 1 Å, it does not change significantly. An inner cut of 1pm and outer cut off of 50/25 pmcorresponding to experimental size of the Zn and O atoms, respectively considered for extractingmean phase shift value.The theoretical mean phase shift value is calculated bythree-dimensional integration of the potential and dividing with the volume bounded by the limits. The peak phase shift values for a microscope resolution of 0.8 Å obtained from literature and multislice calculation as implemented in MacTempas for the atoms in a crystal along with mean phase shift values are given in peak phase values on top of Zn and O columns have been evaluated, and selected columns at three different distances corresponding to different thickness levels from the edge of the specimen for area P are indicatedin the figures. Same columns are considered for the comparisons of two different OEW reconstructed using theCB and SB. The atom numbers corresponding to Zn atom evaluated from the peak and mean phase values are plotted inFig. 5for area P.Difference in the number of Zn atoms between CB and SB is within ±1 and ±3 corresponding to reference peak and mean phase respectively.Another noticeable point is thatsimilar amount of Zn and O atoms are obtained for different areas for area P, suggesting that peak phase values as used from the theoretical model fits well in this case for both light and heavy atoms adjacent to each other. The reconstructed phase values and corresponding atom numbers for both Zn and O atoms in the neighboring columns for area P are given in Fig. 6. It can be seen that the atom numbersare in close agreement with difference ±1 atoms for Zn and O atoms.A Similar comparison is given forarea Q in Fig 7 shows the reconstructed phase and amplitude images. Fig 8 shows a comparison of phase shift as well as the atom numbers for Zn between the CB and SB. It can be seen that there is a gradual increase in the atom number with the thickness for the SB but almost constant (but different than the SB) atom number is obtained 14 from the CB. Thus, for area Q the match is poor between CB and SBbecause of relatively thicker sample area and stronger dynamical effect. D. Inline holography/HRTEM Fig.9 shows the reconstructed phase image of ZnO film along <11-20> orientation from different thickness regions of the sample. The peak phase values from Zn columns are given in the line scan for the columns indicated in the image. The peak phase values remain almost same between thinner and thicker regions of the sample at around 0.18rad which corresponds to~1Zn atom. In case of O peak phase value is around 0.09 rad and corresponds to ~1 atom.Lehmann et. al.[14] first described the difference between in-line and off-axis electron holography at atomic resolution in GaAs crystal along <1-10> Z.A. The phase and amplitude reconstructed from the SB and CB agrees well up to a thickness of 3/2 times the extinction length, but significant deviations observed at lower frequency and thicker specimen area. The agreement between the two methods for the thinner area, is due to the similar wave function reconstructed in the limit of linear imaging with negligible inelastic scattering. However, for thicker area, due to significant inelastic scattering, reconstruction methods corresponding to CB and SB yield two different wave functions. This is because the mathematical formulation of SB contains average OEW function, while CB contains sum of squared OEW function. It is mentioned that the deviation observed in the thicker area between CB and SB reconstructed wave function maybe either due to fundamental quantum mechanical differences or numerically difficult inversion of the imaging process. In the present investigation, our results agree with observation made in HRTEM experiment is not comparable with the CB reconstruction. This could be because of the reconstruction scheme employed in the MacTempas package.Counting of atoms depends on the theoretical reference phase values, i.e., mean or peak values of phase. We obtain ~ 3 times higher atom numbersfor Zn and O using reference mean phase value compared to peak phase value. The experimentally observed higher mean phase values compared to theory could be because of incoherent aberrations or vibrations present in the recorded image. Figure 1 . 1The schematics showing principle of image formation for (a) HRTEM, (b) off-axis electron holography, and (c) off-axis holography with double bi-prism set up. Figure 2 . 24 Figure 3 . 25 Figure 4 . 26 Figure 5 . 27 Figure 6 . 28 Figure 7 . 29 Figure 8 . 30 Figure 9 . 2243254265276287298309(a) & (e) Steps involving in reconstruction methods to extract phase and amplitude from the hologram and HRTEM image series, respectively. (b) Fourier transform of the hologram showing one CB and two SBs. (c)& (d) are the example atomic resolution hologram and HRTEM image of ZnO epitaxial thin film along <11-20> Z.A.. Peak phase shift of (a)Zn and (b) O atoms with increase in the number of atoms in the column calculated using isolated atom model and multislice method considering dynamical scattering. The resolution was set to 0.5 Å.Also plotted mean phase shift with the inner and outer bound ofpotential 10 to 50 pm, respectively. (a) & (c) Amplitude and (b) &(d) phase image of ZnO along <11-20> Z. A for area P obtained through reconstruction CB and SB of off-axis electron hologram, respectively. The corresponding complex wave function can be found in the supplementary Fig.S10[(a) and (b)].Three different arrows are indicated in the phase image along which the peak and mean phase values are extracted. Larger dots and smaller dots are corresponding to Zn and O atoms, respectively. (a) Peak (Φ p ) and mean (Φ m ) phase shift and (b) corresponding atom numbers of Zn atomalong three different arrowsfrom area P reconstructed fromSB and CB.Zn atom number matches well between SB and CB with ±1 atom. However, the number of atoms derived from the mean phase value is three times higher than peak phase value. Comparison of (a) peak phase shift and (b) corresponding atom number with variation of thickness in Zn and O columns for area P. Almost similar number of atoms are obtained for Zn and O atoms at the neighboring sites. (a) & (c) Amplitude and (b) & (d) phase image of ZnO along <01-10> Z. A. for area Q obtained through reconstruction CB and SB of off-axis electron hologram, respectively. The corresponding complex wave function can be found in the supplementary Fig.S10[(c)and (d)].Three different arrows are indicated in the phase image along which the peak and mean phase values are extracted. (a) Peak (Φ p ) phase shift and (b) corresponding atom numbers of Zn atomalong three different arrowsfrom area Q reconstructed fromSB and CB.No agreement is found in the number of Zn atoms between SB and CB. Almost constant number of Zn atoms are obtained from CB reconstruction, however, for SB reconstruction for the first arrow near the edge shows systematic increase in atoms number. Reconstructed phase image of HRTEM image series for different thickness regions. Throughout the sample area almost constant phase and atom number 1 are obtained. Supplementary Document Quantitative atom counting of Zn and O atoms by atomic resolution off-axis and in-line holography U. Bhat 1 , and R. Datta 1 1 International Centre for Materials Science, Chemistry and Physics of Materials Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore 560064, India. S1. HRTEM image before and after digital aberration correction 32 Fig S1. Example of HRTEM reconstructed amplitude, phase and image (a) -(c) before and (d)-(f) after digital aberration correction. Cs=-35µm, Δf=2nm are used for the aberration correction. S2. Phase contrast transfer function (PCTF): Figure S2.(a) PCTF function at 300 kV under optimum lens parameters, Cs=-35µm Δf=8nm with the envelope function (blue dotted line) corresponding to spread in defocus 1nm. The point resolution and information limits are marked as g s and g i respectively. The positive PCTF gives negative phase contrast or white atom contrast. (b) PCTF function corresponding to positive phase contrast or dark atom contrastwith Cs=35um, Δf=-8nm. 35 S 4 . 354phase detection limit significantly. However, there is another limit posed by reconstruction methods where standard deviation in vacuum phase value poses experimental phase detection limit [see section III. of the manuscript]. The calculated phase detection limit as a function of electron dose N, contrast V and p are shown in figure S3. The dashed vertical lines are marked corresponding to the current experimental parameters.Figure S3. dependence of theoretical phase detection limit on area of reconstruction (p 2 ), visibility and electron dose are given. Experimental and theoretical phase shift value of B and N by in-line holography Fig S4. For a BN monolayer, the peak phase shift value of B and N are 0.022/0.09 and 0.026/0.13 with the difference 0.004/0.04 rad by experiment and simulation. Difference between the simulation and experimental values are because of Stobb's factor [39]. S 5. Atomic potential modelThe charge distribution and corresponding atomic potential is calculated by Hartree-Fock procedure[Ref. 19]. The atomic potential in 3D is given by 38 Figure S6 . 39 Figure S7 . 40 Figure S8 . 41 Figure S9 . 42 Figure S10 . 38S639S740S841S942S10mean phase shift with same outer cut off & varying inner cut off, same inner cut off with varying outer cut off for Zn and O atoms S 6. Scattering factor The variation of feq (according to Moliere) with inner and outer bound of the potential. There is no change in feq by changing inner cut off from 0.001 to 0.01 Å(for the same outer cut of 1 Å) but changes to inner cut off of 0.1ang for both Zn and O. On the other hand, keeping the inner cut off fixed(0.01 Å), there is only change in amplitude at small scattering angle (<0.25 Å -1 )by changing the outer cut-off. S 7. Peak phase as a function of cut-off (a) & (c) The peak phase of Zn and O atom as a function of inner cut-off (fixed outer cut-off 1 Å) and (b) & (d) as a function of outer cut-off ( fixed inner cut-off, 0.01Å). The peak phase value does not change with the outer cut off potential from 0.9 to 1.4 Å. Peak phase values also do not change for inner cut-off of 0.001 and 0.01 Å but changes significantly for 0.1 Å. S 8. Peak intensity as a function of cut-off Corresponding intensity plots of Zn and O atoms, similar trend is observed in the intensity plot also with variation of inner and outer cut off potential. S 9. Phase shift and Intensity for different k Phase shift and intensity plot of Zn and O atoms for k=1.25 and 2 Å -1 . The inner and outer cut off are 0.01 and 1 Å respectively. S 10. Reconstructed wave function Reconstructed wave function of area P (a) and (b) and area Q (c) and (d) both from central and side band. & 2.Atomic resolution off-axis electron holography is a recent development where electron interference fringesencode phase information at the sub-Å length scale where object wave is an atomic resolution electron micrograph. Double bi-prism set up atthe special location in the microscope column is important to avoid Fresnel artifact and Vignetting effect particularly at the atomic resolution where the field of view is severely restricted [5 D B B C Williams, Carter, 978-3-319-26651-0Transmission Electron Microscopy Diffraction, Imaging, and Spectrometry. D.B.Williams. B. C. Carter, Transmission Electron Microscopy Diffraction, Imaging, and Spectrometry, 2017. ISBN 978-3-319-26651-0. Electron holography -Basics and applications. H Lichte, M Lehmann, Reports Prog. Phys. 71H. Lichte, M. Lehmann, Electron holography -Basics and applications, Reports Prog. Phys. 71 (2008). Phase retrieval through force variation for ultra-resolution in field-emmission transmission electron microscopy. W Coene, G Jannsen, M O Beeck, D Dyck, Phys. Rev. Lett. 693743W. Coene, G. Jannsen, M.O. d. Beeck, D. V Dyck, Phase retrieval through force variation for ultra-resolution in field-emmission transmission electron microscopy., Phys. Rev. Lett. 69 (1992) 3743. D Van Dyck, M Op De Beeck, New direct methods for phase and structure retrieval in HREM. Proceedings of the XIIth International Congress for Electron Microscopy. 17D. Van Dyck, M. Op de Beeck, New direct methods for phase and structure retrieval in HREM. Proceedings of the XIIth International Congress for Electron Microscopy (1990) 17 A new microscopic principle. D Gabor, Nature. 161777D. Gabor, A new microscopic principle, Nature. 161 (1948) 777. Reconstructed Wavefronts and Communication Theory. E N Leith, J Upatnieks, J. Opt. Soc. Am. 521123E.N. Leith, J. Upatnieks, Reconstructed Wavefronts and Communication Theory, J. Opt. Soc. Am. 52 (1962) 1123. Beobachtungen und Messungen an Biprisma-Interferenzen mit Elektronenwellen. G Möllenstedt, H Düker, Zeitschrift Für Phys. 145377G. Möllenstedt, H. Düker, Beobachtungen und Messungen an Biprisma-Interferenzen mit Elektronenwellen, Zeitschrift Für Phys. 145 (1956) 377. Electron holographic observations of the electrostatic field associated with thin reverse-biased p-n junctions. S Frabboni, G Matteucci, G Pozzi, M Vanzi, Phys. Rev. Lett. 552196S. Frabboni, G. Matteucci, G. Pozzi, M. Vanzi, Electron holographic observations of the electrostatic field associated with thin reverse-biased p-n junctions, Phys. Rev. Lett. 55 (1985) 2196. Direct observation of potential distribution across Si/Si p-n junctions using off-axis electron holography. M R Mccartney, D J Smith, R Hull, J C Bean, E Voelkl, B Frost, Appl. Phys. Lett. 652603M.R. McCartney, D.J. Smith, R. Hull, J.C. Bean, E. Voelkl, B. Frost, Direct observation of potential distribution across Si/Si p-n junctions using off-axis electron holography, Appl. Phys. Lett. 65 (1994) 2603. Quantitative off-axis electron holography of GaAs p-n junctions prepared by focused ion beam milling. D Cooper, R Truche, . C Twitchett-Harrison, R E Dunin-Borkowski, P A Midgley, J. Microsc. 233102D. Cooper, R. Truche, a. C. Twitchett-Harrison, R.E. Dunin-Borkowski, P. a. Midgley, Quantitative off-axis electron holography of GaAs p-n junctions prepared by focused ion beam milling, J. Microsc. 233 (2009) 102. Electron holography with a Cs-corrected transmission electron microscope. D Geiger, H Lichte, M Linck, M Lehmann, Microsc. Microanal. 1468D. Geiger, H. Lichte, M. Linck, M. Lehmann, Electron holography with a Cs-corrected transmission electron microscope, Microsc. Microanal. 14 (2008) 68. Direct structure inversion from exit waves. Part II: A practical example. A Wang, F R Chen, S Van Aert, D Van Dyck, Ultramicroscopy. 11677A. Wang, F.R. Chen, S. Van Aert, D. Van Dyck, Direct structure inversion from exit waves. Part II: A practical example, Ultramicroscopy. 116 (2012) 77. Electron holography approaching atomic resolution. H Lichte, Ultramicroscopy. 20293H. Lichte, Electron holography approaching atomic resolution, Ultramicroscopy. 20 (1986) 293. Holographic focal series: Differences between inline and offaxis electron holography at atomic resolution. T Niermann, M Lehmann, J. Phys. D. Appl. Phys. 49194002T. Niermann, M. Lehmann, Holographic focal series: Differences between inline and off- axis electron holography at atomic resolution, J. Phys. D. Appl. Phys. 49 (2016) 194002. . G C Schaal, Jönsson, Optik (Stuttgart). 24529Optik (Stuttgart)Schaal, G. C. Jönsson, and EF Krimmel, 1966/1967, Optik (Stuttgart) 24, 529. Optik (Stuttgart) 24 (1966) 529. Triple-biprism electron interferometry. K Harada, T Matsuda, A Tonomura, T Akashi, Y Togawa, J. Appl. Phys. 991K. Harada, T. Matsuda, A. Tonomura, T. Akashi, Y. Togawa, Triple-biprism electron interferometry, J. Appl. Phys. 99 (2006) 1. Maximum-likelihood method for focus-variation image reconstruction in high resolution transmission electron microscopy. W M J Coene, A Thust, M Op De, D Beeck, Van Dyck, Ultramicroscopy. 64109W.M.J. Coene, A. Thust, M. Op De Beeck, D. Van Dyck, Maximum-likelihood method for focus-variation image reconstruction in high resolution transmission electron microscopy, Ultramicroscopy. 64 (1996) 109. What is the focus variation method? Is it new? Is it direct?. W O Saxton, Ultramicroscopy. 55171W.O. Saxton, What is the focus variation method? Is it new? Is it direct?, Ultramicroscopy. 55 (1994) 171. Advanced computing in TEM. E J Kirkland, E.J. Kirkland,Advanced computing in TEM, S. Edition(2010). Real space image simulation in high resolution electron microscopy*. R Kilaas, R Gronsky, Ultramicroscopy. 11289R.Kilaas, R. Gronsky, Real space image simulation in high resolution electron microscopy*. Ultramicroscopy. 11 (1983) 289. Sub-angstrom structure characterisation: The Brite-Euram route towards one angstrom. D Van Dyck, H Lichte, K D Van Der, Mast, Ultramicroscopy. 641D. Van Dyck, H. Lichte, K.D. Van Der Mast, Sub-angstrom structure characterisation: The Brite-Euram route towards one angstrom, Ultramicroscopy. 64 (1996) 1. Off-axis and inline electron holography: A quantitative comparison. C T Koch, A Lubk, Ultramicroscopy. 110460C.T. Koch, A. Lubk, Off-axis and inline electron holography: A quantitative comparison, Ultramicroscopy. 110 (2010) 460. Off-axis and inline electron holography: Experimental comparison. T Latychevskaia, P Formanek, C T Koch, A Lubk, Ultramicroscopy. 110472T. Latychevskaia, P. Formanek, C.T. Koch, A. Lubk, Off-axis and inline electron holography: Experimental comparison, Ultramicroscopy. 110 (2010) 472. . C Ozsoy-Keskinbora, C B Boothroyd, R E Dunin-Borkowski, P A Van Aken, C T , C. Ozsoy-Keskinbora, C.B. Boothroyd, R.E. Dunin-Borkowski, P.A. Van Aken, C.T. 19 Hybridization approach to in-line and off-axis (electron) holography for superior resolution and phase sensitivity. Koch, Sci. Rep. 41Koch, Hybridization approach to in-line and off-axis (electron) holography for superior resolution and phase sensitivity, Sci. Rep. 4 (2014) 1. Big Bang' tomography as a new route to atomic-resolution electron tomography. D Van Dyck, F.-R Chen, Nature. 486243D. Van Dyck, F.-R. Chen, 'Big Bang' tomography as a new route to atomic-resolution electron tomography, Nature. 486 (2012) 243. Phase measurement of atomic resolution image using transport of intensity equation. K Ishizuka, B Allman, J. Electron Microsc. (Tokyo). 54191K. Ishizuka, B. Allman, Phase measurement of atomic resolution image using transport of intensity equation, J. Electron Microsc. (Tokyo). 54 (2005) 191. Determination of the 3D shape of a nanoscale crystal with atomic resolution from a single image. C L Jia, S B Mi, J Barthel, D W Wang, R E Dunin-Borkowski, K W Urban, A Thust, Nat. Mater. 131044C.L. Jia, S.B. Mi, J. Barthel, D.W. Wang, R.E. Dunin-Borkowski, K.W. Urban, A. Thust, Determination of the 3D shape of a nanoscale crystal with atomic resolution from a single image, Nat. Mater. 13 (2014) 1044. Quantitative Agreement between Electron-Optical Phase Images of WSe2 and Simulations Based on Electrostatic Potentials that Include Bonding Effects. S Borghardt, F Winkler, Z Zanolli, M J Verstraete, J Barthel, A H Tavabi, R E Dunin-Borkowski, B E , Phys. Rev. Lett. 1181S. Borghardt, F. Winkler, Z. Zanolli, M.J. Verstraete, J. Barthel, A.H. Tavabi, R.E. Dunin- Borkowski, B.E. Kardynal, Quantitative Agreement between Electron-Optical Phase Images of WSe2 and Simulations Based on Electrostatic Potentials that Include Bonding Effects, Phys. Rev. Lett. 118 (2017) 1. Direct structure inversion from exit waves. Part II: A practical example. A Wang, F R Chen, S Van Aert, D Van Dyck, Ultramicroscopy. 11677A. Wang, F.R. Chen, S. Van Aert, D. Van Dyck, Direct structure inversion from exit waves. Part II: A practical example, Ultramicroscopy. 116 (2012) 77. Electron Microscopy of Probability Currents at Atomic Resolution. A Lubk, A Béché, J Verbeeck, Phys. Rev. Lett. 1151A. Lubk, A. Béché, J. Verbeeck, Electron Microscopy of Probability Currents at Atomic Resolution, Phys. Rev. Lett. 115 (2015) 1. Experimental analysis of charge redistribution due to chemical bonding by high-resolution transmission electron microscopy. J C Meyer, S Kurasch, H J Park, V Skakalova, D Künzel, A Grobß, A Chuvilin, G Algara-Siller, S Roth, T Iwasaki, U Starke, J H Smet, U Kaiser, Nat. Mater. 10209J.C. Meyer, S. Kurasch, H.J. Park, V. Skakalova, D. Künzel, A. Grobß, A. Chuvilin, G. Algara-Siller, S. Roth, T. Iwasaki, U. Starke, J.H. Smet, U. Kaiser, Experimental analysis of charge redistribution due to chemical bonding by high-resolution transmission electron microscopy, Nat. Mater. 10 (2011) 209. Absolute Scale Quantitative Off-Axis Electron Holography at Atomic Resolution. F Winkler, J Barthel, A H Tavabi, S Borghardt, B E Kardynal, R E Dunin-Borkowski, Phys. Rev. Lett. 120156101F. Winkler, J. Barthel, A.H. Tavabi, S. Borghardt, B.E. Kardynal, R.E. Dunin-Borkowski, Absolute Scale Quantitative Off-Axis Electron Holography at Atomic Resolution, Phys. Rev. Lett. 120 (2018) 156101. Controlling structural quality of ZnO thin film on c-plane sapphire during pulsed laser deposition. B Loukya, P Sowjanya, K Dileep, R Shipra, S Kanuri, L S Panchakarla, R Datta, J. Cryst. Growth. 32920B. Loukya, P. Sowjanya, K. Dileep, R. Shipra, S. Kanuri, L.S. Panchakarla, R. Datta, Controlling structural quality of ZnO thin film on c-plane sapphire during pulsed laser deposition, J. Cryst. Growth. 329 (2011) 20 Native defects affecting the Li atom distribution tune the optical emission of ZnO:Li epitaxial thin film. R Sahu, K Dileep, B Loukya, R Datta, Appl. Phys. Lett. 104R. Sahu, K. Dileep, B. Loukya, R. Datta, Native defects affecting the Li atom distribution tune the optical emission of ZnO:Li epitaxial thin film, Appl. Phys. Lett. 104 (2014). Electron holography: Optimum position of the biprism in the electron microscope. H Lichte, Ultramicroscopy. 6479H. Lichte, Electron holography: Optimum position of the biprism in the electron microscope, Ultramicroscopy. 64 (1996) 79. Experimental study of amplitude and phase detection limits in electron holography. A Harscher, H Lichte, Ultramicroscopy. 6457A. Harscher, H. Lichte, Experimental study of amplitude and phase detection limits in electron holography, Ultramicroscopy. 64 (1996) 57. . D Cooper, R Truche, P Rivallin, J M Hartmann, F Laugier, F Bertin, A Chabli, J L , D. Cooper, R. Truche, P. Rivallin, J.M. Hartmann, F. Laugier, F. Bertin, A. Chabli, J.L. Medium resolution off-axis electron holography with millivolt sensitivity. Rouviere, Appl. Phys. Lett. 911Rouviere, Medium resolution off-axis electron holography with millivolt sensitivity, Appl. Phys. Lett. 91 (2007) 1. Approaching routine 2π/1000 phase resolution for off-axis type holography. E Voelkl, D Tang, Ultramicroscopy. 110447E. Voelkl, D. Tang, Approaching routine 2π/1000 phase resolution for off-axis type holography, Ultramicroscopy. 110 (2010) 447. Fabrication of a freestanding boron nitride single layer and Its defect assignments. C Jin, F Lin, K Suenaga, S Iijima, Phys. Rev. Lett. 1023C. Jin, F. Lin, K. Suenaga, S. Iijima, Fabrication of a freestanding boron nitride single layer and Its defect assignments, Phys. Rev. Lett. 102 (2009) 3. Accurate Analytical Hartree-Fock Potentials for Neutral Atoms as a Sum of Yukawa Terms (*). T G Strand, T Tietz, Il Nuovo Cimento. 89121T.G. Strand, T.Tietz. Accurate Analytical Hartree-Fock Potentials for Neutral Atoms as a Sum of Yukawa Terms (*). Il Nuovo Cimento, Vol. XLI B,N.1 (1966) 89. 21 On the calculation of electron scattering factors from the free atom formulation. P A Childs, D L Misell, J. Phys. C Solid State Phys. 4158P.A. Childs, D.L. Misell, On the calculation of electron scattering factors from the free atom formulation, J. Phys. C Solid State Phys. 4 (1971) 158. BYATT Analytical Representation of Hartree Potentials and Electron Scattering. W J , Phys. Rev. 5258W. J. BYATT Analytical Representation of Hartree Potentials and Electron Scattering, Phys. Rev. 52 (2009) 58. Anharmonic thermal vibrations in ZnO*. K Kihara, G Donnay, Canadian Mineralogist. 23647K. Kihara and G. Donnay, Anharmonic thermal vibrations in ZnO*, Canadian Mineralogist 23 (1985) 647. Electron Holographic Materials Analysis at Atomic Resolution. M Linck, H Lichte, M Lehmann, Int. J. Mat. Res.(formerly Z. Metallkd.). 890M. Linck, H. Lichte, M. Lehmann, Electron Holographic Materials Analysis at Atomic Resolution, Int. J. Mat. Res.(formerly Z. Metallkd.), 97 (2006) 890. High-Resolution TEM. D B Williams, C B Carter, 978-0-387-76501-3Transmission Electron Microscopy. Boston, MASpringer486Williams D.B., Carter C.B. High-Resolution TEM. In: Transmission Electron Microscopy. Springer, Boston, MA. (2009) 486.ISBN 978-0-387-76501-3.
{'fraction_non_alphanumeric': 0.047344785226654855, 'fraction_numerical': 0.03004611635821737, 'mean_word_length': 4.496181516325401, 'pattern_counts': {'":': 0, '<': 10, '<?xml version=': 0, '>': 9, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'Quantitative atom counting of Zn and O atoms in zinc oxide(ZnO)epitaxial thin film by three different routes; reconstruction of phase from side and central band of atomic resolution off-axis and in-line electron holography are presented. It is found that the reconstructed phase from both side and central band and corresponding atom number for both Zn (Z = 30) and O (Z = 8) atom columns are in close agreementalong with the systematic increase in thickness for thinner sample area.However, complete disagreement is observed for thethicker sample area. On the other hand,the reconstructed phase obtained via in-line holography showsno systematic change with thickness.Phase detection limits and atomic model used to count the atoms are discussed.Corresponding author e-mail address: 2 I.', 'arxivid': '1808.07001', 'author': ['U Bhat \nInternational Centre for Materials Science, Chemistry and Physics of Materials Unit\nJawaharlal Nehru Centre for Advanced Scientific Research\n560064BangaloreIndia\n', 'R Datta \nInternational Centre for Materials Science, Chemistry and Physics of Materials Unit\nJawaharlal Nehru Centre for Advanced Scientific Research\n560064BangaloreIndia\n'], 'authoraffiliation': ['International Centre for Materials Science, Chemistry and Physics of Materials Unit\nJawaharlal Nehru Centre for Advanced Scientific Research\n560064BangaloreIndia', 'International Centre for Materials Science, Chemistry and Physics of Materials Unit\nJawaharlal Nehru Centre for Advanced Scientific Research\n560064BangaloreIndia'], 'corpusid': 96425883, 'doi': '10.1063/1.5075532', 'github_urls': [], 'n_tokens_mistral': 14230, 'n_tokens_neox': 12152, 'n_words': 7646, 'pdfsha': '850e4247d1a33c3fdc3389e1c908514afd92258c', 'pdfurls': ['https://arxiv.org/pdf/1808.07001v1.pdf'], 'title': ['Quantitative atom counting of Zn and O atomsby atomic resolution off-axis and in-line holography', 'Quantitative atom counting of Zn and O atomsby atomic resolution off-axis and in-line holography'], 'venue': []}
arxiv
Raising and lowering operators for angular momentum quantum numbers l in spherical harmonics Q H Liu School for Theoretical Physics, and Department of Applied Physics Hunan University 410082ChangshaChina D M Xun L Shan School for Theoretical Physics Department of Applied Physics Key Laboratory for Micro-Nano Optoelectronic Devices of Ministry of Education, and State Key Laboratory for Chemo/Biosensing and Chemometrics and Hunan University 410082ChangshaChina Raising and lowering operators for angular momentum quantum numbers l in spherical harmonics 1 2angular momentumraising and lowering operatorsquantum numbers PACS numbers: 0365Ca Two vector operators aimed at shifting angular momentum quantum number l in spherical harmonics lm , primarily proposed by Prof. X. L. Ka in 2001, are further studied. For a given magnetic quantum number m , specific states lm in spherical harmonics with the lowest angular momentum quantum numbers l are obtained and the state with minimum angular momentum quantum number in whole set of the spherical harmonics is 0, 0 . How to use these states to generate whole set of spherical harmonics is illustrated. I. INTRODUCTION Testing an idea against a vital theoretical model usually enriches the understanding of both the idea and the model. In quantum mechanics, the ladder operator technique is widely used. For instance, the action of the angular momentum ladder operator L + and L − with definition x y L L iL ± ≡ ± on spherical harmonics lm raises and lowers respectively the magnetic quantum number m by one while leaving the angular momentum quantum number l unaltered. Then is there any ladder operator that shifts the values of l in the spherical harmonics lm ? Looking into literature, we can find that there are indeed results relevant to the solution to this problem. In 1980, Szpikowski and Góźdź pointed out in passing in the appendix A of their paper (Szpikowski and Góźdź 1980) an operator O in the interacting boson nuclear model can diminish both l and m in lm with m=l as , l l to another one ', ' l l , where the operator O is a polynomial of terms containing powers of L + , L − and the tensor operators ( ) k T where superscript k denotes the rank under rotational transformations. In 1994, with help of the tensor operators T , Shanker (Shanker 1994) showed that the raising and lowering operator As far as our knowledge goes, the first attempt to give a direct answer to the problem is due to Prof. Ka in 2001, who presented a derivation and observed that once acting on the spherical harmonics lm two vector operators (Ka 2001), ( ) ( 1) i l l = × + + R N L N , ( ) , i l l = × − Q N L N(1)+ = − ± − ∓ ,(2) 2 1 ( ) ( 1, ) 1, 2 3 z l R l lm b l m l m l + = + + + , 2 1 ( ) ( , ) 1, 2 1 z l Q l lm b l m l m l + = − − − ,(3) where ( , ) ( )( 1) a l m l m l m = + + − ∓ , ( , ) ( )( ) b l m l m l m = + − . (4) Unfortunately, these two vector operators (1) are not full ones because they contain information of the state acted, i.e., the angular momentum quantum number l . Prof. Ka left the operator (1) almost finished. In fact, they become full operators once the following replacement is made in operators (1), 2 2 2 4 / 1 1 4Ł 1 1 2 2 L l + − + − → = ,(5)where 2 2 2 / , Ł / L ≡ ≡ Ł L are respectively dimensionless angular momentum and its square. Explicitly, the vector operators we deal with in this paper take following simple forms, 2 4Ł 1 1 2 i + + = × + R N Ł N , 2 4Ł 1 1 . 2 i + − = × − Q N Ł N(6) These two vector operators have not been reported before, and many properties are possibly unknown to us yet. However, the present paper is mainly concerned with an illustration that they are really raising and lowering operators under looking for. This paper is organized as following. In section II, the fundamental commutation relations are presented, and in Section III, the quantum states of lowest quantum numbers are determined. In last section IV in addition to discussions we will briefly mention some interesting topics of further studies. II. FUNDAMANTAL COMMUTATION RELATIONS It is accustomed (Hall and Mitchell 2002) to assume that operator 2 L L ≡ is hermitian and satisfies 2 [ , ] 0 L L = ，[ , ] 0 L = L . (7) Choosing the simultaneous eigenvalues of two operators 2 , z L L , we have, . L lm l l lm = + Note that a vector operator V by definition satisfies the commutation relations, (Sukurai 1994) , . i j ijk k V L i V ε ⎡ ⎤= ⎣ ⎦ (9) Moreover if 0 = V L i , it can be easily proved (Ka 2001), 2 ( ) ( ) . i L × × = × − V L L V L V( 10) As a consequence of these relations (9)-(10), we have 2 2 2 , 2 ( ) 2 2 . L i i L ⎡ ⎤ × = × × + × = − ⎣ ⎦ N L N L L N L N(11) Using relations (9) and (11), we have immediately following commutation relation, ( ) 2 2 2 , 4 Ł 1 1 . L ⎡ ⎤= + + ⎣ ⎦ R R(12) Acting of this relation (12) on both sides with spherical harmonics lm , we find that operator R really shifts the angular momentum quantum number from l to 1 l + , ( ) 2 2 2 2 2 4 Ł 1 1 ( 1)( 2) . L lm L lm lm l l lm = + + + = + + R R R R(13) In other words, 1, ' , lm l m ∝ + R where magnetic quantum number ' m may differ from the original one m . Similarly, we have for operator Q , ( ) 2 2 2 , 4 Ł 1 1 L ⎡ ⎤= − + − ⎣ ⎦ Q Q ,(15) and it shifts the angular quantum number from l to 1 l − , , 1 , '. l m l m ∝ − Q(16) Next, in order to examine how magnetic quantum number changes on the action of operators R and Q , we need to calculate commutation relations such as[ , a n d , . z z L R R L Q Q ± ± ± ± = ± = ±(18) Eqs. (14) and (17) show that z R and z Q are operators respectively raise and lower the quantum number l in spherical harmonics lm by one while keeping the magnetic quantum number m unchanged. Eqs. (14), (16) and (18) show that R ± and Q ± are operators respectively raise and lower l in lm by one and also move the magnetic quantum number m by 1 ± respectively. Explicitly, we have after some calculations, 2 1 ( 2, ) 1, 1 2 3 l R lm a l m l m l ± + = + ± + ± + , 2 1 ( , ) 1, 1 2 1 l Q lm a l m l m l ± + = − ± − ∓ ,(19) 2 1 ( 1, ) 1, 2 3 z l R lm b l m l m l + = + + + , 2 1 ( , ) 1, 2 1 z l Q lm b l m l m l + = − − − .(20) Note that the operators used here heave nothing to do with the state acted whereas those used in (2) In these cases, subsequent applications of the lowering operators will produce zero kets. In contrast, by acting on these kets with appropriate raising and lowering operators and multiplying by suitable normalization factors, we can produce an infinite even whole set of the kets. For instance, once we know a state Firstly, we solve the differential equation 0 L θ θ ψ θ ϕ ψ θ ϕ θ θ θ θ ⎛ ⎞ ∂ ∂ ⎛ ⎞ ⎛ ⎞ + + = + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ∂ ∂ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ .(26) By standard method of separation of variables and with usual requirement of singlevaluedness of the state function, general solution to Eq.(26) is given by, ( ) ( ) states each of them with its lowest angular momentum quantum number, and only four of them are independent. Careful analysis shows that only two of them, , m m ± , bear the lowest angular momentum quantum numbers for a given the magnetic quantum number of magnitude m , and the state with minimum angular momentum quantum number in whole set of spherical harmonics turns out to be 0, 0 . Starting from this state 0, 0 , we can generate the whole set of the spherical harmonics with appropriate action of the raising and lowering operators. The new operators (6) introduced in present paper may have wider and deeper respects worthy of future explorations. Among of them we mention a connection between the operators (6) and the coherent states defined on the sphere. As reviewed by Hall and Mitchell (Hall and Mitchell 2002), there are different forms of coherent states proposed by substantially different points of view. Among these coherent states that are defined by the eigenfunctions of annihilation or lowering operators, one is introduced by Kowalski and Rembielinski who presented the normalized vector operators (Kowalski and Rembielinski, 2000), ( ) ( ) ( ) -( 3 ) 3are only half-finished. From Eqs. (19)-(20), we see that two pairs of operators R ± andIII. DETERMINATION OF QUANTUM STATES WITH LOWEST QUANTUM NUMBERSFrom Eqs(19)-(20), operators that can lower the angular momentum quantum number are , to solve equations in (22) and discuss the relations between solutions. We will deal with this problem in spherical polar coordinates ( , ) R or Q contain two variables ( , ) θ ϕ , solutions to Eqs. (22) must depend on two quantum numbers. In general, we look for solutions that are simultaneously eigenvalue of z L . ACKNOWLEDGMENTSThis subject is supported by "973" National Key Basic Research Program of China (Grant No. 2007CB310500). The first author is grateful to Prof. X. L. Ka, Beijing Normal University, for help discussions.In spherical polar coordinates, 0hereafter i c ( 1 ,2 ,1 ,2 ,1 ,2 . i z z = + + − − ) denote integration constants. The square integrability requires that the exponent of sine function can not be negative. The final result is simply with a normalization factor z cIn terms of kets, ,, we find that one of the two independent solutions to each of these two equations is known. In fact, when 0 m ≤ ,the zero ket. One can then easily verify that another independent solution to the differential equationIV. CONCLUSIONS AND REMARKSDeveloping one step further of the work of Prof. Ka(Ka 2001), we present the full form of the raising and lowering vector operators R and Q that shift the angular momentum quantum number in spherical harmonics lm . Apparently, three lowering operators give six different 2What we ascertain now is that operators Z , R and Q are formed by combination of × N Ł (or × Ł N ) and N with some coefficients depending on operator 2 4Ł 1 + , and there is no simple linear relation in between. Furthermore, there are certainly relations between the operators (6) and the tensor operators ( ) k T or the Lenz vector operators for they also play roles in shift the angular momentum quantum number. All these issues will be discussed in detail in near future. Lenz vector operations on spherical hydrogen atom eigenfunctions. C E Burkhardt, J J Leventhal, Am. J. Phys. 72Burkhardt, C. E. and Leventhal, J. J. (2004). Lenz vector operations on spherical hydrogen atom eigenfunctions. Am. J. Phys. 72, 1013-1016. Coherent states on spheres. B C Hall, J J Mitchell, J. Math. Phys. 43Hall, B.C. and Mitchell, J.J. (2002). Coherent states on spheres. J. Math. Phys. 43, 1211-1236. X L Ka, Advanced Quantum Mechanics. Beijing2nd ed. Higher EducationKa, X. L. (2001). Advanced Quantum Mechanics, 2nd ed. Higher Education, Beijing. pp.76-77, pp.102-103. Quantum mechanics on a sphere and coherent states. K Kowalski, J Rembielinski, J. Phys. A. 33Kowalski, K. and Rembielinski, J. (2000). Quantum mechanics on a sphere and coherent states, J. Phys. A., 33, 6035-6048. J J Sakurai, Modern Quantum Mechanics. New YorkAddison-WesleySakurai, J. J. (1994). Modern Quantum Mechanics, Addison-Wesley, New York. R Shankar, Principles of Quantum Mechanics. New York2nd ed. PlenumShankar, R. (1994). Principles of Quantum Mechanics, 2nd ed. Plenum, New York. The orthonormal basis for symmetric irreducible representations of O(5) × SU(1, 1) and its application to the interacting boson model. S Szpikowski, A Góźdź, Nucl. Phys. A. 340Szpikowski, S. and Góźdź, A. (1980). The orthonormal basis for symmetric irreducible representations of O(5) × SU(1, 1) and its application to the interacting boson model. Nucl. Phys. A. 340, 76-92.
{'fraction_non_alphanumeric': 0.064052627023284, 'fraction_numerical': 0.031680083095299925, 'mean_word_length': 3.6607503025413473, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 40, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'Two vector operators aimed at shifting angular momentum quantum number l in spherical harmonics lm , primarily proposed by Prof. X. L. Ka in 2001, are further studied. For a given magnetic quantum number m , specific states lm in spherical harmonics with the lowest angular momentum quantum numbers l are obtained and the state with minimum angular momentum quantum number in whole set of the spherical harmonics is 0, 0 . How to use these states to generate whole set of spherical harmonics is illustrated.', 'arxivid': '0912.5146', 'author': ['Q H Liu \nSchool for Theoretical Physics, and Department of Applied Physics\nHunan University\n410082ChangshaChina\n', 'D M Xun ', 'L Shan ', '\nSchool for Theoretical Physics\nDepartment of Applied Physics\nKey Laboratory for Micro-Nano Optoelectronic Devices of Ministry of Education, and State Key Laboratory for Chemo/Biosensing and Chemometrics\nand Hunan University\n410082ChangshaChina\n'], 'authoraffiliation': ['School for Theoretical Physics, and Department of Applied Physics\nHunan University\n410082ChangshaChina', 'School for Theoretical Physics\nDepartment of Applied Physics\nKey Laboratory for Micro-Nano Optoelectronic Devices of Ministry of Education, and State Key Laboratory for Chemo/Biosensing and Chemometrics\nand Hunan University\n410082ChangshaChina'], 'corpusid': 118143767, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 3634, 'n_tokens_neox': 3161, 'n_words': 2062, 'pdfsha': '50d0047d79dd9791fdf3b8ef36c60456a0587d45', 'pdfurls': ['https://export.arxiv.org/pdf/0912.5146v1.pdf'], 'title': ['Raising and lowering operators for angular momentum quantum numbers l in spherical harmonics', 'Raising and lowering operators for angular momentum quantum numbers l in spherical harmonics'], 'venue': []}
arxiv
LARGE MUSIC RECOMMENDATION STUDIES FOR SMALL TEAMS Kyle Robinson [email protected] School of Computer Science University of Waterloo Canada Dan Brown [email protected] School of Computer Science University of Waterloo Canada David R Cheriton School of Computer Science University of Waterloo Canada LARGE MUSIC RECOMMENDATION STUDIES FOR SMALL TEAMS Running live music recommendation studies without direct industry partnerships can be a prohibitively daunting task, especially for small teams. In order to help future researchers interested in such evaluations, we present a number of struggles we faced in the process of generating our own such evaluation system alongside potential solutions. These problems span the topics of users, data, computation, and application architecture. RUNNING A LIVE RECOMMENDATION STUDY There are clearly benefits to evaluating music recommender systems with real users [1]. In our recent paper analysing user perceptions of diversity in music recommendation we found that mere accuracy evaluations are not necessarily good indicators of individual track ratings, and overall list satisfaction is not a function of individual track ratings alone [2]. Insights such as this are not new; a decade and a half ago McNee et al. informally argued that there must be more emphasis put on user-centric recommender system evaluation [3], yet just two years ago Dacrema et al. highlighted a disturbing lack of attention to evaluation even in strictly offline analyses [4]. User studies and online analyses require significantly more resources and time than strictly offline analyses. In the hope of assisting researchers completing live evaluations of their methods, we present some of the struggles faced in developing our recent study, and their resolutions. For further reading on live evaluation of recommender systems we refer readers to the relevant chapters of the Recommender Systems Handbook [1,5]. General Architecture The goals of our system were twofold: to generate up-todate music recommendations for previously unseen participants using the same models described in pre-existing research, and to generate these recommendations on demand. The final system consisted of an online appli- cation which, after obtaining consent, collected participants' listening histories from the LastFM API, fed their data through several recommendation algorithms to obtain top-n lists, obtained music previews and metadata from the Spotify API, and displayed song previews alongside song-specific appropriateness questions and global summary questions. We implemented the system using a Flask backend which served static HTML and Javascript 1 . The study was hosted on a single AWS EC2 server instance using Elastic Beanstalk. Users and Data Up-to-date Training Data A first challenge is that for collaborative filtering recommendation (especially for music) training data must be collected as close to the study as possible in order ensure that participant data is known by the model. Real data is also difficult to locate and obtain. In the domain of music, LastFM continues to provide a useful API for user-song listening event (LE) data collection 2 , though collection is not always a straightforward process. We collected a base data set from pseudorandomly selected users to train our model by crawling the public social graph of friends lists. Contacting authors of prior research utilizing data sets which fit our needs proved to be vital in developing a method of data collection. Although their data was not up-to-date, we were able to develop our own collection method after corresponding. This data was topped-up before each subsequent study, and appropriately randomized. Live Participant Data An especially interesting challenge was that we needed to be able to collect data from participants as they connected to the system. This data also needed to be as current as possible. Our solution to this problem was to use a mediatracking application. LastFM and its associated API also worked well for this purpose. For smaller pools of participants, we helped them register an account and manually monitored it over a collection period of a few weeks. For larger pools of participants we specified in recruitment and consent materials that they must have an exist-1 An un-maintained repository of our application can be found at https://github.com/Stack-Attack/music_rec_div_study 2 The LastFM API documentation can be accessed at https://www.last.fm/api ing account containing some minimum number of listening events/plays/scrobbles. Amazon Mechanical Turk specifically does not allow researchers to ask participants to log into any accounts, but because LastFM accounts are publicly accessible by default, data can be accessed with only a username. It is worth noting that in our case, some participants appear to have created new accounts to complete the study without being prompted to do so. Showing Music Recommendations To evaluate a recommendation, participants need to be able to listen to it! Music previews can typically be accessed without having to authenticate with a music service. We used the Spotify API to obtain 30 second previews with album art in the form of HTML iframes embedded in the page 3 . The relevant track previews were retrieved by searching for tracks using their exact song and artist names as well as the region a participant was connecting form. We discarded the small portion of tracks that did not return any results; this is likely unavoidable. Computation Model Training We trained two different ML models for our project, one more traditional model based on matrix factorization, and one more modern approach based on neural networks. Writing the necessary code to implement models efficiently is time-consuming and error prone. Additionally, training models on huge data sets seems infeasible due to size and dimensionality. Using open-source libraries can save time and help alleviate the risk of errors impacting results, though they may mis-implement key algorithmic features, or be difficult to extend. Comparing multiple implementations online can help, but one must ensure to follow any licenses and reference the source. Data Handling In order to reduce the size of data, we filtered out irrelevant items and users. By filtering out tracks with 10 or fewer LEs we reduced the number of unique tracks by 82% while only decreasing LE count by 6%. Even after filtering, our Variational Autoencoder (MultVAE) was too large to fit in GPU memory, and so we trained multiple model variations concurrently using CPU's in order to make up for lost time. Some models may simply be infeasible without access to High Performance Computing (HPC) resources. Other models may simply not be suited to real-world implementations without significant structural changes and/or preprocessing steps. This is simply the reality of evaluating models on real populations. Complex Architecture and Resource Requirements The size of trained models is too large to fit in memory, especially if a new model is loaded for each server connection. The size of trained models can be very large even after removing unnecessary data (i.e., neural network optimizer information). Our trained MultVAE model was over 6.5GB in size. Luckily, online cloud computing platforms often offer specific instances with large amounts of dedicated memory at the expense of processing power. These instances are a great fit for running user-studies which will inherently have a low number of concurrent users. As of now, simple hosting services such as Heroku will be infeasible due to the memory requirements [6]. AWS Elastic Beanstalk, however, provides very cost effective solutions in the form of memory optimized EC2 instances [7]. Even with the low number of concurrent users, there will still be some asynchronous computation required. The ideal, yet complex, solution to this problem is to decouple the longer tasks (recommendation and data collection) from the main server using a separate worker process or even server. Developing this architecture can be timeconsuming, expensive, and unnecessary for such small temporary applications. We found success by limiting the server to one Python process, and running data collection and recommendation on separate threads using the built in concurrent.futures library 4 . As live user data collection was input/output bound it did not block the server from handling requests, and as most recommendation and processing tasks utilized multi-core optimized libraries these tasks were also handled relatively quickly. This kind of architecture would certainly not work for large-scale applications, but was ideal for our small user-count study due to its simplicity and efficient use of only one compute node. Summary The benefits of evaluating music recommender systems on real users are as intuitive as they are founded in empirical evaluation [1]. Without industry collaboration, and at minimal cost, we were able to develop a music recommendation system which could generate and present recommendations to new users within a single un-moderated interactive session. To assist and encourage future researchers in developing similar systems, we have described some of the challenges and solutions to problems encountered along the way. Among the problems we addressed were training data collection, live user data collection, and obtaining music previews. We also discussed our technical implementation; specifically dealing with issues of memory management and availability. In general, we hope that researchers embrace collaboration with others to better base our analysis of recommender systems in users themselves. Our key message is that independent user studies on music recommendation are both important and achievable. © K. Robinson, and D. Brown. Licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0). Attribution: K. Robinson, and D. Brown, "Large Music Recommendation Studies for Small Teams", in Proc. of the 22nd Int. Society for Music Information Retrieval Conf., Online, 2021. Details on embedding Spotify music previews can be found at https://developer.spotify.com/documentation/widgets/guides/adding-aspotify-embed/ In practice, we used the Flask-Executor python library to manage our futures: https://pypi.org/project/Flask-Executor/ Evaluating recommender systems with user experiments. B P Knijnenburg, M C Willemsen, Recommender Systems Handbook. SpringerSecond EditionB. P. Knijnenburg and M. C. Willemsen, "Evalu- ating recommender systems with user experiments," in Recommender Systems Handbook, Second Edition. Springer, 2015, pp. 309-352. Quantitative User Perceptions of Music Recommendation List Diversity. K Robinson, D Brown, Proceedings of the 22nd International Society for Music Information Retrieval Conference. the 22nd International Society for Music Information Retrieval ConferenceK. Robinson and D. Brown, "Quantitative User Per- ceptions of Music Recommendation List Diversity," in Proceedings of the 22nd International Society for Mu- sic Information Retrieval Conference, 2021. Being accurate is not enough: How accuracy metrics have hurt recommender systems. S M Mcnee, J Riedl, J A Konstan, Conference on Human Factors in Computing Systems -Proceedings. S. M. McNee, J. Riedl, and J. A. Konstan, "Being ac- curate is not enough: How accuracy metrics have hurt recommender systems," in Conference on Human Fac- tors in Computing Systems -Proceedings, 2006, pp. 1097-1101. Are we really making much progress? A worrying analysis of recent neural recommendation approaches. M F Dacrema, P Cremonesi, D Jannach, Proceedings of the 13th ACM Conference on Recommender Systems. the 13th ACM Conference on Recommender SystemsM. F. Dacrema, P. Cremonesi, and D. Jannach, "Are we really making much progress? A worrying anal- ysis of recent neural recommendation approaches," in Proceedings of the 13th ACM Conference on Recom- mender Systems, 2019, pp. 101-109. Evaluating recommender systems. A Gunawardana, G Shani, Recommender Systems Handbook. SpringerSecond EditionA. Gunawardana and G. Shani, "Evaluating recom- mender systems," in Recommender Systems Hand- book, Second Edition. Springer, 2015, pp. 265-308. Dyno Types. Heroku, Heroku, "Dyno Types," https://devcenter.heroku.com/ articles/dyno-types, 2021. EC2 Instance Types. A W Services, A. W. Services, "EC2 Instance Types," https://aws. amazon.com/ec2/instance-types/, 2021.
{'fraction_non_alphanumeric': 0.031212676168484752, 'fraction_numerical': 0.00836053825941556, 'mean_word_length': 4.921735030645921, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 6, 'lorem ipsum': 0, 'www.': 1, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'Running live music recommendation studies without direct industry partnerships can be a prohibitively daunting task, especially for small teams. In order to help future researchers interested in such evaluations, we present a number of struggles we faced in the process of generating our own such evaluation system alongside potential solutions. These problems span the topics of users, data, computation, and application architecture.', 'arxivid': '2301.13388', 'author': ['Kyle Robinson [email protected] \nSchool of Computer Science\nUniversity of Waterloo\nCanada\n', 'Dan Brown [email protected] \nSchool of Computer Science\nUniversity of Waterloo\nCanada\n', 'David R Cheriton \nSchool of Computer Science\nUniversity of Waterloo\nCanada\n'], 'authoraffiliation': ['School of Computer Science\nUniversity of Waterloo\nCanada', 'School of Computer Science\nUniversity of Waterloo\nCanada', 'School of Computer Science\nUniversity of Waterloo\nCanada'], 'corpusid': 245960502, 'doi': '10.48550/arxiv.2301.13388', 'github_urls': ['https://github.com/Stack-Attack/music_rec_div_study'], 'n_tokens_mistral': 2896, 'n_tokens_neox': 2640, 'n_words': 1846, 'pdfsha': '3c19d397062d9f7f7b5549b502e2587d330ee185', 'pdfurls': ['https://export.arxiv.org/pdf/2301.13388v1.pdf'], 'title': ['LARGE MUSIC RECOMMENDATION STUDIES FOR SMALL TEAMS', 'LARGE MUSIC RECOMMENDATION STUDIES FOR SMALL TEAMS'], 'venue': []}
arxiv
Initial boundary value problems for a fractional differential equation with hyper-Bessel operator 21 Nov 2016 March 9, 2022 Fatma Al-Musalhi Department of Mathematics Sultan Qaboos University P.O. Box 36Al-KhoudhOman Nasser Al-Salti Department of Mathematics Sultan Qaboos University P.O. Box 36Al-KhoudhOman Erkinjon Karimov Department of Mathematics Sultan Qaboos University P.O. Box 36Al-KhoudhOman Initial boundary value problems for a fractional differential equation with hyper-Bessel operator 21 Nov 2016 March 9, 2022 Direct and inverse source problems of a fractional diffusion equation with regularized Caputo-like counterpart hyper-Bessel operator are considered. Solutions to these problems are constructed based on appropriate eigenfunction expansion and results on existence and uniqueness are established. To solve the resultant equations, a solution to a non-homogeneous fractional differential equation with regularized Caputo-like counterpart hyper-Bessel operator is also presented. Introduction In this paper, we consider the following fractional differential equation involving hyper-Bessel operator with a source term F : C t θ ∂ ∂t α u(x, t) − u xx (x, t) = F.(1) We study both a direct problem where F = f (x, t) is a known function of space and time and an inverse source problem where F = f (x) is an unknown function of space only. Here C t θ ∂ ∂t α stands for a regularized Caputo-like counterpart hyper-Bessel operator of order 0 < α < 1 (see formula (5)). The hyper-Bessel operator was introduced by Dimovski in [2] and it arises in various problems, such as, fractional relaxation [3] and fractional diffusion models [4]. As an example, authors in [4] used hyper-Bessel operator to describe heat diffusion for fractional Brownian motion. Their analysis based on converting fractional power of hyper-Bessel operator into Erdélyi-Kober (E-K) fractional integral. For more details about fractional Brownian motion, the reader is referred to [4,7]. In fact, expressing hyper-Bessel operator in terms of Erdélyi-Kober fractional integral plays a key role in finding solution to fractional differential equations involving hyper-Bessel operator as illustrated in this paper as well. Some results related to hyper-Bessel operator are given in [3], [6]. In [1], AL-Saqabi and Kiryakova considered Volterra integral equation of second kind and a fractional differential equation, involving (E-K) integral or differential operator. They found explicit solutions to these equations using transmutation method which reduces solutions to known integral solutions of Riemann-Liouville fractional equations. The purpose of this paper is to prove existence and uniqueness of solution to a fractional diffusion equation involving a regularized Caputo-like counterpart hyper-Bessel operator considering both direct and inverse source problems. Preliminaries In this section, we recall some definitions and results related to fractional hyper-Bessel operator which will be used later in this paper. We start by writing down the definition of Erdélyi-Kober fractional integral. Definition 2.1. (see [1,3]) Erdélyi-Kober fractional integral of a function f (t) with arbitrary parameters δ > 0, γ ∈ R and β > 0 is defined as I γ,δ β f (t) = t −β(γ+δ) Γ(δ) t 0 (t β − τ β ) δ−1 τ βγ f (τ ) d(τ β ),(2) which is reduced to the well-known Riemann-Liouville fractional integral when γ = 0 and β = 1 with a power weight. For δ < 0, the interpretation is via integro-differential operator I γ,δ β f (t) = (γ + δ + 1)I γ,δ+1 β f (t) + 1 β I γ,δ+1 β t d dt f (t) . In the following theorem, we present an explicit solution to an integral equation involving E-K fractional integral. Theorem 2.2. (see [1], Theorem 1) The unique solution y(t) ∈ C βµ , µ ≥ max {0, −γ} − 1 to the following fractional integral equation of a second kind : y(t) − λt βδ I γ,δ β y(t) = f (t), or equivalently, y(t) − λt −βγ t 0 (t β − τ β ) δ−1 Γ(δ) τ βγ y(τ ) d(τ β ) = f (t), with f ∈ C βµ , has the explicit form of a convolutional type integral : y(t) = f (t) + λt −βγ t 0 (t β − τ β ) δ−1 E δ,δ λ(t β − τ β ) δ τ βγ f (τ ) d(τ β ).(3) Next, we use E-K integral to define the regularized counterpart hyper-Bessel operator. Definition 2.3. (see [3]) The hyper-Bessel operator of order 0 < α < 1 is defined in terms of E-K integral as follows t θ d dt α f (t) =    (1 − θ) α t −(1−θ)α I 0,−α 1−θ f (t), if θ < 1, (θ − 1) α I −1,−α 1−θ t (1−θ)α f (t), if θ > 1.(4) Note that when θ = 0, the hyper-Bessel operator coincides with Riemann-Liouville fractional derivative. Now, recall that Caputo and Riemann-Liouville fractional derivatives of order 0 < α < 1 are defined as (see [5]): C D α 0|t f (t) = 1 Γ(1 − α) t 0 f ′ (τ ) (t − τ ) α dτ, D α 0|t f (t) = 1 Γ(1 − α) d dt t 0 f (τ ) (t − τ ) α dτ, respectively, and they are related by ( [5]): C D α 0|t f (t) = D α 0|t (f (t) − f (0 + )). Using the above relation, we can express the regularized Caputo-like counterpart hyper-Bessel operator as : C t θ d dt α f (t) = t θ d dt α f (t) − f (0) t −α(1−θ) (1 − θ) −α Γ(1 − α) ,(5) and in terms of E-K fractional integral : C t θ d dt α f (t) = (1 − θ) α t −α(1−θ) I 0,−α 1−θ (f (t) − f (0)) .(6) Also, we need to recall the Mittag-Leffler function of one parameter : E α (z) = ∞ k=0 z k Γ(αk + 1) , Re(α) > 0, z ∈ C, and the Mittag-Leffler of two parameters E α,β * (z) = ∞ k=0 z k Γ(αk + β * ) , Re(α) > 0, Re(β * ) > 0, z ∈ C. Now, we need the following results related to the Mittag-Leffler function Theorem 2.4. (see [11]) If Re(µ) > 0, Re(β * ) > 0, λ is a complex number and f (t) is an integrable function, then x a (x − u) β * −1 E α,β * (λ(x − u) α ) du u a (u − t) µ−1 Γ(µ) f (t)dt = x a (x − t) β * +µ−1 E α, β * +µ (λ(x − t) α ) f (t)dt. Theorem 2.5. (see [10]) Let α < 2, β * ∈ R and πα 2 < µ < min {π, πα} . Then we have the following estimate |E α,β * (z)| ≤ M 1 + |z| , µ ≤ |argz| ≤ π, |z| ≥ 0.(7) Here and in the rest of the paper, M denotes a positive constant. In the following theorem, we present a homogeneous fractional equation with regularized Caputo-like counterpart hyper-Bessel operator and its explicit solution as proved in [3]. Theorem 2.6. ( see [3], Theorem 2.1) The function u(t) = E α − λt α(1−θ) (1 − θ) α , solves the fractional Cauchy problem      C t θ d dt α u(t) = −λu(t), α ∈ (0, 1), θ < 1, t ≥ 0, u(0) = 1. In this paper, we consider a more general case, namely, a non-homogeneous fractional differential equation with a regularized Caputo-like counterpart hyper-Bessel operator presented in the following lemma: Lemma 2.7. Consider the following non-homogeneous fractional differential equation C t θ d dt α u(t) = −λu(t) + f (t), α ∈ (0, 1), θ < 1, t ≥ 0,(8) with u(0) = u 0 , where u 0 is a constant. Then, its solution is given in the integral form u(t) = u 0 E α (λ * t ρα ) + 1 ρ α Γ(α) t 0 (t ρ − x ρ ) α−1 f (x) d(x ρ ) + λ * ρ α t 0 (t ρ − x ρ ) 2α−1 E α,2α (λ * (t ρ − x ρ ) α ) f (x) d(x ρ ),(9) where, ρ = 1 − θ and λ * = − λ ρ α . Moreover, if f = f 0 is constant, then the solution reduces to u(t) = C * E α (λ * t ρα ) + f 0 λ . where C * = u 0 − f 0 λ . In particular, when f = 0, we have u(t) = u 0 E α (λ * t ρα ) . Proof. First, using relation (6), equation (8) can be written as ρ α t −αρ I 0,−α ρ (u(t) − u 0 ) = −λu(t) + f (t), which, on dividing by ρ α t −αρ , becomes I 0,−α ρ (u(t) − u 0 ) = λ * t ρα u(t) + t ρα ρ α f (t), where λ * = − λ ρ α . Using the following property of the inverse of E-K integral (see [8], Theorem 2.7): (I η,α m ) −1 = I η+α,−α m , the above equation can be written as an integral equation, namely, u(t) − λ * I −α,α ρ (t ρα u(t)) = u 0 + 1 ρ α I −α,α ρ (t ρα f (t)) , or equivalently, u(t) − λ * Γ(α) t 0 (t ρ − τ ρ ) α−1 u(τ ) d(τ ρ ) = u 0 + 1 ρ α Γ(α) t 0 (t ρ − τ ρ ) α−1 f (τ ) d(τ ρ ). Whereupon using Theorem 2.2, we have u(t) = f * (t) + λ * t 0 (t ρ − τ ρ ) α−1 E α,α (λ * (t ρ − τ ρ ) α )f * (τ ) d(τ ρ ) + u 0 1 + λ * t 0 (t ρ − τ ρ ) α−1 E α,α (λ * (t ρ − τ ρ ) α ) d(τ ρ ) ,(10)where, f * (t) = 1 ρ α Γ(α) t 0 (t ρ − x ρ ) α−1 f (x) d(x ρ ). The first integral in (10) can be simplified using Theorem 2.4 to the following: λ * ρ α t 0 (t ρ − x ρ ) 2α−1 E α,2α (λ * (t ρ − x ρ ) α ) f (x) d(x ρ ),(11) and the second integral in (10) can be also simplified as follows: u 0 1 + λ * t 0 (t ρ − τ ρ ) α−1 E α,α (λ * (t ρ − τ ρ ) α ) d(τ ρ ) = u 0 1 + ∞ k=0 (λ * ) k+1 Γ(αk + α) t 0 (t ρ − τ ρ ) αk+α−1 d(τ ρ ) = u 0 1 + ∞ k=0 (λ * ) k+1 Γ(α(k + 1) + 1) t ρ α(k+1) = u 0 1 + ∞ m=1 (λ * ) m Γ(αm + 1) t ραm = u 0 E α (λ * t ρα ).(12) Substituting the two simplified forms (11) and (12) into (10), we get the integral solution (9). Now, if f (t) = f 0 is constant, then evaluating the first integral in the expression (9) gives f 0 ρ α Γ(α) t 0 (t ρ − x ρ ) α−1 d(x ρ ) = f 0 t ρα ρ α Γ(α + 1) . Substituting back into (9) and proceeding in a similar way as in (12), the expression of u(t) can be reduced to u(t) = u 0 E α (λ * t ρα ) − f 0 λ ∞ k=1 λ * k t ραk Γ(αk + 1) , which can be rewritten as u(t) = f 0 λ + C * E α (λ * t ρα ) , where C * = u 0 − f 0 λ . Finally, if f (t) = f 0 = 0, the the expression of u(x, t) can be further reduced to u(t) = u 0 E α (λ * t ρα ) , which is consistent with Theorem 2.6. The rest of this paper is devoted for the main results. In the remaining two sections, we present existence and uniqueness results of solutions to direct and inverse source problems involving a regularized Caputo-like counterpart hyper-Bessel operator. A Direct Problem Statement of Problem and Main Result Find a function u(x, t) in a domain Ω = {0 < x < 1, 0 < t < T } satisfying C t θ ∂ ∂t α u(x, t) − u xx (x, t) = f (x, t), (x, t) ∈ Ω,(13) the boundary conditions u(0, t) = 0, u(1, t) = 0, 0 ≤ t ≤ T,(14) and the initial condition u(x, 0) = ψ(x), 0 ≤ x ≤ 1,(15) where f (x, t) is a given function, θ < 1, 0 < α < 1 and C t θ ∂ ∂t α is the regularized Caputolike counterpart hyper-Bessel operator defined in (5). Our aim is to prove the existence and uniqueness of solution to the problem (13) -(15) as stated in the following theorem: Theorem 3.1. Assume that the following conditions hold • ψ(x) ∈ C[0, 1] such that ψ(0) = ψ(1) = 0 and ψ ′ (x) ∈ L 2 (0, 1), • f (·, t) ∈ C 3 [0, 1] such that f (0, t) = f (1, t) = f xx (0, t) = f xx (1, t) = 0, and ∂ 4 ∂x 4 f (·, x) ∈ L(0, 1), then, the problem (13) − (15) has a unique solution given by u(x, t) = ∞ k=1 ψ k E α −k 2 π 2 (1 − θ) α t (1−θ)α + F k (t) sin(kπx), where, F k (t) = 1 (1 − θ) α Γ(α) t 0 t (1−θ) − τ (1−θ) α−1 f k (τ ) d(τ (1−θ) ) − k 2 π 2 (1 − θ) 2α t 0 t (1−θ) − y (1−θ) 2α−1 E α,2α − λ(t (1−θ) − y (1−θ) ) α (1 − θ) α f k (y) d(y (1−θ) ), ψ k = 2 1 0 ψ(x) sin(kπx) dx, k = 1, 2, 3, · · · f k (t) = 2 1 0 f (x, t) sin(kπx) dx, k = 1, 2, 3, · · · Proof of Result Existence of Solution Using separation of variables method for solving the homogeneous equation corresponding to (13) along with the homogeneous boundary conditions (14) yields the following spectral problem:    X ′′ + λX = 0, X(0) = 0, X(1) = 0.(16) It is known that the above problem is self adjoint and has the following eigenvalues λ k = (kπ) 2 , k = 1, 2, 3, · · · and the corresponding eigenfunctions are X k = sin(kπx) k = 1, 2, 3, · · · .(17) Using the fact that the system of eigenfunctions (17) forms an orthogonal basis in L 2 (0, 1) [9], we can write the solution u(x, t) in the form of a series expansion as follows: u(x, t) = ∞ k=1 u k (t) sin(kπx),(18) and f (x, t) = ∞ k=1 f k (t) sin(kπx),(19) where, u k (t) is the unknown to be determined and f k (t) is known and given by f k (t) = 2 1 0 f (x, t) sin(kπx)dx. Substituting (18) and (19) into (13) and (15), we get the linear fractional differential equation C t θ d dt α u k (t) + k 2 π 2 u k (t) = f k (t),(20) with the initial condition u k (0) = ψ k , where, ψ k is the coefficient of the series expansion of ψ(x) in terms of the orthogonal basis (17), i.e., ψ k = 2 1 0 ψ(x) sin(kπx) dx. Whereupon using Lemma 2.7, the solution of equation (20) is given by u k (t) = ψ k E α −k 2 π 2 ρ α t ρα + F k (t), where, ρ = 1 − θ and F k (t) = 1 ρ α Γ(α) t 0 (t ρ − τ ρ ) α−1 f k (τ ) d(τ ρ ) − k 2 π 2 ρ 2α t 0 (t ρ − y ρ ) 2α−1 E α,2α − λ ρ α (t ρ − y ρ ) α f k (y) d(y ρ ). Consequently, the expression of u(x, t) can be written as u(x, t) = ∞ k=1 ψ k E α −k 2 π 2 ρ α t ρα + F k (t) sin(kπx).(21) To complete the proof of existence, we need to prove the uniform convergence of the series representations of u(x, t), C t θ ∂ ∂t α u(x, t), u x (x, t), u xx (x, t). We start with the series representation of u(x, t), rewriting F k (t) as follows F k (t) = 1 k 2 π 2 ρ α Γ(α) t 0 (t ρ − τ ρ ) α−1 f ′′ k (τ ) d(τ ρ ) − 1 ρ 2α t 0 (t ρ − y ρ ) 2α−1 E α,2α − k 2 π 2 ρ α (t ρ − y ρ ) α f ′′ k (y) d(y ρ ), where, f ′′ k (t) = 2 1 0 f xx (x, t) sin(kπx)dx. Now, we estimate the Mittag-Leffler function using inequality (7) : E α,2α − k 2 π 2 ρ α (t ρ − y ρ ) α ≤ Mρ α ρ α + k 2 π 2 |t ρ − y ρ | α , which implies the following estimate for u(x, t): |u(x, t)| ≤ M ∞ k=1 |ψ k | ρ α + k 2 π 2 |t ρ − y ρ | α + 1 k 2 π 2 t 0 |t ρ − τ ρ | α−1 |f ′′ k (τ )|d(τ ρ ) + t 0 |t ρ − y ρ | 2α−1 ρ α + k 2 π 2 |t ρ − y ρ | α |f ′′ k (y)| d(y ρ ) . Since ψ(x) ∈ C[0, 1] and f (·, t) ∈ C 3 [0, 1] , then the above series converges and hence, by Weierstrass M-test the series representation of u(x, t) is uniformly convergent in Ω. Next, we show the uniform convergence of series representation of u xx (x, t), which is given by u xx (x, t) = − ∞ k=1 k 2 π 2 ψ k E α −k 2 π 2 ρ α t ρα + F k (t) sin(kπx). To prove this assertion, we have the following estimate |u xx (x, t)| ≤ M ∞ k=1 k 2 π 2 |ψ k | ρ α + k 2 π 2 (t ρ − y ρ ) α + 1 k 2 π 2 t 0 |t ρ − τ ρ | α−1 |f (4) k (τ )|d(τ ρ ) + t 0 |t ρ − y ρ | 2α−1 ρ α + k 2 π 2 (t ρ − y ρ ) α |f (4) k (y)| d(y ρ ) , where f (4) k (t) = 2 1 0 ∂ 4 dx 4 f (x, t) sin(kπx) dx. Since ψ(0) = ψ(1) = 0 and ∂ 4 f ∂x 4 (·, t) ∈ L(0, 1), then using integration by parts, we arrive at the following estimate |u xx (x, t)| ≤ M ∞ k=1 1 kπ ψ (1) k + 1 k 2 π 2 ≤ M ∞ k=1 1 (kπ) 2 + ∞ k=1 ψ (1) k 2 , where, we have used the inequality 2ab ≤ a 2 + b 2 and ψ (1) k = 2 1 0 ψ ′ (x) cos(kπx) dx. Then, Bessel's inequality for trigonometric functions ∞ k=0 g 2 k ≤ g 2 L 2 (0,1) , implies |u xx (x, t)| ≤ M ∞ k=1 1 (kπ) 2 + ψ ′ (x) 2 L 2 (0,1) . Thus, the series in expression of u xx (x, t) is bounded by a convergent series which implies that its uniformly convergent by Weierstrass M-test. Finally, series representation of C t θ ∂ ∂t α u(x, t) is given by C t θ ∂ ∂t α u(x, t) = − ∞ k=1 k 2 π 2 ψ k E α −k 2 π 2 ρ α + t ρα + F k (t) sin(kπx) + f (x, t), and convergence of the above series follows directly from the uniform convergence of u xx (x, t), which also ensures the uniform convergence of u x (x, t). Uniqueness of Solution: Suppose that u 1 (x, t) and u 2 (x, t) are two solutions of the problem (13) -(15), then u(x, t) = u 1 (x, t) − u 2 (x, t) satisfies the following boundary value problem: C t θ ∂ ∂t α u − ∂ 2 u ∂x 2 = 0, (x, t) ∈ Ω,(22)u(0, t) = 0, u(1, t) = 0, 0 ≤ t ≤ T,(23)u(x, 0) = 0, 0 ≤ x ≤ 1.(24) Define the following function: u k (t) = 2 1 0 u(x, t) sin(kπx)dx.(25) Then, the initial condition (24) implies u k (0) = 0.(26) Applying regularized Caputo-like counterpart hyper-Bessel operator to (25), we get C t θ d dt α u k (t) = 2 1 0 C t θ ∂ ∂t α u(x, t) sin(kπx)dx, = 2 1 0 u xx (x, t) sin(kπx)dx. Then, integrating by parts twice and using boundary conditions (23), we obtain the following fractional differential equation C t θ d dt α u k (t) − (kπ) 2 u k (t) = 0. Using Lemma 2.7, the above equation with the initial condition (26) has the trivial solution u k (t) ≡ 0, and hence we have Therefore, using the completeness property of system (17), we deduce that u(x, t) = 0 in Ω, which implies the uniqueness of solution to the problem (13) -(15). Inverse source problem Here, we consider an inverse source problem of finding a pair of functions {u(x, t), f (x)} in a rectangular domain Ω = {(x, t) : 0 < x < 1, 0 < t < T } , which satisfies the following initialboundary value problem: C t θ ∂ ∂t α u(x, t) − u xx (x, t) = f (x), (x, t) ∈ Ω (27) u(0, t) = 0, u(1, t) = 0, 0 ≤ t ≤ T,(28)u(x, 0) = ψ(x), u(x, T ) = φ(x), 0 ≤ x ≤ 1,(29) where φ and ψ are given functions, such that ψ(0) = ψ(1) = 0, φ(0) = φ(1) = 0, which follows directly from (28) and (29). As in the previous section, we seek solution to problem (27) -(29) in a form of series expansions using the orthogonal system (17) as follows: u(x, t) = ∞ k=1 u k (t) sin(kπx), f (x) = ∞ k=1 f k sin(kπx). where f k , u k are the unknowns to be determined. Substituting the above expressions for u(x, t) and f (x) into (27) and (29) gives the following fractional differential equation: C t θ d dt α u k (t) + k 2 π 2 u k (t) = f k , with the following conditions u k (0) = ψ k , u k (T ) = φ k , where ψ k , φ k are called the Fourier sine coefficients and defined as ψ k = 2 1 0 ψ(x) sin(kπx)dx, φ k = 2 1 0 φ(x) sin(kπx)dx. Solving the above equation, using Lemma 2.7, we obtain u k (t) = C k E α − k 2 π 2 (1 − θ) α t (1−θ)α + f k k 2 π 2 , and using the given initial conditions, we have C k = ψ k − φ k 1 − E α − k 2 π 2 (1 − θ) α T (1−θ)α , f k = k 2 π 2 (ψ k − C k ). Hence, the expressions for u(x, t) and f (x) can be written as, u(x, t) = ∞ k=1 C k E α − k 2 π 2 (1 − θ) α t (1−θ)α sin(kπx) + (ψ k − C k ) sin(kπx) = ψ(x) − ∞ k=1 1 − E α − k 2 π 2 (1 − θ) α t (1−θ)α 1 − E α − k 2 π 2 (1 − θ) α T (1−θ)α (ψ k − φ k ) sin(kπx), and f (x) = ∞ k=1 k 2 π 2 (ψ k − C k ) sin(kπx) = ψ ′′ (x) − ∞ k=1 k 2 π 2 (ψ k − φ k ) 1 − E α − k 2 π 2 (1 − θ) α T (1−θ)α sin(kπx). Appropriate conditions on the given functions ψ(x) and φ(x), see the Theorem 4.1 below, are assumed for establishing the uniform convergence of the series expansions of u(x, t), C t θ ∂ ∂t example, for f (x) we have the following estimate: Assuming that ψ(x) ∈ C 2 [0, 1] and ψ ′′′ (x), φ ′′′ (x) ∈ L 2 (0, 1), then by Weierstrass M-test the series representation of f (x) is uniformly convergent. Also, the series representation of C t θ ∂ ∂t α u(x, t), which is given by |f (x)| ≤ |ψ ′′ (x)| + ∞ k=1 k 2 π 2 (1 − θ) α + k 2 π 2 T (1−θ)α (1 − M)(1 − θ) α + k 2 π 2 T (1−θ)α (|ψ k | + |φ k |) ≤ |ψ ′′ (x)| + M ∞ k=1 1 kπ ψ (3) k + φC t θ ∂ ∂t α u(x, t) = − ∞ k=1 E α − k 2 π 2 (1 − θ) α t (1−θ)α 1 − E α − k 2 π 2 (1 − θ) α T (1−θ)α k 2 π 2 (ψ k − φ k ) sin(kπx), can be estimated as follows: C t θ ∂ ∂t α u(x, t) ≤ M ∞ k=1 k 2 π 2 (1 − θ) α + k 2 π 2 t (1−θ)α (|ψ k | + |φ k |) ≤ M ∞ k=1 1 kπ ψ (1) k + φ (1) k ≤ M ∞ k=1 1 (kπ) 2 + ψ ′ (x) 2 L 2 (0,1) + φ ′ (x) 2 L 2 (0,1) . It is clear that the above series is uniformly convergent. The main result for this section can be summarized in the following theorem: u(x, t) = ψ(x) − ∞ k=1 1 − E α − k 2 π 2 (1 − θ) α t (1−θ)α 1 − E α − k 2 π 2 (1 − θ) α T (1−θ)α (ψ k − φ k ) sin(kπx), f (x) = ψ ′′ (x) − ∞ k=1 k 2 π 2 (ψ k − φ k ) 1 − E α − k 2 π 2 (1 − θ) α T (1−θ)α sin(kπx), x, t) sin(kπx)dx = 0. (x, t), u xx (x, t) and f (x). This can be done in a similar approach as presented earlier. For φ ′′′ (x) cos(kπx) dx. Theorem 4. 1 . 1Assume ψ(x), φ(x) ∈ C 2 [0, 1] such that ψ (i) (0) = ψ (i) (1) = φ (i) (0) = φ (i) and ψ ′′′ (x), φ ′′′ (x) ∈ L 2 (0, 1),then the inverse source problem (27) -(29) has a unique pair of solutions {u(x, t), f (x)} given by x) sin(kπx)dx. Acknowledgements. Authors acknowledge financial support from The Research Council (TRC),Oman. This work is funded by TRC under the research agreement no. ORG/SQU/CBS/13/030. B Al-Saqabi, V Kiryakova, Explicit solutions of fractional integral and differential equations involving Erdélyi-Kober operators. 95B. AL-Saqabi and V. Kiryakova, Explicit solutions of fractional integral and differential equations involving Erdélyi-Kober operators, Applied Mathematics and Computations, 95 (1998), 1-13. Operational calculus of a class of differential operators. I Dimovski, C.R. Acad. Bulg. Sci. 1912I. Dimovski, Operational calculus of a class of differential operators, C.R. Acad. Bulg. Sci., 19 (12) (1966), 1111-1114. Fractional relaxation with time-varying coefficient, Fractional Calculus and Applied Analysis. R Garra, A Giusti, F Mainardi, G Pagnini, 17R. Garra, A. Giusti, F. Mainardi and G. Pagnini, Fractional relaxation with time-varying coefficient, Fractional Calculus and Applied Analysis, 17 (2014), 424-439. Fractional diffusion with time-varying coefficients. R Garra, E Orsingher, F Polito, Journal of Mathematical Physics. 56AIP Conf. Proc.R. Garra, E. Orsingher and F. Polito, Fractional diffusion with time-varying coefficients, Journal of Mathematical Physics, AIP Conf. Proc., 56 (2015) 1-19. Fractional calculus: integral and differential equations of fractional order, Fractals and Fractional Calculus in continuum. R Gorenflo, F Mainardi, Mechanics : A. Carpinteri and F. MainardiSpringer VerlagWien and New YorkR. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equations of frac- tional order, Fractals and Fractional Calculus in continuum Mechanics : A. Carpinteri and F. Mainardi (eds), Springer Verlag, Wien and New York , (1997), 223-276. Generalized Fractional Calculus and Applications. V Kiryakova, Longman-J. Wiley, Harlow-N.YorkV. Kiryakova, Generalized Fractional Calculus and Applications. Longman-J. Wiley, Harlow-N.York, 1994. Fractional Brownian motions, Fractional noises and applications. B B Mandelbrot, J W Van Ness, SIAM Review. 104B. B. Mandelbrot and J.W. Van Ness, Fractional Brownian motions, Fractional noises and applications, SIAM Review, 10 (4) (1968), 422-437. Fractional Powers of a Class of Ordinary Differential Operators. A C Mcbride, Proceedings of the London Mathematical Society. 345A.C. McBride, Fractional Powers of a Class of Ordinary Differential Operators. Proceedings of the London Mathematical Society, 3 (45) (1982), 519-546. On the basis property of systems of sines and cosines. E I Moiseev, Doklady AN SSSR. 2754E. I. Moiseev, On the basis property of systems of sines and cosines, Doklady AN SSSR, 275 (4) (1984) 794-798. Fractional differential equations. I Podlubny, Academic Press IncSan Diego, CAI. Podlubny, Fractional differential equations. Academic Press Inc., San Diego, CA, 1999. A singular integral equation with a generalized Mittag-Leffler function in the kernal. T R Prabhakar, Yokohama Math. J. 19T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernal, Yokohama Math. J., 19 (1971), pp. 7-15.
{'fraction_non_alphanumeric': 0.12623300453212477, 'fraction_numerical': 0.04136674664533902, 'mean_word_length': 3.101877164206306, 'pattern_counts': {'":': 0, '<': 24, '<?xml version=': 0, '>': 8, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 42, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'Direct and inverse source problems of a fractional diffusion equation with regularized Caputo-like counterpart hyper-Bessel operator are considered. Solutions to these problems are constructed based on appropriate eigenfunction expansion and results on existence and uniqueness are established. To solve the resultant equations, a solution to a non-homogeneous fractional differential equation with regularized Caputo-like counterpart hyper-Bessel operator is also presented.', 'arxivid': '1610.05524', 'author': ['Fatma Al-Musalhi \nDepartment of Mathematics\nSultan Qaboos University\nP.O. Box 36Al-KhoudhOman\n', 'Nasser Al-Salti \nDepartment of Mathematics\nSultan Qaboos University\nP.O. Box 36Al-KhoudhOman\n', 'Erkinjon Karimov \nDepartment of Mathematics\nSultan Qaboos University\nP.O. Box 36Al-KhoudhOman\n'], 'authoraffiliation': ['Department of Mathematics\nSultan Qaboos University\nP.O. Box 36Al-KhoudhOman', 'Department of Mathematics\nSultan Qaboos University\nP.O. Box 36Al-KhoudhOman', 'Department of Mathematics\nSultan Qaboos University\nP.O. Box 36Al-KhoudhOman'], 'corpusid': 119140620, 'doi': '10.1515/fca-2018-0013', 'github_urls': [], 'n_tokens_mistral': 10062, 'n_tokens_neox': 8706, 'n_words': 4671, 'pdfsha': 'a4f44f46500b7eaa9073f3b131cd3dc9f239ec97', 'pdfurls': ['https://export.arxiv.org/pdf/1610.05524v2.pdf'], 'title': ['Initial boundary value problems for a fractional differential equation with hyper-Bessel operator', 'Initial boundary value problems for a fractional differential equation with hyper-Bessel operator'], 'venue': []}
arxiv
The distance between a naive cumulative estimator and its least concave majorant May 2018 Hendrik P Lopuhaä DIAM Faculty EEMCS Delft University of Technology Mekelweg 42628 CDDelftThe Netherlands Eni Musta DIAM Faculty EEMCS Delft University of Technology Mekelweg 42628 CDDelftThe Netherlands The distance between a naive cumulative estimator and its least concave majorant May 2018Least concave majorantGrenander-type estimatorLimit distributionCentral limit theorem for L p -distanceBrownian motion with parabolic drift 2010 MSC: 60F0562E20 We consider the process Λ n − Λ n , where Λ n is a cadlag step estimator for the primitive Λ of a nonincreasing function λ on [0, 1], and Λ n is the least concave majorant of Λ n . We extend the results in Lopuhaä (2006, 2008) to the general setting considered in Durot(2007).Under this setting we prove that a suitably scaled version of Λ n − Λ n converges in distribution to the corresponding process for two-sided Brownian motion with parabolic drift and we establish a central limit theorem for the L p -distance between Λ n and Λ n . Introduction Grenander-type estimators are well known methods for estimation of monotone curves. In case of estimating nonincreasing curves, they are constructed by starting with a naive estimator for the primitive of the curve of interest and then take the left-derivative of the least concave majorant (LCM) of the naive estimator. The first example can be found in Grenander (1956) in the context of estimating a nonincreasing density f on [0, ∞) on the basis of an i.i.d. sample from f . The empirical distribution function F n of the sample is taken as a naive estimator for the cumulative distribution function corresponding to f and the Grenander estimator is found by taking the leftderivative f n of the least concave majorant F n . Similar estimators have been developed in other statistical models, e.g., regression (see Brunk (1958)), random censoring (see Huang and Wellner (1995)), or the Cox model (see Lopuhaä and Nane (2013)). Durot (2007) considers Grenander-type estimators in a general setup that incorporates several statistical models. A large part of the literature is devoted to investigating properties of Grenander-type estimators for monotone curves, and somewhat less attention is paid to properties of the difference between the corresponding naive estimator for the primitive of the curve and its LCM. Kiefer and Wolfowitz (1976) show that sup t | F n − F n | = O p ((n −1 log n) 2/3 ). Although the first motivation for this type of result has been asymptotic optimality of shape constrained estimators, it has several important statistical applications. The Kiefer-Wolfowitz result was a key argument in Sen et al. (2010) to prove that the m out of n bootstrap from F n works. Mammen (1991) suggested to use the result to make an asymptotic comparison between a smoothed Grenander-type estimator and an isotonized kernel estimator in the regression context. See also Wang and Woodroofe (2007) for a similar application of their Kiefer-Wolfowitz comparison theorem. An extension to a more general setting was established in Durot and Lopuhaä (2014), which has direct applications in Durot et al. (2013) to prove that a smoothed bootstrap from a Grenander-type estimator works for k-sample tests, and in Groeneboom and Jongbloed (2013) and Lopuhaä and Musta (2017) to extract the pointwise limit behavior of smoothed Grenander-type estimators for a monotone hazard from that of ordinary kernel estimators. To approximate the L p -error of smoothed Grenander-type estimators by that of ordinary kernel estimators, such as in Csörgö and Horváth (1988) for kernel density estimators, a Kiefer-Wolfowitz type result no longer suffices. In that case, results on the L p -distance, between F n and F n are more appropriate, such as the ones in Durot and Tocquet (2003) and Kulikov and Lopuhaä (2008). In this paper, we extend the results in Durot and Tocquet (2003) and Kulikov and Lopuhaä (2008) to the general setting of Durot (2007). Our main result is a central limit theorem for the L p -distance between Λ n and Λ n , where Λ n is a naive estimator for the primitive Λ of a monotone curve λ and Λ n is the LCM of Λ n . As special cases we recover Theorem 5.2 in Durot and Tocquet (2003) and Theorem 2.1 in Kulikov and Lopuhaä (2008). Our approach requires another preliminary result, which might be of interest in itself, i.e., a limit process for a suitably scaled difference between Λ n and Λ n . As special cases we recover Theorem 1 in Wang (1994), Theorem 4.1 in Durot and Tocquet (2003), and Theorem 1.1 in Kulikov and Lopuhaä (2006). Main results We consider the general setting in Durot (2007). Let λ : [0, 1] → R be nonincreasing and assume that we have at hand a cadlag step estimator Λ n of Λ(t) = t 0 λ(u) du, t ∈ [0, 1]. In the sequel we will make use of the following assumptions. (A1) λ is strictly decreasing and twice continuously differentiable on [0, 1] with inf t |λ ′ (t)| > 0. (A2) Let B n be either a Brownian motion or a Brownian bridge. There exists q > 6, C q > 0, L : [0, 1] → R, and versions of M n = Λ n − Λ and B n such that P n 1−1/q sup t∈[0,1] M n (t) − n −1/2 B n • L(t) > x ≤ C q x −q for all x ∈ (0, n]. Moreover, L is increasing and twice differentiable on [0, 1], with sup t |L ′′ (t)| < ∞ and inf t |L ′ (t)| > 0. Note that this setup includes several statistical models, such as monotone density, monotone regression, and the monotone hazard model under random censoring, see Durot (2007)[Section 3]. We consider the distance between Λ n and its least concave majorant Λ n = CM [0,1] Λ n , where CM I maps a function h : R → R into the least concave majorant of h on the interval I ⊂ R. Consider the process A n (t) = n 2/3 Λ n (t) − Λ n (t) , t ∈ [0, 1],(1) and define Z(t) = W (t) − t 2 , ζ(t) = [CM R Z](t) − Z(t),(2) where W denotes a standard two-sided Brownian motion originating from zero. For each t ∈ (0, 1) fixed and t + c 2 (t)sn −1/3 ∈ (0, 1), define ζ nt (s) = c 1 (t)A n t + c 2 (t)sn −1/3 ,(3) where c 1 (t) = |λ ′ (t)| 2L ′ (t) 2 1/3 , c 2 (t) = 4L ′ (t) |λ ′ (t)| 2 1/3 .(4) Our first result is the following theorem, which extends Theorem 1.1 in Kulikov and Lopuhaä (2006). Theorem 1. Suppose that assumptions (A1)-(A2) are satisfied. Let ζ nt and ζ be defined in (3) and (2). Then the process {ζ nt (s) : s ∈ R} converges in distribution to the process {ζ(s) : s ∈ R} in D(R), the space of cadlag function on R. Note that as a particular case ζ nt (0) converges weakly to ζ(0). In this way, we recover Theorem 1 in Wang (1994) and Theorem 4.1 in Durot and Tocquet (2003). The proof of Theorem 1 follows the line of reasoning in Kulikov and Lopuhaä (2006). Let us briefly sketch the argument to prove Theorem 1. Note that A n = D [0,1] [n 2/3 Λ n ] and ζ = D R [Z], where D I h = CM I h − h, for h : R → R. Since D I is a continuous mapping, the main idea is to apply the continuous mapping theorem to properly scaled approximations of the processes Λ n and Z on a suitable chosen fixed interval I. The first step is to determine the weak limit of Λ n , which is given in the following lemma. Lemma 2. Suppose that assumptions (A1)-(A2) are satisfied. Then for t ∈ (0, 1) fixed, the process X nt (s) = n 2/3 Λ n (t + sn −1/3 ) − Λ n (t) − Λ(t + sn −1/3 ) − Λ(t) converges in distribution to the process {W (L ′ (t)s) : s ∈ R}. Since n 2/3 (Λ(t + sn −1/3 ) − Λ(t)) ≈ n 1/3 λ(t)s + λ ′ (t)s 2 /2 and D I is invariant under addition of linear functions, it follows that the process A n can be approximated by a Brownian motion with a parabolic drift. The idea now is to use continuity of D I , for a suitably chosen interval I = [−d, d], to show that D I E nt converges to D I Z t , where E nt (s) = n 2/3 Λ n (t + sn −1/3 ) Z t (s) = W (L ′ (t)s) + λ ′ (t)s 2 /2.(5) In order to relate this to the processes ζ nt and ζ in Theorem 1, note that A n (t + sn −1/3 ) = [D Int E nt ](s), where I nt = [−tn 1/3 , (1 − t)n 1/3 ] , and by Brownian scaling, the process Z(s) has the same distribution as the process c 1 (t)Z t (c 2 (t)s). This means that we must compare the concave majorants of E nt on the intervals I nt and I, as well as the concave majorants of Z t on the interval I and R. Lemma 1.2 in Kulikov and Lopuhaä (2006) shows that, locally, with high probability, both concave majorants of the process Z t coincide on [−d/2, d/2], for large d > 0. A similar result is established for the concave majorants of the process E nt in Lemma 3, which is analogous to Lemma 1.3 in Kulikov and Lopuhaä (2006). As a preparation for Theorem 4, the lemma also contains a similar result for a Brownian motion version of E nt . Let B n be as in assumption (A2) and let ξ n be a N (0, 1) distributed random variable independent of B n , if B n is a Brownian bridge, and ξ n = 0, when B n is a Brownian motion. Define versions W n of a Brownian motion by W n (t) = B n (t) + ξ n t, for t ∈ [0, 1], and define A W n = n 2/3 CM [0,1] Λ W n − Λ W n (6) where Λ W n (t) = Λ(t) + n −1/2 W n (L(t)), with L as in assumption (A2). Furthermore, define E n = √ n(Λ n − Λ), Λ E n = Λ n , A E n = A n . The superscripts E and W refer to the empirical and Brownian motion version. For d > 0, let I nt (d) = [0, 1] ∩ [t − dn −1/3 , t + dn −1/3 ] and, for J = E, W , define the event N J nt (d) = [CM [0,1] Λ J n ](s) = [CM Int(d) Λ J n ](s), for all s ∈ I nt (d/2) .(7) Let I nt = I nt (log n) and N J nt = N J nt (log n). Lemma 3. Assume that assumptions (A1)-(A2) hold. For d > 0, let N J nt (d) be the event defined in (7). There exists C > 0, independent of n, t, d, such that P (N W nt (d)) c = O e −Cd 3 P (N E nt (d)) c = O n 1−q/3 d −2q + e −Cd 3 , where q is from assumption (A2). The proof of Theorem 1 now follows the same line of reasoning as that of Theorem 1.1 in Kulikov and Lopuhaä (2006), see Section 3 for more details. The next step is to deal with the L p norm. Our main result is the following. Theorem 4. Suppose that assumptions (A1)-(A2) are satisfied and let A n and ζ be defined by (1) and (2), respectively. Let µ be a measure on the Borel sets of R, such that (A3) dµ(t) = w(t) dt, where w(t) ≥ 0 is differentiable with bounded derivative on [0, 1]. Then, for all 1 ≤ p < min(q, 2q − 7), (with q as in assumption (A2)), n 1/6 1 0 A n (t) p dµ(t) − m d − → N (0, σ 2 ), where m = E [ζ(0) p ] 1 0 2 p/3 L ′ (t) 2p/3 |λ ′ (t)| p/3 dµ(t) and σ 2 = 1 0 2 (2p+5)/3 L ′ (t) (4p+1)/3 |λ ′ (t)| (2p+2)/3 w 2 (t) dt ∞ 0 cov (ζ(0) p , ζ(s) p ) ds. For the special cases that λ is a probability density or a regression function, we recover Theorem 2.1 in Kulikov and Lopuhaä (2008) and Theorem 5.2 inDurot and Tocquet (2003), respectively. In order to prove Theorem 4 we first need some preliminary results. We aim at approximating the L p -norm of A n by that of the Brownian motion version A W n and then finding the asymptotic distribution for the latter one. To this end, we first need to relate the moments of A n to those of A W n . We start by showing that, for J = E, W, a rescaled version of Λ J n can be approximated by the same process Y nt plus a linear term. This result corresponds to Lemma 4.1 in Kulikov and Lopuhaä (2008). Lemma 5. Suppose that assumptions (A1)-(A2) are satisfied. Then, for t ∈ (0, 1) fixed, for J = E, W, and s ∈ [−tn 1/3 , (1 − t)n 1/3 ], it holds n 2/3 Λ J n (t + n −1/3 s) = Y nt (s) + L J nt (s) + R J nt (s), where L J nt (s) is linear in s and Y nt (s) = n 1/6 W n (L(t + n −1/3 s)) − W n (L(t)) + 1 2 λ ′ (t)s 2 . More- over, for all p ≥ 1, E sup |s|≤log n R W nt (s) p = O n −p/3 (log n) 3p , uniformly in t ∈ (0, 1). If, in addition 1 ≤ p < q (with q as in assumption (A2)), then E sup |s|≤log n R E nt (s) p = O n −p/3+p/q uniformly in t ∈ (0, 1). Since the map D I is invariant under addition of linear terms, Lemma 5 allows us to approximate the moments of A J n (t) = n 2/3 D [0,1] Λ J n by those of [D Hnt Y nt ](0) for some interval H nt , as in Lemma 4.2 in Kulikov and Lopuhaä (2008). Lemma 6. Suppose that assumptions (A1)-(A2) are satisfied. and let Y nt be the process defined in Lemma 5. Define H nt = [−n 1/3 t, n 1/3 (1 − t)] ∩ [− log n, log n]. Then for all p ≥ 1, it holds E A W n (t) p = E [[D Hnt Y nt ] (0) p ] + o n −1/6 , uniformly for t ∈ (0, 1). If, in addition 1 ≤ p < min(q, 2q − 7), with q from condition (A2), then also E A E n (t) p = E [[D Hnt Y nt ] (0) p ] + o n −1/6 , uniformly for t ∈ (0, 1). The process Y nt has the same distribution as Y nt = W n 1/3 L t + n −1/3 s − L(t) + 1 2 λ ′ (t)s 2 ,(8) which is close to the process Z t in (5) by continuity of Brownian motion. Lemma 4.3 in Kulikov and Lopuhaä (2008) is then used to show that the concave majorants at zero are sufficiently close. Note that, with by Brownian scaling, the process c 1 (t)Z t (c 2 (t)s) has the same distribution as the process Z(s). As a consequence of Lemma 6 the moments of A J n (t) can be related to those of the process ζ. This formulated in the next lemma, which corresponds to Lemma 4.4 in Kulikov and Lopuhaä (2008). Lemma 7. Suppose that assumptions (A1)-(A2) are satisfied. Then, for all p ≥ 1, E A W n (t) p = 2L ′ (t) 2 |λ ′ (t)| p/3 E [ζ(0) p ] + o n −1/6 uniformly in t ∈ (n −1/3 log n, 1 − n −1/3 log n) and E A W n (t) p ≤ 2L ′ (t) 2 |λ ′ (t)| p/3 E [ζ(0) p ] + o n −1/6 uniformly in t ∈ (0, 1). If, in addition 1 ≤ p < min(q, 2q − 7), where q is from assumption (A2), then the same (in)equalities hold for A E N (t). In Lemmas 6 and 7 the moments of A E n and A W n are approximated by the moments of the same process. This suggests that the difference between them is of smaller order than n −1/6 . Indeed, on the events N J nt , where A J n = n 2/3 D Int Λ J n , we make use of Lemma 6 and the fact that D I is invariant under addition of linear functions to obtain that sup t∈(0,1) n 2p/3 [D Int Λ E n ](t) − n 2p/3 [D Int Λ W n ](t) ≤ sup t∈(0,1) sup |s|≤log n |R E nt (s)| + |R W nt (s)| , where the processes R J nt converge to zero sufficiently fast. On the other hand, on (N J nt ) c we just need the boundedness of the moments of A J n , which follows by Lemma 7 and the fact that the probability of these events is very small (Lemma 3). Lemma 8. Suppose that assumptions (A1)-(A2) are satisfied. Then, for 1 ≤ p < min(q, 2q − 7), with q from assumption (A2), it holds E A E n (t) p − A W n (t) p = o n −1/6 E A E n (t) − A W n (t) p = o n −1/6 uniformly in t ∈ (0, 1). From Lemma 7 it follows that n 1/6 |m − 1 0 E A W n (t) p dt| → 0, where m is the asymptotic mean in Theorem 4. Moreover, Lemma 8 implies that n 1/6 1 0 A E n (t) p dt − 1 0 A W n (t) p dt ≤ n 1/6 1 0 A E n (t) p − A W n (t) p dt → 0. As a consequence, in order to prove Theorem 4, it suffices to prove asymptotic normality of its Brownian motion version T W n = n 1/6 1 0 A W n (t) p − E A W n (t) p dµ(t). The proof of this is completely similar to that of Theorem 2.1 in Kulikov and Lopuhaä (2008). First, by using Theorem 1 for a Brownian version of ζ nt and the mixing property of A W n (this can be obtained in the same way as Lemma 4.6 in Kulikov and Lopuhaä (2008)), we derive the asymptotic variance of T W n in the following lemma. Lemma 9. Suppose that assumptions (A1)-(A3) are satisfied. Then, for every p ≥ 1, Var n 1/6 1 0 A W n (t) p dµ(t) → 1 0 2 (2p+5)/3 L ′ (t) (4p+1)/3 |λ ′ (t)| (2p+2)/3 w 2 (t) dt ∞ 0 cov (ζ(0) p , ζ(s) p ) ds. The last step is proving the asymptotic normality of T W n . This is done by a big-blocks smallblocks argument, where the contribution of the small blocks to the asymptotic distribution is negligible, while the mixing property of A W n allows us to approximate the sum over the big blocks by a sum of independent random variables which satisfy the assumptions of Lindeberg central limit theorem. Proofs Proof of Lemma 2. The proof is completely similar to that of Lemma 1.1 in Kulikov and Lopuhaä (2006), but this time E n = √ n(Λ n − Λ) and sup t∈[0,1] |E n (t) − B n • L(t)| = O p (n −1/2+1/q ), according to (A2). Similar to the proof of Lemma 1.1 in Kulikov and Lopuhaä (2006), this means that X nt (s) = n 1/6 W n (L(t + sn −1/3 )) − W n (L(t)) + O p (n −1/3+1/q ) d = W (L ′ (t)s) + R n (s), where sup s∈I |R n (s)| → 0 in probability for compact I ⊂ R. From here on the proof is the same as that of Lemma 1.1 in Kulikov and Lopuhaä (2006 Λ W n (t) − at and V W n (a) = n 1/3 L(U W n (a)) − L(g(a)) , where g denotes the inverse of λ. As in the proof of Lemma 1.3 in Kulikov and Lopuhaä (2006)[see (2. 2)], we get P (N W nt (d)) c ≤ P λ W n (t − n −1/3 d) = λ W n (t − n −1/3 d/2) + P λ W n (t + n −1/3 d) = λ W n (t + n −1/3 d/2) .(9) Then, with s = t − dn −1/3 /2, x = d/2, and ǫ n = inf t∈[0,1] |λ ′ (t)|dn −1/3 /8, it holds (see (2.3) in Kulikov and Lopuhaä (2006)), P λ W n (t − n −1/3 d) = λ W n (t − n −1/3 d/2) ≤ P λ W n (s + n −1/3 x) − λ(s + n −1/3 x) > ǫ n + P λ W n (s) − λ(s) < −ǫ n .(10) Moreover, using the switching relation λ W n (t) ≤ a ⇔ U W n (a) ≤ t, we rewrite this probability as P U W n (λ(s + n −1/3 x) + ǫ n ) > s + n −1/3 x = P V W n (λ(s + n −1/3 x) + ǫ n ) > n 1/3 L(s + n −1/3 x) − L(g(λ(s + n −1/3 x) + ǫ n )) = P V W n (λ(s + n −1/3 x) + ǫ n ) > inf t∈[0,1] |λ ′ (t)| inf t∈[0,1] L ′ (t)d 8 sup t∈[0,1] |λ ′ (t)| . It suffices to show that there exists positive constants C 1 , C 2 such that P V W n (a) > x ≤ C 1 e −C2x 3(11) because then it follows that P V W n (λ(s + n −1/3 x) + ǫ n ) > inf t∈[0,1] |λ ′ (t)| inf t∈[0,1] L ′ (t)d 8 sup t∈[0,1] |λ ′ (t)| ≤C 1 e −C2d 3 . Similarly we can also bound the second probabilities in (9) and (10). Then the statement of the lemma follows immediately. Now we prove (11). First write V W n (a) = n 1/3 L argmax t∈[0,1] W (L(t)) + √ n(Λ(t) − at) − L(g(a)) = n 1/3 argmax s∈[L(0),L(1)] W (s) + √ n(Λ L −1 (s)) − aL −1 (s) − L(g(a) ) . Using properties of the argmax functional we obtain that the right hand side is equal to the argmax of the process n 1/6 W n −1/3 s + L(g(a)) − W (L(g(a))) + n 2/3 Λ L −1 n −1/3 s + L(g(a)) − Λ(g(a)) − aL −1 n −1/3 s + L(g(a)) + ag(a) for s ∈ I n (a) = [n 1/3 (L(0)−L(g(a))), n 1/3 (L(1)−L(g(a)))]. By Brownian motion scaling, V W n (a) is equal in distribution to argmax t∈In(a) {W (t)−D a,n (t)}, where W is a standard two-sided Brownian motion originating from zero and D a,n (s) = −n 2/3 Λ L −1 n −1/3 s + L(g(a)) − Λ(g(a)) − aL −1 n −1/3 s + L(g(a)) + ag(a) . By Taylor's formula and the assumptions on λ and L, one can show that there exist a constant c 0 > 0, independent of n, a and t, such that D a,n (t) ≥ c 0 t 2 . Then (11) follows from Theorem 4 in Durot (2002), which proves the first statement. To continue with the second statement, let λ n be the left derivative of Λ n and define the inverse process U n (a) = argmax t∈[0,1] {Λ n (t) − at} , and V n (a) = n 1/3 (U n (a) − g(a)) , where g denotes the inverse of λ. As in (9), we get P (N E nt (d)) c ≤ P λ n (t − n −1/3 d) = λ n (t − n −1/3 d/2) + P λ n (t + n −1/3 d) = λ n (t + n −1/3 d/2) .(12) where similar to (10), P λ n (t − n −1/3 d) = λ n (t − n −1/3 d/2) ≤ P λ n (s + n −1/3 x) − λ(s + n −1/3 x) > ǫ n + P λ n (s) − λ(s) < −ǫ n . Then using the switching relation λ n (t) ≤ a ⇔ U n (a) ≤ t, we rewrite the first probability in (13) as P V n (λ(s + n −1/3 x) + ǫ n ) > inf t∈[0,1] |λ ′ (t)|d 8 sup t∈[0,1] |λ ′ (t)| . According to Lemma 6.4 in Durot et al. (2012), there exists positive constants C 1 , C 2 > 0, independent of n, a, and x, such that P (V n (a) > x) ≤ C 1 n 1−q/3 x 2q + 2e −C2x 3 . It follows that P V n (λ(s + n −1/3 x) + ǫ n ) > inf t∈[0,1] |λ ′ (t)|d 8 sup t∈[0,1] |λ ′ (t)| ≤C 1 n 1−q/3 d 2q + 2e −C2d 3 . Similarly we can also bound the second probabilities in (12) and (13). Then the statement of the lemma follows immediately. Proof Theorem 1. The proof is similar to the proof of Theorem 1.1 inKulikov and Lopuhaä (2006). We briefly sketch the main steps. Arguing as in the proof of Theorem 1.1 inKulikov and Lopuhaä s ∈ I nt = [−tn 1/3 , (1 − t)n 1/3 ], where E nt is defined in (5). To prove convergence in distribution, we show that for any bounded continuous function g : D(K) → R, |E[g(D Int E nt )] − E[g(D R Z t )]| → 0.(14) To this end, we choose d > 0 sufficiently large, such that K ⊂ [−d/2, d/2] ⊂ [−d, d] = I and take n sufficiently large so that I ⊂ I nt . Then, similar to inequality (2.7) in Kulikov and Lopuhaä (2006), the triangular inequality yields |E[g(D Int E nt )] − E[D R Z t ]| ≤ |E[g(D Int E nt )] − E[D I E nt ]| + |E[g(D I E nt )] − E[D I Z t ]| + |E[g(D I Z t )] − E[D R Z t ]| .(15) In the same way as in Kulikov and Lopuhaä (2006), the three terms on the right hand side are shown to go to zero. For the last term on the right hand side of (15), the argument is exactly the same and makes use of their Lemma 1.2. The first term on the right hand side of (15) is bounded similar to their inequality (2.9) and then uses Lemma 3. For the second term on the right hand side of (15), note that from Lemma 2, it follows that Z nt (s) = n 2/3 Λ n (t + sn −1/3 ) − Λ n (t) − Λ(t + sn −1/3 ) − Λ(t) + 1 2 λ ′ (t)s 2 , converges in distribution to Z t . Therefore, because of the continuity of the mapping D I , we get now have E nt (s) = Z nt (s) + n 2/3 Λ n (t) + λ(t)sn 1/3 + R nt (s), where |E[h(D I Z nt )] − E[h(D I Z t )]| → 0,R nt (s) = n 2/3 Λ(t + sn −1/3 ) − Λ(t) − λ(t)sn −1/3 − 1 2 λ ′ (t)s 2 n −2/3 . Similar to the argument leading up to (2.11) in Kulikov and Lopuhaä (2006), from the continuity of D I , its invariance under addition of linear functions, and continuity of λ ′ , it follows that (14) and finishes the proof. |E[g(D I Z nt )] − E[g(D I E nt )]| → 0. This establishes Proof of Lemma 5. By a Taylor expansion, together with (6), we can write n 2/3 Λ W n (t + n −1/3 s) = Y nt (s) + L W nt (s) + R W nt (s), where L W nt (s) = n 2/3 Λ(t) + n 1/6 W n (L(t)) + n 1/3 λ(t)s and R W nt (s) = n 2/3 Λ(t + n −1/3 s) − Λ(t) − n −1/3 λ(t)s − 1 2 n −2/3 λ ′ (t)s 2 = 1 6 n −1/3 λ ′′ (θ 1 )s 3 for some |θ 1 − t| ≤ n −1/3 |s|. Then, from the assumptions (A1)-(A2), it follows that sup |s|≤log n R W nt (s) p = O n −p/3 (log n) 3p , uniformly in t ∈ (0, 1). Similarly, we also obtain n 2/3 Λ E n (t + n −1/3 s) = n 2/3 Λ W n (t + n −1/3 s) + n 1/6 E n (t + n −1/3 s) − B n (L(t + n −1/3 s)) − n 1/6 ζ n L(t) + L ′ (t)n −1/3 s − n 1/6 ζ n L(t + n −1/3 s) − L(t) − L ′ (t)n −1/3 s = Y nt (s) + L E nt (s) + R E nt (s), where L E nt (s) = L W nt (s) − n 1/6 ζ n L(t) − n −1/6 ζ n L ′ (t)s and R E nt (s) = R W nt (s) + n 1/6 E n (t + n −1/3 s) − B n (L(t + n −1/3 s)) − 1 2 n −1/2 ζ n L ′′ (θ 2 )s 2 , for some |θ 2 − t| ≤ n −1/3 |s|. Let S n = sup s∈[0,1] |E n (s) − B n (L(s))|. From assumption (A2) we have P(S n > n −1/2+1/q x) ≤ C q x −q and it follows that E [S p n ] = ∞ 0 P (S p n ≥ x) dx = p ∞ 0 y p−1 P (S n ≥ y) dy = pn −p/2+p/q ∞ 0 x p−1 P S n ≥ n −1/2+1/q x dx ≤ pn −p/2+p/q 1 0 x p−1 dx + C q ∞ 1 x p−1−q dx = O n −p/2+p/q ,(16) if p < q. Consequently E sup |s|≤log n R E nt (s) p = O n −p/3+p/q . Proof of Lemma 6. Note that we can write A J n (t)1 N J nt = n 2/3 [D Int Λ J n ](t)1 N J nt . We have E A J n (t) p = n 2p/3 E [D Int Λ J n ](t) p + E A J n (t) p − n 2p/3 [D Int Λ J n ](t) p 1 (N J nt ) c . To bound the second term on the right hand side, first note that A J n (t) p − n 2p/3 [D Int Λ J n ](t) p ≤ 2A J n (t) p ,(17) because the LCM on [0, 1] always lies above the LCM over I nt . Since Λ is concave, we have that CM [0,1] Λ E n − Λ E n ≤ CM [0,1] Λ E n − [CM [0,1] Λ + Λ E n − Λ + CM [0,1] Λ − Λ = CM [0,1] Λ E n − [CM [0,1] Λ + Λ E n − Λ ≤ 2 sup s∈[0,1] Λ E n (s) − Λ(s) , which means that 0 ≤ A E n (t) p ≤ 2 p n 2p/3 sup s∈[0,1] Λ E n (s) − Λ(s) p . Furthermore, 0 ≤ A W n (t) p ≤ 2 p n 2p/3 Λ(1) + n −1/2 sup s∈[0,1] |W n (s)| p . In contrast to Kulikov and Lopuhaä (2008) it is more convenient to treat both cases separately. For the case J = E, with (17), we find that E A E n (t) p − n 2p/3 [D Int Λ E n ](t) p 1 (N E nt ) c ≤ 2 p+1 n 2p/3 E sup s∈[0,1] Λ E n (s) − Λ(s) p 1 (N J nt ) c , where sup s∈[0,1] Λ E n (s) − Λ(s) p ≤ 2 p sup s∈[0,1] Λ E n (s) − Λ(s) − n −1/2 W n (L(s)) p + n −p/2 sup s∈[0,1] |W n (L(s))| p . For the first term on the right hand side we get with Hölder's inequality n 2p/3 E sup s∈[0,1] Λ E n (s) − Λ(s) − n −1/2 W n (L(s)) p 1 (N E nt ) c ≤ n 2p/3 E sup s∈[0,1] Λ E n (s) − Λ(s) − n −1/2 W n (L(s)) pℓ 1/ℓ P (N E nt ) c 1/ℓ ′ = n 2p/3 O(n −p+p/q )O n 1−q/3 (log n) −2q + e −C(log n) 3 1/ℓ ′ , for any ℓ, ℓ ′ > 1 such that 1/ℓ + 1/ℓ ′ = 1, according to (16) and Lemma 3. When q > 6, then the right hand side is of the order o(n −1/6 ). For the second term, with Hölder's inequality n 2p/3 E sup s∈[0,1] |W n (L(s))| p 1 (N E nt ) c ≤ n 2p/3 E sup s∈[0,L(1)] |W n (s)| pℓ 1/ℓ P (N E nt ) c 1/ℓ ′ for any ℓ, ℓ ′ > 1 such that 1/ℓ + 1/ℓ ′ = 1. Since all moments of sup s∈[0,L(1)] |W n (s)| are finite, it follows from Lemma 3 that the right hand side is of the order n 2p/3 E sup s∈[0,1] |W n (L(s))| p 1 (N E nt ) c ≤ n 2p/3 O n 1−q/3 (log n) −2q + e −C(log n) 3 1/ℓ ′ . Hence, because q > 6 and p < 2q − 7, it follows that |A E n (t) p − n 2p/3 [D Int Λ E n ](t)| = o(n −1/6 ). Next, consider the case J = W . Then with (17) and Cauchy-Schwarz, we find Then as in Lemma 4.2 in Kulikov and Lopuhaä (2008), one can show that This finishes the proof. where A W n is defined in (6) and c 1 (t) and c 2 (t) are defined in (4). According to Theorem 1, ζ nt converges in distribution to ζ, as defined in (2). As in the proof of Lemma 4.7 in Kulikov and Lopuhaä (2008), Lemma 7 yields that, for s, t, and k fixed, the sequence ζ W nt (s) k is uniformly integrable, so that the moments of (ζ W nt (0) k , ζ W nt (s) k ) converge to the corresponding moments of (ζ(0) k , ζ(s) k ). Furthermore, the process {A W n (t) : t ∈ (0, )} is strong mixing, i.e., for d > 0, sup |P (A ∩ B) − P (A)P (B)| = α n (d) = 48e −Cnd 3(20) where C > 0 only depends on λ and L from (A2), and where the supremum is taken over all sets A ∈ σ{A W n (s) : 0 ≤ s ≤ t} and B ∈ σ{A W n (s) : t + d ≤ s < 1}. This can be obtained by arguing completely the same as in the proof of Lemma 4.6 in Kulikov and Lopuhaä (2008). The rest of the proof is the same as that of Lemma 4.7 in Kulikov and Lopuhaä (2008). Proof of Theorem 4. The proof is completely similar to the proof of Theorem 2.1 in Kulikov and Lopuhaä (2008), by using the method of big-blocks small-blocks and the exponential decreasing mixing function α n from (20). ( 2006 ) 2006, it suffices to show that for any compact K ⊂ R, the process {A n (t + sn −1/3 ) : s ∈ K} converges in distribution to the process {[D R Z t ](s) : s ∈ K} on D(K), the space of cadlag functions on K, where Z t is defined in (5). By definition A n (t + sn −1/3 ) = [D Int E nt ](s), for for any h : D(I) → R bounded and continuous. Moreover, we E A W n (t) p − n 2p/3 [D Int Λ W n ](t) p 1Again using that all moments of sup s∈[0,L(1)] |W n (s)| are finite, according to Lemma 3, the right hand side is of the ordern 2p/3 O(e −C(log n) 3 ) = o(n −1/6 ). It follows that for J = E, W , E A J n (t) p = n 2p/3 E [D Int Λ J n ](t) p + o n −1/6 . Moreover, Lemma 5 implies that n 2/3 [D Int Λ J n ](t) = [D Hnt Y nt ](0) + ∆ nt , where ∆ nt = [D Hnt (Y nt + R J nt )](0) − [D Hnt Y nt ](0). From Lemma 5, we have E|∆ nt | p ≤ 2 p E sup |s|≤log n R J nt (s) p = O n −p/3+p/q . E A J n (t) p = E [[D Hnt Y nt (0) p ] + ǫ nt + o n −1/6 = E [[D Hnt Y nt (0) p ] + O n −1/3+1/q (log n) 2p−2 + o n −1/6 = E [[D Hnt Y nt (0) p ] + o n −1/6 . ).Proof Lemma 3. Let λ W n be the left derivative of Λ W n = CM [0,1] Λ W n . Define the inverse process U W n (a) = argmax t∈[0,1] Proof of Lemma 7. The proof is exactly the same as the one for Lemma 4.4 inKulikov and Lopuhaä (2008). Define J nt = [n 1/3 (L(a nt ) − L(t)) /L ′ (t), n 1/3 (L(b nt ) − L(t)) /L ′ (t)], where a nt = max(0, t− n −1/3 log n) and b nt = min(1, t + n −1/3 log n). Furthermore, here we takeAs in the proof of Lemma 4.4 inKulikov and Lopuhaä (2008), it follows that 1 − α n ≤ φ nt (s)/s ≤ 1 + α n , for s ∈ H nt , the interval from Lemma 6, and α n = C 1 n −1/3 log n, with C 1 > 0 only depending on L ′ . Let Z t be the process in (5). ThenwhereỸ nt is defined in(8). Lemma 4.3 inKulikov and Lopuhaä (2008), then allows us to approximate the moments of [D HntỸnt ](0) by the moments of [D Jnt Z t ](0). Completely similar to the proof of Lemma 4.4 inKulikov and Lopuhaä (2008), the result now follows from Lemma 6 and Brownian scaling.Proof of Lemma 8. Let I nt and N J nt be as in Lemma 6 and defineWe bound the two terms on the right hand side, following the same line of reasoning as in Lemma 4.5 inKulikov and Lopuhaä (2008). Using that according to Lemma 3,the second term on the right hand side of (19) is of the order O P(K c nt ) 1/2 = o n −1/6 , because q > 6. On the other hand, the first term on the right hand side of (19) can be bounded bywhere the right hand side is of the order O n −1/3+1/q = o n −1/6 , according to Lemmas 5 and 7.In the same way, we havewhich is of the order O n −p/3+p/q = o n −1/6 , according to Lemma 5.Proof of Lemma 9. The proof is completely similar to the proof of Lemma 4.7 inKulikov and Lopuhaä (2008). For t ∈ (0, 1) fixed, and t + c 2 (t)sn −1/3 ∈ (0, 1), let ζ nt (s) = c 1 (t)A W n (t + c 2 (t)sn −1/3 ), On the estimation of parameters restricted by inequalities. H D Brunk, Ann. Math. Statist. 29Brunk, H. D., 1958. On the estimation of parameters restricted by inequalities. Ann. Math. Statist. 29, 437-454. Central limit theorems for L p -norms of density estimators. M Csörgö, L Horváth, Probab. Theory Related Fields. 802Csörgö, M., Horváth, L., 1988. Central limit theorems for L p -norms of density estimators. Probab. Theory Related Fields 80 (2), 269-291. Sharp asymptotics for isotonic regression. C Durot, Probab. Theory Related Fields. 1222Durot, C., 2002. Sharp asymptotics for isotonic regression. Probab. Theory Related Fields 122 (2), 222-240. On the L p -error of monotonicity constrained estimators. C Durot, Ann. Statist. 353Durot, C., 2007. On the L p -error of monotonicity constrained estimators. Ann. Statist. 35 (3), 1080-1104. Testing equality of functions under monotonicity constraints. C Durot, P Groeneboom, H P Lopuhaä, J. Nonparametr. Stat. 254Durot, C., Groeneboom, P., Lopuhaä, H. P., 2013. Testing equality of functions under monotonicity constraints. J. Nonparametr. Stat. 25 (4), 939-970. The limit distribution of the L ∞ -error of Grenander-type estimators. C Durot, V N Kulikov, H P Lopuhaä, Ann. Statist. 403Durot, C., Kulikov, V. N., Lopuhaä, H. P., 2012. The limit distribution of the L ∞ -error of Grenander-type estimators. Ann. Statist. 40 (3), 1578-1608. A Kiefer-Wolfowitz type of result in a general setting, with an application to smooth monotone estimation. C Durot, H P Lopuhaä, Electron. J. Stat. 82Durot, C., Lopuhaä, H. P., 2014. A Kiefer-Wolfowitz type of result in a general setting, with an application to smooth monotone estimation. Electron. J. Stat. 8 (2), 2479-2513. On the distance between the empirical process and its concave majorant in a monotone regression framework. C Durot, A.-S Tocquet, Ann. Inst. H. Poincaré Probab. Statist. 392Durot, C., Tocquet, A.-S., 2003. On the distance between the empirical process and its concave majorant in a monotone regression framework. Ann. Inst. H. Poincaré Probab. Statist. 39 (2), 217-240. On the theory of mortality measurement. U Grenander, II. Skand. Aktuarietidskr. 39Grenander, U., 1956. On the theory of mortality measurement. II. Skand. Aktuarietidskr. 39, 125-153 (1957). Smooth and non-smooth estimates of a monotone hazard. P Groeneboom, G Jongbloed, From probability to statistics and back: high-dimensional models and processes. Beachwood, OH9Groeneboom, P., Jongbloed, G., 2013. Smooth and non-smooth estimates of a monotone hazard. In: From probability to statistics and back: high-dimensional models and processes. Vol. 9 of Inst. Math. Stat. (IMS) Collect. Inst. Math. Statist., Beachwood, OH, pp. 174-196. Estimation of a monotone density or monotone hazard under random censoring. J Huang, J A Wellner, Scand. J. Statist. 221Huang, J., Wellner, J. A., 1995. Estimation of a monotone density or monotone hazard under random censoring. Scand. J. Statist. 22 (1), 3-33. Asymptotically minimax estimation of concave and convex distribution functions. J Kiefer, J Wolfowitz, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete. 341Kiefer, J., Wolfowitz, J., 1976. Asymptotically minimax estimation of concave and convex distri- bution functions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 34 (1), 73-85. The limit process of the difference between the empirical distribution function and its concave majorant. V N Kulikov, H P Lopuhaä, Statist. Probab. Lett. 7616Kulikov, V. N., Lopuhaä, H. P., 2006. The limit process of the difference between the empirical distribution function and its concave majorant. Statist. Probab. Lett. 76 (16), 1781-1786. Distribution of global measures of deviation between the empirical distribution function and its concave majorant. V N Kulikov, H P Lopuhaä, J. Theoret. Probab. 212Kulikov, V. N., Lopuhaä, H. P., 2008. Distribution of global measures of deviation between the empirical distribution function and its concave majorant. J. Theoret. Probab. 21 (2), 356-377. Smooth estimation of a monotone hazard and a monotone density under random censoring. H P Lopuhaä, E Musta, Statist. Neerlandica. 71Lopuhaä, H. P., Musta, E., 2017. Smooth estimation of a monotone hazard and a monotone density under random censoring. Statist. Neerlandica 71, 58-82. Shape constrained non-parametric estimators of the baseline distribution in Cox proportional hazards model. Scand. H P Lopuhaä, G F Nane, J. Stat. 403Lopuhaä, H. P., Nane, G. F., 2013. Shape constrained non-parametric estimators of the baseline distribution in Cox proportional hazards model. Scand. J. Stat. 40 (3), 619-646. Estimating a smooth monotone regression function. E Mammen, Ann. Statist. 192Mammen, E., 1991. Estimating a smooth monotone regression function. Ann. Statist. 19 (2), 724-740. Inconsistency of bootstrap: the Grenander estimator. B Sen, M Banerjee, M Woodroofe, Ann. Statist. 384Sen, B., Banerjee, M., Woodroofe, M., 2010. Inconsistency of bootstrap: the Grenander estimator. Ann. Statist. 38 (4), 1953-1977. A Kiefer-Wolfowitz comparison theorem for Wicksell's problem. X Wang, M Woodroofe, Ann. Statist. 354Wang, X., Woodroofe, M., 2007. A Kiefer-Wolfowitz comparison theorem for Wicksell's problem. Ann. Statist. 35 (4), 1559-1575. The limit distribution of the concave majorant of an empirical distribution function. Y Wang, Statist. Probab. Lett. 201Wang, Y., 1994. The limit distribution of the concave majorant of an empirical distribution func- tion. Statist. Probab. Lett. 20 (1), 81-84.
{'fraction_non_alphanumeric': 0.11421924811166317, 'fraction_numerical': 0.0492834372038255, 'mean_word_length': 3.04460448367987, 'pattern_counts': {'":': 0, '<': 11, '<?xml version=': 0, '>': 31, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 12, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'We consider the process Λ n − Λ n , where Λ n is a cadlag step estimator for the primitive Λ of a nonincreasing function λ on [0, 1], and Λ n is the least concave majorant of Λ n . We extend the results in Lopuhaä (2006, 2008) to the general setting considered in Durot(2007).Under this setting we prove that a suitably scaled version of Λ n − Λ n converges in distribution to the corresponding process for two-sided Brownian motion with parabolic drift and we establish a central limit theorem for the L p -distance between Λ n and Λ n .', 'arxivid': '1706.05173', 'author': ['Hendrik P Lopuhaä \nDIAM\nFaculty EEMCS\nDelft University of Technology\nMekelweg 42628 CDDelftThe Netherlands\n', 'Eni Musta \nDIAM\nFaculty EEMCS\nDelft University of Technology\nMekelweg 42628 CDDelftThe Netherlands\n'], 'authoraffiliation': ['DIAM\nFaculty EEMCS\nDelft University of Technology\nMekelweg 42628 CDDelftThe Netherlands', 'DIAM\nFaculty EEMCS\nDelft University of Technology\nMekelweg 42628 CDDelftThe Netherlands'], 'corpusid': 88522908, 'doi': '10.1016/j.spl.2018.04.001', 'github_urls': [], 'n_tokens_mistral': 14458, 'n_tokens_neox': 12826, 'n_words': 7201, 'pdfsha': '2ef588e886d32bd2901017f6018b9435f8848262', 'pdfurls': ['https://arxiv.org/pdf/1706.05173v2.pdf'], 'title': ['The distance between a naive cumulative estimator and its least concave majorant', 'The distance between a naive cumulative estimator and its least concave majorant'], 'venue': []}
arxiv
Diffusive Radiation in One-dimensional Langmuir Turbulence 21 May 2007 G D Fleishman A.F. Ioffe Physico-Technical Institute Russian Academy of Sciences 194021St. PetersburgRussia New Jersey Institute of Technology 07102NewarkNJ I N Toptygin State Polytechnical University 195251St. PetersburgRussia Diffusive Radiation in One-dimensional Langmuir Turbulence 21 May 2007 We calculate spectra of radiation produced by a relativistic particle in the presence of onedimensional Langmuir turbulence which might be generated by a streaming instability in the plasma, in particular, in the shock front or at the shock-shock interactions. The shape of the radiation spectra is shown to depend sensitively on the angle between the particle velocity and electric field direction. The radiation spectrum in the case of exactly transverse particle motion is degenerate and similar to that of spatially uniform Langmuir oscillations. In case of oblique propagation, the spectrum is more complex, it consists of a number of power-law regions and may contain a distinct high-frequency spectral peak. The emission process considered is relevant to various laboratory plasma settings and for astrophysical objects as gamma-ray bursts and collimated jets.PACS numbers: 05.10.Gg,52.27.Ny,98.62.Nx,98.70.Rz Various kinds of two-stream instability including longitudinal, transverse, and oblique modes [1, 2, 3, 4] are common for many astrophysical objects [5, 6, 7] and laboratory plasma settings[8,9]. These instabilities are efficient to produce a high level of magnetic and/or electric turbulence in the plasma. We note that the electric turbulence is of primary importance for many applications, in particular, it is capable of charged particle acceleration[7,10,11]in contrast to purely magnetic turbulence, which is only efficient for angular scattering of the charged particle while cannot change the particle energy directly. Remarkably, recent theoretical and numerical studies[12,13,14]demonstrate that the energy density going to the electric (Langmuir) turbulence can be very large, exceeding, for example, the energy density of the initial regular magnetic field in the plasma.Relativistic electrons propagating in the plasma with developed Langmuir turbulence will random walk due to random Lorenz forces produced by these turbulent electric fields. Apparently, during this random diffusive motion the electron will generate electromagnetic emission. We will refer to this radiative process as Diffusive Radiation in Langmuir waves (DRL) to emphasize the key role of the diffusive random walk of the particle in the stochastic electric fields.Curiously, the theory of DRL has not been developed in sufficient detail yet, although a few particular issues related to this radiative process (called also Electrostatic Bremsstrahlung) have been considered[15,16,17,18,19,20]. However, the detailed treatment of the DRL spectral shape in various regimes is currently unavailable. This letter attempts to remedy the situation by calculating the DRL spectrum within the perturbation theory and determines the region of applicability of the perturbative treatment.Electromagnetic emission produced by a charged particle can be calculated within the perturbation theory when the particle moves almost rectilinearly with almost constant velocity. Apparently, the non-zero acceleration of the particle in the external field should be taken into account to obtain a non-zero radiation intensity. This perturbative treatment is widely used because of its simplicity. Frequently, one calculates first the particle acceleration w(t) due to a given field along the rectilinear trajectory and then uses this expression obtained for w(t) to find the radiation spectrum. In case of a random external field, however, when w(t) is also a random function of time t it is more convenient to express the radiation intensity via spatial and temporal spectrum of the external electric and/or magnetic field directly[21]. Accordingly, the spectral and angular distribution of the emission produced by a single particle in a plasma with random field has the form: We calculate spectra of radiation produced by a relativistic particle in the presence of onedimensional Langmuir turbulence which might be generated by a streaming instability in the plasma, in particular, in the shock front or at the shock-shock interactions. The shape of the radiation spectra is shown to depend sensitively on the angle between the particle velocity and electric field direction. The radiation spectrum in the case of exactly transverse particle motion is degenerate and similar to that of spatially uniform Langmuir oscillations. In case of oblique propagation, the spectrum is more complex, it consists of a number of power-law regions and may contain a distinct high-frequency spectral peak. The emission process considered is relevant to various laboratory plasma settings and for astrophysical objects as gamma-ray bursts and collimated jets. Various kinds of two-stream instability including longitudinal, transverse, and oblique modes [1,2,3,4] are common for many astrophysical objects [5,6,7] and laboratory plasma settings [8,9]. These instabilities are efficient to produce a high level of magnetic and/or electric turbulence in the plasma. We note that the electric turbulence is of primary importance for many applications, in particular, it is capable of charged particle acceleration [7,10,11] in contrast to purely magnetic turbulence, which is only efficient for angular scattering of the charged particle while cannot change the particle energy directly. Remarkably, recent theoretical and numerical studies [12,13,14] demonstrate that the energy density going to the electric (Langmuir) turbulence can be very large, exceeding, for example, the energy density of the initial regular magnetic field in the plasma. Relativistic electrons propagating in the plasma with developed Langmuir turbulence will random walk due to random Lorenz forces produced by these turbulent electric fields. Apparently, during this random diffusive motion the electron will generate electromagnetic emission. We will refer to this radiative process as Diffusive Radiation in Langmuir waves (DRL) to emphasize the key role of the diffusive random walk of the particle in the stochastic electric fields. Curiously, the theory of DRL has not been developed in sufficient detail yet, although a few particular issues related to this radiative process (called also Electrostatic Bremsstrahlung) have been considered [15,16,17,18,19,20]. However, the detailed treatment of the DRL spectral shape in various regimes is currently unavailable. This letter attempts to remedy the situation by calculating the DRL spectrum within the perturbation theory and determines the region of applicability of the perturbative treatment. Electromagnetic emission produced by a charged particle can be calculated within the perturbation theory when the particle moves almost rectilinearly with almost constant velocity. Apparently, the non-zero acceleration of the particle in the external field should be taken into account to obtain a non-zero radiation intensity. This perturbative treatment is widely used because of its simplicity. Frequently, one calculates first the particle acceleration w(t) due to a given field along the rectilinear trajectory and then uses this expression obtained for w(t) to find the radiation spectrum. In case of a random external field, however, when w(t) is also a random function of time t it is more convenient to express the radiation intensity via spatial and temporal spectrum of the external electric and/or magnetic field directly [21]. Accordingly, the spectral and angular distribution of the emission produced by a single particle in a plasma with random field has the form: W ⊥ Ω,ω = (2π) 3 Q 2 M 2 c 3 γ 2 V ω ω ′ 2 1 − ω ω ′ γ 2 * + ω 2 2ω ′2 γ 4 * × dq 0 dqδ(ω ′ − q 0 + qv) | F q0,q⊥ | 2 ,(1) where γ * = γ −2 + ω 2 pe ω 2 −1/2 , Q, M , and γ are the charge, mass, and Lorenz-factor of the emitting particle, c is the speed of light, ω pe is the plasma frequency, V is the volume of the emission source, F q0,q⊥ is the temporal and spatial Fourier component of the Lorenz force transverse to the emitting particle velocity, ω ′ = ω 2 γ −2 + θ 2 + ω 2 pe ω 2 ,(2) θ is the emission angle relative to the particle velocity vector v, ω is the frequency of the emitted wave. Contribution W ⊥ Ω,ω (marked with the superscript ⊥) is provided by a component of the particle acceleration transverse to the particle velocity. In case of the electric E (in contrast to magnetic) field, there is also a component of the acceleration along the particle velocity. The corresponding contribution has the form W Ω,ω = 2(2π) 3 Q 4 M 2 c 3 γ 6 V ω ω ′ 3 1 − ω 2ω ′ γ 2 * × dq 0 dqδ(ω ′ − q 0 + qv) | E q0,q | 2 ,(3) which is typically small by a factor γ −2 compared with the transverse contribution. Nevertheless, there exist special cases when the transverse contribution is zero or very small and the parallel contribution comes to play. In particular, for the considered here one-dimensional turbulent electric field the parallel contribution will dominate for a particle moving along the field direction. Let us start with the case when the particle moves at a large angle to the Langmuir turbulence direction ϑ ≫ γ −1 , so the standard transverse contribution dominates. Following the derivation given in [21], but with the Lorenz force F = QE specified by electric E in place of magnetic B field, it is easy to find | F q0,q⊥ | 2 = Q 2 | E q0,q⊥ | 2 = T V (2π) 4 Q 2 δ αβ − v α v β v 2 K αβ (q 0 , q),(4) where K αβ (q 0 , q) = C αβ K(q 0 , q), T is the total time of emission; C αβ describes the longitudinal nature of the Langmuir waves, i.e., C αβ = q α q β /q 2 , while K(q 0 , q) is the temporal and spatial spectrum of the Langmuir turbulence. The developed approach allows for arbitrary anisotropy of the turbulence, although we have to specify the shape of the turbulence spectrum to promote further the calculation of the DRL spectrum. Analytical and numerical studies of the two-stream instabilities suggest that the Langmuir turbulence produced is frequently highly anisotropic [1,3,4,12,13,14], which is confirmed also by available in situ observations, e.g., performed in the Earth magnetosphere [22]. Although this anisotropy can be reduced at later stages of the nonlinear turbulence evolution due to randomization of the wave vector directions, here we assume that the Langmuir turbulence is highly anisotropic, namely, one-dimensional. As an example, we can suppose that all the wave vectors are directed along the shock normal n, therefore, C αβ = n α n β , while the spectrum K(q 0 , q) can be approximated by a power-law over q above certain critical value k 0 : K(q 0 , q) = a ν k ν−1 0 E 2 L (k 2 0 + q 2 ) ν/2 δ(q ⊥ )δ(q 0 − ω pe ).(5) Here, the presence of the second δ-function is related to the assumption that the electric turbulence is composed of Langmuir waves all of which oscillate in time with the same frequency ω pe ; the normalization constant a ν is set up by the condition K(q 0 , q)dq 0 dq = E 2 L , where E 2 L is the mean square of the electric field in the Langmuir turbulence. Now, substituting (4) with (5) into general expression (1), taking the integrals over dq 0 , dq , and dq ⊥ with the use of three available δ-functions and dividing by the total (infinite) time of emission T we find I ⊥ Ω,ω = a ν k ν−1 0 E 2 L Q 4 sin 2 ϑ 2πM 2 c 4 γ 2 | cos ϑ| ω ω ′ 2 × 1 − ω ω ′ γ 2 * + ω 2 2ω ′2 γ 4 * ω ′ − ω pe nv 2 + k 2 0 −ν/2 ,(6) where ϑ is the angle between the particle velocity v and vector n. This expression diverges formally when cos ϑ → 0. Accordingly, for a particle moving transversely to Langmuir turbulence direction we need to recalculate the integrals in Eq. (1) taking into account that δ(ω ′ − q 0 + qv) → δ(ω ′ − q 0 ) for qv = 0, which yields I ⊥ Ω,ω = Q 4 E 2 L 2πM 2 c 3 γ 2 ω ω ′ 2 1 − ω ω ′ γ 2 * + ω 2 2ω ′2 γ 4 * δ(ω ′ −ω pe ) (7) in full agreement with the results of [17] obtained for spatially uniform Langmuir oscillations. This is not a random coincidence. Indeed, since the 1D Langmuir waves experience spatial variations along only one direction (of vector n), thus, the particle moving transversely to this direction "feels" spatially uniform field pattern like that considered in [17]. There is another special geometry when intensity (6) is insufficient to describe the DRL spectrum: it is the case of particle motion along vector n. Indeed, the "parallel" contribution becomes important for ϑ < ∼ γ −1 . Substituting (5) into (3) and taking the integrals similarly to derivation of (6) we find: I Ω,ω = a ν k ν−1 0 E 2 L Q 4 | cos ϑ| πM 2 c 4 γ 6 ω ω ′ 3 × 1 − ω 2ω ′ γ 2 * ω ′ − ω pe nv 2 + k 2 0 −ν/2 .(8) Apparently, spectra (6) and (8) look rather differently compared with the spectrum (7) produced by a particle moving transversely to vector n. Given, in particular, that the frequency ω ′ is directly linked with the emission angle θ, the δ-function δ(ω ′ − ω pe ) permits emitting a single frequency only in each direction. By comparison, no δ-function enters (6) and (8), thus, a continuum spectrum rather than distinct frequencies is emitted along any direction. Clearly, there remains a distinct contribution to the emission intensity when ω ′ ≈ ω pe . However, the range of the parameter space where this resonant condition holds is relatively narrow, so the "non-resonant" contribution from the remaining part of the parameter space where ω ′ = ω pe can easily dominate the resonant contribution. To see this explicitly, consider the radiation intensity into the full solid angle by integration of (6) over dΩ = sin θdθdϕ ≈ 2πd(ω ′ /ω) that yields I ⊥ ω = a ν k ν−1 0 E 2 L Q 4 sin 2 ϑ M 2 c 4 γ 2 | cos ϑ| ∞ 1/2γ 2 * d ω ′ ω ω ω ′ 2 × 1 − ω ω ′ γ 2 * + ω 2 2ω ′2 γ 4 * ω ′ − ω pe nv 2 + k 2 0 −ν/2 .(9) Let us analyze essential properties of the DRL spectrum on the basis of the asymptotic evaluation. At low frequencies ω ≪ ω pe γ 2 , we can discard ω ′ in (9) everywhere in the braces except narrow region of parameters when ω ′ ≈ ω pe . This means that for ω ≪ ω pe γ 2 the integral is composed of two contributions. The first of them, a non-resonant one, arises from integration over the region, where ω ′ ≪ ω pe . Here, the emission is beamed within the characteristic emission angle of ϑ ∼ γ −1 along the particle velocity. The integral converges rapidly, and so it may be taken along the infinite region, which produces a flat radiation spectrum, I ω ∝ ω 0 , or I ω ∝ ω 2 at lower frequencies, ω < ω pe γ. However, as far as ω ′ approaches ω pe , a resonant contribution comes into play. Now, in a narrow vicinity of ω pe , we can adopt ω ′ − ω pe nv 2 + k 2 0 −ν/2 ∝ δ(ω ′ − ω pe ),(10) which results in a single-wave-like contribution, I ω ∝ ω 1 . The full spectrum at ω < ω pe γ 2 , therefore, is just a sum of these two contributions. At high frequencies, ω ≫ ω pe γ 2 , the term ω ′ dominates in the braces, so other terms can be discarded. Thus, a power-law tail, I ω ∝ ω −ν arises in this spectral range. Although this integral cannot be taken analytically in a general case, it is easy to obtain corresponding spectra numerically. Figure 1 displays a numerically integrated example of the DRL spectra for various angles between v and n for highly relativistic particle with γ = 10 6 . The black/solid curve shows the DRL spectrum arising as the particle moves strictly perpendicular to vector n, which coincide with the spectrum arising in the case of spatially uniform Langmuir oscillations [17]. The spectrum consists of a rising region I ω ∝ ω 1 at ω < 2ω pe γ 2 and drops abruptly to zero at ω > 2ω pe γ 2 . Remarkably, there are prominent differences between this spectrum and those generated for oblique particle propagation even though the spectra can be similar to each other in the immediate vicinity of the spectral peak. To find the applicability region of the perturbation theory applied above, we should estimate the characteristic deflection angle of the emitting electron on the emission coherence length l c = 2cγ 2 * /ω, where the elementary emission pattern is formed [21]. Consider a simple source model consisting of uncorrelated cells with the size l 0 = 2πc/ω 0 , each of which contains coherent Langmuir oscillations with the plasma frequency ω pe . Inside each cell the electron velocity can change by the angle θ 0 ∼ ω st /(ω pe γ) if ω 0 < ∼ ω pe , where ω st = Q E 2 L 1/2 /M c (in the other case, ω 0 > ω pe , the results of [21] apply). Then, after traversing N = l c /l 0 cells, the mean square of the deflection angle is θ 2 c = θ 2 0 N ∼ ω 2 st ω 0 /(ωω 2 pe ). The perturbation theory is only applicable if this diffusive deflection angle is smaller than the relativistic beaming angle, γ −1 , i.e., it is always valid at sufficiently high frequencies ω > ω * ≡ ω 2 st ω 0 γ 2 /ω 2 pe . Note, that the bounding frequency ω * increases with ω 0 , while Diffusive Synchrotron Radiation (DSR) in random magnetic field displays the opposite trend. The perturbation theory will be applicable to the entire DRL spectrum if the condition θ 2 c < γ −2 holds for the frequency ω pe γ [21], where the coherence length of the emission has a maximum. This happens for the particles whose Lorenz-factors obey the inequality γ < ω 3 pe /(ω 2 st ω 0 ).(11) Let's compare the DRL spectra calculated numerically ( Figure 1) for the case of a broad turbulence distribution over spatial scales with the DSR spectra in case of stochastic magnetic fields [21]. Apparently, there is a remarkable difference between these two emission mechanisms, especially in the case of the long-wave turbulence, ω 0 | cos ϑ| ≡ k 0 c| cos ϑ| ≪ ω pe . First, we note that the perturbation theory of DRL has a broader applicability region, in particular, it applies for higher energy electrons than the perturbative version of the DSR theory [21] since the criterion (11) is softer than the corresponding criterion in [21]. This happens because in the presence of the Langmuir turbulence the rapid temporal oscillations of the electric field direction substantially compensate angular deflections of the particle, so the average trajectory is much more similar to a straight line than for the case of the random magnetic field with the same ω 0 and ω st . Then, a distinct spectral peak at ω = 2ω pe γ 2 is formed with the linear decrease of the spectrum with frequency, which is not present in case of the DSR. At lower frequencies, however, this falling part of the spectrum gives way to a flat spectrum, which is entirely missing within the one-wave approach [17] and absent for exactly transverse particle motion. Position of the corresponding turning point depends on the ω 0 cos ϑ/ω pe ratio in such a way 1: DRL spectra produced by a particle with γ = 10 6 in a plasma with developed 1d Langmuir turbulence for various particle propagation directions: cos ϑ = 0, solid/black curve; cos ϑ = 10 −3 , dashed/red curve; cos ϑ = 0.5, dashdotted/green curve; cos ϑ = 1, dotted/blue curve. Parameters are given in the Figure. The "parallel" contribution (blue/dotted curve) is very small (10 −12 ) for the highly relativistic particle, although it becomes competing for moderately relativistic particles. that for ω 0 | cos ϑ| > ∼ ω pe the flat spectral region entirely dominates the range from ω pe γ to ω pe γ 2 . It is worth emphasizing that the deviations of the DRL spectrum from the single-wave spectrum (read, from the transverse case, cos ϑ = 0) is prominent even for oblique propagation angles only slightly different from π/2, e.g., cos ϑ = 10 −3 as in Figure 1. Therefore, the presence of a broad turbulence spectrum considered here in detail results in important qualitative change of the emission mechanism, which cannot generally be reduced to a simplified treatment relying on the single-wave approximation with some rms value of the Langmuir electric field. Modern computer simulations of shock wave interactions, especially in the relativistic case, suggest that the energy density of the excited Langmuir turbulence can be very large [13], e.g., far in excess the energy of the initial regular magnetic field. In particular, at the shock wave front the electric field can be as strong as the corresponding wave-breaking limit, i.e., ω st ∼ ω pe [14]. In this case the random walk of relativistic electrons in the stochastic electric field can give rise to powerful contribution in the nonthermal emission of an astrophysical object, entirely dominating full radiation spectrum or some broad part of it. Although any detailed application of the considered emission process is beyond the scope of this letter, we mention that the DRL is a promising mechanism for the gamma-ray bursts and extragalactic jets. In particular, some of the prompt gamma-ray burst emission displays rather hard low-energy spectra with the photon spectral index α up to 0. The DRL spectral asymptote I ω ∝ ω 1 , which appears just below the spectral peak at 2ω pe γ 2 , fits well to those spectra. Remarkably, the flat lowerfrequency asymptote, I ω ∝ ω 0 , can account for the phenomenon of the X-ray excess [23,24] and prompt optical flashes accompanying some GRBs. In addition, this mechanism along with the DSR in random magnetic fields [25] can be relevant to the UV-Xray flattenings observed in some extragalactic jets. For example [26], X-ray observations of the jet in 3C 273 look inconsistent with the standard synchrotron of DSR models. Remarkably, the entire UV-to-X-ray spectrum of 3C 273 might be produced by DRL, which can be much flatter than usual DSR in the range ω Be γ 2 ≪ ω ≪ ω pe γ 2 . PACS numbers: 03.50.-z,05.10.Gg,41.60.-m,52.25.-b,52.27.Ny,52.35.-g,98.62.Nx,98.70.Rz FIG. 1: DRL spectra produced by a particle with γ = 10 6 in a plasma with developed 1d Langmuir turbulence for various particle propagation directions: cos ϑ = 0, solid/black curve; cos ϑ = 10 −3 , dashed/red curve; cos ϑ = 0.5, dashdotted/green curve; cos ϑ = 1, dotted/blue curve. Parameters are given in the Figure. The "parallel" contribution (blue/dotted curve) is very small (10 −12 ) for the highly relativistic particle, although it becomes competing for moderately relativistic particles. AcknowledgmentsThis work was supported in part by the RFBR grants 06-02-16295, 06-02-16859, and 07-02-00245. We have made use of NASA's Astrophysics Data System Abstract Service. . O Buneman, Physical Review Letters. 1O. Buneman, Physical Review Letters 1, 8 (1958). . E S Weibel, Physical Review Letters. 283E. S. Weibel, Physical Review Letters 2, 83 (1959). . Y B Fainberg, V D Shapiro, V I Shevchenko, Sov. Phys. JETP. 30528Y. B. Fainberg, V. D. Shapiro, and V. I. Shevchenko, Sov. Phys. JETP 30, 528 (1970). . A Bret, M E Dieckmann, C Deutsch, Physics of Plasmas. 132109A. Bret, M. E. Dieckmann, and C. Deutsch, Physics of Plasmas 13, 2109 (2006). . Y A Gallant, J Arons, Astrophys. J. 435230Y. A. Gallant and J. Arons, Astrophys. J. 435, 230 (1994). . R Fender, T Belloni, astro- ph/0406483ARA&A. 42317R. Fender and T. Belloni, ARA&A 42, 317 (2004), astro- ph/0406483. . K.-I Nishikawa, P Hardee, G Richardson, R Preece, H Sol, G J Fishman, astro-ph/0409702Astrophys. J. 622K.-I. Nishikawa, P. Hardee, G. Richardson, R. Preece, H. Sol, and G. J. Fishman, Astrophys. J. 622, 927 (2005), astro-ph/0409702. . M Tabak, J Hammer, M E Glinsky, W L Kruer, S C Wilks, J Woodworth, E M Campbell, M D Perry, R J Mason, Physics of Plasmas. 11626M. Tabak, J. Hammer, M. E. Glinsky, W. L. Kruer, S. C. Wilks, J. Woodworth, E. M. Campbell, M. D. Perry, and R. J. Mason, Physics of Plasmas 1, 1626 (1994). . D V Romanov, V Y Bychenkov, W Rozmus, C E Capjack, R Fedosejevs, Physical Review Letters. 93215004D. V. Romanov, V. Y. Bychenkov, W. Rozmus, C. E. Capjack, and R. Fedosejevs, Physical Review Letters 93, 215004 (2004). . A M Bykov, Y A Uvarov, JETP Letters. 57644A. M. Bykov and Y. A. Uvarov, JETP Letters 57, 644 (1993). . J T Frederiksen, C B Hededal, T Haugbølle, A Nordlund, astro-ph/0308104ApJ. 608J. T. Frederiksen, C. B. Hededal, T. Haugbølle, and A. Nordlund, ApJ 608, L13 (2004), astro-ph/0308104. . A Bret, M.-C Firpo, C Deutsch, physics/0502097Physical Review Letters. 94A. Bret, M.-C. Firpo, and C. Deutsch, Physical Review Letters 94, 115002 (2005), physics/0502097. . M E Dieckmann, Physical Review Letters. 94155001M. E. Dieckmann, Physical Review Letters 94, 155001 (2005). L O Silva, Relativistic Jets: The Common Physics of AGN, Microquasars, and Gamma-Ray Bursts. P. A. Hughes and J. N. Bregman856AIP Conf. Proc.L. O. Silva, in AIP Conf. Proc. 856: Relativistic Jets: The Common Physics of AGN, Microquasars, and Gamma-Ray Bursts., edited by P. A. Hughes and J. N. Bregman (2006), pp. 109-128. . A K Gailitis, V N Tsytovich, Soviet Phys. -JETP. 191165A. K. Gailitis and V. N. Tsytovich, Soviet Phys. -JETP 19, 1165 (1964). . S A Colgate, Astrophys. J. 150163S. A. Colgate, Astrophys. J. 150, 163 (1967). . V N Tsytovich, A S Chikhachev, Soviet Astronomy. 13385V. N. Tsytovich and A. S. Chikhachev, Soviet Astronomy 13, 385 (1969). . D B Melrose, Astrophys. Space. Sci. 10197D. B. Melrose, Astrophys. Space. Sci. 10, 197 (1971). S A Kaplan, V N Tsytovich, Plasma astrophysics (International Series of Monographs in Natural Philosophy. OxfordPergamon PressS. A. Kaplan and V. N. Tsytovich, Plasma astrophysics (International Series of Monographs in Natural Philoso- phy, Oxford: Pergamon Press, 1973, 1973). . R Schlickeiser, A&A. 410397R. Schlickeiser, A&A 410, 397 (2003). . G D Fleishman, Astrophys. J. 638348G. D. Fleishman, Astrophys. J. 638, 348 (2006). . A Mangeney, C Lacombe, M Maksimovic, A A Samsonov, N Cornilleau-Wehrlin, C C Harvey, J.-M , A. Mangeney, C. Lacombe, M. Maksimovic, A. A. Samsonov, N. Cornilleau-Wehrlin, C. C. Harvey, J.-M. . P Bosqued, Trávníček, Annales Geophysicae. 243507Bosqued, and P. Trávníček, Annales Geophysicae 24, 3507 (2006). . R D Preece, M S Briggs, G N Pendleton, W S Paciesas, J L Matteson, D L Band, R T Skelton, C A Meegan, Astrophys. J. 473310R. D. Preece, M. S. Briggs, G. N. Pendleton, W. S. Pa- ciesas, J. L. Matteson, D. L. Band, R. T. Skelton, and C. A. Meegan, Astrophys. J. 473, 310 (1996). . T Sakamoto, D Q Lamb, N Kawai, A Yoshida, C Graziani, Astrophys. J. 629311T. Sakamoto, D. Q. Lamb, N. Kawai, A. Yoshida, C. Graziani, and et al., Astrophys. J. 629, 311 (2005). . G D Fleishman, MNRAS. 36511G. D. Fleishman, MNRAS 365, L11 (2006). . S Jester, D E Harris, H L Marshall, K Meisenheimer, astro- ph/0605529Astrophys. J. 648S. Jester, D. E. Harris, H. L. Marshall, and K. Meisen- heimer, Astrophys. J. 648, 900 (2006), astro- ph/0605529.
{'fraction_non_alphanumeric': 0.05679445107398568, 'fraction_numerical': 0.03565035799522673, 'mean_word_length': 4.068418068418069, 'pattern_counts': {'":': 0, '<': 7, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 21, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'We calculate spectra of radiation produced by a relativistic particle in the presence of onedimensional Langmuir turbulence which might be generated by a streaming instability in the plasma, in particular, in the shock front or at the shock-shock interactions. The shape of the radiation spectra is shown to depend sensitively on the angle between the particle velocity and electric field direction. The radiation spectrum in the case of exactly transverse particle motion is degenerate and similar to that of spatially uniform Langmuir oscillations. In case of oblique propagation, the spectrum is more complex, it consists of a number of power-law regions and may contain a distinct high-frequency spectral peak. The emission process considered is relevant to various laboratory plasma settings and for astrophysical objects as gamma-ray bursts and collimated jets.PACS numbers: 05.10.Gg,52.27.Ny,98.62.Nx,98.70.Rz Various kinds of two-stream instability including longitudinal, transverse, and oblique modes [1, 2, 3, 4] are common for many astrophysical objects [5, 6, 7] and laboratory plasma settings[8,9]. These instabilities are efficient to produce a high level of magnetic and/or electric turbulence in the plasma. We note that the electric turbulence is of primary importance for many applications, in particular, it is capable of charged particle acceleration[7,10,11]in contrast to purely magnetic turbulence, which is only efficient for angular scattering of the charged particle while cannot change the particle energy directly. Remarkably, recent theoretical and numerical studies[12,13,14]demonstrate that the energy density going to the electric (Langmuir) turbulence can be very large, exceeding, for example, the energy density of the initial regular magnetic field in the plasma.Relativistic electrons propagating in the plasma with developed Langmuir turbulence will random walk due to random Lorenz forces produced by these turbulent electric fields. Apparently, during this random diffusive motion the electron will generate electromagnetic emission. We will refer to this radiative process as Diffusive Radiation in Langmuir waves (DRL) to emphasize the key role of the diffusive random walk of the particle in the stochastic electric fields.Curiously, the theory of DRL has not been developed in sufficient detail yet, although a few particular issues related to this radiative process (called also Electrostatic Bremsstrahlung) have been considered[15,16,17,18,19,20]. However, the detailed treatment of the DRL spectral shape in various regimes is currently unavailable. This letter attempts to remedy the situation by calculating the DRL spectrum within the perturbation theory and determines the region of applicability of the perturbative treatment.Electromagnetic emission produced by a charged particle can be calculated within the perturbation theory when the particle moves almost rectilinearly with almost constant velocity. Apparently, the non-zero acceleration of the particle in the external field should be taken into account to obtain a non-zero radiation intensity. This perturbative treatment is widely used because of its simplicity. Frequently, one calculates first the particle acceleration w(t) due to a given field along the rectilinear trajectory and then uses this expression obtained for w(t) to find the radiation spectrum. In case of a random external field, however, when w(t) is also a random function of time t it is more convenient to express the radiation intensity via spatial and temporal spectrum of the external electric and/or magnetic field directly[21]. Accordingly, the spectral and angular distribution of the emission produced by a single particle in a plasma with random field has the form:', 'arxivid': '0705.3032', 'author': ['G D Fleishman \nA.F. Ioffe Physico-Technical Institute\nRussian Academy of Sciences\n194021St. PetersburgRussia\n\nNew Jersey Institute of Technology\n07102NewarkNJ\n', 'I N Toptygin \nState Polytechnical University\n195251St. PetersburgRussia\n'], 'authoraffiliation': ['A.F. Ioffe Physico-Technical Institute\nRussian Academy of Sciences\n194021St. PetersburgRussia', 'New Jersey Institute of Technology\n07102NewarkNJ', 'State Polytechnical University\n195251St. PetersburgRussia'], 'corpusid': 25523432, 'doi': '10.1103/physreve.76.017401', 'github_urls': [], 'n_tokens_mistral': 8513, 'n_tokens_neox': 7062, 'n_words': 4528, 'pdfsha': '36d7c1a5e17b9127fa6ebb67adf1f7654479f741', 'pdfurls': ['https://arxiv.org/pdf/0705.3032v1.pdf'], 'title': ['Diffusive Radiation in One-dimensional Langmuir Turbulence', 'Diffusive Radiation in One-dimensional Langmuir Turbulence'], 'venue': []}
arxiv
Magnetic structure of Cd-doped CeCoIn 5 27 Mar 2007 M Nicklas Max Planck Institute for Chemical Physics of Solids Nöthnitzer Str. 4001187DresdenGermany O Stockert Max Planck Institute for Chemical Physics of Solids Nöthnitzer Str. 4001187DresdenGermany Tuson Park Los Alamos National Laboratory 87545Los AlamosNMUSA K Habicht Hahn-Meitner-Institut Glienicker Str. 10014109BerlinGermany K Kiefer Hahn-Meitner-Institut Glienicker Str. 10014109BerlinGermany L D Pham University of California 95616DavisCAUSA J D Thompson Los Alamos National Laboratory 87545Los AlamosNMUSA Z Fisk University of California 92697IrvineCAUSA F Steglich Max Planck Institute for Chemical Physics of Solids Nöthnitzer Str. 4001187DresdenGermany Magnetic structure of Cd-doped CeCoIn 5 27 Mar 2007(Dated: November 2, 2018)numbers: 7470Tx7525+z7127+a7530Mb The heavy fermion superconductor CeCoIn5 is believed to be close to a magnetic instability, but no static magnetic order has been found. Cadmium doping on the In-site shifts the balance between superconductivity and antiferromagnetism to the latter with an extended concentration range where both types of order coexist at low temperatures. We investigated the magnetic structure of nominally 10% Cd-doped CeCoIn5, being antiferromagnetically ordered below TN ≈ 3 K and superconducting below Tc ≈ 1.3 K, by elastic neutron scattering. Magnetic intensity was observed only at the ordering wave vector QAF = ( 1 2 , 1 2 , 1 2 ) commensurate with the crystal lattice. Upon entering the superconducting state the magnetic intensity seems to change only little. The commensurate magnetic ordering in CeCo(In1−xCdx)5 is in contrast to the incommensurate antiferromagnetic ordering observed in the closely related compound CeRhIn5. Our results give new insights in the interplay between superconductivity and magnetism in the family of CeT In5 (T = Co, Rh, and Ir) based compounds. The heavy fermion superconductor CeCoIn5 is believed to be close to a magnetic instability, but no static magnetic order has been found. Cadmium doping on the In-site shifts the balance between superconductivity and antiferromagnetism to the latter with an extended concentration range where both types of order coexist at low temperatures. We investigated the magnetic structure of nominally 10% Cd-doped CeCoIn5, being antiferromagnetically ordered below TN ≈ 3 K and superconducting below Tc ≈ 1.3 K, by elastic neutron scattering. Magnetic intensity was observed only at the ordering wave vector QAF = ( 1 2 , 1 2 , 1 2 ) commensurate with the crystal lattice. Upon entering the superconducting state the magnetic intensity seems to change only little. The commensurate magnetic ordering in CeCo(In1−xCdx)5 is in contrast to the incommensurate antiferromagnetic ordering observed in the closely related compound CeRhIn5. Our results give new insights in the interplay between superconductivity and magnetism in the family of CeT In5 (T = Co, Rh, and Ir) based compounds. In conventional superconductors only small amounts of magnetic impurities destroy the superconducting state, while in the heavy-fermion systems, like CeCu 2 Si 2 , 1 the presence of a dense lattice of magnetic rare-earth atoms is needed to generate unconventional superconductivity (SC). The discovery of SC and antiferromagnetism in the family of CeT In 5 (T = Co, Rh, or Ir) compounds, forming in the tetragonal HoCoGa 5 structure, with CeCoIn 5 2 and CeIrIn 5 3 displaying a superconducting ground state and CeRhIn 5 4 being antiferromagnetically ordered, offers an ideal opportunity to study the peculiar interplay of these two ground states. Though no sign of static magnetic order has been found in CeCoIn 5 , pronounced non-Fermi-liquid (NFL) behavior in the normal state in thermodynamic and transport properties is observed at low temperatures. 2,5,6,7 Commonly, NFL behavior occurs in the close proximity to a quantum critical point (QCP). The search for a magnetic phase in CeCoIn 5 , which could give rise to such a QCP by a continuous suppression of the transition temperature via, e.g., applying external pressure or chemical doping has not been successful until recently. Studies of Cd doping on the In-site in CeCoIn 5 revealed that only a few percent of Cd doping indeed lead to the development of antiferromagnetic (AF) order. 8 Also, reports on the existence of field-induced magnetism in the Abrikosov vortex state in CeCoIn 5 underline its proximity to magnetism. 9,10 Cd-doping continuously suppresses the superconducting transition temperature of CeCoIn 5 (T c = 2.3 K), as shown in the phase diagram depicted in Fig. 1. For a nominal Cd-concentration in excess of 7.5%, AF order has been observed, coexisting with SC at low temperatures. 8 With further increasing Cd-concentration the Néel temperature, T N , increases monotonically and no SC is found above 12.5% Cd anymore. In this report we present neutron scattering data on a CeCo(In 1−x Cd x ) 5 sample with x = 0.1, where x represents the nominal concentration, situated in the concentration range where SC and antiferromagnetism coexist at low temperatures. Single crystals of CeCo(In 0.9 Cd 0.1 ) 5 were grown using a standard In-flux technique with a nominal concentra- tion of 10% Cd in the indium flux. X-ray diffraction confirmed that the samples crystallize in the tetragonal HoCoGa 5 type of crystal structure with lattice parameters a = 4.6122(4)Å and c = 7.5483(9)Å. Microprobe analysis revealed a uniform distribution of the Cd throughout the sample and an actual concentration of only about one percent Cd in the sample, i.e. ∼ 10% of the nominal concentration in the flux. For an easier comparison with literature, we will refer in this paper, too, to the nominal concentration as used in Ref. 8 . The magnetic order was investigated by elastic neutron scattering. The experiments were performed on the cold tripleaxis spectrometer V2 at the BER-II reactor of the Hahn-Meitner-Institut in Berlin, using a wavelength of the incoming neutrons of λ = 2.73Å, corresponding to a neutron energy E = 11 meV. Pyrolytic graphite, PG(002), was used as monochromator and analyzer. The horizontal collimation before the monochromator was given by the 58 Ni guide and was 60' before the sample, before the analyzer and in front of the detector. A tunable PG-filter in the scattered beam reduced the contamination of second-order neutrons. The platelet-like sample of CeCo(In 0.9 Cd 0.1 ) 5 with a mass m ≈ 10 mg and dimensions 2 × 2 × 0.3 mm 3 , 0.3 mm being the thickness along the tetragonal c-axis, was mounted on a copper pin attached to the mixing chamber of a dilution refrigerator. Data were recorded at temperatures between T = 60 mK and 3.2 K along principal and high symmetry directions in the (h h ℓ ) scattering plane. Analyzing the scattered neutrons, i.e. performing elastic scattering, considerably improves the signal-to-background ratio in comparison to (standard) diffraction. This holds especially true in our case since the sample and thus the signal was quite small. In addition to these neutron scattering studies, we conducted heat capacity and electrical resisitivity measurements in a Physical Properties Measurement System (Quantum Design) in the temperature range 350 mK ≤ T ≤ 10 K. In order to correlate the results of the microscopic and the macroscopic studies, all experiments were performed on the same singlecrystalline sample. The specific heat data taken on this sample (see Fig. 2) show two anomalies, one at T N = 3.02 K corresponding to the transition to the antiferromagnetically ordered state and the second one at T c = 1.27 K indicating the phase transition to the superconducting state, in good agreement with literature. 8 Despite this T c value, zero resistance is already observed below 2 K. Similar deviations between thermodynamic and electrical transport results are also known for CeIrIn 5 3 as well as for Ir-rich CeRh 1−x Ir x In 5 . 12 The specific heat exhibits a mean-field like jump ∆C = 1.46 J/mol K at T N . Below the AF transition only 30% of the magnetic entropy (R ln 2) is released suggesting a substantially Kondo-compensated ordered moment. To determine the magnetic structure of CeCo(In 0.9 Cd 0.1 ) 5 we carried out elastic neutron scattering experiments and performed elastic scans along ( 1 2 , 1 2 , ℓ) at 60 mK. Magnetic intensity was detected at Q = ( 1 2 , 1 2 , 1 2 ), cf. Fig. 3, and at symmetry equivalent positions like Q = ( 3 2 , 3 2 , 1 2 ) and Q = ( 3 2 , 3 2 , 3 2 ). Due to the small sample size long counting times (several minutes per point) were required. The magnetic peak at ( 1 2 , 1 2 , 1 2 ) monotonically decreases upon heating the sample and vanishes at T N . Scans along other high symmetry directions revealed no additional intensity, e.g. no intensity was found at (1, 1, 1 2 ), (0, 0, 1 2 ), ( 3 2 , 3 2 , 0), or ( 3 2 , 3 2 , 1). In particular, no magnetic intensity could be detected at ( 1 2 , 1 2 , δ), δ ≈ 0.3, an incommensurate position where CeRhIn 5 , a closely related member of the CeT In 5 (T = Co, Rh, or Ir) family, displays magnetic superstructure peaks. 13 Hence, the commensurate magnetic order in CeCo(In 0.9 Cd 0.1 ) 5 with a propagation vector Q AF = ( 1 2 , 1 2 , 1 2 ) is in marked contrast to the incommensurate order observed in other compounds of the CeTIn 5 family as well as in CeCu 2 Si 2 . 14 It is speculated that rather small changes in the Fermi surface are responsible for this behavior. The integrated magnetic intensity obtained from Gaussian fits to the data (cf. Fig. 3) is depicted in Fig. 4. The magnetic intensity starts to build up below T N ≈ 3 K, in agreement with T N determined from specific heat, increases continuously and eventually saturates below ≈ T c (= 1.27 K), potentially indicating missing magnetic intensity in the superconducting state. No distinct anomaly is resolved at the superconducting transition, in particular, the magnetic intensity does not vanish below T c . Our neutron scattering results clearly demonstrate the existence of AF order well below T c , down to lowest temperatures (T = 60 mK). Nuclear magnetic quadrupole resonance (NQR) experiments give supplementary evidence for the microscopic coexistence of SC and antiferromagnetism, and the estimated moment of approximately 0.7µ B is in line with the observed magnetic intensity, 15 however, we cannot calculate the magnetic moment precisely. As shown by Pham et al., 8 applying pressure reverses the effect of Cd doping. A generalized pressuretemperature (p − T ) phase diagram for CeRhIn 5 and the doping series CeCo(In 1−x Cd x ) 5 (including pure CeCoIn 5 ) 16,17 describing the pressure dependence of T N (p) and T c (p) suggests the same underlying physics. According to this p−T phase diagram, CeCo(In 0.9 Cd 0.1 ) 5 can be considered to correspond to CeRhIn 5 at p = 1.6 GPa. As already mentioned, at atmospheric pressure CeRhIn 5 orders magnetically below T N = 3.8 K with an incommensurate ordering wave vector ( 1 2 , 1 2 , δ), δ = 0.297. Pressure dependent neutron scattering experiments on CeRhIn 5 at 1.8 K show that the incommensurability, δ, and the ordered moment are changing only slightly. 18 In marked contrast to the simple expectation inferred from our present results, no magnetic intensity at 1.6 GPa is reported at ( 1 2 , 1 2 , 1 2 ) in CeRhIn 5 corresponding to CeCo(In 0.9 Cd 0.1 ) 5 at ambient pressure. In CeRhIn 5 AF order can be suppressed by either Co or Ir substitution on the Rh-site. With increasing substitution level a superconducting phase, first coexisting with antiferromagnetism, develops until antiferromagnetism becomes suppressed and only SC survives. 12,19 The corresponding phase diagram is similar to the one of CeCo(In 1−x Cd x ) 5 with CeCoIn 5 being situated on the purely SC side, cf. Fig. 1. In Rh-rich CeRh 1−y Ir y In 5 pressure studies even reveal the same generic p − T phase diagram found for CeCo(In 1−x Cd x ) 5 8 suggesting a close relationship. 20 In striking contrast to the neutron scattering results obtained for CeRhIn 5 under pressure, a commensurate magnetic structure with ordering wave vector ( 1 2 , 1 2 , 1 2 ) develops in the doping series CeRh 1−y Ir y In 5 and CeRh 1−z Co z In 5 at low temperatures, while the same incommensurate ordering wave vector present in the pure system is still observed below T N . 21,22 We speculate that the appearance of the commensurate magnetic ordering wave vector ( 1 2 , 1 2 , 1 2 ) is related to SC. Perhaps this commensurate magnetic ordering has so far been overlooked in superconducting CeRhIn 5 under pressure. Improved experiments with a better signal-to-background ratio are called for to answer this question. In summary, we carried out elastic neutron scattering experiments on CeCo(In 0.9 Cd 0.1 ) 5 . At low temperatures we found magnetic intensity at the commensurate wave vector Q AF = ( 1 2 , 1 2 , 1 2 ). The magnetic intensity is building up below T N , with T N being in good agreement with specific heat data. No indication for additional intensity was observed at incommensurate positions where CeRhIn 5 , the related AF member in the CeT In 5 family, orders. At low temperatures magnetic order is coexisting with SC. A saturation of the magnetic intensity below T c possibly reveals missing magnetic intensity in the superconducting state. We acknowledge useful discussions with P. Thalmeier and G. Knebel. We would like to thank A. D. Bianchi for the X-ray diffraction and M. Meissner for his assistance with the low temperature equipment at HMI. Work at Los Alamos was performed under the auspices of the U.S. DOE/Office of Science. Work at UC Davis and UC Irvine has been supported by NSF Grant No. DMR 053360. FIG. 1 : 1Doping -temperature (x − T ) phase diagram of CeCo(In1−xCdx)5, where x is the nominal Cd concentration. Diamonds indicate the Néel temperature, TN, and circles the superconducting transition temperature, Tc, determined by specific heat measurements. The arrow indicates the concentration investigated in this work, with transition temperatures as indicated by the solid symbols. Open symbols represent data taken from Ref. 8. FIG. 2 : 2Temperature dependence of the magnetic contribution to the specific heat, Cmag, for CeCo(In0.9Cd0.1)5. To obtain Cmag, the contribution of the non-magnetic reference compound LaCoIn5 11 was subtracted from the specific heat of CeCo(In0.9Cd0.1)5. The Néel temperature, TN, and the superconducting transition temperature, Tc, are indicated by arrows. Inset shows the electrical resistivity measured on the same sample. In0.9Cd0.1)5 at different temperatures. No magnetic intensity can be resolved at T = 3.22 K (> TN, as determined by specific heat). The solid lines are Gaussian fits to the data. FIG. 4 : 4Temperature dependence of the integrated magnetic intensity as obtained by Gaussian fits to the scans across Q = PACS numbers: 74.70.Tx, 75.25.+z, 71.27.+a, 75.30.Mb * Electronic address: [email protected]. * Electronic address: [email protected] . F Steglich, J Aarts, C D Bredl, W Lieke, D Meschede, W Franz, H Schäfer, Phys. Rev. Lett. 431892F. Steglich, J. Aarts, C. D. Bredl, W. Lieke, D. Meschede, W. Franz, and H. Schäfer, Phys. Rev. Lett. 43, 1892 (1979). . C Petrovic, P G Pagliuso, M F Hundley, R Movshovich, J L Sarrao, J D Thompson, Z Fisk, P Monthoux, J. Phys. Cond. Mat. 13337C. Petrovic, P. G. Pagliuso, M. F. Hundley, R. Movshovich, J. L. Sarrao, J. D. Thompson, Z. Fisk, and P. Monthoux, J. Phys. Cond. Mat. 13, L337 (2001). . C Petrovic, R Movshovich, M Jaime, P G Pagliuso, M F Hundley, J L Sarrao, Z Fisk, J D Thompson, Europhys. Lett. 53354C. Petrovic, R. Movshovich, M. Jaime, P. G. Pagliuso, M. F. Hundley, J. L. Sarrao, Z. Fisk, and J. D. Thompson, Europhys. Lett. 53, 354 (2001). . H Hegger, C Petrovic, E G Moshopoulou, M F Hundley, J L Sarrao, Z Fisk, J D Thompson, Phys. Rev. Lett. 844986H. Hegger, C. Petrovic, E. G. Moshopoulou, M. F. Hund- ley, J. L. Sarrao, Z. Fisk, and J. D. Thompson, Phys. Rev. Lett. 84, 4986 (2000). . Y Kohori, Y Yamato, Y Iwamoto, T Kohara, E D Bauer, M B Maple, J L Sarrao, Phys. Rev. B. 64134526Y. Kohori, Y. Yamato, Y. Iwamoto, T. Kohara, E. D. Bauer, M. B. Maple, and J. L. Sarrao, Phys. Rev. B 64, 134526 (2001). . A Bianchi, R Movshovich, I Vekhter, P G Pagliuso, J L Sarrao, Phys. Rev. Lett. 91257001A. Bianchi, R. Movshovich, I. Vekhter, P.G. Pagliuso, and J. L. Sarrao, Phys. Rev. Lett. 91, 257001 (2003). . J Paglione, M A Tanatar, D G Hawthorn, E Boaknin, R W Hill, F Ronning, M Sutherland, L Taillefer, Phys. Rev. Lett. 91246405J. Paglione, M. A. Tanatar, D.G. Hawthorn, E. Boaknin, R.W. Hill, F. Ronning, M. Sutherland, and L. Taillefer, Phys. Rev. Lett. 91, 246405 (2003). . L D Pham, Tuson Park, S Maquilon, J D Thompson, Z Fisk, Phys. Rev. Lett. 9756404L. D. Pham, Tuson Park, S. Maquilon, J. D. Thompson, and Z. Fisk, Phys. Rev. Lett. 97, 056404 (2006). . V F Mitrović, M Horvatić, C Berthier, G Knebel, G Lapertot, J Flouquet, Phys. Rev. Lett. 97117002V. F. Mitrović, M. Horvatić, C. Berthier, G. Knebel, G. Lapertot, and J. Flouquet, Phys. Rev. Lett. 97, 117002 (2006). . B.-L Young, R R Urbano, N J Curro, J D Thompson, J L Sarrao, A B Vorontsov, M J Graf, Phys. Rev. Lett. 9836402B.-L. Young, R. R. Urbano, N. J. Curro, J. D. Thompson, J. L. Sarrao, A. B. Vorontsov, and M. J. Graf, Phys. Rev. Lett. 98, 036402 (2007). . M F Hundley, private communicationM. F. Hundley, private communication. . P G Pagliuso, C Petrovic, R Movshovich, D Hall, M F Hundley, J L Sarrao, J D Thompson, Z Fisk, Phys. Rev. B. 64100503P. G. Pagliuso, C. Petrovic, R. Movshovich, D. Hall, M. F. Hundley, J. L. Sarrao, J. D. Thompson, and Z. Fisk, Phys. Rev. B 64, 100503(R) (2001). . W Bao, P G Pagliuso, J L Sarrao, J D Thompson, Z Fisk, J W Lynn, R W Erwin, Phys. Rev. B. 6214621W. Bao, P. G. Pagliuso, J. L. Sarrao, J. D. Thompson, Z. Fisk, J. W. Lynn, and R. W. Erwin, Phys. Rev. B 62, R14621 (2000); . W Bao, P G Pagliuso, J L Sarrao, J D Thompson, Z Fisk, J W Lynn, R W Erwin, Phys. Rev. B. 63219901W. Bao, P. G. Pagliuso, J. L. Sarrao, J. D. Thompson, Z. Fisk, J. W. Lynn, and R. W. Erwin, Phys. Rev. B 63, 219901(E) (2001). . O Stockert, E Faulhaber, G Zwicknagl, N Stüßer, H S Jeevan, M Deppe, R Borth, R Küchler, M Loewenhaupt, C Geibel, F Steglich, Phys. Rev. Lett. 92136401O. Stockert, E. Faulhaber, G. Zwicknagl, N. Stüßer, H. S. Jeevan, M. Deppe, R. Borth, R. Küchler, M. Loewenhaupt, C. Geibel, and F. Steglich, Phys. Rev. Lett. 92, 136401 (2004). . R R Urbano, B.-L Young, N J Curro, J D Thompson, L D Pham, Z Fisk, cond-mat/0702008R. R. Urbano, B.-L. Young, N. J. Curro, J. D. Thompson, L. D. Pham, and Z. Fisk, cond-mat/0702008 (2007). . M Nicklas, R Borth, E Lengyel, P G Pagliuso, J L Sarrao, V A Sidorov, G Sparn, F Steglich, J D Thompson1, J. Phys. Condens. Matter. 13905M. Nicklas, R. Borth, E. Lengyel, P. G. Pagliuso, J. L. Sarrao, V. A. Sidorov, G. Sparn, F. Steglich, and J. D. Thompson1, J. Phys. Condens. Matter 13, L905 (2001). . V A Sidorov, M Nicklas, P G Pagliuso, J L Sarrao, Y Bang, A V Balatsky, J D Thompson, Phys. Rev. Lett. 89157004V. A. Sidorov, M. Nicklas, P.G. Pagliuso, J. L. Sarrao, Y. Bang, A.V. Balatsky, and J. D. Thompson, Phys. Rev. Lett. 89, 157004 (2002). . A Llobet, J S Gardner, E G Moshopoulou, J.-M , A. Llobet, J. S. Gardner, E. G. Moshopoulou, J.-M. . M Mignot, W Nicklas, N O Bao, P G Moreno, I N Pagliuso, J L Goncharenko, J D Sarrao, Thompson, Phys. Rev B. 6924403Mignot, M. Nicklas, W. Bao, N. O. Moreno, P. G. Pagliuso, I. N. Goncharenko, J. L. Sarrao, and J. D. Thompson, Phys. Rev B 69, 024403 (2004). . V S Zapf, E J Freeman, E D Bauer, J Petricka, C Sirvent, N A Frederick, R P Dickey, M B Maple, Phys. Rev. B. 6514506V. S. Zapf, E. J. Freeman, E. D. Bauer, J. Petricka, C. Sirvent, N. A. Frederick, R. P. Dickey, and M. B. Maple, Phys. Rev. B 65, 014506 (2001). . M Nicklas, V A Sidorov, H A Borges, P G Pagliuso, J L Sarrao, J D Thompson, Phys. Rev. B. 7020505M. Nicklas, V. A. Sidorov, H. A. Borges, P. G. Pagliuso, J. L. Sarrao, and J. D. Thompson, Phys. Rev. B, 70, 020505(R) (2004). . A D Christianson, A Llobet, Wei Bao, J S Gardner, I P Swainson, J W Lynn, J.-M Mignot, K Prokes, P G Pagliuso, N O Moreno, J L Sarrao, J D Thompson, A H Lacerda, Phys. Rev. Lett. 95217002A. D. Christianson, A. Llobet, Wei Bao, J. S. Gardner, I. P. Swainson, J.W. Lynn, J.-M. Mignot, K. Prokes, P. G. Pagliuso, N. O. Moreno, J. L. Sarrao, J. D. Thompson, and A. H. Lacerda, Phys. Rev. Lett. 95, 217002 (2005). . M Yokoyama, H Amitsuka, K Matsuda, A Gawase, N Oyama, I Kawasaki, K Tenya, H Yoshizawa, J. Phys. Soc. Jpn. 75103703M. Yokoyama, H. Amitsuka, K. Matsuda, A. Gawase, N. Oyama, I. Kawasaki, K. Tenya, and H. Yoshizawa, J. Phys. Soc. Jpn. 75, 103703 (2006).
{'fraction_non_alphanumeric': 0.06926365339353475, 'fraction_numerical': 0.04284547768864979, 'mean_word_length': 3.786206896551724, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 12, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'The heavy fermion superconductor CeCoIn5 is believed to be close to a magnetic instability, but no static magnetic order has been found. Cadmium doping on the In-site shifts the balance between superconductivity and antiferromagnetism to the latter with an extended concentration range where both types of order coexist at low temperatures. We investigated the magnetic structure of nominally 10% Cd-doped CeCoIn5, being antiferromagnetically ordered below TN ≈ 3 K and superconducting below Tc ≈ 1.3 K, by elastic neutron scattering. Magnetic intensity was observed only at the ordering wave vector QAF = ( 1 2 , 1 2 , 1 2 ) commensurate with the crystal lattice. Upon entering the superconducting state the magnetic intensity seems to change only little. The commensurate magnetic ordering in CeCo(In1−xCdx)5 is in contrast to the incommensurate antiferromagnetic ordering observed in the closely related compound CeRhIn5. Our results give new insights in the interplay between superconductivity and magnetism in the family of CeT In5 (T = Co, Rh, and Ir) based compounds.', 'arxivid': 'cond-mat/0703703', 'author': ['M Nicklas \nMax Planck Institute for Chemical Physics of Solids\nNöthnitzer Str. 4001187DresdenGermany\n', 'O Stockert \nMax Planck Institute for Chemical Physics of Solids\nNöthnitzer Str. 4001187DresdenGermany\n', 'Tuson Park \nLos Alamos National Laboratory\n87545Los AlamosNMUSA\n', 'K Habicht \nHahn-Meitner-Institut\nGlienicker Str. 10014109BerlinGermany\n', 'K Kiefer \nHahn-Meitner-Institut\nGlienicker Str. 10014109BerlinGermany\n', 'L D Pham \nUniversity of California\n95616DavisCAUSA\n', 'J D Thompson \nLos Alamos National Laboratory\n87545Los AlamosNMUSA\n', 'Z Fisk \nUniversity of California\n92697IrvineCAUSA\n', 'F Steglich \nMax Planck Institute for Chemical Physics of Solids\nNöthnitzer Str. 4001187DresdenGermany\n', 'M Nicklas \nMax Planck Institute for Chemical Physics of Solids\nNöthnitzer Str. 4001187DresdenGermany\n', 'O Stockert \nMax Planck Institute for Chemical Physics of Solids\nNöthnitzer Str. 4001187DresdenGermany\n', 'Tuson Park \nLos Alamos National Laboratory\n87545Los AlamosNMUSA\n', 'K Habicht \nHahn-Meitner-Institut\nGlienicker Str. 10014109BerlinGermany\n', 'K Kiefer \nHahn-Meitner-Institut\nGlienicker Str. 10014109BerlinGermany\n', 'L D Pham \nUniversity of California\n95616DavisCAUSA\n', 'J D Thompson \nLos Alamos National Laboratory\n87545Los AlamosNMUSA\n', 'Z Fisk \nUniversity of California\n92697IrvineCAUSA\n', 'F Steglich \nMax Planck Institute for Chemical Physics of Solids\nNöthnitzer Str. 4001187DresdenGermany\n'], 'authoraffiliation': ['Max Planck Institute for Chemical Physics of Solids\nNöthnitzer Str. 4001187DresdenGermany', 'Max Planck Institute for Chemical Physics of Solids\nNöthnitzer Str. 4001187DresdenGermany', 'Los Alamos National Laboratory\n87545Los AlamosNMUSA', 'Hahn-Meitner-Institut\nGlienicker Str. 10014109BerlinGermany', 'Hahn-Meitner-Institut\nGlienicker Str. 10014109BerlinGermany', 'University of California\n95616DavisCAUSA', 'Los Alamos National Laboratory\n87545Los AlamosNMUSA', 'University of California\n92697IrvineCAUSA', 'Max Planck Institute for Chemical Physics of Solids\nNöthnitzer Str. 4001187DresdenGermany', 'Max Planck Institute for Chemical Physics of Solids\nNöthnitzer Str. 4001187DresdenGermany', 'Max Planck Institute for Chemical Physics of Solids\nNöthnitzer Str. 4001187DresdenGermany', 'Los Alamos National Laboratory\n87545Los AlamosNMUSA', 'Hahn-Meitner-Institut\nGlienicker Str. 10014109BerlinGermany', 'Hahn-Meitner-Institut\nGlienicker Str. 10014109BerlinGermany', 'University of California\n95616DavisCAUSA', 'Los Alamos National Laboratory\n87545Los AlamosNMUSA', 'University of California\n92697IrvineCAUSA', 'Max Planck Institute for Chemical Physics of Solids\nNöthnitzer Str. 4001187DresdenGermany'], 'corpusid': 119330463, 'doi': '10.1103/physrevb.76.052401', 'github_urls': [], 'n_tokens_mistral': 7310, 'n_tokens_neox': 6178, 'n_words': 3465, 'pdfsha': '5000224f2aec5732ab82994486fd850bd9b6ca57', 'pdfurls': ['https://arxiv.org/pdf/cond-mat/0703703v1.pdf'], 'title': ['Magnetic structure of Cd-doped CeCoIn 5', 'Magnetic structure of Cd-doped CeCoIn 5', 'Magnetic structure of Cd-doped CeCoIn 5', 'Magnetic structure of Cd-doped CeCoIn 5'], 'venue': []}
arxiv
PERSONALIZED STUDENT ATTRIBUTE INFERENCE A PREPRINT Moustapha Khalid [email protected]@uqam.ca Université du Québec à Montréal Montreal QCCanada Marie-Jean Askia Université du Québec à Montréal Montreal QCCanada Meurs Université du Québec à Montréal Montreal QCCanada PERSONALIZED STUDENT ATTRIBUTE INFERENCE A PREPRINT Big data · Educational data mining · Knowledge tracing · Machine learning Accurately predicting their future performance can ensure students successful graduation, and help them save both time and money. However, achieving such predictions faces two challenges, mainly due to the diversity of students' background and the necessity of continuously tracking their evolving progress. The goal of this work is to create a system able to automatically detect students in difficulty, for instance predicting if they are likely to fail a course. We compare a naive approach widely used in the literature, which uses attributes available in the data set (like the grades), with a personalized approach we called Personalized Student Attribute Inference (PSAI). With our model, we create personalized attributes to capture the specific background of each student. Both approaches are compared using machine learning algorithms like decision trees, support vector machine or neural networks. Introduction As all academic institutions aim to improve the quality of education, the success of their students is essential. To make university affordable and worthwhile, it is hence important to ensure that most of the students enrolled in a program succeed it and graduate on time. Therefore, early interventions for students who most likely will fail their courses can help them save both time and money. A possible solution towards this end is to build an automatic system that would successfully predict their future outcome. However, predicting students' performance is complex. The attributes frequently used by researchers are the Grade Point Average (GPA), internal assessment and students' demographic (gender, age etc). The issue with those attributes is that they tell nothing valuable about the student background and what s/he has been through. For that reason, predicting methods need to incorporate a way to capture students' background along with the historical student accomplishments (grades, credits obtained, GPA). We developed a personalized model called Personalized Student Attribute Inference (PSAI), which creates personalized attributes to capture the specific background of each student. We compare our model with a naive approach, which uses directly the attributes available in the data set (like the grades, credits obtained, GPA, etc.). The next Section explains our approach. Section 3 describes our experimental process and results, and finally, Section 4 discusses limitations and concludes. Personnalized Models We focus on personalized models, which take into account as much as possible the specifics of each student profile, and therefore emphasizes the student's background. Grades, GPA, credits obtained, etc. are not sufficient since they cannot model a student's knowledge. For example, students SA and SB have both a GPA of 3.7 but SA took only easy courses (3 courses in total) and SB took the most difficult ones (5 courses in total). Both have the same GPA so we cannot automatically determine who is the most talented. It shows that static attributes (the ones that are recorded directly like the grades) do not actually provide much profile details about a student. Thus, for accurately predicting student performance, one should consider other attributes as for instance the difficulty of the courses. To do so, we grouped courses by similar level of difficulty then we assigned them a weight, increasing with difficulty. Also, we assigned a score to each student depending on the total number of courses s/he took, their difficulty and the grades s/he obtained. Personalized Student Attribute Inference (PSAI) For assigning weights to courses according to their difficulty, we take a scale which limits are the weights of "extremely" easy courses and those of courses "extremely" difficult. We experimentally assign weights as follows: "extremely" easy courses get a weight of 0.5 and "extremely" difficult courses get a weight of 2. An "extremely" difficult course is hence 4 times more difficult than an "extremely" easy course. By analyzing University marks system (where our data came from), for the "extremely" easy courses , we consider an average mark of 4.15 (between A (4.0) and A + (4.3) ) and for the "extremely" difficult courses, an average of 1.15 (between D (1.0) and D + (1.3)) . Subsequently, in order to be able to assign a weight to a course according to the average mark obtained, we must consider a parametric function that will take this average mark as input and output the associated weight in accordance with the limits established above. The function must also be decreasing, i.e. if the input (the average mark) increases, the output (the weight) must necessarily decrease. Let β × exp (−αx) be an exponential function where β and α are parameters to be determined, and x is the average of the marks obtained by the students who took the course. To estimate the parameters β and α, we use the limits we fixed. Solving the following equations system: β × exp (−1.15α) = 2 β × exp (−4.15α) = 0.5 provides: α = ln(4) 3 and β = 2 exp ( 1.15 ln (4) 3 ) Making use of this function that assigns a weight to a course according to its difficulty, we present hereafter our algorithm to create a personalized data set, which will be used to train machine learning algorithms and make performance predictions. PSAI Algorithm Algorithm 1 PSAI Algorithm for course A Input: Prior information on courses (average mark in the course, marks obtained) took by students that took course A 1: α ← ln(4) 3 2: β ← 2 exp ( 1.15 ln(4) 3 ) 3: For each student that took course A: 4: For each course i taken before course A: 5: Let m i be the average mark in the course i (according to all students that took this course) 6: Let n i be the mark of the current student in course i 7: Compute the score of the student in the course i: S i = n i × β × exp (−αm i ) 8: End For 9: Compute the total score of the student: S = mean of S i 10: End For 11: Compute the weight of the course A: W A = β × exp (−αm A ) where m A is the average mark in course A Output: A score for each student and the computed weight for course A Algorithm 1 provides a dataset that will be used to train the prediction model. This dataset contains a score for each student and the weight of the course for which we want to predict the student performance. We also add as an attribute the overall success rate in the course. Experiments and Results For the sake of brevity, we present our results only for the following course (acronyms changed for non-disclosure reasons): ABC2222 : 6483 students with 5256 success and 1227 failures. The question asked to our models is the following: Will a given student fail the course? The method of training and testing in our experiments is the cross validation Kohavi et al. [1995]. We used the following machine learning algorithms : Decision trees Safavian and Landgrebe [1991], K-Nearest Neighbors Cover and Hart [1967], Support Vector Machine (SVM) Cortes andVapnik [1995], Random Forest Breiman [2001], an ensemble learning model (AdaBoost) Dietterich [2002] and Neural Network Sarle [1994]. The evaluation metric is the F-measure (or F1-score) Sasaki et al. [2007]. Table 1 shows the results obtained with several machine learning algorithms. We compare our results with those obtained using a direct (naive) method (the standard method) which only uses the attributes related to the course and the students present in the database. In our case these attributes are: admission base, citizenship, previous program, legal status, college program, age, gender, number of course credits obtained and GPA. Conclusion PSAI outperforms standard methods. Despite the imbalanced nature of the data set including fewer failures than successes, our approach detects many of these failures. It proves the need to consider several "hidden" aspects including the difficulty of the courses taken. The main limitation of the approach is related to the data set itself. Some essential data are missing to improve the model. We do not have the information of students before their first registration at the university. Hence, our model can only be used from the second course taken by the student. In addition, as often when dealing with real data collected over a long period of time, a large part of the records is unusable because of missing data and errors/noises. Reproducibility. This project is publicly released as an open source software in the following repository: https://gitlab.labikb.ca/khalid/psai Table 1 : 1Failure prediction in ABC2222 resultsABC2222 Algorithm Naive method PSAI F-measure(%) F-measure(%) Neural network 32,87 68,47 Decision tree 44,37 66,37 Adaboost 45,70 69,57 k-NN 37,07 64,31 Random Forest 47,36 58,30 SVM 27,81 62,75 A study of cross-validation and bootstrap for accuracy estimation and model selection. Ron Kohavi, Ijcai. Montreal, Canada14Ron Kohavi et al. A study of cross-validation and bootstrap for accuracy estimation and model selection. In Ijcai, volume 14, pages 1137-1145. Montreal, Canada, 1995. A survey of decision tree classifier methodology. David S Rasoul Safavian, Landgrebe, IEEE transactions on systems, man, and cybernetics. 213S Rasoul Safavian and David Landgrebe. A survey of decision tree classifier methodology. IEEE transactions on systems, man, and cybernetics, 21(3):660-674, 1991. Nearest neighbor pattern classification. Thomas Cover, Peter Hart, IEEE transactions on information theory. 131Thomas Cover and Peter Hart. Nearest neighbor pattern classification. IEEE transactions on information theory, 13(1): 21-27, 1967. Leo Breiman. Random forests. Corinna Cortes, Vladimir Vapnik, Machine learning. 203Machine learningCorinna Cortes and Vladimir Vapnik. Support-vector networks. Machine learning, 20(3):273-297, 1995. Leo Breiman. Random forests. Machine learning, 45(1):5-32, 2001. Ensemble learning. The handbook of brain theory and neural networks. S Thomas G Dietterich ; Warren, Sarle, Neural networks and statistical models. 2Thomas G Dietterich. Ensemble learning. The handbook of brain theory and neural networks, 2:110-125, 2002. Warren S Sarle. Neural networks and statistical models. 1994. The truth of the f-measure. Yutaka Sasaki, Teach Tutor mater. 15Yutaka Sasaki et al. The truth of the f-measure. Teach Tutor mater, 1(5):1-5, 2007.
{'fraction_non_alphanumeric': 0.04203314507915934, 'fraction_numerical': 0.024997685399500046, 'mean_word_length': 4.614345114345114, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 1, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': "Accurately predicting their future performance can ensure students successful graduation, and help them save both time and money. However, achieving such predictions faces two challenges, mainly due to the diversity of students' background and the necessity of continuously tracking their evolving progress. The goal of this work is to create a system able to automatically detect students in difficulty, for instance predicting if they are likely to fail a course. We compare a naive approach widely used in the literature, which uses attributes available in the data set (like the grades), with a personalized approach we called Personalized Student Attribute Inference (PSAI). With our model, we create personalized attributes to capture the specific background of each student. Both approaches are compared using machine learning algorithms like decision trees, support vector machine or neural networks.", 'arxivid': '2212.14682', 'author': ['Moustapha Khalid [email protected]@uqam.ca \nUniversité du Québec à Montréal Montreal\nQCCanada\n', 'Marie-Jean Askia \nUniversité du Québec à Montréal Montreal\nQCCanada\n', 'Meurs \nUniversité du Québec à Montréal Montreal\nQCCanada\n'], 'authoraffiliation': ['Université du Québec à Montréal Montreal\nQCCanada', 'Université du Québec à Montréal Montreal\nQCCanada', 'Université du Québec à Montréal Montreal\nQCCanada'], 'corpusid': 218540055, 'doi': '10.1007/978-3-030-47358-7_56', 'github_urls': [], 'n_tokens_mistral': 2852, 'n_tokens_neox': 2517, 'n_words': 1673, 'pdfsha': '4de4458825ec5f59b5a5c0e88e42e0d9b96757d3', 'pdfurls': ['https://export.arxiv.org/pdf/2212.14682v1.pdf'], 'title': ['PERSONALIZED STUDENT ATTRIBUTE INFERENCE A PREPRINT', 'PERSONALIZED STUDENT ATTRIBUTE INFERENCE A PREPRINT'], 'venue': []}
arxiv
Secure Multiparty Computation with Partial Fairness 25 Nov 2010 November 29, 2010 Amos Beimel Eran Omri Ilan Orlov Department of Computer Science Department of Computer Science Ben Gurion University Be'er Sheva Israel Department of Computer Science Bar Ilan University Ramat Gan Israel Ben Gurion University Be'er ShevaIsrael Secure Multiparty Computation with Partial Fairness 25 Nov 2010 November 29, 2010 A protocol for computing a functionality is secure if an adversary in this protocol cannot cause more harm than in an ideal computation where parties give their inputs to a trusted party which returns the output of the functionality to all parties. In particular, in the ideal model such computation is fairall parties get the output. Cleve (STOC 1986) proved that, in general, fairness is not possible without an honest majority. To overcome this impossibility, Gordon and Katz (Eurocrypt 2010) suggested a relaxed definition -1/p-secure computation -which guarantees partial fairness. For two parties, they construct 1/p-secure protocols for functionalities for which the size of either their domain or their range is polynomial (in the security parameter). Gordon and Katz ask whether their results can be extended to multiparty protocols.We study 1/p-secure protocols in the multiparty setting for general functionalities. Our main result is constructions of 1/p-secure protocols when the number of parties is constant provided that less than 2/3 of the parties are corrupt. Our protocols require that either (1) the functionality is deterministic and the size of the domain is polynomial (in the security parameter), or (2) the functionality can be randomized and the size of the range is polynomial. If the size of the domain is constant and the functionality is deterministic, then our protocol is efficient even when the number of parties is O(log log n) (where n is the security parameter). On the negative side, we show that when the number of parties is super-constant, 1/p-secure protocols are not possible when the size of the domain is polynomial.A protocol for computing a functionality is secure if an adversary in this protocol cannot cause more harm than in an ideal computation where parties give their inputs to a trusted party which returns the output of the functionality to all parties. This is formalized by requiring that for every adversary in the real world, there is an adversary in the ideal world, called simulator, such that the output of the real-world adversary and the simulator are indistinguishable in polynomial time. Such security can be achieved when there is a majority of honest parties[16]. Secure computation is fair -all parties get the output. Cleve[9]proved that, in general, fairness is not possible without an honest majority.To overcome the impossibility of [9], Gordon and Katz[22]suggested a relaxed definition -1/p-secure computation -which guarantees partial fairness. Informally, a protocol is 1/p-secure if for every adversary in the real world, there is a simulator running in the ideal world, such that the output of the real-world adversary and the simulator cannot be distinguished with probability greater than 1/p. For two parties, Gordon and Katz construct 1/p-secure protocols for functionalities whose size of either their domain or their range is polynomial (in the security parameter). They also give impossibility results when both the domain and range are super-polynomial. Gordon and Katz ask whether their results can be extended to multiparty protocols. We give positive and negative answers to this question.Previous Results. Cleve [9] proved that any protocol for coin-tossing without an honest majority cannot be fully secure, specifically, if the protocol has r rounds, then it is at most 1/r-secure. Protocols with partial fairness, under various definitions and assumptions, have been constructed for coin-tossing [9, 10, 24, 4], for contract signing/exchanging secrets[6,23,12,5,11,7], and for general functionalities[27,13,2,17,25,14,22]. We next describe the papers that are most relevant to our paper. Moran, Naor, and Segev [24] construct 2-party protocols for coin tossing that are 1/r-secure (where r is the number of rounds in the protocol). Gordon and Katz [22] define 1/p-security and construct 2-party 1/p-secure protocols for every functionality whose size of either the domain or the range of the functionality is polynomial. Finlay, in a previous work [4] we construct multiparty protocols for coin tossing that are O(1/r)-secure provided that the fraction of bad parties is slightly larger than half. In particular, our protocol is O(1/r)-secure when the number of parties is constant and the fraction of bad parties is less than 2/3. Gordon et al. [20] showed that complete fairness is possible in the two party case for some functions. Gordon and Katz [19] showed similar results for the multiparty case. The characterization of the functions that can be computed with full fairness without honest majority is open. Completeness for fair computations has been studied in[21]. Specifically, they show a specific function that is complete for fair two-party computation; this function is also complete for 1/p-secure two-party computation.Our ResultsWe study 1/p-secure protocols in the multiparty setting. We construct two protocols for general functionalities assuming that the fraction of corrupt parties is less than 2/3. The first protocol is efficient when(1)The number of parties is constant, the functionality is deterministic, and the size of the domain of inputs is at most polynomial in the security parameter, or (2) The number of parties is O(log log n) (where n is the security parameter), the functionality is deterministic, and the size of the domain of inputs is constant. The second protocol is efficient when the number of parties is constant, the functionality can be randomized, and the size of the range of the functionality is at most polynomial in the security parameter. Our second protocol does not provide correctness, i.e., in a case of premature termination, with probability of 1/ poly(n), the remaining active parties output a value which might be inconsistent with their inputs. In contrast, our first protocol provides correctness. CASE I: 1 ≤ i < i ⋆ . For every j ∈ J the dealer sets x j = x j and for every j / ∈ J it chooses x j independently with uniform distribution from the domain X n ; it computes the output σ i J ← f n ( x 1 , . . . , x m ).CASE II: i ⋆ ≤ i ≤ r. The dealer sets σ i J = w.The dealer T interacts with the parties in rounds, where in round i, for 1 ≤ i ≤ r, there are of three phases:The peeking phase. The dealer sends to the adversary all the values σ i J such that all parties in Q J are corrupted.The abort and premature termination phase. The adversary sends to T the identities of the parties that abort in the current round. If there are less than t + 1 active parties, then T sends σ i−1 J to the active parties, where Q J is the set of the active parties when parties can also abort during this phase (see exact details inFigure 1). The honest parties return this output and halt.The main phase. If at least t + 1 parties are active, T notifies the active parties that the protocol proceeds normally.If after r rounds, there are at least t + 1 active parties, T sends w to all active parties and the honest parties output this value.Example 3.1 As an example, assume that m = 5 and t = 3. In this case the dealer computes a value σ i J for every set of size 2 or 3. Consider an execution of the protocol where p 1 aborts in round 4 and p 3 and p 4 abort in round 100. In this case, T sends σ 99 {2,5} to p 2 and p 5 , which return this output. value of the set that contains all of them, i.e., σ i−1 J . In the special case of premature termination already in the first round, the remaining active parties engage in a fully secure protocol (with honest majority) to compute f n .The use of the outer secret-sharing scheme with threshold t + 1 plays a crucial role in eliminating the online dealer. On the one hand, it guarantees that an adversary, corrupting at most t parties, cannot reconstruct the shares of round i before round i. On the other hand, at least m − t parties must abort to prevent the reconstruction of the outer secret-sharing scheme (this is why we cannot proceed after m−t parties aborted). Furthermore, since t ≤ 2m/3, when at least m − t corrupt parties aborted, there is an honest majority. To see this, assume that at least m − t corrupt parties aborted. Thus, at most t − (m − t) = 2t − m corrupt parties are active. There are m − t honest parties (which are obviously active), therefore, as 2t−m < m − t (since t < 2m/3), an honest majority is achieved when m − t parties abort. In this case we can execute a protocol with full security for the reconstruction.Finally, we replace the off-line dealer by using a secure-with-abort and cheat-detection protocol computing the functionality computed by the dealer, that is, Functionality MultiShareGen r . Obtaining the outputs of this computation, an adversary is unable to infer any information regarding the input of honest parties or the output of the protocol (since it gets t shares of a (t + 1)-out-of-m secret-sharing scheme). The adversary, however, can prevent the execution, at the price of at least one corrupt party being detected cheating by all other parties. In such an event, the remaining parties will start over without the detected cheating party. This goes on either until the protocol succeeds or there is an honest majority and a fully secure protocol computing f n is executed.A formal description of the protocol appears inFigure 3. The reconstruction functionality used in this protocol (when at least m − t parties aborted) appears inFigure 4. The details of how to construct a protocol secure-with-abort and cheat-detection with O(1) rounds are given in[4].Comparison with the multiparty coin-tossing protocol of [4].Our protocol combines ideas from the protocols of[22,4]. However, there are some important differences between our protocol and the protocol of[4]. In the coin-tossing protocol of [4], the bits σ i J are shared using a threshold scheme where the threshold is smaller than the size of the set Q J . This means that a proper subset of Q J containing corrupt parties can reconstruct σ i J . In coin-tossing this is not a problem since there are no inputs. However, when computing functionalities with inputs, such σ i J might reveal information on the inputs of honest parties in Q J , and we share σ i J with threshold |Q J |. As a result, we use more sets Q J than in [4] and the bias of the protocol is increased (put differently, to keep the same security, we need to increase the number of rounds in the protocol). For example, the protocol of [4] has small bias when there are polynomially many parties and t = m/2. Our protocol is efficient only when there are constant number of parties. As explained in Section 4, this difference is inherent as a protocol for general functionalities with polynomially many parties and t = m/2 cannot have a small bias.A 1/p-Secure Protocol for Polynomial RangeUsing an idea of [22], we modify our protocol such that it will have a small bias when the size of the range of the functionality F is polynomially bounded (even if F is randomized and has a big domain of inputs). The only modification is the way that each σ i J is chosen prior to round i ⋆ : with probability 1/(2p) we choose σ i J as a random value in the range of f n and with probability 1 − 1/(2p) we choose it as inFigure 2. Formally, in the model with the dealer, in the preprocessing phase of MPCWithD r described inFigure 1, we replaceStep(5)with the following step:• For each i ∈ {1, . . . , i ⋆ − 1} and for each J ⊆ [m] \ D 0 s.t. m − t ≤ |J| ≤ t,Proof:We start with the case of a deterministic functionality F. Recall that x 1 , . . . , x m are the inputs used by the dealer to obtain w = f n (x 1 , . . . , x m ) and σ i ⋆ J = w for each J ⊆ [m] s.t. m − t ≤ |J| ≤ t. Let J be such that the adversary obtains σ i J in round i < i ⋆ . Recall that x 1 , . . . , x m are the inputs used by the dealer to obtain σ i J , that is, σ i J = f n ( x 1 , . . . , x m ), where x j = x j for each j ∈ J and x j is selected uniformly at random from x j for every j / ∈ J. We bound the probability that σ i J = w by the probability that x j = x j for all j / ∈ J. The probability that x j = x j is 1/d. Therefore, the probability that both sets are the same is (1/d) m−|J| > (1/d) m .In each round of the protocol, A obtains the value σ i J for each subset Q J s.t. J ⊆ [m] and m − t ≤ |J| ≤ t, therefore, A obtains less than 2 t values. For each such two values σ i J and σ i J ′ obtained by A in round i < i ⋆ , the sets of inputs { x j : j / ∈ J} and { x j : j / ∈ J ′ } are totally independent. Therefore, the probability that all the values that the adversary sees in round i < i ⋆ are equal to w = f n (x 1 , . . . , x m ) is at least (1/d m ) 2 t −1 .For randomized functionality F, we think of the evaluation of f n ( x 1 , . . . , x m ) as two steps: first x j is randomly chosen from X n for every j ∈ J and then the randomized functionality is evaluated. Therefore, as A obtains less than 2 t values in each round i < i ⋆ , that the probability that all the values that the adversary sees in each round i < i ⋆ are equal to the specific w is at least (1/d m ) 2 t −1 · ǫ 2 t −1 .In the next lemma, we prove the correctness of the simulation by using the previous two lemmas. Introduction Our protocols combine ideas from the protocols of Gordon and Katz [22] and our paper [4], both of which generalize the protocol of Moran, Naor, and Segev [24]. Specifically, our protocols proceed in rounds, where in each round values are given to subsets of parties. There is a special round i ⋆ in the protocol. Prior to round i ⋆ , the values given to a subset of parties are values that can be computed from the inputs of the parties in this subset; staring from round i ⋆ the values are the "correct" output of the functionality. The values given to a subset are secret shared such that only if all parties in the subset cooperate they can reconstruct the value. If in some round many (corrupt) parties have aborted such that there is a majority of honest parties among the active parties, then the set of active parties reconstructs the value given to this set in the previous round. 1 Similar to the protocols of [24,22,4], the adversary can cause harm (e.g., bias the output of the functionality) only if it guesses i ⋆ ; we show that in our protocols this probability is small and the protocols are 1/p-secure. The values in our protocols are chosen similar to [22]. The mechanism to secret share the values is similar to [4], however, there are important differences in this sharing, as the sharing mechanism of [4] is not appropriate for 1/p-secure computations of functionalities which depend on inputs. To complete the picture, we prove interesting impossibility results. We show that, in general, when the number of parties is super-constant, 1/p-secure protocols are not possible without honest majority when the size of the domain is polynomial. This impossibility result justifies the fact why in our protocols the number of parties is constant. We also show that, in general, when the number of parties is ω(log n), 1/p-secure protocols are not possible without honest majority even when the size of the domain is 2. The proof of the impossibility result is rather simple and follows from an impossibility result of [22]. Our impossibility results should be contrasted with the coin-tossing protocol of [4] which is an efficient 1/p-secure protocol even when m(n), the number of parties, is polynomial in the security parameter and the number of bad parties is m(n)/2 + O(1). Our results show that these parameters are not possible for general 1/p-secure protocols even when the size of the domain of inputs is 2. Open Problems. In both our impossibility results the size of the range is super-polynomial. It is open if there is an efficient 1/p-secure protocol when the number of parties is not constant and the size of both the domain and range is polynomial. In addition, the impossibility results do not rule out that the doubleexponential dependency on the number of parties can be improved. The protocols of [22] are private -the adversary cannot learn any information on the inputs of the honest parties (other than the information that it can learn in the ideal world of computing F). The adversary can only bias the output. Our first protocol is not private (that is, the adversary can learn extra information). However, we do not know whether the second protocol is private. 2 It is open if there are general multiparty 1/p-secure protocols that are also private. Preliminaries A multi-party protocol with m parties is defined by m interactive probabilistic polynomial-time Turing machines p 1 , . . . , p m . Each Turning machine, called party, has the security parameter 1 n as a joint input and a private input y j . The computation proceeds in rounds. In each round, the active parties broadcast and receive messages on a common broadcast channel. The number of rounds in the protocol is expressed as some function r(n) in the security parameter (typically, r(n) is bounded by a polynomial). At the end of the protocol, the (honest) parties should hold a common value w (which should be equal to an output of a predefined functionality). In this work we consider a corrupt, static, computationally-bounded (i.e., non-uniform probabilistic polynomial-time) adversary that is allowed to corrupt some subset of parties. That is, before the beginning of the protocol, the adversary corrupts a subset of the parties and may instruct them to deviate from the protocol in an arbitrary way. The adversary has complete access to the internal state of each of the corrupted parties and fully controls the messages that they send throughout the protocol. The honest parties follow the instructions of the protocol. The parties communicate via a synchronous network, using only a broadcast channel. The adversary is rushing, that is, in each round the adversary hears the messages broadcast by the honest parties before broadcasting the messages of the corrupted parties for this round (thus, broadcast messages of the corrupted parties can depend on the broadcast messages of the honest parties in this round). Notation. For an integer ℓ, define [ℓ] = {1, . . . , ℓ}. For a set J ⊆ [m], define Q J = {p j : j ∈ J}. An m-party functionality F = {f n } n∈N is a sequence of polynomial-time computable, randomized mappings f n : (X n ) m → Z n , where X n = {0, 1} ℓ d (n) and Z n = {0, 1} ℓr(n) are the domain of inputs of each party and the range respectively; ℓ d , ℓ r : N → N are some fixed functions. We denote the size of the domain and the range of F by d(n) and g(n) respectively, that is, d(n) = 2 ℓ d (n) and g(n) = 2 ℓr(n) . For a randomized mapping f n , the assignment w ← f n (x 1 , . . . , x m ) denotes the process of computing f n with the inputs x 1 , . . . , x m and with uniformly chosen random coins and assigning the output of the computation to w. If F is deterministic, we sometimes call it a function. We sometime omit n from functions of n (for example, we write d instead of d(n)). The Real vs. Ideal Paradigm The security of multiparty computation protocols is defined using the real vs. ideal paradigm. In this paradigm, we consider the real-world model, in which protocols are executed. We then formulate an ideal model for executing the task. This ideal model involves a trusted party whose functionality captures the security requirements from the task. Finally, we show that the real-world protocol "emulates" the ideal-world protocol: For any real-life adversary A there exists an ideal-model adversary S (called simulator) such that the global output of an execution of the protocol with A in the real-world model is distributed similarly to the global output of running S in the ideal model. In both models there are m parties p 1 , . . . , p m holding a common input 1 n and private inputs y 1 , . . . , y m respectively, where y j ∈ X n for 1 ≤ j ≤ m. The Real Model. Let Π be an m-party protocol computing F. Let A be a non-uniform probabilistic polynomial time adversary that gets the input y j of each corrupted party p j and the auxiliary input aux. Let REAL Π,A(aux) ( y, 1 n ), where y = (y 1 , . . . , y m ), be the random variable consisting of the view of the adversary (i.e., the inputs of the corrupted parties and the messages it got) and the output of the honest parties following an execution of Π. The Ideal Model. The basic ideal model we consider is a model without abort. Specifically, there is an adversary S which has corrupted a subset B of the parties. The adversary S has some auxiliary input aux. An ideal execution for the computing F proceeds as follows: Send inputs to trusted party: The honest parties send their inputs to the trusted party. The corrupted parties may either send their received input, or send some other input of the same length (i.e., x j ∈ X n ) to the trusted party, or abort (by sending a special " abort j " message). Denote by x 1 , . . . , x m the inputs received by the trusted party. If p j does not send an input, then the trusted party selects x j ∈ X n with uniform distribution. 3 Trusted party sends outputs: The trusted party computes f n (x 1 , . . . , x m ) with uniformly random coins and sends the output to the parties. Outputs: The honest parties output the value sent by the trusted party, the corrupted parties output nothing, and S outputs any arbitrary (probabilistic polynomial-time computable) function of its view (its inputs, the output, and the auxiliary input aux). Let IDEAL F ,S(aux) ( y, 1 n ) be the random variable consisting of the output of the adversary S in this ideal world execution and the output of the honest parties in the execution. 1/p-Indistinguishability and 1/p-Secure Computation As explained in the introduction, some ideal functionalities for computing F cannot be implemented when there is no honest majority. We use 1/p-secure computation, defined by [22], to capture the divergence from the ideal worlds. Definition 2.1 (1/p-indistinguishability) A function µ(·) is negligible if for every positive polynomial q(·) and all sufficiently large n it holds that µ(n) < 1/q(n). A distribution ensemble X = {X a,n } a∈Dn,n∈N is an infinite sequence of random variables indexed by a ∈ D n and n ∈ N, where D n is a domain that might depend on n. For a fixed function p(n), two distribution ensembles X = {X a,n } a∈Dn,n∈N and Y = {Y a,n } a∈Dn,n∈N are computationally 1/p-indistinguishable, denoted X 1/p ≈ Y , if for every non-uniform polynomial-time algorithm D there exists a negligible function µ(·) such that for every n and every a ∈ D n , Pr[D(X a,n ) = 1] − Pr[D(Y a,n ) = 1] ≤ 1 p(n) + µ(n). Two distribution ensembles are computationally indistinguishable, denoted X C ≡ Y , if for every c ∈ N they are computationally 1 n c -indistinguishable. We next define the notion of 1/p-secure computation [22]. The definition uses the standard real/ideal paradigm [15,8], except that we consider a completely fair ideal model (as typically considered in the setting of honest majority), and require only 1/p-indistinguishability rather than indistinguishability. Definition 2.2 (1/p-secure computation [22]) Let p = p(n) be a function. An m-party protocol Π is said to 1/p-securely compute a functionality F where there are at most t(n) corrupt parties, if for every nonuniform probabilistic polynomial-time adversary A in the real model controlling at most t(n) parties, there exists a non-uniform probabilistic polynomial-time adversary S in the ideal model, controlling the same parties as A, such that the following two distribution ensembles are computationally 1/p-indistinguishable IDEAL F ,S(aux) ( y, 1 n ) aux∈{0,1} * , y∈(Xn) m ,n∈N 1/p ≈ REAL Π,A(aux) ( y, 1 n ) aux∈{0,1} * , y∈(Xn) m ,n∈N . We next define statistical distance between two random variables and the notion of perfect 1/p-secure computation, which implies the notion of 1/p-secure computation. Definition 2.3 (statistical distance) We define the statistical distance between two random variables A and B as the function SD (A, B) = 1 2 α Pr[A = α] − Pr[B = α] . Definition 2.4 (perfect 1/p-secure computation) An m-party protocol Π is said to perfectly 1/p-secure compute a functionality F if for every non-uniform adversary A in the real model, there exists a polynomialtime adversary S in the ideal model such that for every n ∈ N, for every y ∈ (X n ) m , and for every aux ∈ {0, 1} * SD IDEAL F ,S(aux) ( y, 1 n ), REAL Π,A(aux) ( y, 1 n ) ≤ 1 p(n) . Security with abort and cheat detection is defined in Appendix A. The cryptographic tools we use are described in Appendix B. The Multiparty Secure Protocols In this section we present our protocols. We start with a protocol that assumes that either the functionality is deterministic and the size of the domain is polynomial, or that the functionality is randomized and both the domain and range of the functionality are polynomial. We then present a modification of the protocol that is 1/p-secure for (possibly randomized) functionalities if the size of the range is polynomial (even if the size of the domain of F is not polynomial). The first protocol is more efficient for deterministic functionalities with polynomial-size domain. Furthermore, the first protocol has full correctness, while in the modified protocol, correctness is only guaranteed with probability 1 − 1/p. Formally, we prove the following two theorems. Theorem 1 Let F = {f n : (X n ) m → Z n } be randomized functionality where the size of domain is d(n) and the size of the range is g(n), and let p(n) be a polynomial. If enhanced trap-door permutations exist, then for any m and t such that m/2 ≤ t < 2m/3, and for any polynomial p(n) there is an r(n)round m-party 1/p(n)-secure protocol computing F tolerating up to t corrupt parties where r(n) = p(n) · (2 · d(n) m · g(n) · p(n)) 2 t , provided that r(n) is bounded by a polynomial in n. If F is deterministic, then there is a r(n)-round 1/p(n)-secure protocol for r(n) = p(n)·d(n) m·2 t , provided that r(n) is bounded by a polynomial in n. Theorem 2 Let F = {f n : (X n ) m → Z n } be randomized functionality where the size of the range g(n) is polynomial in n and m is constant, and let p(n) be a polynomial. If enhanced trap-door permutations exist, then for t such that m/2 ≤ t < 2m/3 and for any polynomial p(n) there is an r(n)-round m-party 1/p(n)secure protocol computing F tolerating up to t corrupt parties where r(n) = (2p(n)) 2 t +1 · g(n) 2 t . Following [24,4], we present the first protocol in two stages. We first describe in Section 3.1 a protocol with a dealer and then in Section 3.2 present a protocol without this dealer. The goal of presenting the protocol in two stages is to simplify the understanding of the protocol and to enable to prove the protocol in a modular way. In Section 3.3, we present a modification of the protocol which is 1/p-secure if the size of the range is polynomial (even if the size of the domain of f is not polynomial). The Protocol for Polynomial-Size Domain with a Dealer We consider a network with m parties where at most t of them are corrupt such that m/2 ≤ t ≤ 2m/3. In this section we assume that there is a special trusted on-line dealer, denoted T . This dealer interacts with the parties in rounds, sending messages on private channels. We assume that the dealer knows the set of corrupt parties. In Section 3.2, we show how to remove this dealer and construct a protocol without a dealer. In our protocol the dealer sends in each round values to subsets of parties; the protocol proceeds with the normal execution as long as at least t + 1 of the parties are still active. If at some round i, there are at most t active parties, then the active parties reconstruct the value given to them in round i− 1, output this value, and halt. Following [24], and its follow up works [22,4], the dealer chooses at random with uniform distribution a special round i ⋆ . Prior to this round the adversary gets no information and if the corrupt parties abort the execution prior to i ⋆ , then they cannot bias the output of the honest parties or cause any harm. After round i ⋆ , the output of the protocol is fixed, and, also in this case the adversary cannot affect the output of the honest parties. The adversary cause harm only if it guesses i ⋆ and this happens with small probability. We next give a verbal description of the protocol. This protocol is designed such that the dealer can be removed from it in Section 3.2. A formal description is given in Figure 1. At the beginning of the protocol each party sends its input y j to the dealer. The corrupted parties may send any values of their choice. Let x 1 , . . . , x m denote the inputs received by the dealer. If a corrupt party p j does not send its input, then the dealer sets x j to be a random value selected uniformly from X n . In a preprocessing phase, the dealer T selects uniformly at random a special round i ⋆ ∈ [r]. The dealer computes w ← f n (x 1 , . . . , x m ). Then, for every round 1 ≤ i < r and every J ⊆ {1, . . . , m} such that m − t ≤ |J| ≤ t, the dealer selects an output, denoted σ i J , as follows (this output is returned by the parties in Q J = {p j : j ∈ J} if the protocol terminates in round i + 1 and Q J is the set of the active parties): Inputs: Each party p j holds a private input y j ∈ X n and the joint input: the security parameter 1 n , the number of rounds r = r(n), and a bound t on the number of corrupted parties. Instructions for each honest party p j : (1) After receiving the " start " message, send input y j to the dealer. (2) If the premature termination step is executed with i = 1, then send its input y j to the dealer. (3) Upon receiving output z from the dealer, output z. (Honest parties do not send any other messages throughout the protocol.) Instructions for the (trusted) dealer: The preprocessing phase: 1. Set D 0 = ∅ and send a " start " message to all parties. 2. Receive an input, denoted x j , from each party p j . For every p j that sends an " abort j " message, notify all parties that party p j aborted, select x j ∈ X n with uniform distribution, and update D 0 = D 0 ∪ {j}. 3. Let D = D 0 . If |D| ≥ m − t, go to premature termination with i = 1. 4. Set w ← f n (x 1 , . . . , x m ) and select i ⋆ ∈ {1, . . . , r} with uniform distribution. 5. For each 1 ≤ i < i ⋆ , for each J ⊆ [m] \ D 0 s.t. m − t ≤ |J| ≤ t: for each j ∈ J set x j = x j , for each j ∈ J select uniformly at random x j ∈ X n , and set σ i J ← f n ( x 1 , . . . , x m ). 6. For each i ⋆ ≤ i ≤ r and for each J ⊆ [m] \ D 0 s.t. m − t ≤ |J| ≤ t, set σ i J = w. 7. Send " proceed " to all parties. Interaction rounds: In each round 1 ≤ i ≤ r, interact with the parties in three phases: • The peeking phase: For each J ⊆ [m] \ D 0 s.t. m − t ≤ |J| ≤ t, if Q J contains only corrupt parties, send the value σ i J to all parties in Q J . • The abort phase: Upon receiving an " abort j " message from a party p j , notify all parties that party p j aborted (ignore all other types of messages) and update D = D ∪ {j}. If |D| ≥ m − t, go to premature termination step. • The main phase: Send " proceed " to all parties. Premature termination step: • If i = 1, then: Receive an input, denoted x j ′ , from each active party p j . For every party p j that sends an " abort j " message, update D = D ∪ {j} and select x j ′ ∈ X n with uniform distribution. Set w ′ ← f n (x 1 ′ , . . . , x m ′ ). • Else, if i > 1, then: For each " abort j " message received from a party p j , update D = D ∪ {j}. Set w ′ = σ i−1 J for J = [m] \ D. • Send w ′ to each party p j s.t. j / ∈ D 0 and halt. Normal termination: If the last round of the protocol is completed, send w to to each party p j s.t. j / ∈ D 0 . The formal proof of the 1/p-security of the protocol appears in Appendix C. We next hint why for deterministic functionalities, any adversary can cause harm in the above protocol by at most O(d O(1) /r), where d = d(n) is the size of the domain of the inputs and the number of parties, i.e., m, is constant. As in the protocols of [24,22,4], the adversary can only cause harm by causing the protocol to terminate in round i ⋆ . In our protocol, if in some round there are two values σ i J and σ i J ′ that the adversary can obtain such that σ i J = σ i J ′ , then the adversary can deduce that i < i ⋆ . Furthermore, the adversary might have some auxiliary information on the inputs of the honest parties, thus, the adversary might be able to deduce that a round is not i ⋆ even if all the values that it gets are equal. However, there are less than 2 t values that the adversary can obtain in each round (i.e., the values of subsets of the t corrupt parties of size at least m − t). We will show that for a round i such that i < i ⋆ , the probability that all these values are equal to a fixed value is 1/d O(1) for a deterministic function f n (for a randomized functionality this probability also depends on the size of the range). By [22,Lemma 2], the protocol is d O(1) /r-secure. Eliminating the Dealer of the Protocol We eliminate the trusted on-line dealer in a few steps using a few layers of secret-sharing schemes. First, we change the on-line dealer, so that, in each round i, it shares the value σ i J of each subset Q J among the parties of Q J using a |J|-out-of-|J| secret-sharing scheme -called inner secret-sharing scheme. As in Protocol MPCWithD r described in Figure 1, the adversary is able to obtain information on σ i J only if it controls all the parties in Q J . On the other hand, the honest parties can reconstruct σ i−1 J (without the dealer), where Q J is the set of active parties containing the honest parties. In the reconstruction, if an active (corrupt) party does not give its share, then it is removed from the set of active parties Q J . This is possible since in the case of a premature termination an honest majority among the active parties is guaranteed (as further explained below). Next, we convert the on-line dealer to an off-line dealer. That is, we construct a protocol in which the dealer sends only one message to each party in an initialization stage; the parties interact in rounds using a broadcast channel (without the dealer) and in each round i each party learns its shares of the ith round inner secret-sharing schemes. In each round i, each party p j learns a share of σ i J in a |J|-out-of-|J| secret-sharing scheme, for every set Q J such that j ∈ J and m − t ≤ |J| ≤ t (that is, it learns the share of the inner scheme). For this purpose, the dealer computes, in a preprocessing phase, the appropriate shares for the inner secret-sharing scheme. For each round, the shares of each party p j are then shared in a 2-out-of-2 secret-sharing scheme, where p j gets one of the two shares (this share is a mask, enabling p j to privately reconstruct its shares of the appropriate σ i J although messages are sent on a broadcast channel). All other parties get shares in a t-out-of-(m − 1) Shamir secret-sharing scheme of the other share of the 2-out-of-2 secret-sharing. See Construction B.1 for a formal description. We call the resulting secret-sharing scheme the outer scheme. To prevent corrupt parties from cheating, by say, sending false shares and causing reconstruction of wrong secrets, every message that a party should send during the execution of the protocol is signed in the preprocessing phase (together with the appropriate round number and with the party's index). In addition, the dealer sends a verification key to each of the parties. To conclude, the off-line dealer gives each party the signed shares for the outer secret sharing scheme together with the verification key. A formal description of the functionality of the off-line dealer, called Functionality MultiShareGen, is given in Figure 2. The protocol with the off-line dealer proceeds in rounds. In round i of the protocol all parties broadcast their (signed) shares in the outer (t + 1)-out-of-m secret-sharing scheme. Thereafter, each party can unmask the message it receives (with its share in the appropriate 2-out-of-2 secret-sharing scheme) to obtain its shares in the |J|-out-of-|J| inner secret-sharing of the values σ i J (for the appropriate sets Q J 's to which the party belongs). If a party stops broadcasting messages or broadcasts improperly signs messages, then all Joint input: The security parameter 1 n , the number of rounds in the protocol r = r(n), a bound t on the number of corrupted parties, and the set of indices of aborted parties D 0 . Private input: Each party p j , where j / ∈ D 0 , has an input x j ∈ X n . Computing default values and signing keys 1. For every j ∈ D 0 , select x j with uniform distribution from X n . 2. Select i ⋆ ∈ [r] with uniform distribution and compute w ← f n (x 1 , . . . , x m ). Outputs: Each party p j such that j / ∈ D 0 receives • The verification key K ver . For each 1 ≤ i < i ⋆ , for each J ⊆ [m] \ D 0 s.t. m − t ≤ |J| ≤ t, (a) For each j ∈ J, set x j = x j . (b) For each j ∈ J, select uniformly at random x j ∈ X n . (c) Set σ i J ← f n ( x 1 , . . . , x m ). 4. For each i ⋆ ≤ i ≤ r and for each J ⊆ [m] \ D 0 s.t. m − t ≤ |J| ≤ t, set σ i J = w. 5. Compute (K sign , K ver ) ← Gen(1 n ). • The messages M j,1 , . . . , M j,r that p j broadcasts during the protocol. • p j 's private masks mask j (R i,J j ) produced in Step (7) other parties consider it as aborted. If m − t or more parties abort, the remaining parties reconstruct the Inputs: Each party p j holds the private input y j ∈ X n and the joint input: the security parameter 1 n , the number of rounds in the protocol r = r(n), and a bound t on the number of corrupted parties. Preliminary phase: 1. D 0 = ∅ 2. If |D 0 | < m − t, (a) The parties in {p j : j ∈ [m] \ D 0 } execute a secure-with-abort and cheat-detection protocol computing Functionality MultiShareGen r . Each honest party p j inputs y j as its input for the functionality. (b) If a party p j aborts, that is, the output of the honest parties is " abort j ", then, set D 0 = D 0 ∪ {j}, chose x j uniformly at random from x j , and goto Step (2). (c) Else (no party has aborted), denote D = D 0 and proceed to the first round. 3. Otherwise (|D 0 | ≥ m − t), the premature termination is executed with i = 1. In each round i = 1, . . . , r do: 4. Each party p j broadcasts M j,i (containing its shares in the outer secret-sharing scheme). (c) The active parties output the output of this protocol, and halt. For every At the end of round r: 8. Each active party p j broadcasts the signed shares R r,J j for each J such that j ∈ J. 9. Let J ⊆ [m] \ D be the lexicographical first set such that all the parties in Q J broadcast properly signed shares R r,J j . Each active party reconstructs the value σ r J , outputs σ r J , and halts. -with probability 1/(2p), select uniformly at random z i J ∈ Z n and set σ i J = z i J . -with the remaining probability 1 − 1/(2p), 1. For every j ∈ J select uniformly at random x j ∈ X n and for each j ∈ J, set x j = x j . 2. Compute σ i J ← f n ( x 1 , . . . , x m ). Similarly, in the protocol without the dealer, Protocol MPC r , we replace Step (3) in MultiShareGen r (described in Figure 2) with the above step. Denote the resulting protocols with and without the dealer models by MPCWithDForRange and MPCForRange r , respectively. The idea why this change improves the protocol is that now the probability that all values held by the adversary are equal prior to round i ⋆ is bigger, thus, the probability that the adversary guesses i ⋆ is smaller. This modification, however, can cause the honest parties to output a value that is not possible given their inputs, and, in general, we cannot simulate the case (which happens with probability 1/(2p)) when the output is chosen with uniform distribution from the range. Impossibility of 1/p-secure Computation with Non-Constant Number of Parties For deterministic functions, our protocol is efficient when the number of parties m is constant and the size of the domain or range is polynomial (in the security parameter n) or when the number of parties is O(log log n) and the size of the domain is constant. We next show that, in general, there is no efficient protocol when the number of parties is m(n) = ω(1) and the size of the domain is polynomial and when m(n) = ω(log n) and the size of the domain of each party is 2. This is done using the following impossibility result of Gordon and Katz [22]. Theorem 3 ([22]) For every ℓ(n) = ω(log n), there exists a deterministic 2-party functionality F with domain and range {0, 1} ℓ(n) that cannot be 1/p-securely computed for p ≥ 2 + 1/ poly(n). We next state and prove our impossibility results. Theorem 4 For every m(n) = ω(log n), there exists a deterministic m(n)-party functionality F ′ with domain {0, 1} that cannot be 1/p-securely computed for p ≥ 2 + 1/ poly(n) without an honest majority. Proof: Let ℓ(n) = m(n)/2 (for simplicity, assume m(n) is even). Let F = {f n } n∈N be the functionality guaranteed in Theorem 3 for ℓ(n). Define an m(n)-party deterministic functionality F ′ = {f ′ n } n∈N , where in f ′ n party p j gets the jth bit of the inputs of f n and the outputs of f n and f ′ n are equal Assume that F ′ can be 1/p-securely computed by a protocol Π ′ assuming that t(n) = m(n)/2 parties can be corrupted. This implies a 1/p-secure protocol Π for F with two parties, where the first party simulates the first t(n) parties in Π ′ and the second party simulates the last t(n) parties. The 1/p-security of Π is implied by the fact that any adversary A for the protocol Π can be transformed into an adversary A ′ for Π ′ controlling m(n)/2 = t(n) parties; as A ′ cannot violate the 1/p-security of Π ′ , the adversary A cannot violate the 1/p-security of Π. Theorem 5 For every m(n) = ω(1), there exists a deterministic m(n)-party functionality F ′′ with domain {0, 1} log n that cannot be 1/p-securely computed for p ≥ 2 + 1/ poly(n) without an honest majority. Proof: Let ℓ(m) = 0.5m(n) log n and let F = {f n } n∈N be the functionality guaranteed in Theorem 3 for ℓ(m). We divide the 2ℓ(n) bits of the inputs of f n into m(n) blocks of length log n. Define an m(n)-party deterministic functionality F ′′ = {f ′′ n } n∈N , where in f ′′ n party p j gets the jth block of the inputs of f n and the outputs of f n and f ′′ n are equal. As in the proof of Theorem 4, a 1/p-secure protocol for F ′′ implies a 1/p-secure protocol for F contradicting Theorem 3. The above impossibility results should be contrasted with the coin-tossing protocol of [4] which is an efficient 1/p-secure protocol even when m is polynomial in the security parameter and the number of bad parties is m(n)/2 + O(1). Notice that in both our impossibility results the size of the range is superpolynomial (as we consider the model where all parties get the same output). It is open if there is an efficient 1/p-secure protocol when the number of parties is not constant and the size of both the domain and range is polynomial. A Security with Abort and Cheat Detection We next present a definition of secure multiparty computation that is more stringent than standard definitions of secure computation with abort. This definition extends the definition for secure computation as given by Aumann and Lindell [1]. Roughly speaking, the definition requires that one of two events is possible: (1) The protocol terminates normally, and all parties receive their outputs, or (2) Corrupted parties deviate from the prescribed protocol; in this case the adversary obtains the outputs of the corrupted parties (but nothing else), and all honest parties are given an identity of one party that has aborted. The formal definition uses the real vs. ideal paradigm as discussed in Section 2.1. We next describe the appropriate ideal model. Send inputs to trusted party: The honest parties send their inputs to the trusted party. The corrupted parties may either send their received input, or send some other input of the same length (i.e., x j ∈ X n ) to the trusted party, or abort (by sending a special " abort j " message). Denote by x 1 , . . . , x m the inputs received by the trusted party. If the trusted party receives an " abort j " message, then it sends " abort j " to all honest parties and terminates (if it received " abort j " from more than one j, then it uses the minimal such j). Trusted party sends outputs to adversary: The trusted party computes w ← f n (x 1 , . . . , x m ) and sends the output w to the adversary. Adversary instructs the trusted party to continue or halt: A sends either a " continue " message or " abort j " to the trusted party for some corrupt party p j , i.e., j ∈ B. If it sends a " continue " message, the trusted party sends w to all honest parties. Otherwise, if the adversary sends " abort j ", then the trusted party sends " abort j " to all honest parties. Outputs: An honest party always outputs the value w it obtained from the trusted party. The corrupted parties output nothing. The adversary A outputs any (probabilistic polynomial-time computable) function of the auxiliary input aux, the inputs of the corrupt parties, and the value w obtained from the trusted party. We let IDEAL CD F ,S(aux) ( y, 1 n ) and REAL Π,A(aux) ( y, 1 n ) be defined as in Section 2.1 (where in this case IDEAL CD F ,S(aux) ( y, 1 n ) refers to the above execution with cheat-detection of F). This ideal model is different from that of [15] in that in the case of an "abort", the honest parties get output " abort j " and not a ⊥ symbol. This means that the honest parties know an identity of a corrupted party that causes the abort. This cheat-detection is achieved by most multiparty protocols, including that of [16], but not all (e.g., the protocol of [18] does not meet this requirement). Using this notation we define secure computation with abort and cheat-detection. Definition A.1 (security-with-abort and cheat-detection) Let F and Π be as in Definition 2.2. A protocol Π is said to securely compute F against at most t(n) corrupt parties with abort and cheat-detection if for every non-uniform polynomial-time adversary A in the real model controlling at most t(n) parties, there exists a non-uniform polynomial-time adversary S in the ideal model controlling the same parties, such that IDEAL CD F ,S(aux) ( y, 1 n ) aux∈{0,1} * , y∈(Xn) m ,n∈N C ≡ REAL Π,A(aux) ( y, 1 n ) aux∈{0,1} * , y∈(Xn) m ,n∈N . B Cryptographic Tools Signature Schemes. Informally, a signature on a message proves that the message was created by its presumed sender, and its content was not altered. A signature scheme is a triple (Gen, Sign, Ver) containing the key generation algorithm Gen, which outputs a pair of keys, the signing key K S and the verification key K v , the signing algorithm Sign, and the verifying algorithm Ver. We assume that it is infeasible to produce signatures without holding the signing key. For formal definition see [15]. Secret Sharing Schemes. An α-out-of-m secret-sharing scheme is a mechanism for sharing data among a set of parties such that every set of size α can reconstruct the secret, while any smaller set knows nothing about the secret. In this paper, we use two schemes: the XOR-based m-out-of-m scheme (i.e., in this scheme α = m) and Shamir's α-out-of-m secret-sharing scheme [26] which is used when α < m. In both schemes, for every α − 1 parties, the shares of these parties are uniformly distributed and independent of the secret. Furthermore, given such α − 1 shares and a secret s, one can efficiently complete them to m shares of the secret s. In our protocols we sometimes require that a single party learns the value of a secret that is shared among all parties. Since all messages are sent over a broadcast channel, we use two layers of secret sharing to obtain the above requirements as described below. Construction B.1 (secret sharing with respect to a certain party) Let s be a secret taken from some finite field F. We share s among m parties with respect to a (special) party p j in an α-out-of-m secret-sharing scheme as follows: 1. Choose shares (s (1) , s (2) ) of the secret s in a two-out-of-two secret-sharing scheme (that is, select s (1) ∈ F uniformly at random and compute s (2) = s − s (1) ). Denote these shares by mask j (s) and comp(s), respectively. (λ (1) , . . . , λ (j−1) , λ (j+1) , . . . , λ (m) ) of the secret comp(s) in an (α − 1)-out-of-(m − 1) Shamir's secret-sharing scheme. For each ℓ = j, denote comp ℓ (s) = λ (ℓ) . Compute shares Output: • The share of party p j is mask j (s). We call this share "p j 's masking share". • The share of each party p ℓ , where ℓ = j, is comp ℓ (s). We call this share "p ℓ 's complement share". In the above scheme, we share the secret s among the parties in P in an α-out-of-m secret-sharing scheme where only sets of size α that contain p j can reconstruct the secret. In this construction, for every β < α parties, the shares of these parties are uniformly distributed and independent of the secret. Furthermore, given such β < α shares and a secret s, one can efficiently complete them to m shares of the secret s. In addition, given β shares and a secret s, one can efficiently select uniformly at random a vector of shares competing the β shares to m shares of s. C Proof of 1/p-Security of the Protocols with a Dealer In this section we prove that our protocols described in Section 3 that assume an trusted dealer are perfect 1/ poly-secure implementations of the ideal functionality F. We start by presenting in Appendix C.1 a simulator for Protocol MPCWithD r . In Appendix C.2, we prove the correctness of the simulation by showing the the global output in the ideal-world is distributed within 1/ poly statistical distance from the global output in the real-world. In Appendix C.3, we describe the required modifications to the simulator for the protocol for F that has a polynomial-size range, and argue that the modified simulation is correct. C.1 The Simulator for Protocol MPCWithD r We next present a simulator S T for Protocol MPCWithD r , described in Figure 1. Let B be the set of indices of corrupted parties in the execution. The simulator S T invokes A on the set of inputs {y j : j ∈ B}, the security parameter 1 n , and the auxiliary input aux, playing the role of the trusted dealer in the interaction with A. Simulating the preprocessing phase: 1. D 0 = ∅. 2. The simulator S T sends a " start " message to all corrupt parties. 3. S T receives a set of inputs {x j : j ∈ B} that A submits to the computation of the dealer. If A does not submit an input on behalf of p j , i.e., A sends an " abort j " message, then, the simulator S T notifies all corrupted parties that party p j aborted and updates D 0 = D 0 ∪ {j}. 4. S T sets D = D 0 . If |D| ≥ m − t, the simulator sets i = 1 and proceeds to simulate the premature termination step. 5. S T selects i ⋆ ∈ {1, . . . , r} with uniform distribution. 6. For each i ∈ {1, . . . , i ⋆ − 1} and for each . . . , x m ). 7. The simulator S T sends " proceed " to all corrupt parties. J ⊆ B \ D 0 s.t. m − t ≤ |J| ≤ t do (a) For each j ∈ [m], if j ∈ J, then S T sets x j = x j , else, S T selects uniformly at random x j ∈ X n . (b) S T sets σ i J ← f n ( x 1 , Simulating interaction rounds: In each round 1 ≤ i ≤ r, the simulator S T interacts in three phases with the parties {p j : j ∈ B \ D 0 }, i.e., the corrupt parties which are active so far: • The peeking phase: -If i = i ⋆ , the simulator S T sends the set of inputs {x j : j ∈ B \ D 0 } to the trusted party computing F and receives w S . -For each J ⊆ B \ D 0 s.t. m − t ≤ |J| ≤ t do 1. If i ∈ {1, . . . , i ⋆ − 1}, the simulator S T sends the value σ i J (prepared in the simulation of the preprocessing phase) to all parties in Q J (i.e., to the adversary). 2. Else, if i ∈ {i ⋆ , . . . , r}, S T sends the value w S to all parties in Q J (i.e., to the adversary). • The abort phase: Upon receiving an " abort j " message from a party p j , 1. S T notifies all corrupted parties that party p j aborted. 2. S T updates D = D ∪ {j}. 3. If at least m − t parties have aborted so far, that is |D| > m − t, the simulator S T proceeds to simulate the premature termination step. • The main phase: S T sends " proceed " to all corrupt parties. Simulating the premature termination step: • If the premature termination step occurred in round i = 1, -The simulator S T receives a set of inputs {x j ′ : j ∈ B \ D} that A submits to the computation of the dealer. If A does not submit an input on behalf of p j , i.e., sends an " abort j " message, then, the simulator S notifies all corrupted parties that party p j aborted and updates D = D ∪ {j}. -The simulator S T sends the set of inputs {x j ′ : j ∈ B \ D} to the dealer and receives w S . • If the premature termination step occurred in round 1 < i < i ⋆ , 1. Upon receiving an " abort j " message from a party p j , the simulator S T updates D = D ∪ {j}. 2. The simulator S T sends the set of inputs {x j : j ∈ B \ D} to the trusted party computing F and receives w S . • (⋄ If the premature termination step occurred in round i ⋆ ≤ i ≤ r, then S T already has w S ⋄) • S T sends the value w S to each party in {p j : j ∈ B \ D 0 }. Simulating normal termination: If the last round of the protocol is completed, then S T sends w S to each party in {p j : j ∈ B \ D 0 }. At the end of the interaction with A, the simulator will output the sequence of messages exchanged between the simulator and the corrupted parties. C.2 Proof of the Correctness of the Simulation for MPCWithD r In order to prove the correctness of the simulation described in Appendix C.1, we consider the two random variables from Section 2.1, both of the form (V, C), where V describes a possible view of A, and C describes a possible output of the honest parties (i.e., C ∈ Z n ). The first random variable REAL MPCWithDr,A(aux) ( y, 1 n ) describes the real world -an execution of Protocol MPCWithD, where V describes the view of the adversary A in this execution, and C is the output of the honest parties in this execution. The second random variable IDEAL F ,S T (aux) ( y, 1 n ) describes the ideal world -an execution with the trusted party computing F (this trusted party is denoted by T F ), where V describes the output of the simulator S T in this execution, and C is the output of the honest parties in this execution. For the rest of this section, we simplify notations and denote the above two random variables by REAL = (V REAL , C REAL ) and IDEAL = (V IDEAL , C IDEAL ) respectively. We consider the probability of a given pair (v, c) according to the two different random variables. We compare the two following probabilities: (1) The probability that v is the view of the adversary A in an execution of Protocol MPCWithD r and c is the output of the honest parties in this execution, where the probability is taken over the random coins of the dealer T . (2) The probability that v is the output of the simulator S T in an ideal-world execution with the trusted party T F and c is the output of the honest parties in this execution, where the probability is taken over the random coins of the simulator S T and the random coins of the ideal-world trusted party T F . In Lemma C.3 we prove the correctness of the simulation by showing that the two random variables are within statistical distance 1/ poly. For the proof of the lemma we need the following claim from [22]. MPCWithD r and let x 1 , . . . , x m be a set of inputs. Assume that for every possible output w obtained by the dealer using this set of inputs the probability that in a round i < i ⋆ all the values that the adversary sees are equal to w is at least α. Then, the probability that A guesses i ⋆ (i.e., causes premature termination in round i ⋆ ) is at most 1/αr. Claim C.1 ([22, Lemma 2]) Let A be an adversary in Protocol As the adversary might have some auxiliary information on the inputs of the honest parties and know the value of f n (x 1 , . . . , x m ), the adversary might be able to deduce that a round is not i ⋆ if not all the values that it gets are equal to this value (or a possible value for randomized functionalities). Specifically, in the worst case scenario, the adversary knows the inputs of all the honest parties. In the next claim we show a lower bound on the probability that all the values that the adversary obtains in a round i < i ⋆ of Protocol MPCWithD r are all equal to a fixed value. Claim C.2 Let d(n) and g(n) be the size of the domain and range, respectively, of a randomized functionality F computed by the protocol MPCWithD r . Let ǫ be a number such that Pr[f n (x 1 , . . . , x m ) = w ℓ ] ≥ ǫ for every set of inputs x 1 , . . . , x m and for each w ℓ from the range of f n (x 1 , . . . , x m ). Then, the probability that in a round i < i ⋆ all the values that the adversary sees are equal to a specific w is at least (ǫ/d(n) m ) 2 t −1 . Furthermore, if F is deterministic, then, this probability is at least (1/d(n) m ) 2 t −1 . Lemma C.3 Let F be a (possibly randomized) functionality, A be a non-uniform polynomial-time adversary corrupting t < 2m/3 parties in an execution of Protocol MPCWithD, and S T be the simulator described in Appendix C.1 (where S T controls the same parties as A). Then, for every n ∈ N, for every y ∈ (X n ) m , and for every aux ∈ {0, 1} * SD REAL MPCWithDr,A(aux) ( y, 1 n ),IDEAL F ,S T (aux) ( y, 1 n ) ≤ 2g(n)d(n) m / (r(n)) 2 t , where d(n) and g(n) are the sizes of the range and the domain of F, respectively, and r(n) be the number of rounds in the protocol. Furthermore, if F is deterministic, then, the statistical distance between these two random variables is at most (d(n) m ) 2 t /r(n). Proof: Our goal here is to show that the statistical distance between the above two random variables is at most as described in lemma. The flow of our proof is as follows. We first bound the statistical distance between the two random variables by the probability that the adversary A guesses the special round i ⋆ . We do this by showing that, conditioned on the event that the adversary fails to guess round i ⋆ , the two random variables are identically distributed. Then, we bound the probability of guessing i ⋆ in time using Claim C.1 and Claim C.2. Observe that, in the simulation, S T follows the same instructions as the trusted party T in Protocol MPCWithD r , except for two changes. First, S T does not compute the output w S , but rather gets w S externally from T F . The simulator obtains this value either in the premature termination phase (if i < i ⋆ ) or in the peeking stage when i = i ⋆ . The second difference is that in the case of a premature termination, S T will always use w S as its message to the corrupt parties, while T will use the value from round i ⋆ − 1 of the appropriate subset Q J as its message. We analyze the probabilities of (v, c) in the two random variables according to weather the premature termination occurred before, during, or after the special round i ⋆ . Premature termination before round i ⋆ . We argue that in this case, both in the real protocol and in the simulation, the view of A is identically distributed in the two worlds. S T follows the same random process in interacting with A (before sending the last message in the premature termination) as does T in the real-world execution. The view of the adversary consists of values which are outputs of evaluations of the function f n on the same input distributions. The adversary does not learn anything about the inputs of the honest parties, hence, its decision to abort does not depend on any new information it obtains during the interaction rounds so far. In addition, in both worlds, the output of the honest parties is the evaluation of the function f n on the same set of inputs for the active parties and uniformly selected random inputs for the aborted parties. Premature termination after round i ⋆ or never occurs. Here v must contain σ i ⋆ J for some J, which, in the real-world execution, is equal to the output value of all sets for any round i > i ⋆ (recall that the output value of the honest parties will be determined by one such value), and in the simulation it equals w S . Thus, in both scenarios, v must be consistent with i ⋆ and with c, hence, v completely determines C. Again, since S T follows the same random process in interacting with A as does T in the real-world execution the probabilities are the same. Premature termination in round i ⋆ . This is the interesting case, which causes the statistical distance. In the real world, the output of the honest parties is σ i ⋆ −1 J for some J, while in the ideal world their output is w S ← f n (x 1 , . . . , x m ). In the first case the output is independent of the adversary's view, while in the It was explained in the proof of Claim C.2 that in each round of the protocol, A obtains less than 2 t values. Therefore, we conclude that he probability that all the values that A obtains in round i < i ⋆ are all equal to w is at least (1/(2p · g)) 2 t . By applying the Claim C.1 we conclude that the probability of the adversary guessing i ⋆ correctly in Protocol MPCWithDForRange r is at most (2p · g) 2 t /r. In case of a premature termination in round i < i ⋆ , with probability 1 − 1/(2p) in both the ideal world and real world, the value that the honest parties output is the evaluation of f n on the inputs of the active parties and random inputs for the parties that aborted. However, with probability 1/(2p), if premature termination occurs prior to round i ⋆ , the output of the honest parties Protocol MPCWithDForRange r is a random value from the range of f n ; the simulator fails to simulate the execution in this case and outputs ⊥. Thus, SD (IDEAL, REAL) ≤ Pr[Premature termination in round i ⋆ ] + (1/2p) · Pr[Premature termination before round i ⋆ ] ≤ (2p · g) 2 t /r + (1/2p). Therefore, the statistical distance is as claimed. D Proof of Security for the Protocols without the Dealer D.1 The Simulator for Protocol MPC r We next prove that Protocol MPC r is a secure real-world implementation of the (ideal) functionality of Protocol MPCWithD r . By Lemma C.3, when r(n) is sufficiently large, Protocol MPCWithD r is a 1/psecure protocol for F. Thus, together we get that Protocol MPC r is a 1/p-secure protocol for F. according to the definition appears in Appendix A. We analyze Protocol MPC r in a hybrid model where there are 3 ideal functionalities: Functionality MultiShareGenWithAbort r . This functionality is an (ideal) execution of Functionality MultiShareGen r in the secure-with-abort and cheat-detection model. That is, the functionality gets a set of inputs. If the adversary sends " abort j " for some corrupt party p j , then this message is sent to the honest parties and the execution terminates. Otherwise, Functionality MultiShareGen r is executed. Then, the adversary gets the outputs of the corrupt parties. Next, the adversary decides whether to halt or to continue: If the adversary decides to continue, it sends a " proceed " message and the honest parties are given their outputs. Otherwise, the adversary sends " abort j " for some corrupt party p j , and this message is sent to the honest parties. Functionality FairMPC. This functionality computes the value f n (x 1 , . . . , x m ). That is, the functionality gets a set of inputs. If a party p j sends " abort j " message then x j selected from X n with uniform distribution, computes an output of the randomized functionality f n for them, and gives it to all parties. When this functionality is executed, an honest majority is guaranteed, hence, the functionality can be implemented with full security (e.g., with fairness). Functionality Reconstruction. This functionality is described in Figure 4; this functionality is used in the premature termination step in Protocol MPC r for reconstructing the output value from the shares of the previous round. When this functionality is executed, an honest majority is guaranteed, hence, the functionality can be implemented with full security (e.g., with fairness). Lemma D.2 Let t < 2m/3. If enhanced trap-door permutations exist, then Protocol MPCForRange r described in Section 3.3, is a computationally-secure implementation (with full security) of the dealer functionality in Protocol MPCWithDForRange r . Proof: Recall that the only difference between Protocol MPC r and Protocol MPCForRange r is in the way that the values that the parties see prior round i ⋆ are produced, i.e., the difference is in Functionality MultiShareGen r . Specifically, in Section 3.3 we presented a modification in Step (3) in Functionality MultiShareGen r in order to get Protocol MPC r from Protocol MPCForRange. Now, observe that the simulator presented above does not refer to Step (3) of Functionality MultiShareGen r in any step. Therefore, the simulator presented in Appendix D.1 for Protocol MPC r is also a simulator for Protocol MPCForRange r . Claim C.5 and Lemma D.2 imply Theorem 2. Figure 1 : 1Protocol MPCWithD r . Computing signed shares of the inner secret-sharing scheme6. For each i ∈ {1, . . . , r} and for each J ⊆[m] \ D 0 s.t. m − t ≤ |J| ≤ t,(a) Create shares of σ i J in a |J|-out-of-|J| secret-sharing scheme for the parties in Q J . For each party p j ∈ Q J , let S i,J j be its share of σ i J .(b) Sign each share S i,J j : compute R i,J j ← (S i,J j , i, J, j, Sign((S i,J j , i, J, j), K sign )).Computing shares of the outer secret-sharing scheme7. For each i ∈ [r], for each J ⊆ [m] \ D 0 s.t. m − t ≤ |J| ≤ t,and each j ∈ J, share R i,J j using a (t + 1)-out-of-m secret-sharing scheme with respect to p j as defined in Construction B.1: compute one masking share mask j (R i,J j ) and m − 1 complement shares (comp 1 (R i,J j ), . . . , comp j−1 (R i,J j ), comp j+1 (R i,J j ), . . . , comp m (R i,J j )).Signing the messages of all parties8. For every 1 ≤ q ≤ m, compute the message m q,i that p q ∈ P broadcasts in round i by concatenating (1) q, (2) i, and (3) the complement shares comp q (R i,J j ) produced in Step(7)for p q (for all J ⊆ [m] \ D 0 s.t. m − t ≤ |J| ≤ t and all j = q s.t. j ∈ J), and compute M q,i ← (m q,i , Sign(m q,i , K sign )). , for each 1 ≤ i ≤ r and each J ⊆ [m] \ D 0 s.t. m − t ≤ |J| ≤ t and j ∈ J. Figure 2 : 2The initialization functionality MultiShareGen r . p j s.t. Ver(M j,i , K ver ) = 0 or if p j broadcasts an invalid or no message, then all parties mark p j as inactive, i.e., set D = D ∪ {j}. If |D| ≥ m − t, premature termination is executed. Premature termination step 6. If i = 1, the active parties use a multiparty secure protocol (with full security) to compute f n : Each honest party inputs y j and the input of each inactive party is chosen uniformly at random from X n . The active parties output the result, and halt. 7. Otherwise, (a) Each party p j reconstructs R i−1,J j , the signed share of the inner secret-sharing scheme produced in Step (6) of Functionality MultiShareGen r , for each J ⊆ [m] \ D 0 s.t. m − t ≤ |J| ≤ t and j ∈ J. (b) The active parties execute a secure multiparty protocol with an honest majority to compute Functionality Reconstruction, where the input of each party p j isR i−1,J j for every J ⊆ [m] \ D 0 s.t. m − t ≤ |J| ≤ t and j ∈ J. Figure 3 : 3The m-party protocol MPC r for computing F.Joint Input: The round number i, the indices of inactive parties D, a bound t on the number of corrupted parties, and the verification key, K ver . Private Input of p j : A set of signed shares R i−1,J j for each J ⊆ [m] \ D 0 s.t. m − t ≤ |J| ≤ t and j ∈ J. Computation: 1. For each p j , if p j 's input is not appropriately signed or malformed, then D = D∪{j}. 2. Set J = [m] \ D. 3. Reconstruct σ i−1 J from the shares of all the parties in Q J . Outputs: All parties receive the value σ i−1 J (as their output). Figure 4 : 4Functionality Reconstruction for reconstructing the output in the premature termination step. Execution in the ideal model. Let B ⊆ [m] denote the set of indices of corrupted parties controlled by an adversary A. The adversary A receives an auxiliary input denoted aux. An ideal execution proceeds as follows: As parties can abort during this reconstruction, they actually reconstruct the value of a subset of this set.2 The problem in our protocols is that the adversary can keep one corrupted party active, thus, the adversary can get the output of the honest parties. For the simplicity of the presentation of our protocols, we present a slightly different ideal world than the traditional one. In our model there is no a default input in case of an "abort". However, the protocol can be presented in the traditional model, where a predefined default input is used if a party aborts. For example, there might not be possible inputs of the corrupt parties causing the honest parties to output such output. These shares are temporary and will later be open to the actual values obtained from T MPCWithD during the interaction rounds using the properties of Shamir's secret-sharing scheme. In Steps 2-5, the simulator S constructs the messages of the honest parties in order to allow the corrupted parties in each J ∈ J to reconstruct τ i J . Case 1:The adversary guesses i ⋆ with some light w. Since there are at most g possible values of f n (x 1 , . . . , x m ), the probability of this event, by the union bound, is at most 1/p 0 .Case 2:The adversary guesses i ⋆ with some heavy w. Thus, by Claim C.2 where ǫ = p 0 · g, the probability of w = σ i J for all values that the adversary sees in round i < i ⋆ is at least (1/d m · p 0 · g) 2 t −1 . By Claim C.1, the probability that the adversary guesses i ⋆ conditioned on the w being heavy is at most (d m · p 0 · g) 2 t −1 /r.We take p 0 = r 2 −t /(g · d m ); the total probability that the adversary guesses i ⋆ in the two cases is at mostTherefore, by Equation(1), the statistical distance between the two random variables in the randomized case is as claimed in the lemma. The case that F is deterministic is simpler. By combining Claim C.1 and Claim C.2 we get that the probability that A guesses i ⋆ is at most (r/d(n) m ) 2 t −1 . By applying Equation (1), we get the bound on statistical distance between the two random variables for the deterministic case as claimed in the lemma.C.3 The Simulator for the Protocol with the Dealer for Polynomial RangeLemma C.4 Let F be a (possibly randomized) functionality. For every non-uniform polynomial-time adversary A corrupting t < 2m/3 parties in an execution of Protocol MPCWithDForRange, there exists a simulator S T in the ideal model, that simulates the execution of A (where S T controls the same parties as A). That is, for every n ∈ N, for every y ∈ (X n ) m , and for every aux ∈ {0, 1} *where g(n) is the size of the range of F, with probability 1/(2p(n)) each value σ i J in round i < i ⋆ is selected uniformly at random from the range, and r(n) be the number of rounds in the protocol.Proof: The simulators and their proofs for Protocol MPCWithDForRange and Protocol MPCWithD are similar; we only present (informally) the differences between the two simulators and the two proofs.The modified simulator. Recall that the protocols MPCWithD and MPCWithDForRange are different only in Step (3) of the share generation step. In MPCWithDForRange, each value σ i J prior to round i ⋆ is chosen with probability 1/(2p) as a random value from the range of f n and with probability 1 − 1/(2p) it is chosen just like inFigure 1. There are two modifications to the simulator. The first modification in the simulator is in Step(6)in the simulation of the preprocessing phase, i.e., in the computation of σ i J for i < i ⋆ . The step that replaces Step (6) appears below.The second modification is less obvious. Recall that both random variables appearing in the lemma contain the output of the honest parties. In the ideal world, the honest parties always output f n applied to their inputs. In the real world, in a premature termination in round i < i ⋆ , with probability 1/(2p), the honest parties output a random value from the range of f n . It is hard to simulate the output of the honest parties in first case.4We simply modify the simulator such that with probability 1/(2p) the simulator returns ⊥, i.e., it announces that the simulation has failed. The new premature termination step appears below.Simulating the premature termination step:• If the premature termination step occurred in round i < i ⋆ ,-With probability 1/(2p), for each j ∈ B \ D 0 send " abort j " to the trusted party computing F and return ⊥. -With the remaining probability 1 − 1/(2p), execute the original simulation of the premature termination step (appearing in Appendix C.1). • Else (i ≥ i ⋆ ), execute the original simulation of the premature termination step (appearing in Appendix C.1).The modified proof. The proof to the simulator for MPCWithDForRange remains basically the same, except for two changes. We first modify Claim C.2 below and prove a slightly different claim, which changes the probability of the adversary guessing i ⋆ .Claim C.5 Let g(n) be the size of the range of the (possibly randomized) functionality F computed by the protocol MPCWithDForRange r and w ∈ Z n . Then, the probability that in a round i < i ⋆ all the values that the adversary sees are equal to w is at least (1/2p(n) · g(n)) 2 t .Proof: According to the protocol, there are two different ways to produce each value σ i J in round i < i ⋆ : (1) Compute f n on a set of inputs and a set of uniformly selected values from the domain of the functionality, and (2) Set σ i J as a uniformly selected value from the range of the functionality. We ignore the first case. In the second option, with probability 1/2p, the value σ i J is uniformly selected from the range. Hence, the probability that σ i J is equal to a specific value is at least 1/(2p · g).Simulating interaction rounds:Let J be the collection of subsets J ⊆ B \ D 0 s.t. m − t ≤ |J| ≤ t. I.e., J is the collection of sets of indices of active corrupt parties after the simulation of the executions of MultiShareGenWithAbort r To simulate round i for i = 1, . . . , r, the simulator S proceeds as follows: gets a bit τ i J for each J ∈ J . 6 2. The simulator S selects shares for the inner secret-sharing scheme for corrupted parties: For every J ∈ J , the simulator S selects uniformly at random shares of τ i J in a |J|-out-of-|J| Shamir secret sharing scheme. Denote these shares by X i,J j : p j ∈ Q J . For each p j ∈ Q J , let Y i,J j ← (X i,J j , i, J, j, Sign((X i,J j , i, J, j), K sign )). 3. The simulator S selects complementary shares for all honest parties: For every J ∈ J and for each j ∈ B \ D 0 ,S selects uniformly at random m − t shares of α j uniformly at random over all possible selections of m − t shares that are shares of α j together with the |B \ D 0 | − 1 sharesStep(3)Then, S signs m ′ q,i , i.e., S computes M ′ q,i ← (m ′ q,i , Sign(m ′ q,i , K sign )). 6. The simulator S sends all the message M ′ q,i on behalf of each honest party p q to A. 7. For every j ∈ B \ D 0 s.t. A sends an invalid or no message on behalf of p j , the simulator S sends " abort j " to T MPCWithD :(c) Otherwise, the simulator S proceeds to the next round.Simulating the premature termination step:• If i = 1, then S simulates A's interaction with Functionality FairMPC as follows:1. S receives from A the inputs of the active corrupt parties. 2. For every j ∈ B \ D: If p j does not send an input, then S sends " abort j " to T MPCWithD else, S sends p j 's input to T MPCWithD .Simulating normal termination at the end of round r:1. The simulator gets w from the trusted party T MPCWithD .2. S constructs all the singed shares of the inner secret-sharing scheme for each J ⊆ [m] \ D 0 s.t. m − t ≤ |J| ≤ t and for each honest party p j ∈ Q J as follows.For each J / ∈ J , the simulator S selects uniformly at random |J \ B| shares of w uniformly at random over all possible selections of |J \ B| shares that together with the |J ∩ B| given shares R i,J j : j ∈ B (produced inStep(2)in the simulation of the preliminary phase) are a sharing of w in a |J|-out-of-|J| secret sharing scheme. (This is possible according to the property of Shamir's scheme) Denote these shares by X r,J j .For each share X r,J j , the simulator concatenates the corresponding identifying details, and signs them to obtain: Y r,J j ← (X r,J j , r, J, j, Sign((X r,J j , r, J, j), K sign )). 3. For each honest party p j , the simulator S sends to A the shares Y r,J j for all subsets J, such that p j ∈ Q J . 4. The simulator S outputs the sequence of messages exchanged between S and the adversary A and halts.D.2 Proving the Correctness of Protocol MPC r and Protocol MPCForRange rIt can be proved that Protocol MPC r is a secure implementation of the (ideal) functionality of the dealer's in Protocol MPCWithD r . That is, Lemma D.1 Let t < 2m/3. If enhanced trap-door permutations exist, then Protocol MPC r presented in Section 3.2, is a computationally-secure implementation (with full security) of the dealer functionality in Protocol MPCWithD r .In[3], a similar framework to the one used in this paper is used: first a protocol with a dealer for the coin-tossing problem is presented and, then, a real-world protocol that is a computationally-secure implementation (with full security) of the dealer functionality is described. In[3], a simulator for this protocol is given. This simulator is similar to the simulator described in Appendix D.1, than a full proof for the simulator is provided. As the proof is very similar to the proof of our simulator, we omit the proof.To conclude the proof, as MPCWithD r is a 1/p-secure implementation of F and MPC r is a secure implementation of the (ideal) functionality of the dealer in Protocol MPCWithD r , by the composition theorem of Canetti[8]we conclude that MPC r 1/p-secure implementation of F. That is, Theorem 1 is proved.Next, we claim that MPCForRange r is a secure implementation of the (ideal) functionality of the dealer in Protocol MPCWithDForRange r . That is, Security against covert adversaries: Efficient protocols for realistic adversaries. Y Aumann, Y Lindell, Proc. of the Fourth Theory of Cryptography Conference -TCC. S. Vadhanof the Fourth Theory of Cryptography Conference -TCCSpringer-Verlag4392Y. Aumann and Y. Lindell. Security against covert adversaries: Efficient protocols for realistic ad- versaries. In S. Vadhan, editor, Proc. of the Fourth Theory of Cryptography Conference -TCC 2006, volume 4392 of Lecture Notes in Computer Science, pages 137-156. Springer-Verlag, 2007. Multiparty computation with faulty majority. D Beaver, S Goldwasser, Proc. of the 30th IEEE Symp. on Foundations of Computer Science. of the 30th IEEE Symp. on Foundations of Computer ScienceD. Beaver and S. Goldwasser. Multiparty computation with faulty majority. In Proc. of the 30th IEEE Symp. on Foundations of Computer Science, pages 468-473, 1989. Protocols for multiparty coin toss with dishonest majority. Full version of. A Beimel, E Omri, I Orlov, 4A. Beimel, E. Omri, and I. Orlov. Protocols for multiparty coin toss with dishonest majority. Full version of [4]. Protocols for multiparty coin toss with dishonest majority. A Beimel, E Omri, I Orlov, Advances in Cryptology -CRYPTO 2010. T. RabinSpringer-Verlag6223A. Beimel, E. Omri, and I. Orlov. Protocols for multiparty coin toss with dishonest majority. In T. Rabin, editor, Advances in Cryptology -CRYPTO 2010, volume 6223 of Lecture Notes in Computer Science, pages 538-557. Springer-Verlag, 2010. A fair protocol for signing contracts. M Ben-Or, O Goldreich, S Micali, R Rivest, Proceedings of the 12th Colloquium on Automata, Languages and Programming. the 12th Colloquium on Automata, Languages and ProgrammingSpringer-VerlagM. Ben-Or, O. Goldreich, S. Micali, and R. Rivest. A fair protocol for signing contracts. In Proceedings of the 12th Colloquium on Automata, Languages and Programming, pages 43-52. Springer-Verlag, 1985. How to exchange (secret) keys. M Blum, ACM Trans. Comput. Syst. 12M. Blum. How to exchange (secret) keys. ACM Trans. Comput. Syst., 1(2):175-193, 1983. Timed commitments. D Boneh, M Naor, Advances in Cryptology -CRYPTO 2000. M. BellareSpringer-Verlag1880D. Boneh and M. Naor. Timed commitments. In M. Bellare, editor, Advances in Cryptology -CRYPTO 2000, volume 1880 of Lecture Notes in Computer Science, pages 236-254. Springer-Verlag, 2000. Security and composition of multiparty cryptographic protocols. R Canetti, J. of Cryptology. 131R. Canetti. Security and composition of multiparty cryptographic protocols. J. of Cryptology, 13(1):143-202, 2000. Limits on the security of coin flips when half the processors are faulty. R Cleve, Proc. of the 18th STOC. of the 18th STOCR. Cleve. Limits on the security of coin flips when half the processors are faulty. In Proc. of the 18th STOC, pages 364-369, 1986. Controlled gradual disclosure schemes for random bits and their applications. R Cleve, Advances in Cryptology -CRYPTO '89. G. BrassardSpringer-Verlag435R. Cleve. Controlled gradual disclosure schemes for random bits and their applications. In G. Brassard, editor, Advances in Cryptology -CRYPTO '89, volume 435 of Lecture Notes in Computer Science, pages 573-588. Springer-Verlag, 1990. Practical and provably secure release of a secret and exchange of signatures. I Damgård, J. of Cryptology. 84I. Damgård. Practical and provably secure release of a secret and exchange of signatures. J. of Cryp- tology, 8(4):201-222, 1995. A randomized protocol for signing contracts. S Even, O Goldreich, A Lempel, CACM. 286S. Even, O. Goldreich, and A. Lempel. A randomized protocol for signing contracts. CACM, 28(6):637-647, 1985. Cryptographic computation: Secure fault-tolerant protocols and the public-key model. Z Galil, S Haber, M Yung, Advances in Cryptology -CRYPTO '87. C. PomeranceSpringer-Verlag293Z. Galil, S. Haber, and M. Yung. Cryptographic computation: Secure fault-tolerant protocols and the public-key model. In C. Pomerance, editor, Advances in Cryptology -CRYPTO '87, volume 293 of Lecture Notes in Computer Science, pages 135-155. Springer-Verlag, 1988. Resource fairness and composability of cryptographic protocols. J A Garay, P D Mackenzie, M Prabhakaran, K Yang, Proc. of the Third Theory of Cryptography Conference -TCC. S. Halevi and T. Rabinof the Third Theory of Cryptography Conference -TCC3876J. A. Garay, P. D. MacKenzie, M. Prabhakaran, and K. Yang. Resource fairness and composability of cryptographic protocols. In S. Halevi and T. Rabin, editors, Proc. of the Third Theory of Cryptog- raphy Conference -TCC 2006, volume 3876 of Lecture Notes in Computer Science, pages 404-428. . Springer-Verlag, Springer-Verlag, 2006. Foundations of Cryptography, Voume II Basic Applications. O Goldreich, Cambridge University PressO. Goldreich. Foundations of Cryptography, Voume II Basic Applications. Cambridge University Press, 2004. How to play any mental game. O Goldreich, S Micali, A Wigderson, Proc. of the 19th ACM Symp. on the Theory of Computing. of the 19th ACM Symp. on the Theory of ComputingO. Goldreich, S. Micali, and A. Wigderson. How to play any mental game. In Proc. of the 19th ACM Symp. on the Theory of Computing, pages 218-229, 1987. Fair computation of general functions in presence of immoral majority. S Goldwasser, L Levin, Advances in Cryptology -CRYPTO '90. A. J. Menezes and S. A. VanstoneSpringer-Verlag537S. Goldwasser and L. Levin. Fair computation of general functions in presence of immoral majority. In A. J. Menezes and S. A. Vanstone, editors, Advances in Cryptology -CRYPTO '90, volume 537 of Lecture Notes in Computer Science, pages 77-93. Springer-Verlag, 1991. Secure computation without agreement. S Goldwasser, Y Lindell, DISC '02: Proceedings of the 16th International Conference on Distributed Computing. London, UKSpringer-VerlagS. Goldwasser and Y. Lindell. Secure computation without agreement. In DISC '02: Proceedings of the 16th International Conference on Distributed Computing, pages 17-32, London, UK, 2002. Springer-Verlag. Complete fairness in multi-party computation without an honest majority. D Gordon, J Katz, Proc. of the Sixth Theory of Cryptography Conference -TCC 2009. of the Sixth Theory of Cryptography Conference -TCC 2009Berlin, HeidelbergSpringer-VerlagD. Gordon and J. Katz. Complete fairness in multi-party computation without an honest majority. In Proc. of the Sixth Theory of Cryptography Conference -TCC 2009, pages 19-35, Berlin, Heidelberg, 2009. Springer-Verlag. Complete fairness in secure two-party computation. S D Gordon, C Hazay, J Katz, Y Lindell, Proc. of the 40th ACM Symp. on the Theory of Computing. of the 40th ACM Symp. on the Theory of ComputingS. D. Gordon, C. Hazay, J. Katz, and Y. Lindell. Complete fairness in secure two-party computation. In Proc. of the 40th ACM Symp. on the Theory of Computing, pages 413-422, 2008. On complete primitives for fairness. S D Gordon, Y Ishai, T Moran, R Ostrovsky, A Sahai, Proc. of the Seventh Theory of Cryptography Conference -TCC 2010. D. Micciancioof the Seventh Theory of Cryptography Conference -TCC 2010Springer-Verlag5978S. D. Gordon, Y. Ishai, T. Moran, R. Ostrovsky, and A. Sahai. On complete primitives for fairness. In D. Micciancio, editor, Proc. of the Seventh Theory of Cryptography Conference -TCC 2010, volume 5978 of Lecture Notes in Computer Science, pages 91-108. Springer-Verlag, 2010. Partial fairness in secure two-party computation. S D Gordon, J Katz, Advances in Cryptology -EUROCRYPT 2010. Henri GilbertSpringer-Verlag6110S. D. Gordon and J. Katz. Partial fairness in secure two-party computation. In Henri Gilbert, editor, Advances in Cryptology -EUROCRYPT 2010, volume 6110 of Lecture Notes in Computer Science, pages 157-176. Springer-Verlag, 2010. How to simultaneously exchange a secret bit by flipping a symmetrically-biased coin. M Luby, S Micali, C Rackoff, Proc. of the 24th IEEE Symp. on Foundations of Computer Science. of the 24th IEEE Symp. on Foundations of Computer ScienceM. Luby, S. Micali, and C. Rackoff. How to simultaneously exchange a secret bit by flipping a symmetrically-biased coin. In Proc. of the 24th IEEE Symp. on Foundations of Computer Science, pages 11-21, 1983. An optimally fair coin toss. T Moran, M Naor, G Segev, Proc. of the Sixth Theory of Cryptography Conference -TCC 2009. of the Sixth Theory of Cryptography Conference -TCC 2009T. Moran, M. Naor, and G. Segev. An optimally fair coin toss. In Proc. of the Sixth Theory of Cryptography Conference -TCC 2009, pages 1-18, 2009. Fair secure two-party computation. B Pinkas, Advances in Cryptology -EU-ROCRYPT 2003. E. BihamSpringer-Verlag2656B. Pinkas. Fair secure two-party computation. In E. Biham, editor, Advances in Cryptology -EU- ROCRYPT 2003, volume 2656 of Lecture Notes in Computer Science, pages 87-105. Springer-Verlag, 2003. How to share a secret. A Shamir, Communications of the ACM. 22A. Shamir. How to share a secret. Communications of the ACM, 22:612-613, 1979. second case, the view determines the output. Thus, in this case the probabilities of the views are different. However, we will show that the event of premature termination in round i ⋆ happens with small probability. Since the probabilities of (v, c) in the first two cases are equal, the statistical distance between the two. A C Yao ; ≤ Pr, Proc. of the 27th IEEE Symp. on Foundations of Computer Science. of the 27th IEEE Symp. on Foundations of Computer ScienceSD (IDEAL, REALHow to generate and exchange secrets. Premature termination in round i ⋆A. C. Yao. How to generate and exchange secrets. In Proc. of the 27th IEEE Symp. on Foundations of Computer Science, pages 162-167, 1986. second case, the view determines the output. Thus, in this case the probabilities of the views are different. However, we will show that the event of premature termination in round i ⋆ happens with small probability. Since the probabilities of (v, c) in the first two cases are equal, the statistical distance between the two , SD (IDEAL, REAL) ≤ Pr[Premature termination in round i ⋆ ]. We next use Claim C.1 and Claim C.2 to bound the probability that the adversary guesses i ⋆ . However, there might be values such that Pr. w = f n (x 1 , . . . , x m )We next use Claim C.1 and Claim C.2 to bound the probability that the adversary guesses i ⋆ . However, there might be values such that Pr[w = f n (x 1 , . . . , x m )] The simulator S receives a set of inputs {x j : j ∈ B \ D 0 } that A submits to Functionality MultiShareGenWithAbort r . If a party p j for j ∈ B \ D 0 does not submit an input, i.e., sends an " abort j " message, then, (a) S sends. abort j " to the trusted party T MPCWithDThe simulator S receives a set of inputs {x j : j ∈ B \ D 0 } that A submits to Functionality MultiShareGenWithAbort r . If a party p j for j ∈ B \ D 0 does not submit an input, i.e., sends an " abort j " message, then, (a) S sends " abort j " to the trusted party T MPCWithD . Otherwise (|D 0 | ≥ m − t), simulate premature termination with i = 1. Otherwise (|D 0 | ≥ m − t), simulate premature termination with i = 1. The simulator S sets σ i J = 0 for every J ⊆ [m] \ D 0 s.t. m − t ≤ |J| ≤ t and for all i ∈ {1, . . . , r}. the simulator S sends to A: • The verification key K ver . • The masking shares mask j (R i,J j ) for each i ∈ {1. S prepares outputs for the corrupted parties for Functionality MultiShareGenWithAbort r. r} and for every J ⊆ [m] \ D 0 s.t. messages M j,1 , . . . , M j,rS prepares outputs for the corrupted parties for Functionality MultiShareGenWithAbort r : The simulator S sets σ i J = 0 for every J ⊆ [m] \ D 0 s.t. m − t ≤ |J| ≤ t and for all i ∈ {1, . . . , r}. the simulator S sends to A: • The verification key K ver . • The masking shares mask j (R i,J j ) for each i ∈ {1, . . . , r} and for every J ⊆ [m] \ D 0 s.t. messages M j,1 , . . . , M j,r . (a) S sends. S , Then , abort j " to the trusted party T MPCWithDS, then, (a) S sends " abort j " to the trusted party T MPCWithD . Otherwise (|D 0 | ≥ m − t), go to simulating premature termination with i = 1. Otherwise (|D 0 | ≥ m − t), go to simulating premature termination with i = 1. Otherwise (A sends a " continue " message to S), (a) The simulator S denotes D = D 0. Otherwise (A sends a " continue " message to S), (a) The simulator S denotes D = D 0 . The simulator sends x j to T MPCWithD for every j ∈ B \ D 0 (and gets as response a " proceed " message). The simulator sends x j to T MPCWithD for every j ∈ B \ D 0 (and gets as response a " proceed " message). That is, S • If i > 1, then S simulates A's interaction with Functionality Reconstruction as follows: 1. S receives from A the inputs of the active corrupt parties, i.e., p j s.t. j ∈ B \ D. 2. If an active corrupt party p j , does not send an input, or its input is not appropriately signed or malformed, then S sends. S gets from the trusted party T MPCWithD the values that the corrupted parties see. abort j " to T MPCWithDS gets from the trusted party T MPCWithD the values that the corrupted parties see. That is, S • If i > 1, then S simulates A's interaction with Functionality Reconstruction as follows: 1. S receives from A the inputs of the active corrupt parties, i.e., p j s.t. j ∈ B \ D. 2. If an active corrupt party p j , does not send an input, or its input is not appropriately signed or malformed, then S sends " abort j " to T MPCWithD .
{'fraction_non_alphanumeric': 0.05906971701181828, 'fraction_numerical': 0.014463840399002495, 'mean_word_length': 3.6946452204010996, 'pattern_counts': {'":': 0, '<': 39, '<?xml version=': 0, '>': 6, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 57, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'A protocol for computing a functionality is secure if an adversary in this protocol cannot cause more harm than in an ideal computation where parties give their inputs to a trusted party which returns the output of the functionality to all parties. In particular, in the ideal model such computation is fairall parties get the output. Cleve (STOC 1986) proved that, in general, fairness is not possible without an honest majority. To overcome this impossibility, Gordon and Katz (Eurocrypt 2010) suggested a relaxed definition -1/p-secure computation -which guarantees partial fairness. For two parties, they construct 1/p-secure protocols for functionalities for which the size of either their domain or their range is polynomial (in the security parameter). Gordon and Katz ask whether their results can be extended to multiparty protocols.We study 1/p-secure protocols in the multiparty setting for general functionalities. Our main result is constructions of 1/p-secure protocols when the number of parties is constant provided that less than 2/3 of the parties are corrupt. Our protocols require that either (1) the functionality is deterministic and the size of the domain is polynomial (in the security parameter), or (2) the functionality can be randomized and the size of the range is polynomial. If the size of the domain is constant and the functionality is deterministic, then our protocol is efficient even when the number of parties is O(log log n) (where n is the security parameter). On the negative side, we show that when the number of parties is super-constant, 1/p-secure protocols are not possible when the size of the domain is polynomial.A protocol for computing a functionality is secure if an adversary in this protocol cannot cause more harm than in an ideal computation where parties give their inputs to a trusted party which returns the output of the functionality to all parties. This is formalized by requiring that for every adversary in the real world, there is an adversary in the ideal world, called simulator, such that the output of the real-world adversary and the simulator are indistinguishable in polynomial time. Such security can be achieved when there is a majority of honest parties[16]. Secure computation is fair -all parties get the output. Cleve[9]proved that, in general, fairness is not possible without an honest majority.To overcome the impossibility of [9], Gordon and Katz[22]suggested a relaxed definition -1/p-secure computation -which guarantees partial fairness. Informally, a protocol is 1/p-secure if for every adversary in the real world, there is a simulator running in the ideal world, such that the output of the real-world adversary and the simulator cannot be distinguished with probability greater than 1/p. For two parties, Gordon and Katz construct 1/p-secure protocols for functionalities whose size of either their domain or their range is polynomial (in the security parameter). They also give impossibility results when both the domain and range are super-polynomial. Gordon and Katz ask whether their results can be extended to multiparty protocols. We give positive and negative answers to this question.Previous Results. Cleve [9] proved that any protocol for coin-tossing without an honest majority cannot be fully secure, specifically, if the protocol has r rounds, then it is at most 1/r-secure. Protocols with partial fairness, under various definitions and assumptions, have been constructed for coin-tossing [9, 10, 24, 4], for contract signing/exchanging secrets[6,23,12,5,11,7], and for general functionalities[27,13,2,17,25,14,22]. We next describe the papers that are most relevant to our paper. Moran, Naor, and Segev [24] construct 2-party protocols for coin tossing that are 1/r-secure (where r is the number of rounds in the protocol). Gordon and Katz [22] define 1/p-security and construct 2-party 1/p-secure protocols for every functionality whose size of either the domain or the range of the functionality is polynomial. Finlay, in a previous work [4] we construct multiparty protocols for coin tossing that are O(1/r)-secure provided that the fraction of bad parties is slightly larger than half. In particular, our protocol is O(1/r)-secure when the number of parties is constant and the fraction of bad parties is less than 2/3. Gordon et al. [20] showed that complete fairness is possible in the two party case for some functions. Gordon and Katz [19] showed similar results for the multiparty case. The characterization of the functions that can be computed with full fairness without honest majority is open. Completeness for fair computations has been studied in[21]. Specifically, they show a specific function that is complete for fair two-party computation; this function is also complete for 1/p-secure two-party computation.Our ResultsWe study 1/p-secure protocols in the multiparty setting. We construct two protocols for general functionalities assuming that the fraction of corrupt parties is less than 2/3. The first protocol is efficient when(1)The number of parties is constant, the functionality is deterministic, and the size of the domain of inputs is at most polynomial in the security parameter, or (2) The number of parties is O(log log n) (where n is the security parameter), the functionality is deterministic, and the size of the domain of inputs is constant. The second protocol is efficient when the number of parties is constant, the functionality can be randomized, and the size of the range of the functionality is at most polynomial in the security parameter. Our second protocol does not provide correctness, i.e., in a case of premature termination, with probability of 1/ poly(n), the remaining active parties output a value which might be inconsistent with their inputs. In contrast, our first protocol provides correctness. CASE I: 1 ≤ i < i ⋆ . For every j ∈ J the dealer sets x j = x j and for every j / ∈ J it chooses x j independently with uniform distribution from the domain X n ; it computes the output σ i J ← f n ( x 1 , . . . , x m ).CASE II: i ⋆ ≤ i ≤ r. The dealer sets σ i J = w.The dealer T interacts with the parties in rounds, where in round i, for 1 ≤ i ≤ r, there are of three phases:The peeking phase. The dealer sends to the adversary all the values σ i J such that all parties in Q J are corrupted.The abort and premature termination phase. The adversary sends to T the identities of the parties that abort in the current round. If there are less than t + 1 active parties, then T sends σ i−1 J to the active parties, where Q J is the set of the active parties when parties can also abort during this phase (see exact details inFigure 1). The honest parties return this output and halt.The main phase. If at least t + 1 parties are active, T notifies the active parties that the protocol proceeds normally.If after r rounds, there are at least t + 1 active parties, T sends w to all active parties and the honest parties output this value.Example 3.1 As an example, assume that m = 5 and t = 3. In this case the dealer computes a value σ i J for every set of size 2 or 3. Consider an execution of the protocol where p 1 aborts in round 4 and p 3 and p 4 abort in round 100. In this case, T sends σ 99 {2,5} to p 2 and p 5 , which return this output. value of the set that contains all of them, i.e., σ i−1 J . In the special case of premature termination already in the first round, the remaining active parties engage in a fully secure protocol (with honest majority) to compute f n .The use of the outer secret-sharing scheme with threshold t + 1 plays a crucial role in eliminating the online dealer. On the one hand, it guarantees that an adversary, corrupting at most t parties, cannot reconstruct the shares of round i before round i. On the other hand, at least m − t parties must abort to prevent the reconstruction of the outer secret-sharing scheme (this is why we cannot proceed after m−t parties aborted). Furthermore, since t ≤ 2m/3, when at least m − t corrupt parties aborted, there is an honest majority. To see this, assume that at least m − t corrupt parties aborted. Thus, at most t − (m − t) = 2t − m corrupt parties are active. There are m − t honest parties (which are obviously active), therefore, as 2t−m < m − t (since t < 2m/3), an honest majority is achieved when m − t parties abort. In this case we can execute a protocol with full security for the reconstruction.Finally, we replace the off-line dealer by using a secure-with-abort and cheat-detection protocol computing the functionality computed by the dealer, that is, Functionality MultiShareGen r . Obtaining the outputs of this computation, an adversary is unable to infer any information regarding the input of honest parties or the output of the protocol (since it gets t shares of a (t + 1)-out-of-m secret-sharing scheme). The adversary, however, can prevent the execution, at the price of at least one corrupt party being detected cheating by all other parties. In such an event, the remaining parties will start over without the detected cheating party. This goes on either until the protocol succeeds or there is an honest majority and a fully secure protocol computing f n is executed.A formal description of the protocol appears inFigure 3. The reconstruction functionality used in this protocol (when at least m − t parties aborted) appears inFigure 4. The details of how to construct a protocol secure-with-abort and cheat-detection with O(1) rounds are given in[4].Comparison with the multiparty coin-tossing protocol of [4].Our protocol combines ideas from the protocols of[22,4]. However, there are some important differences between our protocol and the protocol of[4]. In the coin-tossing protocol of [4], the bits σ i J are shared using a threshold scheme where the threshold is smaller than the size of the set Q J . This means that a proper subset of Q J containing corrupt parties can reconstruct σ i J . In coin-tossing this is not a problem since there are no inputs. However, when computing functionalities with inputs, such σ i J might reveal information on the inputs of honest parties in Q J , and we share σ i J with threshold |Q J |. As a result, we use more sets Q J than in [4] and the bias of the protocol is increased (put differently, to keep the same security, we need to increase the number of rounds in the protocol). For example, the protocol of [4] has small bias when there are polynomially many parties and t = m/2. Our protocol is efficient only when there are constant number of parties. As explained in Section 4, this difference is inherent as a protocol for general functionalities with polynomially many parties and t = m/2 cannot have a small bias.A 1/p-Secure Protocol for Polynomial RangeUsing an idea of [22], we modify our protocol such that it will have a small bias when the size of the range of the functionality F is polynomially bounded (even if F is randomized and has a big domain of inputs). The only modification is the way that each σ i J is chosen prior to round i ⋆ : with probability 1/(2p) we choose σ i J as a random value in the range of f n and with probability 1 − 1/(2p) we choose it as inFigure 2. Formally, in the model with the dealer, in the preprocessing phase of MPCWithD r described inFigure 1, we replaceStep(5)with the following step:• For each i ∈ {1, . . . , i ⋆ − 1} and for each J ⊆ [m] \\ D 0 s.t. m − t ≤ |J| ≤ t,Proof:We start with the case of a deterministic functionality F. Recall that x 1 , . . . , x m are the inputs used by the dealer to obtain w = f n (x 1 , . . . , x m ) and σ i ⋆ J = w for each J ⊆ [m] s.t. m − t ≤ |J| ≤ t. Let J be such that the adversary obtains σ i J in round i < i ⋆ . Recall that x 1 , . . . , x m are the inputs used by the dealer to obtain σ i J , that is, σ i J = f n ( x 1 , . . . , x m ), where x j = x j for each j ∈ J and x j is selected uniformly at random from x j for every j / ∈ J. We bound the probability that σ i J = w by the probability that x j = x j for all j / ∈ J. The probability that x j = x j is 1/d. Therefore, the probability that both sets are the same is (1/d) m−|J| > (1/d) m .In each round of the protocol, A obtains the value σ i J for each subset Q J s.t. J ⊆ [m] and m − t ≤ |J| ≤ t, therefore, A obtains less than 2 t values. For each such two values σ i J and σ i J ′ obtained by A in round i < i ⋆ , the sets of inputs { x j : j / ∈ J} and { x j : j / ∈ J ′ } are totally independent. Therefore, the probability that all the values that the adversary sees in round i < i ⋆ are equal to w = f n (x 1 , . . . , x m ) is at least (1/d m ) 2 t −1 .For randomized functionality F, we think of the evaluation of f n ( x 1 , . . . , x m ) as two steps: first x j is randomly chosen from X n for every j ∈ J and then the randomized functionality is evaluated. Therefore, as A obtains less than 2 t values in each round i < i ⋆ , that the probability that all the values that the adversary sees in each round i < i ⋆ are equal to the specific w is at least (1/d m ) 2 t −1 · ǫ 2 t −1 .In the next lemma, we prove the correctness of the simulation by using the previous two lemmas.', 'arxivid': '1011.5567', 'author': ['Amos Beimel ', 'Eran Omri ', 'Ilan Orlov ', "\nDepartment of Computer Science\nDepartment of Computer Science\nBen Gurion University Be'er Sheva\nIsrael\n", '\nDepartment of Computer Science\nBar Ilan University Ramat Gan\nIsrael\n', "\nBen Gurion University\nBe'er ShevaIsrael\n"], 'authoraffiliation': ["Department of Computer Science\nDepartment of Computer Science\nBen Gurion University Be'er Sheva\nIsrael", 'Department of Computer Science\nBar Ilan University Ramat Gan\nIsrael', "Ben Gurion University\nBe'er ShevaIsrael"], 'corpusid': 7386786, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 26604, 'n_tokens_neox': 24308, 'n_words': 16541, 'pdfsha': 'a40016ede4993f26e92991d4f16f1d39927f9370', 'pdfurls': ['https://arxiv.org/pdf/1011.5567v1.pdf'], 'title': ['Secure Multiparty Computation with Partial Fairness', 'Secure Multiparty Computation with Partial Fairness'], 'venue': []}
arxiv
THREE-DIFFEOMORPHISM CONFORMAL SPACE OVER LORENTZIAN MANIFOLD 18 Aug 2017 Andrzej Lukasz Glinka THREE-DIFFEOMORPHISM CONFORMAL SPACE OVER LORENTZIAN MANIFOLD 18 Aug 2017arXiv:1708.07731v1 [math.GM] Through making use of a Borel measure and a piecewise-Riemannian inner scalar product, it is shown that over a Lorentzian manifold every three diffeomorphisms generate a conformal space, whose elements are smooth vector-valued functions equipped with compact supports. Few examples, in particular a diffeoinvariant measure, are provided with respect to an arbitrary smooth function introduced as into consideration as a multiplier to a local scale factor. Introduction Let us consider a Lorentzian manifold M of D = d + 1 dimensions and the Sylvester signature (d, 1, 0), or s = d − 1. Let g µν (x) be a metric tensor on M of dimension D × D and properties g(x) = det g µν (x) < 0, g µ α = δ µ α , g µ µ = D, where δ µ α = diag[1, . . . , 1] is the Kronecker D × D unit square matrix. In analysis of such an object one can make use of the theory of function spaces, particularly the class of vector-valued Banach functional spaces, Cf. e.g. the Refs. [1]- [3], known for some modern applications, Cf. e.g. the Refs. [4]- [8]. In this article, M is concisely approached by the topological techniques of a Borel measure of integration and a piecewise-Riemannian inner scalar product enriched by an arbitrary smooth function which is a multiplier to a local scale function. As a result, any three diffeomorphisms are done the reason to a local scale factor which generates a conformal space C(M) over M whose elements are smooth vector-valued functions equipped with compact supports. Furthermore, as the particular case, it is shown that a diffeoinvariant measure over C(M) can be reached with a help of a certain special choice of an arbitrary smooth function, or a local scale function. Additionally, basic topological, settheoretical, and analytic properties of C(M) are briefly discussed. Borel measure Let us consider a coordinate chart x µ ∈ R D on M, and a diffeomorphism x µ →x α (x) into a new non-singular coordinate chartx α (x). Then, a squared arc length on M is called diffeoinvariant if (1) ds 2 = g µν (x)dx µ dx ν =g αβ (x)dx α dx β , where usual summation convention holds, and one gets well-known relations g µν (x) =g αβ (x) ∂x α ∂x µ ∂x β ∂x ν , (2a) g(x) =g(x) det ∂x α (x) ∂x µ 2 . (2b) Considering the generic Borel measure in a coordinate chartx α (x) (3) d Dx |g(x)| 1/2 = dx 0 ∧ . . . ∧ dx d |g(x)| 1/2 det ∂x α (x) ∂x µ −1 , and performing a change of variables x µ → X α (x) such that infinitesimally (4) dx µ = λ(x)dX µ (x), where λ(x) = 0 is a local scale factor function, one obtains a new Borel measure (5) DX(x) = dX 0 (x) ∧ . . . ∧ dX d (x)[f (x)] 1/2 , where an arbitrary smooth function f (x) is of the form (6) f (x) = [λ(x)] 2D |g(x)| det ∂x α (x) ∂x µ −2 , and conversely (7) λ(x) = f (x) |g(x)| det ∂x α (x) ∂x µ . is determined up to an arbitrary smooth function f (x) > 0. This construction shows a non-trivial role of a distinguished collection of D smooth functions which are components of a smooth vector field (8) X µ (x) = x x0 [λ(x ′ )] −1 dx ′µ , where (x 0 ) µ is an arbitrary reference point on a M, and x ′ stands for a variable of integration. Whenever one takes into account the standard definition of support as a closure of the set of non-zero values of a function, then it becomes obvious that all these components are equipped with individual compact supports, and, therefore, the vector-valued smooth function X µ (x) in itself is equipped with a compact support. Conformal space C(M) Naturally, a new Borel measure induces a vector-valued function space spanned by a collection of smooth vector fields {X µ (x)}. Let Q µν (X) be a metric tensor on this function space. By virtue of diffeoinvariance, a change of variables x µ → X α (x µ ) (9) g µν (x)dx µ dx ν = Q αβ (X(x))dX α (x)dX β (x), leads to the relations Q αβ (X(x)) = g µν (x) ∂x µ ∂X α (x) ∂x ν ∂X β (x) , (10a) Q(X(x)) = g(x) det ∂x µ ∂X α (x) 2 ,(10b) with Q(X(x)) = det Q µν (X(x)). Meanwhile, for a coordinate chart given by the Eq. (8) one obtains ∂x α ∂X µ (x) = ∂X µ (x) ∂x α −1 = λ(x)δ α µ , (11a) det ∂x α ∂X µ (x) = [λ(x)] D ,(11b) and, consequently Q αβ (X(x)) = [λ(x)] 2 g αβ (x), (12a) Q(X(x)) = [λ(x)] 2D g(x),(12b) what shows that Q(X(x)) < 0 and that C(M) is a conformal space over M. As a result due to the Eq. (10b), one has the Jacobian determinant (13) det ∂x α ∂X µ (x) = f (x) g(x) 1/2 det ∂x α (x) ∂x µ , which is compatible with the Eq. (11b). Applying the Jacobian determinant of the composition (14) det ∂x µ ∂X α (x) det ∂X α (x) ∂x β (x) det ∂x β (x) ∂x µ = 1, one gets the relation (15) det ∂X α (x) ∂x β (x) = g(x) f (x) 1/2 det ∂x β (x) ∂x µ −2 , which leads to a non-diffeoinvariant Borel measure (16) DX|Q(X)| 1/2 = M d D x ′ |f (x ′ )g(x ′ )| 1/2 . Examples For f (x) = 1, which corresponds to the scale factor (17) λ(x) = |g(x)| −1 det ∂x α (x) ∂x µ 2 1/(2D) , one obtains the canonical diffeoinvariant Borel measure (18) DX|Q(X)| 1/2 = M d D x ′ |g(x ′ )| 1/2 , along with the formula (19) Q(X(x)) = − det ∂x α (x) ∂x µ 2 . On the other hand, for f (x) = |g(x)| −1/2 , or (20) λ(x) = |g(x)| −3/2 det ∂x α (x) ∂x µ 2 1/(2D) , a new Borel measure corresponds with a flat Lorentzian manifold (21) DX|Q(X)| 1/2 = M d D x ′ , whereas (22) Q(X(x)) = −|g(x)| −1/2 det ∂x α (x) ∂x µ 2 . For a more sophisticated choice (23) f (x) = |g(x)| det ∂x α (x) ∂x µ −2 , which gives the scale factor λ(x) = 1, one has identically Q αβ (X(x)) = g αβ (x) and Q(X(x)) = g(x). Because in this case a new Borel measure is (24) DX|Q(X)| 1/2 = M d D x ′ |g(x ′ )| det ∂x α (x ′ ) ∂x ′µ −1 , one sees that only for (25) det ∂x α (x) ∂x µ = |g(x)| 1/2 ,X, Y = DX|Q(X)| 1/2 |Q µν X µ Y ν | DX|Q(X)| 1/2 , (27a) ||X|| =     DX|Q(X)| 1/2 |Q µν X µ X ν | DX|Q(X)| 1/2     1/2 , (27b) d(X, Y ) =     DX|Q(X)| 1/2 |Q µν (X µ − Y µ )(X µ − Y µ )| DX|Q(X)| 1/2     1/2 . (27c) Consequently, one can summarize Theorem. Every three non-singular charts on M generate a conformal space C(M) equipped with a diffeoinvariant Borel measure and a piecewise-Riemannian inner product. Making use of the standard topological axioms, it can be shown directly that C(M) is a vector space, a function space, a real inner product space, a normed space, a real Hilbert space, and a metric space. By virtue of a piecewise-Riemannian nature, C(M) induced by the Borel measure is complete, as a result of the Cauchy-Schwarz inequality for inner product and the triangle inequality for distance. Because C(M) is a normed metric space, it is a Hausdorff topological space with topology given by unions of open balls. Because all elements of a C(M) are the Cartesian products of the set of reals, C(M) is a finite topological space because the underlying set is finite. Because every finite topological space is a compact space, C(M) is a compact metric space, that is a metrizable space. Because a distance induces the Tychonoff topology, C(M) is completely metrizable space, that is a metrically topological complete space, and is homeomorphic to a complete metric space which has a countable dense subset. Consequently, C(M) is a Polish space, a Luzin space, a Suslin space, and a Radon space. Among many other topological properties, it can be shown that C(M) is a separable Banach space, a perfectly normal Hausdorff space, a locally convex space, a bornological space, a reflexive space, a nuclear space, a barelled space, a perfect space, an Alexandroff space, a Baire space, a Borel space, a Dieudonné complete space, a Fréchet space, a Grothendieck space, a Mackey space, a Montel space, an Orlicz space, a Pták space, a Riesz space, a Schwartz space, a Sobolev space, an Urysohn space. the Eq. (18) is obtained, while the Eq. (21) is reached for(26) det ∂x α (x) ∂x µ = |g(x)|.For this reason, one can induce a piecewise-Riemannian inner scalar product, norm and distance over C(M) 1/(2D) Topological Vector Spaces: Second Edition. L Narici, &amp; E Beckenstein, 1219.46001Pure and Applied Mathematics. 296Chapman and Hall/CRC PressL. Narici & E. Beckenstein, Topological Vector Spaces: Second Edition, Pure and Applied Mathematics 296, Chapman and Hall/CRC Press, Boca Raton, 2011, Zbl 1219.46001. R J Fleming, &amp; J E Jamison, 1139.46001Isometries on Banach Spaces: Vector-Valued Function Spaces. Boca Raton; Boca RatonChapman and Hall/CRC Press2R. J. Fleming & J. E. Jamison, Isometries on Banach Spaces: Vector-Valued Function Spaces, Vol. 2, Monographs and Surveys in Pure and Applied Mathematics 138, Boca Raton, Chapman and Hall/CRC Press, Boca Raton, 2007, Zbl 1139.46001. History of Banach Spaces and Linear Operators. A Pietsch, 1112.46002Birkhäuser, BostonA. Pietsch, History of Banach Spaces and Linear Operators, Birkhäuser, Boston, 2007, Zbl 1112.46002. Weak Convergence of Stochastic Processes: With Applications to Statistical Limit Theorems, De Gruyer Graduate. V S Mandrekar, 1358.60002De GruyerBerlinV. S. Mandrekar, Weak Convergence of Stochastic Processes: With Applications to Sta- tistical Limit Theorems, De Gruyer Graduate, De Gruyer, Berlin, 2016, Zbl 1358.60002. C Alabiso, &amp; I Weiss, 1317.46001A Primer on Hilbert Space Theory: Linear Spaces, Topological Spaces, Metric Spaces, Normed Spaces, and Topological Groups. ChamSpringerC. Alabiso & I. Weiss, A Primer on Hilbert Space Theory: Linear Spaces, Topological Spaces, Metric Spaces, Normed Spaces, and Topological Groups, Springer, Cham, 2015, Zbl 1317.46001. Functional Analysis: An Introduction to Metric Spaces, Hilbert Spaces, and Banach Algebras. J Muscat, 1312.46002SpringerChamJ. Muscat, Functional Analysis: An Introduction to Metric Spaces, Hilbert Spaces, and Banach Algebras, Springer, Cham, 2014, Zbl 1312.46002. V Kanovei, M Sabok, &amp; J Zapletal, 1278.03001Canonical Ramsey Theory on Polish Spaces. CambridgeCambridge University Press202V. Kanovei, M. Sabok, & J. Zapletal, Canonical Ramsey Theory on Polish Spaces, Cambridge Tracts in Mathematics 202, Cambridge University Press, Cambridge, 2013, ZBl 1278.03001. J Kakol, W Kubiś, &amp; M López-Pellicer, 1231.46002Descriptive Topology in Selected Topics of Functional Analysis. BerlinSpringer24J. Kakol, W. Kubiś, & M. López-Pellicer, Descriptive Topology in Selected Topics of Functional Analysis, Developments in Mathematics 24, Springer, Berlin, 2011, Zbl 1231.46002. E-mail address: laglinka@gmail. E-mail address: [email protected]
{'fraction_non_alphanumeric': 0.09755403868031855, 'fraction_numerical': 0.035551763367463025, 'mean_word_length': 3.964235294117647, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'Through making use of a Borel measure and a piecewise-Riemannian inner scalar product, it is shown that over a Lorentzian manifold every three diffeomorphisms generate a conformal space, whose elements are smooth vector-valued functions equipped with compact supports. Few examples, in particular a diffeoinvariant measure, are provided with respect to an arbitrary smooth function introduced as into consideration as a multiplier to a local scale factor.', 'arxivid': '1708.07731', 'author': ['Andrzej Lukasz ', 'Glinka '], 'authoraffiliation': [], 'corpusid': 119576851, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 4160, 'n_tokens_neox': 3540, 'n_words': 1821, 'pdfsha': '02f6d4f817f008d53c8a1444661cf8c771eb5699', 'pdfurls': ['https://arxiv.org/pdf/1708.07731v1.pdf'], 'title': ['THREE-DIFFEOMORPHISM CONFORMAL SPACE OVER LORENTZIAN MANIFOLD', 'THREE-DIFFEOMORPHISM CONFORMAL SPACE OVER LORENTZIAN MANIFOLD'], 'venue': []}
arxiv
Quantum Gravity Effect on the Tunneling Particles from 2+1 dimensional New-type Black Hole 25 Oct 2017 Ganim Gecim [email protected] Department of Physics Faculty of Science Akdeniz University 07058AntalyaTurkey Yusuf Sucu [email protected] Department of Physics Faculty of Science Akdeniz University 07058AntalyaTurkey Quantum Gravity Effect on the Tunneling Particles from 2+1 dimensional New-type Black Hole 25 Oct 2017 We investigate the Generalized Uncertainty Principle (GUP) effect on the Hawking temperature for the 2+1 dimensional New-type black hole by using the quantum tunneling method for both the spin-1/2 Dirac and the spin-0 scalar particles. In computation of the GUP correction for the Hawking temperature of the black hole, we modified Dirac and Klein-Gordon equations. We observed that the modified Hawking temperature of the black hole depends not only on the black hole properties, but also on the graviton mass and the intrinsic properties of the tunneling particle, such as total angular momentum, energy and mass. Also, we see that the Hawking temperature was found to be probed by these particles in different manners. The modified Hawking temperature for the scalar particle seems to be lower compared to its standard Hawking temperature. Also, we find that the modified Hawking temperature of the black hole caused by Dirac particle's tunnelling rised by the total angular momentum of the particle. It is diminishable by the energy and mass of the particle and graviton mass as well. These intrinsic properties of the particle, except total angular momentum for the Dirac particle, and graviton mass may cause screening for the black hole radiation. Introduction Black hole radiation is theoretically very important phenomenon for researchers who attempts to merge the gravitation with the thermodynamics and the quantum mechanics [1,2,3,4,5,6,7,8]. With the formulation of the quantum field theory in curved spacetime based on the framework of the standard Heisenberg uncertainty principle, it was proved that a black hole can emit particles that are created by the quantum vacuum fluctuation near its outer horizon [6,7,8]. Since then, many alternative methods have been proposed to derive the black hole radiation known as Hawking radiation in the literature. For instance, the semi-classical method, based on quantum tunneling process of a particle across the outer horizon of a black hole from inside to outside, can be used to derive the Hawking radiation. The method implies two different approaches to compute the imaginary part of the action (S), which is the classically forbidden trajectory of a particle across the outher horizon: the null geodesic [9,10,11,12] and the Hamilton-Jacobi [13,14,15]. In both approaches, the tunneling probability of a particle from a black hole, Γ, is defined in terms of the classical action, as Γ = e − 2 h ImS [9,10,11,12,13,14,15]. By using the semiclassical method, a lot of studies about the Hawking radiation of a black hole as quantum tunneling process of a point-like particle have been carried out in the literature [18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36]. On the other hand, above mentioned studies on this method provide no specific information on type of the particle that is tunnelled from a black hole. That is because the Hawking radiation does not dependent on the intrinsic properties, such as mass, total (orbital+spin) angular momentum, energy and charge, of the tunneling point-like particle. The existence of a minimal observable length which can be identified by the order of the Planck scale is a characteristic of the candidate theories of quantum gravity, such as string theory, loop quantum gravity and noncommutative geometry [37,38,39,40,41]. This length lead us to a generalized uncertainty principle (GUP) instead of the standard Heisenberg uncertainty principle. Because a particle is not a point-like particle in the context of these candidate theories anymore. Therefore, the uncertainty on the momentum of a particle increases and thus the standard Heisenberg uncertainty principle can be generalized as follows; ∆x∆p ≥h 2 1 + β(∆p) 2 ,(1) where β = β 0 /M 2 p , the M 2 p is the Planck mass, β 0 is the dimensionless parameter [42,43,44,45]. The commutation relations between a particle position, x, and momentum, p, are modified in the following way; [x i , p j ] = ihδ ij 1 + βp 2 ,(2) where x i and p j represent the modified position and momentum operators, respectively, and their definitions are as follows; x i = x 0i , p j = p 0j (1 + βp 2 0 ).(3) The x 0i and p 0j =−ih∂ j in Eq.(3) are the standard position and momentum operators, respectively, and p 2 0 =p 0j p 0j [53]. Then, the modified energy expression becomes E = E 1 − βE 2 = E 1 − β p 2 + m 2 0 ,(4) for which the energy mass shell condition, E 2 =p 2 + m 2 0 , is used. From these relations, the square of the momentum operator can be derived by the following way, p 2 = p i p i ≃ −h 2 ∂ i ∂ i − 2β∂ j ∂ j ∂ i ∂ i ,(5) where the higher order terms of the β parameter are neglected. The GUP relations are of great help to understand the nature of a black hole since quantum effects are the essential effects near the event horizon of a black hole. Recently, to investigate the quantum effects under the GUP relations, the thermodynamics properties of various black holes have been studied by using the quantum tunneling process of particles with various spins [46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64]. These studies indicate that the modify Hawking radiation depends not only on the black hole's properties but also on the intrinsic properties of the tunneling particle. The New-type black hole is one of the important results of the New Massive Gravity, which is 2 + 1 dimensional gravity and graviton in this theory has a mass [65]. In the framework of the standard Heisenberg uncertainty principle, the Hawking radiation of the New-type black holes had been studied by using the quantum tunneling process of the scalar, Dirac and vector boson particles [24,36]. In these studies, it was shown that the Hawking radiation only depends on the properties of the black hole and is independent from the properties of the tunneling point-particles, that is, all these particles tunnel from the black holes in the same way. This indicate that, even if an observer in enough or safe (i.e infinite) distant from a black hole may detect Hawking radiation of the black hole, the observer can not determine what kind of particles compose of the radiation. Therefore, in this study, we will investigate whether the properties of the tunneling particles will affect the Hawking radiation of the black hole by using the quantum tunneling process of both the scalar and Dirac particles in the framework of the GUP. The organization of this work are as follows: In the Section 2, we modify the Dirac equation with respect to the GUP relations. Subsequently, using the modified Dirac equation, we calculate the tunneling possibility of the Dirac particle by using the Hamilton-Jacobi method, and, then, we find the modify Hawking temperature of the black hole. In the Section 3, the standard Klein-Gordon equation is rewritten under the GUP for the 2 + 1 dimensional New-type black hole. Subsequently, the tunneling probability of the scalar particle from the black hole and the modify Hawking temperature of the black hole are calculated, respectively. In conclusion, we evaluate and summarize the results. Dirac particle's tunneling in the New-type Black Hole Using the GUP relations, the standard Dirac equation given in Ref. [66] can be modified as follows; (6) and its explicit form is −iσ 0 (x)∂ 0 Ψ = iσ i (x)∂ i − iσ µ (x)Γ µ − m 0 h 1 + βh 2 ∂ j ∂ j − βm 2 0 Ψ,iσ 0 (x)∂ 0 Ψ + iσ i (x) 1 − βm 2 0 ∂ i Ψ + iβh 2 σ i (x)∂ i ∂ j ∂ j Ψ − m 0 h 1 − βm 2 0 Ψ −m 0 βh∂ j ∂ j Ψ − iσ µ (x)Γ µ 1 + βh 2 ∂ j ∂ j − βm 2 0 Ψ = 0,(7) where the Ψ is the modify Dirac spinor, m 0 is mass of the Dirac particle, σ µ (x) are the spacetime dependent Dirac matrices, the Γ µ (x) are the spin affine connection for spin-1/2 particle [66]. The spacetime background of the New-type black hole is given by ds 2 = L 2 f (r) dt 2 − 1 f (r) dr 2 − r 2 dφ 2 ,(8) where L is the AdS 3 radius defined as L 2 = 1 2m 2 = 1 2Λ where Λ is cosmological constant and m is graviton mass, and f (r)=(r − r + )(r − r − ) is defined in terms of the outer, r + , and inner, r − , horizons radius, respectively. The black hole's horizons are located at r ± = 1 2 (−b ± √ b 2 − 4c) , where b and c are two constant parameters [24,65]. Using the Eq.(8), the spinorial affine connections are derived as follows [24]; Γ 0 = − i 4 f ′ (r)σ 3 σ 1 , Γ 1 = 0 , Γ 2 = 1 2 f (r)σ 1 σ 2 .(9) To calculate the tunneling probability of a Dirac particle from the black hole, we use the following ansatz for the modified wave function; Ψ(x) = exp ī h S (t, r, φ) A (t, r, φ) B (t, r, φ)(10) where the A (t, r, φ) and B (t, r, φ) are the functions of space-time. The S(t, r, φ) is the classical action term for particle trajectory. Inserting the Eqs. (9) and (10) in Eq. (7), we obtain the following equations for the leading order inh and β as A   1 L √ f ∂S ∂t + m 0 1 − βm 2 0 + βm 0 L 2 r 2 ∂S ∂φ 2 + βm 0 f L 2 ∂S ∂r 2   +B   i √ f (1 − βm 2 0 ) L ∂S ∂r + (1 − βm 2 0 ) Lr ∂S ∂φ + i βf 3/2 L 3 ∂S ∂r 3   +B   i β √ f L 3 r 2 ∂S ∂r ∂S ∂φ 2 + βf L 3 r ∂S ∂φ ∂S ∂r 2 + β L 3 r 3 ∂S ∂φ 3   = 0, A   −i √ f (1 − βm 2 0 ) L ∂S ∂r + (1 − βm 2 0 ) Lr ∂S ∂φ − i βf 3/2 L 3 ∂S ∂r 3   +A   −i β √ f L 3 r 2 ∂S ∂r ∂S ∂φ 2 + βf L 3 r ∂S ∂φ ∂S ∂r 2 + β L 3 r 3 ∂S ∂φ 3   +B   1 L √ f ∂S ∂t − m 0 1 − βm 2 0 − βm 0 L 2 r 2 ∂S ∂φ 2 − βm 0 f L 2 ∂S ∂r 2   = 0. (11) These two equations have nontrivial solutions for the A (t, r, φ) and B (t, r, φ) when the determinant of the coefficient matrix is vanished. Accordingly, neglecting of the terms containing higher order of the β parameter, provides β   2m 4 0 − 4f L 4 r 2 ∂S ∂r 2 ∂S ∂φ 2 − 2 L 4 r 4 ∂S ∂φ 4 − 2f 2 L 4 ∂S ∂r 4   + 1 L 2 f ∂S ∂t 2 − f L 2 ∂S ∂r 2 − 1 L 2 r ∂S ∂φ 2 − m 2 0 = 0.(12) Due to the commuting Killing vectors (∂ t ) and (∂ φ ), we can separate the S (t, r, φ), in terms of the variables t, r and φ, as S (t, r, φ) = −Et+jφ+K(r), where, E and j are the energy and angular momentum of the particle, respectively, and K(r) = K 0 (r) + βK 1 (r) [57]. Using these definition in Eq. (12), the integral of the radial equation, K(r), becomes K ± (r) = ± E 2 − f (r) (m 2 0 L 2 + j 2 /r 2 ) f (r) [1 + βχ] dr,(13) where χ is an abbreviation and it is χ = E 2 [E 2 − 2m 2 0 L 2 f (r)] L 2 f (r) [E 2 − f (r) (m 2 0 L 2 + j 2 /r 2 )] . Then, by integrating the radial equation, K ± (r) are obtained as K ± (r) = ±iπ E r + − r − [1 + βΠ] ,(14) where the abbreviation Π is Π = (r + − r − ) 2 3L 2 m 2 0 r 2 + − j 2 + 4E 2 r 2 + 2L 2 r 2 + (r + − r − ) 4 . On the other hand, the tunneling probabilities of particles crossing the outer horizon are given by P out = exp − 2 h ImK + (r) , P in = exp − 2 h ImK − (r) .(15) Hence, the tunneling probability of the Dirac particle is given by Γ = e − 2 h ImS = P out P in = exp − 4πĒ h (r + − r − ) [1 + βΠ] ,(16) where ImS (t, r, φ)=ImK + (r) − ImK − (r) [67,68] and ImK + (r)=−ImK − (r). Then, the modify Hawking temperature of the Dirac particle, T D H , is obtained as T D H =h (r + − r − ) 4π 1 [1 + βΠ D ] ,(17) where to find the temperature it is used the the following relation [69,70,71,72]: Γ = e − 2 h ImS = e − E T H .(18) If the T D H are at first expanded in terms of the β powers and second neglected the higher order of the β terms, then the modify Hawking temperature of the black hole is obtained as T D H ≃h (r + − r − ) 4π [1 − βΠ](19) From these results, we see that the modified Hawking temperature includes not only the mass parameter of the black hole, but also the AdS 3 radius, L, (and, hence, the graviton mass) and the angular momentum, energy and mass of the tunnelled Dirac particle. On the other hand, in the case of β = 0, the modified Hawking temperature is reduced to the standard temperature obtained by quantum tunneling process of the point particles with spin-0, spin-1/2 and spin-1, respectively [24,36]. Scalar particle's tunneling in the New-type Black Hole To investigate the quantum gravity effects on the tunneling process of the scalar particles from the black hole, by using the GUP relations, the standard Klein-Gordon equation are modified as − (ih) 2 ∂ t ∂ t Φ = (−ih) 2 ∂ i ∂ i − M 2 0 1 − 2β −h 2 ∂ i ∂ i + M 2 0 Φ,(20) and its explicit form of the modified Klein-Gordon equation is written as follows; h 2 ∂ t ∂ t Φ +h 2 ∂ i ∂ i Φ + 2βh 4 ∂ i ∂ i (∂ i ∂ i Φ) + M 2 0 (1 − 2βM 2 0 ) Φ = 0,(21) where Φ and M 0 are the modify wave function and mass of the scalar particle, respectively. Then, the modified Klein-Gordon equation in the New-type black hole background becomes as follows: h 2 f ∂ 2 Φ ∂t 2 −h 2 r 2 ∂ 2 Φ ∂φ 2 − 2βh 4 f ∂ 2 ∂r 2 − f L 2 ∂ 2 Φ ∂r 2 − 2βh 4 r 2 ∂ 2 ∂φ 2 − 1 L 2 r ∂ 2 Φ ∂φ 2 −h 2 f ∂ 2 Φ ∂r 2 + M 2 0 L 2 1 − 2βM 2 0 Φ = 0.(22) To consider the tunneling radiation of the black hole with the Eq. (22), we employ the modify wave function of the scalar particle as, Φ (t, r, φ) = A exp ī h S (t, r, φ) ,(23) where A is a constant. Substituting the Eq.(23) into the Eq. (22) and neglecting the the higher order terms ofh, we get the equation of motion of the scalar particle as ∂S ∂t 2 − f (r) 2 ∂S ∂r 2 − f (r) r 2 ∂S ∂φ 2 − M 2 0 L 2 f (r) − β 2f (r) r 4 L 2 ∂S ∂φ 4 +β   M 4 0 L 2 f (r) − 2f (r) 3 L 2 ∂S ∂r 4   = 0(24) Using S (t, r, φ) = −Et + jφ + W (r), where E and j are the energy and angular momentum of the particle, respectively, and W (r) = W 0 (r) + βW 1 (r) [57], then, the radial integral, W (r), becomes as follows; W ± (r) = ± E 2 − f (r) (M 2 0 L 2 + j 2 /r 2 ) f (r) [1 + βΩ] dr,(25) where the abbreviation Ω is Ω = f (r) 2 (M 4 0 L 4 − j 4 /r 4 ) − [E 2 − f (r) (M 2 0 L 2 + j 2 /r 2 )] 2 L 2 f (r) [E 2 − f (r) (m 2 0 L 2 + j 2 /r 2 )] . And, W ± (r) are computed as W ± (r) = ±iπ E r + − r − [1 + βΣ] ,(26) where the abbreviation Σ is Σ = (r + − r − ) 2 3L 2 M 2 0 r 2 + + 3j 2 + 4E 2 r 2 + 2L 2 r 2 + (r + − r − ) 4 , where W + (r h ) is outgoing and W − (r h ) is incoming solutions of the radial part. Then, using the Eq.(15), the tunneling probability of the scalar particle is calculated as Γ = exp − 4πĒ h (r + − r − ) [1 + βΣ](27) and, subsequently, using the Eq.(18), the modify Hawking temperature of the scalar particle, T KG H , becomes as T KG H =h (r + − r − ) 4π 1 [1 + βΣ] .(28) Furthermore, as neglecting the higher order of the β terms in the expanding form of T KG H in terms of the β, we find the modify Hawking temperature of the black hole as follows; T KG H ≃h (r + − r − ) 4π [1 − βΣ](29) From these results, it can seen that the modified Hawking temperature is related not only to the mass parameter of the black hole, but also to the AdS 3 radius, L, (and, hence, to the graviton mass) and angular momentum, energy and mass of the tunneling scalar particle. Furthermore, as can be seen from Eq. (19 and Eq.(29), the Hawking temperature probed by a Dirac particle is higher than that of a scalar particle: T D H =T KG H + βh j 2 m 2 πr 2 + (r + −r − ) for M 0 =m 0 and L 2 = 1 2m 2 . On the other hand, in the case of β = 0, the modify Hawking temperature reduced to the standard temperature obtained by quantum tunneling process of the point particles with spin-0, spin-1/2 and spin-1, respectively [24,36]. Concluding remarks In this study, we investigated the quantum gravity effect on the tunneled both spin-0 scalar and spin-1/2 Dirac particles from New-type black hole in the context of 2+1 dimensional New Massive Gravity. For this, at first, using the GUP relations, we modified the Klein-Gordon and Dirac equations that describe the spin-0 scalar and spin-1/2 Dirac particles, respectively. Then, using the Hamilton-Jacobi method, the tunneling probabilities of the these particles are derived, and subsequently, the corrected Hawking temperature of the black hole is calculated. We find that the modified Hawking temperature not only depends on the black hole's properties, but also depends on the emitted particle's mass, energy and total angular momentum. Also, it is worth to mention that, the modified Hawking temperature depends on mass of the graviton, i.e. quantum particle which mediates gravitational radiation in the context of New Massive Gravity. As can be seen from Eq. (19), the Hawking temperature of the Dirac particle increase by the total angular momentum of the particle while it decreases by the energy and mass of the particle and the graviton mass. In addition, we can summarize some important results as follows: • In Eq.(29), the modify Hawking temperature of the scalar particle is lower than the standard Hawking temperature. • However, in Eq. (19), as 4E 2 r 2 + + 3m 2 0 2m 2 r 2 + (r + − r − ) 2 < j 2 (r + − r − ) 2 , the modify Hawking temperature of the Dirac particle is higher than the standard Hawking temperature. Furthermore, when 4E 2 r 2 + + 3m 2 0 2m 2 r 2 + (r + − r − ) 2 > j 2 (r + − r − ) 2 , the modify Hawking temperature is lower than the standard Hawking temperature. If 4E 2 r 2 + + 3m 2 0 2m 2 r 2 + (r + − r − ) 2 =j 2 (r + − r − ) 2 , then the GUP effect is canceled, and the Hawking temperature of the Dirac particle reduces to the standard Hawking temperature. • According to Eq. (19) and Eq.(29), the modify Hawking temperature of the Newtype black hole probed by tunneling Dirac particle is higher than that of scalar particle: T D H = T KG H + βh j 2 m 2 πr 2 + (r + − r − ) . where we adopt that the mass of the Dirac particle is equivalent to the mass of the scalar particle, i.e. m 0 = M 0 . • The New-type black hole is classified as six classes according to the signatures of the parameters b and c, and, hence, it exhibits different physical and mathematical properties. For example, it reduced to the static BTZ black hole in the case of b = 0 and c < 0. In this context, according to tunneling of the scalar and Dirac particles, the modify Hawking temperature of the static BTZ black hole is respectively. Here, r + = −r − = |c| is used and the T H =h √ |c| 2π is the standard Hawking temperature of the static BTZ black hole in the context of the 2+1 dimensional New Massive Gravity theory [24]. • In the absence of the quantum gravity effect, i.e. β=0, the modify Hawking temperature is reduced to the standard temperature obtained by quantum tunneling of the massive spin-0, spin-1/2 and spin-1 point particles [24,36]. Finally, in the context of GUP, we have seen that the graviton and the tunneling particle masses have an affect decreasing the Hawking temperature in both scalar and Dirac particle tunneling proses. On the other hand, the total angular momentum has different effect on the Hawking temperature for a type of tunneling particle. For a scalar particle, it results in decrease in the temperature whereas it provides an increase in the temperature for a Dirac particle. These results show that the intrinsic properties of the particle, except total angular momentum for the Dirac particle, and graviton mass may cause screening for the black hole radiation. AcknowledgementsThis work was supported by Akdeniz University, Scientific Research Projects Unit. . J M Greif, Princeton UniversityJunior thesisJ. M. Greif, Junior thesis, Princeton University,(unpublished) (1969). Rigidity of a black hole. B Carter, Nature Physical Science. 238B. Carter, "Rigidity of a black hole", Nature Physical Science, vol. 238, pp.71-72, 1972. Black Holes and Entropy. J D Bekenstein, Physical Review D. 78J.D. Bekenstein, "Black Holes and Entropy", Physical Review D, vol.7, no.8, pp.2333-2346, 1973. Generalized second law of thermodynamics in black-hole physics. J D Bekenstein, Physical Review D. 912J.D. Bekenstein, "Generalized second law of thermodynamics in black-hole physics", Physical Review D, vol.9, no.12, pp.3292-3300, 1974. The Four Laws of Black Hole Mechanics. J M Bardeen, B Carter, S W Hawking, Communications in Mathematical Physics. 31J.M. Bardeen, B. Carter and S.W. Hawking, "The Four Laws of Black Hole Mechanics", Communications in Mathematical Physics, vol.31, pp.161-170, 1973. Black hole explosions. S W Hawking, Nature. 248S.W. Hawking, "Black hole explosions", Nature, vol.248, pp.30-31, 1974. Particle creation by black holes. S W Hawking, Communications in Mathematical Physics. 43S.W. Hawking, "Particle creation by black holes", Communications in Mathematical Physics, vol.43, pp.199-220, 1975. Black holes and thermodynamics. S W Hawking, Physical Review D. 132S.W. Hawking, "Black holes and thermodynamics", Physical Review D, vol.13, no.2, pp.191-197, 1976. Self-interaction correction to black hole radiance. P Kraus, F Wilczek, Nuclear Physics B. 433P. Kraus and F. Wilczek, "Self-interaction correction to black hole radiance", Nuclear Physics B, vol.433, pp.403-420, 1995. Effect of self-interaction on charged black hole radiance. P Kraus, F Wilczek, Nuclear Physics B. 437P. Kraus and F. Wilczek, "Effect of self-interaction on charged black hole radiance", Nuclear Physics B, vol.437, pp.231-242, 1995. Hawking Radiation As Tunneling. M K Parikh, F Wilczek, Physical Review Letters. 8524M.K. Parikh and F. Wilczek, "Hawking Radiation As Tunneling", Physical Review Letters, vol.85, no.24, pp.5042-5045, 2000. New coordinates for de Sitter space and de Sitter radiation. M K Parikh, Physics Letters B. 546M.K. Parikh, "New coordinates for de Sitter space and de Sitter radiation", Physics Letters B, vol.546, pp.189-195, 2002. Hawking radiation as tunneling for extremal and rotating black holes. M Angheben, M Nadalini, L Vanzo, S Zerbini, Journal of High Energy Physics. 514M. Angheben, M. Nadalini, L. Vanzo and S. Zerbini, "Hawking radiation as tunneling for extremal and rotating black holes", Journal of High Energy Physics, vol. 2005, no. 5, article 014, 2005. Particle production and complex path analysis. K Srinivasan, T Padmanabhan, Physical Review D. 60K. Srinivasan and T. Padmanabhan, "Particle production and complex path analysis", Physical Review D, vol.60, Article no.024007, pp.1-20, 1999. Method of complex paths and general covariance of Hawking radiation. S Shankaranarayanan, K Srinivasan, T Padmanabhan, Modern Physics Letters A. 169S. Shankaranarayanan, K. Srinivasan and T. Padmanabhan, "Method of complex paths and general covariance of Hawking radiation", Modern Physics Letters A, vol.16, no.9, pp.571-578, 2001. Generalization of the KKW analysis for black hole radiation. E C Vagenas, Physics Letters B. 559E.C. Vagenas, "Generalization of the KKW analysis for black hole radiation", Physics Letters B, vol.559, pp.65-73, 2003. A note on the Hawking radiation calculated by quasi-classical tunneling method. Y P Hu, J Y Zhang, Z Zhao, Modern Physics Letters A. 254Y.P. Hu, J.Y. Zhang and Z. Zhao, "A note on the Hawking radiation calculated by quasi-classical tunneling method", Modern Physics Letters A, vol.25, no.4, pp.295-308, 2010. Black hole radiation of massive spin-2 particles in (3+1) dimensions. I Sakalli, A Ovgun, The European Physical Journal Plus. 13184I. Sakalli and A. Ovgun, "Black hole radiation of massive spin-2 particles in (3+1) dimensions", The European Physical Journal Plus, vol.131, Article no. 84, pp.1-13, 2016. Gravitinos tunneling from black holes. A Yale, R B Mann, Physics Letters B. 673A. Yale and R.B. Mann, "Gravitinos tunneling from black holes", Physics Letters B, vol. 673, pp.168-172, 2009. Tunnelling, temperature, and Taub-NUT black holes. R Kerner, R B Mann, Physical Review D. 73104010R. Kerner and R.B. Mann, "Tunnelling, temperature, and Taub-NUT black holes", Physical Review D vol. 73, Article no. 104010, pp1-11, 2006. Fermions tunnelling from black holes. R Kerner, R B Mann, Classical and Quantum Gravity. 259R. Kerner and R.B. Mann, "Fermions tunnelling from black holes", Classical and Quantum Gravity, vol. 25, no.9, pp.1-17, 2008. Tunnelling from Godel black holes. R Kerner, R B Mann, Physical Review D. 75084022R. Kerner and R.B. Mann, "Tunnelling from Godel black holes", Physical Review D, vol. 75, Article no. 084022, pp.1-9, 2007. G Gecim, Y Sucu, arXiv:1406.0290v2Hawking Radiation of Topological Massive Warped-AdS3 Black Hole Families. G. Gecim and Y. Sucu, "Hawking Radiation of Topological Massive Warped-AdS3 Black Hole Families", arXiv:1406.0290v2, 2014. Tunnelling of relativistic particles from new type black hole in new massive gravity. G Gecim, Y Sucu, Journal of Cosmology and Astroparticle Physics. 201302G. Gecim and Y. Sucu, "Tunnelling of relativistic particles from new type black hole in new massive gravity", Journal of Cosmology and Astroparticle Physics, vol. 2013, no. 02, Article no. 023, 2013. Dirac and scalar particles tunnelling from topological massive warped-AdS 3 black hole. G Gecim, Y Sucu, Astrophysics and Space Science. 357105G. Gecim and Y. Sucu, "Dirac and scalar particles tunnelling from topological massive warped- AdS 3 black hole", Astrophysics and Space Science, vol. 357, article 105, 2015. Black hole radiation of spin-1 particles in (1 + 2) dimensions. S I Kruglov, Modern Physics Letters A. 2939S.I. Kruglov, "Black hole radiation of spin-1 particles in (1 + 2) dimensions", Modern Physics Letters A, vol. 29, no. 39, Article no. 1450203, pp.1-5, 2014. Black hole emission of vector particles in (1+1) dimensions. S I Kruglov, International Journal of Modern Physics A. 2922S.I. Kruglov, "Black hole emission of vector particles in (1+1) dimensions", International Journal of Modern Physics A, vol. 29, no. 22, Article no.1450118, pp.1-10, 2014. Hawking Radiation of Massive Vector Particles From Warped AdS3 Black Hole. H Gursel, I Sakalli, Canadian Journal of Physics. 942H. Gursel and I. Sakalli, "Hawking Radiation of Massive Vector Particles From Warped AdS3 Black Hole", Canadian Journal of Physics, vol. 94, no. 2, pp. 147-149, 2016. Hawking radiation of spin-1 particles from three dimensional rotating hairy black hole. I Sakalli, A Ovgun, Journal of Experimental and Theoretical Physics. 1213I. Sakalli and A. Ovgun, "Hawking radiation of spin-1 particles from three dimensional rotating hairy black hole", Journal of Experimental and Theoretical Physics, vol. 121, no. 3, pp. 404-407, 2015. Vector particles tunneling from BTZ black holes. G R Chen, S Zhou, Y C Huang, International Journal of Modern Physics D. 241550005G.R. Chen, S. Zhou and Y.C. Huang, "Vector particles tunneling from BTZ black holes", International Journal of Modern Physics D, vol. 24, Article no. 1550005, pp. 1-7, 2015. Hawking radiation of vector particles as tunneling from the apparent horizon of Vaidya black holes. G R Chen, Y C Huang, International Journal of Modern Physics A. 3015G.R. Chen, Y.C. Huang, "Hawking radiation of vector particles as tunneling from the apparent horizon of Vaidya black holes", International Journal of Modern Physics A, vol. 30, no. 15, Article no. 1550083, pp. 1-9, 2015. Massive vector particles tunneling from Kerr and Kerr-Newman black holes. X Q Li, G R Chen, Physics Letters B. 751X.Q. Li and G.R. Chen, " Massive vector particles tunneling from Kerr and Kerr-Newman black holes", Physics Letters B, vol.751, pp.34-38, 2015. Quantum tunneling of massive spin-1 particles from non-stationary metrics. I Sakalli, A Ovgun, General Relativity and Gravitation. 481I. Sakalli and A. Ovgun, "Quantum tunneling of massive spin-1 particles from non-stationary metrics", General Relativity and Gravitation, vol. 48, no. 1,Article no. 1, 2016. Vector particles tunneling from four-dimensional Schwarzschild black holes. G R Chen, S Zhou, Y C Huang, Astrophysics and Space Science. 35751G.R. Chen, S. Zhou and Y.C. Huang, "Vector particles tunneling from four-dimensional Schwarzschild black holes", Astrophysics and Space Science, vol. 357, article 51, 2015. Hawking radiation of massive vector particles from the linear dilaton black holes. R Li, J Zhao, The European Physical Journal Plus. 1317R. Li and J. Zhao, " Hawking radiation of massive vector particles from the linear dilaton black holes", The European Physical Journal Plus, vol. 131, no. 7, Article no. 249, 2016. Massive vector bosons tunnelled from the (2+1)-dimensional black holes. G Gecim, Y Sucu, The European Physical Journal Plus. 1323G. Gecim and Y. Sucu, "Massive vector bosons tunnelled from the (2+1)-dimensional black holes", The European Physical Journal Plus, vol. 132, no. 3, Article no. 105, 2017. Maximal localization in the presence of minimal uncertainties in positions and in momenta. H Hinrichsen, A Kempf, Journal of Mathematical Physics. 375H. Hinrichsen and A. Kempf, "Maximal localization in the presence of minimal uncertainties in positions and in momenta", Journal of Mathematical Physics, vol. 37, no. 5, pp. 2121-2137, 1996. On quantum field theory with nonzero minimal uncertainties in positions and momenta. A Kempf, Journal of Mathematical Physics. 383A. Kempf, "On quantum field theory with nonzero minimal uncertainties in positions and momenta", Journal of Mathematical Physics, vol. 38, no. 3, pp. 1347-1372, 1997. Discreteness of space from the generalized uncertainty principle. A F Ali, S Das, E C Vagenas, Physics Letters B. 678A.F. Ali, S. Das and E.C. Vagenas, "Discreteness of space from the generalized uncertainty principle", Physics Letters B, vol. 678, pp. 497-499, 2009. Discreteness of space from GUP II: Relativistic wave equations. S Das, E C Vagenas, A F Ali, Physics Letters B. 690Erratum ibid. 692 (2010) 342S. Das, E.C. Vagenas and A.F. Ali, "Discreteness of space from GUP II: Relativistic wave equations", Physics Letters B, vol. 690, pp. 407-412, 2010. [Erratum ibid. 692 (2010) 342]. Sub-Planckian black holes and the Generalized Uncertainty Principle. B J Carr, J Mureika, P Nicolini, Journal of High Energy Physics. 201507B.J. Carr, J. Mureika and P. Nicolini, "Sub-Planckian black holes and the Generalized Uncertainty Principle", Journal of High Energy Physics, vol. 2015, no.07, Article no. 052, 2015. Hilbert space representation of the minimal length uncertainty relation. A Kempf, G Mangano, R B Mann, Physical Review D. 522A. Kempf, G. Mangano and R.B. Mann, "Hilbert space representation of the minimal length uncertainty relation", Physical Review D, vol. 52, no. 2, pp. 1108-1118, 1995. Signatures in the Planck Regime. S Hossenfelder, Physics Letters B. 575S. Hossenfelder et al., "Signatures in the Planck Regime", Physics Letters B, vol 575, pp. 85-99, 2003. Can we measure structures to a precision better than the Planck length?. S Hossenfelder, Classical and Quantum Gravity. 2911S. Hossenfelder, "Can we measure structures to a precision better than the Planck length?", Classical and Quantum Gravity, vol. 29, no. 11, Article no. 115011, 2012. Deformation of the Dirac equation. M Faizal, S I Kruglov, International Journal of Modern Physics D. 251M. Faizal and S.I. Kruglov, "Deformation of the Dirac equation", International Journal of Modern Physics D, vol. 25, no.1, Article no. 1650013, 2016. Natural cutoffs and quantum tunneling from black hole horizon. K Nozari, S Saghafi, Journal of High Energy Physics. 201211K. Nozari and S. Saghafi, "Natural cutoffs and quantum tunneling from black hole horizon", Journal of High Energy Physics, vol. 2012, no. 11, Article no. 005, 2012. Exact solution of the Klein Gordon equation in the presence of a minimal length. T K Jana, P Roy, Physics Letters A. 373T.K. Jana and P. Roy, "Exact solution of the Klein Gordon equation in the presence of a minimal length", Physics Letters A, vol. 373, pp.1239-1242, 2009. Minimal Length and Generalized Dirac Equation. K Nozari, M Karami, Modern Physics Letters A. 2040K. Nozari and M. Karami, "Minimal Length and Generalized Dirac Equation", Modern Physics Letters A, vol. 20, no. 40, pp. 3095-3103, 2005. Massive vector particles tunneling from black holes influenced by the generalized uncertainty principle. X Li, Physics Letters B. 763X.Q Li, "Massive vector particles tunneling from black holes influenced by the generalized uncertainty principle", Physics Letters B, vol. 763, pp. 80-86, 2016. Fermions Tunnelling with Effects of Quantum Gravity. D Chen, H Wu, H Yang, Advances in High Energy Physics. 2013D. Chen, H. Wu and H. Yang, "Fermions Tunnelling with Effects of Quantum Gravity", Advances in High Energy Physics, vol. 2013, Article no. 432412, 2013. Remnants, fermions' tunnelling and effects of quantum gravity. D Chen, Q Q Jiang, P Wang, H Yang, Journal of High Energy Physics. 201311D. Chen, Q.Q. Jiang, P. Wang and H. Yang, "Remnants, fermions' tunnelling and effects of quantum gravity", Journal of High Energy Physics, vol. 2013, no. 11, Article no. 176, 2013. Observing remnants by fermions' tunneling. D Y Chen, H W Wu, H Yang, Journal of Cosmology and Astroparticle Physics. 201403D.Y. Chen, H.W. Wu and H. Yang, "Observing remnants by fermions' tunneling", Journal of Cosmology and Astroparticle Physics, vol. 2014, no. 03, Article no. 036, 2014. Effects of quantum gravity on black holes. D Chen, H Wu, H Yang, S Yang, International Journal of Modern Physics D. 2926D. Chen, H. Wu, H. Yang and S. Yang, "Effects of quantum gravity on black holes", International Journal of Modern Physics D, vol. 29, no. 26, Article no. 1430054, 2014. Quantum gravity corrections to fermions tunnelling radiation in the Taub-NUT spacetime. X X Zeng, Y Chen, General Relativity and Gravitation. 474X.X. Zeng and Y. Chen, "Quantum gravity corrections to fermions tunnelling radiation in the Taub-NUT spacetime", General Relativity and Gravitation, vol. 47, no. 4, Article no. 47, 2015. Quantum tunneling from a high dimensional Gdel black hole. H L Li, Z W Feng, X T Zu, General Relativity and Gravitation. 482H.L. Li, Z.W. Feng and X.T. Zu, "Quantum tunneling from a high dimensional Gdel black hole", General Relativity and Gravitation, vol. 48, no.2, Article no. 18, 2016. Quantum gravity corrections to the tunneling radiation of scalar particles. P Wang, H Yang, S Ying, International Journal of Theoretical Physics. 55526332642P. Wang, H. Yang and S. Ying, "Quantum gravity corrections to the tunneling radiation of scalar particles", International Journal of Theoretical Physics, vol. 55, no. 5, pp. 26332642, 2016. Fermions Tunnelling with Quantum Gravity Correction. Z Y Liu, J R Ren, Communications in Theoretical Physics. 626Z.Y. Liu and J.R. Ren, "Fermions Tunnelling with Quantum Gravity Correction", Communications in Theoretical Physics, vol. 62, no. 6, pp. 819-823, 2014. Minimal Length Effects on Tunnelling from Spherically Symmetric Black Holes. B Mu, P Wang, H Yang, ID 898916Advances in High Energy Physics. 2015B. Mu, P. Wang and H. Yang, "Minimal Length Effects on Tunnelling from Spherically Symmetric Black Holes", Advances in High Energy Physics, vol.2015, Article ID 898916, 2015. Scalar Particles Tunneling and Effect of Quantum Gravity. G Li, X Zu, Journal of Applied Mathematics and Physics. 03G. Li and X. Zu, "Scalar Particles Tunneling and Effect of Quantum Gravity", Journal of Applied Mathematics and Physics, vol. 03, pp. 134-139, 2015. Quantum-corrected self-dual black hole entropy in tunneling formalism with GUP. M A Anacleto, F A Brito, E Passos, Physics Letters B. 749M. A. Anacleto, F.A. Brito and E. Passos, "Quantum-corrected self-dual black hole entropy in tunneling formalism with GUP", Physics Letters B, vol. 749, pp. 181-186, 2015. Black hole remnant and quantum tunnelling in three-dimensional Gdel spacetime. H Li, X Zu, Astrophysics and Space Science. 3576H. Li and X. Zu, "Black hole remnant and quantum tunnelling in three-dimensional Gdel spacetime", Astrophysics and Space Science, vol. 357, Article no. 6, 2015. The remnants in Reissner-Nordstrm-de Sitter quintessence black hole. Z Feng, L Zhang, X Zu, Modern Physics Letters A. 2926Z. Feng, L. Zhang and X. Zu, "The remnants in Reissner-Nordstrm-de Sitter quintessence black hole", Modern Physics Letters A, vol. 29, no. 26, Article no. 1450123, 2014. G Gecim, Y Sucu, arXiv:1704.03537v2The GUP effect on tunnelling of massive vector bosons from the 2+1 dimensional black hole. G. Gecim and Y. Sucu, "The GUP effect on tunnelling of massive vector bosons from the 2+1 dimensional black hole", arXiv:1704.03537v2, (2017). The GUP effect on Hawking Radiation of the 2+1 dimensional Black Hole. G Gecim, Y Sucu, arXiv:1704.03536v1Physics Letters B. 773G. Gecim and Y. Sucu, "The GUP effect on Hawking Radiation of the 2+1 dimensional Black Hole", Physics Letters B, vol. 773, pp. 391-394, [arXiv:1704.03536v1], 2017. Quasinormal modes for new type black holes in new massive gravity. Y Kwon, S Nam, J Park, S H Yi, Classical and Quantum Gravity. 2814Y. Kwon, S. Nam, J. Park and S.H. Yi, "Quasinormal modes for new type black holes in new massive gravity", Classical and Quantum Gravity, vol. 28, no. 14, Article no. 145006, 2011. Exact solution of Dirac equation in 2+1 dimensional gravity. Y Sucu, N , Journal of Mathematical Physics. 485Y. Sucu and N. Unal, "Exact solution of Dirac equation in 2+1 dimensional gravity", Journal of Mathematical Physics, vol. 48, no. 5, Article no. 052503, 2007. A simpler method for researching fermions tunneling from black holes. K Lin, S Z Yang, Chinese Physics B. 2011K. Lin and S.Z. Yang, "A simpler method for researching fermions tunneling from black holes", Chinese Physics B, vol. 20, no. 11, Article no. 110403, 2011. A note on the Hawking radiation calculated by the quasiclassical tunneling method. Y P Hu, J Y Zhang, Z Zhao, Modern Physics Letters A. 25Y.P. Hu, J.Y.Zhang and Z. Zhao, "A note on the Hawking radiation calculated by the quasiclassical tunneling method", Modern Physics Letters A, vol. 25, pp. 295-308, 2010. Fermion tunneling from dynamical horizons. R , Di Criscienzo, L Vanzo, Europhysics Letters. 826R. Di Criscienzo and L. Vanzo, "Fermion tunneling from dynamical horizons", Europhysics Letters, vol. 82, no. 6, Article no. 60001, 2008. Exotic properties of superfluid 3He. G E Volovik, World ScientificSingaporeG.E. Volovik, Exotic properties of superfluid 3He, World Scientific, Singapore, 1992. Simulation of Panleve-Gullstrand black hole in thin 3He-A film. G E Volovik, Journal of Experimental and Theoretical Physics Letters. 69G.E. Volovik, "Simulation of Panleve-Gullstrand black hole in thin 3He-A film", Journal of Experimental and Theoretical Physics Letters, vol. 69, pp. 705-713, 1999. G E Volovik, The Universe in a Helium Droplet. OxfordClarendon PressG.E. Volovik, The Universe in a Helium Droplet, Clarendon Press, Oxford, 2003.
{'fraction_non_alphanumeric': 0.07410370905711999, 'fraction_numerical': 0.051751284102934725, 'mean_word_length': 3.8987229738272853, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 16, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': "We investigate the Generalized Uncertainty Principle (GUP) effect on the Hawking temperature for the 2+1 dimensional New-type black hole by using the quantum tunneling method for both the spin-1/2 Dirac and the spin-0 scalar particles. In computation of the GUP correction for the Hawking temperature of the black hole, we modified Dirac and Klein-Gordon equations. We observed that the modified Hawking temperature of the black hole depends not only on the black hole properties, but also on the graviton mass and the intrinsic properties of the tunneling particle, such as total angular momentum, energy and mass. Also, we see that the Hawking temperature was found to be probed by these particles in different manners. The modified Hawking temperature for the scalar particle seems to be lower compared to its standard Hawking temperature. Also, we find that the modified Hawking temperature of the black hole caused by Dirac particle's tunnelling rised by the total angular momentum of the particle. It is diminishable by the energy and mass of the particle and graviton mass as well. These intrinsic properties of the particle, except total angular momentum for the Dirac particle, and graviton mass may cause screening for the black hole radiation.", 'arxivid': '1710.09125', 'author': ['Ganim Gecim [email protected] \nDepartment of Physics\nFaculty of Science\nAkdeniz University\n07058AntalyaTurkey\n', 'Yusuf Sucu [email protected] \nDepartment of Physics\nFaculty of Science\nAkdeniz University\n07058AntalyaTurkey\n'], 'authoraffiliation': ['Department of Physics\nFaculty of Science\nAkdeniz University\n07058AntalyaTurkey', 'Department of Physics\nFaculty of Science\nAkdeniz University\n07058AntalyaTurkey'], 'corpusid': 55352508, 'doi': '10.1155/2018/8728564', 'github_urls': [], 'n_tokens_mistral': 13716, 'n_tokens_neox': 11353, 'n_words': 6731, 'pdfsha': '3cfcc176f51aca2626c4b43022e94d25c324423c', 'pdfurls': ['https://arxiv.org/pdf/1710.09125v1.pdf'], 'title': ['Quantum Gravity Effect on the Tunneling Particles from 2+1 dimensional New-type Black Hole', 'Quantum Gravity Effect on the Tunneling Particles from 2+1 dimensional New-type Black Hole'], 'venue': []}
arxiv
The dark Stodolsky effect: constraining effective dark matter operators with spin-dependent interactions 25 May 2023 Guillaume Rostagni [email protected] Institute for Particle Physics Phenomenology Department of Physics Durham University DurhamUK Jack D Shergold [email protected] Institute for Particle Physics Phenomenology Department of Physics Durham University DurhamUK The dark Stodolsky effect: constraining effective dark matter operators with spin-dependent interactions 25 May 2023Prepared for submission to JCAP We present a comprehensive discussion of the Stodolsky effect for dark matter (DM), and discuss two techniques to measure the effect and constrain the DM parameter space. The Stodolsky effect is the spin-dependent shift in the energy of a Standard Model (SM) fermion sitting in a bath of neutrinos. This effect, which scales linearly in the effective coupling, manifests as a small torque on the SM fermion spin and has historically been proposed as a method of detecting the cosmic neutrino background. We generalise this effect to DM, and give expressions for the induced energy shifts for DM candidates from spin-0 to spin-3 2 , considering all effective operators up to mass dimension-6. In all cases, the effect scales inversely with the DM mass, but requires an asymmetric background. We show that a torsion balance experiment is sensitive to energy shifts of ∆E Á 10´2 8 eV, whilst a more intricate setup using a SQUID magnetometer is sensitive to shifts of ∆E Á 10´3 2 eV. Finally, we compute the energy shifts for a model of scalar DM, and demonstrate that the Stodolsky effect can be used to constrain regions of parameter space that are not presently excluded. Introduction There is now overwhelming evidence for dark matter (DM) on both galactic [1][2][3][4][5] and cosmological [6,7] distance scales, which is estimated to constitute " 26% of the total energy density of the universe [8]. Despite this, the exact nature of DM remains a mystery, with all evidence for its existence coming from its gravitational interactions with visible matter. Nevertheless, the possibility remains that DM could interact with Standard Model (SM) fields non-gravitationally, which could allow us to better study its nature. In order for new fields in a SM extension to be considered as DM candidates, they must be cold in the present epoch and capable of reproducing the observed relic density, electrically neutral, and unable to decay into SM particles over cosmological timescales. This leaves an overwhelmingly large number of DM candidate theories, which are tedious to constrain individually. Effective field theories (EFTs) are an incredibly powerful tool to constrain DM in a model-independent way [9][10][11][12]; by making use of the symmetries of the interaction Lagrangian, EFTs reduce the landscape of underlying theories to a finite number of permitted operators. These operators are typically classified by the spin of the DM particle, along with their mass dimension and coupling to the SM, which can then be constrained and mapped onto the candidate DM theory on a case-by-case basis. To that end, several experiments have already been set up or proposed to directly search for DM using a variety of techniques, each of which have sensitivity to different ranges of parameter space: scattering on ultracold nuclei [13][14][15]; scattering in Xenon time projection chambers [16][17][18]; axion telescopes [19][20][21][22][23]; scattering in particle accelerators [24]; atom interferometers [25][26][27]. At the same time, DM has also been indirectly constrained using a variety of astrophysical [28][29][30][31][32][33][34][35] and cosmological [7,8,36,37] probes. In this paper, we propose two experiments to observe spin-dependent energy shift induced by a DM background, which is more commonly known as the Stodolsky effect, and has historically been discussed in the context of cosmic neutrino background (CνB) detection [38][39][40][41]. The Stodolsky effect has several features which make a promising avenue for DM detection. First, unlike scattering, the magnitude of the energy shift depends on the DM-SM coupling linearly rather than quadratically, leading to an effect which is less suppressed by tiny coupling constants. Second, whilst many detection techniques depend heavily on the mass of the DM particle under consideration, the Stodolsky effect depends primarily on the velocity of the background particle. For neutrinos, this leads to an energy shift that is largely independent of the neutrino mass [41], which if also true for dark matter would allow us to probe a wide region of parameter space. On the contrary, the Stodolsky effect for neutrinos requires either a neutrino-antineutrino or left-right helicity asymmetry in the background, the former of which is expected to be absent in the standard CνB scenario. As we will see, analogous requirements persist for DM backgrounds, potentially restricting the range of models that can give rise to the Stodolsky effect. Even so, both chiral and asymmetric [42] models of DM exist, which alongside models with finite chemical potential generate an asymmetry during DM production. We additionally note that there are several mechanisms (e.g. finite chemical potential, DM reflection at the surface of the Earth [43], gravitational potentials [44]) through which either asymmetry may develop post-production. The remainder of this paper will be structured as follows. In Section 2 we will review the Stodolsky effect for neutrinos and introduce the general formalism that will be used throughout. Following this, in Section 3 we will compute the magnitude of the Stodolsky effect for all effective DM operators ranging from spin-0 to spin- 3 2 , up to dimension-6. Finally, we will discuss the experimental signatures of the Stodolsky effect and the feasibility of this technique for DM detection in Section 4, before concluding in Section 5. The Stodolsky effect We begin by reviewing the Stodolsky effect for the CνB, which has been discussed in several previous works [38][39][40][41]. This will closely follow the formalism of [41], with the exception that we will more carefully treat the external states as partially localised wavepackets, rather than eigenstates of definite momentum. Additionally, we will assume that the neutrinos are monochromatic in the CνB reference frame, which is a good approximation when their momentum distribution is narrow. Working in the mass basis, the effective low energy Hamiltonian density for neutrinoelectron interactions after applying a Fierz transformation is H int pxq " G F ? 2 ÿ i,jν i γ µ p1´γ 5 qν jē γ µ pV ij´Aij γ 5 qe, (2.1) where G F is the Fermi constant, V ij and A ij are the effective vector and axial couplings, respectively, and i, j P t1, 2, 3u denote the neutrino mass eigenstate. To leading order in H int , the energy shift of electron helicity state h e is given by ∆E e p⃗ p e , h e q " ÿ ν,i,hν ÿ Nν xe he , ν i,hν | ż d 3 x H int pxq|e he , ν i,hν y, (2.2) where h e and h ν denote the electron and neutrino helicities, respectively, whilst ř ν is the instruction to sum over neutrinos and antineutrinos. Similarly, ř Nν is a sum over all neutrinos in the background with the degrees of freedom specified by the preceding sum. The external states are incoherent superpositions of momentum eigenstates, defined by [45][46][47] |ψpp ψ , x ψ , h ψ qy " ż d 3 q ψ p2πq 3 1 a 2E q ψ ω ψ pp ψ , q ψ qe´i ⃗ q ψ¨⃗ x ψ |tq ψ , h ψ uy, (2.3) with ψ P te, νu, where E q ψ is the energy of the momentum eigenstate with momentum q ψ and ω ψ is a wavepacket function centred on the momentum p ψ . The wavepacket states are normalised to unity, which also sets the normalisation of ω ψ . We use relativistic normalisation for the momentum eigenstates |tp ψ , h ψ uy " a 2E p a : ψ p⃗ p ψ , h ψ q|0y, (2.4) where a : ψ p⃗ p, hq is the particle creation operator for species ψ with momentum and helicity ⃗ p and h, respectively, whilst its Hermitian conjugate is the corresponding annihilation operator. We denote the antiparticle creation and annihilation operators with b : ψ and b ψ , respectively. These satisfy the anticommutation relations ! a i p⃗ p, hq, a : j p⃗ q, h 1 q ) " ! b i p⃗ p, hq, b : j p⃗ q, h 1 q ) " p2πq 3 δ p3q p⃗ p´⃗ qqδ ij δ hh 1 ,(2.ω ν i pp ν i , q ν i qων i pp ν i , q 1 ν i qe´i p⃗ qν i´⃗ q 1 ν i q¨⃗ xν î xtq 1 e , h e u, tq 1 ν i , h ν i u|H int pxq|tq ν i , h ν i u, tq e , h e uy,(2.8) where we have introduced the shorthand dΠ " d 3 q e p2πq 3 d 3 q 1 e p2πq 3 d 3 q ν i p2πq 3 d 3 q 1 ν i p2πq 3 1 a 2E qe 1 a 2E q 1 e 1 a 2E qν i 1 b 2E q 1 ν i . (2.9) In line with [45,47], we now average ∆E e over the regions in which the wavepackets are localised, i.e. we take ∆E e p⃗ p e , h e q Ñ 1 V 2 ż d 3 x e d 3 x ν i ∆E e p⃗ p e , h e q,(2.ÿ Nν 1 4V 2 ż d 3 x d 3 q e d 3 p ν i p2πq 6 E e E ν i |ω e pp e , q e q| 2 |ω ν i pp ν i , q ν i q| 2 xH int y,(2.11) where xH int y " xtq e , h e u, tq ν i , h ν i u|H int pxq|tq ν i , h ν i u, tq e , h e uy. (2.12) Recalling the normalisation ş d 3 q p2πq 3 |ωpp, qq| 2 " 1 allows us to identify |ωpp, qq| 2 {V as the phase space density for a single particle. The sum over all particles in the background can therefore be used to replace the wavepacket functions with momentum distribution functions ÿ Nν |ω ν i pp ν i , q ν i q| 2 V " n ν pν i,hν qf ν i p⃗ q ν i q, |ω e pp e , q e q| 2 V " 1 V p2πq 3 δ p3q p⃗ p e´⃗ q e q,(2.13) where n ν pν i,hν q is the number density of background neutrino eigenstate i with helicity h ν . Finally, after noting that nothing in xH int y depends on position and considering an electron at rest in the lab frame, we find ∆E e p ⃗ 0, h e q " 1 4m e ÿ ν,i,hν n ν pν i,hν q ż d 3 p ν i p2πq 3 f ν i p⃗ p ν i q 1 E ν i xH int yˇˇ| ⃗ pe|"0 " 1 4m e ÿ ν,i,hν n ν pν i,hν q B 1 E ν i xH int y F ,(2.14) where m e is the electron mass and the outermost angled brackets denote an averaged quantity, which must be done in order to account for the relative motion of the Earth to the CνB reference frame. The averaging procedure differs slightly between the CνB and DM, as we do not know the velocity of the former. We therefore use the flux averages from [41] for the CνB, whilst those for DM are discussed at length in Appendix A. Expanding out the external states, applying the appropriate anticommutation relations and taking the traces of Dirac spinor chains yields [41] xH int y " 2 ? 2G F A ii m e h e " m ν i h ν i pS e¨Sν i q´pS e¨pν i q ı`f pV ii q,(2.15) where h "˘1 denotes the particle spin eigenvalue, m ν i denotes the neutrino mass and f pV ii q contains terms that do not depend on the electron spin, which will not contribute to the Stodolsky effect. Note that (2.15) takes the opposite sign for external antineutrino states, whilst for external Majorana neutrino states the expectation value is twice as large. The spin vector for massive fermions is given by S µ "ˆ⃗ p¨⃗ s m , ⃗ s`p ⃗ p¨⃗ sq⃗ p mpE`mq˙,(2.16) for a particle with spin vector ⃗ s in its own reference frame. By inspection, we see that S satisfies pp¨Sq " 0. If we restrict our discussion to helicity eigenstates then ⃗ s will be directed along ⃗ p, such that (2.16) reduces to S µ "ˆ| ⃗ p| m , E m ⃗ p |⃗ p|˙,(2.17) and we instead identify h "˘1 with the particle helicity 1 . Naturally, we cannot use (2.17) for a particle at rest. The energy splitting between the two electron spin states is then found by taking the difference between the energy shifts for each spin state, which after performing the flux averaging on (2.15) gives ∆E D e " ? 2G F 3 | ⃗ β C | ÿ i A ii " 2 ÿ sν p2´| ⃗ β ν i | 2 qpn ν pν D i,sν q´n ν pν D i,sν q 1 | ⃗ β ν i | p3´| ⃗ β ν i | 2 qpn ν pν D i,L q´n ν pν D i,R q`n ν pν D i,R q´n ν pν D i,L qq ı , (2.18) for a Dirac neutrino background, where the subscripts L and R denote left and right helicity neutrinos, respectively, with R{L corresponding to h ν i "˘1 p¯1q for (anti)neutrinos. Additionally, ⃗ β C is the relative velocity between the Earth and CνB reference frame, which may be time dependent, and ⃗ β ν i is the lab frame neutrino velocity. For completeness, we note that whilst the term scaling as | ⃗ β ν i |´1 appears divergent, it in fact tends to zero as | ⃗ β ν i | Ñ 0 as a consequence of a vanishing helicity asymmetry for slow neutrinos 2 . Similarly, we find for a Majorana neutrino background ∆E M e " 2 ? 2G F 3 | ⃗ β C | ÿ i A ii | ⃗ β ν i | p3´| ⃗ β ν i | 2 qpn ν pν M i,L q´n ν pν M i,R qq. (2.19) We immediately see that the Stodolsky effect for neutrinos requires either a non-zero neutrinoantineutrino or helicity asymmetry, but depends only on the neutrino velocity and scales linearly with G F . These features allow an experiment utilising the Stodolsky effect to probe a vast region of DM parameter space, as the effect is less suppressed than scattering in weakly coupled regions, whilst only depending on the dark matter velocity, | ⃗ β DM | » 1.2ˆ10´3 [48], independent of the DM mass. We are now ready to move onto the Stodolsky effect for DM, which we will henceforth refer to as the dark Stodolsky effect (DSE) to distinguish it from the effect for neutrinos. By analogy with (2.2), the energy shift of an at rest SM fermion ψ in a DM background will be given by ∆E ψ p ⃗ 0, h ψ q " 1 4m ψ ÿ d.o.f. n DM B 1 E DM xH int y F ,(2.20) where the sum runs over the DM degrees of freedom. For the remainder of this paper we will focus on the object appearing inside the angled brackets, which will typically be some kinematic structure depending on the effective DM operator under consideration. When evaluating these expectation values we will only keep the terms that depend on S ψ , as no other terms will contribute to the DSE. The energy splitting of the two SM fermion spin states can then found by starting with our master equation (2.20), and then taking the difference in the energy shifts for the two spin states. This will typically enter as an overall factor of two. Effective dark matter operators We now turn our attention to the rich landscape of effective DM operators that can give rise to the DSE. For the remainder of this work, we will consider an effective DM Lagrangian of the form L DM " L SM`Lkin`Lint , (3.1) where L SM is the complete SM Lagrangian, L kin contains the kinetic and mass terms for the DM field, and L int contains effective SM-DM interaction operators. This will take the form L int "´g ψχ Λ d´4 O µν... DM O SM µν... ,(3.2) where g ψχ denotes the coupling between the SM fermion and DM field, Λ is the new physics scale, and d is the combined mass dimension of the SM and DM effective operators, O SM and O DM , respectively. We will only work with Lagrangians that are Lorentz invariant, Hermitian, invariant under the SM gauge group and irreducible by the equations of motion, the procedure for which is discussed in Appendix B. By inspection of the expectation value, we immediately see that in order for an operator to contribute to the DSE it must contain at least two copies of the field operator corresponding to each external field. For bosonic DM, this gives a minimum combined mass dimension for O SM and O DM of d " 5, whilst for fermionic DM, the minimum mass dimension is d " 6. As such, we will include all effective DM operators up to d " 6. However, we will not consider DM operators with d ą 6, which become increasingly suppressed by the new physics scale Λ with increasing d. For an operator O SM "ψΓ µν... ψ, after expanding out the field operators and external states, and applying the appropriate (anti)commutation relations, we will find the general form for the expectation value containing a trace over the SM fermion Dirac structure xH int y " g ψχ Λ d´4 P µν... χ Trru ψūψ Γ µν... s,(3.3) where P χ contains details of the DM kinematics, which may itself contain Dirac traces, Γ denotes some string of gamma matrices, and we have used the shorthand u ψ " u ψ pp ψ , s ψ q. The trace can be simplified in a basis independent way be making use of the identities upp, hqūpp, h 1 q " 1 2 p { p`mqp1`hγ 5 { Sqδ hh 1 , (3.4) vpp, hqvpp, h 1 q " 1 2 p { p´mqp1`hγ 5 { Sqδ hh 1 ,(3.5) with { A " γ µ A µ for some general four vector A. There are a total of five independent gamma matrix structures that can be included in the fermion trace 1, γ 5 , γ µ , γ µ γ 5 , σ µν , (3.6) where σ µν " i 2 rγ µ , γ ν s, γ 5 " i 4! ε αβµν γ α γ β γ µ γ ν and ε αβµν is the Levi-Civita symbol. Of these, only some will give rise to expectation values that depend on the SM fermion spin, and the remainder can be neglected. Explicitly, we find Trru ψūψ Γ µν... s " $ ' ' ' ' ' ' & ' ' ' ' ' ' % 2m ψ , Γ " 1, 0, Γ " γ 5 , 2p µ ψ , Γ " γ µ , 2m ψ h ψ S µ ψ , Γ " γ µ γ 5 , 2h ψ ε αβµν p α ψ S β ψ , Γ " σ µν ,(3.7) such that of the five independent gamma matrix structures appearing in (3.6), we only need to consider γ µ γ 5 and σ µν . There is an additional Lorentz invariant structure that we need to consider, ε αβµν P αβ χūψ σ µν u ψ , (3.8) which will clearly depend on S ψ . This can be rewritten in terms of γ 5 aś 2iP µν χūψ σ µν γ 5 u ψ , (3.9) and so we will consider the structureψσ µν γ 5 ψ as an additional 'independent' operator throughout. Finally, we note that there are several operator combinations, e.g. O DM O SM " |ϕ| 2ψ γ 5 ψ,(3.10) containing some complex scalar DM field ϕ, that couple left to right-chiral SM fermions and appear to be dimension-5. However, in order for the SM component to be gauge invariant under SUp2q L , we require an additional insertion of the SM fermion mass. As a result, if Λ " m ψ , the operator will effectively scale as one of dimension-6. However, as we do not specify the new physics scale, we will treat such operators as dimension-5 throughout. By extension, we will define the dimension of any operator considered in the remainder of this work as the sum of the mass dimensions of its field content and the number of derivatives. Spin-0 New scalar fields are popular candidates for DM [49][50][51], which typically take the form of axion or Higgs-like particles. Axions are well motivated DM candidates, naturally arising in any extension to the SM where an approximate global symmetry is spontaneously broken, where they play the role of the pseudo Nambu-Goldstone boson associated with the symmetry breaking. On the other hand, Higgs-like extensions to the SM require very few additional parameters. In fact, a real singlet scalar coupled to the SM Higgs is the minimal renormalisable extension to the SM capable of explaining DM [52]. In our EFT approach, we will make no reference to the underlying theory and simply consider some complex scalar field ϕ, for which the corresponding field decompositions are 12) and the analogous field decomposition for a real scalar DM candidate is found by setting b " a. Unlike neutrinos, the creation and annihilation operators for bosonic DM follow commutation relations " a i p⃗ pq, a : j p⃗ qq ı " ϕpxq " ż d 3 p p2πq 3 1 a 2E p`a p⃗ pqe´i p¨x`b: p⃗ pqe ip¨x˘, (3.11) ϕ˚pxq " ż d 3 p p2πq 3 1 a 2E p`a : p⃗ pqe ip¨x`b p⃗ pqe´i p¨x˘,(3." b i p⃗ pq, b : j p⃗ qq ı " p2πq 3 δ p3q p⃗ p´⃗ qqδ ij , (3.13) with all other commutators equal to zero. As it turns out, there is only one scalar operator up to dimension-6 that gives rise to the DSE, with interaction Lagrangian L ϕ int "´i g ψϕ Λ 2 pϕ˚Ð Ñ B µ ϕqpψγ µ γ 5 ψq,(3.14) where ϕ˚Ð Ñ B µ ϕ " ϕ˚pB µ ϕq´pB µ ϕ˚qϕ. The corresponding Hamiltonian density is found via a Legendre transformation H ϕ int " ÿ ϕ 9 ϕ BL ϕ int B 9 ϕ´L ϕ int " ig ψϕ Λ 2´ϕ˚p ⃗ ∇ϕq´p ⃗ ∇ϕ˚qϕ¯¨`ψ ⃗ γγ 5 ψ˘, (3.15) where the sum runs over ϕ and ϕ˚. In a background of pure ϕ scalars, the relevant expectation value that contributes to the DSE can be computed using the appropriate field decompositions and commutators to find xH ϕ int y "´2 g ψϕ Λ 2 ⃗ p ϕ¨`ūψ ⃗ γγ 5 u ψ˘"´4 g ψϕ Λ 2 m ψ h ψ p⃗ p ϕ¨⃗ S ψ q, (3.16) where in going from the first to the second equality we have used the trace identity given in (3.7). If a background of pure ϕ˚scalars is considered instead, the expectation value (3.16) takes the opposite sign. Plugging this into our master equation (2.20), we therefore find the energy shift of the SM fermion state with spin h ψ ∆E ϕ ψ p ⃗ 0, h ψ q "´g ψϕ Λ 2 h ψ C p⃗ p ϕ¨⃗ S ψ q E ϕ G pn ϕ pϕq´n ϕ pϕ˚qq,(3.17) with n ϕ pϕq and n ϕ pϕ˚q the number densities of background species ϕ and ϕ˚, respectively. Replacing the average with the expression given in (A.7) yields ∆E ϕ ψ p ⃗ 0, h ψ q "´2 g ψϕ Λ 2 h ψ β C pn ϕ pϕq´n ϕ pϕ˚qq, (3.18) where β C is the magnitude of the relative velocity between the laboratory and DM reference frames. By taking the difference between the energy shift for each SM fermion spin state, we find an energy splitting ∆E ϕ ψ " ∆E ϕ ψ p ⃗ 0, 1q´∆E ϕ ψ p ⃗ 0,´1q "´4 g ψϕ Λ 2 β C pn ϕ pϕq´n ϕ pϕ˚qq. (3.19) The energy splitting (3.19) is therefore independent of the DM kinematics, potentially allowing us to constrain scalar DM with masses ranging over many orders of magnitude. Notably, however, we still require a matter-antimatter asymmetry in order to generate a DSE for scalar DM. The culprit in this case is the derivative appearing between the scalar fields in (3.14), which generates an overall minus sign between the positive and negative frequency field modes. This differs from the neutrino case presented in Section 2, where the asymmetry results from the anticommutation relations for fermionic operators. Finally, for completeness we note that the corresponding energy splittings for a real scalar DM background are found by setting n ϕ pϕq " n ϕ pϕ˚q, such that (3.19) should vanish identically. Label Table 1. Lorentz invariant, Hermitian, gauge invariant and irreducible spin-1 2 DM operators contributing to the DSE up to dimension-6, along with their corresponding expectation values in a background of Dirac fermions and antifermions, denoted by |χy and |χy, respectively. We leave the global factors of the coupling, new physics scale and SM fermion spin eigenvalue, h ψ , implicit. O DM O SM Background xH int y O χ 1 pχγ µ χqpψγ µ γ 5 ψq |χy 4m ψ pp χ¨Sψ q |χy´4m ψ pp χ¨Sψ q O χ 2 pχγ µ γ 5 χqpψγ µ γ 5 ψq |χy 4m ψ m χ h χ pS χ¨Sψ q |χy 4m ψ m χ h χ pS χ¨Sψ q O χ 3 pχσ µν χqpψσ µν ψq |χy 8h χ " pp χ¨Sψ qpS χ¨pψ q pp χ¨pψ qpS χ¨Sψ q ‰ |χy´8 h χ " pp χ¨Sψ qpS χ¨pψ q pp χ¨pψ qpS χ¨Sψ q ‰ O χ 4 ipχσ µν χqpψσ µν γ 5 ψq |χy´8h χ ε αβµν p α χ p β ψ S µ χ S ν ψ |χy 8h χ ε αβµν p α χ p β ψ S µ χ S ν ψ Spin-1 2 We now turn our attention to spin-1 2 dark matter, popular candidates for which include sterile neutrinos [53][54][55], which may also explain short baseline anomalies [56], and neutralinos [57,58], which naturally arise from supersymmetric models. As we have already seen for neutrinos, the DSE for spin-1 2 backgrounds differs considerably to the effect for scalar DM, as it can additionally depend on the helicity composition of the background. Furthermore, as the product of four fermion field operators has mass dimension-6, there can be no derivative couplings for fermions at the order considered here. As such, we only need to consider Lorentz structures containing products of linearly independent gamma matrices (3.6) and Levi-Civita symbols, the latter of which can be treated as an additional gamma matrix structure, σ µν γ 5 . In all cases, the absence of derivative couplings, along with the anticommutators for fermionic operators will necessarily lead to energy splittings that require a background asymmetry. Considering a spin-1 2 DM candidate χ, we tabulate all irreducible operators contributing the DSE up to dimension-6, along with their corresponding expectation values in Table 1. For each, we consider the case where the background consists of Dirac χ and anti-χ, which we denote by |χy and |χy, respectively. The corresponding expectation values in Majorana χ backgrounds are found by summing those in |χy and |χy backgrounds. In addition to the operators shown in Table 1, we could also have considered the operators O χ 5 " ipχσ µν γ 5 χqpψσ µν ψq and O χ 6 " pχσ µν γ 5 χqpψσ µν γ 5 ψq. However, we show in Appendix B that these are exactly equal to O χ 4 and O χ 3 , respectively. We have already seen the operators O χ 1 and O χ 2 in Section 2, which gave rise to the neutrino-antineutrino and helicity asymmetry terms for neutrinos, respectively. The third operator in Table 1 (3.20) leads to an energy shift , O χ 3 , with interaction Lagrangian L χ 3 int "´g ψχ Λ 2 pχσ µν χqpψσ µν ψq,∆E χ 3 ψ p ⃗ 0, h ψ q " 2g ψχ m ψ Λ 2 h ψ ÿ hχ h χ « B pp χ¨Sψ qpS χ¨pψ q E χ F´B pp χ¨pψ qpS χ¨Sψ q E χ F ff pn χ pχ hχ q´n χ pχ hχ qq,(3.21) which after replacing the averages with (A.10) and (A.11) yields ∆E χ 3 ψ p ⃗ 0, h ψ q " 7g ψχ 4Λ 2 h ψ pn χ pχ R q´n χ pχ L q´n χ pχ L q`n χ pχ R qq`O`β 2 C , 1´β r˘, (3.22) where β r " β C {β c » 1 is the ratio of the relative frame velocity and galactic circular velocity, β c . The resulting energy splitting between the SM fermion spin states has magnitude ∆E χ 3 ψ " 7g ψχ 2Λ 2 h ψ pn χ pχ R q´n χ pχ L q´n χ pχ L q`n χ pχ R qq ,(3.23) to leading order in small quantities, where the subscripts L and R denote the number densities of left and right helicity DM fermions, which satisfy n χ pχ R q`n χ pχ L q " n χ pχq and n χ pχ L q`n χ pχ R q " n χ pχq, respectively. Remarkably, this is energy shift is not suppressed by the velocity scale provided that β r » 1. Despite this, energy shifts of the form (3.23) are exceedingly difficult to generate; whilst the helicity asymmetry requirement of (3.23) naively appears comparable to that of (2.18) for neutrinos, this is not the case. The first difference is seen when considering Majorana fermions, for which n χ pχ R q " n χ pχ L q and n χ pχ L q " n χ pχ R q. This leads to (3.23) vanishing identically, whilst (2.18) becomes (2.19), which importantly is non-zero. The second difference is more subtle. A chiral theory such as the weak interaction will naturally lead to scenarios in which n χ pχ R q » n χ pχ L q ‰ n χ pχ L q » n χ pχ R q, (3.24) in particular when the DM fermion is produced relativistically, such that its helicity and chirality coincide 3 . This helicity profile is sufficient to generate a DSE through the operator O χ 2 , but not O χ 3 , which requires a further fermion-antifermion asymmetry (e.g. through a chemical potential) to give a non-zero energy splitting. This significantly restricts the number of models that can generate a DSE through operators of the form O χ 3 . Finally, we note that as discussed in Appendix B of [41], background helicity asymmetries vanish for very cold DM as a consequence of the relative frame velocity. This is true irrespective of the DM spin, and so should be taken into account whenever an operator requires a non-zero helicity asymmetry to contribute to the DSE. The final operator appearing in Table 1, O χ 4 , generates an energy shift scaling with ∆E χ 4 ψ p ⃗ 0, s ψ q " ε αβµν p α χ p β ψ S µ χ S ν ψ ,(3.25) which considering only helicity eigenstates and making use of the identities given in Appendix C vanishes identically. We note, however, that this operator may give rise to a non-zero DSE for an alternative experimental setup where the SM fermion is not at rest in the lab frame. Spin-1 Vector bosons remain popular in many models of DM, with candidates including additional U p1q gauge bosons [60][61][62][63][64], superpartners to neutrinos [57,58], and Kaluza-Klein states in theories with extra dimensions [58,65,66]. It is also entirely possible to generate dark hadronic vector states in non-Abelian extensions to the SM [67]. The DSE for vector bosons is similar to that for scalar bosons, and may depend on either the total background DM density or require an asymmetry in the presence of derivative couplings. They differ, however, in the fact that vector bosons carry an additional Lorentz index, which expands the number of contributing operators. Here we consider a massive 4 vector field X µ , with field decomposition X µ pxq " ż d 3 p p2πq 3 1 a 2E p ÿ l`a p⃗ p, lqϵ µ pp, lqe´i p¨x`b: p⃗ p, lqϵμpp, lqe ip¨x˘, (3.26) Xμpxq " ż d 3 p p2πq 3 1 a 2E p ÿ l`a : p⃗ p, lqϵμpp, lqe ip¨x`b p⃗ p, lqϵ µ pp, lqe´i p¨x˘, (3.27) where the creation and annihilation operators satisfy the commutation relations (3.13), whilst ϵ µ pp, lq " ϵ µ l is the polarisation vector with polarisation l P t´1, 0, 1u. Considering the helicity eigenstates for a state with momentum along the`z direction, these take the form ϵ µ pp, 1q " ϵ μ " 1 ? 2¨0 1 i 0‹ ‹ ‚ , ϵ µ pp,´1q " ϵ μ " ϵ˚μ , ϵ µ pp, 0q " ϵ µ L " 1 m¨| ⃗ p| 0 0 E‹ ‹ ‚ , (3.28) which we will refer to as the right, left and longitudinal polarisation states, respectively, together satisfying ϵpp, lq¨ϵpp, l 1 q˚"´δ ll 1 . The polarisation vectors for momenta along other directions are found by applying the appropriate rotation matrix. These will need to be considered in order to perform the averaging appropriately. We tabulate all irreducible operators for vector DM contributing to the DSE up to dimension-6 in Table 2. As before, we consider each of the cases where the background consists of a complex vector field, X µ and its conjugate, Xμ, which we denote by |Xy and |X˚y, respectively. The corresponding expectation values in real X backgrounds are found by summing those in |Xy and |X˚y backgrounds. The first operator in Table 2, O X 1 , is analogous to the one appearing in L ϕ int . This has already been discussed in detail in 3.1; the only difference here is the overall sign of the energy shift, generated by the contraction of two polarisation vectors. As a result, the energy splitting between the two SM fermion spin states will be the same for O X 1 as for its scalar counterpart, and sensitivity to the individual energy shifts is required to distinguish between the two operators. The second operator in Table 2, O X 2 , generates an energy shift ∆E X 2 ψ p ⃗ 0, h ψ q " g ψX 2m ψ Λ h ψ ÿ l X B 1 E X Im " ε αβµν p α ψ S β ψ ϵ˚µ l X ϵ ν l X ı F`n X pX l X q´n X pXl X q˘, (3.29) for charged vector bosons, and zero otherwise. We immediately see that the longitudinal modes of X µ with real polarisation vectors do not contribute to the DSE. This leaves the remaining two polarisation states, which after substituting in the average (A.13) and taking the difference between the two energy shifts gives an SM fermion energy splitting Label O DM O SM Background xH int y O X 1 ipXα Ð Ñ B µ X α qpψγ µ γ 5 ψq |Xy 4m ψ p⃗ p X¨⃗ S ψ q |X˚y´4m ψ p⃗ p X¨⃗ S ψ q O X 2 iXμX ν pψσ µν ψq |Xy 2 Im " ε αβµν ϵ˚α l X ϵ β l X p µ ψ S ν ψ ı |X˚y´2 Im " ε αβµν ϵ˚α l X ϵ β l X p µ ψ S ν ψ ı O X 3 XμX ν pψσ µν γ 5 ψq |Xy 4 Im " pϵl X¨S ψ qpϵ l X¨p ψ q ı |X˚y´4 Im " pϵl X¨S ψ qpϵ l X¨p ψ q ı O X 4 i " X µ˚p B µ X ν q´pB µ Xν qX µ ‰ pψγ ν γ 5 ψq |Xy´4m ψ Re " pϵl X¨S ψ qp⃗ ϵ l X¨⃗ p X q ı |X˚y 4m ψ Re " pϵl X¨S ψ qp⃗ ϵ l X¨⃗ p X q ı O X 5 " X µ˚p B µ X ν q`pB µ Xν qX µ ‰ pψγ ν γ 5 ψq |Xy 4m ψ Im " pϵl X¨S ψ qp⃗ ϵ l X¨⃗ p X q ı |X˚y 4m ψ Im " pϵl X¨S ψ qp⃗ ϵ l X¨⃗ p X q ı O X 6 ipX˚µX ν`X µ Xν qpψ Ð Ñ B µ γ ν γ 5 ψq |Xy´8m ψ Re " pϵl X¨S ψ qp⃗ ϵ l X¨⃗ p ψ q ı |X˚y´8m ψ Re " pϵl X¨S ψ qp⃗ ϵ l X¨⃗ p ψ q ı O X 7 iε αβµν " XαpB β X µ q´pB β XμqX α ‰ pψγ ν γ 5 ψq |Xy 0 |X˚y 0 O X 8 ε αβµν " XαpB β X µ q`pB β XμqX α ‰ pψγ ν γ 5 ψq |Xy 4m ψ Im " ε αβiν ϵ˚α l X ϵ β l X p i X S ν ψ ı |X˚y 4m ψ Im " ε αβiν ϵ˚α l X ϵ β l X p i X S ν ψ ı∆E X 2 ψ " 7g ψX 8m X Λ`n X pX´q´n X pX`q´n X pX˚q`n X pX˚q˘, (3.30) to leading order. Similar to (3.23), the energy shift from O X 2 is not suppressed by the velocity scale, but requires both a polarisation and matter-antimatter symmetry in order to give a non-zero contribution to the DSE. The former requirement is more difficult for vector bosons than fermions, which permit chiral Lagrangians that preferentially produce fermions of a single helicity at high energies. For vector DM, a polarisation asymmetry must therefore be generated through another mechanism such as scattering on a polarised fermionic background. The third operator in Table 2 gives rise to an energy shift ∆E X 3 ψ p ⃗ 0, s ψ q " B 1 E X Im " pp ψ¨ϵl X qpS ψ¨ϵl X q ‰ F ,(3.31) which vanishes for an SM fermion at rest in the lab frame. We note, however, that there may be a contribution to the energy shift from the right and left polarisation states for other experimental setups. On the other hand, the longitudinal state cannot contribute for any setup as its polarisation vector is real. The fourth operator in Table 2 is unique, and leads to an energy splitting ∆E X 4 ψ " 2g ψX Λ 2 ÿ l X B 1 E X Re " pϵl X¨S ψ qp⃗ ϵ l X¨⃗ p X q ‰ F`n X pX l X q´n X pXl X q"´4 g ψX Λ 2 β C pn X pX L q´n X pXLqq ,(3.32) which depends solely on the density of longitudinally polarised background states. This energy splitting is most closely related to the one generated by O X 1 , which instead depends on the total asymmetry between X µ and its conjugate. As such, it must always be the case that ∆E X 1 ψ ě ∆E X 4 ψ , which may serve to distinguish the two. Of the remaining operators, all have vanishing contributions to the Stodolsky effect for our experimental setup: the contribution from O X 5 is proportional to the imaginary part of the kinematic structure found in (3.32), which is real valued after averaging over background momenta; the contributions from O X 6 is proportional to ⃗ p ψ , which is zero for our setup; the contribution due to O X 7 vanishes at the kinematic level, as A H X 7 int E " ε αβµν Re " ϵ˚α l X ϵ β l X ı " 0,(3.33) whilst the energy splitting due to O X 8 scales with A H X 8 int E " ε αβiν ϵ˚α l X ϵ β l X p i X S ν ψ ,(3.34) which is zero for the longitudinal states since ϵL " ϵ L , and for the right and left helicity states as only their spatial components are non-zero. Notice that the operator giving rise the Zeeman effect, O F " F µνψ σ µν ψ, where F µν is the field strength tensor, does not appear in Table 2. This is because it only contains a single copy of the vector field, and as a result has a zero expectation value for incoherent background DM states (2.3). Instead, the Zeeman effect occurs in a coherent background, defined as the minimum uncertainty state and by extension the state which is the closest to a classical background. Importantly, bosonic field operators have non-zero expectation values in coherent backgrounds. Such coherent states can be formed by any boson, leading to SM fermion spin-dependent energy shifts that are generated by lower dimension operators than those for incoherent states. It is possible, therefore, that the energy shifts arising from coherent states are significantly larger than those considered here. It is also worth noting that none of the operators in Table 2 describe U p1q gauge bosons, but that the wider class of operators generating energy splittings for coherent backgrounds can. The operator O F is such an example. We will explore these states in a future work. Spin-3 2 With the exception of the gravitino [58], there are no known spin- 3 2 fermions in renormalisable theories [68]. Despite this, spin- 3 2 DM has been shown capable of reproducing the observed relic density [69], and can be produced as bound states in non-Abelian extensions to the SM. In particular, the spin- 3 2 baryons are the lightest states of a dark SU p3q with a single quark flavour [67,70]. Whilst sharing many properties with spin-1 2 fermions, the additional spin degrees of freedom carried by Rarita-Schwinger (RS) fermions give rise to operators with richer Lorentz structures. This is turn leads a larger number of operators that generate a DSE, the energy shift from which will depend on up to four helicity states. As we will see, the contribution to the energy shift from each helicity state will differ in both sign and magnitude for RS fermions, which may serve as an additional tool to help distinguish them from spin-1 2 fermions. In this section, we consider a spin-3 2 fermion Ψ with field decomposition Ψ µ pxq " ż d 3 p p2πq 3 1 a 2E p ÿ λ`a pp, λqξμ pp, λqe´i p¨x`b: pp, λqξμ pp, λqe ip¨x˘, (3.35) Ψ µ pxq " ż d 3 p p2πq 3 1 a 2E p ÿ λ`a : pp, λqξμ pp, λqe ip¨x`b pp, λqξμ pp, λqe´i p¨x˘,(3.36) where λ P t 3 2 , 1 2 ,´1 2 ,´3 2 u is the helicity of the RS fermion, and again we set a " b for Majorana fermions, whilst with the sum running over the values of l P t´1, 0, 1u and h "˘1 for which l`h 2 " λ. Finally, the Clebsch-Gordan coefficients for an RS field can be found in [60], and are given by ξμ pp, λq " ÿ tl,su C λ l,h ϵ µ pp, lqupp, hq,(3.λ "`3 2 : C 3 2 1,1 " 1, λ "`1 2 : C 1 2 1,´1 " c 1 3 , C 1 2 0,1 " c 2 3 , (3.39) λ "´3 2 : C´3 2 1,´1 " 1, λ "´1 2 : C´1 2 1,1 " c 1 3 , C´1 2 0,´1 " c 2 3 ,(3.40) with all other coefficients equal to zero. Once more, we tabulate all irreducible operators for RS fermion DM contributing to the DSE up to dimension-6 in Table 3. As before, we consider backgrounds of RS fermions, Ψ and anti-RS fermions,Ψ, which we denote by |Ψy and |Ψy, respectively. The corresponding expectation values in backgrounds of RS fermions that satisfy the Majorana condition are found by summing those in |Ψy and |Ψy backgrounds. We additionally introduce the shorthand ÿ C f pl Ψ , h Ψ q " ÿ tl Ψ ,h Ψ u´C λ Ψ l Ψ ,h Ψ¯2 f pl Ψ , h Ψ q,(3.41) with f some arbitrary function depending on the helicity structure of the background, and note that the argument used to exclude the operators O χ 5 and O χ 6 in Section 3.2 applies here to the equivalent operators with spin-3 2 fields. Label Table 3. Lorentz invariant, Hermitian, gauge invariant and irreducible spin-3 2 DM operators contributing to the DSE up to dimension-6, along with their corresponding expectation values in a background of RS and anti-RS fermions, denoted by |Ψy and |Ψy, respectively. We leave the global factors of the coupling, new physics scale and SM fermion spin eigenvalue, h ψ , implicit. O DM O SM Background xH int y O Ψ 1 pΨ α γ µ Ψ α qpψγ µ γ 5 ψq |Ψy´4m ψ pp Ψ¨Sψ q |Ψy 4m ψ pp Ψ¨Sψ q O Ψ 2 pΨ α γ µ γ 5 Ψ α qpψγ µ γ 5 ψq |Ψy´4m ψ m Ψ ř C h Ψ pS Ψ¨Sψ q |Ψy´4m ψ m Ψ ř C h Ψ pS Ψ¨Sψ q O Ψ 3 ipΨ µ Ψ ν qpψσ µν ψq |Ψy 4m Ψ ř C Im " ε αβµν p α ψ S β ψ ϵ˚µ l Ψ ϵ ν l Ψ ı |Ψy´4m Ψ ř C Im " ε αβµν p α ψ S β ψ ϵ˚µ l Ψ ϵ ν l Ψ ı O Ψ 4 pΨ α σ µν Ψ α qpψσ µν ψq |Ψy´8 ř C h Ψ " pp Ψ¨Sψ qpS Ψ¨pψ q pp Ψ¨pψ qpS Ψ¨Sψ q ‰ |Ψy 8 ř C h Ψ " pp Ψ¨Sψ qpS Ψ¨pψ q pp Ψ¨pψ qpS Ψ¨Sψ q ‰ O Ψ 5 pΨ µ Ψ ν qpψσ µν γ 5 ψq |Ψy 8m Ψ ř C Im " pS ψ¨ϵl Ψ qpp ψ¨ϵl Ψ q ı |Ψy´8m Ψ ř C Im " pS ψ¨ϵl Ψ qpp ψ¨ϵl Ψ q ı O Ψ 6 ipΨ α σ µν Ψ α qpψσ µν γ 5 ψq |Ψy 8 ř C h Ψ ε αβµν p α Ψ p β ψ S µ Ψ S ν ψ |Ψy´8 ř C h Ψ ε αβµν p α Ψ p β ψ S µ Ψ S ν ψ The first operator, O Ψ 1 , gives the same energy shift as the similar spin-1 2 operator O χ 1 up to an overall sign, which results from the contraction of two polarisation vectors. It therefore only requires a matter-antimatter asymmetry in order to generate a DSE, but cannot tell us anything about the helicity structure of the background. This naturally makes it difficult to distinguish from O χ 1 . The remaining operators are far more interesting. Consider O Ψ 2 , which gives rise to an energy shift ∆E Ψ 2 ψ p ⃗ 0, h ψ q "´g ψΨ Λ 2 m Ψ h ψ ÿ λ Ψ ÿ C h Ψ B 1 E Ψ pS Ψ¨Sψ q F`n Ψ pΨ λ Ψ q`n Ψ pΨ λ Ψ q" 7g ψΨ 8Λ 2 h ψ "`n Ψ pΨ``q`n Ψ pΨ``q´n Ψ pΨ´´q´n Ψ pΨ´´q1 3`n Ψ pΨ`´q`n Ψ pΨ`´q´n Ψ pΨ´`q´n Ψ pΨ´`q˘ı, (3.42) where the subscripts˘˘and˘¯refer to the˘3 2 and˘1 2 helicity states, respectively. Taking the difference between the energy shifts for each spin state gives the energy splitting due to O Ψ 2 ∆E Ψ 2 ψ " 7g ψΨ 4Λ 2 "`n Ψ pΨ``q`n Ψ pΨ``q´n Ψ pΨ´´q´n Ψ pΨ´´q1 3`n Ψ pΨ`´q`n Ψ pΨ`´q´n Ψ pΨ´`q´n Ψ pΨ´`q˘ı, (3.43) which requires a non-zero helicity asymmetry in order to generate a DSE, akin to O χ 2 . This is easily achieved in a chiral theory similar to the weak interaction. Owing to the Clebsch-Gordan coefficients, however, the contribution to the DSE from the˘1 2 helicity states is suppressed by a factor of three, which for the same total DM density leads to a reduced energy shift. As such, if the mass, and by extension the number density of the DM is known, the reduced energy splitting could serve as a tool to distinguish between spin-1 2 and spin-3 2 DM backgrounds. Although difficult to observe, we also note that the energy shifts of the individual spin states differ by an overall sign between O Ψ 2 and O χ 2 . The operator O Ψ 3 yields a similarly suppressed energy splitting ∆E Ψ 3 ψ " 7g ψΨ 4Λ 2 "`n Ψ pΨ``q´n Ψ pΨ``q´n Ψ pΨ´´q`n Ψ pΨ´´q1 3`n Ψ pΨ`´q´n Ψ pΨ`´q´n Ψ pΨ´`q`n Ψ pΨ´`q˘ı, ,(3.44) which is only non-zero in a background with both a fermion-antifermion and helicity asymmetry. In this case, we note the analogous lower spin operator is in fact bosonic, O X 3 , which should result in a slightly larger splitting for the same background density. However, the biggest difference is in the generation of (3.30) and (3.44); as previously discussed, a helicity asymmetry cannot arise at the Lagrangian level for bosons, but are possible in chiral theories of fermions which if relativistic at production prefer a given helicity. Consequently, it is much easier to generate the DSE from O Ψ 3 . The remaining three operators, O Ψ 4 , O Ψ 5 and O Ψ 6 are analogous to O χ 3 , O X 3 and O χ 4 , respectively, such that only the first contributes a DSE for the experimental setup considered here. In the same way as (3.43), the energy shifts due to O Ψ 4 differ from their analogues by an overall sign and a small suppression factor from the Clebsch-Gordan coefficients. Finally, we have omitted the pseudoscalar analogues of O Ψ 3 and O Ψ 5 , proportional to Ψ µ γ 5 Ψ ν , from Table 3 as the expectation values of their Hamiltonians vanish trivially using (3.7). The discussion here is easily extended to higher spin states, which we naively expect will differ only in the overall sign and magnitude of their DSEs. In particular, the magnitude of the DSE for most operators should decrease with increasing spin, as progressively smaller Clebsch-Gordan coefficients will suppress the contribution from the intermediate helicity states. Experimental feasibility Observing the tiny energy splittings induced by the DSE directly is a remarkable challenge due to their small magnitude. Take for example the splitting due to the CνB, whose magnitude is expected to be of order |∆E ψ | " G F β C n ν,0 » 5ˆ10´3 9 eV,(4.1) assuming maximal neutrino-antineutrino asymmetry, where we have used β C » 10´3 and n ν,0 " 56 cm´3 is the predicted relic neutrino density per degree of freedom. This is approximately thirty orders of magnitude smaller than the energy splitting due to the Zeeman effect in a 1 G magnetic field. Clearly then, this effect is nigh impossible to observe on the scale of a single target. To that end, we identify two methods utilising macroscopic targets through which the DSE may be observed. Both of these rely on the same property; as a result of the energy splitting due to the DM background, the SM fermion Hamiltonian, H ψ , and spin operators orthogonal to the DM wind, S K , no longer commute, leading to a spin precession dS K dt " irH ψ , S K s " Op∆E ψ q,(4.2) which can equivalently be interpreted as a torque. A ferromagnet with polarisation transverse to the DM wind will therefore experience a macroscopic acceleration as a result of the spin precession, which can be observed with a Cavendish-style torsion balance. Alternatively, a target initially polarised along an external magnetic field will develop some transverse magnetisation as a consequence of the DM background, which may measurable with a SQUID magnetometer. We will explore each of these methods in turn. Torsion balance The possibility of using a torsion balance to observe the tiny energy splittings due to the CνB was first identified by Stodolsky in [38] and has since been discussed in several works [39][40][41]. A single SM fermion interacting with the DM background will experience a torque τ ψ » |∆E ψ |, such that a macroscopic target consisting of N ψ fermions with degree of polarisation P will experience a total torque τ tot » P N ψ |∆E ψ | " N A m A P A Mˆ# Z|∆E e |, ψ " e, |∆E N |, ψ " N, (4.3) where N denotes an atomic nucleus, N A is the Avogadro number, whilst M , A and Z denote the total mass, the mass number and atomic number of the target, respectively. We have additionally introduced the "Avogadro mass" m A " 1 g mol´1. To estimate the sensitivity of a torsion balance to this energy splitting, we consider the same setup as [41] using a torsion balance consisting of N m spherical, uniformly dense ferromagnets a distance R away from some central axis. To maximise the sensitivity, we additionally assume that opposing ferromagnets are polarised antiparallel to one another. For this setup, the torsion balance will experience a linear acceleration a » N A m A P A N m Rˆ# Z|∆E e |, ψ " e, |∆E N |, ψ " N. (4.4) As such, if accelerations as small as a 0 can be measured, the experiment is sensitive to energy splittings |∆E ψ | Á a 0 m A N A A P R N mˆ# 1 Z , ψ " e, 1, ψ " N, " p5.2¨10´2 8 eVq " a 0 10´1 5 cm s´2 ı " R 1 cm ȷ " 2 N m ȷ A Pˆ# 1 Z , ψ " e, 1, ψ " N, (4.5) where for our reference sensitivity we have used a 0 " 10´1 5 cm s´2, which has recently been achieved in torsion balance tests of the weak equivalence principle [71]. By comparison with (4.1), we see that this torsion balance experiment is insensitive to the CνB, but may still be able to observe DM for which the background number density n DM " n ν,0 . In particular, as the background DM number density scales as n DM " ρ DM {m DM , where ρ DM » 0.4 GeV cm´3 is the local dark matter energy density [72], low mass DM scenarios are ideal candidates for detection using this method. Finally, we note that a torsion balance consisting test masses suspended by superconducting magnets has been considered in [73], which has an estimated sensitivity to accelerations as small as a 0 » 10´2 3 cm s´2. This, in turn, would allow us to probe energy splittings of order 10´3 6 eV. SQUID magnetometer The DM wind resulting from the relative motion of the Earth through the background can acts similarly to a magnetic field, leading to the spin precession (4.2). As such, if the target spins are initially aligned along some fixed external magnetic field ⃗ B ext that is not colinear with the DM wind, the presence of the background will cause the spins to shift away from the axis of ⃗ B ext and give rise to a small transverse magnetisation. The spins will then precess around the combined magnetic field and DM wind with some characteristic frequency, which can be detected using a highly sensitive SQUID magnetometer. This idea has previously been discussed in the context of axion DM in [74], and is the basis of the CASPEr experiment [75]. Following the calculations in Appendix D, we find that the transverse magnetisation of a target consisting of N ψ spins evolves as |M K ptq| " 2ρN A m A P A |R sin`ω ψ,0 2 t˘| 1`R 2 c 1`R 2 cos 2´ω ψ,0 2 t¯ˆ# Zµ e , ψ " e, µ N , ψ " N, (4.6) where ρ is the mass density of the target, µ ψ denotes the magnetic moment of species ψ, R " ∆E ψ {∆E ψ,B is the ratio of the DM and Zeeman energy splittings, and ω ψ,0 " ∆E ψ,B ? 1`R 2 . In (4.6) we have assumed that the DM wind is exactly perpendicular to ⃗ B ext , which maximises the transverse magnetisation, and that both the external magnetic field and DM wind directions are constant in time 5 . We give the full expression for |M K ptq| and discuss the time dependence in Appendix D. The transverse magnetisation has a maximum of |M K pt max q| " 2ρN A m A P A |R| 1`R 2ˆ# Zµ e , ψ " e, µ N , ψ " N, t max " p2k`1qπ ω ψ,0 , (4.7) for R ď 1, with k P t0, 1, 2, . . . u. Supposing that a magnetometer can precisely measure transverse magnetic fields with magnitude B 0 , we will have sensitivity to energy splittings with magnitude |∆E ψ | Á B 0 | ⃗ B ext | m A ρN A A Pˆ# 1 Z , ψ " e, 1, ψ " N, " p1.0¨10´3 2 eVq " B 0 10´1 6 T ȷ « | ⃗ B ext | 10´1 0 T ff " 7.9 g cm´3 ρ ȷ A Pˆ# 1 Z , ψ " e, 1, ψ " N, (4.8) Figure 1. Evolution of the transverse magnetic field generated by the DM background, normalised by the SQUID sensitivity B 0 " 10´1 6 T. From bottom to top, the blue, orange and green curves correspond to energy splittings ∆E " 2¨10´3 2 eV, 2¨10´3 0 eV and 2¨10´2 8 eV, respectively, whilst the solid and dotted curves correspond in turn to the cases where β C,∥ " 0 and β C,∥ " 1{ ? 3. Finally, we assume an applied magnetic | ⃗ B ext | " 10´1 0 T, and an iron target with magnetic moment µ ψ " 3.15¨10´8 eV T´1, equal to the nuclear magneton. for R ! 1, where we have used ∆E ψ,B " 2µ ψ | ⃗ B ext |. For our reference scenario we have chosen B 0 " 5¨10´1 4 T, corresponding to the SQUID magnetometer discussed in [76], and used the density of iron in place of ρ. It is clear that the SQUID magnetometer setup is at the very least as sensitive as the torsion balance setup discussed in Section 4.1, but can be made more sensitive by decreasing the applied magnetic field. The ideal setup would therefore be to initially apply a strong external magnetic field to align the target spins, and then steadily decrease the applied field to maximise the acquired transverse polarisation. One should also notice that ω ψ,0 " ω ψ,L ? 1`R 2 , where ω ψ,L " 2πf ψ,L is the angular Larmor frequency of the system. The overall magnetisation of the system would therefore initially precess about the applied magnetic field with frequency ω ψ,L , increasing to ? 2ω ψ,L as the applied field is turned down. This effect could also be interpreted as a field dependent gyromagnetic ratio. Given, however, that R ! 1 for most reasonable scenarios, we do not expect this to have an observable effect on the signal. We show the time dependence of the SQUID magnetometer signal in Figure 1, including the case where the DM wind is not exactly orthogonal to the applied magnetic field. In particular, we show the signal when the fraction of the relative frame velocity along ⃗ B ext , β C,∥ " 1{ ? 3, or equivalently when the relative velocity is split equally along each direction. Importantly, this does not have a drastic effect on magnitude of the signal, and so should not severely impact the sensitivity of this method outside the extreme case where β C,∥ Ñ 1. Example: a scalar DM model To give a rough estimate of the constraints that can be placed on DM using this method, we consider the two component DM model given in [51], which features a heavy, leptophilic dark vector mediator Z 1 µ and complex scalar ϕ with interaction Lagrangian L Z 1 " g 2 ϕ Z 1 µ Z 1µ |ϕ| 2´i g ϕ ϕ˚Ð Ñ B µ ϕZ 1µ´Z1 µl γ µ pg L P L`gR P R ql,(4.9) where l P te, µ, τ u, g ϕ , g L and g R are dimensionless couplings, and P R{L " p1˘γ 5 q{2 are the right and left chirality projection operators. Focusing on the case with l " e, and integrating out the heavy Z 1 leads to the effective low energy Lagrangian L Z 1 "´i g ϕ pg R`gL q 2m 2 Z 1´ϕ˚Ð Ñ B µ ϕ¯ēγ µ e´i g ϕ pg R´gL q 2m 2 Z 1´ϕ˚Ð Ñ B µ ϕ¯ēγ µ γ 5 e`. . . . (4.10) Of interest to us is the second term, which by comparison with (3.19) generates an electron energy splitting with magnitude |∆E e | " 2g ϕ |g R´gL | m 2 Z 1 β C |n ϕ pϕq´n ϕ pϕ˚q|. (4.11) Next, rewriting |n ϕ pϕq´n ϕ pϕ˚q| " |δ ϕ |ρ DM {m ϕ , where δ ϕ P r´1, 1s parameterises the asymmetry between ϕ and ϕ˚, and considering purely axial couplings, g R "´g L " g A , we find |∆E e | " 4 Λ 2 Z 1 ρ DM m ϕ β C |δ ϕ |. " p1.2¨10´3 4 eVq « Λ´1 Z 1 356 TeV´1 ff 2 " 10 MeV m ϕ ȷ |δ ϕ |,(4.12) for β C " 7.6ˆ10´4 [77], where we have defined the effective new physics scale Λ Z 1 " m Z 1 { a g ϕ |g A | and assumed that ϕ makes up the entire local relic density, ρ DM » 0.4 GeV cm´3. If we instead assume production via freeze-out, we can estimate the local DM density of ϕ in terms of Λ Z 1 and m ϕ , which for m 2 Z 1 " m 2 ϕ " m 2 e has a different scaling to (4.12) |∆E e | " p1.2¨10´3 4 eVq « 356 TeV´1 Λ´1 Z 1 ff 2 " 10 MeV m ϕ ȷ 3 |δ ϕ |. (4.13) In both cases, the reference value, Λ´1 Z 1 " 356 TeV corresponds to the approximate value required to reproduce the relic density for m ϕ " 10 MeV. More generally, we require Λ´1 Z 1 Á p356 TeV´1q " 10 MeV m ϕ ȷ 1 2 ,(4.14) so as not to overclose the universe. It is instructive to recast both (4.12) and (4.13) in terms of the constraints that can be placed on the effective new physics scale using the DSE. Given a sensitivity to energy shifts |∆E 0 | Á 10´3 2 eV, corresponding to the SQUID magnetometer considered in (4.8), we find the constraint on the effective new physics scale Λ´1 Z 1 Á p38.8 TeV´1q " 10 MeV m ϕ ȷ 3 2 " 10´3 2 eV |∆E 0 | ȷ 1 2 b |δ ϕ |,(4. 15) Figure 2. Constraint projections on the effective DM coupling, Λ´1 Z 1 " ? g ϕ g A {m Z 1 , from the SQUID magnetometer for the generic (green) and freeze-out (orange) production scenarios, where we assume δ ϕ " 1. We compare these with the constraints from direct detection experiments [78][79][80] (blue), assuming constant DM form factors, and anomalous supernova cooling constraints (red), which we compute following the method of [51] for the 18 M @ progenitor discussed in [81]. For comparison, we show the combination of parameters that reproduce the local relic density for a freeze-out scenario with the black curve, corresponding to the saturation of (4.14). assuming the energy splitting from freeze-out production (4.13), which for Op1q values of the asymmetry parameter, i.e. supposing that the dark sector matter-antimatter asymmetry follows that of the visible sector, is just one order of magnitude away from being able to probe Λ Z 1 that reproduces the measured relic density at m ϕ " 10 MeV. If we instead assume that ϕ makes up the entirety of dark matter independent of m ϕ and Λ Z 1 , corresponding to the energy splitting (4.12), we find the constraint Λ´1 Z 1 À p3.3¨10 3 TeV´1q " m ϕ 10 MeV ı 1 2 " |∆E 0 | 10´3 2 eV ȷ 1 2 1 a |δ ϕ | ,(4.16) which is once more roughly an order of magnitude away from the freeze-out band (4.14). We show the constraints that could be placed on Λ´1 Z 1 using a SQUID magnetometer in Figure 2 as a function of m ϕ for both the freeze-out (FO) and unspecified production scenarios, and compare these with the existing constraints from direction detection experiments [78][79][80] and anomalous supernova cooling, computed following the method of [51]. As expected, this experiment significantly outperforms existing direct detection experiments for m ϕ À 30 MeV. Additionally, if freeze-out is assumed, the SQUID magnetometer experiment is instead able to place constraints on the minimum value of Λ´1 Z 1 , owing to the linear scaling of the DSE with the effective coupling. Importantly, this includes regions that are currently unconstrained by SN 1987a. Aside, notice that the energy splitting due to ϕ backgrounds far exceeds that expected from the CνB for the parameter ranges considered here, assuming the same asymmetry for both. It is therefore entirely possible that the DSE completely washes out the Stodolsky effect for neutrinos. One could also envisage scenarios in which the opposite is true, and the DSE is overwhelmed by the CνB, or those in which one acts as a significant background to the other. This should be taken into consideration when using this technique, especially as it is difficult to distinguish between the operators responsible for the DSE. Nevertheless, the observation of either the DSE or Stodolsky effect for neutrinos would be a strong indicator of as-yet-unobserved physics. Conclusions Despite comprising " 26% of the energy density of the universe, detecting DM is an incredible challenge that has yet to be accomplished. Here we have explored the possibility of constraining DM models using the DSE: tiny energy splittings between the spin states of SM fermions induced by an incoherent DM background. Throughout, we have used an EFT formalism and identified all effective DM operators up to dimension-6, for DM candidates with spin-0 to spin- 3 2 , that can give rise to the DSE. Our key finding is that the energy splittings due to the DSE scale linearly with the effective DM coupling, inversely with the DM mass, and are roughly independent of the DM kinematics. Importantly, this differs from traditional DM direct detection experiments, where the sensitivity typically decreases with decreasing DM mass. On the other hand, every operator discussed here requires either a particle-antiparticle or helicity asymmetry in the background to give a non-zero contribution to the DSE. This technique therefore favours chiral models and those with a sizeable chemical potential during production, however we note that either asymmetry may develop post-production through several mechanisms e.g. DM reflection at surface of the Earth, scattering on polarised backgrounds. In this work, we have identified two methods through which these tiny energy splittings can be observed. The first utilises an extremely sensitive, polarised torsion balance, which experiences a torque due to the energy splittings induced by the DM background. For a conservative setup, this experiment is sensitive to energy splittings of ∆E ψ » 10´2 8 eV, but could have a sensitivity to splittings as small as ∆E ψ » 10´3 6 eV for a more optimistic setup. The second utilises a SQUID magnetometer to detect the time-varying magnetisation of a target due to the DM background, which acts similarly to an external magnetic field on the target. We estimate that this experiment will be sensitive to splittings of ∆E ψ » 10´3 2 eV. Finally, we have explored a scalar DM model, considering both the case where the new scalar constitutes the entire local DM density regardless of the model parameters, and the more realistic scenario where it is produced via freeze-out. In both scenarios, we showed the SQUID magnetometer proposal is able to exclude regions of parameter space that are not already ruled out by direct detection experiments or SN 1987a, provided that there is a sizeable asymmetry in the DM background. For the range of parameters considered, we also demonstrated that the DSE for the scalar DM model far exceeded the Stodolsky effect for neutrinos, provided that the asymmetry in both backgrounds was comparable. Clearly, the DSE is a powerful tool to constrain DM models in otherwise difficult-to-test regions of parameter space. version of this work. We are also grateful to Yuber F. Perez-Gonzalez, Lucien Heurtier and Animesh Datta for some helpful comments during the preparation of this manuscript. Jack D. Shergold is supported by an STFC studentship under the STFC training grant ST/T506047/1. A Lab frame averaging In this appendix we will describe the averaging procedure used to compute the energy shifts in the lab frame. We begin by assuming that the DM is described by an isothermal spherical halo, with galaxy frame velocity distribution f p⃗ pq "ˆ2 π m 2 DM σ 2˙3 2 e´| ⃗ p| 2 2m 2 DM σ 2 , (A.1) where ⃗ p is the DM momentum in the galactic reference frame, m DM is its mass and σ is the velocity dispersion. The normalisation factor is found by requiring that ş d 3 p p2πq 3 f p⃗ pq " 1. As a result of the frame transformation, DM particles in the lab frame will not follow (A.1) but instead the transformed distribution function f lab , such that the average of some lab frame quantity X lab will be given by xX lab y " ż d 3 p p2πq 3 X lab f lab p⃗ pq " 1 p2πq 3 ż X lab f lab p⃗ pq|⃗ p| 2 sin θ d|⃗ p| dθ dϕ. (A.2) To find f lab p⃗ pq, we first note that since all velocities involved are small, the momentum of the DM particle in the lab frame ⃗ p lab can be written in terms of the relative frame velocity ⃗ β C as ⃗ p lab » ⃗ p`m DM ⃗ β C " |⃗ p|¨c os ϕ sin θ sin ϕ sin θ cos θ‚`m DM β C¨0 0 1‚ , (A.3) where β C " | ⃗ β C |, and we have chosen ⃗ β C ||z for simplicity. This choice makes no difference at the level of averaging, but becomes important when considering experimental setups. We will therefore write our final expressions for averaged quantities in terms of a general orientation of ⃗ β C . Next, since f lab p⃗ p lab q " f p⃗ pq, the lab frame distribution function will satisfy f lab p⃗ pq " f p⃗ p´m DM ⃗ β C q "ˆ2 π m 2 DM σ 2˙3 2 e´| ⃗ p| 2`m2 DM β C 2m 2 DM σ 2 e |⃗ p|β C cos θ m DM σ 2 , (A.4) which can be readily plugged into (A.2) to compute averaged lab frame quantities. In addition to the distribution function, we must also write the lab frame polarisation vectors in terms of DM reference frame quantities. To do so, we rotate the polarisation vectors (3.28) to point along an arbitrary axis, and then use ⃗ p lab to rewrite angles in the lab frame in terms of those in the DM frame, yielding ϵ μ " pϵ μ q˚" 1 ? 2¨0 1 |⃗ p lab | cos ϕ p|⃗ p| cos θ`β C m DM q´i sin ϕ 1 |⃗ p lab | sin ϕ p|⃗ p| cos θ`β C m DM q`i cos φ |⃗ p| |⃗ p lab | sin θ‹ ‹ ‹ ‚ , (A.5) ϵ µ L "¨| ⃗ p lab | m DM |⃗ p| |⃗ p lab | cos ϕ sin θ |⃗ p| |⃗ p lab | sin ϕ sin θ 1 |⃗ p lab | p|⃗ p| cos θ`β C m DM q‹ ‹ ‹ ‹ ‚ , (A.6) again assuming ⃗ β C ||z. Relaxing the assumption ⃗ β C ||z, we find the averages relevant to the operators considered in this work B 1 E DM p⃗ p DM¨⃗ S ψ q F " 2β C s ψ,∥ , (A.7) B 1 E DM pp DM¨Sψ q F "´2β C s ψ,∥ , (A.8) B 1 E DM pS DM¨Sψ q F " " p1´8β 2 r q 8β 2 r Erf p2β r q´1 2 ? πβ r e´4 β 2 r ȷ s ψ,∥ m DM »´7 8 s ψ,∥ m DM`O p1´β r q, (A.9) B 1 E DM pp DM¨Sψ qpS DM¨pψ q F " « p1´16β 2 r´6 4β 4 r q 16β 4 r Erf p2β r q p1`8β 2 r q 4 ? πβ r e´4 β 2 r ff β 2 c m ψ s ψ,∥ »´5β 2 c m ψ s ψ,∥`O p1´β r q, (A.10) B 1 E DM pp DM¨pψ qpS DM¨Sψ q F " " p1´8β 2 r q 8β 2 r Erf p2β r q´1 2 ? πβ r e´4 β 2 r ȷ m ψ s ψ,∥ »´7 8 m ψ s ψ,∥`O p1´β r q, (A.11) B 1 E DM ε αβµν p α DM p β ψ S µ DM S ν ψ F " 0, (A.12) B 1 E DM ε αβµν p α ψ S β ψ ϵ˚μ ϵ ν F "˘i " p1´8β 2 r q 8β 2 r Erf p2β r q´1 2 ? πβ r e´4 β 2 r ȷ m ψ m DM s ψ,∥ »¯7 i 8 m ψ m DM s ψ,∥`O p1´β r q, (A.13) B 1 E DM ε αβµν p α ψ S β ψ ϵ˚µ L ϵ ν L F " 0, (A.14) B 1 E DM pp ψ¨ϵ˘q pS ψ¨ϵ˚q F " 0, (A.15) B 1 E DM pp ψ¨ϵL qpS ψ¨ϵL q F "´2 m ψ m DM β C s ψ,∥ , (A.16) B 1 E DM p⃗ p X¨⃗ ϵ˘qpϵ˚¨S ψ q F " 0, (A.17) B 1 E DM p⃗ p X¨⃗ ϵ L qpϵL¨S ψ q F "´2β C s ψ,∥ , (A.18) B 1 E DM ε αβiν ϵ˚α ϵ β p i X S ν ψ F " 0, (A.19) B 1 E DM ε αβiν ϵ˚α L ϵ β L p i X S ν ψ F " 0, (A.20) with s ψ,∥ " p ⃗ β C¨⃗ s ψ q{β C and β r " β C {β c , where β c " ? 2σ is the circular velocity of the galaxy. The presence of s ψ,∥ indicates that only the spin state directed along the DM wind experiences an energy shift. We will not include this factor explicitly in the main text. B Operator basis Here we outline the identities used to reduce the effective DM operator bases to those appearing in Section 3. We begin by noting that excluding field indices, the Lorentz structures that can enter into our effective DM operators are the ones already given in (3.6), along with the partial derivative, B µ , and Levi-Civita tensor, ε αβµν . Considering operators up to dimension-6 with a fermionic SM part, we can therefore have at most a single derivative entering. As such, for spin-0 and spin-1 2 DM particles, the only way that the Levi-Civita tensor can enter a Lagrangian is through operators that contain at least three gamma matrices; which can then be reduced to simpler Lorentz structures via the Chisholm identity and the definition of γ 5 γ α γ β γ µ " η αβ γ µ`ηβµ γ α´ηαµ γ β´i ε σαβµ γ σ γ 5 , (B.1) γ 5 " i 4! ε αβµν γ α γ β γ µ γ ν , (B.2) where η µν is the metric tensor. These can also be used to derive the identity ε αβµν σ µν "´2iσ αβ γ 5 , (B.3) which generates the additional Lorentz structure given in (3.9), allowing us to express operators containing a Levi-Civita tensors in terms of operators containing the more convenient σ µν γ 5 structure. The trace of the product of fermion spinors with this structure is Tr " u ψūψ σ µν γ 5 ‰ " 2i´p µ ψ S ν ψ´S µ ψ p ν ψ¯. (B.4) Additionally, we can use this to demonstrate that O χ 5 " O χ 4 via ipχσ µν γ 5 χqpψσ µν ψq "´1 2 ε µναβ pχσ αβ χqpψσ µν ψq " ipχσ αβ χqpψσ αβ γ 5 ψq, (B.5) and that O χ 6 " O χ 3 using pχσ µν γ 5 χqpψσ µν γ 5 ψq "´1 4 ε µναβ ε µνδγ pχσ αβ χqpψσ δγ ψq Many operators containing derivatives can be reduced using symmetry currents, and through integration by parts. Take, for example, the scalar operator B µ |ϕ| 2˘ψ γ µ γ 5 ψ " B µ " |ϕ| 2ψ γ µ γ 5 ψ ‰´| ϕ| 2 B µ`ψ γ µ γ 5 ψ˘. (B.7) The first term on the right-hand side is a total derivative and so will not contribute to the classical action, whilst the second term contains the derivative of the axial current which can be re-expressed as 2im ψ |ϕ| 2ψ γ 5 ψ using the equations of motion for spin-1 2 fields. We can therefore perform the operator reductioǹ B µ |ϕ| 2˘ψ γ µ γ 5 ψ ÝÑ |ϕ| 2ψ γ 5 ψ, (B.8) which as per (3.7) does not contribute to the DSE. Note that similar structures containing vector currents vanish from the requirement B µ pψγ µ ψq " 0. Further reductions in the effective operator basis are obtained using equations of motion. In particular, we make use of the spin-1 and spin-3 2 equations of motion, which lead to the constraints B µ X µ " 0, B µ Ψ µ " 0, γ µ Ψ µ " 0, (B.9) allowing us to eliminate or simplify operators in which spin-1 fields share an index with a derivative. Manipulations using the third identity, { Ψ " 0, yields the basis given in Table 3. The operator bases used throughout this work are those in which the equations of motion have been applied maximally. This avoids the need to apply Hamilton's equations to the Hamiltonian when computing the energy shifts, which is a far more involved task than using the Euler-Lagrange equations. For completeness, we also specify the spin-independent members of our operator bases: at spin-0, these are |ϕ| 2ψ ψ, i|ϕ| 2ψ γ 5 ψ, and ipϕ : Ð Ñ B µ ϕqpψγ µ ψq; for spin-1 DM, we have |X| 2ψ ψ, i|X| 2ψ γ 5 ψ, along with the vector current analogues of each axial-vector operator appearing in Table 2, hermitianised appropriately with factors of the imaginary unit. Finally, the full spin-1 2 basis includes products of fermion bilinears not given in Table 1, whilst the complete basis for RS fermions is given in [68]. The same spin-0 basis, along with a similar spin-1 basis can also be found in [82]. C Levi-Civita identity Here we derive an identity that can be used to evaluate contractions of four-vectors and a Levi-Civita tensor of the form ε αβµν A α B β C µ D ν , (C.1) in a more practical manner, where A, B, C and D are some unspecified four-vectors. First we note that this contraction will always contain at least three spatial components, and so will carry a global factor of p´1q 3 regardless of the four-vectors considered. Next, recalling that we can write a contraction of the Levi-Civita symbol and a series of vectors as a matrix determinant, we have ε αβµν A α B β C µ D ν " p´1q 3ˇA 0 A 1 A 2 A 3 B 0 B 1 B 2 B 3 C 0 C 1 C 2 C 3 D 0 D 1 D 2 D 3ˇ, (C.2) which can be neatly re-expressed in terms of scalar triple products as ε αβµν A α B β C µ D ν "´A 0 r ⃗ B, ⃗ C, ⃗ Ds`B 0 r ⃗ A, ⃗ C, ⃗ Ds´C 0 r ⃗ A, ⃗ B, ⃗ Ds`D 0 r ⃗ A, ⃗ B, ⃗ Cs, (C.3) where r ⃗ A, ⃗ B, ⃗ Cs " ⃗ A¨p ⃗ Bˆ⃗ Cq is the scalar triple product which is unchanged by cyclic permutations, and antisymmetric under the interchange of any two elements. One could also choose to take the determinant in other ways which may better suit the experimental setup. To demonstrate the use of this identity, we consider a contraction that may occur between a dark fermion χ and a SM fermion ψ, which have four-momenta and spin, p and S, respectively ε αβµν p α χ p β ψ S µ χ S ν ψ . (C.4) If the SM fermion is at rest in the lab frame, we can use (C.3) to reduce (C.4) to ε αβµν p α χ p β ψ S µ χ S ν ψ " m ψ r⃗ p χ , ⃗ S χ , ⃗ S ψ s. (C.5) In the most general case, this can then be evaluated by explicitly plugging in values for the relevant spins and momenta. However, if we instead consider the DM to be in a helicity eigenstate, we will have ⃗ S χ ∥ ⃗ p χ , from which it follows that ε αβµν p α χ p β ψ S µ χ S ν ψ " 0, (C.6) by making use of the antisymmetric property of the scalar triple product. D Fermion spin precession Here we derive the spin precession of an SM fermion in a combined magnetic and DM background field that gives rise to the transverse magnetisation (4.6). To do so, we need to set up the differential equation that governs the evolution of the SM fermion spin. There will be two components to this: the precession due to the DM background, and the precession due to an external magnetic field. Both of these are due to the same effect, a non-diagonal Hamiltonian resulting from the energy splittings due to background fields. We begin with the time-dependent Schrödinger equation, which for our system takes the form i B Bt ψpx, tq " pH kin pxq`V DM`VB qψpx, tq, (D.1) where ψpx, tq is the fermion wavefunction, H kin pxq is its kinetic Hamiltonian, which is spin and time-independent, whilst V DM and V B are the potentials due to the DM background and applied magnetic field, respectively, which are spin-dependent and we will treat as constant in time here 6 . This motivates the factorisation ψpx, tq " XpxqT ptq, (D.2) where Xpxq is a scalar, containing the spatial components of the wavefunction, and T ptq is an eigenspinor of the form T ptq "ˆT`p tq T´ptq˙, |T ptq| 2 " 1. (D.3) This factorisation makes (D.1) separable, but it is easier to note that H kin pxqXpxq " E kin Xpxq, (D.4) such that we can absorb E kin as a time-independent, spin-diagonal contribution to the potential. The overall factor of Xpxq can then be factored out, allowing us to writê i B Bt´H˙T ptq " 0, (D. 5) where H is the total Hamiltonian, including the spin-diagonal contribution from H kin . If the magnetic field is defined such that it points along z, then the z oriented spin state will experience an energy shift. Additionally, the up and down spin states should experience a shift of opposite sign. The potential due to the magnetic field should therefore be proportional to the spin operator along z, that is V B " ∆E ψ,B 2 S z " ∆E ψ,B 2ˆ1 0 0´1˙, (D.6) where ∆E B is the energy shift due to the magnetic field. We see that this has the desired properties, as if we act on an S z eigenstate with eigenvalue 7 s z "˘1, we get the eigenvalue s z ∆E B {2. Next, we seek to do the same for the potential due to the DM, which should be directed along the DM wind. Explicitly, V DM " ∆E ψ 2 pβ C,x S x`βC,y S y`βC,z S z q " ∆E ψ 2ˆβ C,z β C,x´i β C,y β C,x`i β C,y´βC,z˙, (D.7) where β C,i " p ⃗ β C¨⃗ e i q{β C P r´1, 1s is the fraction of the relative frame velocity along the direction i. We should also include a diagonal term due to the spin-independent effects from the DM, however this can simply be absorbed into E kin . The total Hamiltonian is then H " 1 2ˆ2 E kin`∆ E ψ,B`∆ E ψ β C,z ∆E ψ pβ C,x´i β C,y q ∆E ψ pβ C,x`i β C,y q 2E kin´∆ E ψ,B´∆ E ψ β C,z˙, (D. 8) such that the solution to (D.5) is given by T˘ptq " « s˘cos´ω ψ 2 t¯´p β C,x¯i β C,y qRs¯˘p1˘β C,z Rqsȃ 1`2β C,z R`R 2 i sin´ω ψ 2 t¯ff e´i E kin t , (D.9) where ω ψ " ∆E ψ,B a 1`2β C,z R`R 2 is the (angular) precession frequency of the system, proportional to the Larmor frequency, s˘" T˘p0q are the initial values of the SM fermion eigenspinor, and R " ∆E ψ {∆E ψ,B P r´1, 1s is the ratio of the energy shifts due to each of the background potentials. We further note that s`is always real, whilst s´may be complex. To compute the spin precession using T˘, we note that the time derivative of some operator O is given by Heisenberg's equation of motion Plugging in (D.9), we find ds x dt "˜β C,x p1`β C,z Rq a 1`2β C,z R`R 2 sinpω ψ tq`β C,y cospω ψ tq¸∆E ψ , (D.12) ds y dt "˜β C,y p1`β C,z Rq a 1`2β C,z R`R 2 sinpω ψ tq´β C,x cospω ψ tq¸∆E ψ , (D.13) ds z dt "˜´p 1´β 2 C,z qR a 1`2β C,z R`R 2 sinpω ψ tq¸∆E ψ , (D.14) where we have related s˘to the initial values of s x , s y and s z , using s x,0 " T p0q : S x T p0q " 2s`Re ps´q " 0, (D.15) s y,0 " T p0q : S y T p0q " 2s`Im ps´q " 0, (D. 16) s z,0 " T p0q : S z T p0q " |s`| 2´| s´| 2 " 1, (D. 17) and assumed that the spins are initially aligned with the external magnetic field. Notice that all three of (D.12), (D.13) and (D.14), especially those along x and y, are proportional to the energy splitting due to the background DM field, and so will vanish in its absence. These equations are readily solved to find the expectation values of the SM fermion spin as a function of time s x ptq " 2R a 1`2β C,z R`R 2 « β C,x p1`β C,z Rq a 1`2β C,z R`R 2 sin 2´ω ψ 2 t¯`β C,y 2 sin pω ψ tq ff , (D.18) s y ptq " 2R a 1`2β C,z R`R 2 « β C,y p1`β C,z Rq a 1`2β C,z R`R 2 sin 2´ω ψ 2 t¯´β C,x 2 sin pω ψ tq ff , (D. 19) s z ptq " 1´2 R 2 p1´β 2 C,z q 1`2β C,z R`R 2 sin 2´ω ψ 2 t¯, (D.20) such that the magnitude of the spin along the transverse direction evolves according to |s K ptq| " b s x ptq 2`s y ptq 2 " 2|R sin`ω ψ 2 t˘| 1`2β C,z R`R 2 b 1´β 2 C,ẑ c 1`2β C,z R`R 2 " cos 2´ω ψ 2 t¯`β 2 C,z sin 2´ω ψ 2 t¯ı, (D. 21) which vanishes identically when |β C,z | " 1, or equivalently when the DM wind is colinear with the magnetic field. Consequently, the expression equivalent to (D.21) for a general magnetic field orientation is found by making the replacement β C,z Ñ β C,∥ , where β C,∥ is the fraction of the relative frame velocity along the external magnetic field direction. The corresponding transverse magnetisation is simply |M K ptq| " n ψ µ ψ |s K ptq|, where n ψ is the number density of SM fermions in the target, and µ ψ is their magnetic moment. Given that ω ψ,0 " ∆E ψ,B ? 1`R 2 , this reduces to (4.6) when β C,∥ " 0. For completeness, we note that (D.21) has a maximum of |s K pt max q| " 2|R| b 1´β 2 C,∥ b 1`β C,∥ R`β 2 C,∥ R 2 1`β C,∥ R`R 2 , t max " p2k`1qπ ω ψ , (D.22) where k P t0, 1, 2, . . . u. This recovers (4.7) for β C,∥ " 0. δ β¯pχ σ αβ χqpψσ δγ ψq " pχσ αβ χqpψσ αβ ψq.(B.6) the time derivative of each of the expectation values, s i , is ds i dt " T ptq :ˆd S i dt˙T ptq. (D.11) Table 2 . 2Lorentz invariant, Hermitian, gauge invariant and irreducible spin-1 DM operators contributing to the DSE up to dimension-6, along with their corresponding expectation values in a background of complex vector bosons and the conjugate field, denoted by |Xy and |X˚y, respectively. We leave the global factors of the coupling, new physics scale and SM fermion spin eigenvalue, h ψ , implicit. For simplicity, we will only consider helicity eigenstates for the remainder of this paper.2 The apparent divergence is an artefact of the frame transformation, and is discussed at length in Section 5.2 and Appendix B of[41]. As helicity is a good quantum number, it is conserved in time. The helicity profile of the DM background today should therefore be the same as at production in the absence of significant late time interactions. See[41] and[59] for the argument as applied to the CνB. In order to be cold, the dark matter background must be massive. For a given choice of axes, at least one of either the DM wind or magnetic field direction must have some time dependence due to the evolution of the relative velocity between the laboratory and DM reference frames. In truth, at least one of these must be time-dependent. If we fix our coordinate system in the lab frame, then due to the relative motion of the Earth to the DM reference frame, the direction of the background wind will change in time. However, this can alternatively be accounted for by weighting the collected data by the projection of the relative velocity onto the magnetic field direction. See supplementary material S10.1 of[83] for details of the weighting, and[84] for a full parametrisation of the relevant coordinate systems. We adopt the convention Si " σi, with i P tx, y, zu and σ denoting a Pauli matrix. AcknowledgmentsWe would like to thank Martin Bauer for some very useful comments about the effective operator basis, and Xiao-Dong Ma for highlighting two redundant operators in a previous Rotation of the Andromeda Nebula from a Spectroscopic Survey of Emission Regions. V C Rubin, W K FordJr, 10.1086/150317Astrophys. J. 159V. C. Rubin and W. K. Ford, Jr., Rotation of the Andromeda Nebula from a Spectroscopic Survey of Emission Regions, Astrophys. J. 159 (1970) 379-403. The rotation curve and geometry of M31 at large galactocentric distances. M S Roberts, R N Whitehurst, 10.1086/153889Astrophys. J. 201M. S. Roberts and R. N. Whitehurst, The rotation curve and geometry of M31 at large galactocentric distances., Astrophys. J. 201 (1975) 327-346. Rotational properties of 21 SC galaxies with a large range of luminosities and radii, from NGC 4605 /R = 4kpc/ to UGC 2885 /R = 122 kpc. V C Rubin, N Thonnard, W K FordJr, 10.1086/158003Astrophys. J. 238471V. C. Rubin, N. Thonnard and W. K. Ford, Jr., Rotational properties of 21 SC galaxies with a large range of luminosities and radii, from NGC 4605 /R = 4kpc/ to UGC 2885 /R = 122 kpc/, Astrophys. J. 238 (1980) 471. 21-cm line studies of spiral galaxies. 2. The distribution and kinematics of neutral hydrogen in spiral galaxies of various morphological types. A Bosma, 10.1086/113063Astron. J. 861825A. Bosma, 21-cm line studies of spiral galaxies. 2. The distribution and kinematics of neutral hydrogen in spiral galaxies of various morphological types., Astron. J. 86 (1981) 1825. A direct empirical proof of the existence of dark matter*. D Clowe, M Bradač, A H Gonzalez, M Markevitch, S W Randall, C Jones, 10.1086/508162Astrophys. J. 648109D. Clowe, M. Bradač, A. H. Gonzalez, M. Markevitch, S. W. Randall, C. Jones et al., A direct empirical proof of the existence of dark matter*, Astrophys. J. 648 (2006) L109. The Evolution of Large Scale Structure in a Universe Dominated by Cold Dark Matter. M Davis, G Efstathiou, C S Frenk, S D M White, 10.1086/163168Astrophys. J. 292M. Davis, G. Efstathiou, C. S. Frenk and S. D. M. White, The Evolution of Large Scale Structure in a Universe Dominated by Cold Dark Matter, Astrophys. J. 292 (1985) 371-394. The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: cosmological analysis of the DR12 galaxy sample. S Alam, BOSS collaboration10.1093/mnras/stx7211607.03155Mon. Not. Roy. Astron. Soc. 470BOSS collaboration, S. Alam et al., The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: cosmological analysis of the DR12 galaxy sample, Mon. Not. Roy. Astron. Soc. 470 (2017) 2617-2652, [1607.03155]. N Aghanim, Planck collaboration10.1051/0004-6361/2018339101807.06209Planck 2018 results. VI. Cosmological parameters. 6416Planck collaboration, N. Aghanim et al., Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641 (2020) A6, [1807.06209]. Effective field theory of dark matter: a global analysis. S Liem, G Bertone, F Calore, R Ruiz De Austri, T M P Tait, R Trotta, 10.1007/JHEP09(2016)0771603.05994JHEP. 0977S. Liem, G. Bertone, F. Calore, R. Ruiz de Austri, T. M. P. Tait, R. Trotta et al., Effective field theory of dark matter: a global analysis, JHEP 09 (2016) 077, [1603.05994]. Effective Operator Bases for Beyond Standard Model Scenarios: An EFT compendium for discoveries. U Banerjee, J Chakrabortty, S Prakash, S U Rahaman, M Spannowsky, 10.1007/JHEP01(2021)028JHEP. 0128U. Banerjee, J. Chakrabortty, S. Prakash, S. U. Rahaman and M. Spannowsky, Effective Operator Bases for Beyond Standard Model Scenarios: An EFT compendium for discoveries, JHEP 01 (2021) 028, [2008.11512]. A complete effective field theory for dark matter. J C Criado, A Djouadi, M Perez-Victoria, J Santiago, 10.1007/JHEP07(2021)0812104.14443JHEP. 0781J. C. Criado, A. Djouadi, M. Perez-Victoria and J. Santiago, A complete effective field theory for dark matter, JHEP 07 (2021) 081, [2104.14443]. Dark matter effective field theory and an application to vector dark matter. J Aebischer, W Altmannshofer, E E Jenkins, A V Manohar, 10.1007/JHEP06(2022)0862202.06968JHEP. 0686J. Aebischer, W. Altmannshofer, E. E. Jenkins and A. V. Manohar, Dark matter effective field theory and an application to vector dark matter, JHEP 06 (2022) 086, [2202.06968]. First results from the cryogenic dark matter search in the Soudan Underground Lab. D S Akerib, CDMS collaboration10.1103/PhysRevLett.93.211301astro-ph/0405033Phys. Rev. Lett. 93211301CDMS collaboration, D. S. Akerib et al., First results from the cryogenic dark matter search in the Soudan Underground Lab, Phys. Rev. Lett. 93 (2004) 211301, [astro-ph/0405033]. Dark Matter Search Results from the CDMS II Experiment. Z Ahmed, CDMS-II collaboration10.1126/science.1186112Science. 3270912.3592CDMS-II collaboration, Z. Ahmed et al., Dark Matter Search Results from the CDMS II Experiment, Science 327 (2010) 1619-1621, [0912.3592]. Search for Low-Mass Weakly Interacting Massive Particles with SuperCDMS. R Agnese, SuperCDMS collaboration10.1103/PhysRevLett.112.2413021402.7137Phys. Rev. Lett. 112241302SuperCDMS collaboration, R. Agnese et al., Search for Low-Mass Weakly Interacting Massive Particles with SuperCDMS, Phys. Rev. Lett. 112 (2014) 241302, [1402.7137]. First results from the LUX dark matter experiment at the Sanford Underground Research Facility. D S Akerib, LUX collaboration10.1103/PhysRevLett.112.091303Phys. Rev. Lett. 112913031310.8214LUX collaboration, D. S. Akerib et al., First results from the LUX dark matter experiment at the Sanford Underground Research Facility, Phys. Rev. Lett. 112 (2014) 091303, [1310.8214]. Search for New Physics in Electronic Recoil Data from XENONnT. E Aprile, XENON collaboration10.1103/PhysRevLett.129.1618052207.11330Phys. Rev. Lett. 129161805XENON collaboration, E. Aprile et al., Search for New Physics in Electronic Recoil Data from XENONnT, Phys. Rev. Lett. 129 (2022) 161805, [2207.11330]. A direct dark matter search in XMASS-I. K Abe, XMASS collaboration10.1016/j.physletb.2018.10.0701804.02180Phys. Lett. B. 789XMASS collaboration, K. Abe et al., A direct dark matter search in XMASS-I, Phys. Lett. B 789 (2019) 45-53, [1804.02180]. New CAST Limit on the Axion-Photon Interaction. V Anastassopoulos, CAST collaboration10.1038/nphys41091705.02290Nature Phys. 13CAST collaboration, V. Anastassopoulos et al., New CAST Limit on the Axion-Photon Interaction, Nature Phys. 13 (2017) 584-590, [1705.02290]. Extended Search for the Invisible Axion with the Axion Dark Matter Experiment. T Braine, ADMX collaboration10.1103/PhysRevLett.124.1013031910.08638Phys. Rev. Lett. 124101303ADMX collaboration, T. Braine et al., Extended Search for the Invisible Axion with the Axion Dark Matter Experiment, Phys. Rev. Lett. 124 (2020) 101303, [1910.08638]. Search for Invisible Axion Dark Matter in the 3.3-4.2 µeV Mass Range. C Bartram, ADMX collaboration10.1103/PhysRevLett.127.261803Phys. Rev. Lett. 1272618032110.06096ADMX collaboration, C. Bartram et al., Search for Invisible Axion Dark Matter in the 3.3-4.2 µeV Mass Range, Phys. Rev. Lett. 127 (2021) 261803, [2110.06096]. A new experimental approach to probe QCD axion dark matter in the mass range above 40 µeV. P Brun, MADMAX collaboration10.1140/epjc/s10052-019-6683-x1901.07401Eur. Phys. J. C. 79186MADMAX collaboration, P. Brun et al., A new experimental approach to probe QCD axion dark matter in the mass range above 40 µeV, Eur. Phys. J. C 79 (2019) 186, [1901.07401]. . E Armengaud, IAXO collaboration ; IAXO10.1088/1475-7516/2019/06/0471904.09155Physics potential of the International Axion Observatory. 0647JCAPIAXO collaboration, E. Armengaud et al., Physics potential of the International Axion Observatory (IAXO), JCAP 06 (2019) 047, [1904.09155]. Light Dark Matter Annihilation and Scattering in LHC Detectors. M Bauer, P Foldenauer, P Reimitz, T Plehn, 10.21468/SciPostPhys.10.2.030SciPost Phys. 1030M. Bauer, P. Foldenauer, P. Reimitz and T. Plehn, Light Dark Matter Annihilation and Scattering in LHC Detectors, SciPost Phys. 10 (2021) 030, [2005.13551]. AION: An Atom Interferometer Observatory and Network. L Badurina, 10.1088/1475-7516/2020/05/0111911.11755JCAP. 0511L. Badurina et al., AION: An Atom Interferometer Observatory and Network, JCAP 05 (2020) 011, [1911.11755]. AEDGE: Atomic Experiment for Dark Matter and Gravity Exploration in Space. Y A Aedge Collaboration, El-Neaj, 10.1140/epjqt/s40507-020-0080-01908.00802EPJ Quant. Technol. 76AEDGE collaboration, Y. A. El-Neaj et al., AEDGE: Atomic Experiment for Dark Matter and Gravity Exploration in Space, EPJ Quant. Technol. 7 (2020) 6, [1908.00802]. Matter-wave Atomic Gradiometer InterferometricSensor (MAGIS-100) at Fermilab. J Coleman, 10.22323/1.340.00211812.00482PoS. 201821MAGIS-100 collaborationMAGIS-100 collaboration, J. Coleman, Matter-wave Atomic Gradiometer InterferometricSensor (MAGIS-100) at Fermilab, PoS ICHEP2018 (2019) 021, [1812.00482]. The Alpha Magnetic Spectrometer (AMS) on the international space station: Part II -Results from the first seven years. M Aguilar, AMS collaboration10.1016/j.physrep.2020.09.003Phys. Rept. 894AMS collaboration, M. Aguilar et al., The Alpha Magnetic Spectrometer (AMS) on the international space station: Part II -Results from the first seven years, Phys. Rept. 894 (2021) 1-116. M C Weisskopf, H D Tananbaum, L P Van Speybroeck, S L O&apos;dell, 10.1117/12.391545astro-ph/0004127Chandra x-ray observatory (cxo):overview, Proc. SPIE Int. 4012M. C. Weisskopf, H. D. Tananbaum, L. P. van Speybroeck and S. L. O'Dell, Chandra x-ray observatory (cxo):overview, Proc. SPIE Int. Soc. Opt. Eng. 4012 (2000) 2, [astro-ph/0004127]. The European Photon Imaging Camera on XMM-Newton: The pn-CCD camera. L Struder, 10.1051/0004-6361:20000066Astron. Astrophys. 365L. Struder et al., The European Photon Imaging Camera on XMM-Newton: The pn-CCD camera, Astron. Astrophys. 365 (2001) L18-26. Combined searches for dark matter in dwarf spheroidal galaxies observed with the MAGIC telescopes, including new data from Coma Berenices and Draco. V A Acciari, MAGIC collaboration10.1016/j.dark.2021.100912Phys. Dark Univ. 351009122111.15009MAGIC collaboration, V. A. Acciari et al., Combined searches for dark matter in dwarf spheroidal galaxies observed with the MAGIC telescopes, including new data from Coma Berenices and Draco, Phys. Dark Univ. 35 (2022) 100912, [2111.15009]. Search for dark matter signals towards a selection of recently detected DES dwarf galaxy satellites of the Milky Way with H. H Abdallah, H.E.S.S. collaboration ; .E.S.S.10.1103/PhysRevD.102.062001Phys. Rev. D. 102620012008.00688H.E.S.S. collaboration, H. Abdallah et al., Search for dark matter signals towards a selection of recently detected DES dwarf galaxy satellites of the Milky Way with H.E.S.S., Phys. Rev. D 102 (2020) 062001, [2008.00688]. The VERITAS Dark Matter Program. B Zitzer, VERITAS collaboration10.22323/1.301.09041708.07447PoS. 2017904VERITAS collaboration, B. Zitzer, The VERITAS Dark Matter Program, PoS ICRC2017 (2018) 904, [1708.07447]. Constraints on dark photon dark matter using data from LIGO's and Virgo's third observing run. Ligo Scientific, Kagra, R Virgo Collaboration, Abbott, 10.1103/PhysRevD.105.063030Phys. Rev. D. 105630302105.13085LIGO Scientific, KAGRA, Virgo collaboration, R. Abbott et al., Constraints on dark photon dark matter using data from LIGO's and Virgo's third observing run, Phys. Rev. D 105 (2022) 063030, [2105.13085]. A direct empirical proof of the existence of dark matter. D Clowe, M Bradac, A H Gonzalez, M Markevitch, S W Randall, C Jones, 10.1086/508162astro-ph/0608407Astrophys. J. Lett. 648D. Clowe, M. Bradac, A. H. Gonzalez, M. Markevitch, S. W. Randall, C. Jones et al., A direct empirical proof of the existence of dark matter, Astrophys. J. Lett. 648 (2006) L109-L113, [astro-ph/0608407]. Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Parameter Results. G Hinshaw, WMAP collaboration10.1088/0067-0049/208/2/191212.5226Astrophys. J. Suppl. 20819WMAP collaboration, G. Hinshaw et al., Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Parameter Results, Astrophys. J. Suppl. 208 (2013) 19, [1212.5226]. The 6dF Galaxy Survey: Baryon Acoustic Oscillations and the Local Hubble Constant. F Beutler, C Blake, M Colless, D H Jones, L Staveley-Smith, L Campbell, 10.1111/j.1365-2966.2011.19250.xMon. Not. Roy. Astron. Soc. 4161106.3366F. Beutler, C. Blake, M. Colless, D. H. Jones, L. Staveley-Smith, L. Campbell et al., The 6dF Galaxy Survey: Baryon Acoustic Oscillations and the Local Hubble Constant, Mon. Not. Roy. Astron. Soc. 416 (2011) 3017-3032, [1106.3366]. Speculations on Detection of the Neutrino Sea. L Stodolsky, 10.1103/PhysRevLett.34.110Phys. Rev. Lett. 34110L. Stodolsky, Speculations on Detection of the Neutrino Sea, Phys. Rev. Lett. 34 (1975) 110. Expected signals in relic neutrino detectors. G Duda, G Gelmini, S Nussinov, 10.1103/PhysRevD.64.122001hep-ph/0107027Phys. Rev. D. 64G. Duda, G. Gelmini and S. Nussinov, Expected signals in relic neutrino detectors, Phys. Rev. D 64 (2001) 122001, [hep-ph/0107027]. Detection prospects for the Cosmic Neutrino Background using laser interferometers. V Domcke, M Spinrath, 10.1088/1475-7516/2017/06/0551703.08629JCAP. 0655V. Domcke and M. Spinrath, Detection prospects for the Cosmic Neutrino Background using laser interferometers, JCAP 06 (2017) 055, [1703.08629]. . M Bauer, J D Shergold, 2207.12413M. Bauer and J. D. Shergold, Limits on the cosmic neutrino background, 2207.12413. Review of asymmetric dark matter. K Petraki, R R Volkas, 10.1142/S0217751X133002871305.4939Int. J. Mod. Phys. A. 281330028K. Petraki and R. R. Volkas, Review of asymmetric dark matter, Int. J. Mod. Phys. A 28 (2013) 1330028, [1305.4939]. The Cosmic Neutrino Background on the Surface of the Earth. A Arvanitaki, S Dimopoulos, 2212.00036A. Arvanitaki and S. Dimopoulos, The Cosmic Neutrino Background on the Surface of the Earth, 2212.00036. Evolution of primordial neutrino helicities in cosmic gravitational inhomogeneities. G Baym, J.-C Peng, 10.1103/PhysRevD.103.1230192103.11209Phys. Rev. D. 103123019G. Baym and J.-C. Peng, Evolution of primordial neutrino helicities in cosmic gravitational inhomogeneities, Phys. Rev. D 103 (2021) 123019, [2103.11209]. M Ghosh, Y Grossman, W Tangarife, X.-J Xu, B Yu, 2209.07082Neutrino forces in neutrino backgrounds. M. Ghosh, Y. Grossman, W. Tangarife, X.-J. Xu and B. Yu, Neutrino forces in neutrino backgrounds, 2209.07082. On neutrino-mediated potentials in a neutrino background. D Blas, I Esteban, M C Gonzalez-Garcia, J Salvado, 2212.03889D. Blas, I. Esteban, M. C. Gonzalez-Garcia and J. Salvado, On neutrino-mediated potentials in a neutrino background, 2212.03889. Neutrino bound states and bound systems. A Y Smirnov, X.-J Xu, 10.1007/JHEP08(2022)1702201.00939JHEP. 08170A. Y. Smirnov and X.-J. Xu, Neutrino bound states and bound systems, JHEP 08 (2022) 170, [2201.00939]. Dark matter direct-detection experiments. T , Marrodán Undagoitia, L Rauch, 10.1088/0954-3899/43/1/0130011509.08767J. Phys. G. 4313001T. Marrodán Undagoitia and L. Rauch, Dark matter direct-detection experiments, J. Phys. G 43 (2016) 013001, [1509.08767]. Axions as Dark Matter Particles. L D Duffy, K Van Bibber, 10.1088/1367-2630/11/10/105008New J. Phys. 111050080904.3346L. D. Duffy and K. van Bibber, Axions as Dark Matter Particles, New J. Phys. 11 (2009) 105008, [0904.3346]. Scalar dark matter candidates. C Boehm, P Fayet, 10.1016/j.nuclphysb.2004.01.015hep-ph/0305261Nucl. Phys. B. 683C. Boehm and P. Fayet, Scalar dark matter candidates, Nucl. Phys. B 683 (2004) 219-263, [hep-ph/0305261]. Scalar dark matter candidates revisited. C Boehm, X Chu, J.-L Kuo, J Pradler, 10.1103/PhysRevD.103.075005Phys. Rev. D. 103750052010.02954C. Boehm, X. Chu, J.-L. Kuo and J. Pradler, Scalar dark matter candidates revisited, Phys. Rev. D 103 (2021) 075005, [2010.02954]. The Minimal model of nonbaryonic dark matter: A Singlet scalar. C P Burgess, M Pospelov, T Veldhuis, 10.1016/S0550-3213(01)00513-2hep-ph/0011335Nucl. Phys. B. 619C. P. Burgess, M. Pospelov and T. ter Veldhuis, The Minimal model of nonbaryonic dark matter: A Singlet scalar, Nucl. Phys. B 619 (2001) 709-728, [hep-ph/0011335]. Sterile-neutrinos as dark matter. S Dodelson, L M Widrow, 10.1103/PhysRevLett.72.17hep-ph/9303287Phys. Rev. Lett. 72S. Dodelson and L. M. Widrow, Sterile-neutrinos as dark matter, Phys. Rev. Lett. 72 (1994) 17-20, [hep-ph/9303287]. Sterile neutrinos, dark matter, and the pulsar velocities in models with a Higgs singlet. A Kusenko, 10.1103/PhysRevLett.97.241301hep-ph/0609081Phys. Rev. Lett. 97241301A. Kusenko, Sterile neutrinos, dark matter, and the pulsar velocities in models with a Higgs singlet, Phys. Rev. Lett. 97 (2006) 241301, [hep-ph/0609081]. Dark-matter sterile neutrinos in models with a gauge singlet in the Higgs sector. K Petraki, A Kusenko, 10.1103/PhysRevD.77.065014Phys. Rev. D. 77650140711.4646K. Petraki and A. Kusenko, Dark-matter sterile neutrinos in models with a gauge singlet in the Higgs sector, Phys. Rev. D 77 (2008) 065014, [0711.4646]. Light Sterile Neutrinos: A White Paper. K N Abazajian, K. N. Abazajian et al., Light Sterile Neutrinos: A White Paper, 1204.5379. Supersymmetric dark matter. G Jungman, M Kamionkowski, K Griest, 10.1016/0370-1573(95)00058-5hep-ph/9506380Phys. Rept. 267G. Jungman, M. Kamionkowski and K. Griest, Supersymmetric dark matter, Phys. Rept. 267 (1996) 195-373, [hep-ph/9506380]. Dark Matter Candidates from Particle Physics and Methods of Detection. J L Feng, 10.1146/annurev-astro-082708-1016591003.0904Ann. Rev. Astron. Astrophys. 48J. L. Feng, Dark Matter Candidates from Particle Physics and Methods of Detection, Ann. Rev. Astron. Astrophys. 48 (2010) 495-545, [1003.0904]. Detecting non-relativistic cosmic neutrinos by capture on tritium: phenomenology and physics potential. A J Long, C Lunardini, E Sabancilar, 10.1088/1475-7516/2014/08/0381405.7654JCAP. 0838A. J. Long, C. Lunardini and E. Sabancilar, Detecting non-relativistic cosmic neutrinos by capture on tritium: phenomenology and physics potential, JCAP 08 (2014) 038, [1405.7654]. . P A Zyla, Particle Data Group collaboration10.1093/ptep/ptaa104Review of Particle Physics. 2020PTEPParticle Data Group collaboration, P. A. Zyla et al., Review of Particle Physics, PTEP 2020 (2020) 083C01. The Physics of Heavy Z 1 Gauge Bosons. P Langacker, 10.1103/RevModPhys.81.1199Rev. Mod. Phys. 810801.1345P. Langacker, The Physics of Heavy Z 1 Gauge Bosons, Rev. Mod. Phys. 81 (2009) 1199-1228, [0801.1345]. Hunting All the Hidden Photons. M Bauer, P Foldenauer, J , 10.1007/JHEP07(2018)0941803.05466JHEP. 0794M. Bauer, P. Foldenauer and J. Jaeckel, Hunting All the Hidden Photons, JHEP 07 (2018) 094, [1803.05466]. The Dark Photon. M Fabbrichesi, E Gabrielli, G Lanfranchi, M. Fabbrichesi, E. Gabrielli and G. Lanfranchi, The Dark Photon, 2005.01515. Dark photon limits: A handbook. A Caputo, A J Millar, C A J O&apos;hare, E Vitagliano, 10.1103/PhysRevD.104.0950292105.04565Phys. Rev. D. 10495029A. Caputo, A. J. Millar, C. A. J. O'Hare and E. Vitagliano, Dark photon limits: A handbook, Phys. Rev. D 104 (2021) 095029, [2105.04565]. Is the lightest Kaluza-Klein particle a viable dark matter candidate?. G Servant, T M P Tait, 10.1016/S0550-3213(02)01012-Xhep-ph/0206071Nucl. Phys. B. 650G. Servant and T. M. P. Tait, Is the lightest Kaluza-Klein particle a viable dark matter candidate?, Nucl. Phys. B 650 (2003) 391-419, [hep-ph/0206071]. Kaluza-Klein dark matter. H.-C Cheng, J L Feng, K T Matchev, 10.1103/PhysRevLett.89.211301hep-ph/0207125Phys. Rev. Lett. 89211301H.-C. Cheng, J. L. Feng and K. T. Matchev, Kaluza-Klein dark matter, Phys. Rev. Lett. 89 (2002) 211301, [hep-ph/0207125]. Dark QCD matters. R Garani, M Redi, A Tesi, 10.1007/JHEP12(2021)1392105.03429JHEP. 12139R. Garani, M. Redi and A. Tesi, Dark QCD matters, JHEP 12 (2021) 139, [2105.03429]. Spin 3/2 Particle as a Dark Matter Candidate: an Effective Field Theory Approach. R Ding, Y Liao, 10.1007/JHEP04(2012)0541201.0506JHEP. 0454R. Ding and Y. Liao, Spin 3/2 Particle as a Dark Matter Candidate: an Effective Field Theory Approach, JHEP 04 (2012) 054, [1201.0506]. Case for decaying spin-3/2 dark matter. M A G Garcia, Y Mambrini, K A Olive, S Verner, 10.1103/PhysRevD.102.083533Phys. Rev. D. 102835332006.03325M. A. G. Garcia, Y. Mambrini, K. A. Olive and S. Verner, Case for decaying spin-3/2 dark matter, Phys. Rev. D 102 (2020) 083533, [2006.03325]. Accidental Composite Dark Matter. O Antipin, M Redi, A Strumia, E Vigiani, 10.1007/JHEP07(2015)0391503.08749JHEP. 0739O. Antipin, M. Redi, A. Strumia and E. Vigiani, Accidental Composite Dark Matter, JHEP 07 (2015) 039, [1503.08749]. Torsion-balance tests of the weak equivalence principle. T A Wagner, S Schlamminger, J H Gundlach, E G Adelberger, 10.1088/0264-9381/29/18/1840021207.2442Class. Quant. Grav. 29184002T. A. Wagner, S. Schlamminger, J. H. Gundlach and E. G. Adelberger, Torsion-balance tests of the weak equivalence principle, Class. Quant. Grav. 29 (2012) 184002, [1207.2442]. The Local Dark Matter Density. J I Read, 10.1088/0954-3899/41/6/0631011404.1938J. Phys. G. 4163101J. I. Read, The Local Dark Matter Density, J. Phys. G 41 (2014) 063101, [1404.1938]. Cosmic neutrinos and their detection. C Hagmann, astro-ph/9905258) Meeting of the Division of Particles and Fields (DPF 99. APS1C. Hagmann, Cosmic neutrinos and their detection, in American Physical Society (APS) Meeting of the Division of Particles and Fields (DPF 99), 1, 1999, astro-ph/9905258. New Observables for Direct Detection of Axion Dark Matter. P W Graham, S Rajendran, 10.1103/PhysRevD.88.0350231306.6088Phys. Rev. D. 8835023P. W. Graham and S. Rajendran, New Observables for Direct Detection of Axion Dark Matter, Phys. Rev. D 88 (2013) 035023, [1306.6088]. Proposal for a Cosmic Axion Spin Precession Experiment (CASPEr). D Budker, P W Graham, M Ledbetter, S Rajendran, A Sushkov, 10.1103/PhysRevX.4.0210301306.6089Phys. Rev. X. 421030D. Budker, P. W. Graham, M. Ledbetter, S. Rajendran and A. Sushkov, Proposal for a Cosmic Axion Spin Precession Experiment (CASPEr), Phys. Rev. X 4 (2014) 021030, [1306.6089]. Solid state systems for electron electric dipole moment and other fundamental measurements. S K Lamoreaux, 10.1103/PhysRevA.66.022109nucl-ex/0109014Phys. Rev. A. 6622109S. K. Lamoreaux, Solid state systems for electron electric dipole moment and other fundamental measurements, Phys. Rev. A 66 (2002) 022109, [nucl-ex/0109014]. The first VERA astrometry catalog. T Hirota, VERA collaboration10.1093/pasj/psaa018Publ. Astron. Soc. Jpn. 72502002.03089VERA collaboration, T. Hirota et al., The first VERA astrometry catalog, Publ. Astron. Soc. Jpn. 72 (2020) 50, [2002.03089]. SENSEI: Direct-Detection Results on sub-GeV Dark Matter from a New Skipper-CCD. L Barak, SENSEI collaboration10.1103/PhysRevLett.125.171802Phys. Rev. Lett. 125171802SENSEI collaboration, L. Barak et al., SENSEI: Direct-Detection Results on sub-GeV Dark Matter from a New Skipper-CCD, Phys. Rev. Lett. 125 (2020) 171802, [2004.11378]. New Constraints and Prospects for sub-GeV Dark Matter Scattering off Electrons in Xenon. R Essig, T Volansky, T.-T Yu, 10.1103/PhysRevD.96.0430171703.00910Phys. Rev. D. 9643017R. Essig, T. Volansky and T.-T. Yu, New Constraints and Prospects for sub-GeV Dark Matter Scattering off Electrons in Xenon, Phys. Rev. D 96 (2017) 043017, [1703.00910]. Light Dark Matter Search with Ionization Signals in XENON1T. E Aprile, XENON collaboration10.1103/PhysRevLett.123.2518011907.11485Phys. Rev. Lett. 123251801XENON collaboration, E. Aprile et al., Light Dark Matter Search with Ionization Signals in XENON1T, Phys. Rev. Lett. 123 (2019) 251801, [1907.11485]. Probing axions with the neutrino signal from the next galactic supernova. T Fischer, S Chakraborty, M Giannotti, A Mirizzi, A Payez, A Ringwald, 10.1103/PhysRevD.94.0850121605.08780Phys. Rev. D. 9485012T. Fischer, S. Chakraborty, M. Giannotti, A. Mirizzi, A. Payez and A. Ringwald, Probing axions with the neutrino signal from the next galactic supernova, Phys. Rev. D 94 (2016) 085012, [1605.08780]. FCNC B and K meson decays with light bosonic Dark Matter. X.-G He, X.-D Ma, G Valencia, 10.1007/JHEP03(2023)0372209.05223JHEP. 0337X.-G. He, X.-D. Ma and G. Valencia, FCNC B and K meson decays with light bosonic Dark Matter, JHEP 03 (2023) 037, [2209.05223]. Constraints on bosonic dark matter from ultralow-field nuclear magnetic resonance. A Garcon, 10.1126/sciadv.aax45391902.04644Sci. Adv. 54539A. Garcon et al., Constraints on bosonic dark matter from ultralow-field nuclear magnetic resonance, Sci. Adv. 5 (2019) eaax4539, [1902.04644]. On Diurnal and Annual Variations of Directional Detection Rates of Dark Matter. A Bandyopadhyay, D Majumdar, 10.1088/0004-637X/746/1/1071006.3231Astrophys. J. 746107A. Bandyopadhyay and D. Majumdar, On Diurnal and Annual Variations of Directional Detection Rates of Dark Matter, Astrophys. J. 746 (2012) 107, [1006.3231].
{'fraction_non_alphanumeric': 0.06460081432095341, 'fraction_numerical': 0.05667554956757774, 'mean_word_length': 3.7518712046321143, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 78, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'We present a comprehensive discussion of the Stodolsky effect for dark matter (DM), and discuss two techniques to measure the effect and constrain the DM parameter space. The Stodolsky effect is the spin-dependent shift in the energy of a Standard Model (SM) fermion sitting in a bath of neutrinos. This effect, which scales linearly in the effective coupling, manifests as a small torque on the SM fermion spin and has historically been proposed as a method of detecting the cosmic neutrino background. We generalise this effect to DM, and give expressions for the induced energy shifts for DM candidates from spin-0 to spin-3 2 , considering all effective operators up to mass dimension-6. In all cases, the effect scales inversely with the DM mass, but requires an asymmetric background. We show that a torsion balance experiment is sensitive to energy shifts of ∆E Á 10´2 8 eV, whilst a more intricate setup using a SQUID magnetometer is sensitive to shifts of ∆E Á 10´3 2 eV. Finally, we compute the energy shifts for a model of scalar DM, and demonstrate that the Stodolsky effect can be used to constrain regions of parameter space that are not presently excluded.', 'arxivid': '2304.06750', 'author': ['Guillaume Rostagni [email protected] \nInstitute for Particle Physics Phenomenology\nDepartment of Physics\nDurham University\nDurhamUK\n', 'Jack D Shergold [email protected] \nInstitute for Particle Physics Phenomenology\nDepartment of Physics\nDurham University\nDurhamUK\n'], 'authoraffiliation': ['Institute for Particle Physics Phenomenology\nDepartment of Physics\nDurham University\nDurhamUK', 'Institute for Particle Physics Phenomenology\nDepartment of Physics\nDurham University\nDurhamUK'], 'corpusid': 258170489, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 38330, 'n_tokens_neox': 32227, 'n_words': 18285, 'pdfsha': 'ad2e587b5b30d65b7a0a5d5f5060834ded3825ec', 'pdfurls': ['https://export.arxiv.org/pdf/2304.06750v2.pdf'], 'title': ['The dark Stodolsky effect: constraining effective dark matter operators with spin-dependent interactions', 'The dark Stodolsky effect: constraining effective dark matter operators with spin-dependent interactions'], 'venue': []}
arxiv
Asymptotic Independence of the Quadratic form and Maximum of Independent Random Variables with Applications to High-Dimensional Tests Dachuan Chen Nankai University Long Feng Nankai University Asymptotic Independence of the Quadratic form and Maximum of Independent Random Variables with Applications to High-Dimensional Tests Asymptotic independenceHigh dimensional dataLarge p, small nOne-sample testTwo-sample test This paper establishes the asymptotic independence between the quadratic form and maximum of a sequence of independent random variables. Based on this theoretical result, we find the asymptotic joint distribution for the quadratic form and maximum, which can be applied into the high-dimensional testing problems. By combining the sum-type test and the max-type test, we propose the Fisher's combination tests for the one-sample mean test and two-sample mean test. Under this novel general framework, several strong assumptions in existing literature have been relaxed. Monte Carlo simulation has been done which shows that our proposed tests are strongly robust to both sparse and dense data. Introduction Independence is a very important property in statistical inference. In this paper, we develop the asymptotic independence between the quadratic form z Az and the maximum max 1≤i≤p |z i | of a 1 arXiv:2204.08628v1 [stat.ME] 19 Apr 2022 sequence of independent sub-Gaussian random variables z = (z 1 , · · · , z p ) , where A is a symmetric matrix. The benefits of this theoretical result will be reflected in the application of highdimensional tests, including the one-sample mean test and two-sample mean test. Yet little research has been done on the asymptotic joint distribution between the the quadratic form and the maximum of a sequence random variables. This is the first paper on this topic. In contrast, the majority of the existing literature are focusing on the development of the asymptotic independence between the sum p i=1 X i and the maximum max 1≤i≤p X i of a sequence of random variables {X i } p i=1 . Here we provide a brief review for the related literature. Chow and Teugels (1978) derives the asymptotic independence between the sum and maximum by assuming {X i } p i=1 to be independent and identically distributed (i.i.d., hereafter). There are two main streams of statistical work which have relaxed the i.i.d. assumption. On one hand, Anderson and Turkman (1991, 1993, 1995 and Hsing (1995) extends the theoretical results by assuming the random vari- ables {X i } p i=1 satisfying the strong mixing condition. On the other hand, Ho and Hsing (1996), Ho and McCormick (1999), McCormick and Qi (2000) and Peng and Nadarajah (2003) establishes the asymptotic independence between the sum and the maximum based on the assumption that {X i } p i=1 is a stationary Gaussian sequence. Feng et al. (2022) further investigates the asymptotic independence between the sum and the maximum of the squares of the dependent random variables without imposing the stationary assumption. In this paper, the asymptotic joint distribution of the quadratic form z Az and the maximum max 1≤i≤p |z i | of independent sub-Gaussian random variables has been derived, where the random variables {z i } p i=1 are assumed to have mean zero and variance one. The asymptotic normality of the standardized quadratic form z Az−tr(A) σ A is guaranteed by the assumption that A is symmetric and sup i p j=1 |a ij | < K. The asymptotic distribution of the maximum max 1≤i≤p |z i | is derived based on the assumption that pP (|z i | > l p (y)) → h(y) for all i where l p (y) → ∞ and h(y) is bounded. The asymptotic independence between the standardized quadratic form and the maximum is mainly based on the assumption that the smallest eigenvalue of A is bounded away from zero and the largest eigenvalue of A is bounded. Our theoretical result on asymptotic independence is novel and different from the existing ones. As a consequence, this general framework provides a new way for us to look at the theoretical foundation of the high-dimensional testing problems, including the one-sample mean test and the two-sample mean test. The research of high-dimensional hypothesis tests has been evolved rapidly in the last two decades, which has been widely applied to a range of areas, including genomics, neuroscience, finance, economics and so on. In general, high-dimensionality means that the data dimension can be larger than the sample size, or the data dimension can also grow to infinity in asymptotics. Under this dimension setting, the classical statistical testing theories no longer applicable. For example, the traditional Hotelling's T 2 test cannot work when the data dimension exceeds the sample size because of the singularity of sample covariance matrix. Consequently, the high dimensional testing problems are grounded on a new theoretical foundation comparing with that of the classical ones. By replacing the sample covariance matrix in Hotelling's T 2 test with the identity matrix or the diagonal matrix of the sample covariance matrix, several sum-type tests has been proposed for the high-dimensional mean test problem, see e.g., Bai and Saranadasa (1996), Srivastava and Du (2008), Srivastava (2009), Chen and Qin (2010) and Srivastava et al. (2013). However, due to the low performance of the sum-type test under the sparse alternative, where there are a few nonzero elements in the mean vector or the mean difference, many efforts have been made to improve the sum-type tests, see, for example, Zhong et al. (2013), Fan et al. (2015 and Chen et al. (2019). Different from the sum-type tests, the other solution for sparse alternative is the max-type tests proposed by Cai et al. (2014), which is particularly powerful for the sparse data but cannot work well with the dense data. In real world, it is usually difficult to identify whether the data is sparse or not. Therefore, it becomes necessary to develop a test which can work well for both sparse and dense alternatives. The power enhancement test proposed by Fan et al. (2015) is one candidate for this purpose which adding a screening statistic to the sum-type test statistic. The other candidate is the adaptive test proposed by Xu et al. (2016) which studied the asymptotic independence between the max-type test and the sum-of-powers tests and then combined them together based on their p-values. He et al. (2021) is another candidate for the same purpose which combined the max-type test with a set of finite-order U-Statistics based on their asymptotic independence. In this paper, based on the novel theoretical result on the asymptotic independence between the quadratic form and maximum of independence random variables, the aforementioned problem has been solved, and at the same time, several strong assumptions made by Xu et al. (2016) and He et al. (2021) have been relaxed. We propose two Fisher's combination tests by combining the max-type tests and sum-type tests for the one-sample mean test and two-sample mean test. The simulation results show that our proposed tests are strongly robust to both sparse data and dense data. The main contributions of this paper are listed as follows: 1. We show the asymptotic independence between the quadratic form and the maximum of independent sub-Gaussian random variables, which is novel in existing literature; 2. Based on the above theoretical results, we have proposed the Fisher's combination test by combining the sum-type test and the max-type test for two types of high-dimensional testing problems: one-sample mean test and two-sample mean test. Simulation results show that our proposed tests are robust to both sparse and dense data; 3. The development of these two applications reflects the theoretical benefits of our general framework in proving the asymptotic independence between the max-type tests and sumtype tests: (1) the strong assumptions on the population covariance structure (i.e., the αmixing condition or the diagonal assumption) in existing literature have been relaxed; (2) besides the sub-Gaussian-type tails, our theoretical development also allows the polynomialtype tail for the sample distribution; 4. By switching the alternative hypothesis to the special local alternative, for example, sparse transformed mean in one-sample mean test or sparse transformed mean difference in twosample mean test, we could obtain the asymptotic independence between the max-type test and the sum-type test under the alternative hypothesis. As a consequence, the expression of the power of our proposed test has been derived, which is the first result on this topic. The organization of this paper is as follows. In Section 2, we provide the basic definition about the distribution of the random variables and state the theoretical result about the asymptotic independence between the quadratic form and maximum of the independent random variables. In Section 3, we apply the theoretical result into two types of tests of high dimensional data. In Section 4, the proposed tests are compared with some existing ones via Monte Carlo simulation. The mathematical proofs of our theoretical results are collected in Section 5. 2 Asymptotic Independence of the Quadratic form and Maxi- mum of Independent Random Variables In this section, we provide the central theoretical results: the asymptotic independence between the quadratic form and maximum of independent sub-Gaussian random variables. The statement will start with the definition of the sub-Gaussian random variable. DEFINITION 2.1 A random variable X with mean µ = E[X] is σ 2 -sub-Gaussian if there is a positive number σ such that E e λ(X−µ) ≤ e σ 2 λ 2 /2 for all λ ∈ R(1) The constant σ is referred to as the sub-Gaussian parameter; for instance, we say that X is σ 2sub-Gaussian when the condition (1) holds. Naturally, any Gaussian variable with variance σ 2 is sub-Gaussian with parameter σ. THEOREM 1 Assume z 1 , · · · , z p are independent σ 2 -sub-Gaussian random variables with E(z i ) = 0 and var(z i ) = 1. Suppose the following assumptions hold: (i) There exist two parameters l p (y) and h(y) satisfies pP (|z i | > l p (y)) → h(y) for all i where l p (y) → ∞ and h(y) is bounded; (ii) A is symmetric and sup i p j=1 |a ij | < K; (iii) There exist a constant c > 0 satisfies c −1 < λ min (A) ≤ λ max (A) < c. Then, we have P z Az − tr(A) σ A ≤ x, max 1≤i≤p |z i | ≤ l p (y) → Φ(x)F h (y)(2) where σ 2 A = 2tr(A 2 ) + p i=1 a 2 ii (E(z 4 i ) − 3) and F h (y) = e −h(y) . Theorem 1 showed the asymptotic joint distribution of the standardized quadratic form and the maximum of independent sub-Gaussian random variables. Specifically, the assumption (i) is used to derive the asymptotic distribution of the maximum; the assumption (ii) guarantees the asymptotic normality of the standardized quadratic form; and finally the asymptotic independence between the standardized quadratic form and the maximum is mainly based on assumption (iii). REMARK 2.1 It is worth to mention that if all |z i |'s in Theorem 1 and in its proof are replaced by z i , the updated theorem and proof still hold. That is, we could show the asymptotic independence between the quadratic form z Az−tr(A) σ A and the maximum max 1≤i≤p z i based on the similar framework as that of Theorem 1. High Dimensional tests In this section, we apply the theoretical results obtained in Section 2 into two types of highdimensional hypothesis tests: one-sample mean test and two-sample mean test. One-sample problem Let X 1 , · · · , X n with X i = (X i1 , · · · , X ip ) for each i ∈ {1, · · · , n} be a sequence of p dimensional independent and identically distributed (iid) observations from a multivariate distribution with mean vector µ and covariance matrix Σ. Our interest is in testing the one sample mean hy- potheses H 0 : µ = 0 versus H 1 : µ = 0(3) The hypothesis test on population mean is a classic and important topic in multivariate statistics, which has been developed in a large statistical literature, see, e.g., Anderson (1962), Eaton (1983 and Muirhead (2009) for classical theory. The most famous methodology is the Hotelling's T 2 test, see Hotelling (1931). In asymptotics, these classical theories assume the dimension p to be fixed as the sample size n goes to infinity. However, when the dimension p is larger than the sample size n, these earlier methods cannot really work. For example, the Hotelling's T 2 test requires the inverse of the sample covariance matrix. Actually, the sample covariance matrix is non-invertible when p > n. To overcome this difficulty, many high-dimensional mean tests have been proposed by allowing p ≥ n and letting both n and p go to infinity in asymptotics. In this subsection, we focus on the hypothesis testing problem in (3) under the setting of p ≥ n. Srivastava and Du (2008) developed the sum-type test statistics for the multivariate normal observations while Srivastava (2009) showed that this sum-type test statistics can also work for the non-normal observations under some certain moment assumptions. In general, as proposed in Srivastava (2009), the sum-type test statistics can be expressed as follows: T SR = nX D −1 sX − (n − 1)p/(n − 3) 2 trR 2 − p 2 /(n − 1) 1 2 1 + trR 2 /p 3 2 1 2 (4) whereR = D − 1 2 s SD − 1 2 s , D s = diag (s 11 , . . . , s pp ) , X = 1 n n i=1 X i , S = (s ij ) 1≤i,j≤p = 1 n − 1 n i=1 (X i −X)(X i −X) . As pointed out by Cai et al. (2014), the aforementioned sum-type test cannot work well when the mean vector is sparse, for example, there are a few nonzero elements in the mean vector. To improve the sum-type test under the sparse alternative, Zhong et al. (2013) M A = (n − 1) max 1 i p δ A i 2 b i,i .(5) Similar to Cai et al. (2014), we proposed the following max-type test statistic MΩ 1/2 by choosing A asΩ 1/2 . HereΩ is a good estimator of the precision matrix Ω = Σ −1 . In what follows, we state the required assumptions for the development of the asymptotic distribution of the max-type test statistic MΩ 1/2 . (C1) X i = µ + Σ 1/2 ε i where ε i = (ε i1 , · · · , ε ip ) and ε ij are independently distributed with E(ε ij ) = 0, var(ε ij ) = 1 and E(ε 4 ij ) < c for some positive constant c. (C2) C −1 0 λ min (Σ) λ max (Σ) C 0 for some constant C 0 > 0. (C3) We assume that the estimatorΩ = (ω ij ) has at least a logarithmic rate of convergence Ω − Ω L 1 = o p 1 log(p) , max 1 i p |ω i,i − ω i,i | = o p 1 log(p) (C4) Suppose the following condition (i) or (ii) hold: (i)(sub-Gaussian-type tails) There exist some constant η > 0, K > 0 such that E(exp(ηε 2 ij )) ≤ K. And log p = o(n 1/4 ); (ii) (polynomial-type tails) Suppose that for some constants γ 0 , c 1 > 0, p ≤ c 1 n γ 0 and for some positive constant , K, E(|ε ij | 2γ 0 +2+ ) ≤ K. The limiting null distribution of the max-type test statistic MΩ 1/2 is given in following theorem. THEOREM 2 Suppose that conditions (C1)-(C4) hold. For any x ∈ R, P H 0 [MΩ 1/2 − 2 log(p) + log{log(p)} x] → exp − 1 √ π exp − x 2 , as p → ∞ Given the above description, we know that the sum-type test can only work well under the dense alternative, while the max-type test can only work well under the sparse alternative. In order to achieve good performance under both sparse and dense alternatives, it is a good choice to propose a combination of the sum-type test and the max-type test based on their asymptotic independence. The existing literature on this combination approach includes Xu et al. (2016) and He et al. (2021). Xu et al. (2016) showed the asymptotic independence between the max-type test and a family of the sum-of-powers tests and constructed the combined test by picking up the minimum of the p-values of these tests in order to reach the maximum power. He et al. (2021) developed the asymptotic independence between the max-type test statistics and the finite-order U-statistics and combined these tests based on the minimum p-values or the Fisher's method, see e.g., Littell and Folks (1971). Compared with these existing methodology, our theoretical contribution is that several strong assumptions in existing literature have been relaxed. First, both Xu et al. (2016) and He et al. (2021) only assume the sub-Gaussian-type tail for the sample distribution. In contrast, our theoretical framework also allows for the polynomial-type tail, e.g., see condition (C4). Second, we impose weaker assumptions for the covariance structure than those of Xu et al. (2016) and He et al. (2021). The detailed assumption and discussion are provided as follows. In the following condition, we state the additional assumption about the covariance structure, which is required to show the asymptotic independence between the sum-type test and max-type test. (C5) sup i p j=1 |a ij | < K where A = Σ 1/2 D −1 Σ 1/2 = (a ij ) 1≤i,j≤p and D is the diagonal matrix of Σ. In existing literatures related to the asymptotic independence between max-type tests and sumtype tests, the assumptions about covariance structure are relatively strong. For example, the Theorem 1 of Xu et al. (2016) is based on the assumption of α-mixing condition. Moreover, the Theorem 2.3 of He et al. (2021) assumed the covariance structure to be diagonal. In contrast, our assumption about the covariance structure (as stated in condition (C5)) is more general. Up to now, based on the result in Theorem 1, we could show the asymptotic independence between T SR and MΩ 1/2 under the null hypothesis. THEOREM 3 Suppose that conditions (C1)-(C5) hold. For any x, y ∈ R, P H 0 [T SR ≤ x, MΩ 1/2 − 2 log(p) + log{log(p)} y] → Φ(x)F (y)(6) where F (y) = exp(− 1 √ π e −y/2 ). To combine the proposed max-type and sum-type tests, we propose the Fisher's combination test, based on the asymptotic independence between T SR and MΩ 1/2 . Specifically, let p (1) MAX . = 1 − F {MΩ 1/2 − 2 log(p) + log{log(p)}} and p (1) SUM . = 1 − Φ (T SR ) denote the p-values with respect to the test statistics T (1) MAX = MΩ 1/2 and T (1) SUM = T SR respectively. Based on p (1) MAX and p (1) SUM , the proposed Fisher's combination test rejects H 0 at the significance level α, if T (1) FC . = −2 log p (1) MAX − 2 log p (1) SUM (7) is larger than c α , i.e. the 1 − α quantile of the chi-squared distribution with 4 degrees of freedom. Based on Theorem 3, we immediately have the following result for T We consider the following alternative hypothesis: H 1 :μ i = 0, i ∈ M, |M| = m, m = o(p 1/2 ), µ = δ (np) 1/2(8) whereμ = Ω 1/2 µ and δ is a vector of constants and δ D −1 δ ≤ Cp for some constant C. It is easy to see that the local alternative H 1 in (8) is a special case of the original alternative hypothesis H 1 : µ = 0. Under this special local alternative, we could further obtain the asymptotic independence between the sum-type test and max-type test as follows. THEOREM 4 Suppose that conditions (C1)-(C5) hold. Under the special local alternative H 1 stated in (8), for any x, y ∈ R, P [T SR ≤ x, MΩ 1/2 − 2 log(p) + log{log(p)} ≤ y] → P [T SR ≤ x] P [MΩ 1/2 − 2 log(p) + log{log(p)} ≤ y] . Define a minimal p-values test T (1) min = min(p (1) SUM , p(1) MAX ). According to Theorem 3, we reject the null hypothesis if p (1) SUM ≤ 1 − √ 1 − γ ≈ γ/2 or p (1) MAX ≤ 1 − √ 1 − γ ≈ γ/2. According to the results in Folks (1971, 1973), we have the power of Fisher combination test β T (1) F C is larger than the power of the minimal p-values test β T (1) min . Thus, we have β T (1) F C ≥ β T (1) min ≥ β T (1) SUM ,γ/2 + β T (1) MAX ,γ/2 − β T (1) SUM ,γ/2 β T (1) MAX ,γ/2 where the last inequality is based on the inclusion-exclusion principle and the result of Theorem 4, and β T (1) SUM ,γ is the power function of the sum-type test T (1) SUM at significant value γ. So does β T (1) MAX ,γ . Two-sample problem Assume that {X i1 , · · · , X in i } for i = 1, 2 are two independent random samples with the sizes n 1 and n 2 , from p-variate distributions F (x − µ 1 ) and G(x − µ 2 ) located at p-variate centers µ 1 and µ 2 . Let n = n 1 + n 2 . We wish to test H 0 : µ 1 = µ 2 versus H 1 : µ 1 = µ 2 ,(9) when their common covariances Σ is unknown. The famous Hotelling's T 2 test statistic for the two-sample problem also requires the dimension p to be fixed in asymptotics. Because it relies on the inverse of the pooled sample covariance matrix, Hotelling's T 2 test cannot work when p ≥ n. To deal with the problem of high-dimensional setting, i.e., when p ≥ n, several sum-type tests have been proposed. Bai and Saranadasa (1996) proposes a test by replacing the pooled sample covariance matrix in Hotelling's T 2 test statistic with the identity matrix. Srivastava and Du (2008) developed a test for two set of multivariate normal observations which sharing the same population covariance matrix. Under some mild assumptions on the covariance structure, Chen and Qin (2010) derived a test statistic and relaxed the assumption p/n → c ∈ (0, ∞) in Bai and Saranadasa (1996) by allowing the arbitrarily large p which can be independent with the sample size n. For the case of unequal covariance matrix, T SKK = X 1 −X 2 D −1 X 1 −X 2 − p pσ 2 SKK c p,n(10)whereσ 2 SKK = 2tr(R 2 ) p − 2 p(n 1 − 1)n 2 1 tr(D −1 S 1 ) 2 − 2 p(n 2 − 1)n 2 2 tr(D −1 S 2 ) 2 , c p,n =1 + trR 2 p 3/2 ,X i = 1 n i n i l=1 X il for i = 1, 2 where S 1 and S 2 are the sample covariance matrices of {X 1l } n 1 l=1 and {X 2l } n 2 l=1 , respectively.D = 1 n 1D 1 + 1 n 2D 2 whereD i , i = 1, 2 is the diagonal matrix of S i . AndR =D −1/2 1 n 1Ŝ 1 + 1 n 2Ŝ 2 D −1/2 . Similar to the discussion in the one-sample problem, the aforementioned sum-type tests cannot work well under the sparse alternatives. Here, sparse alternative means that the difference of the population means is sparse. To overcome this difficulty, Chen et al. (2019) proposed a test based on the thresholding technique and data transformation, which can be regard as the extension of the method in Zhong et al. (2013). For the sparse alternative, Cai et al. (2014) proposed the following max-type test statistics: WΩ 1/2 = n 1 n 2 n 1 + n 2 max 1≤i≤pW 2 i(11) whereW = (W 1 , · · · ,W p ) . =Ω 1/2 (X 1 −X 2 ). To facilitate the description of theoretical results in two-sample mean test, we switch the condition (C1) in previous section to a new one as follows. (C1 ) For k = 1, 2, X ki = µ k + Σ 1/2 ε ki where ε ki = (ε ki1 , · · · , ε kip ) and ε kij are independently distributed with E(ε kij ) = 0, var(ε kij ) = 1 and E(ε 4 kij ) < c for some positive constant c. The limiting null distribution of the max-type test statistic WΩ 1/2 is as follows. THEOREM 5 Suppose that conditions (C1 ), (C2)-(C4) hold. For any x ∈ R, P H 0 [WΩ 1/2 − 2 log(p) + log{log(p)} x] → exp − 1 √ π exp − x 2 , as p → ∞ Because it is difficult to tell whether the data is sparse or not in the real world, we need to develop a test which is robust to both sparse and dense alternatives at the same time. The main idea of the solution is to combine the sum-type test and max-type test based on the asymptotic independence between them, which is similar to those of Xu et al. (2016) and He et al. (2021). To achieve the asymptotic independence between sum-type test and max-type test, we now apply the theoretical results stated in Section 2 into the two-sample mean test as follows. THEOREM 6 Suppose that conditions (C1 ), (C2)-(C5) hold. For any x, y ∈ R, P H 0 [T SKK ≤ x, WΩ 1/2 − 2 log(p) + log{log(p)} y] → Φ(x)F (y)(12) It is clear that in the two-sample problem, the result of asymptotic independence shares the same theoretical benefits as that of the one-sample problem. For example, compared with Xu et al. Based on the asymptotic independence between T SKK and WΩ 1/2 , we propose the Fisher's combination test which utilizing the max-type and sum-type tests: T (2) FC . = −2 log p(2) MAX − 2 log p We consider the following alternative hypothesis: H 1 :μ i = 0, i ∈ M, |M| = m, m = o(p 1/2 ), µ 1 − µ 2 = δ (np) 1/2(14) whereμ = Ω 1/2 (µ 1 − µ 2 ) and δ is a vector of constants and δ D −1 δ ≤ Cp for some constant C. As the special case of the original alternative hypothesis H 1 : µ 1 = µ 2 , the special local alternative H 1 in (14) (14), for any x, y ∈ R, P [T SKK ≤ x, WΩ 1/2 − 2 log(p) + log{log(p)} ≤ y] → P [T SKK ≤ x] P [WΩ 1/2 − 2 log(p) + log{log(p)} ≤ y] . Similar to the one-sample problem, based on Theorems 6 and 7 and the results in Folks (1971, 1973), and by defining the minimal p-values test as T MAX ), we have the following relationship among the powers of different tests: β T (2) F C ≥ β T (2) min ≥ β T (2) SUM ,γ/2 + β T (2) MAX ,γ/2 − β T (2) SUM ,γ/2 β T (2) MAX ,γ/2 where β T (2) SUM ,γ is the power function of the sum-type test T SUM at significant value γ. So does β T (2) MAX ,γ . Simulation One-sample problem For the one-sample problem, we compare our Fisher's combination test T FC in (7) (abbreviated as FC) with • the sum-type test T SR in (4) proposed by Srivastava (2009) where k = 1, . . . , [p/2], and a i,j = 0 otherwise. Ω = D 1/2 Ω 1/2 Ω 1/2 D 1/2 and Σ = Ω −1 (f) Model 6 (non-sparse case): Σ * = σ * i,j where σ * i,i = 1, σ * i,j = 0.8 for 2(k − 1) + 1 i = j 2k, where k = 1, . . . , [p/2], and σ * i,j = 0 otherwise. Σ = D 1/2 Σ * D 1/2 + E + δI with δ = λ min D 1/2 Σ * D 1/2 + E +0.05, where E is a symmetric matrix with the support of the off-diagonal entries chosen independently according to the Bernoulli(0.3) distribution with the values of the non-zero entries drawn randomly from Unif (−0.2, 0.2). (g) Model 7 (non-sparse case): Σ * = σ * i,j where σ * i,i = 1 and σ * i,j = |i − j| −5 /2 for i = j Σ = D 1/2 Σ * D 1/2 (h) Model 8 (non-sparse case): Σ = D 1/2 (F + u 1 u 1 + u 2 u 2 + u 3 u 3 ) D 1/2 , where F = (f i,j ) is a p × p matrix with f i,i = 1, f i,i+1 = f i+1,i = 0 .5 and f i,j = 0 otherwise, and u i are orthonormal vectors for i = 1, 2, 3. For the generation of errors ε i = (ε i1 , · · · , ε ip ) , we consider three settings of ε ij 's: For power comparison, we consider µ = κ(1/σ 1/2 11 , · · · , 1/σ 1/2 mm , 0, · · · , 0) where κ is chosen as ||µ|| 2 = 0.5. Figures 1-3 (1) Normal distribution: ε ij i.i.d ∼ N (0, 1); (2) standardized t 5 distribution: ε ij i.i.d ∼ t(5)/ 5/3 (3) standardized mixture normal distribution: ε ij i.i.d ∼ {0.9N (0, 1) + 0.1N (0, 9)}/ √ 1.8 Two-sample problem We compare our Fisher's combination test T We generate X i = µ + Σ 1/2 z i and Y i = Σ 1/2 ξ i where Σ also generated from the eight models and z i , ξ i has the three scenarios as ε i in the above subsection. Under the null hypothesis, we set µ = 0. Under the alternative hypothesis, we set µ = Σ 1/2 θ where θ = (θ 1 / √ m, · · · , θ m / √ m, 0, · · · , 0), θ i ∼ 2B(1, 0.5) − 1 are independent binomial random variables. Table 3-4 report the empirical sizes of these tests with n 1 = n 2 = 60, p = 100, 200, 300. The SKK, MΩ 1/2 , FC and AD tests can control the empirical sizes very well in most settings of the covariance structure. However, we also observe that ( increasing from 1 to 20. When the number of nonzero elements inμ = θ is small, our proposed FC test is as powerful as the max-type tests. When the number of nonzero elements inμ = θ is large, the power of our FC test is exactly higher than that of all other tests. In real world, it is impossible to identify whether the data is sparse or not. Thus, the above results demonstrate that our FC test is good in any case. Appendix First, we restate the Central Limit Theorem for Linear Quadratic (Kelejian and Prucha (2001), Theorem 1, p.227). THEOREM 8 Consider the following linear quadratic form Q p = ε Aε + b ε = p i=1 p j=1 a ij ε i ε j + p i=1 b i ε i where {ε i , i = 1,and sup i p j=1 |a ij | < K. Also p −1 p i=1 |b i | 2+ε 0 < K for some ε 0 > 0. (iii): sup i E |ε i | 4+ε 0 < K for some ε 0 > 0. Then, assuming that p −1 Var (Q p ) ≥ c for some c > 0 Q p − E (Q p ) Var (Q p ) → d N (0, 1) Proof of Theorem 1 Define B i = {|z i | > l p } and A p (x) = z Az−tr(A) σ A ≤ x . We first proof the following important lemma. LEMMA 5.1 Under the assumption of Theorem 1, for each d ≥ 1, we have lim p→∞ H(d, p) ≤ 1 d! h d (y) < ∞(15) where H(d, p) . = 1≤i 1 <···<i d ≤p P (B i 1 · · · B i d ). And then, we have 1≤i 1 <···<i d ≤p |P (A p (x)B i 1 · · · B i d ) − P (A p (x)) · P (B i 1 · · · B i d )| → 0(16) as p → ∞. Proof. Because pP (B i ) → h(y), we have pP (B i ) < h(y) + for any > 0 as p → ∞. By the independence of z i , we have H(d, p) = 1≤i 1 <···<i d ≤p P (B i 1 · · · B i d ) = 1≤i 1 <···<i d ≤p d k=1 P (B i k ) ≤C d p {p −1 (h(y) + )} d ≤ 1 d! (h(y) + ) d So, by letting → 0, we have lim p→∞ H(d, p) ≤ 1 d! h d (y) < ∞ by assumption (i) in Theorem 1. Here we prove (15). Define z = (z 1 , z 2 ) where z 1 = (z 1 , · · · , z d ) and z 2 = (z d+1 , · · · , z p ). And A = A 11 A 12 A 21 A 22 So, z Az = z 1 A 1 z 1 + 2z 1 A 12 z 2 + z 2 A 2 z 2 . Next, we will show that P z 1 A 1 z 1 > σ A ≤ p −t for > 0. Because z i is sub-Gaussian random variables, there exist η > 0 and K > 0 such that E(exp(ηz 2 i )) ≤ K. Because λ max (A 1 ) ≤ λ max (A) < c, P z 1 A 1 z 1 > σ A ≤P cz 1 z 1 > σ A =P η d i=1 z 2 i > c −1 η σ A ≤ exp −c −1 η σ A E(e η d i=1 z 2 i ) = exp −c −1 η σ A {E(e ηz 2 i )} d ≤K d exp −c −1 η σ A By the assumption (iii), we have σ 2 A ≥ 2tr(A 2 ) ≥ 2c −2 p. So P z 1 A 1 z 1 > σ A ≤ K d exp − √ 2c −2 η p 1/2 .(17) Define A = Q ΛQ where Q = (q ij ) 1≤i,j≤p is an orthogonal matrix and Λ = diag {λ 1 , . . . , λ p } , λ i , i = 1, . . . , p are the eigenvalues of A. Note that 1≤j≤p a 2 ij is the i th diagonal element of A 2 = Q Λ 2 Q, we have 1≤j≤p a 2 ij = p l=1 q 2 li λ 2 l ≤ c 2 according to Assumption (iii). Next, define θ = 2η dc 2 σ 2 , we have P z 1 A 12 z 2 ≥ σ A ≤ exp (−θ σ A ) E exp(θz 1 A 12 z 2 ) = exp (−θ σ A ) E(e θ d i=1 p j=d+1 a ij z i z j ) ≤ exp (−θ σ A ) E(E(e θ p j=d+1 ( d i=1 a ij z i )z j |z 1 )) = exp (−θ σ A ) E p j=d+1 E(e (θ d i=1 a ij z i )z j |z 1 ) ≤ exp (−θ σ A ) E   p j=d+1 exp   σ 2 θ 2 2 d i=1 a ij z i 2     = exp (−θ σ A ) E   exp   σ 2 θ 2 2 p j=d+1 d i=1 a ij z i 2     ≤ exp (−θ σ A ) E exp dσ 2 θ 2 2 p j=d+1 d i=1 a 2 ij z 2 i ≤ exp (−θ σ A ) E exp dc 2 σ 2 θ 2 2 d i=1 z 2 i = exp (−θ σ A ) E exp η d i=1 z 2 i ≤K d exp (−θ σ A ) ≤ K d exp − √ 2c −1 θ p 1/2 So P z 1 A 12 z 2 ≥ σ A ≤K d exp − 4η dc 4 σ 2 p 1/2(18) Similarly, we also can prove that P (−z 1 ) A 12 z 2 ≥ σ A ≤K d exp − 4η dc 4 σ 2 p 1/2 (19) Let Θ p = z 1 A 1 z 1 + 2z 1 A 12 z 2 . P (|Θ p | > σ A ) ≤P z 1 A 1 z 1 > σ A /2 + P |z 1 A 12 z 2 | > σ A /4 ≤P z 1 A 1 z 1 > σ A /2 + P z 1 A 12 z 2 > σ A /8 + P −z 1 A 12 z 2 > σ A /8 So, by (17), (18) and (19), there exist a constant c > 0, P (|Θ p | > σ A ) ≤ K d exp(−c p 1/2 )(20)P (A p (x)B 1 · · · B d ) =P z 2 A 2 z 2 − tr(A) + Θ p σ A ≤ x, B 1 · · · B d ≤P z 2 A 2 z 2 − tr(A) + Θ p σ A ≤ x, |Θ p | ≤ σ A , B 1 · · · B d + P (|Θ p | > σ A ) ≤P z 2 A 2 z 2 − tr(A) σ A ≤ x + , B 1 · · · B d + K d exp(−c p 1/2 ) =P z 2 A 2 z 2 − tr(A) σ A ≤ x + P (B 1 · · · B d ) + K d exp(−c p 1/2 ) ≤ P z 2 A 2 z 2 − tr(A) σ A ≤ x + , |Θ p | ≤ σ A + P (|Θ p | > σ A ) P (B 1 · · · B d ) + K d exp(−c p 1/2 ) ≤P z 2 A 2 z 2 − tr(A) + Θ p σ A ≤ x + 2 P (B 1 · · · B d ) + 2K d exp(−c p 1/2 ) =P (A p (x + 2 )) P (B 1 · · · B d ) + 2K d exp(−c p 1/2 ) Similarly, we can prove that P (A p (x)B 1 · · · B d ) ≥ P (A p (x − 2 )) P (B 1 · · · B d ) − 2K d exp(−c p 1/2 ) So, we have |P (A p (x)B 1 · · · B d ) − P (A p (x)) · P (B 1 · · · B d )| ≤ ∆ p, · P (B 1 · · · B d ) + 2K d exp(−c p 1/2 )(21) where ∆ p, = |P (A p (x)) − P (A p (x + 2 ))| + |P (A p (x)) − P (A p (x − 2 ))| = P (A p (x + 2 )) − P (A p (x − 2 )) Obviously, the inequality (21) holds for all i 1 , · · · , i d . Thus, 1≤i 1 <···<i d ≤p |P (A p (x)B i 1 · · · B i d ) − P (A p (x)) · P (B i 1 · · · B i d )| ≤ 1≤i 1 <···<i d ≤p ∆ p, · P (B i 1 · · · B i d ) + 2K d exp(−c p 1/2 ) ≤ ∆ p, · H(d, p) + p d · 2K d exp(−c p 1/2 ) By Theorem 8, we have P (A p (x)) → Φ(x) as p → ∞. So ∆ p, → Φ(x + 2 ) − Φ(x − 2 ). By letting → 0, we have ∆ p, → 0. By (15), we have lim p→∞ H(d, p) < ∞. Additionally, p d · 2K d exp(−c p 1/2 ) → 0 as p → ∞. So we can obtain (16). Proof of Theorem 1 First, we show that P max 1≤i≤p |z i | ≤ l p (y) → F (y).(22) Because pP (|z i | > l p (y)) → h(y), we have h(y) − < pP (|z i | > l p (y)) < h(y) + for any > 0 as p → ∞. In fact, by the independence of z i , we have P max 1≤i≤p |z i | ≤ l p (y) =P (|z i | ≤ l p (y), 1 ≤ i ≤ p) = p i=1 {P (|z i | ≤ l p (y))} = p i=1 (1 − P (|z i | > l p (y))) ≤ (1 − (h(y) − )p −1 ) p → e −h(y)+ . Similarly, we have P max 1≤i≤p |z i | ≤ l p (y) = p i=1 (1 − P (|z i | > l p (y))) ≥ (1 − (h(y) + )p −1 ) p → e −h(y)− . So e −h(y)− ≤ P max 1≤i≤p |z i | ≤ l p (y) ≤ e −h(y)+ . By letting → 0, we obtain the result (22). Additionally, by Theorem 8, we know that P z Az − tr(A) σ A ≤ x → Φ(x)(23) To show (2), we only need to show that P z Az − tr(A) σ A ≤ x, max 1≤i≤p |z i | > l p (y) → Φ(x)(1 − F (y))(24) Recall the notations in Lemma 5.1, we have P z Az − tr(A) σ A ≤ x, max 1≤i≤p |z i | > l p = P p i=1 A p B i .(25) Here the notation A p B i stands for A p ∩ B i and we brief A p (x) as A p . From the inclusion-exclusion principle, P p i=1 A p B i ≤ 1≤i 1 ≤p P (A p B i 1 ) − 1≤i 1 <i 2 ≤p P (A p B i 1 B i 2 ) + · · · + 1≤i 1 <···<i 2k+1 ≤p P (A p B i 1 · · · B i 2k+1 )(26) and P p i=1 A p B i ≥ 1≤i 1 ≤p P (A p B i 1 ) − 1≤i 1 <i 2 ≤p P (A p B i 1 B i 2 ) + · · · − 1≤i 1 <···<i 2k ≤p P (A p B i 1 · · · B i 2k )(27) for any integer k ≥ 1. Define H(p, d) = 1≤i 1 <···<i d ≤p P (B i 1 · · · B i d ) for d ≥ 1. From (15) we know lim d→∞ lim sup p→∞ H(p, d) = 0.(28) Set ζ(p, d) = 1≤i 1 <···<i d ≤p P (A p B i 1 · · · B i d ) − P (A p ) · P (B i 1 · · · B i d ) for d ≥ 1. By Lemma 5.1, lim p→∞ ζ(p, d) = 0(29) for each d ≥ 1. The assertion (26) implies that P p i=1 A p B i ≤ P (A p ) 1≤i 1 ≤p P (B i 1 ) − 1≤i 1 <i 2 ≤p P (B i 1 B i 2 ) + · · · − 1≤i 1 <···<i 2k ≤p P (B i 1 · · · B i 2k ) + 2k d=1 ζ(p, d) + H(p, 2k + 1) ≤ P (A p ) · P p i=1 B i + 2k d=1 ζ(p, d) + H(p, 2k + 1),(30) where the inclusion-exclusion formula is used again in the last inequality, that is, P p i=1 B i ≥ 1≤i 1 ≤p P (B i 1 ) − 1≤i 1 <i 2 ≤p P (B i 1 B i 2 ) + · · · − 1≤i 1 <···<i 2k ≤p P (B i 1 · · · B i 2k ) for all k ≥ 1. By the definition of l p and (22), P p i=1 B i → 1 − F (y) as p → ∞. By (23), P (A p ) → Φ(x) as p → ∞. From (25), (29) and (30), by fixing k first and sending p → ∞ we obtain that lim sup p→∞ P z Az − tr(A) σ A ≤ x, max 1≤i≤p |z i | > l p ≤ Φ(x) · [1 − F (y)] + lim p→∞ H(p, 2k + 1). Now, by letting k → ∞ and using (28) we have lim sup p→∞ P z Az − tr(A) σ A ≤ x, max 1≤i≤p |z i | > l p ≤ Φ(x) · [1 − F (y)].(31) By applying the same argument to (27), we see that the counterpart of (30) becomes P p i=1 A p B i ≥ P (A p ) 1≤i 1 ≤p P (B i 1 ) − 1≤i 1 <i 2 ≤p P (B i 1 B i 2 ) + · · · + 1≤i 1 <···<i 2k−1 ≤p P (B i 1 · · · B i 2k−1 ) + 2k−1 d=1 ζ(p, d) − H(p, 2k) ≥ P (A p ) · P p i=1 B i + 2k−1 d=1 ζ(p, d) − H(p, 2k). where in the last step we use the inclusion-exclusion principle such that P p i=1 B i ≤ 1≤i 1 ≤p P (B i 1 ) − 1≤i 1 <i 2 ≤p P (B i 1 B i 2 ) + · · · + 1≤i 1 <···<i 2k−1 ≤p P (B i 1 · · · B i 2k−1 ) for all k ≥ 1. Review (25) and repeat the earlier procedure to see lim inf p→∞ P z Az − tr(A) σ A ≤ x, max 1≤i≤p |z i | > l p ≥ Φ(x) · [1 − F (y)] by sending p → ∞ and then sending k → ∞. This and (31) yield (24). The proof is completed. Proof of Theorem 2 Taking the same procedure as Theorem 4 in Cai et al. (2014), we have P (M Ω 1/2 − 2 log(p) + log log(p) ≤ x) → exp − 1 √ π exp − x 2(32) where M Ω 1/2 = max 1≤i≤p ν 2 i , where ν i = 1 √ n n k=1 ε ki . Let t = √ nX we have ||Ω 1/2 t|| ∞ − ||Ω 1/2 t|| ∞ ≤ ||(Ω 1/2 − Ω 1/2 )t|| ∞ ≤ ||Ω 1/2 t|| ∞ ||(Ω 1/2 Ω −1/2 − I p )|| L 1 By (32), we have ||Ω 1/2 t|| ∞ = O p (log(p)). By condition (C3), we have ||( Ω 1/2 Ω −1/2 − I p )|| L 1 = o p (log −1 (p)). So ||Ω 1/2 t|| ∞ − ||Ω 1/2 t|| ∞ = o p (1). Here we obtain the result. Proof of Theorem 3 According to the proof of Theorem 3.1 in Srivastava (2009), we have T SR = nX D −1X − p 2tr(R 2 ) + o p (1). And by the proof of Theorem 2, we have MΩ 1/2 = M Ω 1/2 + o p (1). So, by Lemma 7.10 in Feng et al. (2022), we only need to show that P H 0 nX D −1X − p 2tr(R 2 ) ≤ x, M Ω 1/2 − 2 log(p) + log{log(p)} y → Φ(x)F (y).(33) By Theorem 1, we only need to show that ν i is independent sub-Gaussian random variables. Obviously, E exp λ √ n n k=1 ε ki = E n e λ √ n ε ki ≤ E(1 + λ √ n ε ki + λ 2 n ε 2 ki ) + o(n −1 ) n ≤ e Cλ 2 for large enough n and some positive constant C. So we obtain the result. Proof of Theorem 4 Define ν M be the sub-vector of ν = (ν 1 , · · · , ν p ) corresponding to i ∈ M. So does ν M c . And T SR = n(X − µ) D −1 (X − µ) − p 2tr(R 2 ) + δ D −1 δ p 2tr(R 2 ) + o p (1) = 1 2tr(R 2 ) (ν M A M ν M ) + 2 2tr(R 2 ) (ν M A MM c ν M c ) + 1 2tr(R 2 ) (ν M c A M c ν M c − p) + δ D −1 δ p 2tr(R 2 ) + o p (1) Additionally, by the proof of Lemma 5.1 and m = o(p 1/2 ), we have P ν M A MM c ν M c ≥ 2tr(R 2 ) ≤ K m exp(−c p 1/2 ) → 0 P ν M A M ν M ≥ 2tr(R 2 ) ≤ K m exp(−c p 1/2 ) → 0 where K and C are two positive constant which dependent on . So T SR = 1 2tr(R 2 ) (ν M c A M c ν M c − p) + δ D −1 δ p 2tr(R 2 ) + o p (1) Similar to the proof of Theorem 2, we have MΩ 1/2 = M Ω 1/2 + o p (1) where M Ω 1/2 = max 1≤i≤p (ν i +μ i ) 2 = max{max i∈M (ν i +μ i ) 2 , max i∈M c ν 2 i }. Under Condition (C1), we have ν M c A M c ν M c is independent of max i∈M (ν i +μ i ) 2 . By Theorem 3, ν M c A M c ν M c is asymptotically independent of max i∈M c ν 2 i . So we obtain that T SR is asymptotically independent of MΩ 1/2 . Proof of Theorem 5,6 and 7 Proof of Theorem 5 Similar to the proof of Theorem 2, we have WΩ 1/2 = W Ω 1/2 + o p (1). By Theorem 1 in Cai et al. (2014), we have P (W Ω 1/2 − 2 log(p) + log log(p) ≤ x) → exp − 1 √ π exp − x 2 So we obtain the result. Proof of Theorem 6 According to the proof of Theorem 1.1 in Srivastava et al. (2013), we have T SKK = 1 2tr(R 2 ) u Au − p + o p (1)(34) where A is defined in condition (C5) and u = n 1 n 2 n 1 +n 2 1 n 1 n 1 l=1 ε 1l − 1 n 2 n 2 l=1 ε 2l . Similar to the proof of Theorem 3, we can also prove that u is sub-Gaussian random variables. So by Theorem 8, we can obtain the result. Proof of Theorem 7 The proof is similar to the proof of Theorem 4. So we omit it here. Error (1) Error (2) Error ( Error (1) Error (2) Error ( Error (1) Error (2) Error ( Error (1) Error (2) Error ( established a new test by thresholding two sum-type tests based on the sample means and then maximizing over a range of thresholding levels. The other effort is made by Fan et al. (2015), which proposed a linear combination between the standard Wald statistic and a power enhancement component as the test statistic, where the power enhancement component equals zero with probability converging to one under null and diverges in probability under some specific regions of alternatives. In contrast, the max-type test statistics proposed by Cai et al. (2014) might be most powerful under the sparse alternative. For convenience, we write the expression of the max-type test statistic for the onesample mean test as follows. For a given invertible p×p matrix A, the null hypothesis H 0 : µ = 0 is equivalent to H 0 : Aµ = 0. Set δ A = δ A 1 , . . . , δ A p := AX . Denote the sample covariance matrix of AX by B = (b i,j ) and define the test statistic ( 2016 ) 2016and He et al. (2021), our result relies on the weaker assumption on the covariance structure and allows for more possibilities for the assumptions about the sample distributions. .. 2 2= 1 − Φ (T SKK )denote the p-values with respect to the test statistics T Assume the same conditions as in Theorem 6, then we have T(2) FC d → χ 2 4 as n, p → ∞. max-type tests M Ip , MΩ 1/2 and MΩ based on (5) (abbreviated as MAX1, MAX2 and MAX3, respectively);• the higher criticism test T HC byZhong et al. (2013) (abbreviated as HC);• the power enhancement test J byFan et al. (2015) (abbreviated as PE).The specific models for the covariance structure are following the settings inCai et al. (2014).For convenience, we collected them as follows. Let D = (d i,j ) be a diagonal matrix with diagonal elements d i,i = Unif(1, 3) for i = 1, . . . , p. Denote by λ min (A) the minimum eigenvalue of a symmetric matrix A. 1 (block diagonal Ω): Σ = (σ i,j ) where σ i,i = 1, σ i,j = 0.8 for 2(k − 1) + 1 i = j 2k where k = 1,. . . , [p/2] and σ i,j = 0 otherwise. (b) Model 2 ('bandable' Σ): Σ = (σ i,j ) where σ i,j = 0.6 |i−j| for 1 i, j p. ( c ) cModel 3 (banded Ω) : Ω = (ω i,j ) where ω i,i = 2 for i = 1, . . . , p, ω i,i+1 = 0.8 for i = 1, . . . , p− 1, ω i,i+2 = 0.4 for i = 1, . . . , p − 2, ω i,i+3 = 0.4 for i = 1, . . . , p − 3, ω i,i+4 = 0.2 for i = 1, . . . , p − 4, ω i,j = ω j,i for i, j = 1, . . . , p and ω i,j = 0 otherwise. (d) Model 4 (sparse Σ): Ω = (ω i,j ) where ω i,j = 0.6 |i−j| for 1 i, j p.Σ = D 1/2 Ω −1 D 1/2 .(e) Model 5 (sparse Σ): Ω 1/2 = (a i,j ) where a i,i = 1, a i,j = 0.8 for 2(k − 1) + 1 i = j 2k, ( 2 ) 2FC in (13) (abbreviated as FC) with • the sum-type test T SKK in (10) proposed by Srivastava et al. (2013) (abbreviated as SKK); • the max-type tests M Ip , MΩ 1/2 and MΩ proposed by Cai et al. (2014) (abbreviated as MAX1, MAX2 and MAX3, respectively); • the higher criticism test T HC by Chen et al. (2019) (abbreviated as HC); • the adaptive test T AD by Xu et al. (2016) (abbreviated as AD). i) the empirical sizes of HC are a little smaller than the nominal level under the Models 1, 3, 4, 6, 7 and 8 of the covariance structure; (ii) the empirical sizes of M Ip are a little smaller than the nominal level under the Models 1 and 5; (iii) the empirical sizes of MΩ are a little higher than the nominal level under the Models 2, 3 an 4.Figures 4-6 reports the power curves of each test with n 1 = n 2 = 60, p = 100 for different settings of the covariance structure. For the settings of error distribution, Figures 4, 5 and 6 report the power curves for the normal distribution, t(5) distribution and mixture normal distribution, respectively. In general, the powers of the SKK and HC tests are always staying around 0.5 under different choices of m (ranging from 1 to 20) while the powers of other tests tend to decrease as m 2, . . . , p} are real valued random variables, and a ij and b i denote real valued coefficients of the quadratic and linear forms. Suppose the following assumptions hold: (i): ε i , for i = 1, 2, . . . , p, have zero means and are independently distributed across i. (ii): A is symmetric let A M , A M c be the sub-matrix of A corresponding to M, M c , respectively. And A MM c is the sub-matrix between the vector ν M and ν M c . According to the proof of Theorem 4.1 in Srivastava (2009), we have Figure 1 :Figure 2 :Figure 3 : 123Power of tests with different numbers of nonzero alpha at n = 120, p = 200 with normal errors. (MAX1 means M Ip ; MAX2 means MΩ 1/2 ; MAX3 means MΩ.) Power of tests with different numbers of nonzero alpha at n = 120, p = 200 with t(5) errors.(MAX1 means M Ip ; MAX2 means MΩ 1/2 ; MAX3 means MΩ.) Power of tests with different numbers of nonzero alpha at n = 120, p = 200 with mixture normal errors.(MAX1 means M Ip ; MAX2 means MΩ 1/2 ; MAX3 means MΩ.) Figure 4 :Figure 5 :Figure 6 : 456Power of tests with different numbers of nonzero alpha at n 1 = n 2 = 120, p = 100 with normal errors.(MAX1 means M Ip ; MAX2 means MΩ 1/2 ; MAX3 means MΩ.) Power of tests with different numbers of nonzero alpha at n 1 = n 2 = 120, p = 100 with t(5) errors.(MAX1 means M Ip ; MAX2 means MΩ 1/2 ; MAX3 means MΩ.) Power of tests with different numbers of nonzero alpha at n 1 = n 2 = 120, p = 100 with mixture normal errors.(MAX1 means M Ip ; MAX2 means MΩ 1/2 ; MAX3 means MΩ.) enables us to obtain the asymptotic independence between the sum-type testT SKK and the max-type test WΩ 1/2 under the alternative hypothesis. The main result is stated in the following theorem. THEOREM 7 Suppose that conditions (C1 ), (C2)-(C5) hold. Under the special local alternative H 1 stated in Table 1 - 12 report the empirical sizes of these tests with n = 120, p = 100, 200, 300. We found that SR, M Ip , MΩ 1/2 , MΩ and FC can control the empirical sizes very well. The empirical sizes of HC test are a little smaller than the nominal level. However, PE test can not control the empirical sizes in most case. So we do not compare it in the alternative hypothesis. report the power curves of each test with n = 120, p = 200 for different settings of the covariance structure. For the settings of error distribution, Figures 1, 2 and 3 report the power curves for the normal distribution, t(5) distribution and mixture normal distribution, respectively. In general, the powers of the SR and HC tests are always very close to 0.25 under different choices of m (ranging from 1 to 20). In contrast, under the most settings of the covariance structure, the powers of M Ip , MΩ 1/2 , MΩ and FC tests tends to decrease as m increasing from 1 to 20 (except that under the Models 3 and 4, the powers of MΩ 1/2 , MΩ and FC tests are very close to 1 for difference choices of m). It is natural because the max-type tests can work better for the sparse case than the non-sparse case. In most scenarios, our proposed FC test is as powerful as the max-type tests when the number of variables with nonzero means is small and is more powerful than the max-type tests when the number of variables with nonzero means is large. This indicates that our FC test can work well in any case, which implies that our FC test is robust to the real data because it is not possible to tell if a dataset is sparse or not. Table 1 : 1Sizes of tests under model 1-4 in one-sample test. Table 2 : 2Sizes of tests under model 5-8 in one-sample test. Table 3 : 3Sizes of tests under model 1-4 in two-sample test. Table 4 : 4Sizes of tests under model 5-8 in two-sample test. The joint limiting distribution of sums and maxima of stationary sequences. C Anderson, K Turkman, Journal of applied probability. 281C. Anderson and K. Turkman. The joint limiting distribution of sums and maxima of stationary sequences. Journal of applied probability, 28(1):33-44, 1991. Limiting joint distributions of sums and maxima in a statistical context. C Anderson, K Turkman, Theory of Probability & Its Applications. 37C. Anderson and K. Turkman. Limiting joint distributions of sums and maxima in a statistical context. Theory of Probability & Its Applications, 37(2):314-316, 1993. Sums and maxima of stationary sequences with heavy tailed distributions. C Anderson, K Turkman, Sankhyā: The Indian Journal of Statistics, Series A. C. Anderson and K. Turkman. Sums and maxima of stationary sequences with heavy tailed distri- butions. Sankhyā: The Indian Journal of Statistics, Series A, pages 1-10, 1995. An introduction to multivariate statistical analysis. T W Anderson, WileyNew YorkT. W. Anderson. An introduction to multivariate statistical analysis. Wiley New York, 1962. Effect of high dimension: by an example of a two sample problem. Z Bai, H Saranadasa, Statistica Sinica. Z. Bai and H. Saranadasa. Effect of high dimension: by an example of a two sample problem. Statistica Sinica, pages 311-329, 1996. Two-sample test of high dimensional means under dependence. T T Cai, W Liu, Y Xia, Journal of the Royal Statistical Society: Series B (Statistical Methodology). 762T. T. Cai, W. Liu, and Y. Xia. Two-sample test of high dimensional means under dependence. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 76(2):349-372, 2014. A two-sample test for high-dimensional data with applications to geneset testing. S X Chen, Y.-L Qin, The Annals of Statistics. 382S. X. Chen and Y.-L. Qin. A two-sample test for high-dimensional data with applications to gene- set testing. The Annals of Statistics, 38(2):808-835, 2010. Two-sample and anova tests for high dimensional means. S X Chen, J Li, P.-S Zhong, The Annals of Statistics. 473S. X. Chen, J. Li, and P.-S. Zhong. Two-sample and anova tests for high dimensional means. The Annals of Statistics, 47(3):1443-1474, 2019. The sum and the maximum of iid random variables. T L Chow, J L Teugels, Proceedings of the 2nd Prague Symposium on Asymptotic Statistics. the 2nd Prague Symposium on Asymptotic Statistics45T. L. Chow and J. L. Teugels. The sum and the maximum of iid random variables. In Proceedings of the 2nd Prague Symposium on Asymptotic Statistics, volume 45, pages 394-403, 1978. Multivariate statistics: a vector space approach. M L Eaton, John Wiley & SonsNew YorkM. L. Eaton. Multivariate statistics: a vector space approach. John Wiley & Sons, New York, 1983. Power enhancement in high-dimensional cross-sectional tests. J Fan, Y Liao, J Yao, Econometrica. 834J. Fan, Y. Liao, and J. Yao. Power enhancement in high-dimensional cross-sectional tests. Econo- metrica, 83(4):1497-1541, 2015. Asymptotic independence of the sum and maximum of dependent random variables with applications to high-dimensional tests. L Feng, T Jiang, X Li, B Liu, 2022Working PaperL. Feng, T. Jiang, X. Li and B. Liu. Asymptotic independence of the sum and maximum of dependent random variables with applications to high-dimensional tests. Working Paper, 2022. Asymptotically independent u-statistics in high-dimensional testing. Y He, G Xu, C Wu, W Pan, The Annals of Statistics. 491Y. He, G. Xu, C. Wu, and W. Pan. Asymptotically independent u-statistics in high-dimensional testing. The Annals of Statistics, 49(1):154-181, 2021. On the asymptotic joint distribution of the sum and maximum of stationary normal random variables. H.-C Ho, T Hsing, Journal of applied probability. 331H.-C. Ho and T. Hsing. On the asymptotic joint distribution of the sum and maximum of stationary normal random variables. Journal of applied probability, 33(1):138-145, 1996. Asymptotic distribution of sum and maximum for gaussian processes. H.-C Ho, W P Mccormick, Journal of applied probability. 364H.-C. Ho and W. P. McCormick. Asymptotic distribution of sum and maximum for gaussian processes. Journal of applied probability, 36(4):1031-1044, 1999. The Generalization of Student's Ratio. H Hotelling, The Annals of Mathematical Statistics. 23H. Hotelling. The Generalization of Student's Ratio. The Annals of Mathematical Statistics, 2(3): 360 -378, 1931. A note on the asymptotic independence of the sum and maximum of strongly mixing stationary random variables. The Annals of Probability. T Hsing, T. Hsing. A note on the asymptotic independence of the sum and maximum of strongly mixing stationary random variables. The Annals of Probability, pages 938-947, 1995. On the asymptotic distribution of the moran i test statistic with applications. H H Kelejian, I R Prucha, Journal of Econometrics. 1042H. H. Kelejian and I. R. Prucha. On the asymptotic distribution of the moran i test statistic with applications. Journal of Econometrics, 104(2):219-257, 2001. Asymptotic optimality of fisher's method of combining independent tests. R C Littell, J L Folks, Journal of the American Statistical Association. 66336R. C. Littell and J. L. Folks. Asymptotic optimality of fisher's method of combining independent tests. Journal of the American Statistical Association, 66(336):802-806, 1971. Asymptotic optimality of fisher's method of combining independent tests ii. R C Littell, J L Folks, Journal of the American Statistical Association. 68341R. C. Littell and J. L. Folks. Asymptotic optimality of fisher's method of combining independent tests ii. Journal of the American Statistical Association, 68(341):193-194, 1973. Asymptotic distribution for the sum and maximum of gaussian processes. W Mccormick, Y Qi, Journal of applied probability. 374W. McCormick and Y. Qi. Asymptotic distribution for the sum and maximum of gaussian pro- cesses. Journal of applied probability, 37(4):958-971, 2000. Aspects of multivariate statistical theory. R J Muirhead, John Wiley & SonsR. J. Muirhead. Aspects of multivariate statistical theory. John Wiley & Sons, 2009. On the joint limiting distribution of sums and maxima of stationary normal sequence. Z Peng, S Nadarajah, Theory of Probability & Its Applications. 47Z. Peng and S. Nadarajah. On the joint limiting distribution of sums and maxima of stationary normal sequence. Theory of Probability & Its Applications, 47(4):706-709, 2003. A test for the mean vector with fewer observations than the dimension under non-normality. M S Srivastava, Journal of Multivariate Analysis. 1003M. S. Srivastava. A test for the mean vector with fewer observations than the dimension under non-normality. Journal of Multivariate Analysis, 100(3):518-532, 2009. A test for the mean vector with fewer observations than the dimension. M S Srivastava, M Du, Journal of Multivariate Analysis. 993M. S. Srivastava and M. Du. A test for the mean vector with fewer observations than the dimension. Journal of Multivariate Analysis, 99(3):386-402, 2008. A two sample test in high dimensional data. M S Srivastava, S Katayama, Y Kano, Journal of Multivariate Analysis. 114M. S. Srivastava, S. Katayama, and Y. Kano. A two sample test in high dimensional data. Journal of Multivariate Analysis, 114:349-358, 2013. An adaptive two-sample test for high-dimensional means. G Xu, L Lin, P Wei, W Pan, Biometrika. 1033G. Xu, L. Lin, P. Wei, and W. Pan. An adaptive two-sample test for high-dimensional means. Biometrika, 103(3):609-624, 2016. Tests alternative to higher criticism for high-dimensional means under sparsity and column-wise dependence. P.-S Zhong, S X Chen, M Xu, The Annals of Statistics. 416P.-S. Zhong, S. X. Chen, and M. Xu. Tests alternative to higher criticism for high-dimensional means under sparsity and column-wise dependence. The Annals of Statistics, 41(6):2820-2851, 2013.
{'fraction_non_alphanumeric': 0.0901669758812616, 'fraction_numerical': 0.04289424860853432, 'mean_word_length': 3.2232233800830525, 'pattern_counts': {'":': 0, '<': 51, '<?xml version=': 0, '>': 41, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 84, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': "This paper establishes the asymptotic independence between the quadratic form and maximum of a sequence of independent random variables. Based on this theoretical result, we find the asymptotic joint distribution for the quadratic form and maximum, which can be applied into the high-dimensional testing problems. By combining the sum-type test and the max-type test, we propose the Fisher's combination tests for the one-sample mean test and two-sample mean test. Under this novel general framework, several strong assumptions in existing literature have been relaxed. Monte Carlo simulation has been done which shows that our proposed tests are strongly robust to both sparse and dense data.", 'arxivid': '2204.08628', 'author': ['Dachuan Chen \nNankai University\n\n', 'Long Feng \nNankai University\n\n'], 'authoraffiliation': ['Nankai University\n', 'Nankai University\n'], 'corpusid': 248239869, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 19847, 'n_tokens_neox': 17092, 'n_words': 10678, 'pdfsha': 'be50acca15e1e5f629caa99a49eeec2f1c932d9d', 'pdfurls': ['https://arxiv.org/pdf/2204.08628v1.pdf'], 'title': ['Asymptotic Independence of the Quadratic form and Maximum of Independent Random Variables with Applications to High-Dimensional Tests', 'Asymptotic Independence of the Quadratic form and Maximum of Independent Random Variables with Applications to High-Dimensional Tests'], 'venue': []}
arxiv
Transportation cost inequalities for stochastic reaction diffusion equations on the whole line R May 2023 Yue Li Shijie Shang Tusheng Zhang Transportation cost inequalities for stochastic reaction diffusion equations on the whole line R May 2023and Phrases: Transportation cost inequalitiesstochastic partial differential equationsreaction diffusion equationsconcentration of measuremoment estimates for stochastic convolutions AMS Subject Classification: Primary 60H15; Secondary 35R60 In this paper, we established quadratic transportation cost inequalities for solutions of stochastic reaction diffusion equations driven by multiplicative space-time white noise on the whole line R. Since the space variable is defined on the unbounded domain R, the inequalities are proved under a weighted L 2 -norm and a weighted uniform metric in the so called L 2 tem , C tem spaces. The new moments estimates of the stochastic convolution with respect to space-time white noise play an important role. In addition, the transportation cost inequalities are also obtained for the stochastic reaction diffusion equations with random initial values. Introduction Let (E, d) be a metric space equipped with the Borel σ-field B. Let P(E) be the class of all the probability measures on E. The L p -Wasserstein distance between µ, ν ∈ P(E) is defined by W p (ν, µ) := inf E×E d(x, y) p π(dx, dy) 1 p , where the infimum is taken over all probability measures π on the product space E × E with marginals µ and ν. In the study of Monge-Kontorovich optimal transportation problem, this distance is interpreted as the minimal cost to transport distribution ν into µ when the transportation cost from x to y is measured by the distance d on E. In applications, to estimate the unknown optimal transportation, we usually need some other quantities to control it, for instance, the relative entropy. The relative entropy(or Kullback information) of ν with respect to µ is defined by H(ν|µ) := E log dν dµ dν, if ν is absolutely continuous with respect to µ, and +∞ otherwise. Definition 1.1. We say that the measure µ satisfies the L p -transportation cost inequality if there exists a constant C > 0 such that for all probability measures ν, W p (ν, µ) ≤ 2CH(ν|µ).(1) The case p = 2 is referred to as the quadratic transportation cost inequality, also Talagrand inequality. Consider the following stochastic reaction diffusion equation:    ∂u ∂t (t, x) = 1 2 ∂ 2 u ∂x 2 (t, x) + b(u(t, x)) + σ(u(t, x)) ∂ 2 W ∂t∂x (t, x), t ∈ [0, ∞), x ∈ R, u(0, x) = u 0 (x), x ∈ R,(2) where the initial value u 0 is a deterministic function, ∂ 2 W ∂t∂x (t, x) is a space-time white noise on some filtrated probability space (Ω, F, {F t } t≥0 , P), here {F t } t≥0 is the filtration satisfying the usual conditions generated by the white noise. The coefficients b(·), σ(·) : R → R are deterministic measurable functions. The purpose of this paper is to establish the quadratic transportation cost inequality for the law of the solution of the stochastic reaction diffusion equation (2) on the so called L 2 tem , C tem spaces (defined below). In 1996, Talagrand [22] established the quadratic transportation cost inequality with sharp constant C = 1 for Gaussian measures on R k . Since then, transportation cost inequalities and their applications have been widely studied. The transportation cost inequalities also have close connections with other functional inequalities like log-Sobolev inequality, Poincaré inequality, the concentration of measure phenomenon, optimal transport problem, and large deviations, see [12,15,14,22,23,1,2,7,9,24,26] and references therein. We note that the transportation cost inequality depends on the underlying metric. The stronger the metric, the stronger the inequality. The Talagrand type transportation cost inequality in [22] was generalized by D. Feyel and A. S.Üstünel [8] to the framework of the abstract Wiener space. It has also been established for the laws of some stochastic processes on the path spaces. Considering diffusion processes, H. Djellout, A. Guillin and L. Wu [6] obtained the quadratic transportation cost inequality under the L 2 ([0, T ], R k )-metric for stochastic differential equations(SDEs) on R k . Later, the quadratic transportation cost inequality under some uniform metric was obtained in [29] also for SDEs on R k . There are now many articles on transportation cost inequalities for various models, see e.g. [30] for stochastic evolution equations on some Hilbert space H under the metric L 2 ([0, T ], H), [16] for multidimensional semi-martingales including time inhomogeneous diffusions, [28,13,17] for SDEs with pure jump, Lévy or fractional noises. Concerning stochastic reaction diffusion equations on bounded domains, there exist several works on the quadratic transportation cost inequality. Let us recall some of them. B. Boufoussi and S. Hajji [3] obtained the quadratic transportation cost inequality for stochastic reaction diffusion equations driven by space-time white noise and driven by fractional noise under the L 2 -metric: d 2 (ξ, γ) = ( T 0 1 0 |ξ(t, x) − γ(t, x)| 2 dtdx) 1 2 . In [11], the authors established the quadratic transportation cost inequality for stochastic reaction diffusion equations under the L 2 -metric d 2 (ξ, γ) = ( T 0 1 0 |ξ(t, x) − γ(t, x)| 2 dtdx) 1 2 and under the uniform metric ||ξ − γ|| = sup 0≤t≤T,0≤x≤1 |ξ(t, x) − γ(t, x)| in the case of additive noise. In [19], the authors established the quadratic transportation cost inequality for stochastic reaction diffusion equations driven by multiplicative noise under the uniform metric ||ξ − γ||. Later the authors in [18] obtained the quadratic transportation cost inequality under the uniform norm for stochastic reaction diffusion equations driven by time-white and space-colored Gaussian noise. The purpose of this paper is to establish the quadratic transportation cost inequality for stochastic reaction diffusion equations driven by multiplicative space-time white noise defined on the whole real line R. Since the space domain is unbounded, we are forced to work on the so called L 2 tem and C tem spaces (see the definitions (4), (5) below ). To obtain the quadratic transportation cost inequality, as a crucial step we need to establish the precise lower order (more difficult than the higher order) moment estimates of stochastic convolution with respect to the space-time white noise under the weighted L 2 -norm and the weighted supremum norm, which is of independent interest. The rest of the paper is organized as follows. In Section 2, we recall the setup for stochastic reaction diffusion equations and state the main result of the paper. In Section 3 we establish the new moment estimates for stochastic convolutions with respect to the spacetime white noise. Section 4 and Section 5 contain the proofs of the quadratic transportation cost inequalities. In Section 6 we prove the quadratic transportation cost inequality for stochastic reaction diffusion equations with random initial conditions. Some inequalities regarding heat kernel are presented in the Appendix. Convention on constants. Throughout the paper the constants denoted by c 1 , c 2 , . . . are positive and their precise values are not important. The dependence of constants on parameters if needed will be indicated, e.g., C T , Θ λ . Statement of the main results In this section, we will recall the setup of stochastic reaction diffusion equations driven by space-time white noise and state the main result of the paper. Let C 2 0 (R) denote the set of continuous functions with compact supports whose second derivatives are also continuous. We say that an adapted, continuous random field {u(t, x) : (t, x) ∈ [0, ∞) × R} is a solution to the stochastic partial differential equation (SPDE) (2) if for t ≥ 0, φ ∈ C 2 0 (R) R u(t, x)φ(x)dx = R u 0 (x)φ(x)dx + 1 2 t 0 ds R u(s, x)φ ′′ (x)dx + t 0 ds R b(u(s, x))φ(x)dx + t 0 R σ(u(s, x))φ(x)W (ds, dx), P-a.s. It was shown in [21] that u is a solution to SPDE (2) if and only if u satisfies the following integral equation u(t, x) =P t u 0 (x) + t 0 R p t−s (x, y)b(u(s, y))dyds + t 0 R p t−s (x, y)σ(u(s, y))W (ds, dy), P-a.s.(3) where p t (x, y) : = 1 √ 2πt e − (x−y) 2 2t , and {P t } t≥0 is the corresponding heat semigroup on R. Next we introduce some spaces used in the paper. For given λ > 0, the space L 2 λ := f : R → R is measurable and R |f (x)| 2 e −2λ|x| dx < ∞ equipped with the inner product f, g L 2 λ = R |f (x)g(x)|e −2λ|x| dx, f, g ∈ L 2 λ is a Hilbert space, and denote the induced norm by · L 2 λ . Define E λ := f ∈ C(R) : sup x∈R |f (x)|e −λ|x| < ∞ , for given λ > 0, and equipped with the metric ̺ λ (f, g) := sup x∈R |f (x) − g(x)|e −λ|x| , f, g ∈ E λ , E λ is a Polish space. We also recall that the L 2 tem and C tem spaces are defined by L 2 tem : = f ∈ C(R) : R |f (x)| 2 e −2λ|x| dx < ∞ for all λ > 0 ,(4)C tem : = f ∈ C(R) : sup x∈R |f (x)|e −λ|x| < ∞ for all λ > 0 ,(5) and endowed with the metrics respectively defined by ρ(f, g) : = ∞ n=1 1 2 n min 1, f − g L 2 1/n , f, g ∈ L 2 tem , ̺(f, g) : = ∞ n=1 1 2 n min 1, ̺ 1/n (f, g) , f, g ∈ C tem . Then f n → f in L 2 tem iff for every λ > 0, f n − f L 2 λ → 0, and f n → f in C tem iff for every λ > 0, ̺ λ (f n , f ) → 0. It is known that C tem and L 2 tem are Polish spaces. Let (E, d) be a Polish space. Consider a continuous Markov process on E with a given transition kernel P t (x, ·). For T > 0 and µ ∈ P(E), let P µ denote the distribution of the Markov process with initial distribution µ, i.e., P µ is the unique probability measure on the path space C([0, T ], E) equipped with the metricd(ϕ, ψ) := sup t∈[0,T ] d (ϕ(t), ψ(t)) , When µ = δ x , the Dirac measure at x ∈ E, we simply denote P µ = P x . Then P µ = E P x µ(dx), µ ∈ P(E). We introduce the following hypotheses. (H1) There exists a constant L b such that for all x, y ∈ R, |b(x) − b(y)| ≤ L b |x − y|. (H2) There exist constants K σ and L σ such that for all x, y ∈ R, (H2(a)) |σ(x)| ≤ K σ , (H2(b)) |σ(x) − σ(y)| ≤ L σ |x − y|. It is well known that under the hypotheses (H1) and (H2), SPDE (2) admits a unique random field solution u(t, x). In fact, for the existence and uniqueness the diffusion coefficient σ(·) does not need to be bounded, the stronger assumption (H2) is needed for proving the transportation cost inequality. The proof of Proposition 2.2 is similar to that of Proposition 2.1. We omit the details. Here are the main results. Using the approach in [27] we derive the following transportation inequality for the stochastic reaction diffusion equation with random initial values. Corollary 2.5. Suppose the hypotheses (H1) and (H2) hold, and µ ∈ P(L 2 tem ).Then W 2 (Q, P µ ) 2 ≤ c 1 H(Q|P µ ), Q ∈ P(C([0, T ], L 2 tem )) holds for some constant c 1 > 0 if and only if W 2 (ν, µ) 2 ≤c 1 H(ν|µ), ν ∈ P(C tem ) holds for some constantc 1 > 0. Corollary 2.6. Suppose the hypotheses (H1) and (H2) hold, and µ ∈ P(L 2 tem ∩C tem ).Then W 2 (Q, P µ ) 2 ≤ c 2 H(Q|P µ ), Q ∈ P(C([0, T ], L 2 tem ∩ C tem )) holds for some constant c 2 > 0 if and only if W 2 (ν, µ) 2 ≤c 2 H(ν|µ), ν ∈ P(L 2 tem ∩ C tem ) holds for some constantc 2 > 0. Let E = L 2 tem or E = C tem . Let µ be the law of the random field solution u(·, ·) of SPDE (2), viewed as a probability measure on C([0, T ], E). For the proof of Theorem 2.3 and 2.4, we recall a lemma describing the probability measure ν that is absolutely continuous with respect to µ. Let ν ≪ µ on C([0, T ], E). Define a new probability measure Q on the filtered probability space (Ω, F, {F t } 0≤t≤T , P) by dQ := dν dµ (u) dP.(6) Denote the Radon-Nikodym derivative restricted on F t by M t := dQ dP Ft , t ∈ [0, T ]. Then M t , t ∈ [0, T ] forms a P-martingale. The following Girsanov transformation lemma can be found in [11]. Lemma 2.7. There exists an adapted random field h = {h(s, x), (s, x) ∈ [0, T ] × R} such that Q-a.s. for all t ∈ [0, T ], t 0 R |h(s, x)| 2 dxds < ∞ and W (dt, dx) := W (dt, dx) − h(t, x) dxdt, is a space-time white noise on [0, T ] × R. Moreover, M t = exp t 0 R h(s, x) W (ds, dx) − 1 2 t 0 R |h(s, x)| 2 dxds , Q-a.s., and H(ν|µ) = 1 2 E Q t 0 R |h(s, x)| 2 dxds ,(7) where E Q stands for the expectation under the measure Q. Moment estimates In the proof of the main results, the moments estimates (see Lemma 3.2 and 3.3) for stochastic convolution against space-time white noise will play a crucial role. Before we give the estimates, we prove the Burkholder-type inequality for Hilbert-space-valued stochastic integral against space-time white noise. We present some estimates associated with the heat kernal p t (x, y) = 1 √ 2πt e − (x−y) 2 2t , whose proofs are straightforward so we omit them here. For any x ∈ R, t > 0 and η ∈ R, R p t (x, y)e η|y| dy ≤ 2e η 2 t 2 e η|x| ,(8)R p t (x, y) 2 e η|y| dy ≤ 1 √ πt e η 2 t 4 e η|x| .(9) For a function f ∈ L 2 λ , λ > 0, t > 0, P t f L 2 λ ≤ √ 2e λ 2 t f L 2 λ .(10) Lemma 3.1 (Burkholder-Davis-Gundy's inequality). Let H be a Hilbert space with inner product and norm denoted by ·, · H and | · | H , respectively. Φ(t, x) is a H-valued adapted random field such that the following stochastic integral with respect to the space-time white noise is well-defined. Assume that for p > 0, E t 0 R |Φ(r, z)| 2 H dzdr p 2 < ∞. Then there exists a constant c p > 0 such that for t > 0, E sup s∈[0,t] s 0 R Φ(r, z)W (dr, dz) p H ≤ c p E t 0 R |Φ(r, z)| 2 H dzdr p 2 . Proof. It is known (see [5,Chapter 4], also [4]) that Walsh's stochastic integral against the space-time white noise can be formulated as an Itô integral w.r.t. a cylindrical Wiener process. Let us make it precise in the current setting. For t ≥ 0, define an operator Φ(t) : L 2 (R) → H by [Φ(t)](f ) := R Φ(t, x)f (x)dx, for f ∈ L 2 (R). Then, Φ(t) is a Hilbert-Schmidt operator, i.e. Φ(t) ∈ L 2 (L 2 (R), H) and |Φ(t)| 2 L 2 (L 2 (R),H) = R |Φ(t, x)| 2 H dx. Indeed, let {e n } n≥1 be an orthonormal basis of L 2 (R), and if {g j } j≥1 is an orthonormal basis of H we have that |Φ(s)| 2 L 2 (L 2 (R),H) = ∞ n=1 |Φ(s)e n | 2 H = ∞ n=1 R Φ(s, y)e n (y)dy 2 H = ∞ n=1 ∞ j=1 R Φ(s, y)e n (y)dy, g j 2 H = ∞ j=1 ∞ n=1 R Φ(s, y), g j H e n (y)dy 2 = ∞ j=1 R Φ(s, y), g j 2 H dy = R |Φ(s, y)| 2 H dy. Now, for r ≥ 0, we expand Φ(r, ·) ∈ L 2 (R) in the basis {e n } n≥1 and write s 0 R Φ(r, z)W (dr, dz) = s 0 R ∞ n=1 Φ(r, ·), e n L 2 (R) e n (z)W (dr, dz) = s 0 ∞ n=1 Φ(r, ·), e n L 2 (R) dβ n (r) = s 0 ∞ n=1 Φ(r)e n dβ n (r) = s 0 Φ(r)dW r , where β n (t) := t 0 R e n (z)W (dr, dz), t ≥ 0, n ≥ 1 are independent 1-dimensional Brownian motions, and W r := ∞ n=1 e n β n (r), r ≥ 0 is a L 2 (R)-cylindrical Wiener process. According to [5,Theorem 4.36], there exists a constant c p such that E sup s∈[0,t] s 0 R Φ(r, z)W (dr, dz) p H =E sup s∈[0,t] s 0 Φ(r)dW r p H ≤c p E t 0 |Φ(s)| 2 L 2 (L 2 (R),H) ds p 2 =c p E t 0 R |Φ(s, y)| 2 H dyds p 2 . Recall the Hilbert space L 2 λ defined in Section 2 with inner product ·, · L 2 λ and induced norm · L 2 λ . Lemma 3.2. Let {σ(s, y) : (s, y) ∈ [0, T ] × R} be an adapted random field such that the following stochastic convolution with respect to the space-time white noise is well defined and σ(t, ·) ∈ L 2 λ for every t, P-a.s. Then for any p > 8, T > 0, there exists a constant C λ,T,p increasing w.r.t. λ such that E sup t≤T R t 0 R p t−s (x, y)σ(s, y)W (ds, dy) 2 e −2λ|x| dx p 2 ≤C λ,T,p E T 0 σ(r, ·) p L 2 λ dr.(11) Proof. We adopt the factorization method(see e.g. [5]). For 0 < α < 1 8 , set J σ (s, y) : = s 0 R (s − r) −α p s−r (y, z)σ(r, z)W (dr, dz),(12)J α−1 f (t, x) : = sin(πα) π t 0 R (t − s) α−1 p t−s (x, y)f (s, y)dsdy.(13) Then we have t 0 R p t−s (x, y)σ(s, y)W (ds, dy) = J α−1 (J α σ) (t, x).(14) Now we consider the left-hand side of (11). Using (14), Minkowski's inequality, (10) and Hölder's inequality, we have for p > 8 E sup t≤T R t 0 R p t−s (x, y)σ(s, y)W (ds, dy) 2 e −2λ|x| dx p 2 =E sup t≤T t 0 R p t−s (·, y)σ(s, y)W (ds, dy) p L 2 λ ≤ sin(πα) π p E sup t≤T t 0 R (t − s) α−1 p t−s (·, y)J α σ(s, y)dyds p L 2 λ ≤π −p E sup t≤T t 0 (t − s) α−1 R p t−s (·, y)J α σ(s, y)dy L 2 λ ds p ≤π −p 2 p 2 e pλ 2 T E sup t≤T t 0 (t − s) α−1 J α σ(s, ·) L 2 λ ds p ≤π −p 2 p 2 e pλ 2 T T 0 s (α−1) p p−1 ds p−1 T 0 E J α σ(s, ·) p L 2 λ ds ≤C ′ λ,T,p T 0 E J α σ(s, ·) p L 2 λ ds,(15) where the condition α > 1 p is used for a finite integral. By (12) and Lemma 3.1 we have E J α σ(s, ·) p L 2 λ =E s 0 R (s − r) −α p s−r (·, z)σ(r, z)W (dr, dz) p L 2 λ ≤c p E s 0 R (s − r) −2α p s−r (·, z)σ(r, z) 2 L 2 λ dzdr p 2 . With this estimate, (15) is bounded by E sup t≤T R t 0 R p t−s (x, y)σ(s, y)W (ds, dy) 2 e −2λ|x| dx p 2 ≤C ′′ λ,T,p T 0 E s 0 R (s − r) −2α p s−r (·, z)σ(r, z) 2 L 2 λ dzdr p 2 ds.(16) We continue to estimate the right-hand side of (16). By (9) and Hölder's inequality we have T 0 E s 0 R (s − r) −2α p s−r (·, z)σ(r, z) 2 L 2 λ dzdr p 2 ds = T 0 E s 0 R (s − r) −2α R |p s−r (y, z)σ(r, z)| 2 e −2λ|y| dy dzdr p 2 ds ≤E T 0 s 0 R (s − r) −2α− 1 2 π − 1 2 e λ 2 s e −2λ|z| |σ(r, z)| 2 dzdr p 2 ds ≤π − p 4 e p 2 λ 2 T E T 0 s 0 (s − r) −(2α+ 1 2 ) p p−2 dr p−2 2 s 0 σ(r, ·) 2× p 2 L 2 λ dr ds ≤2 − p−2 2 π − p 4 e p 2 λ 2 T p − 2 p( 1 4 − α) − 1 p−2 2 E T 0 s p( 1 4 −α)−1 s 0 σ(r, ·) p L 2 λ dr ds ≤2 − p−2 2 π − p 4 e p 2 λ 2 T T p( 1 4 −α) p − 2 p( 1 4 − α) − 1 p−2 2 1 p( 1 4 − α) E T 0 σ(r, ·) p L 2 λ dr = : C ′′′ λ,T,p E T 0 σ(r, ·) p L 2 λ dr.(17) In the last step we changed the order of integration, and the condition α < 1 4 − 1 p is used. Let C λ,T,p := min 1 p <α< 1 4 − 1 p C ′′ λ,T,p · C ′′′ λ,T,p . Then (11) follows from (16) and (17). and σ(t, ·) ∈ L 2 λ for every t, P-a.s.. Then for any 0 < p ≤ 8, T, ǫ > 0, there exists a constant C ǫ,λ,T,p increasing w.r.t. λ such that E sup t≤T R t 0 R p t−s (x, y)σ(s, y)W (ds, dy) 2 e −2λ|x| dx p 2 ≤ǫ E sup t≤T σ(t, ·) p L 2 λ + C ǫ,λ,T,p E T 0 σ(t, ·) p L 2 λ dt, Proof. The proof is inspired by [19]. Step 1: We show that for q > 8, η > 0, P sup t≤T t 0 R p t−s (x, y)σ(s, y)W (ds, dy) L 2 λ > η ≤P T 0 σ(s, ·) q L 2 λ ds > η q + c λ,T,q η q E min η q , T 0 σ(s, ·) q L 2 λ ds .(18) Set Ω η := ω : T 0 σ(r, ·) q L 2 λ dr ≤ η q . Then we have by Chebyshev's inequality that In view of the above equality, we can use the bound in Lemma 3.2: P sup t≤T t 0 R p t−s (·, y)σ(s, y)W (ds, dy) L 2 λ > η ≤P(Ω/ Ω η ) + P sup t≤T t 0 R p t−s (·, y)σ(s, y)W (ds, dy) L 2 λ ½ Ωη > η ≤P T 0 σ(s, ·) q L 2 λ ds > η q + 1 η q E supE sup t≤T t 0 R p t−s (·, y)σ(s, y)W (ds, dy) L 2 λ ½ Ωη q ≤E sup t≤T t 0 R p t−s (·, y)σ(s, y)W (ds, dy) q L 2 λ ≤C λ,T,q E T 0 σ(s, ·) q L 2 λ ds ≤C λ,T,q E min{η q , T 0 σ(s, ·) q L 2 λ ds},(20) where C λ,T,q is the constant appeared in (11) with q replacing p. Now (18) follows immediately from (19) and (20). Step 2: For 0 < p ≤ 8, we follow Lemma A.2 in [19] to obtain that E sup t≤T t 0 R p t−s (·, y)σ(s, y)W (ds, dy) p L 2 λ = ∞ 0 pη p−1 P sup t≤T t 0 R p t−s (·, y)σ(s, y)W (ds, dy) p L 2 λ > η dη ≤ ∞ 0 pη p−1 P T 0 σ(s, ·) q L 2 λ ds > η q dη + C λ,T,q ∞ 0 pη p−1−q E min η q , T 0 σ(s, ·) q L 2 λ ds dη ≤C λ,T,p,q E t 0 σ(s, ·) q L 2 λ ds p q ≤C λ,T,p,q E sup s≤T σ(s, ·) (q−p)p q L 2 λ T 0 σ(s, ·) p L 2 λ ds p q ≤ǫ E sup s≤T σ(s, ·) p L 2 λ + C ǫ,λ,T,p E T 0 σ(s, ·) p L 2 λ ds, where C λ,T,p,q = 1 + q q−p C λ,T,q , and Young's inequality is used in the last step. Consider the solution of the following SPDE: v(t, x) =P t u 0 (x) + t 0 R p t−s (x, y)b(v(s, y)) dyds + t 0 R p t−s (x, y)σ(v(s, y)) W (ds, dy).(22) By Lemma 2.7 it follows that under the measure Q, the law of (v, u) forms a coupling of (µ, ν). Therefore by the definition of the Wasserstein distance, W 2 (ν, µ) 2 ≤ E Q sup t∈[0,T ] d(u(t), v(t)) 2 . Here d stands for the metric of space E. More precisely, we have W 2 (ν, µ) 2 ≤ E Q sup t∈[0,T ] ∞ n=1 1 2 n min 1, u(t) − v(t) L 2 1/n 2 ≤ E Q sup t∈[0,T ] ∞ n=1 1 2 n v(t) − u(t) 2 L 2 1/n ≤ ∞ n=1 1 2 n E Q sup t∈[0,T ] R |v(t, x) − u(t, x)| 2 e − 2 n |x| dx . In view of (7), to prove the quadratic transportation cost inequality W 2 (ν, µ) ≤ 2CH(ν|µ),(23) it is sufficient to show that exists a constant C independent of n such that for any n ∈ N, E Q sup t∈[0,T ] R |u(t, x) − v(t, x)| 2 e − 2 n |x| dx ≤ CE Q T 0 R h 2 (s, y) dyds(24) when the right-hand side of (24) is finite. To this end, for λ > 0 let us estimate E Q sup t∈[0,T ] R |u(t, x) − v(t, x)| 2 e −2λ|x| dx . For convenience, in the sequel we denote E Q still by the symbol E. From (21) and (22) it follows that E sup t≤T R |u(t, x) − v(t, x)| 2 e −2λ|x| dx ≤3 E sup t≤T R t 0 R p t−s (x, y)[b(u(s, y)) − b(v(s, y))]dsdy 2 e −2λ|x| dx + 3 E sup t≤T R t 0 R p t−s (x, By Hölder's inequality and (H1) we have that I 1 ≤E sup t≤T R t 0 R p t−s (x, y)dyds × t 0 R p t−s (x, y)|b(u(s, y)) − b(v(s, y))| 2 dyds e −2λ|x| dx ≤T L 2 b E sup t≤T R t 0 R p t−s (x, y)|u(s, y) − v(s, y)| 2 dyds e −2λ|x| dx ≤2T e 2λ 2 T L 2 b E T 0 R |u(s, y) − v(s, y)| 2 e −2λ|y| dyds ≤2T e 2λ 2 T L 2 b T 0 E sup r≤s R |u(r, y) − v(r, y)| 2 e −2λ|y| dy ds.(26) In the third inequality, Fubini's theorem and (8) were used. According to Lemma 3.3, for ǫ > 0, E sup t≤T R t 0 R p t−s (x, y)[σ(u(s, y)) − σ(v(s, y))] W (ds, dy) 2 e −2λ|x| dx ≤ǫ E sup s≤T R |σ(u(s, y)) − σ(v(s, y))| 2 e −2λ|y| dy + C ǫ,λ,T T 0 E R |σ(u(s, y)) − σ(v(s, y))| 2 e −2λ|y| dy ds ≤ǫL 2 σ E sup s≤T R |u(s, y) − v(s, y)| 2 e −2λ|y| dy + C ǫ,λ,T L 2 σ T 0 E sup r≤s R |u(r, y) − v(r, y)| 2 e −2λ|y| dy ds.(27) For the third term I 3 , by the boundedness of σ(see (H2)), Minkowski's inequality for integrals, (10) and Hölder's inequality we have I 3 ≤K 2 σ E sup t≤T t 0 P t−s h(s, ·)ds 2 L 2 λ ≤K 2 σ E sup t≤T t 0 P t−s h(s, ·) L 2 λ ds 2 ≤K 2 σ E sup t≤T t 0 √ 2e λ 2 t h(s, ·) L 2 λ ds 2 ≤2e 2λ 2 T T K 2 σ E T 0 h(s, ·) 2 L 2 λ ds ≤2e 2λ 2 T T K 2 σ E T 0 R |h(t, x)| 2 dxdt.(28) Substituting (26)- (28) into (25), we have (1 − 3ǫL 2 σ )E sup t≤T R |u(t, x) − v(t, x)| 2 e −2λ|x| dx ≤C ǫ,λ,T,L b ,Lσ T 0 E sup r≤s R |u(s, y) − v(s, y)| 2 e −2λ|y| dy ds + 6T e 2λ 2 T K 2 σ E T 0 R |h(t, x)| 2 dxdt. Taking any 0 < ǫ < 1 3L 2 σ and applying the Gronwall's inequality we obtain E sup t≤T R |u(t, x) − v(t, x)| 2 e −2λ|x| dx ≤ 6T e 2λ 2 T K 2 σ 1 − 3ǫL 2 σ exp T C ǫ,λ,T,L b ,Lσ 1 − 3ǫL 2 σ E T 0 R |h(t, x)| 2 dxdt. Taking infimum over ǫ, we get E sup t≤T R |u(t, x) − v(t, x)| 2 e −2λ|x| dx ≤ C λ,T,L b ,Lσ,Kσ E T 0 R |h(t, x)| 2 dxdt. Noting that C λ,T,L b ,Lσ,Kσ is increasing with respect to λ, it follows that for each n ∈ N, E sup t≤T R |u(t, x) − v(t, x)| 2 e − 2 n |x| dx ≤ C 1,T,L b ,Lσ,Kσ E T 0 R |h(t, x)| 2 dxdt, This proves (24) and hence the proof of Theorem 2.3 is complete. Proof of Theorem 2.4 In the proof of the main results, the following moments estimates for stochastic convolution against space-time white noise obtained in [20] will play a crucial role. Lemma 5.1. Let h : R + −→ R + be an increasing function. Let {σ(s, y) : (s, y) ∈ R + × R} be an adapted random field such that the following stochastic convolution with respect to space-time white noise is well defined. Let τ be a stopping time. (i) Then for any p > 10 and T > 0, there exists a constant Θ p,h(T ),T > 0 such that E sup (t,x)∈[0,T ∧τ ]×R t 0 R p t−s (x, y)σ(s, y)W (ds, dy) e −h(t)|x| p ≤Θ p,h(T ),T E T ∧τ 0 R |σ(t, x)| p e −ph(t)|x| dxdt.(29) (ii) Then for any ǫ, T > 0 and 0 < p ≤ 10, there exists a constant Θ ǫ,p,h(T ),T > 0 such that E sup (t,x)∈[0,T ∧τ ]×R t 0 R p t−s (x, y)σ(s, y)W (ds, dy) e −h(t)|x| p ≤ǫ E sup (t,x)∈[0,T ∧τ ]×R |σ(t, x)|e −h(t)|x| p + Θ ǫ,p,h(T ),T E T ∧τ 0 R |σ(t, x)| p e −ph(t)|x| dxdt.(30) Here Θ p,h(T ),T and Θ ǫ,p,h(T ),T are increasing w.r.t. h(T ) and w.r.t. T . Proof of Theorem 2.4. Let E = C tem . As in the proof Theorem 2.3, take ν ≪ µ on C([0, T ], E). Define the corresponding measure Q by (6). Let h(t, x) be the corresponding random field appeared in Lemma 2.7. Then the solution u(t, x) of equation (2) satisfies the following SPDE under the measure Q, u(s, y)) W (ds, dy) u(t, x) =P t u 0 (x) + t 0 R p t−s (x, y)b(u(s, y)) dyds + t 0 R p t−s (x, y)σ(+ t 0 R p t−s (x, y)σ(u(s, y))h(s, y) dyds.(31) Consider the solution of the following SPDE: v(t, x) =P t u 0 (x) + t 0 R p t−s (x, y)b(v(s, y)) dyds + t 0 R p t−s (x, y)σ(v(s, y)) W (ds, dy).(32) By Lemma 2.7 it follows that under the measure Q, the law of (v, u) forms a coupling of (µ, ν). Therefore by the definition of the Wasserstein distance, W 2 (ν, µ) 2 ≤ E Q sup t∈[0,T ] d(u(t), v(t)) 2 . Here d stands for the metric of space E. More precisely, we have W 2 (ν, µ) 2 ≤ E Q   sup t∈[0,T ] ∞ n=1 1 2 n min 1, ̺ 1/n (u(t), v(t)) 2   ≤ E Q sup t∈[0,T ] ∞ n=1 1 2 n (̺ 1/n (u(t), v(t))) 2 ≤ ∞ n=1 1 2 n E Q sup (t,x)∈[0,T ]×R |u(t, x) − v(t, x)| 2 e − 2 n |x| Similarly, to prove the quadratic transportation cost inequality, it is sufficient to show that exists a constant Θ independent of n such that for any n ∈ N, E Q sup (t,x)∈[0,T ]×R |u(t, x) − v(t, x)| 2 e − 2 n |x| ≤ ΘE Q T 0 R h 2 (s, y) dyds(33) when the right-hand side of (33) is finite. To this end, for λ > 0, we consider Y (T ) := E Q sup (t,x)∈[0,T ]×R |u(t, x) − v(t, x)| 2 e −2λ|x| . Again we still denote E Q by the symbol E for convenience. From (31) and (32) it follows that u(s, y))h(s, y)dyds 2 e −2λ|x| = : 3(I + II + III). Y (T ) ≤3E sup (t,x)∈[0,T ]×R t 0 R p t−s (x, y) b(v(s, y)) − b(u(s, y)) dyds 2 e −2λ|x| + 3E sup (t,x)∈[0,T ]×R t 0 R p t−s (x, y) σ(v(s, y)) − σ(u(s, y)) W (ds, dy) 2 e −2λ|x| + 3E sup (t,x)∈[0,T ]×R t 0 R p t−s (x, y)σ( By the assumption (H1) and (8), the term I can be estimated as follows: I ≤L 2 b E sup (t,x)∈[0,T ]×R t 0 R p t−s (x, y)|u(s, y) − v(s, y)|dyds 2 e −2λ|x| ≤L 2 b E sup (t,x)∈[0,T ]×R t 0 sup y∈R |u(s, y) − v(s, y)| 2 e −2λ|y| R p t−s (x, y)e 2λ|y| dyds e −2λ|x| ≤2e 2λ 2 T L 2 b E t 0 sup (r,y)∈[0,s]×R |u(r, y) − v(r, y)| 2 e −2λ|y| ds =2e 2λ 2 T L 2 b t 0 Y (s)ds.(35) By the boundedness of σ and using Hölder's inequality and (9) the term III can be bounded as follows, III ≤K 2 σ E sup (t,x)∈[0,T ]×R t 0 R p 2 t−s (x, y)dyds · t 0 R |h(s, y)| 2 dyds ≤ 1 √ π K 2 σ E sup t∈[0,T ] t 0 1 √ t − s ds t 0 R |h(s, y)| 2 dyds ≤ 2 √ T √ π K 2 σ E T 0 R |h(s, y)| 2 dyds.(36) For the term II, by the assumption (H2) and the estimate (30) we obtain that for any ǫ > 0, II ≤ǫL 2 σ E sup (t,x)∈[0,T ]×R |u(t, x) − v(t, x)| 2 e −2λ|x| + Θ ǫ,λ,T L 2 σ E T 0 R |u(t, x) − v(t, x)| 2 e −2λ|x| dxdt.(37) It remains to give an estimate for the following integral. E T 0 R |u(t, x) − v(t, x)| 2 e −2λ|x| dxdt. Lemma 5.2. For any λ > 0, 0 ≤ t ≤ T , there exists a constant Θ λ,T,L b ,Lσ,Kσ increasing w.r.t. λ such that E T 0 R |u(t, x) − v(t, x)| 2 e −2λ|x| dxdt ≤ Θ λ,T,L b ,Lσ,Kσ E T 0 R |h(t, x)| 2 dxdt.(38) Proof. Define F (t) := E R |u(t, x) − v(t, x)| 2 e −2λ|x| dx, t ∈ [0, T ]. By (31) and (32), we have F (t) ≤3E R t 0 R p t−s (x, y) [b(u(s, y)) − b(v(s, y))] dyds 2 e −2λ|x| dx + 3E R t 0 R p t−s (x, y)σ(u(s, y))h(s, y)dyds 2 e −2λ|x| dx + 3E R t 0 R p t−s (x, y) [σ(u(s, y)) − σ(v(s, y))]W (ds, dy) 2 e −2λ|x| dx = : 3J 1 + 3J 2 + 3J 3 .(39) For the term J 1 , using the assumption (H1), Hölder's inequality and (8) we obtain J 1 ≤L 2 b E R t 0 R p t−s (x, y)dyds · t 0 R p t−s (x, y)|u(s, y) − v(s, y)| 2 dyds e −2λ|x| dx ≤2te 2λ 2 t L 2 b E t 0 R |u(s, y) − v(s, y)| 2 e −2λ|y| dy ds =2te 2λ 2 t L 2 b t 0 F (s)ds,(40) where we used the Fubini theorem in the last step. And also we can estimate J 2 as follows, J 2 ≤K 2 σ E R t 0 R p t−s (x, y)dyds · t 0 R p t−s (x, y)|h(s, y)| 2 dyds e −2λ|x| dx ≤tK 2 σ E R t 0 R p t−s (x, y)|h(s, y)| 2 dydse −2λ|x| dx ≤2te 2λ 2 t K 2 σ E t 0 R |h(s, y)| 2 dyds,(41) where the boundedness of σ and (8) were used. As for the term J 3 , by the Fubini theorem(see [25]) and the Bukrholder-Davis-Gundy's inequality (see [10]), then using the assumption (H2) and (9) we have J 3 ≤ 8 R E t 0 R p t−s (x, y) 2 |σ(u(s, y)) − σ(v(s, y))| 2 dyds e −2λ|x| dx ≤8e λ 2 t L 2 σ E t 0 R 1 π(t − s) |u(s, y) − v(s, y)| 2 e −2λ|y| dyds = 8 √ π e λ 2 t L 2 σ t 0 F (s) √ t − s ds,(42) where in the last step we used the Fubini theorem. Substituting (40)-(42) into (39) leads to F (t) ≤6te 2λ 2 t K 2 σ E t 0 R |h(s, y)| 2 dyds + 6te 2λ 2 t L 2 b t 0 F (s)ds + 24 √ π e λ 2 t L 2 σ t 0 F (s) √ t − s ds.(43) Then iterating (43) and using Gronwall's inequality yield that for any t ∈ [0, T ], E R |u(t, x) − v(t, x)| 2 e −2λ|x| dx ≤ Θ λ,T,L b ,Lσ,Kσ E T 0 R |h(s, y)| 2 dyds. Then (38) follows from the preceding inequality. Now we come back to estimate II. Combining (37) with (38) leads to II ≤ ǫL 2 σ Y (T ) + L 2 σ Θ ǫ,λ,T Θ λ,T,L b ,Lσ,Kσ E T 0 R |h(s, y)| 2 dyds.(44) Then putting (34), (35), (36) and (44) together, we obtain Y (T ) ≤6e 2λ 2 T L 2 b t 0 Y (s)ds + 3ǫL 2 σ Y (T ) + Θ ǫ,λ,T,L b ,Lσ,Kσ E T 0 R |h(t, x)| 2 dxdt.(45) Note that Θ ǫ,λ,T,L b ,Lσ,Kσ is increasing with respect to λ and with repsect to T . Recall that for any λ > 0, (see [21]) E sup (t,x)∈[0,T ]×R |u(t, x)| 2 e −2λ|x| < ∞, E sup (t,x)∈[0,T ]×R |v(t, x)| 2 e −2λ|x| < ∞. Hence Y (T ) < ∞ for any T > 0. Clearly, (45) still holds if we replace T with any t ∈ [0, T ]. Next, taking any 0 < ǫ < 1 3L 2 σ and applying the Gronwall's inequality we obtain Y (T ) ≤ Θ ǫ,λ,T,L b ,Lσ,Kσ 1 − 3ǫL 2 σ exp 6T e 2λ 2 T L 2 b 1 − 3ǫL 2 σ E T 0 R |h(t, x)| 2 dxdt. Since 0 < ǫ < 1 3L 2 σ is arbitrary, we get Y (T ) ≤ Θ λ,T,L b ,Lσ,Kσ E T 0 R |h(t, x)| 2 dxdt. By the definition of Y (T ), taking λ = 1 n , n ∈ N in the above inequality, and noting that Θ λ,T,L b ,Lσ,Kσ is increasing with respect to λ, we have for each n ∈ N, E sup (t,x)∈[0,T ]×R |u(t, x) − v(t, x)| 2 e − 2 n |x| ≤ Θ λ,T,L b ,Lσ,Kσ E T 0 R |h(t, x)| 2 dxdt. This proves (33) and hence the proof of Theorem 2.4 is complete. 6 Proof of Corollary 2.5 and 2.6 In this section we generalize the transportation cost inequalities from deterministic initial values to random initial values. We will follow the same approach as in the proof of Theorem 3.1 in [27]. Proof of Corollary 2.5 In fact, the result of Theorem 2.3 can be written in the form: W 2 (Q, P u 0 ) 2 ≤ c 1 H(Q|P u 0 ), Q ∈ P(C([0, T ], L 2 tem )), u 0 ∈ L 2 tem .(46) According to Theorem 2.1 in [27], it suffices to show that the Warssenstein distance between the laws of solutions is Lipschitz continuous w.r.t. the initial values, which is stated in the following proposition. Proposition 6.1. There exists a constant c > 0 such that W 2 (P f , P g ) ≤ cρ(f, g), f, g ∈ L 2 tem .(47) Proof. We denote by u f and u g the two solutions of SPDE (2) starting from f, g ∈ L 2 tem . For λ > 0, we have E sup t≤T R |u f (t, x) − u g (t, x)| 2 e −2λ|x| dx ≤3 sup t≤T R R p t (x, y)(f (y) − g(y))dy 2 e −2λ|x| dx + 3 E sup t≤T R t 0 R p t−s (x, y) b(u f (s, y)) − b(u g (s, y)) dyds 2 e −2λ|x| dx + 3 E sup t≤T R t 0 R p t−s (x, y) σ(u f (s, y)) − σ(u g (s, y)) W (ds, dy) 2 e −2λ|x| dx .(48) By Hölder's inequality and (8), we have sup t≤T R R p t (x, y)(f (y) − g(y))dy 2 e −2λ|x| dx ≤ sup t≤T R R p t (x, y)dy R p t (x, y)|f (y) − g(y)| 2 dy e −2λ|x| dx ≤2e 2λ 2 T R |f (y) − g(y)| 2 e −2λ|y| dy =2e 2λ 2 T f − g 2 L 2 λ .(49) Similarly, E sup t≤T R t 0 R p t−s (x, y) b(u f (s, y)) − b(u g (s, y)) dyds 2 e −2λ|x| dx ≤2e 2λ 2 T T L 2 b T 0 E sup r≤s R |u f (r, y) − u g (r, y)| 2 e −t≤T R t 0 R p t−s (x, y) σ(u f (s, y)) − σ(u g (s, y)) W (ds, dy) 2 e −2λ|x| dx ≤ǫL 2 σ E sup s≤T R |u f (s, y) − u g (s, y)| 2 e −2λ|y| dy + C ǫ,λ,T L 2 σ T 0 E sup r≤s R |u f (r, y) − u g (r, y)| 2 e −2λ|y| dy ds,(51) Then combining (49)-(51) with (48) together and using Gronwall's inequality, it follows that there exists some constant c > 0 dependent on ǫ, λ, T, L b , L σ such that E sup t≤T u f (t, ·) − u g (t, ·) L 2 λ ≤ c f − g L 2 λ . Note that the constant c is increasing w.r.t. λ. Hence there exists a constant c independent of n such that for all n ≥ 1, The proof of this proposition is complete. With this proposition and Theorem 2.3, we complete the proof of Corollary 2.5. Proof of Corollary 2.6 From Theorem 2.3 and Theorem 2.4, we see that for u 0 ∈ L 2 tem ∩ C tem , W 2 (Q, P u 0 ) 2 ≤ CH(Q|P u 0 ), Q ∈ P(C([0, T ], L 2 tem ∩ C tem )), P u 0 is the law of the solution of the SPDE (2) started at u 0 . Also, according to Theorem 2.1 in [27], to prove Corollary 2.6 we only need to establish the following proposition. Remark 6.2. In fact, in C tem the Lipschitz property like Proposition 6.1 does not exist, unless we consider the equation (2) in C tem ∩ L 2 tem . Proposition 6.3. There exists a constantc > 0 such that W 2 (P f , P g ) ≤cρ(f, g), f, g ∈ L 2 tem ∩ C tem . Proof. Let u f (u g , respectively) be the unique solution of SPDE (2) with u 0 = f ∈ L 2 tem ∩ C tem (g, respectively). For λ > 0, by (3) and Gronwall's inequality we can obtain ̺ λ (u f (t, ·), u g (t, ·)) ≤ c ′′ (̺ λ (f, g) + f − g L 2 λ ). min 1, ̺ 1/n (u f (t, ·), u g (t, ·)) ≤ ∞ n=1 1 2 n c ′′ min 1, ̺ 1/n (f, g) + f − g L 2 1/n ≤c ′′ (̺(f, g) + ρ(f, g)). Consequently we have Together with (52), we obtain ̺(u f (t, ·), u g (t, ·)) + ρ(u f (t, ·), u g (t, ·)) ≤c(̺(f, g) + ρ(f, g)). The proof of this Proposition is complete. According to Theorem 2.1 in [27], we have completed the proof of Corollary 2.6. Proposition 2.1 ([21]). Assume that (H1) and (H2(b)) hold and u 0 ∈ C tem . Then there exists a random field solution to the stochastic reaction diffusion equation (2) with sample paths a.s. in C([0, T ], C tem ). Proposition 2 . 2 . 22Assume that (H1) and (H2(b)) hold and u 0 ∈ L 2 tem . Then there exists a random field solution to the stochastic reaction diffusion equation (2) with sample paths a.s. in C([0, T ], L 2 tem ). Theorem 2. 3 . 3Suppose the hypotheses (H1) and (H2) hold. If u 0 ∈ L 2 tem , then, the law of the solution u(·, ·) of SPDE (2) satisfies the quadratic transportation cost inequality (1) on the space C([0, T ], L 2 tem ). Theorem 2.4. Suppose the hypotheses (H1) and (H2) hold. If u 0 ∈ C tem , then, the law of the solution u(·, ·) of SPDE (2) satisfies the quadratic transportation cost inequality (1) on the space C([0, T ], C tem ). Lemma 3. 3 . 3Let {σ(s, y) : (s, y) ∈ [0, T ] × R} be an adapted random field such that the following stochastic convolution with respect to the space-time white noise is well defined By the local property of the stochastic integral(see Lemma A.1 in[19]), −s (x, y)σ(s, y)W (ds, dy) = ½ Ωη t 0 R p t−s (x, y)σ(s, y)W (ds, dy), P-a.s. E = L 2 tem . Take ν ≪ µ on C([0, T ], E). Define the corresponding measure Q by(6). Let h(t, x) be the corresponding random field appeared in Lemma 2.7. Then the solution u(t, x) of equation(2)satisfies the following SPDE under the measure Q, u(t, x) =P t u 0 −s (x, y)σ(u(s, y))h(s, y) dyds. y)[σ(u(s, y)) − σ(v(s, y))] W (ds, dy) 1 + I 2 + I 3 ). u f (t, ·) − u g (t, ·) min 1, u f (t, ·) − u g (t, ·) u f (t, ·) − u g (t, ·) ( 0 0−s (x, y) σ(u f (s, y)) − σ(u g (s, y) W (ds, dy) λ,T,L b sup x∈R |f (x) − g(x)| 2 e −2λ|x| + ǫ c λ,T,L b ,Lσ E sup (t,x)∈[0,T ]×R u f (t, x) − u g (t, (t, x) − u g (t, x) 2 e −2λ|x| dx ≤ c λ,T,L b ,Lσ R |f (y) − g(y)| 2 e −2λ|y| dy.(56)Then by(55)and(56)and taking some appropriate ǫ we obtainE sup (t,x)∈[0,T ]×R u f (t, x) − u g (t, x) 2 e −2λ|x| ≤c ′ sup x∈R |f (x) − g(x)| 2 e −2λ|x| + R |f (y) − g(y)| 2 e −2λ|y| dy .for some constant c ′ increasing w.r.t. λ. Hence, there exists a constant c ′′ > 0, such that for all λ ∈ E sup t∈[0,T ] ̺(u f (t, ·), u g (t, ·)) =E sup t∈[0,T ] ∞ n=1 1 2 n min 1, ̺ 1/n (u f (t, ·), u g (t, Hypercontractivity of Hamilton-Jacobi equations. S Bobkov, I Gentil, M Ledoux, J. Math. Pures Appl. 80Bobkov, S., Gentil, I. and Ledoux, M. Hypercontractivity of Hamilton-Jacobi equa- tions. J. Math. Pures Appl. 80:669-696, 2001. Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. S G Bobkov, F Götze, J. Funct. Anal. 163Bobkov, S. G. and Götze, F. Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163:1-28, 1999. Transportation inequalities for stochastic heat equations. B Boufoussi, S Hajji, Statist. Probab. Lett. 139Boufoussi, B. and Hajji, S. Transportation inequalities for stochastic heat equations. Statist. Probab. Lett. 139:75-83, 2018. Stochastic integrals for spde's: A comparison. R C Dalang, L Quer-Sardanyons, Expo. Math. 29Dalang, R. C. and Quer-Sardanyons, L. Stochastic integrals for spde's: A comparison. Expo. Math. 29:67-109, 2011. Stochastic equations in infinite dimensions. G Da Prato, J Zabczyk, Cambridge University Press152Cambridge2nd ed. Encyclopedia of Mathematics and its ApplicationsDa Prato, G. and Zabczyk, J. Stochastic equations in infinite dimensions, 2nd ed. Encyclopedia of Mathematics and its Applications, 152. Cambridge University Press, Cambridge, 2014. Transportation cost-information inequalities and applications to random dynamical systems and diffusions. H Djellout, A Guillin, L Wu, Ann. Probab. 32Djellout, H., Guillin, A. and Wu, L. Transportation cost-information inequalities and applications to random dynamical systems and diffusions. Ann. Probab. 32:2702-2732, 2004. Transportation cost inequalities on path and loop groups. S Fang, J Shao, J. Funct. Anal. 218Fang, S. and Shao, J. Transportation cost inequalities on path and loop groups. J. Funct. Anal. 218:293-317, 2005. Monge-Kantorovitch measure transportation and Monge-Ampère equation on Wiener space. D Feyel, A S , Probab. Theory Related Fields. 128Feyel, D. andÜstünel, A. S. Monge-Kantorovitch measure transportation and Monge- Ampère equation on Wiener space. Probab. Theory Related Fields. 128:347-385, 2004. A new characterization of Talagrand's transport entropy inequalities and applications. N Gozlan, C Roberto, P.-M Samson, Ann. Probab. 39Gozlan, N., Roberto, C.and Samson, P.-M. A new characterization of Talagrand's transport entropy inequalities and applications. Ann. Probab. 39:857-880, 2011. Analysis of stochastic partial differential equations. D Khoshnevisan, CBMS Regional Conference Series in Mathematics. ProvidenceAmerican Mathematical Society119Khoshnevisan, D. Analysis of stochastic partial differential equations. CBMS Regional Conference Series in Mathematics, 119. American Mathematical Society, Providence, 2014. Talagrand concentration inequalities for stochastic partial differential equations. D Khoshnevisan, A Sarantsev, Stoch. Partial Differ. Equ. Anal. Comput. 7Khoshnevisan, D. and Sarantsev, A. Talagrand concentration inequalities for stochastic partial differential equations. Stoch. Partial Differ. Equ. Anal. Comput. 7:679-698, 2019. The concentration of measure phenomenon. Mathematical Surveys and Monographs. M Ledoux, American Mathematical Society89ProvidenceLedoux, M. The concentration of measure phenomenon. Mathematical Surveys and Monographs, 89. American Mathematical Society, Providence, 2001. Transportation inequalities for stochastic differential equations with jumps. Y Ma, Stochastic Process. Appl. 120Ma, Y. Transportation inequalities for stochastic differential equations with jumps. Stochastic Process. Appl. 120:2-21, 2010. Bounding d-distance by informational divergence: a method to prove measure concentration. K Marton, Ann. Probab. 24Marton, K. Bounding d-distance by informational divergence: a method to prove mea- sure concentration. Ann. Probab. 24:857-866, 1996. Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. F Otto, C Villani, J. Funct. Anal. 173Otto, F. and Villani, C. Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173:361-400, 2000. Concentration for multidimensional diffusions and their boundary local times. Probab. Theory Related Fields. S Pal, 154Pal, S. Concentration for multidimensional diffusions and their boundary local times. Probab. Theory Related Fields. 154:225-254, 2012. Transportation inequalities for stochastic differential equations driven by a fractional Brownian motion. B Saussereau, Bernoulli. 18Saussereau, B. Transportation inequalities for stochastic differential equations driven by a fractional Brownian motion. Bernoulli 18:1-23, 2012. Transportation inequalities under uniform metric for a stochastic heat equation driven by time-white and space-colored noise. S Shang, R Wang, Acta Appl. Math. 170Shang, S. and Wang, R. Transportation inequalities under uniform metric for a stochas- tic heat equation driven by time-white and space-colored noise. Acta Appl. Math. 170:81-97, 2020. Talagrand concentration inequalities for stochastic heat-type equations under uniform distance. S Shang, T Zhang, Electron. J. Probab. 24Shang, S. and Zhang, T. Talagrand concentration inequalities for stochastic heat-type equations under uniform distance. Electron. J. Probab. 24:1-15, 2019. Global well-posedness to stochastic reaction-diffusion equations on the real line R with superlinear drifts driven by multiplicative space-time white noise. S Shang, T Zhang, arXiv:2106.02879Shang, S. and Zhang, T. Global well-posedness to stochastic reaction-diffusion equa- tions on the real line R with superlinear drifts driven by multiplicative space-time white noise. arXiv:2106.02879. Two contrasting properties of solutions for one-dimensional stochastic partial differential equations. T Shiga, Canad. J. Math. 46Shiga, T. Two contrasting properties of solutions for one-dimensional stochastic partial differential equations. Canad. J. Math. 46:415-437, 1994. Transportation cost for Gaussian and other product measures. M Talagrand, Geom. Funct. Anal. 6Talagrand, M. Transportation cost for Gaussian and other product measures. Geom. Funct. Anal. 6:587-600, 1996. New concentration inequalities in product spaces. M Talagrand, Invent. Math. 126Talagrand, M. New concentration inequalities in product spaces. Invent. Math. 126:505- 563, 1996. An introduction to analysis on Wiener space. A S Üstünel, Lecture Notes in Mathematics. Springer-VerlagÜstünel, A. S. An introduction to analysis on Wiener space. Lecture Notes in Mathe- matics, 1610. Springer-Verlag, Berlin, 1995. An introduction to stochastic partial differential equations. J B Walsh, Lecture Notes in Math. SpringerWalsh, J. B. An introduction to stochastic partial differential equations. Lecture Notes in Math., 1180. Springer, Berlin, 1986. Probability distance inequalities on Riemannian manifolds and path spaces. F.-Y Wang, J. Funct. Anal. 206Wang, F.-Y. Probability distance inequalities on Riemannian manifolds and path spaces. J. Funct. Anal. 206:167-190, 2004. Talagrand inequality on free path space and application to stochastic reaction diffusion equations. F.-Y Wang, T.-S Zhang, Acta Math. Appl. Sin. Engl. Ser. 36Wang, F.-Y. and Zhang, T.-S. Talagrand inequality on free path space and application to stochastic reaction diffusion equations. Acta Math. Appl. Sin. Engl. Ser. 36:253-261, 2020. Transportation inequalities for stochastic differential equations of pure jumps. L Wu, Ann. Inst. Henri Poincaré Probab. Stat. 46Wu, L. Transportation inequalities for stochastic differential equations of pure jumps. Ann. Inst. Henri Poincaré Probab. Stat. 46:465-479, 2010. Talagrand's T 2 -transportation inequality w.r.t. a uniform metric for diffusions. L Wu, Z Zhang, Acta Math. Appl. Sin. Engl. Ser. 20Wu, L. and Zhang, Z. Talagrand's T 2 -transportation inequality w.r.t. a uniform metric for diffusions. Acta Math. Appl. Sin. Engl. Ser. 20: 357-364, 2004. Talagrand's T 2 -transportation inequality and log-Sobolev inequality for dissipative SPDEs and applications to reaction-diffusion equations. L Wu, Z Zhang, Chinese Ann. Math. Ser. B. 27Wu, L. and Zhang, Z. Talagrand's T 2 -transportation inequality and log-Sobolev in- equality for dissipative SPDEs and applications to reaction-diffusion equations. Chinese Ann. Math. Ser. B 27:243-262, 2006.
{'fraction_non_alphanumeric': 0.1261461072367669, 'fraction_numerical': 0.04312335657939865, 'mean_word_length': 3.2163308589607635, 'pattern_counts': {'":': 0, '<': 23, '<?xml version=': 0, '>': 50, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 43, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'In this paper, we established quadratic transportation cost inequalities for solutions of stochastic reaction diffusion equations driven by multiplicative space-time white noise on the whole line R. Since the space variable is defined on the unbounded domain R, the inequalities are proved under a weighted L 2 -norm and a weighted uniform metric in the so called L 2 tem , C tem spaces. The new moments estimates of the stochastic convolution with respect to space-time white noise play an important role. In addition, the transportation cost inequalities are also obtained for the stochastic reaction diffusion equations with random initial values.', 'arxivid': '2305.19739', 'author': ['Yue Li ', 'Shijie Shang ', 'Tusheng Zhang '], 'authoraffiliation': [], 'corpusid': 258988043, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 19099, 'n_tokens_neox': 16636, 'n_words': 9070, 'pdfsha': 'c27bb3b95c62745a136a560371565ad34324e812', 'pdfurls': ['https://export.arxiv.org/pdf/2305.19739v1.pdf'], 'title': ['Transportation cost inequalities for stochastic reaction diffusion equations on the whole line R', 'Transportation cost inequalities for stochastic reaction diffusion equations on the whole line R'], 'venue': []}
arxiv
Lorentz violation and higher-derivative gravity 8 Sep 2014 C A Hernaski H Belich Departamento de Física e Química Indiana University Center for Spacetime Symmetries 47405BloomingtonIndianaUSA Universidade Federal do Espírito Santo Avenida Fernando Ferrari, 514, Goiabeiras29060-900VitóriaESBrazil Lorentz violation and higher-derivative gravity 8 Sep 2014numbers: 0460-m0450Kd1130Er Keywords: Lorentz BreakingGravity ModelsHigher Derivatives * Electronic address: chernask@indianaedu † Electronic address: belichjr@gmailcom In this work, we analyze a gravity model with higher derivatives including a CPT-even Lorentzviolating term. In principle, the model could be a low-energy limit of a Lorentz-invariant theory presenting the violation of Lorentz symmetry as a consequence of a spontaneous symmetry-breaking mechanism if a decoupling between the metric and the Nambu-Goldstone modes is assumed. We have set up a convenient operator basis for the expansion of wave operators for symmetric secondrank tensors in the presence of a background vector. By using this set of operators, the particle content is obtained, and its consistency, regarding the conditions for stability and unitarity, is discussed. We conclude that this extra Lorentz noninvariant contribution is unable to address the problems of stability and unitarity of higher-derivative gravity models. I. INTRODUCTION Problems involving the consistent quantization of the gravitational interaction have been widely discussed and are well known. The standard quantization of the Einstein-Hilbert (EH) Lagrangian using the formalism of perturbative quantum field theory (QFT) turns out to be inadequate, since it results in a non-renormalizable model [1]. Attempts to formulate a consistent quantum gravity model in four spacetime dimensions led to the proposal of additional terms involving higher derivatives. These terms modify the propagating structure of the excitations (including the graviton) in order to improve the ultraviolet (UV) behavior of the scattering amplitudes [2][3][4]. However, invariably, they also introduce negative-norm excitations called ghost particles that damage the unitarity of the S matrix. As discussed in Ref. [5], this is a general feature of gravitational models built up with diffeomorphismcovariant terms and points to an incompatibility between renormalizability and unitarity of the S matrix. Many attempts to try to circumvent this incompatibility were discussed in the literature [2][3][4][5][6]. An interesting possibility that evades some previous assumptions is to consider the Lorentz and diffeomorphism-symmetry-breaking effects on these gravity models, which, in turn, should describe the phenomenological aspects of general relativity (GR) in a low-energy limit. The interest in models that present the breaking of the Lorentz symmetry has increased after Kostelecký and Samuel showed that string theory, one of the leading candidates to handle the issue of consistent quantization of gravity, may present some phases where Lorentz symmetry is violated through a spontaneous symmetry-breaking mechanism triggered by the appearance of nonvanishing expectation vacuum values of nontrivial Lorentz tensors [7]. In this case, the models with Lorentz symmetry breaking are considered effective, and the analysis of their phenomenological aspects at low energies may provide information and impose restrictions on the fundamental theory from which they stem. A general framework for testing the low-energy manifestations of CPT and Lorentz symmetry breaking is the Standard-Model Extension (SME) [8]. In this framework, the effective Lagrangian corresponds to the usual Lagrangian of the Standard Model (SM), to which is added SM operators of any dimensionality contracted with Lorentz-violating (LV) tensorial background coefficients. The effective Lagrangian is written in a Lorentz-invariant form under coordinate transformations to guarantee the observer independence of physics. However, the physically relevant transformations are those that affect only the dynamical fields of the theory. These changes are called particle transformations, whereas the coordinate transformations (including the background tensor) are called observer transformations. In Refs. [9,10], these concepts are more deeply analyzed. To contemplate the possible consequences of CPT and LV for experiments where the gravitational interaction plays an important role, gravity was also included in the general framework of the SME [11]. One of the most striking consequences of this fusion is the requirement of a dynamic mechanism for the diffeomorphism/Lorentz-invariance violation -that is, the violation of these spacetime symmetries should be spontaneous to be consistent with the geometrical aspects of the gravitational interaction [12]. In practice, this means that the tensor coefficients appearing in the diffeomorphism/Lorentz-violating operators must be replaced by dynamical fields, and these, in turn, under a specific dynamic mechanism, acquire nonvanishing vacuum expectation values. As compared with the scenario of explicit breaking, one finds extra Nambu-Goldstone and massive modes that are responsible for restoring the particle diffeomorphism/Lorentz invariance of the model [13,14]. Concerning the experimental searches for the CPT/Lorentz-violation signals, the generality of the SME has provided the basis for many investigations. In the flat spacetime limit, the empirical studies include muons [15], mesons [16,17], baryons [18], photons [19], electrons [20], neutrinos [21], and the Higgs [22] sector. The gravity sector was also explored in Refs. [23][24][25][26]. In Ref. [27], one can find the current limits on coefficients for Lorentz violation. The aim of this work is to investigate if, by considering a diffeomorphism/Lorentzviolating CPT-even term, one can get a model with the spectrum free from instabilities and, at the same time, accompanied by higher derivatives, since an improved UV behavior of the scattering amplitudes is expected in the presence of such terms. For the discussion of these issues, we have analyzed the properties of the particle propagators that show up in this model. For this task, we have concentrated on the quadratic Lagrangian resulting from the expansion of the dynamical fields around a vacuum solution to the equations of motion. For simplicity, we have decided to focus only on the propagating modes that come from the dynamical metric, leaving aside the Nambu-Goldstone and massive modes that should appear as the result of the spontaneous symmetry-breaking mechanism. This choice relies on the fact that one can write interacting Lagrangians where these extra modes decouple from the metric modes, and one can study the properties of the latter independently of the former. However, one should be aware that the full consistency of the interacting system should include these extra modes. We hope that this inconsistency does not appear perturbatively at the quadratic level, but this possibility should be explicitly investigated [28]. Assuming the validity of such a decoupling, we work out the propagators of the model using a suitable operator basis for the wave operator that highlights the physical degrees of freedom (d.o.f.) propagating in the model, as well as keeping track of the role of the Lorentz breaking in providing extra couplings between these d.o.f. In the following, we perform a general analysis of the stability and unitarity, pointing out the extra subtleties brought by the LV. Our conclusion is that the extra room obtained by removing the Lorentz invariance is not enough to render the higher-derivative model consistent. Unless we consider the model from a effective point of view and assume some parameters of the Lagrangian to be of the same order as some assumed cutoff, the model has insurmountable consistency difficulties even in the infrared regime. This paper is organized as follows: In Sec. II, we set up the notation and conventions to be used, and we start with a general Lagrangian containing the above mentioned LV term in addition to the EH and curvature-square terms. The collection and analysis of propagators is carried out in Sec. II A. The conditions for stability of the spectrum are attained in Sec. II B. In Sec. III, we present our discussions and conclusions. II. GRAVITATION WITH HIGHER DERIVATIVES AND VIOLATION OF LORENTZ SYMMETRY In Ref. [29] is carried out an analysis of the spectral consistency of a gravitational model in 3 + 1 dimensions that contains, in addition to the EH term, terms with higher derivatives (R 2 and R ab R ab ) plus two CPT-odd terms which manifest the breaking of the CPT and diffeomorphism/Lorentz symmetry through a constant background vector: a Chern-Simons-like and a Ricci-Cotton-like term. As in the case of gravitation without diffeomorphism/Lorentz breaking, we concluded that the higher-derivative terms should be avoided to get a stable model. The only consistent combination is the EH Lagrangian added to the Chern-Simons-type term with a timelike background vector. In this work, we address the analysis of the compatibility of unitarity with higher-derivative terms by considering a CPTeven diffeomorphism/Lorentz-violating operator. We consider the following Lagrangian: L = √ −g αR + βR ab R ab + γR 2 + 1 4 κ abcd R abf g R f g cd .(1) The parameters α, β, and γ are arbitrary, and the background tensor κ abcd is responsible for the breakdown of the diffeomorphism invariance that, whenever we consider the linearized model, will lead to the breaking of Lorentz symmetry. To simplify the discussion, we will make the additional assumption that this tensor is constructed with a background vector b a [7]. Then, we have κ abcd = 1 2 κ ac g bd + κ bd g ac − κ bc g ad − κ ad g bc ,(2) with κ ab = κ b a b b − 1 4 g ab b · b(3) and κ = 4 3 κ ab b a b b .(4) The conventions adopted for the Riemann tensor, Ricci and scalar curvature are those followed by Ref. [32]. That is, R a bcd = ∂ c Γ a bd − ∂ d Γ a bc + Γ a ce Γ e bd − Γ a de Γ e bc ,(5)R bc = R a bac , R = g ab R ab ,(6) and the Christoffel symbol Γ a bc is given by Γ a bc = 1 2 g ad (∂ b g dc + ∂ c g db − ∂ d g bc ) .(7) The background tensor given in Eq. (2) has the same symmetries as the Riemann tensor, as can easily be seen by the structure of the breaking term in Eq. (1). We can assume that the origin of the background vector, b a , is due to a spontaneous breaking of Lorentz symmetry in a more fundamental theory, as for example in string theory [7]. The extra modes coming from the spontaneous symmetry-breaking mechanism are not discussed, and we expect that for some classes of models this can be done independently without disturbing the present analysis, as discussed in the Introduction. Furthermore, the nature of the background vector, if it is space-, time-, or lightlike, is not assumed a priori. We discuss the consistency of the model, case by case, in the following sections. The values of the constants appearing in Eq. (1) must be prescribed in such a way to obtain a model with a particle content free of ghosts and tachyons. The Minkowski metric corresponds to a possible solution of the Euler-Lagrange equations obtained from Eq. (1). Therefore, we can consider a perturbation of the metric field around the Minkowski metric, g ab = η ab + h ab ,(8) where η ab = diag (1, −1, −1, −1). In terms of h ab , the Lagrangian [Eq. (1)] establishes the dynamics for this field in an anisotropic spacetime scenario. Furthermore, we will assume that the background vector is constant in the asymptotic Minkowski coordinates ∂ a b c = 0.(9) We should highlight that this condition is not equivalent to the covariant constancy of the background vector, b µ , under general diffeomorphisms. The (covariant) constancy of b µ does not hold for arbitrary manifolds; it rather imposes a constraint on the curved space [11]. For that reason, we are not assuming b µ to be covariantly constant. We relax this more stringent condition and, in simply considering metric fluctuations (weak-field approximation), it is legitimate to consider constancy in the asymptotic Minkowskian sense, as stated above in Eq. (9). The particle content described by this model can be examined considering the propagators, whose structures depend only on the quadratic Lagrangian in h ab . Up to total derivatives, it is given by (L) 2 = α 2 1 2 h µν h µν − 1 2 h h + h∂ µ ∂ ν h µν − h µν ∂ µ ∂ λ h λ ν + β 2 h 2 h − 2h ∂ µ ∂ ν h µν + h µν ∂ µ ∂ ν ∂ κ ∂ λ h κλ + γ 8 h µν 2 h µν − 2h ∂ µ ∂ ν h µν + 2h µν ∂ µ ∂ ν ∂ κ ∂ λ h κλ + h 2 h + κb a b c 4 2h ag ∂ f ∂ d ∂ c ∂ g h f d − h d g ∂ f ∂ a ∂ c ∂ g h f d − 2h af ∂ d ∂ c h f d + h d f ∂ a ∂ c h f d − h ag ∂ f ∂ g h f c + h af 2 h f c − κb · b 8 h ag ∂ f ∂ d ∂ a ∂ g h f d + h d f 2 h f d − 2h af ∂ d ∂ a h f d .(10) It is important to notice that, although the background vector breaks the diffeomorphism invariance of the Lagrangian [Eq. (1)], the quadratic Lagrangian has the gauge invariance h ′ ab = h ab + ∂ a ξ b + ∂ a ξ b .(11) This is due to the fact that the model in Eq. (1) is constructed using only scalar and tensor curvatures. Under general diffeomorphism transformations, these quantities transform covariantly (or invariantly in the case of scalars). When considering the linear contribution in the field h ab , these terms are invariant under the gauge transformation [Eq. (11)] arising from the linearized diffeomorphisms x ′µ = x µ + ξ µ (x) .(12) As in GR, the gauge invariance in Eq. (11) is responsible for inhibiting the propagation of a spin-1 and a spin-0 mode [30]. Moreover, dynamically, the graviton keeps its two helicity d.o.f, as we shall see in the next section. A. Obtaining the propagators To characterize the particle content that propagates in this spacetime scenario, we analyze the propagator structure of the excitations. Then, we first write Eq. (10) as follows: (L) 2 = 1 2 h ab O ab;cd h cd ,(13) where O is the wave operator. The task of obtaining the propagators is equivalent to finding the inverse of this operator. For this, we will use an algebraic method by finding an operator basis in terms of which we can expand it. The two main criteria for the construction of this basis will be algebraic simplicity and easy physical interpretation. As we will see later, the first criterion is fulfilled if we build an operator basis consisting of projectors and mappers. The vector space of the field in question is then divided into subspaces defined by projectors, and those with the same dimension can be mapped in each other by means of the mappers. Our interest is in the particle content described by the model. Thus, in order to satisfy the second criterion, we define the subspaces in such a way as to associate a well-defined spin to the d.o.f. of the fields that reside in these subspaces. In models with Lorentz invariance, the kinetic terms of the Lagrangian are built using only metrics and derivatives. Thus, the kinetic term for the field h ab , if it is Lorentz invariant, should be such that the wave operator could be expanded in terms of Barnes-Rivers operators [31]. These, in turn, are given by P (2) ab;cd = 1 2 (θ ac θ bd + θ ad θ bc ) − 1 3 θ ab θ cd ),(14)P (1) ab;cd = 1 2 (θ ac ω bd + θ ad ω bc + θ bc ω ad + θ bd ω ac ),(15)P 11 (0) ab;cd = 1 3 θ ab θ cd ,(16)P 22 (0) ab;cd = ω ab ω cd ,(17)P 12 (0) ab;cd = 1 √ 3 θ ab ω cd ,(18)P 21 (0) ab;cd = 1 √ 3 ω ab θ cd .(19) However, the presence of background tensors allows extra couplings to the fields in such a way that the Barnes-Rivers operators are, in general, insufficient to expand the wave operators containing these structures. We can arrange the terms of the quadratic Lagrangian [Eq. (10)] into two distinct classes. One is composed of terms in which at least one of the h fields is contracted with the background vector, and the other presents terms in which the background vectors are contracted with each other or with the derivatives. Thus, we rewrite Eq. (13) as follows: (L) 2 = 1 2 h ab O 1 ab;cd h cd + 1 2 h ab O 2 ab;cd h cd ,(20) with O 1 ab;cd = 1 2 κp 2 [− (v · p) (v d η bc p a − v a ω cd p b + v a η bc p d −v d ω ab p c ) + p 2 (v a v d η bc − v a v d ω bc )] + · · ·(21) and O 2 = αp 2 1 2 η ac η bd − 1 2 η ab η cd − η ac ω bd + η ab ω cd + βp 4 1 2 η ac η bd + 11 6 η ab η cd + 7 3 ω ab ω cd − η ac ω bd − 11 3 η ab ω cd + 2 3 γp 4 (η ab η cd − 2η ab ω cd + ω ab ω cd ) − 1 4 κv 2 p 4 η ac η bd + 13 3 ω ab ω cd − 2η ac ω bd + 10 3 η ab η cd − 20 3 η ab ω cd + 1 2 κp 2 (v · p) 2 (η bc η ad − η ad ω bc ) + · · · .(22) The ellipses in the above expressions refer to terms with the same prior structure, but switching a ←→ b, b ↔ c, ab ↔ cd in such a way that the wave operator is symmetric under these exchanges. The fundamental structures that effectively contribute to the operational character of O 2 correspond to metrics and derivatives or metrics and momenta in Fourier space. Thus, we expect to expand O 2 in terms of the Barnes-Rivers operators [Eqs. (14) - (19)]. In fact, one can show that O 2 ab;cd = 1 2 p 2 α + β − 1 2 κb 2 p 2 P (2) + 6β + 2γ − 11 4 κb 2 p 2 − α P 11 (0) ab;cd + 1 2 κp 2 (b · p) 2 P (2) + P 11 (0) + 1 2 P (1) ab;cd .(23) Once the O 1 operator contains in its internal structure the background vector, b a , we cannot expand it in terms of the Barnes-Rivers operators. The direction in spacetime defined by b a breaks Lorentz symmetry, promoting the coupling between the d.o.f. of the distinct spins defined by Eq. (14) - (17). In the cases of spins 1 and 2, which define subspaces with dimension 3 and 5, respectively, the background vector yields a complete splitting into onedimensional subspaces. Furthermore, the structure of the terms in Eq. (21) b a = b · p p 2 p a + p 2 * p 2 e 3 a ,(24) where p 2 * = (b · p) 2 − b 2 p 2(25) and e 3 a is a spacelike vector orthogonal to p. If we define e 1 and e 2 such that e i · e j = −δ ij , e i · p = 0,(26) we can split the transverse operator as θ = ρ + σ + τ,(27) with ρ ab = −e 1 a e 1 b , σ ab = −e 2 a e 2 b , τ ab = −e 3 a e 3 b .(28) which act in each one of the spin subspaces. Also, we can define mappers among the subspaces defined by these projectors. Any operator, projector, or mapper can be written, up to changes of basis by unitary transformations, as P (IJ) (ij) = (−) R+S ψ (I) (i) ψ (J) (j) ,(29) where ψ (I) (i) is the ith eigenvector of the Barnes-Rivers spin projector P (I). R, S can assume the values 0 and 1, corresponding to the spin parities +1 or −1, respectively. By convention, we choose the subscripts i and j to range from 1 to 5 for the spin 2 (I, J = 2), to assume the values 6 and 7 for the 0-spins (I, J = 0) defined by Eqs. (16) and (17), respectively, and to range from 8 to 10 for the spin 1 (I, J = 1). In terms of the vectors in Eq. (26), we can write the ψ eigenvectors as ψ(2) (1)ab = 1 √ 2 (e 1a e 2b + e 2a e 1b ) ,(30)ψ(2) (2)ab = 1 √ 2 (e 1a e 3b + e 3a e 1b ) ,(31)ψ(2) (3)ab = 1 √ 2 (e 2a e 3b + e 3a e 2b ) ,(32)ψ(2) (4)ab = 1 √ 2 (ρ ab − σ ab ) ,(33)ψ(2) (5)ab = 1 √ 6 (ρ ab + σ ab − 2τ ab ) ,(34)ψ(0) (6)ab = 1 √ 3 θ ab ,(35)ψ(0) (7)ab = ω ab ,(36)ψ(1) (8)ab = − 1 √ 2 e 1 a p b p 2 + e 1 b p a p 2 ,(37)ψ(1) (9)ab = − 1 √ 2 e 2 a p b p 2 + e 2 b p a p 2 ,(38)ψ(1) (10)ab = − 1 √ 2 e 3 a p b p 2 + e 3 b p a p 2 .(39) Using Eq. (26), the normalization of the ψ eigenvectors is given by ψ (I) (i)ab ψ (J) ab (j) = (−) P δ IJ δ ij .(40) The operators of the type P ij (JM) are objects that map the subspace defined by P ii (JJ) in the subspace defined by P jj (MM). The algebra that these operators satisfy is orthonormal and complete in the following sense: P ij (JM) ab;f g P kl (NP ) f g cd = δ jk δ M N P il (JP ) ab;cd ,(41)i,J P ii (JJ) = 1.(42) In terms of these operators, we can write O 1 as follows: O 1 = 1 2 κp 2 1 2 (b · p) 2 + b 2 p 2 (P 22 (2 − 2) + P 33 (2 − 2)) + 1 3 (b · p) 2 + 2b 2 p 2 P 55 (2 − 2) + (b · p) 2 (P 11 (2 − 2) + P 44 (2 − 2)) + 1 3 2 (b · p) 2 + b 2 p 2 P 11 (0 − 0) + √ 2 3 ((b · p) 2 − b 2 p 2 ) (P 15 (0 − 2) + P 51 (2 − 0)) .(43) The expansion of the wave operator in terms of the operators in Eq. (41) The sum over the JM indices is already contained in the sum over the ij ones, according to our notation in Eqs. (30) -(39). In the case where the matrices a(JM) are invertible, the propagator saturated with the emission and absorption sources of particles is given by Π = i ij a −1 (JM) ij J * ab P ij (JM) ab;cd J cd .(45) However, as already discussed, the Lagrangian of Eq. (10) has the gauge symmetry of Eq. (11), which implies the absence of spin-1 and spin-0 sectors, as defined by Eqs. (15) and (17), respectively. In this case, the wave operator is not invertible. Nevertheless, the gauge invariance also requires that the sources of emission and absorption satisfy conditions such that there is no emission and absorption of the modes that depend on the gauge. Thus, the contractions of the spin-1 and spin-0 operators [Eqs. (15) and (17) a (02) =              A 0 0 0 0 0 0 B 0 0 0 0 0 0 B 0 0 0 0 0 0 A 0 0 0 0 0 0 C F 0 0 0 0 F D              ,(46)with A = 1 2 p 2 α + βp 2 + κ (b · p) 2 − 1 2 b 2 p 2 ,(47)B = 1 2 p 2 α + βp 2 + 1 2 κ (b · p) 2 ,(48)C = 1 2 p 2 α + βp 2 + 1 3 κ (b · p) 2 + 1 2 b 2 p 2 ,(49)D = p 2 (6β + 2γ) p 2 − α + 1 6 κp 2 2 (b · p) 2 − 31 2 b 2 p 2 ,(50)F = √ 2 6 κp 2 (b · p) 2 − b 2 p 2 .(51) The constraints satisfied by the sources are given by p a J ab = p a J ba = 0,(52)τ ab J bc = τ ab J cb = 0,(53) where the last condition is valid only on the mass shell. These identities are responsible for the inhibition of the spin-1 and spin-0 modes associated with the gauge invariance of the model. As we can see, the matrix a(02) assumes a block-diagonal form. The lower-right block points to a coupling between the spin 0 and the fifth component of the spin 2. The other components of the spin 2 do not couple to the other spins. However, the distinction of the coefficients, A and B, indicates that the propagation of components 1 and 4 have dynamics independent of the components 2 and 3. This splitting of the spin-2 matrix into direct-sumof-U (1) complex-conjugate-related representations is compatible with the CPT invariance and the breaking of the Lorentz symmetry by a background vector. The structure of the a(02) matrix suggests that we can split it into three distinct blocks: This condition assures positive-energy modes for any value of the momentum p, preventing the arising of instabilities due to the absence of a minimum energy. a 14 (2) =   A 0 0 A   ,(54)a 23 (2) =   B 0 0 B   ,(55)a 56 (02) =   C F F D   .(56) For LV, the poles of the propagators may take forms more general than p 2 = m 2 , due to the presence of the background tensors. In this case, the discussion of stability is a much more subtle issue. The positivity of energy for all p and the observer Lorentz invariance are insufficient to prevent the appearance of spacelike momenta in the dispersion relations. In a strongly boosted frame, this implies the reemergence of negative energies, spoiling the supposed stability of the model. So, besides requiring the positivity of the energy for all p in one frame, one must also impose that spacelike momenta are absent or, equivalently, that the energy is positive in all frames. For a general quadratic form built from the energy and momentum, these requirements imply that the particle and antiparticle energy solutions of the dispersion relations should correspond to positive and negative branches of a hyperbola whose asymptotes lie inside or on the light cone in the Minkowski causal diagram. In spite of the possible solution to the stability problem in the way we have described, this pattern of dispersion relation raises other problems related to microcausality [33][34][35][36]. In fact, as discussed in Ref. [33], the microcausality condition, that expresses that fields should commute for spacelike separations of their arguments, imposes that the modulus of the group velocity of the wave packets, formed out from the superposition of plane waves that satisfy the associated dispersion relations, must be lower than or at most equal to 1 (in natural units). But this requirement, in general, clashes with the stability constraint, since there, the slopes of the asymptotes are the limit of the group velocity and, as discussed above, should be greater than or at least equal to 1. Apparently, the only exception is the limiting situation where the asymptotes lie exactly on the light cone. The mentioned difficulty in simultaneously attaining stability and causality in LV models originates in the role that these assumptions play in the search for a Lorentz-invariant description of quantum theory. For the moment, we must only discuss the conditions of stability and unitarity and shall comment where the microcausal problem can appear. A detailed investigation of this question should be considered, taking into account the role of the extra modes arising from the SSB mechanism responsible for the restoration of the Lorentz symmetry. Another criterion analyzed in this work concerns the probabilistic character of processes in QFT. The unitarity of the scattering S matrix is a compulsory requirement to achieve any reasonable interpretation of the results of scattering processes. Since the residue of the propagator evaluated at the poles provides information on the norms of the states associated with the propagating mode, we impose that these norms must be positive definite. Considering what was said about stability and unitarity, our investigation in this work concerns the implementation of the following conditions for the absence of tachyons and ghosts: p 0 = f p, m 2 > 0, p 2 ≥ 0 ∀ p,(57)IRes Π p 0 =f ( p) > 0,(58) where f ( p, m 2 ) is an arbitrary function of the momenta with the masses of modes, m 2 , depending on the constants of the model, including the background vector. These functions are obtained as roots of the polynomials defined by the determinants of matrices the in Eqs. (54)-(56). Using Eq. (29), we can rewrite the propagator as Π = i (−) R+S ij S * i A −1 ij JM, m 2 S j p 0 − f p, m 2 −1 ,(59) where S i = ψ ab i J ab and A −1 ij (JM, m 2 ) is the matrix a (JM) −1 with the pole f ( p, m 2 ) extracted. Thus, the positivity condition [Eq. (58)] for arbitrary sources is guaranteed if the eigenvalues of A −1 ij (JM, m 2 ) are positive (negative) defined for the case when R + S = 0 (1). Moreover, one can show that A −1 ij (JM, m 2 ) has only one non-null eigenvalue whenever evaluated at the pole, so that a positive (negative) eigenvalue is ensured by the positivity (negativity) of the trace A −1 ij (JM, m 2 ) evaluated at the pole. We can then rewrite the condition for the absence of ghosts as (−1) R+S trA −1 ij JM, m 2 | p 0 =f ( p,m 2 ) > 0.(60) Taking the inverse of the matrices in Eqs. (54)-(56), we can see that each one of them presents a simple massless pole (p 2 = 0) besides simple massive poles in the 14 and 23 sectors and a double massive pole in the 56 sector. The massive poles, which appear in matrices of the 14 and 23 sectors, can be put in the following general quadratic form: A ij p 2 0 + B ij p 2 + C ij | p|p 0 + D ij = 0,(61) where each pair of subindeces (i, j) can assume the values (1,4) and (2,3), referring to the respective alluded sectors. These sets of parameters are given by A 14 = β + 1 2 κ b 2 0 + b 2 ,(62)B 14 = − β − 1 2 κ b 2 0 − b 2 − κ b 2 cos 2 θ ,(63)C 14 = −2κb 0 b cos θ,(64)D 14 = α,(65)A 23 = β + 1 2 κb 2 0 ,(66)B 23 = −β + 1 2 κ b 2 cos 2 θ,(67)C 23 = −κb 0 | p| cos θ,(68)D 23 = α.(69) For general values of the parameters, Eq. (61) may describe ellipses, parabolas, or hyperbolas on the plane (p 0 ,| p|) with the center at the origin or any of its degenerate situations. For physical reasons, and in agreement with the condition for the absence of tachyons, we demand that it have two real roots for p 0 and that each one of these roots have single-valued positivity. Following our previous considerations, the positivity of energy is ensured in all reference frames if spacelike four-momenta are avoided. This can be reached by imposing the existence of asymptotes constrained to lie inside the light cone. This is only possible for the hyperbolic case. In terms of the parameters of the quadratic form, we have C 2 ij − 4A ij B ij > 0,(70)|m ± | = − C ij 2A ij ± C ij 2A ij − D ij A ij ≥ 1,(71)D ij /A ij < 0,(72) where m ± are the slopes of the two asymptotes of the hyperbola. The poles of the 56 sector are the solutions of the quartic equation of the form λ 1 p 4 0 + λ 2 p 4 + λ 3 | p| 3 p 0 + λ 4 | p|p 3 0 + λ 5 p 2 0 + λ 6 p 2 + λ 7 p 2 p 2 0 + λ 8 = 0,(73) where the coefficients (λ 1 , . . . , λ 8 ) depend on the Lagrangian parameters and on the angle θ between the vector b and the particle momentum p. In this case, there are many other possible situations for the curves in the plane (p 0 ,| p|) as compared with the previous one. Obviously, there are physically reasonable possibilities among them. As a simple example, we have the interesting situation in which the quartic form is separable in the product of two quadratic ones. Within this decomposition, we can demand that each one correspond to a hyperbola with the same desired properties of the previous discussion. Due to the complexity of analyzing this dispersion relation in general, let us concentrate on the other sectors and return to this case when we discuss the unitarity of the model. κ > 0, β < 0 b 2 > −2 β κ , b 2 0 + b 2 < 2 β κ , α > 0,(74)κ < 0, β > 0 b 2 > −2 β κ , b 2 0 + b 2 < 2 β κ , α < 0,(75) 23 Sector : κ > 0, β < 0, b 2 0 < 2 β κ , α > 0,(76)κ < 0, β > 0, b 2 0 < 2 β κ , α < 0,(77) in order to avoid instabilities coming from these sectors. It is worthwhile emphasizing that the same set of conditions is required for any choice of the background vector: b 2 > 0, b 2 < 0, or b 2 = 0. We can assume the Sun-centered reference frame (SCRF) to be the one where the com- J * ab 1 2 α + κ (v · p) 2 (η ac η bd + η ad η bc ) − η ab η cd p 2 =0 J cd > 0.(78) We recognize the structure inside the brackets as the residue matrix of the graviton propagator. In comparison with the EH model, we have the appearance of the κ (v · p) 2 term following the Newton constant α. This contribution to the Newton constant, due to the LV term, denotes a better UV behavior, at least on the mass-shell analysis, and could have interesting consequences in the searches for renormalizable gravity models. To assure the positivity of the residue of the propagator for the massless mode [Eq. (78)], we must impose α + κ (v · p) 2 > 0.(79) For any type of background vector, this condition is satisfied for all p if we require that κ > 0,(80)α > 0.(81) By reanalysing the conditions for stability given in Eqs. (74)-(77), we conclude that only the conditions in Eqs. (74) and (77) are compatible with these new constraints. One important point to notice is that, even if we relax the constraint on the positivity of κ by requiring the positivity of the combination α + κ (v · p) 2 up to some momentum cutoff, we still have problems with the conditions in Eqs. (75) and (77), since they require α < 0, whereas low-energy unitarity requires α > 0. For the massive poles of the ij sectors, the condition on the residues of the propagators demands that p 0 − f n ij ( p) tra −1 ij p 0 =f n ij > 0,(82)f 14 = κb 0 b · p β + 1 2 κ b 2 0 + b 2 +      κb 0 b · p β + 1 2 κ b 2 0 + b 2   2 + β − 1 2 κb 2 p 2 − κ b · p 2 − α β + 1 2 κ b 2 0 + b 2    1/2 ,(83)f 23 = κb 0 b · p 2 β + 1 2 κb 2 0 +      κb 0 b · p 2 β + 1 2 κb 2 0   2 + β p 2 − 1 2 κ b · p 2 − α β + 1 2 κb 2 0    1/2 .(84) Taking the inverse of the matrices (54) and (55) with the coefficients in Eqs. (47)- (51) and using the condition in Eq. (82), we get the following inequalities for the 14 and 23 sectors, respectively: 2 κb 0 b · p      κb 0 b · p 2 β + 1 2 κb 2 0   2 + β p 2 − 1 2 κ b · p 2 − α β + 1 2 κb 2 0    1/2 > (85) −2 κb 0 b · p 2 β + 1 2 κ b 2 0 + b 2 + κb 2 0 p 2 + κ b · p 2 + α, κb 0 b · p      κb 0 b · p 2 β + 1 2 κb 2 0   2 + β p 2 − 1 2 κ b · p 2 − α β + 1 2 κb 2 0    1/2 > (86) − κb 0 b · p 2 2 β + 1 2 κb 2 0 + 1 2 κ b · p 2 + α 2 + 1 2 κb 2 0 p 2 . From the condition in Eq. (74), we see that α > 0, β < 0, κ > 0, and β + 1 2 κ b 2 0 + b 2 < 0. So, the right-hand side of expression (86) is positive definite, and so should be the left-hand side, but this is false, since b 0 and b · p have no definite sign. The same reasoning may be applied to the expression (87) with the aid of the conditions in Eq. (76), and we again reach a contradiction. The conclusion of this analysis is that the propagation of these modes violates the unitarity of the S matrix. The only way to try to fix this problem is inhibiting the propagation of these modes. However, if we look for the quadratic form [Eq. (61)], we note that to force the absence of these modes in any reference frame we must impose that A = B = C = 0, and from Eqs. Considering our previous conditions for unitarity and stability, we must impose that γ > 0 in order to get a stable spin-0 mode. The residue matrix becomes only 1 2γp 2 , and the positivity of γ also ensures that this is a nonghost mode. With these choices, the model with Lagrangian αR + γR 2 propagates two modes: a massless spin 2, identified as the graviton, and a massive spin 0. This is a well-known model in the context of higher-derivative gravity [37]. In spite of the presence of the higher derivatives, the model must be complemented with another (R µν R µν ) 2 term to attain renormalizability, but as we have discussed in the Introduction, this insertion spoils the unitarity. III. DISCUSSION AND CONCLUSIONS We started with a general higher-derivative gravity model added with a LV CPT-even term also containing higher derivatives, since we intended to analyze the role of the Lorentzinvariance assumption in the incompatibility of renormalizability and unitarity of higherderivative gravity models. Our guide to propose such a term was grounded on the general framework of the Standard-Model Extension, where this kind of contribution arises as a spontaneous breaking of the Lorentz symmetry at some high energy in a fundamental theory. The assumption of spontaneous instead of explicit symmetry breaking is particularly mandatory in the gravity sector of the SME if we do not want to abandon the most usual geometrical interpretation of the gravitational interaction. The most striking signal of spontaneous symmetry breaking is the appearance of extra massless and massive modes as a result of the breaking. Our position was only to discuss the metric modes by hoping that this situation would correspond to scenarios where the interaction that promotes the coupling between the metric and extra modes does not affect the quadratic Lagrangian. To our mind, this assumption should be further investigated, since the extra modes are responsible for making the whole model consistent. An explicit example of a potential triggering the spontaneous breaking of the Lorentz symmetry and its complete analysis would shed some light on this question and also could change our conclusions. Another point to be highlighted is that, as an effective model, the SME only aims to provide lowenergy effects coming from some more fundamental theory, and as such, it cannot be pushed to extremely high energies without expecting that some inconsistencies will appear. In our investigation, we tried to discuss Lorentz violation independent of this restriction, but without clashing with it, and our general considerations of stability and unitarity point to the conclusion that the extra Lorentz noninvariant term is unable to fix the mentioned problem of the incompatibility of renormalizability and unitarity. Within the philosophy of effective field theories, the terms considered here can be accepted if we constrain the masses of the ghost particles to be of the same order of the cutoff energy of the model, and the new d.o.f of a more fundamental theory should be able to fix the situation. However, an effective discussion of the model should contemplate other possibilities for the violating terms that respect the same structure as the one considered here, and it would also be interesting to investigate them. establishes the possible coupling between these d.o.f. In order to present the projectors that compose the Barnes-Rivers operators, we firstly write the background vector in terms of the momentum and of a spacelike vector as presents, in general, coefficients a (JM) organized in matrix blocks that show a coupling between the J and M spins. To avoid redundancy in the notation, we denote the coefficient matrices associated with operators of the J and M spins simply as a (JM), with a random order in the appearance of the J and M letters. Thus, the a(JJ), a(MM), and a(MJ) coefficients are parts of the same matrix a(JM) in such a way that a wave operator is generally written as O = ij a ij (JM) P ij (JM) . The matrices a14 and a 23 only carry information about the dynamics of the spin 2, while the matrix a 56 shows the conjunction of the spin-0 d.o.f. with one of the d.o.f. of the spin 2. B. Analysis of the particle content To analyze whether the particle content described by the model in Eq. (20) is tachyon and ghost free, we explore the structure of the propagator [Eq. (45)] expressed in terms of the inverses of the matrices in Eqs. (54)-(56). The saturated propagator has poles whenthe determinants of these matrices vanish. In a Lorentz-invariant model, the zeros of these determinants always occur for values of p 2 such that p 2 = m 2 , with m 2 being defined as the particle mass. The condition for the absence of tachyons in this case is given by m 2 > 0. By inspecting the conditions in Eqs. (70)-(72) and using the two sets of parameters in Eqs. (62)-(65) and Eqs. (66)-(69), we conclude that the parameters of the Lagrangian [Eq. (1)] must satisfy the conditions 14 Sector : ponents of the background vector are sufficiently small in such a way as to match the many experiments that are in agreement with Lorentz invariance. In this sense, the relations in Eqs. (74)-(77) impose limits on the possible boosted reference frames related to the SCRF.Apparently, this dependence of the physical consistency on some restricted set of inertial observers clashes with the observer invariance of the starting Lagrangian. Nevertheless, one should be aware that our discussion about physical consistency relies on the validity of the perturbative approach to QFT and, for enough strongly boosted frames, the LV parameters are not supposed to be small anymore, jeopardizing the conclusions obtained with perturbation theory.Let us turn to the discussion of the fulfillment of the condition in Eq. (58). As we already stated, this is needed in order to get a unitary model. For the massless pole, we can use the source constraints [Eq. (53)], the inverse of matrices (54)-(56), and the condition for the absence of ghosts[Eq. (58)] at the pole p 2 = 0 to get where f n ij means the nth positive energy root of the polynomial dispersion relation of the ij sector. Only for the 56 sector can the n index be nontrivial, since there can be two positive energy solutions for the fourth-degree polynomial [Eq. (73)]. Solving Eq. (61) with the two sets of parameters [Eqs. (62)-(69)] and imposing the conditions in Eqs. (70)-(72), we get the two positive energy solutions (62)-(69) we see that this is only possible if κ = 0 and β = 0. That is, the LV term in the Lagrangian [Eq. (1)] brings unavoidable inconsistencies to the quantum perturbative description of the model expanded around the flat Minkowski metric and must be switched off in order to give rise a healthy particle spectrum. With this considerable change in the scenario, the 56 sector becomes approachable. By making κ = β = 0, the 0 component of the spin 2 decouples from the spin 0 in the matrix [Eq. (46)], and the other coefficients become equal (A = B = C = D), setting up the coefficient of the spin-2 projector. Furthermore, the expression (73) becomes αγp 2 − 1 2 α 2 = 0. ] with the sources should be null, resulting in the gauge invariant propagator given by Eq. (45) with only the inverses of the largest nondegenerate submatrices.In our model, the complete wave operator [Eq. (44)] presents only a nonvanishing matrix, that is given by AcknowledgmentsWe would like to express our thanks to V.A. Kostelecký for useful discussions. C.H. wishes also to thank the International Center for Spacetime Symmetries for their kind hospitality. This work has been supported by CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior-Brazil) and CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico, Brazil). S Weinberg, General Relativity: An Einstein Centenary Survey. S. W. Hawking and W. IsraelCambridge, EnglandCambridge University PressS. Weinberg, General Relativity: An Einstein Centenary Survey, edited by S. W. Hawking and W. Israel (Cambridge University Press, Cambridge, England, 1980). . D E Neville, Phys. Rev. D. 183535D. E. Neville, Phys. Rev. D 18 3535 (1978). . D E Neville, Phys. Rev. D. 21867D. E. Neville, Phys. Rev. D 21 867 (1980). . K S Stelle, Phys. Rev. D. 16953K. S. Stelle, Phys. Rev. D 16, 953 (1977). . E Sezgin, P Van Nieuwenhuizen, Phys. Rev. D. 213269E. Sezgin and P. van Nieuwenhuizen, Phys. Rev. D 21, 3269 (1980). . R Utiyama, Phys. Rev. 1011597R. Utiyama, Phys. Rev. 101, 1597 (1956); . T W B Kibble, J. Math. Phys. 2212T. W. B. Kibble, J. Math. Phys. 2, 212 (1961). . V A Kostelecký, S Samuel, Phys. Rev. D. 39683V. A. Kostelecký and S. Samuel, Phys. Rev. D 39, 683 (1989). . D Colladay, V A Kostelecký, Phys. Rev. D. 556760D. Colladay and V. A. Kostelecký, Phys. Rev. D 55, 6760 (1997). . D Colladay, V A Kostelecký, Phys. Rev. D. 58116002D. Colladay and V. A. Kostelecký, Phys. Rev. D 58, 116002 (1998). . H Belich, T Costa-Soares, M A Santos, M T D Orlando, Rev. Bras. Ensino Fís. 291H. Belich, T. Costa-Soares, M. A. Santos, and M. T. D. Orlando, Rev. Bras. Ensino Fís. 29, 1 (2007). . V A Kostelecký, Phys. Rev. D. 69105009V. A. Kostelecký, Phys. Rev. D 69, 105009 (2004). Explicit symmetry breaking is permitted if more general geometrical frameworks are considered in the description of gravity. In this case, the geometrical entities of the manifold implement directional dependences at each spacetime point corresponding to nonzero Lorentzviolating coefficients. Further discussion can be found in V. A Kostelecký, N Russell, R Tso, Phys. Lett. B. 716470Explicit symmetry breaking is permitted if more general geometrical frameworks are con- sidered in the description of gravity. In this case, the geometrical entities of the manifold implement directional dependences at each spacetime point corresponding to nonzero Lorentz- violating coefficients. Further discussion can be found in V. A. Kostelecký, N. Russell, and R. Tso, Phys. Lett. B 716, 470 (2012). . R Bluhm, Shu-Hong Fung, V A Kostelecký, Phys. Rev. D. 7765020R. Bluhm, Shu-Hong Fung, and V. A. Kostelecký, Phys. Rev. D 77, 065020 (2008). . R Bluhm, V A Kostelecký, Phys. Rev. D. 7165008R. Bluhm and V. A. Kostelecký, Phys. Rev. D 71, 065008 (2005). . V W Hughes, Phys. Rev. Lett. 87111804V. W. Hughes et al., Phys. Rev. Lett. 87, 111804 (2001); . R Bluhm, V A Kostelecký, C , R. Bluhm, V. A. Kostelecký, and C. . D Lane, Phys. Rev. Lett. 841098D. Lane, Phys. Rev. Lett. 84, 1098 (2000); . E O Iltan, J. High Energy Phys. 0616E. O. Iltan, J. High Energy Phys. 06 (2003) 016. . H Nguyen, KTeV CollaborationY B Hsiung, KTeV CollaborationarXiv:hep-ex/0112046Nucl. Phys. B, Proc. Suppl. 86312H. Nguyen (KTeV Collaboration), arXiv:hep-ex/0112046; Y. B. Hsiung et al., Nucl. Phys. B, Proc. Suppl. 86, 312 (2000); . J M Link, FOCUS CollaborationPhys. Lett. B. 5567J. M. Link et al. (FOCUS Collaboration), Phys. Lett. B 556, 7 (2003); . R Ackerstaff, OPAL CollaborationZ. Phys. C. 76401R. Ackerstaff et al. (OPAL Collaboration), Z. Phys. C 76, 401 (1997); . M Feindt, DELPHI CollaborationNo. 97-98 CONF 80DELPHI reportM. Feindt et al.(DELPHI Collaboration), DELPHI report No. 97-98 CONF 80 (1997); . K Abe, BELLE CollaborationPhys. Rev. Lett. 863228K. Abe et al.(BELLE Collaboration), Phys. Rev. Lett. 86, 3228 (2001); . B Aubert, BaBar CollaborationarXiv:hep-ex/0303043arXiv:hep-ex/0607103B. Aubert et al.(BaBar Collaboration), arXiv:hep-ex/0303043; arXiv:hep-ex/0607103. . V A Kostelecký, R Potting, Phys. Rev. D. 513923V. A. Kostelecký and R. Potting, Phys. Rev. D 51, 3923 (1995); . D Colladay, V A Kostelecký, Phys. Lett. B. 344259D. Colladay and V. A. Kostelecký, Phys. Lett. B 344, 259 (1995); . Phys. Rev. D. 526224Phys. Rev. D 52, 6224 (1995); . V A Kostelecký, R J Van Kooten, Phys. Rev. D. 545585V. A. Kostelecký and R. J. Van Kooten, Phys. Rev. D 54, 5585 (1996); . O Bertolami, Phys. Lett. B. 395178O. Bertolami et al., Phys. Lett. B 395, 178 (1997); . N Isgur, Phys. Lett. B. 515333N. Isgur et al., Phys. Lett. B 515, 333 (2001); . V A Kostelecký, Phys. Rev. V. A. Kostelecký, Phys. Rev. . Lett, 801818Lett. 80, 1818 (1998); . V A Kostelecký, Phys. Rev. D. 6116002V. A. Kostelecký, Phys. Rev. D 61, 016002 (1999); . V A Kostelecký, Phys. Rev. D. 6476001V. A. Kostelecký, Phys. Rev. D 64, 076001 (2001); . V A Kostelecký, R J Van Kooten, Phys. Rev. D. 82101702V. A. Kostelecký and R. J. Van Kooten, Phys. Rev. D 82, 101702(R) (2010). . D Bear, R E Stoner, R L Walsworth, V A Kostelecký, C Lane, Phys. Rev. Lett. 855038D. Bear, R. E. Stoner, R.L. Walsworth, V. A. Kostelecký, and C. D Lane, Phys. Rev. Lett. 85, 5038 (2000); . M A Humphrey, D F Phillips, R L Walsworth, Phys. Rev. A. 6263405M. A. Humphrey, D.F. Phillips, and R. L. Walsworth, Phys. Rev. A 62, 063405 (2000); . D F Phillips, M A Humphrey, E M Mattison, R E Stoner, R F C Vessot, R L Walsworth, Phys. Rev. D. 63111101D. F. Phillips, M. A. Humphrey, E. M. Mattison, R. E. Stoner, R. F. C. Vessot, and R. L. Walsworth, Phys. Rev. D 63, 111101 (2001); . M A Humphrey, Phys. Rev. A. V. A. Kostelecký and C. D. Lane68116010Phys. Rev. DM. A. Humphrey et al., Phys. Rev. A 68, 063807 (2003). V. A. Kostelecký and C. D. Lane, Phys. Rev. D 60, 116010 (1999); . V A Kostelecký, C D Lane, J. Math. Phys. 406245V. A. Kostelecký and C. D. Lane, J. Math. Phys. 40, 6245 (1999); . R Bluhm, V A Kostelecký, C , R. Bluhm, V.A. Kostelecký, C. . D Lane, N Russell, Phys. Rev. Lett. 8890801D. Lane, and N. Russell, Phys. Rev. Lett. 88, 090801 (2002); . F Canè, Phys. Rev. Lett. 93230801F. Canè et al., Phys. Rev. Lett. 93, 230801 (2004); . P Wolf, F Chapelet, S Bize, A Clairon, Phys. Rev. Lett. 9660801P. Wolf, F. Chapelet, S. Bize, and A. Clairon, Phys. Rev. Lett. 96, 060801 (2006). . S M Carroll, G B Field, R Jackiw, Phys. Rev. D. 411231S. M. Carroll, G. B. Field, and R. Jackiw, Phys. Rev. D 41, 1231 (1990); . V A Kostelecký, M Mewes, Phys. Rev. Lett. 87251304V. A. Kostelecký and M. Mewes, Phys. Rev. Lett. 87, 251304 (2001); . J A Lipa, J A Nissen, S Wang, D , J. A. Lipa, J. A. Nissen, S. Wang, D. . A Stricker, D Avaloff, Phys. Rev. Lett. 9060403A. Stricker, and D. Avaloff, Phys. Rev. Lett. 90, 060403 (2003); . Q G Bailey, V A , Q. G. Bailey and V. A. . Kostelecký, Phys. Rev. D. 7076006Kostelecký, Phys. Rev. D 70, 076006 (2004); . R Lehnert, R Potting, Phys. Rev. Lett. 93110402R. Lehnert and R. Potting, Phys. Rev. Lett. 93, 110402 (2004); . Phys. Rev. D. 70125010Phys. Rev. D 70, 125010 (2004); . B Feng, M Li, J.-Q Xia, X Chen, X Zhang, Phys. Rev. Lett. 96221302B. Feng, M. Li, J.-Q. Xia, X. Chen, and X. Zhang, Phys. Rev. Lett. 96, 221302 (2006); . V A Kostelecký, M Mewes, Phys. Rev. V. A. Kostelecký and M. Mewes, Phys. Rev. . Lett, Phys. Rev. Lett. 97140401Lett., Phys. Rev. Lett. 97, 140401 (2006); . B , Phys.Rev.Lett. 9841603B. Altschul, Phys.Rev.Lett. 98, 041603 (2007); . V , V. . A Kostelecký, M Mewes, Astrophys. J. Lett. 6891A. Kostelecký and M. Mewes, Astrophys. J. Lett. 689, L1 (2008); . Phys. Rev. D. 8015020Phys. Rev. D 80, 015020 (2009); . Phys. Rev. Lett. 110201601Phys. Rev. Lett. 110, 201601 (2013). . H Dehmelt, R Mittleman, R S Van Dyck, Jr , P Schwinberg, Phys. Rev. Lett. 834694H. Dehmelt, R. Mittleman, R. S. Van Dyck, Jr., and P. Schwinberg, Phys. Rev. Lett. 83, 4694 (1999); . R K Mittleman, I I Ioannou, H G Dehmelt, N Russell, Phys. Rev. R. K. Mittleman, I. I. Ioannou, H. G. Dehmelt, and N. Russell, Phys. Rev. . Lett, 832116Lett. 83, 2116 (1999); . G Gabrielse, Phys. Rev. Lett. 823198G. Gabrielse et al., Phys. Rev. Lett. 82, 3198 (1999); . R Bluhm, V , R. Bluhm, V. . A Kostelecký, N Russell, Phys. Rev. Lett. 822254A. Kostelecký, and N. Russell, Phys. Rev. Lett. 82, 2254 (1999); . Phys. Rev. Lett. 791432Phys. Rev. Lett. 79, 1432 (1997); . R Bluhm, V A Kostelecký, N Russell, Phys. Rev. D. 573932R. Bluhm, V. A. Kostelecký and N. Russell, Phys. Rev. D 57, 3932 (1998); . C , C. D. . Lane, Phys. Rev. D. 7216005Lane, Phys. Rev. D 72, 016005 (2005); . L.-S Hou, W.-T Ni, Y.-C M Li, Phys. Rev. Lett. 90201101L.-S. Hou, W.-T. Ni, and Y.-C.M. Li, Phys. Rev. Lett. 90, 201101 (2003); . R Bluhm, V A Kostelecký, Phys. Rev. Lett. 841381R. Bluhm and V. A. Kostelecký, Phys. Rev. Lett. 84, 1381 (2000); . B , B. . R Heckel, Phys. Rev. Lett. 9721603R. Heckel et al., Phys. Rev. Lett. 97, 021603 (2006); . H Müller, S Herrmann, A Saenz, A , H. Müller, S. Herrmann, A. Saenz, A. . C Peters, Lämmerzahl, Phys. Rev. D. 68116006Peters, and C. Lämmerzahl, Phys. Rev. D 68, 116006 (2003); . R Lehnert, J. Math. Phys. 453399R. Lehnert, J. Math. Phys. 45, 3399 (2004); . B , Phys. Rev. Lett. 96201101B. Altschul, Phys. Rev. Lett. 96, 201101 (2006). . V Barger, S Pakvasa, T J Weiler, K Whisnant, Phys. Rev. Lett. 855055V. Barger, S. Pakvasa, T. J. Weiler, and K. Whisnant, Phys. Rev. Lett. 85, 5055 (2000); . J , J. . N Bahcall, Phys. Lett. B. 534114N. Bahcall et al., Phys. Lett. B 534, 114 (2002); . V A Kostelecký, M Mewes, Phys. Rev. D. 7031902V. A. Kostelecký and M. Mewes, Phys. Rev. D 70, 031902 (2004); . V A Kostelecký, M Mewes, Phys. Rev. D. 7076002V. A. Kostelecký and M. Mewes, Phys. Rev. D 70, 076002 (2004); . V A Katori, R Kostelecký, Tayloe, Phys. Rev. D. 74105009Katori, V. A. Kostelecký, and R. Tayloe, Phys. Rev. D 74, 105009 (2006); . V A Kostelecký, M ; J S Mewes, V A Díaz, M Kostelecký, Mewes, Phys. Rev. D. 6976007Phys. Rev. DV. A. Kostelecký and M. Mewes, Phys. Rev. D 69, 016005 (2004). J. S. Díaz, V. A. Kostelecký, and M. Mewes, Phys. Rev. D 80, 076007 (2009); . V A Kostelecký, M Mewes, Phys. Rev. D. 8596005V. A. Kostelecký and M. Mewes, Phys. Rev. D 85, 096005 (2012); . J S Díaz, V A Kostelecký, Phys. Rev. D. 8516013J. S. Díaz and V. A. Kostelecký, Phys. Rev. D 85, 016013 (2012); . J S Díaz, V A , J. S. Díaz, V. A. . R Kostelecký, Lehnert, Phys. Rev. D. 8871902Kostelecký, and R. Lehnert, Phys. Rev. D 88, 071902(R) (2013); . J S Díaz, V A Kostelecký, M Mewes, Phys. Rev. D. 8943005J. S. Díaz, V. A. Kostelecký, and M. Mewes, Phys. Rev. D 89, 043005 (2014). . D L Anderson, M Sher, I Turan, Phys. Rev. D. 7016001D. L. Anderson, M. Sher, and I. Turan, Phys. Rev. D 70, 016001 (2004). . Q G Bailey, V A Kostelecký, Phys. Rev. D. 7445001Q. G. Bailey and V. A. Kostelecký, Phys. Rev. D 74, 045001 (2006). . V A Kostelecký, N Russell, J D Tasson, Phys. Rev. Lett. 100111102V. A. Kostelecký, N. Russell, and J. D. Tasson, Phys. Rev. Lett. 100, 111102 (2008). . V A Kostelecký, J D Tasson, Phys. Rev. Lett. 10210402V. A. Kostelecký and J. D. Tasson, Phys. Rev. Lett. 102, 010402 (2009). . V A Kostelecký, J D Tasson, Phys. Rev. Lett. 10210402V. A. Kostelecký and J. D. Tasson, Phys. Rev. Lett. 102, 010402 (2009). . V A Kostelecký, N Russell, Rev. Mod. Phys. 8311V. A. Kostelecký and N. Russell, Rev. Mod. Phys. 83, 11 (2011). The discussion of the stability and unitarity of a Lorentz-violating gravity model without higher derivatives, but taking into account the extra modes coming from the spontaneous symmetry-breaking mechanism was performed in R. V Maluf, C A S Almeida, R Casana, M M Ferreira, Jr , arXiv:1402.3554The discussion of the stability and unitarity of a Lorentz-violating gravity model without higher derivatives, but taking into account the extra modes coming from the spontaneous symmetry-breaking mechanism was performed in R. V. Maluf, C. A. S. Almeida, R. Casana, and M. M. Ferreira, Jr. arXiv:1402.3554. . B Pereira-Dias, C A Hernaski, J A Helayël-Neto, Phys. Rev. D. 8384011B. Pereira-Dias, C. A. Hernaski, and J. A. Helayël-Neto, Phys. Rev. D 83, 084011 (2011). . S Weinberg, Phys. Rev. 138988S. Weinberg, Phys. Rev. 138, B988 (1965). . R J Rivers, Il Nuovo Cimento. 34387R. J. Rivers, Il Nuovo Cimento 34, 387 (1964). C W Misner, K S Thorne, J A Wheeler, Gravitation. San FranciscoW. H. Freeman and CompanyC. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman and Company, San Francisco, 1973). . V A Kostelecký, R Lehnert, Phys. Rev. D. 6365008V. A. Kostelecký and R. Lehnert, Phys. Rev. D 63, 065008 (2001). . C Adam, F R Klinkhamer, Nucl. Phys. 607247C. Adam and F. R. Klinkhamer, Nucl. Phys. B607, 247 (2001). . C M Reyes, Phys. Rev. D. 82125036C. M. Reyes, Phys. Rev. D 82, 125036 (2010). . R Bluhm, N L Gagne, R Potting, A Vrublevskis, Phys.Rev. D. 77125007R. Bluhm, N. L. Gagne, R. Potting, and A.Vrublevskis, Phys.Rev. D 77, 125007 (2008). . P Van Nieuwenhuizen, Nucl. Phys. 60478P. van Nieuwenhuizen, Nucl. Phys. B60, 478 (1973).
{'fraction_non_alphanumeric': 0.0751525751711819, 'fraction_numerical': 0.053494343554629356, 'mean_word_length': 3.5099437777964253, 'pattern_counts': {'":': 0, '<': 15, '<?xml version=': 0, '>': 25, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 96, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'In this work, we analyze a gravity model with higher derivatives including a CPT-even Lorentzviolating term. In principle, the model could be a low-energy limit of a Lorentz-invariant theory presenting the violation of Lorentz symmetry as a consequence of a spontaneous symmetry-breaking mechanism if a decoupling between the metric and the Nambu-Goldstone modes is assumed. We have set up a convenient operator basis for the expansion of wave operators for symmetric secondrank tensors in the presence of a background vector. By using this set of operators, the particle content is obtained, and its consistency, regarding the conditions for stability and unitarity, is discussed. We conclude that this extra Lorentz noninvariant contribution is unable to address the problems of stability and unitarity of higher-derivative gravity models.', 'arxivid': '1409.5742', 'author': ['C A Hernaski ', 'H Belich ', '\nDepartamento de Física e Química\nIndiana University Center for Spacetime Symmetries\n47405BloomingtonIndianaUSA\n', '\nUniversidade Federal do Espírito Santo\nAvenida Fernando Ferrari, 514, Goiabeiras29060-900VitóriaESBrazil\n'], 'authoraffiliation': ['Departamento de Física e Química\nIndiana University Center for Spacetime Symmetries\n47405BloomingtonIndianaUSA', 'Universidade Federal do Espírito Santo\nAvenida Fernando Ferrari, 514, Goiabeiras29060-900VitóriaESBrazil'], 'corpusid': 119167949, 'doi': '10.1103/physrevd.89.104027', 'github_urls': [], 'n_tokens_mistral': 19605, 'n_tokens_neox': 16074, 'n_words': 9810, 'pdfsha': 'c785d33dfdd0ab7b906657ec11a99070ca74ec2d', 'pdfurls': ['https://arxiv.org/pdf/1409.5742v1.pdf'], 'title': ['Lorentz violation and higher-derivative gravity', 'Lorentz violation and higher-derivative gravity'], 'venue': []}
arxiv
Robust Topological Inference in the Presence of Outliers Siddharth Vishwanath Bharath K Sriperumbudur Kenji Fukumizu The Institute of Statistical Mathematics TokyoJapan Satoshi Kuriki The Institute of Statistical Mathematics TokyoJapan Department of Statistics The Pennsylvania State University University ParkPAUSA Robust Topological Inference in the Presence of Outliers The distance function to a compact set plays a crucial role in the paradigm of topological data analysis. In particular, the sublevel sets of the distance function are used in the computation of persistent homologya backbone of the topological data analysis pipeline. Despite its stability to perturbations in the Hausdorff distance, persistent homology is highly sensitive to outliers. In this work, we develop a framework of statistical inference for persistent homology in the presence of outliers. Drawing inspiration from recent developments in robust statistics, we propose a median-of-means variant of the distance function (MoM Dist), and establish its statistical properties. In particular, we show that, even in the presence of outliers, the sublevel filtrations and weighted filtrations induced by MoM Dist are both consistent estimators of the true underlying population counterpart, and their rates of convergence in the bottleneck metric are controlled by the fraction of outliers in the data. Finally, we demonstrate the advantages of the proposed methodology through simulations and applications. arXiv:2206.01795v1 [math.ST] 3 Jun 2022 computation of persistence diagrams comes from the celebrated stability of persistence diagrams . In a nutshell, the stability result for persistence diagrams guarantees that (i) the persistence diagrams resulting from two compact sets X and Y are close whenever the sets themselves are close in the Hausdorff distance, and, (ii) the functional persistence diagrams resulting from two filter functions f and g are close whenever f and g are close w.r.t. the · ∞ metric.In the statistical setting, one has access to X only through samples X n = {X 1 , X 2 , . . . , X n } obtained using a probability distribution P which is supported on the (unknown) set X. The objective, in a statistical inference framework, is to use the samples X n to infer the true population persistence diagram Dgm(V X ). The offset X n (r) and filter function f n , constructed using the sample points, are themselves random quantities associated with their population counterparts X(r) and f X , respectively, and these may be used to construct a sample estimator Dgm(V Xn ). To this end, several existing works have studied the statistical properties of Dgm(V Xn ), e.g., constructing confidence bands and characterizing the convergence rate of Dgm(V Xn ) to Dgm(V X ) in the space of persistence diagrams(Fasy et al., 2014;Chazal et al., 2015aChazal et al., ,b, 2017Vishwanath et al., 2020).ContributionsIn practical settings, real-world data is likely subject to measurement errors and the presence of outliers. While some assumptions may be imposed on the noise and the outliers, in the most baneful settings, the given data may be subject to adversarial contamination. In this setting, for m < n/2, we assume that the samples X n , which we have access to, contain only n − m points obtained from the probability distribution P with supp(P) = X, and make no further assumptions on the remaining m points. In principle, the m outliers may be carefully chosen by an adversary after examining the remaining n − m points. The overarching objective of this paper is to construct an estimator of the (unknown) population quantity Dgm(V[X]) using the corrupted sample points X n which is, both, statistically consistent and computationally efficient.While the stability of persistence diagrams guarantees that small perturbations in the sample points induce only small changes in the resulting persistence diagrams, even a few outliers in the samples can lead to deleterious effects. This issue is further exacerbated in the adversarial setting, where the adversary is free to place the m points where it may drastically impact the resulting topological inference.In this paper, we introduce MoM Dist, denoted by d n,Q , as an outlier-robust variant of the empirical distance function which is constructed using the median-of-means principle, and we establish its theoretical properties. Notably the MoM Dist relies on a tuning parameter Q which is easy to interpret. While the persistence diagram resulting from the sublevel filtration of d n,Q is a valid candidate for statistical inference, it can be expensive to compute in practice. To overcome this, we use the weighted filtrations introduced by Buchet et al.(2016)and Anai et al. (2019) to construct d n,Q -weighted filtrations, V [X n , d n,Q ], as computationally efficient estimators of Dgm(V [X]). Our main contributions are the following: (I) We show that sublevel set persistence diagrams of d n,Q are consistent estimators of the sublevel set persistence diagram of the true population counterpart d X even in the presence of outliers (Theorem 3.1).(II) We establish a stability result for the the d n,Q -weighted filtrations, V [X n , d n,Q ], and we show that they are stable w.r.t. adversarial contamination (Theorem 3.2).(III) Furthermore, we show that the persistence diagram Dgm(V[X n , d n,Q ]) is both a computationally efficient and statistically consistent estimator of Dgm(V[X]), and we establish its convergence rate (Theorem 3.3).(IV) Next, in a sensitivity analysis framework, we quantify the gain in robustness achieved when using the d n,Q -weighted filtrations vis-à-vis its non-robust d n -weighted counterpart (Theorem 3.4). Introduction Given a compact set X ⊂ R d , its persistence diagram encodes the subtle geometric and topological features which underlie X as a multiscale summary, and forms the cornerstone of topological data analysis. Persistent homology serves as the backbone for computing persistence diagrams, and encodes the homological features underlying X at different resolutions. The computation of persistent homology is typically achieved by constructing a filtration V X , i.e., a nested sequence of topological spaces, which captures the evolution of geometric and topological features as the resolution varies. The persistent homology, which is encoded in its persistence module, V X , extracts the homological information from the filtration V X . This is then summarized in a persistence diagram Dgm(V X ). Broadly speaking, there are two different methods for obtaining filtrations. The first, and, arguably more classical method is obtained by examining the union of balls of radius r centered on the points of X called the r-offset of X, denoted X(r), for each resolution r > 0. The resulting filtration V [X] = {X(r) : r > 0}, depends only on the metric properties of X. The second, and more general approach is based on constructing a filter function f X , which reflects the topological features underlying X. The resulting filtration V [f X ], in this case, is obtained by probing the sublevel sets f −1 X ((−∞, r]) or the superlevel sets f −1 X ([r, ∞)) associated with f X . While these two methods are vastly different, in principle, they both attempt to explore the topological features underlying X. In this context, the distance function d X to the set X plays a special role in topological data analysis, and satisfies the property that V [X] = V [d X ]. That is, the sublevel sets of the distance function encode the same topological information as the filtration from its offsets. The appeal of using the distance function in the (V) Lastly, we propose a data-driven procedure for adaptively selecting the tuning parameter Q using Lepski's method. For the data-driven choice Q, we show that the resulting estimator Dgm V[X n , d n, Q ] is statistically consistent and establish its convergence rate (Theorem 3.5). Related Work Several approaches have been proposed in existing literature to overcome the sensitivity of persistence diagrams to noise. The prevailing ideas in these approaches rely on constructing a filter function, f P , which reflects both the topological information and the distribution of mass underlying the support supp(P) = X. Replacing the population probability measure P with the empirical measure P n associated with the samples X n results in an empirical estimator f Pn . Some notable examples include the distance-to-measure (Chazal et al., 2011), the kernel distance (Phillips et al., 2015), and kernel density estimators (Fasy et al., 2014). While these approaches mitigate, to some extent, the influence of noise on the resulting persistence diagrams, they are not without their drawbacks. For starters, while it may be argued that Dgm(V[f Pn ]) is more resilient to noise, ultimately, this sample estimator corresponds to the population quantity Dgm(V[f P ]), which may, nevertheless, omit some subtle geometric and topological features present in Dgm(V[X]). Furthermore, from a statistical perspective, if X n comprises of only n − m points from P and the remaining m points constitute outliers, then the sample estimator V [f Pn ], obtained using X n , will no longer be a valid estimator of the population quantity Dgm(V[f P ]) which we wish to infer. Lastly, the exact computation of these estimators can be prohibitively expensive, if not impossible in practice. For instance, the exact computation of the distance-to-measure requires computing an order-k Voronoi diagram. Moreover, in the general setting, the sublevel/superlevel filtrations arising from these approaches are computed using cubical homology, which relies on a (nuisance) grid resolution parameter. If this resolution is too coarse, then some subtle topological features are affected. On the flipside, if the resolution is too fine, then the accuracy is still impacted, as noted in Fasy et al. (2014). In the high-dimensional setting, cubical homology also falls victim to the curse of dimensionality, i.e., for a fixed grid resolution, the number of simplices in the resulting cubical complex grows exponentially with the dimension of the ambient space. In order to overcome these computational drawbacks, Buchet et al. (2016) and Anai et al. (2019) propose weighted filtrations, V [X n , f Xn ], using power distances. While the weighted filtrations circumvent the need for constructing grid-based approximations, they come at the expense of exact inference, i.e., the weighted filtrations V [X n , f Xn ] only approximate V [f Xn ] and do not provide valid statistical inference, even in the absence of outliers. More recently, Vishwanath et al. (2020) propose robust persistence diagrams which are resilient to outliers using kernel density estimators (KDE), and also develop a principled framework for characterizing the sensitivity to outliers using an analogue of influence functions. Although Vishwanath et al. (2020, Theorem 1) describes the gain in robustness by considering the robust KDE f n ρ,σ using the persistence influence function, Vishwanath et al. (2020, Theorems 2 & 3) together establish that as n → ∞ and σ → 0, the persistence diagram Dgm f n ρ,σ recovers the same information which underlies the sample points X n . However, if the underlying distribution is contaminated, e.g., P * = (1 − π)P signal + πP noise , then the topological inference we hope to target is that of P signal and not that of P * . Finally, with a similar objective of mitigating the impact of noise in topological inference, recent approaches have considered multi-parameter persistent homology as a robust tool for inferring the topological features underlying X n (Carlsson and Zomorodian, 2009). While some recent results have demonstrated some promise (e.g., Vipond et al. 2021), they are, nevertheless, computationally infeasible for most applications, in addition to being hard to interpret (Otter et al., 2017;Bjerkevik et al., 2020). On the statistical front, robust statistics was founded on the seminal works of Tukey (1960) and Huber (1964) with the objective of developing a framework of statistical inference stable to model misspecification and the presence of extraneous errors. Over the past few decades, robust counterparts for several inference tasks have been explored in literature (Huber, 2004;Hampel et al., 2011). More recently, in the landscape of big-data and high-dimensional statistics, the field of robust statistics has witnessed a renewed interest in the statistics and computer science literature (Diakonikolas et al., 2017). In particular, the classical problem of mean and covariance estimation has been revisited in several works (Audibert and Catoni, 2011;Minsker, 2015;Devroye et al., 2016;Joly and Lugosi, 2016) with the objective of easing model assumptions to, either, the regularity of the data generating mechanism, or, the presence of outliers. See Lugosi and Mendelson (2019a) for a recent survey. A common theme underlying these works is the constant struggle to achieve a Goldilocks equilibrium: the right balance of statistical optimality, computational efficiency and robustness to model misspecification. In this regard, the median-of-means estimator, and, more broadly, the median-of-means principle (Lecué and Lerasle, 2020), has emerged as a powerful tool for "robustifying" an existing estimator in near linear time. Although this comes slightly at the expense of statistical optimality, median-of-means estimators are, nevertheless, easier to compute than statistically optimal and robust methods such as the tournament estimators introduced by Lugosi and Mendelson (2019b). However, computing the median in high dimensions is not a welldefined task, and can be computationally burdensome. To make matters worse, robust topological summaries naïvely employing the median-of-means principle require estimating the median in infinite-dimensional space, which can be hopeless to achieve in a computationally tractable fashion. Our work overcomes this limitation by proposing a pointwise median-of-means estimator which, although computationally tractable, exhibits a concentration of measure phenomenon with respect to the true target population counterpart in the · ∞ metric. Organization. The remainder of this paper is organized as follows. In Section 2 we present the necessary background on persistent homology and robust statistics. We first introduce the proposed methodology in Section 3.1, and then present the main results in the remainder of the section. We establish the statistical properties of the proposed estimator in Section 3.2, and we present the influence analysis in Section 3.4. Numerical results supporting the theory are provided in Section 4. The proofs of all the results are collected in Section 6. Preliminaries The following subsections introduce the essential ingredients used for the remaining of the paper. Definitions and Notations. For two sets A and B ⊆ A, id : B → A given by b → b denotes the identity map. For n ∈ Z + , we use the notation [n] = {1, 2, . . . , n}, and for real-valued functions f and g we employ the notation f (n) g(n) if f (n) = O g(n) . Given a metric space (M, ρ) with metric ρ : M × M → R ≥0 , the ball of radius r centered at x ∈ M is denoted B ρ (x, r) . For a compact set X ⊂ M, the r-offset of X w.r.t the metric ρ is given by X ρ (r) = x∈X B ρ (x, r). The distance function w.r.t. the compact set X plays a central role in extracting the geometric and topological features underlying X. When r < 0 we explicitly define Bρ(x, r) = ∅. Definition 2.1 (Distance function). For a metric space (M, ρ) and a compact set X ⊆ M, the distance function to the set X, denoted as d X , is given by d X (y) = · inf x∈X ρ(x, y), for all y ∈ M. For a finite collection of points X n , the distance function d Xn is simply denoted as d n . For two compact sets X, Y ⊂ (M, ρ) the Hausdorff distance between X and Y is given by H ρ (X, Y) = · max sup x∈X d Y (x), sup y∈Y d X (y) = inf { > 0 : X ⊆ Y ρ ( ), Y ⊆ X ρ ( )}, and metrizes the space of all compact subsets of (M, ρ). Throughout the paper we assume that (M, ρ) = (R d , · ) is the usual Euclidean space with the 2 metric, and omit the subscript ρ. However, the results here should extend to general metric spaces (M, ρ) with simple modifications along the lines of Chazal et al. (2015b) and Buchet et al. (2016). We use P(X) to denote the set of Borel probability measures defined on R d with support X ⊆ R d , and for x ∈ R d , δ x is used to denote a Dirac measure at x. A key assumption used throughout the paper is a regularity condition for the data generating mechanism. For a, b > 0, the probability measure satisfies the (a, b)−standard condition if P B(x, r) > 1 ∧ ar b for all r > 0.(1) We denote by P(X, a, b) the subset of P(X) which satisfies the (a, b)−standard condition in Eq. (1) for a, b > 0. This regularity assumption is standard in the domain of statistical shape analysis (e.g., Cuevas and Rodríguez-Casal 2004;Chazal et al. 2015b. Throughout the paper, we assume that the samples X n are obtained in an adversarial contamination setting (S), as defined below. S S (S) . The data comprises of n samples X n = {X 1 , X 2 , . . . , X n }, where m < n/2 samples are contaminated with unknown outliers. No distributional assumption is made on these outliers. The remaining n−m samples are observed iid from a distribution P ∈ P(X, a, b), for compact X ⊂ R d and a, b > 0. A glossary of notations for additional definitions and notations introduced in the subsequent sections is provided in Appendix A. Background on Persistent Homology In this section we provide the necessary background on persistent homology arising from single parameter filtrations. We refer the reader to ; Edelsbrunner and Harer (2010) for a detailed introduction. Given a compact set X, the building block of any topological data analysis pipeline to extract meaningful information from X begins with a nested sequence of filtered topological spaces called a filtration, simply denoted by V . The sequence of spaces are parametrized by a resolution parameter t. There are several approaches for constructing a filtration using X. One approach is to consider the collection of offsets built on top of X, i.e., V t = V t [X] = X(t). For s < t, the offsets are nested V s ⊆ V t , and V [X] = · V t [X] : t ∈ R is a nested sequence of topological spaces and defines the filtration built using the offsets of X. The second approach to constructing a filtration is using a filter function f X : R d → R which carries the topological information underlying X. In this scenario, one typically constructs the filtration from the sublevel sets associated with f X , given by V t = f −1 X (−∞, t] for each resolution t. Again, for s < t, V s ⊂ V t and the sequence V [f X ] = V t [f X ] : t ∈ R constitutes the sublevel filtration from f X . Mutatis mutandis a similar notion holds for the superlevel filtration. In general, the filtration V [X] can be very different from V [f X ], although the prevailing objective is for V [f X ] to encode the same information as in V [X]. In this context, the distance function d X plays a special role owing to the fact that its sublevel filtration is the same filtration associated with the offsets, i.e., V [d X ] = V [X]. This fact plays an important role in motivating the MoM Dist estimator introduced in Section 3.1, and follows by noting that for every resolution t > 0, d −1 X (−∞, t] = x ∈ R d : d X (x) ≤ t = x∈X B(x, t). Let V = V t : t ∈ R denote a generic filtration and let ι t s : V s → V t denote the inclusion map between the filtered spaces at resolutions s < t. For each resolution t, let V t = H * V t ; F be the homology of V t with coefficients in a field F. As the resolution t varies, the evolution of topological features is captured by V . Roughly speaking, new cycles (i.e., connected components, loops, holes, and higher dimensional analogues) are born, or existing cycles can merge and disappear. The collection of cycles in V t at each resolution t is encoded as a vector space in V t . The inclusion maps ι t s : V s → V t induce linear maps φ t s : V s → V t between the vector spaces V s and V t . As such, the collection V can be described more succinctly as the category V = V t , ι t s : s ≤ t with the inclusion maps ι t s representing the morphisms for s ≤ t. The image of V under the homology functor Hom * : V → V, gives us the persistence module V = · V t , φ t s : s ≤ t , where the induced maps φ t s : V s → V t are homomorphisms between two vector spaces. For r < s < t, the persistence module can equivalently be represented as . . . V r V s V t . . . φ t s φ s r Informally, a new topological feature is born at resolution b ∈ R if the cycle associated with that feature is not present in V b− for all > 0. The same feature is said to die at resolution d > b if the cycle associated with this feature disappears from V d+ for all > 0, resulting in the (ordered) persistence pair (b, d). By collecting all the persistence pairs, the persistence module V may be succinctly represented by a persistence diagram, Dgm(V) = · (b, d) ∈ R 2 : b ≤ d ≤ ∞ . Interleaving of Persistence Modules Given two persistence modules V = V t , φ t s s≤t and W = W t , ψ t s s≤t , they are said to be equivalent (or isomorphic) if there exists a family of isomorphisms {ξ t } t∈R such that each ξ t : V t → W t is an isomorphism. This notion can be extended to define two collection of maps {α t : t ∈ R} and {β t : t ∈ R} which weave the two persistence modules together. Where, as per convention, the order of homology, denoted by * , is an arbitrary non-negative integer. Definition 2.2 (Interleaving of persistence modules). Given two persistence modules V and W, and two monotone increasing maps α, β : R → R, V and W are said to be (α, β)-interleaved if the following diagrams commute for all s ≤ t V s V t V β(s) V β(t) W α(s) W α(t) W s W t V β(t) V t V β•α(t) W t W α•β(t) W α(t) αs φ t s ψ α(t) α(s) αt βt α β(t) ψ α•β(t) t βs ψ t s φ β(t) β(s) βt φ β•α(t) t αt β α(t) Remark 2.1. The persistence modules V and W are purely algebraic objects, and their underlying filtrations V and W are not necessarily compatible. However, when the filtrations V and W arise as filtered subsets of the same underlying space (e.g., R d ), we can similarly define an (α, β)−interleaving between the filtrations V and W by replacing all linear maps in Definition 2.2 by inclusion maps. The resulting persistence diagrams Dgm(V) and Dgm(W) are elements of the space of persistence diagrams Ω = {(x, y) : x ≤ y}, which is endowed with the family of q−Wasserstein metrics W q (·, ·) for 1 ≤ q ≤ ∞. We refer the reader to Edelsbrunner and Harer (2010); Mileyko et al. (2011) for more details. In special case of q = ∞, the resulting metric W ∞ is commonly referred to as the bottleneck distance, and is given as follows. Definition 2.3 (Bottleneck distance). Given two persistence diagrams D 1 , D 2 ∈ Ω, the bottleneck distance is given by W ∞ (D 1 , D 2 ) = · inf γ∈Γ sup p∈D 1 ∪∆ p − γ(p) ∞ , where Γ = {γ : D 1 ∪ ∆ → D 2 ∪ ∆} is the set of all multi-bijections from D 1 to D 2 including the diagonal ∆ = {(x, y) : x = y} with infinite multiplicity. Although the space of persistence diagrams (Ω, W q ), together with the q−Wasserstein distance, presents a challenging mathematical structure for refined statistical analyses (Mileyko et al., 2011;Turner et al., 2014), the stability of persistence diagrams provides a handle on this space by allowing us to directly work on the space generating the filtrations. Lemma 2.1 (Stability of persistence diagrams; Cohen-Steiner et al. 2007;. Given two compact sets X, Y ⊂ R d , W ∞ Dgm(V[X]), Dgm(V[Y]) ≤ H(X, Y). Alternatively, for two filter functions f, g : R d → R, W ∞ Dgm(V[f ]), Dgm(V[g]) ≤ f − g ∞ . Remark 2.2. (i) When the interleaving maps (α, β) are additive, i.e., of the form α : t → t + and β : t → t + δ, then persistence diagrams Dgm(V) and Dgm(W) obtained from the persistence modules satisfy the following relationships: Dgm(V) ∈ Dgm(W) ⊕ [−δ, ] 2 and Dgm(W) ∈ Dgm(V) ⊕ [− , δ] 2 , where ⊕ denotes the Minkowski sum in R 2 . A coarser bound is obtained from the stability theorem (Cohen-Steiner et al., 2007) which guarantees that W ∞ Dgm(V), Dgm(W) ≤ max { , δ}. (ii) Furthermore, when the interleaving maps are identical, i.e., α ≡ β : t → t + , this notion can be extended to define an interleaving pseudo-distance between persistence modules, d I (V, W) = · inf > 0 : V and W are (α, α) − interleaved for α : t → t + . From the isometry theorem the interleaving distance is identical to the bottleneck distance, i.e., W ∞ Dgm(V), Dgm(W) = d I (V, W). In such cases, it is equivalent to say that V and W are (α, α)-interleaved or d I (V, W) ≤ . Similarly, for filtrations V and W comprising of subsets of R d , d I (V, W ) = · inf > 0 : V t ⊆ W t+ and W t ⊆ V t+ .(2) By functoriality, d I (V, W ) ≤ =⇒ d I (V, W) ≤ =⇒ W ∞ Dgm(V), Dgm(W) ≤ . Weighted Rips Filtrations In practice, given a compact set X ⊂ R d or a filter function f , the persistence modules V[X] and V[f ] are computed using simplicial complexes. In particular: (i) For each t ∈ R, one may use the Čech or Alpha complex to compute the nerve of the cover, nerve{B(x, t) : x ∈ X}. Since the Nerve lemma (Edelsbrunner and Harer, 2010) guarantees that V t [X] ∼ = nerve{B(x, t) : x ∈ X}, the resulting persistence module V[X] may be computed exactly using simplicial homology. (ii) In the case of V[f ], this is typically achieved by choosing a grid resolution parameter , and constructing a cubical complex K on the underlying space. The function f : R d → R may be extended to define f : K → R, and at each resolution t ∈ R, the sublevel sets V t [f X ] can be approximated using the lower-star filtration K t = {σ ∈ K : max x∈σ f (x) ≤ t}. Therefore, the filtration V[f ] can be approximated by the filtration K t : t ∈ R , and the resulting persistence module is computed using cubical homology. Note that (i) is able to compute the exact persistence module in practice, but is unable to weight points according to f . On the other hand, (ii) is only an approximate computation and depends on the nuisance parameter . Furthermore, the size of the cubical complex is |K | = O( −d ), making it scale poorly in high dimensions. To overcome this limitation, Buchet et al. (2016) proposed the f -weighted filtrations, which was subsequently generalized by Anai et al. (2019). Given a non-negative weight function f : R d → R ≥0 and power 1 ≤ p ≤ ∞, the weighted radius function of resolution t > 0 at x is given by (a) V t [Xn] unweighted (b) V t [Xn, f ] for p = 1 (c) V t [Xn, f ] for p = ∞r f,x (t) = · (t p − f (x) p ) 1/p if t ≥ f (x) −∞ if t < f (x). Consequently, B f,ρ (x, t) is the weighted ball of resolution t at x w.r.t. the metric ρ, which is illustrated in Figure 1, and is given by B f,ρ (x, t) = · B ρ (x, r f,x (t)) = y ∈ R d : ρ(x, y) ≤ r f,x (t) . Given X ⊆ R d , the collection of weighted balls V t [X, f ] = {B f (x, t) : x ∈ X}, is called the weighted cover of X n . The f -weighted offset at resolution t is given by the union of balls in V t [X, f ], V t [X, f ] = · x∈X B f (x, t). Together with the inclusion maps ι t s : V s [X, f ] → V t [X, f ], the f −weighted filtration is given by V [X, f ] = · V t [X, f ], ι t s : s ≤ t . The image of V [X, f ] under the homology functor Hom * : V [X, f ] → V[X, f ], results in the weighted persistence module V[X, f ] = · V t [X, f ], φ t s : s ≤ t , where the induced maps φ t s : V s [X, f ] → V t [X, f ] are linear maps between vector spaces. The weighted-simplicial complexes C t [X, f ] = nerve V t [X, f ] and R t [X, f ] = Rips V t [X, f ] denote the weighted-Čech complex and weighted-Rips complex associated with the weighted cover V t [X, f ] respectively. Without loss of generality V t [X, f ] = H * V t [X, f ] is the homology of the offset V t [X, f ], which, by the nerve lemma, is the same as the homology of the weighted-Čech complex. Furthermore, if f (x) ≡ 0 for all x ∈ R d then the resulting filtrations are the usual unweighted filtrations. In particular, V [X n ] ∼ = C [X n , f ] and R[X n , f ] correspond to Čech and Rips filtrations, respectively. The following structural results appear in Anai et al. (2019), and serve as analogues of the stability result for f -weighted filtrations. (Anai et al., 2019, Propositions 3.2 & 3.3). Given X ⊂ R d and f, g : Lemma 2.2 X → R + (i) V[X, f ] and V[X, g] are (α, α)-interleaved for α : t → t + f − g ∞ . Additionally, given Y ⊂ R d and h : X ∪ Y → R + , if h is L-Lipschitz and H(X, Y) ≤ , then (ii) V[X, h] and V[Y, h] are (β, β)-interleaved for β : t → t + (1 + L p ) 1/p . Median-of-means Estimators Median-of-means (MoM) estimators have gained popularity in the robust machine learning owing to recent success, both theoretically and experimentally. See, for example, Devroye et al. (2016); Lugosi and Mendelson (2019a); Lecué and Lerasle (2020). The background for MoM estimators in the context of mean estimation is as follows: samples X n = {X 1 , X 2 , . . . , X n } are observed and we wish to construct an estimator for the population mean θ. The sample meanθ = X n is known to achieve sub-Gaussian estimation error only when the samples X n themselves are observed from a sub-Gaussian distribution. Robust statistics deals with two important relaxations to this model: (i) the samples X n are observed iid from P, but P is no longer sub-Gaussian and is assumed to have heavy tails; and (ii) a fraction π < 1 2 of the samples are assumed to be contaminated with outliers, and the remaining (1 − π)n samples are observed from a well-behaved distribution P. The median-of-means estimatorθ MOM , originally introduced by Nemirovskij and Yudin (1983), addresses these relaxations by constructing a robust estimator of location as follows: For 1 ≤ Q ≤ n, the sample X n is partitioned into subsets {S 1 , S 2 , . . . , S Q } such that each subset S q ⊂ {1, 2, . . . , n} with |S q | = n/Q . The MoM estimatorθ MOM is, then, defined aŝ θ MOM = · median θ 1 ,θ 2 , . . . ,θ Q , where {θ q : q ∈ [Q]} are the sample means computed for each subset {S q : q ∈ [Q]}. Audibert and Catoni (2011) showed that, in the univariate setting,θ MOM achieves sub-Gaussian rates of convergence for heavy tailed data. Minsker (2015) and Devroye et al. (2016) extend these results to the multivariate setting by considering the geometric median. The MoM idea has subsequently been extended in several other directions, e.g., U-statistics (Joly and Lugosi, 2016), kernel mean embeddings (Lerasle et al., 2019) and general Mestimators (Lecué and Lerasle, 2020) among others. Most importantly, these extensions move away from the heavy-tailed framework and provide significant insights on howθ MOM can overcome the second relaxation, i.e., estimation in the presence of outlying contamination. While the MoM estimators are not unique in their ability to recover the signal under heavy tailed noise, or in the presence of contamination, they are very simple to construct in most cases, and provide a clear characterization of the effect of noise on the estimation error. Main Results In the following, we present a MoM estimator to obtain outlier robust persistence diagrams in Section 3.1, and its statistical properties along with the influence analysis are presented in Sections 3.2-3.4. In Section 3.5 we present a method for adaptively calibrating the MoM tuning parameter using a data-driven procedure. The proofs for all results are deferred to Section 6. Empirical distance function using the Median-of-Means principle Let X n = {X 1 , X 2 , . . . , X n } ⊂ R d be a sample of n observations. We assume that the samples are obtained under sampling setting (S). We emphasize that this setting encompasses the following scenarios: (a) The samples X n are obtained i.i.d. from P ∈ P(X, a, b) for compact X ⊂ R d . (b) The samples are obtained from a distribution P = (1 − π)P signal + πP noise , where π ∈ (0, 1/2) and P signal ∈ P(X, a, b). (c) X 1 , X 2 , . . . , X n is first sampled i.i.d. from P ∈ P(X, a, b), and then handed over to an adversary. The adversary is then free to examine the n points, and replace any m < n/2 of them with some points of their choice. The modified dataset, X n , is then shuffled and handed to the topologist for inference, who has no prior knowledge of the original X 1 , X 2 , . . . , X n . The central objective is to derive a statistically consistent and computationally efficient estimator of Dgm(V[X]) which is robust to the misspecification scenarios detailed above, using the samples X n . To this end, the MoM Distance (MoM Dist) function d n,Q is defined as follows. Definition 3.1 (MoM Dist). Given a collection of points X n ⊂ R d and 1 ≤ Q ≤ n, let {S 1 , S 2 , . . . S Q } be a partition of X n into Q disjoint blocks, such that each subset S q ⊂ X n comprises of |S q | = n/Q samples . The MoM distance function d n,Q : R d → R ≥0 is defined to be d n,Q (y) = · median d n,Sq (y) : q ∈ [Q] = median inf x∈Sq x − y : q ∈ [Q] .(3) The proposed outlier robust persistence diagram Dgm(V[X n , d n,Q ]) is then obtained using d n,Q -weighted filtration V [X n , d n,Q ]. Note that we recover the usual empirical distance function, i.e., d n,1 ≡ d n when Q = 1. Remark 3.1. For each block S q , distance function d n,q ∈ L ∞ (R d ) can be viewed as the Kuratowski embedding of S q . The most natural generalization of the multivariate median-of-means estimators proposed by Minsker (2015) and Lerasle et al. (2019) would suggest the following estimator as the natural candidate for MoM Dist: d n,Q = arg inf f ∈L∞(R d ) Q q=1 f − d n,Sq ∞ , where the median under consideration corresponds to the geometric median in L ∞ (R d ). Although d n,Q has its appeal from a theoretical perspective, the computation of d n,Q involves an infinite-dimensional optimization problem, making it infeasible in practice. In contrast, the proposed estimator in Definition 3.1, is a pointwise median-of-means estimator with a tractable computational cost. This has the promise of being highly modular, and widely applicable in many practical settings. The technical difficulty arises in showing that the pointwise estimator d n,Q achieves an exponential concentration bound around d X in the L ∞ (R d ) metric. Similar to the proposed methodology in Definition 3.1, the procedure of partitioning the data X n into smaller subsets, and then aggregating them as an estimator of persistent homology has been shown to satisfy several favorable properties by Solomon et al. (2021) and Gómez and Mémoli (2021), albeit in a different context. We argue that a similar principle, in our setting, also leads to provably robust estimators. Computational considerations. Given a weighting function f , the first step in constructing the f -weighted filtration begins with estimating the weights associated with the sample points, i.e., w i = f (X i ) for all i ∈ [n]. After this step, the computational complexity of constructing the f -weighted filtration V [X n , f ] is independent Without loss of generality, we may assume that n is divisible by Q, so that n/Q ∈ Z+ Method Pre-processing Evaluation Provably robust? Table 1 compares the computational complexity for three robust filtrations: (i) the MoM Dist d n,Q , (ii) the distance-to-measure δ n,k (DTM, Anai et al., 2019), and (iii) the robust kernel density estimator f n σ (RKDE, Vishwanath et al., 2020). Given a test point x ∈ R d , the distance from x to each block S q is optimally computed using a k−d tree. The pre-processing step, which involves the construction of the k−d tree (Wald and Havran, 2006) The distance-to-measure with parameter m requires the evaluation of the distance to the kth nearest neighbor for k = mn . This is, again, optimally computed using a k−d tree; however, unlike d n,Q , the k−d tree needs to be constructed for all n samples, resulting in a time complexity of O(n log n) for pre-processing. Thereafter, the evaluation time takes O(k log n) for each query point, resulting in O(n · k log n) for evaluation over n samples. The robust KDE f n σ , on the other hand, requires O(n 2 ) time to compute the Gram-matrix in each iteration of the KIRWLS algorithm, and takes O(n 2 ) for outer loops. After this pre-processing step, the coefficients of f n σ may be used to evaluate each query in O(n) time. The three weighted filtrations V [X n , d n,Q ], V [X n , δ n,k ] and V [X n , f n σ ] are illustrated in Figure 2. We conclude this section with the following result, which establishes that MoM Dist is 1−Lipschitz. V [X n , d n,Q ] (MoM Dist-filtration) O n Q log(n/Q) O n · (Q + log n/Q) Yes V [X n , δ n,k ] (Anai et al., 2019, DTM-filtration) O(n log n) O(kn log n) No V [X n , f n σ ] (, typically has time complexity O(|S q | log |S q |) for each block q ∈ [Q] with |S q | = n/Q. Thereafter O(log |S q |) time is(a) V t [Xn, d n,Q ], MoM Dist (b) V t [Xn, δ n,k ], DTM (c) V t [Xn, f n σ ], RKDE Lemma 3.1. Given samples X n = {X 1 , X 2 , . . . , X n } and Q < n, |d n,Q (x) − d n,Q (y)| ≤ x − y , for all x, y ∈ R d . Statistical properties of V[d n,Q ] We begin our analysis by characterizing the persistence diagrams obtained using the sublevel filtration of d n,Q . The following result (proved in Section 6.3), establishes that Dgm(V[d n,Q ]) is a statistically consistent estimator of target population quantity Dgm(V[X]) under sampling setting (S), and establishes its rate of convergence in the W ∞ metric. Theorem 3.1 (Sublevel filtration). Suppose P ∈ P(X, a, b) is a probability distribution with support X satisfying the (a, b)−standard condition, and X n is obtained under sampling condition (S). For 2m < Q < n and for all δ < e −(1+b)Q , P W ∞ Dgm(V[d n,Q ]), Dgm(V[X]) ≤ g(n, Q, a, b) ≥ 1 − δ,(4) where g(n, Q, a, b) = Q log(n/Q) an + 4Q log(1/δ) a(Q − 2m)n 1/b . Furthermore, if the number of outliers grows with n as m = cn for c > 0 and ∈ [0, 1) then E W ∞ Dgm(V[d n,Q ]), Dgm(V[X]) log n n 1− 1/b .(5) Remark 3.2. The following salient observations can be made from Proposition 3.1. (i) In addition to characterizing the uniform rate of convergence of d n,Q , Eq. (4) also provides a uniform confidence band for Dgm(V[X]) in the presence of outliers. The two terms appearing in g(n, Q, a, b) may be interpreted as follows: The first term is similar to the term appearing in Chazal et al. (2015b, Theorem 2) with an effective sample size of n/Q instead of n, which is a consequence of the Median-of-Means procedure. The second term incorporates the desired confidence level δ adaptive to the volume dimension b > 0, with an effective sample size of n/Q. Notably, as the number of outliers m increases, the number of blocks Q must also increase; thereby widening the resulting confidence band. (ii) The complex inter-dependence of the parameters m, Q and δ in Eq. (4) is simplified in Eq. (5). In the absence of outliers, i.e., when m = 0 and Q = 1, we recover the same convergence rate as in Chazal et al. (2015b, Theorem 4), E W ∞ Dgm(V[d n,Q ]), Dgm(V[X]) log n n 1/b .(6) Specifically, it becomes apparent that accommodating for more adverse noise conditions comes at the price of an attenuated rate of convergence. (iii) The admissible confidence level δ for constructing the confidence band is implicitly dependent on the parameter Q. This phenomenon is unavoidable with estimators based on the median-of-means principle. We refer the reader to Lugosi and Mendelson (2019a, Section 2.4) for a comprehensive discussion on how robustness must come at the price of the confidence level δ being restricted. The proof of Proposition 3.1 relies on the following lemma, which allows us to control the deviation of a pointwise median-of-means estimator from its uncontaminated population counterpart in terms of a Binomial tail probability. Lemma 3.2. Suppose P ∈ P(X) for X ⊂ R d and X n = X * n-m ∪ Y m is obtained under sampling condition (S) with X * n-m observed i.i.d. from P. Let P n denote the empirical measure associated with X n and for 2m < Q < n, let P q be the empirical measure associated with the block S q for all q ∈ [Q]. Given a statistical functional T : P(R d ) → L ∞ (R d ), let T Q (P n ) ∈ L ∞ (R d ) be the pointwise MoM estimator given by T Q (P n )(x) = median T (P q )(x) : q ∈ [Q] , for all x ∈ R d . Then, for t > 0 P T Q (P n ) − T (P) ∞ > t ≤ P   q∈A ξ q (t; n, Q) > Q − 2m 2   , where A = {q ∈ [Q] : S q ∩ Y m = ∅} are the indices for the blocks containing no outliers, and ξ q (t; n, Q) = · 1 T (P q ) − T (P) ∞ > t for all q ∈ A. The statement of Lemma 3.2 holds for empirical processes arising from general classes of pointwise medianof-means estimators. In particular, by taking T (P q ) = d s,q to be the distance function w.r.t. block S q , the estimator d n,Q satisfies the conditions of Lemma 3.2. We also point out that the exponential concentration bound in Proposition 3.1 is strictly better than similar bounds appearing in other pointwise MoM estimators, e.g., Humbert et al. (2020, Theorem 2). This is owing to the Chernoff bound (instead of a Hoeffding bound) used for bounding the Binomial tail probability appearing in Lemma 3.2. This provides a significant gain for Binomial random variables with shrinking probability (Hagerup and Rüb, 1990). Statistical properties of V [X n , d n,Q ] In practice, the sublevel filtration V [d n,Q ] cannot be computed exactly, and one must rely on approximations using cubical homology. To this end, we now turn our attention to d n,Q -weighted filtrations computed on the sample points directly. Before we study the statistical properties of the d n,Q -weighted filtration, we provide a useful characterization of the persistence diagram obtained using the sublevel sets of d n,Q . Lemma 3.3. Given samples X n and Q < n, V [d n,Q ] and V [R d , d n,Q ] are (id, α)−interleaved for α : t → 2 p−1 p t for all p ≥ 1. In particular, V [d n,Q ] = V [R d , d n,Q ] when p = 1. We now turn our attention to the d n,Q -weighted filtration V [X n , d n,Q ]. The following result establishes that the persistence module V[X n , d n,Q ] is sufficiently regular. Next, in order to establish that Dgm(V[X n , d n,Q ]) is a consistent estimator of Dgm(V[X]) and to construct uniform confidence bands in the space of persistence diagrams (Ω, W ∞ ), we need a tighter control for how the two persistence modules are interleaved. To this end, Lemmas 3.5 and 3.6 will be of assistance, and serve as generalizations of Anai et al. (2019, Lemma 4.8 & Proposition 4.9). The following result, which holds for a general metric space (M, ρ) and an arbitrary weight function f , provides a handle for the interleavings between f -weighted filtrations computed on two nested sets using the same function f . Lemma 3.5. Given a metric space (M, ρ), two compact subsets X, Y of M such that X ⊆ Y, and a weight function f : M → R ≥0 , let V ρ [X, f ] and V ρ [Y, f ] be their respective f -weighted filtrations. If f satisfies the property that inf x∈X ρ(x, y) ≤ f (y) + a, for a > 0 and for all y ∈ Y, then the filtrations are (id, α)-interleaved, i.e., V t [X, f ] ⊆ V t [Y, f ] ⊆ V α(t) [X, f ], for α : t → 2 1− 1 p t + a + sup x∈X f (x). Since map α appearing in Lemma 3.5 is not purely a translation map, it does not lead to a bound in the interleaving metric as per Eq. (2), and, therefore, a bound in the W ∞ metric cannot be characterized using Lemma 3.5 alone. The next result, which is stated only for the Euclidean space (R d , · ), establishes that for sufficiently large values of t, the map α may be replaced by a translation map. Lemma 3.6. Let (M, ρ) = (R d , · ). Suppose X, Y ⊂ R d are compact sets such that X ⊆ Y, and f satisfies the same conditions as in Lemma 3.5 for a > 0. Let t(X) be the filtration value for the simplex corresponding to X in nerve V ρ [X, f ] , i.e., t(X) = · inf t > 0 : x∈X B f,ρ (x, t) = ∅ , and β : t → t + c(X) be a non-decreasing map with c(X) = · a + sup x∈X f (x) + 1 − 1 p t(X). Then for all t ≥ t(X), the homomorphisms φ β(t) t : V t ρ [X, f ] → V β(t) ρ [X, f ] are trivial, i.e., Im φ β(t) t ∼ = F if V t ρ [X, f ] = H 0 V t ρ [X, f ] {0} if V t ρ [X, f ] = H k V t ρ [X, f ] , k > 0 . Furthermore, the bottleneck distance between the resulting f -weighted persistence diagrams is bounded above as Figure 3: Illustration of Lemmas 3.5 and 3.6 for p = 2, t(X) = 3 and a + sup x∈X f (x) = 1. The interleaving maps α and β are illustrated in blue and red, respectively. When t < t(X), the interleaving map is α from Proposition 3.5. For t ≥ t(X), the map β is obtained using Lemma 3.6. Extending β along the black line yields the interleaving bound. W ∞ Dgm V ρ [X, f ] , Dgm V ρ [Y, f ] ≤ c(X).α : t → 2 1− 1 p t + 1, t < t(X) β : t → t + 1 + 1 − 1 p , t ≥ t(X) In essence, the preceding two results enable us to control the filtrations in two separate stages, and, then, "stitch" the results together. See Figure 3 for an illustration. This forms the crux of the next result, which establishes an analogue of the stability result for d n,Q -weighted filtrations, but unlike the stability for the usual distance function d n , it is also robust to outliers. W ∞ V[X n , d n,Q ], V[X * n-m , d n−m ] ≤ sup x∈X * n-m d n,Q (x) + d n,Q − d n−m ∞ + 1 − 1 p t(X * n-m ), where t(X * n-m ) is the filtration value of the simplex associated with the inliers X * n-m in the filtration V [X * n-m , d n,Q ]. In particular, when p = 1 we have W ∞ V[X n , d n,Q ], V[X * n-m , d n−m ] ≤ sup x∈X * n-m d n,Q (x) + d n,Q − d n−m ∞ .(7) Remark 3.4. The following observations follow from Theorem 3.2. (i) In contrast to what would follow from Lemma 2.2 (ii) for the standard unweighted filtration, the term appearing in the r.h.s. of Eq. (7) completely eliminates the dependence on the Hausdorff distance between X n and X * n-m in the d n,Q −filtration. More generally, the same bound in Proposition 3.2 holds even when V [X n , d n,Q ] is replaced by V [M, d n,Q ] for any set M ⊇ X * n-m . (ii) Notably, V [X n , d n,Q ] remains resilient to outliers. To see this, observe that the first term appearing in the r.h.s. of Eq. (7) may be bounded as sup x∈X * n-m d n,Q (x) = sup x∈X * n-m |d n,Q (x) − d X (x)| ≤ d n,Q − d X ∞ , where the first equality follows from the fact that d X (x) = 0 for all x ∈ X * n-m . Therefore, from the proof of Theorem 3.1, the r.h.s. of Eq. (7) vanishes with high probability for sufficiently large sample sizes. (iii) For p = 1, a similar analysis for the DTM-filtrations appears in Anai et al. (2019, Theorem 4.5) and the bottleneck distance is bounded above as W ∞ Dgm(V[X n , δ n,k ]), Dgm(V[X * n-m , δ n−m,k ]) ≤ n k W 2 (X * n-m , X n ) + sup x∈X * n-m δ n−m,k . While the last term on the r.h.s. converges to the uncontaminated population analogue with high probability, the first term involving the Wasserstein distance W 2 (X * n-m , X n ) can be large even for a few extreme outliers. In contrast, the r.h.s. of Eq. (7) converges to zero with high probability with no assumptions on the outliers Y m . With this background we are now in a position to state our main result, which characterizes the rate of convergence for the d n,Q -weighted filtration on the contaminated sample points, V [X n , d n,Q ], to the counterfactual population analogue V [X] in the W ∞ metric. Theorem 3.3 (d n,Q -weighted filtration). Let p = 1. Suppose P ∈ P(X, a, b) is a probability distribution with support X satisfying the (a, b)−standard condition, and X n = X * n-m ∪ Y m is obtained under sampling condition (S). Then, for 2m < Q < n and for all δ ∈ (0, 1), P W ∞ V[X * n-m ∪ Y m , d n,Q ], V[X] ≤ f(n, m, Q, δ 1 , δ 2 ) ≥ 1 − δ, where f(n, m, Q, a, b) = · Q log(n/Q) a(n/Q) + 4Q log(1/δ 1 ) a(Q − 2m)n 1/b + log(n − m) a(n − m) + 4 log(1/δ 2 ) a(n − m) 1/b , for δ 1 , δ 2 ∈ (0, 1) such that δ 1 ≤ e −(1+b)Q and δ 1 + δ 2 = δ. In particular, if m n = cn for 0 ≤ < 1, then E W ∞ V[X * n-m ∪ Y m , d n,Q ], V[X] log n n 1− 1/b .(8) Remark 3.5. We make the following observations from Theorem 3.3. (i) The term appearing in the r.h.s. of Eq. (8) is identical to the term appearing in the r.h.s. of Eq. (6) in Theorem 3.1. Therefore, the d n,Q -weighted filtration and the d n,Q sublevel filtration converge to the same population limit with identical convergence rates. They both differ from the minimax rate without outliers (Chazal et al., 2015b, Theorem 4) by a factor of n − /b . (ii) The uniform confidence band we obtain from Theorem 3.3 can, in principle, be computed for any confidence level δ ∈ (0, 1). However, the restriction on δ 1 makes the confidence band obtained using V [X n , d n,Q ] wider than that obtained using Proposition 3.1. This is, ultimately, the price we have to pay for choosing the computationally tractable d n,Q -weighted filtration as the estimator as opposed to the d n,Q sublevel filtration. We conclude this section with the following result, which relates the sublevel filtration V [d n,Q ] to V [X n , d n,Q ]. W ∞ Dgm(V[d n,Q ]), Dgm(V[X n , d n,Q ]) ≤ sup x∈X * n-m d n,Q (x). The above result characterizes the error incurred when using V [X n , d n,Q ] to approximate the sublevel filtration V [d n,Q ]. In light of Remark 3.4 (ii), this error vanishes with increasing sample size. In contrast, the approximation error for the DTM-filtration is non-vanishing (Anai et al., 2019, Proposition 4.6). Influence analysis The statistical analysis in the previous sections establishes that, even in the presence of outliers, as the number of samples increases we can eventually mitigate the effect of the outliers. In this section, we provide a more precise characterization for the influence the outliers have on the resulting d n,Q -weighted filtrations, in contrast to the non-robust counterpart-the d n -weighted filtrations. Given a probability measure P ∈ P(X, a, b), Vishwanath et al. (2020, Definition 4.1) characterized the influence an outlier at x ∈ R d has on a persistence diagram Dgm(V[f P ])-obtained using the sublevel sets of f P -using the persistence influence function Ψ(f P ; x) = · lim →0 W ∞ Dgm V[f P x ] , Dgm(V[f P ]) ,(9) where P x = (1 − )P + δ x is the perturbation curve w.r.t. x in the space of probability measures. The persistence influence is a generalization of the influence function in robust statistics (Hampel et al., 2011) to general metric spaces. The analysis in this section is similar in spirit to the analysis based on the persistence influence, but differs in two important aspects. First, the d n,Q -weighted filtration is computed purely on the sample points-by partitioning the samples into Q disjoint blocks-and, therefore, the notion of persistence influence is adapted to the samples, in contrast to Eq. (9), which is based on the data-generating distribution P. Additionally, unlike the case of the persistence influence function-where the influence of outliers in the resulting persistence diagram is quantified in terms of the bottleneck distance-here we directly examine the influence the outlying point has on the resulting persistence diagram itself. This provides a more tractable interpretation for how outliers impact the resulting topological inference. The discussion in the previous section focused on the weighted filtrations, which can be approximated using the weighted-Čech complex. Here, we will explicitly restrict ourselves to the case of the weighted Rips filtrations. Firstly, a majority of the computational applications of persistent homology are performed using the Rips complex, with several optimized implementations widely available, e.g. Ripser, Gudhi, GiottoTDA. Furthermore, since the Rips complex R t [X, f ] is defined to be the flag complex associated with the 1-skeleton ofČ t [X, f ], the weighted Rips persistence diagram is entirely characterized by its 0− and 1−simplices. With this background, we now introduce the empirical persistence influence framework. Suppose we are given a collection of observations X n , which is sampled i.i.d. from a probability distribution P of interest. Let Dgm(V[X n , f n ]) be its weighted-Rips persistence diagram, where the weight function f n is constructed using the samples X n . Suppose X n is contaminated with m < n 2 outliers to obtain the contaminated dataset X n+m . In particular, we may assume that the m-points are placed at an outlying location x 0 , i.e., X n+m = X n m j=1 {x 0 } , such that the factor m and the location x 0 together control the relative influence the outliers have. This is similar to the role played by the factor in the perturbation curve associated with the persistence influence. Note that when m = 0, the influence of the outliers is non-existent in the dataset. Let Dgm(V[X, f n+m ]) be the weighted-Rips persistence diagram constructed on X n+m using the weight function f n+m . This gives rise to a collection of spurious topological features in the resulting persistence diagram. If b n ({x 0 }) is the birth time associated with a hypothetical topological feature with mass 0 at x 0 (i.e. 0δ x 0 ) in Dgm(V[X n , f n ]), and b n+m ({x 0 }) is the birth time associated with the observed topological feature associated with the m-points at x 0 (i.e. mδ x 0 ), then the empirical persistence influence of x 0 can be characterized by influence(b; X n , f n , m, x 0 ) = ∆b n,m ({x 0 }) = b n ({x 0 }) − b n+m ({x 0 }).(10)Indeed, when b n ({x 0 }) − b n+m ({x 0 }) is small,; X n , f n , m, x 0 ) = W ∞ Dgm(V[f m+n ]), Dgm(V[f n ]) ≤ f n+m − f n ∞ .(11) The following result establishes that, under some mild conditions and with high probability, the d n,Q -weighted Rips persistence diagrams are more robust than their non-robust counterpart. Theorem 3.4 (Influence analysis of d n,Q -weighted filtrations). For X n observed i.i.d. from P ∈ P(X, a, b) and x 0 ∈ R d , let X n+m be given by X n+m = X n m j=1 {x 0 } .(x 0 ) = · cd X (x 0 ) b > log n Q n Q + 4(1 + b) 2 Q 3 n Q ,(I) then, for all δ ∈ (0, 1) satisfying (1 + b) 2 Q 2 ≤ log(2/δ) ≤ n Q (x 0 ) − log n Q 4Q ,(II) with probability greater than 1 − δ, d n+m − d n ∞ − d n+m,Q − d n ∞ ≥ 2 log n Q an Q + 8 log(2/δ) an Q 1/b . Remark 3.6. The result from Theorem 3.4 may be interpreted as follows. (i) The first part guarantees that the d n,Q -weighted persistence diagram always has a smaller influence on the birth time in comparison to the non-robust counterpart. Since the Rips persistence diagram is entirely determined by the filtration values associated with the 0− and 1− simplices, this provides a partial picture for the influence x 0 has on the resulting persistence diagrams. Characterizing the influence on the 1−simplices is far more challenging owing to the combinatorial complexity in characterizing their lifetimes. (ii) The second part compares the upper bounds on the empirical persistence influence from Eq. (11). When conditions (I) and (II) hold, then with high probability, persistence diagrams obtained using d n,Q are closer to the truth than those obtained using d n . Therefore, the interplay between n, m and x 0 is better understood by characterizing when conditions (I) and (II) hold. (iii) For fixed n observe that (I) is satisfied whenever d X (x 0 ) is sufficiently large, i.e., x 0 is sufficiently far away from the support. On the other hand, if x 0 is fixed, then (I) is satisfied when log n Q /n Q is sufficiently small, i.e., n is sufficiently large. Together, this implies that for condition (I) to be satisfied, either (a) we need the outliers to be sufficiently well-separated from the support X such that we are able to distinguish outliers x 0 from the inliers X n , or (b) for outliers placed very close to the support X we need sufficiently many inliers n for us to be able to distinguish them from the outliers. On the other hand, note that if n and m are fixed, then the r.h.s. of (I) is directly proportional to Q. Although Q can take any values between 2m < Q < (n + m), choosing a value of Q much larger than 2m + 1 will likely breach condition (I) for a fixed x 0 . Equivalently, for a suboptimal choice of Q, we need the outliers to be sufficiently far away from the inliers in order to be able to distinguish them. (iv) The l.h.s. of (II) is equivalent to the constraint that δ ≤ e −(1+b)Q , which appears in Theorems 3.1 and 3.3. The r.h.s. of (II) specifies a lower-bound on the confidence level δ. Condition (I) guarantees that the admissible values of δ ∈ (0, 1) satisfying (II) is nonempty. For fixed m, Q and x 0 , the r.h.s. of (II) is directly proportional to n, i.e., the lower bound vanishes as n → ∞. (v) When conditions (I) and (II) are satisfied, we have the following lower bound from the l.h.s. of (II): d n+m − d n ∞ − d n+m,Q − d n ∞ log(n + m/Q) a(n + m)/Q + Q 2 (n + m)/Q 1/b .(12) In the regime when n, m → ∞, and for the optimal choice of Q, i.e., Q = km for k > 2, the r.h.s. of Eq. (12) is non-trivial when m = Ω(n 1/3 ). Therefore, under conditions (I) and (II), when there are sufficiently many outliers, there is greater evidence to support the robustness of d n,Q . Auto-tuning the parameter Q The result in Theorem 3.3 relies on the crucial assumption that the number of outliers m * is known a priori. While this assumption may hold in certain adversarial settings, in general, this information may be unavailable. In order to make Theorem 3.3 more useful in practical settings, we discuss two solutions for calibrating the parameter Q. The first procedure is based on Lepski's method (Lepski, 1991), which is a powerful data-driven method for adaptive parameter selection. In this case, we also provide theoretical guarantees for the adaptively tuned estimator. The second procedure-which is based on some heuristic observations regarding the sample estimator V[X n , d n,Q ]works well in practice, and may be used as a precursor to Lepski's method. When the number of outliers m * is known, choosing Q * = 2m * + 1 results in the rate of convergence in Theorem 3.3. However, without access to m * , Lepski's method provides a systematic procedure for selecting a parameter Q which provides the same error guarantees as Q * (Birgé, 2001). The procedure is as follows. Let m min and m max be two coarse bounds on (unknown) m * such that m min ≤ m * ≤ m max . For a choice of θ > 1, let m(j) = θ j m min and define J = · j ≥ 1 : m min ≤ m(j) < θm max . For P ∈ P(X, a, b) and X n obtained under sampling condition (S), let V n (j) = V[X n , d n,Q(j) ] be the persistence module obtained using the MoM Dist-weighted filtration with Q(j) = 2m(j) + 1. For δ ∈ (0, 1) and δ max = δ − e −(1+b)(2mmax+1) , let h(n, m, δ) be defined as follows: h(n, m, δ) = 2 2m + 1 an W 0 ne 4(1+b)(2mmax+1) 2m + 1 1/b + 1 a(n − m) W 0 (n − m)e 4 log(1/δmax) 1/b , where for z > 0, W 0 (z) is the Lambert W 0 function given by the identity W 0 (z)e W 0 (z) = z. With this background, let be the output of the following procedure:  = · min j ∈ J : W ∞ V n (j), V n (j ) ≤ 2h(n, m(j ), δ) for all j ∈ J , j > j ,(13) the resulting weighted persistence module V n = V n () = V[X n , d n,Q() ] is the Lepski estimator for V[X]. The following result establishes that the adaptive selection of Q results in an estimator with the same convergence guarantees as in Theorem 3.3. Theorem 3.5 (Adaptive d n,Q -weighted filtration). Suppose X n is obtained under sampling condition (S) for P ∈ P(X, a, b), and suppose m min and m max are known such that unknown number of outliers m * ∈ [m min , m max ] and m * < n/2. For a chosen θ > 1 let be the output of data-driven procedure in Eq. (13) and let V n = V n (). Then, for all δ ∈ (0, 1), P W ∞ Dgm V n , Dgm V[X] ≤ 3h(n, θm * , δ) ≥ 1 − δ log θ θm min m max . Remark 3.7. We make the following useful observations from Theorem 3.5. (i) We make the distinction that the output V n of Lepski's method does not necessarily correspond to the optimal choice V * n if m * were known. Instead, Theorem 3.5 guarantees that error associated with V n is of the same order (up to constants) as that of V * n . (ii) While Lepski's method guarantees optimal errors for the adaptive estimator without any knowledge of the true m * ; in practice, however, the empirical performance depends on several factors. Since the procedure in Theorem 3.5 is designed to match the guarantee of Theorem 3.3, the success of the procedure crucially depends on the tightness of the bound f(n, m, Q, δ 1 , δ 2 ) in Theorem 3.3. Furthermore, the implementation described in Eq. (13) requires knowledge of the parameters a, b > 0 arising from the (a, b)−standard condition. While the calibration of a and b in practice is more of an art and beyond the scope of the paper, we emphasize here that it is possible to construct a statistically consistent estimator of the true population quantity V[X] in a purely data-adaptive fashion, even in the presence of adversarial contamination. (iii) Unlike a standard grid search, Lepski's method adapts to the true noise level m * in an efficient manner. Given a reasonable estimate for m min and m max , Lepski's method has a computational cost of O(log 2 θ (m max /m min )). However, the choice of θ > 1 must also be made judiciously, e.g., replacing θ with √ θ for the procedure in Eq. (13) will require ∼ 4 times more computational time. (iv) In the worst case, when there are no reasonable estimates for m min and m max , choosing m min = 1 and m max = n/2 requires O(log 2 θ (n)) computational time. Notably, more than just the additional computational price, a suboptimal choice of m min and m max leads to poor performance. To see this, note that the term h(n, m, δ) is a lower bound for the term f(n, m, Q, δ 1 , δ 2 ) in Theorem 3.3 when Q = 2m + 1 and δ 1 = e −(1+b)(2mmax+1) ≤ e −(1+b)Q . Therefore, when the number of outliers grows with n as m * = cn for c > 0 and ∈ [0, 1), a similar analysis to that in Theorem 3.1 and Theorem 3.3 yields that E W ∞ Dgm V n , Dgm V[X] log n n/m max 1/b . Therefore, if the bound m max is not tight, i.e., m max = Cn β for < β, then, asymptotically, the output of Lepski's method is not adaptive to the true noise m * , and, instead, reflects the suboptimal choice of m max . In a similar vein, Lepski's method may be used to adaptively select the parameter Q to obtain a statistically consistent sublevel set persistence module. The following result outlines a data-driven procedure to obtain  ∈ J such that the resulting sublevel persistence module V n = V n () = V[d n,Q() ] has the same convergence guarantee as Theorem 3.1. Theorem 3.6 (Adaptive sublevel filtration). For P ∈ P(X, a, b), suppose X n is obtained under sampling condition (S), and suppose m min and m max are known such that unknown number of outliers m * ∈ [m min , m max ] and m * < n/2. Let W n (j) = V[d n,Q(j) ] be the sublevel persistence module obtained using d n,Q(j) with Q(j) = 2m(j) + 1 for all j ∈ J . For a chosen θ > 1, let be the output of data-driven procedure, = min j ∈ J : W ∞ V n (j), V n (j ) ≤ 2p(n, m(j ), δ) for all j ∈ J , j > j , where p(n, m, δ) = 2m + 1 an W 0 ne (1+b) log(1/δ) 2m + 1 1/b . Then, for all δ ≤ e −(1+b)(2mmax+1) and V n = V n (), P W ∞ Dgm V n , Dgm V[X] ≤ 3h(n, θm * , δ) ≥ 1 − δ log θ θm min m max . The proof is identical to that of Theorem 3.5, and is, therefore, omitted. The success of Lepski's method depends on the tightness of the probabilistic bounds, knowledge of the (nuisance) parameters (i.e. a, b) appearing in these bounds, and a prudent choice for m min and m max . While the calibration of a is beyond the scope of this paper, in R d a conservative choice for b would be the dimension d of the ambient space. We refer the reader to (Chazal et al., 2015b, Section 4) for further details. To address the last bottleneck in Lepski's method, we describe a heuristic method to select the parameter Q, which may be used to obtain reasonable choices for m min and m max . The method is based on the observation that the blocks {S q : q ∈ [Q]} may be resampled by shuffling the sample points X n prior to partitioning it. The resulting estimator V[X n , d n,Q ] is an unbiased estimator of the same population quantity when 2m < Q < n. Therefore, we may choose the smallest value of Q for which the pairwise bottleneck distance over permutations of the data is minimized. Specifically, suppose X σ n = X σ(1) , X σ(2) , . . . , X σ(n) is a permutation of X n , then Q R = arg min Q≥1 1≤i<j≤N W ∞ V[X σ i n , d n,Q ], V[X σ j n , d n,Q ] , where, for a chosen number of replicates N , σ i , σ j are permutations of [n] for each i, j ∈ [N ]. Furthermore, for m R = Q R /2 and for a constant C > 1, the bounds m min and m max may be taken to be C −1 m R and C m R , respectively. Experiments In the following section, we supplement the theory through illustration of the performance of the robust filtrations V[d n,Q ] and V[X n , d n,Q ] in synthetic experiments. The tools for data-adaptive construction of d n,Q -weighted filtrations, in addition to the code for all experiments, are made publicly available in the RobustTDA.jl Julia package . In all experiments, the persistence diagrams are computed using the Ripserer.jl backend (Čufar, 2020), and we set the parameter p = 1 for the weighted-filtrations. Adaptive calibration of Q For n = 500, K = 30 replicates and for each i ∈ [K], point clouds X Comparison of V[d n,Q ] and V[X n , d n,Q ] The objective of this experiment is to illustrate that the d n,Q -weighted filtration V [X n , d n,Q ] reasonably approximates the sublevel filtration V [d n,Q ]. For the same setup as 4.1, X n comprises of n = 550 points obtained by sampling 500 points on a circle with additive Gaussian noise (σ = 0.01) and m = 50 outliers added from a Matérn cluster process. For Q = Q selected using Lepski's method, Figure 5 (a) depicts the MoM Dist function d n,Q . Figure 5 (b) illustrates the scatter plot for X n with the points colored by the weights d n,Q (x i ) for each x i ∈ X n . The shaded regions show the d n,Q -weighted offsets V t [X n , d n,Q ] for t ∈ {1.5, 1.75, 2, 2.25} colored from white to blue. Figure 5 High dimensional topological inference In this experiment, we illustrate the advantage of using d n,Q -weighted filtrations for high dimensional topological inference. Points are uniformly sampled in R 3 from two interlocked circles. Using a random rotation matrix Q ∈ SO(100), the points are transformed to an arbitrary configuration in R 100 . The samples X n ⊂ R 100 are obtained by replacing 12.5% of the points in R 100 with outliers sampled from Uniform [−0.2, 0.2] 100 . A scatterplot for X n projected to 3 arbitrary coordinates is shown in Figure 6 (a). Since the point cloud is embedded in R 100 , computing sublevel filtrations using cubical homology with the same resolution as earlier requires (10/0.5) 100 ≈ 10 131 simplices to be stored in memory. In contrast, computing the d n,Q -weighted filtrations requires is less intensive. Figure 6 (b) shows the persistence diagram Dgm( V n ) obtained using d n,Q -weighted filtrations, where the parameter Q is adaptively selected using Lepski's method. The two 1st order homological features underlying the interlocked circles are recovered. Figure 6 (c) illustrates the persistence diagram Dgm(V[X n , δ n,k ]) obtained using DTM-weighted filtrations. Since the DTM parameter k ∈ [1, n] results in a smoothing similar to the parameter Q ∈ [1, n] for the MoM Dist, the parameter k is set to the value of Q obtained using Lepski's method. Recovering the true signal under adversarial contamination In this experiment, we illustrate how V[X n , d n,Q ] can be used to recover the true topological features in the presence of adversarial contamination. In Figure 7 (a), we consider a 28 × 28 image for the digit "6" from the MNIST database (Deng, 2012). We consider the setting in which an adversary is allowed to manipulate 10% of the image by modifying the pixel intensities. Figure 7 (b) depicts the adversarially contaminated version of the image by transforming the "6" to an "8". For each pixel p with pixel intensity ι(p), we convert the image to a point cloud X n ⊂ R 2 by sampling 10 * ι(p) points uniformly from the region enclosed by the pixel. Figures 7(d, e) illustrate the point clouds obtained from the true and contaminated images with n − m ≈ 1100 and n ≈ 1300, respectively. The persistence diagrams constructed using the distance function d n for the two point clouds are reported in Figures 7(g, h). The persistence diagram in Figure 7 (h) indicates the presence of the additional loop introduced by the adversary. To account for the adversarial contamination, we compute the MoM Dist function d n,Q with the parameter Q selected using the contamination budget, i.e., Q = 1 + 2(1100 × 10%) = 221. Figure 7 (f) shows the adversarially contaminated point cloud with each point x i ∈ X n colored by the value of d n,Q (x i ). The resulting d n,Q -weighted persistence diagram Dgm(V[X n , d n,Q ]) is reported in Figure 7 (f). We note that Dgm(V[X n , d n,Q ]) recovers the prominent features of Figure 7 (g) up to a rescaling. Additionally, for each pixel p we compute a rescaled version of d n,Q , given by f n,Q (p) = max x d n,Q (x) − d n,Q (p) max x d n,Q (x) , as a proxy for the pixel intensity obtained using d n,Q . In Figure 7 (c), we plot the level sets {p : f n,Q = t} on the original image for t ≥ 0.8. Empirical influence analysis In this experiment, we examine the influence of outliers on d n,Q -weighted filtrations. For n = 500, points X n are sampled uniformly from a circle. We compute the unweighted persistence diagram D n = Dgm(V[X n ]). d n+m,ρ,σ (x) = · K σ (·, x) − f n+m ρ,σ H = 1≤i,j≤n+m w i w j K σ (X i , X j ) + K σ (x, x) − 2f n+m ρ,σ (x). The RKDE-weighted persistence diagram D RKDE n+m,ρ,σ = Dgm(V[X n ∪ Y m , d n+m,ρ,σ ]) is then computed using the d n+m,ρ,σ -weighted filtration on the composite sample. The bandwidth of the kernel and the parameters for the Hampel loss function are selected using the same approach as in Vishwanath et al. (2020). For each diagram, we compute the birth time b({x 0 }) for the first outlier x 0 ∈ Y m , and the bottleneck influence W ∞ (D n+m , D n ), as described in Section 3.4. We generate 10 such samples for each value of m, and report the average in Figure 8. From Figure 8 (a), we note that D M oM n+m,Q and D DT M n+m,k show similar behavior, although the outliers consistently appear earlier in the DTM persistence diagram D DT M n+m,k . Since the birth time b({x 0 }) alone does not fully characterize the impact an outlier has on inferring the topological feature underlying the circle, we also compute the maximum persistence for the first order persistence diagram in Figure 8 (b). We point out that the behavior of b({x 0 }) w.r.t. m largely reflects the influence an outlier has on the relevant topological signal. Furthermore, for D M oM n+m,Q , we observe the sharp transition which occurs between m = 50 and m = 80, which is due to the fact that the theoretical guarantees for d n,Q from Theorem 3.4 are valid only when 2m < Q = 100. Similarly, from Theorem 3.3, the outliers are guaranteed to have little influence on D M oM n+m,Q whenever m ≤ 50, as seen in Figure 8 (c). On the other hand, while the RKDE remains resilient to uniform outliers, we note that D RKDE n+m,ρ,σ is significantly impacted by the outliers placed at a single point in center of the circle. This is evidenced by the sharp transitions for D RKDE n+m,ρ,σ in Figures 8 (b, c). However, unlike d n+m,Q and δ n+m,k , by construction d n+m,ρ,σ ∞ ≤ sup x 2K σ (x, x) < ∞. Therefore, the impact the outliers have on D RKDE n+m,ρ,σ are bounded; and despite being more sensitive to the outliers, the resulting influence the outliers have on D RKDE n+m,ρ,σ in Figures 8 (a, b, c) is bounded. Conclusion & Discussion In this paper, we introduce a methodology for constructing filtrations which are computationally efficient, provably robust, and statistically consistent even in the presence of outliers. To our knowledge, our results are the first of this type. To elaborate, we introduced MoM Dist, d n,Q , as a computationally efficient and outlier-robust variant of the distance function based on the median-of-means principle, and established some of its theoretical properties. In particular, when the samples contain outliers in the adversarial contamination setting, we (i) showed that the d n,Q -weighted filtrations are statistically consistent estimators of the true (uncontaminated) population counterpart, (ii) characterized its convergence rate in the bottleneck metric, and (iii) provided uniform confidence bands in the space of persistence diagrams. Furthermore, we used an empirical influence analysis framework to quantify the robustness of the d n,Q −filtrations, and provide a framework for selecting the parameter Q. Topological inference in the presence of outliers is a topic which has received considerable attention in recent years, and with good reason. We would like to highlight that the objective in this paper has been to develop a framework of topological inference in which the population target is the persistence diagram Dgm(V[X]). Therefore, the proposed methodology disregards, to a large extent, the distribution of mass on the support. As a future direction, we would like to explore a framework of inference which incorporates information from, both, the geometry of the underlying space and the structure of the probability measure generating the data. As noted in Anai et al. (2019, Section 5), their results follow only from a few simple properties of the distance-to-measure. We build off their foundation to provide some useful generalizations which we hope will be useful in the analysis of other estimators using this framework. Proofs In this section, we present the proofs for the results in Section 3. Proof for Lemma 3.1 We begin by noting that for each q ∈ [Q], the distance function d n,Sq associated with the block S q is 1−Lipschitz (Boissonnat et al., 2018, Chapter 9.1). Thus, for each q ∈ [Q] and for all x, y ∈ R d we have that 0 ≤ d n,q (x) ≤ d n,q (y) + x − y , and, therefore, it follows that median{d n,q (x) : q ∈ [Q]} ≤ median{d n,q (y) : q ∈ [Q]} + x − y . As a result, we obtain that d n,Q (x) ≤ d n,Q (y) + x − y . Exchanging x and y in the steps above yields the desired result. Proof for Lemma 3.2 For t > 0, define two events E 1 = T Q (P n ) − T (P) ∞ ≤ t , and E 2 = # q ∈ [Q] : T (P q ) − T (P) ∞ > t ≤ Q 2 . First, we show that E 2 ⊆ E 1 . To this end for any ω ∈ E 2 , we have ω ∈ E 2 =⇒ ω ∈ # q ∈ [Q] : T (P q ) − T (P) ∞ > t ≤ Q 2 =⇒ ω ∈ # q ∈ [Q] : T (P q ) − T (P) ∞ ≤ t > Q − Q 2 =⇒ ω ∈ # q ∈ [Q] : ∀x ∈ R d , T (P)(x) − t ≤ T (P q )(x) ≤ T (P)(x) + t > Q 2 =⇒ ω ∈ ∀x ∈ R d , T (P)(x) − t ≤ median T (P q )(x) : q ∈ [Q] ≤ T (P)(x) + t =⇒ ω ∈ ∀x ∈ R d , T (P)(x) − t ≤ T Q (P n )(x) ≤ T (P)(x) + t =⇒ ω ∈ T Q (P n ) − T (P) ∞ ≤ t =⇒ ω ∈ E 1 . Therefore, we have E 2 ⊆ E 1 . Next, note that E 2 can be written as E 2 =    Q q=1 ξ q (t; n, Q) ≤ Q 2    , where, for each q ∈ [Q], ξ q (t; n, Q) = · 1 T (P q ) − T (P) ∞ > t . Since 0 ≤ ξ q (t; n, Q) ≤ 1 a.s., we have that Q q=1 ξ q (t; n, Q) = q∈A ξ q (t; n, Q) + q∈A c ξ q (t; n, Q) ≤ q∈A ξ q (t; n, Q) + |A c | ≤ q∈A ξ q (t; n, Q) + m. As a result, we can further bound the probability of E 2 from below as P(E 2 ) ≥ P   q∈A ξ q (t; n, Q) ≤ Q 2 − m   .(14) Combining Eq. (14) with the fact that E 2 ⊆ E 1 , we obtain P T Q (P n ) − T (P) ∞ > t = P(E c 1 ) ≤ P(E c 2 ) ≤ P   q∈A ξ q (t; n, Q) > Q 2 − m   , which gives us the desired result. Proof of Theorem 3.1 First, we note from the stability of persistence diagrams that, P W ∞ Dgm(d n,Q ), Dgm(d X ) > t ≤ P d n,Q − d X ∞ > t .(15) Therefore, it suffices to control the probability of the event d n,Q − d X ∞ > t . To this end, let A = {q ∈ [Q] : S q ∩ Y m = ∅} be the blocks which contain no outliers. From the assumption on Q, i.e., 2m < Q < n, it follows that, and |A| > Q/2. For q ∈ [Q], let ξ q (t; n, Q) be given by ξ q (t; n, Q) = 1 d n,q − d X ∞ > t . On application of Lemma 3.2 to the estimator d n,Q , it follows that P d n,Q − d X ∞ > t ≤ P   q∈A ξ q (t; n, Q) > Q 2 − m   .(16) Since S q ⊆ X * n-m for all q ∈ A, it follows that {ξ q (t; n, Q) : q ∈ A} are i.i.d. Bernoulli p(t; n, Q) random variables, where p(t; n, Q) = E(ξ q (t; n, Q)) = P d n,q − d X ∞ > t . For the remainder of the proof we need two key ingredients: (i) we need an upper bound for E(ξ q (t; n, Q)), and (ii) we need a tight bound for the binomial tail probability in Eq. (16). Bound for p(t; n, Q). From Chazal et al. (2015b, Theorem 2), under the (a, b)−standard condition it follows that p(t; n, Q) ≤ 2 b at b exp − n Q at b = exp − n Q at b − log at b + b log 2 .(17) Binomial tail probability bound. For 0 < < 1, using the Chernoff-Hoeffding bound from Lemma B.2 yields, P   1 |A| q∈A ξ q (t; n, Q) >   ≤ exp |A| 2 e + log p(t; n, Q) . Using the bound for p(t; n, Q) from Eq. (17), we obtain P   1 |A| q∈A ξ q (t; n, Q) >   ≤ exp |A| 2 e + b log 2 − n Q at b − log at b ≤ exp |A| 1 + b − n Q at b − log at b ≤ exp |A| 1 + b − Ω(t, n/Q) , where, in the last line we use Ω(t, n/Q) = · (n/Q)at b + log at b for brevity. When t satisfies the condition that Ω(t, n/Q) ≥ 2(1 + b ) ,(18) then it implies that 1 + b − Ω(t, n/Q) ≤ − 2 Ω(t, n/Q), and we get P   1 |A| q∈A ξ q (t; n, Q) >   ≤ exp − |A| 2 Ω(t, n/Q) . By setting δ equal to the r.h.s. of the inequality above, we obtain Ω(t, n/Q) = 2 log(1/δ) |A| .(19) When δ ≤ e −(1+b)Q , using the fact that Q > |A| and 0 < < 1, it follows that Ω(t, n/Q) = 2 log(1/δ) |A| ≥ 2(1 + b)Q |A| ≥ 2(1 + b ) , and, therefore, the condition in Eq. (18) is satisfied. Consequently, for δ ≤ e −(1+b)Q , on rearranging the terms in Eq. (19) we obtain P   q∈A ξ q (t; n, Q) > 2 log(1/δ) Ω(t, n/Q)   ≤ δ.(20) Comparing Eq. (16) with Eq. (20) we conclude that P   q∈A ξ q (t; n, Q) > Q − 2m 2   = P   q∈A ξ q (t; n, Q) > 2 log(1/δ) Ω(t, n/Q)   ≤ δ, by setting 2 log(1/δ) Ω(t, n/Q) = Q − 2m 2 ⇐⇒ Ω(t, n/Q) = 4log(1/δ) Q − 2m . Since Ω(t, n/Q) = n Q at b + log at b , this is equivalent to exp n Q at b n Q at b = n Q exp 4log(1/δ) Q − 2m . Moreover, using the fact that the Lambert W 0 function is given by the identity W 0 (x)e W 0 (x) = x (Hoorfar and Hassani, 2008), we obtain that t = Q an W 0 n Q exp 4log(1/δ) Q − 2m 1/b .(21) Therefore, from Eq. (15) and (16), for t satisfying Eq. (21) and for all τ ≥ t we have that P W ∞ Dgm(d n,Q ), Dgm(d X ) > τ ≤ P W ∞ Dgm(d n,Q ), Dgm(d X ) > t ≤ δ.(22) Since δ ≤ e −(1+b)Q , observe that 4 log(1/δ) Q − 2m ≥ 4(1 + b)Q Q − 2m ≥ 4(1 + b) > 1. Furthermore, using the fact that W 0 (z) ≤ log(z) for z > e (Hoorfar and Hassani, 2008, Eq. 1.1), we may take τ to be t = Q an W 0 n Q exp 4log(1/δ) Q − 2m 1/b ≤ Q an log n Q exp 4log(1/δ) Q − 2m 1/b = Q log(n/Q) an + 4Qlog(1/δ) a(Q − 2m)n 1/b = · τ. Plugging this into Eq. (22), we obtain the desired result. For the second claim in the theorem, by inverting the relationship between t and δ in Eq. (21) and using the fact that W 0 (z) is an increasing function for z > 0, observe that the constraint on δ equivalently specifies a constraint on t, i.e., δ ≤ e −(1+b)Q ⇐⇒ t ≥ Q an W 0 n Q exp 4(1 + b)Q Q − 2m 1/b . A sufficient condition for this to hold is that t ≥ t(n, Q) = · Q log(n/Q) an + 4(1 + b)Q 2 a(Q − 2m)n 1/b . Therefore, from Eq. (22) we have that for all t ≥ t(n, Q) P W ∞ Dgm(d n,Q ), Dgm(d X ) > t ≤ exp − Q − 2m 4 Ω(t, n/Q) . By taking P W ∞ Dgm(d n,Q ), Dgm(d X ) > t to be its maximum value of 1 in the interval [0, t(n, Q)] we have E W ∞ Dgm(d n,Q ), Dgm(d X ) = ∞ 0 P W ∞ Dgm(d n,Q ), Dgm(d X ) > t dt ≤ t(n, Q) + ∞ t(n,Q) exp − Q − 2m 4 Ω(t, n/Q) dt. By taking w = Ω(t, n/Q) and setting r n = 4(1 + b)Q/(Q − 2m), we further obtain E W ∞ Dgm(d n,Q ), Dgm(d X ) t(n, Q) + Q n 1/b ∞ rn e −w/4 W 0 n Q e w 1/b w + 1 dw (ii) t(n, Q) + log(n/Q) n/Q 1/b ∞ rn e −w/4 w + 1 dw a + Q n 1/b ∞ rn e −w/4 w 1/b w + 1 dw b ,(23) where (ii) follows from the fact that W 0 (z) ≤ log(z) for z > e together with, either, an application of Lemma B.1 (iii) when b ≥ 1, or Lemma B.1 (i) with the additional factor 2 1/b−1 being absorbed in the symbol when b < 1. The term a can be bounded above using the incomplete Γ function as, a = Similarly, using the fact that w + 1 > 1, the term b may be bounded above as, b = ∞ rn e −w/4 w 1/b w + 1 dw ≤ ∞ rn e −w/4 w 1/b dw ≤ Γ(1 + b −1 ) 4 1+1/b ∞ rn π(v)dv ≤ Γ(1 + b −1 ) 4 1+1/b < ∞, where π is the probability density function of the distribution Γ 1 + b −1 , 1/4 . Therefore, the inequality in Eq. (23) becomes E W ∞ Dgm(d n,Q ), Dgm(d X ) log(n/Q) n/Q + Q 2 (Q − 2m)n 1/b + Q n 1/b . When the number of outliers grows with n as m n = cn where 0 ≤ < 1, let the number of blocks be Q n = 3cn β , where ≤ β < 1. Therefore, E W ∞ Dgm(d n,Q ), Dgm(d X ) inf ≤β<1 log n/n β n/n β + n 2β (3n β − 2n )n 1/b + n β n 1/b log n n 1− 1/b , which gives us the desired result. Proof of Lemma 3.3 For simplicity, let f = d n,Q denote the MoM Dist function. By definition, V [f ] and V [R d , f ] are (id, α)−interleaved if the following relationship holds V t [f ] ⊆ V t [R d , f ] ⊆ V α(t) [f ]. The first inclusion is straightforward since V t [f ] ⊆ x∈V t [f ] B f (x, t) = x∈R d B f (x, t) = V t [R d , f ]. For the second inclusion, suppose x ∈ V t [R d , f ], i.e., there exists y ∈ R d such that x − y ≤ r f,y (t). It suffices to show that x ∈ V α(t) [f ]. To this end, note that since d n,Q is 1−Lipschitz by Lemma 3.1 it follows that f (x) ≤ f (y) + x − y ≤ f (y) + r f,y (t) = f (y) + (t p − f (y) p ) 1 p (i) ≤ 2 p−1 p (f (y) p + (t p − f (y) p )) 1 p = 2 p−1 p t, where (i) follows from an application of Lemma B.1 (iii). Since f (x) ≤ 2 p−1 p t = α(t), it implies that x ∈ V α(t) and the result follows. When p = 1, note that α(t) = t, and therefore V [f ] = V [R d , f ]. Proof of Lemma 3.5 Since X ⊆ Y, the inclusion V t ρ [X, f ] ⊆ V t ρ [Y, f ] holds trivially. For the next part, let V t = V t ρ [X, f ] and U t = V t ρ [Y, f ] denote the respective f -weighted filtrations, so as to avoid the notational overload. In order to show the second inclusion, i.e., U t ⊆ V α(t) , consider z ∈ U t . Then, there exists y ∈ Y such that z ∈ B f,ρ (y, t). If y ∈ X ⊂ Y, then it immediately follows that z ∈ V t ⊆ V α(t) . In what remains, for y ∈ Y \ X, it is sufficient to show that there exists x ∈ X such that z ∈ B f,ρ (x, α(t)). To this end, let x * y = arg inf x∈X ρ(x, y) be the projection of y onto X via ρ. Then two following cases arise: (I) ρ(x * y , z) ≤ ρ(x * y , y), and (II) ρ(x * y , z) ≥ ρ(x * y , y) (see Figure 9). Case I. The distance between x * y and z will satisfy ρ(x * y , z) ≤ ρ(x * y , y) (i) ≤ f (y) + a (ii) ≤ t p − ρ(y, z) p 1 p + a ≤ t + a where (i) follows from the assumption on f , and (ii) follows from the fact that if z ∈ B f,ρ (y, t), then ρ(y, z) ≤ r f,y (t) = (t p − f (y) p ) 1/p . Furthermore, from Lemma B.1 (vi) we obtain ρ(x * y , z) ≤ (t + a + f (x * y )) p − f (x * y ) p 1 p ≤ t + a + sup x∈X f (x) p − f (x * y ) p 1 p ≤ 2 1− 1 p t + a + sup x∈X f (x) p − f (x * y ) p 1 p = α(t) p − f (x * y ) p 1 p = r f,x * y (α(t)), where the last inequality holds because 2 1− 1 p ≥ 1. The last line implies that z ∈ B f,ρ x * y , α(t) ⊆ V α(t) . Case II. For r = ρ(x * y , y) let y be the projection of z onto ∂B x * y , r , i.e., y = arg inf x ∈∂B(x * y ,r) ρ(x , z). 3 The interleavings α(t) (Lemma 3.5) and β(t) are combined to provide the bound in W ∞ . Claim 1 . Let y ∈ Y \ X. We need to show that there exists x ∈ X such that B f,ρ (x, β(t)) ∪ B f,ρ (y, t) is star-shaped around x 0 , i.e., for any z ∈ B f,ρ (y, t) the curve Γ[x 0 , z] is contained inside the set B f,ρ (x, β(t)) ∪ B f,ρ (y, t). See Figure 10. To this end, let x = arg inf z∈X ρ(z, y) be the projection of y onto X. Note that, from the definition of x 0 , x 0 ∈ B f,ρ (x, t) for all t ≥ t(X). For simplicity, let S t = B f,ρ (x, β(t)) ∪ B f,ρ (y, t). Additionally, let π(x) and π(y) be the projection of x and y onto Γ[x 0 , z], respectively, i.e., z] ρ(x , x), mutatis mutandis, the same for π(y). By definition, ρ(x, π(x)) ≤ ρ(x, x 0 ) and ρ(y, π(y)) ≤ ρ(y, z), and consequently, π(y) ∈ B f,ρ (y, t). This implies that Γ[π(y), z] ⊆ S t . What remains to be established is that Γ[x 0 , π(y)] ⊆ S t . In order to show this, note that it is sufficient to show that π(y) ∈ B f,ρ (x, β(t)). Indeed, if this holds, then Γ[x 0 , π(y)] ⊆ B f,ρ (x, β(t)) ⊆ S t , and it will follow that Γ[x 0 , π(y)] ∪ Γ[π(y), z] = Γ[x 0 , z] ⊆ S t . π(x) = arg inf x ∈Γ[x 0 , Let τ = ρ(y, π(y)). Since π(y) ∈ B f,ρ (y, t), when ρ(x, y) > a it follows that τ ≤ r f,y (t) ≤ t p − f (y) p 1 p ≤ t p − ρ(x, y) − a p 1 p , where the last inequality follows from the assumption on f . Thus, we have ρ(x, y) ≤ t p − τ p 1 p + a.(26) Alternatively, when ρ(x, y) ≤ a, Eq. (26) holds trivially. Since π(x) ∈ B f,ρ (x, t) and ρ(x, π(x)) ≤ ρ(x, x 0 ), it follows that ρ(x, π(x)) ≤ t(X). Since ρ = · , Anai et al. (2019, Lemma B.2) holds, which, combined with Eqs. (26) and (27) yields ρ(x, π(y)) 2 (i) ≤ t p − τ p 1 p + a 2 + τ (2t(X) − τ ) ≤ t p − τ p 2 p + τ (2t(X) − τ ) + a 2 + 2a t p − τ p 1 p (ii) ≤ (t + κt(X)) 2 + a 2 + 2at ≤ (t + κt(X)) 2 + a 2 + 2a(t + κt(X)) = t + a + κt(X) 2 , where ( and noting that t p − τ p ≤ t p since τ ≤ t, and κ = (1 − 1 p ). Additionally, from Lemma B.1 (vi) we obtain ρ(x, π(y)) ≤ t + a + κt(X) ≤ t + a + κt(X) + sup x∈X f (x) p − f (x) p 1 p = r f,x (β(t)). This implies that π(y) ∈ B f,ρ (x, β(t)), and establishes claim 1 . For claim 2 , note that since W t ∼ {x 0 }, for the kth homology group W t , we have that W t F for k = 0, and W t {0} for k > 0. Therefore, the map w t : W t → U β(t) is trivial, and consequently, so is the linear map U t → U β(t) . In order to show claim 3 , observe that the persistence modules U and V are (id, α)-interleaved for all t and for α : t → 2 1− 1 p t + a + sup x∈X f (x) (id, β)-interleaved for t ≥ t(X) and for β : t → t + a + κt(X) + sup x∈X f (x) . α(t) = t + 2 1− 1 p − 1 t + a + sup x∈X f (x) ≤ t + κt(X) + a + sup x∈X f (x) = β(t). Thus, α(t) ≤ β(t) for t ≤ t(X). Since β : t → t + c(X) is an additive interleaving for c(X) = κt(X) + a + sup x∈X f (x), this implies that W ∞ Dgm(U), Dgm(V) ≤ c(X), which establishes claim 3 . Proof of Theorem 3.2 We begin by establishing the following result: W ∞ V[X n , d n,Q ], V[X * n-m , d n,Q ] ≤ sup x∈X * n-m d n,Q (x) + 1 − 1 p t(X * n-m ). Observe that from Lemma 3.5 and Lemma 3.6, it suffices to show that for every y ∈ Y m the MoM-Dist function d n,Q satisfies the property that inf x∈X * n-m x − y ≤ d n,Q (y). When p = 1, the term b ≤ d n,Q − d X ∞ using Lemma 2.2 (i), and the term c ≤ H(X * n-m , X) using Lemma 2.2 (ii). The term a is bounded above by taking p = 1 in Eq. (28) (from the proof of Theorem 3.2) to give a = sup x∈X * n-m d n,Q (x) ( ) ≤ sup x∈X d n,Q (x) ( †) ≤ d n,Q − d X ∞ + sup x∈X d X (x) ( ‡) = d n,Q − d X ∞ , where ( ) follows from the fact X * n-m ⊂ X, ( †) uses the identity f (x) ≤ f − g ∞ + g(x) for all x ∈ X, and ( ‡) follows from Eq. (29). Plugging in the bounds for the bottleneck distance we obtain W ∞ V[X * n-m ∪ Y m , d n,Q ], V[X] ≤ 2 d n,Q − d X ∞ + H(X * n-m , X). By noting that the Hausdorff distance H(X * n-m , X) = d n−m − d X ∞ , for t 1 , t 2 such that t 1 + t 2 = t we may bound the tail probability for the bottleneck distance as follows. P W ∞ V[X * n-m ∪ Y m , d n,Q ], V[X] > t ≤ P 2 d n,Q − d X ∞ > t 1 + P d n−m − d X ∞ > t 2 ≤ δ 1 + δ 2 = δ,(30) where the relationship between δ 1 , δ 2 and t 1 , t 2 is given by Eq. (21), i.e., δ 1 ≤ e −(1+b)Q from the condition in Theorem 3.1, δ 2 = δ − δ 1 , t 1 = 2 Q an W 0 n Q exp 4log(1/δ 1 ) Q − 2m 1/b , and t 2 = 1 an W 0 ne 4log(1/δ 2 ) 1/b . Furthermore, using the bound for the Lambert W 0 function W 0 (z) ≤ log z for z > e, we have t 1 ≤ 2 Q log(n/Q) a(n/Q) + 4Q log(1/δ 1 ) a(Q − 2m)n 1/b , t 2 ≤ log(n − m) a(n − m) + 4 log(1/δ 2 ) a(n − m) 1/b , and t = t 1 + t 2 ≤ f(n, m, Q, δ 1 , δ 2 ). Therefore, the bound in Eq. (30) yields P W ∞ V[X * n-m ∪ Y m , d n,Q ], V[X] ≤ f(n, m, Q, a, b) ≥ P W ∞ V[X * n-m ∪ Y m , d n,Q ], V[X] > t 1 + t 2 ≥ 1 − δ, which gives the desired result. The second part of the theorem follows directly using the identical procedure as that used in the proof of Theorem 3.1 in Section 6.3. Proof of Proposition 3.1 We begin by noting from Lemma 3. Using an identical argument, but reversing the order, we have that V [X * n-m , d n,Q ] and V [X n , d n,Q ] are (id, η)−interleaved. We can now apply the "triangle inequality" for generalized interleavings (Bubenik et al., 2015, Proposition 3.11) to obtain that V [d n,Q ] and V [X n , d n,Q ] are (id • η • id, α • id • η)−interleaved. On simplifying the interleaving maps, we obtain that the two filtrations are (η, ξ)−interleaved for ξ(t) = α • η(t) = 2 p−1 p η(t). Proof of Theorem 3.4 The birth time of a connected component at x 0 in V [X, f ] is given by b f ({x 0 }) = f (x 0 ). Therefore, ∆b n,m ({x 0 }) is given by ∆b n,m ({x 0 }) = b n ({x 0 }) − b n+m ({x 0 }) = d n (x 0 ) − d n+m (x 0 ) = d n (x 0 ), where the last equality follows from the fact that d n+m (x 0 ) = 0, since x 0 ∈ X n+m . On the other hand, from the proof of Proposition 3.2, d n (x 0 ) = inf x∈Xn x − x 0 ≤ d n+m,Q (x 0 ). Therefore, we have that ∆b n,m,Q ({x 0 }) = d n (x 0 ) − d n+m,Q (x 0 ) ≤ ∆b n,m ({x 0 }), and the result follows. For the second part, we begin by observing that d n+m − d n ∞ can be bounded from below as follows: d n+m − d n ∞ ≥ d n (x 0 ) − d n+m (x 0 ) = d n (x 0 ) − 0 = inf x∈Xn x − x 0 ≥ inf x∈X x − x 0 = d X (x 0 ). Furthermore, for δ ≤ e −(1+b)Q and k = · max 1, 2 b−1 b , with probability greater than 1 − δ, log n Q + 4Q log(2/δ) n Q d n+m,Q − d n ∞ (i) ≤ d n+m,Q − d X ∞ + d n − d X ∞( 1/b = · η(n, m, Q, δ), where, for n Q = (n + m)/Q, (i) is a consequence of the triangle inequality and (ii) follows from the proofs of Theorem 3.1 and Theorem 3.3, (iii) uses Lemma B.1, (iv) follows from the fact that W 0 (z) < log(z) for z > e, and (v) uses the fact that n Q < n and (Q − 2m) −1 ≤ 1 for 2m > Q. Observe that if 2η(n, m, Q, δ) ≤ d X (x 0 ), then with probability greater than 1 − δ, d n+m − d n ∞ − d n+m,Q − d n ∞ ≥ d X (x 0 ) − η(n, m, Q, δ) ≥ η(n, m, Q, δ),(31) and the result follows. Therefore, in order to establish the claim for the second part it suffices to check that 2η(n, m, Q, δ) ≤ d X (x 0 ) under conditions (I) and (II). To this end, note that d X (x 0 ) ≥ 2η(n, m, Q, δ) ⇐⇒ (x 0 ) ≥ log n Q + 4Q log(2/δ) n Q , which is satisfied whenever δ satisfies the r.h.s. of condition (II), i.e., log(2/δ) ≤ n Q (x 0 ) − log n Q 4Q . Furthermore, the l.h.s. of condition (II), i.e., δ ≤ e −(1+b)Q , is satisfied only when (1 + b) 2 Q 2 ≤ n Q (x 0 ) − log n Q 4Q , or, equivalently, when condition (I) is satisfied: (x 0 ) ≥ log n Q n Q + 4(1 + b) 2 Q 3 n Q . The result now follows from Eq. (31). Proof of Theorem 3.5 Let j * = min {j ∈ J : m(j) > m * }. By definition of J we have that |J | ≤ 1 + log θ (m max /m min ) and m(j * ) < θm * for θ > 1. The outline of the proof is as follows. First, we show that h(n, m, δ) is non-decreasing in m, from which it follows that h(n, m(j), δ) ≤ h(n, m(j + 1), δ). Next, we show that the event { ≤ j * } contains the event E given by E = {j∈J :j≥j * } W ∞ V n (j), V[X] ≤ h(n, m(j), δ) . Then, using a standard procedure for obtaining the Lepski bound (e.g., Theorem 5.1 of Minsker 2018 and Theorem 3.1 of Chen and Zhou 2020), we show that the event E, and, therefore the event { ≤ j * }, holds with probability at least 1 − δ log θ (m max /m min ). Lastly, we use the bound on the event { ≤ j * } to obtain the desired result. d X : Distance function to a compact set X given by d X (y) = inf x∈X x − y 1 p ≤ x/y ≤ (x/y) p for p ≥ 1. By rearranging the terms, we get x ≤ y 1−p x p and x ≥ y 1− 1 p x 1 p . A Glossary of Notations Part (vi). We have x = (x + y − y) = (x + y − y) p 1 p . From Part (ii) we have (x + y − y) p ≤ (x + y) p − y p , which, on rearrangement, yields x ≤ (x + y) p − y p 1 p . Lemma B.2 (Chernoff-Hoeffding bound simplified). Suppose Z 1 , Z 2 , . . . Z N are i.i.d. Bernoulli(p) random variables. Then, for 0 < < 1, P   1 N 1≤i≤N Z i >   ≤ exp N 2 e + log(p) . Proof. For 0 < < 1, using the Chernoff-Hoeffding bound for binomial random variables (Hoeffding, 1963, Theorem 1) we have P   1 N 1≤i≤N Z i >   ≤ exp −N · KL Ber( )||Ber(p) ,(34) where Ber( ) and Ber(p) are Bernoulli distributions with parameters and p respectively, and KL(P||Q) is the Kullback-Leibler divergence of Q w.r.t P. Simplifying the quantity in the exponent, we get Figure 1 : 1Illustration of offsets for t = 0.5 and f (x) = inf y∈S 1 x − y . Vishwanath et al., 2020, RKDE-filtration) O(n 2 ) O(n 2 ) No n = #samples, Q = #blocks, k = mn = DTM parameter, σ =RKDE bandwidth, and = #iterations of KIRWLS algorithm of the choice of the weighting function f . needed for a single query(Cormen et al., 2009, Chapter 10). The results for each block q ∈ [Q] are then aggregated to compute the median, which takes an additional O(Q) time per query. This results in a total evaluation time of O(n · (Q + log n/Q)) for n samples. Figure 2 : 2Comparison of V t [Xn, f ] for the median filtration value t = median{w1, w2, . . . , wn} and p = ∞. Lemma 3. 4 ( 4Regularity). For X n obtained under sampling setting (S) and d n,Q defined in Eq. (3), the persistence module V[X n , d n,Q ] is q−tame and pointwise finite-dimensional. The proof of Lemma 3.4 is a direct consequence of Anai et al. (2019, Proposition 3.1), and ensures that the persistence diagram Dgm(V[X n , d n,Q ]) is well-defined. Remark 3. 3 . 3Unlike Lemma 3.5, which is stated for general metric spaces, restricting ourselves to the Euclidean space (R d , · ) in Lemma 3.6 is sufficient for the objective of this work. However, as outlined in the proof, the only issue arises whenAnai et al. (2019, Lemma B.1) is invoked. WhileAnai et al. (2019, Lemma B.1) (which holds for affine spaces satisfying the parallelogram identity) extends naturally to Banach spaces, the extension to general metric spaces will require some care on a case-by-case basis. Theorem 3. 2 . 2(Stability & robustness of d n,Q -weighted filtrations) Let X n = X * n-m ∪ Y m be a collection of points obtained under sampling condition (S). For Q > 2m let d n,Q be the MoM Dist function computed on the contaminated points X n and let d n−m be the distance function w.r.t. the inliers X * n-m . Then Proposition 3. 1 . 1Given samples X n = {X 1 , X 2 , . . . , X n } and Q < n, the filtrations V [d n,Q ] andV [X n , d n,Q ] are (η, ξ)−interleaved, where η : t → 2 p−1 p t + sup x∈X * n-m d n,Q (x), ξ : t → 2 p−1 p η(t),and p ≥ 1. Specifically, when p = 1, For 2m < Q < n + m, let d n+m and d n+m,Q denote the distance and MoM distance function w.r.t. X n+m , and let ∆b n,m ({x 0 }) and ∆b n,m,Q ({x 0 }) be as defined in Eq. (10) for d n+m and d n+m,Q respectively. Then ∆b n,m,Q ({x 0 }) ≤ ∆b n,m ({x 0 }) a.s. Furthermore, for n Q = (n + m)/Q and c = min a2 −(1+b) , a2 −2b , if generated on a circle, and m (i) ∼ Unif([50, 150]) outliers added from a Matérn cluster process. This is illustrated inFigure 4(a). Taking m min = 20, m max = 200 and θ = 1.07, the adaptive estimate m (i) is computed using Lepski's method, and m (i) R is computed using the heuristic method described in Section 3.5 with N = 50. For a single replicate i ∈ [K],Figure 4(b) plots 1≤i<j≤N W ∞ V[X σ i n , d n,Q ], V[X σ j n , d n,Q ] vs. Q.In most cases, we have observed that the resampled bottleneck distance criterion stabilizes shortly before the optimal value of m.Figure 4 (c) shows a boxplot for the relative errors m (i) − m (i) /m (i) : i ∈ [K] and m (i)R − m (i) /m (i) : i ∈ [K]for Lepski's method and the heuristic procedure, respectively. Lepski's method is fairly robust to the choice of the hyperparameters, and, consistently selects m (i) ≥ m (i) . In contrast, since the resampled bottleneck distance from the heuristic procedure often stabilizes before m (i) , we observe that m (i) R < m (i) . Figure 4 : 4(c) depicts the sublevel persistence diagram Dgm(V[d n,Q ]) computed using cubical homology on a grid of resolution 0.5. As expected by the result of Proposition 3.1, the d n,Q -weighted persistence diagram Dgm(V[X n , d n,Q ]) in Figure 5 (d) captures the essential topological information in Dgm(V[d n,Q ]).https://www.github.com/sidv23/RobustTDA.jl/ Comparison of Lepski's method and the heuristic procedure for selecting the parameter Q. ( a )Figure 5 : a5MoM Dist function d n,Q (b) V t [Xn, d n,Q ] (c) Dgm V[d n,Q ] (d) Dgm V[Xn, d n,Q ]Comparison of sublevel filtrations with the dn,Q-weighted filtration. ( a )Figure 6 : a6Xn projected to coordinates 11, 53 & 91 (b) MOM diagram Dgm( Vn)(c) DTM diagram Dgm V[Xn, δ n,k ] Robust persistence diagrams for interlocked circles in R 100 using dn,Q and δ n,k weighted filtrations. In a small neighborhood around the center of the circle, outliers Y m are sampled uniformly from [−0.1, 0.1] 2 . For the composite sample X n ∪ Y m and a fixed value of Q = 100 & k = 50, we compute the MoM Dist weighted persistence diagram D M oM n+m,Q= Dgm(V[X n ∪ Y m , d n+m,Q ]), the DTM weighted persistence diagram D DT M n+m,k = Dgm(V[X n ∪ Y m , δ n+m,k ]), and the RKDE weighted persistence diagram D RKDE n+m,ρ,σ from the RKDE f n+m ρ,σ using the Hampel loss ρ and a Gaussian kernel K σ . i K σ (·, X i ) does not behave like a distance function, we convert f n+m ρ,σ to a distance-like function d n+m,ρ,σ using a similar approach as Phillips et al.(2015)to obtain Figure 7 :Figure 8 : 78Xn (f) Xn colored by d n,Q (g) Dgm(V[Xn-m]) (h) Dgm(V[Xn]) (i) Dgm V[Xn, d n,Q ] Recovering the topological information underlying the signal in the presence of adversarial contamination. (a) influence(b; Xn, fn, m, x 0 ) (b) max Persistence for the first order diagram (c) influence(W∞; Xn, fn, m, x 0 ) Influence analysis for dn,Q-weighted filtrations vis-à-vis DTM-based filtrations and unweighted filtrations. v −1 e −v dv = e 1/4 Γ 0, (r n − 1)/4 < ∞. Figure 9 : 9Illustration of Case I (Left) and Case II (Right). Figure 10 : 10Illustration of Claim 1 . i) is a consequence of Anai et al. (2019, Lemma B.2), (ii) follows from Anai et al. (2019, Lemma B.3) When t ≤ t(X), from Anai et al. (2019, Lemma B.1), r f,x (t) : The f -weighted radius function of resolution t at x. r f,x (t) = (t p − f (x) p ) 1 p B f,ρ (x, t) : f -weighted ball at x with radius r f,x (t) w.r.t the metric ρ. V t [X, f ] : The f -weighted offset of X at resolution t given by V t [X, f ] = x∈X B f (x, t) V [X, f ] : f -weighted filtration, i.e., · · · → V t 1 [X, f ] → V t 2 [X, f ] → . . . V tn [X, f ] → . . . V[X, f ] : f -weighted persistence module, i.e., V[X, f ] = Hom(V [X, f ])Dgm(V) : Persistence diagram associated with the persistence module V θ n,Q : MoM-estimator, median{θ 1 , . . . ,θ Q }, whereθ q is the estimator from block S q .d n,Q : MoM Dist function given by d n,Q (x) = median inf y∈Sq x − y : q ∈ [Q] last inequality uses the fact that x log(x) ≥ −1/e for all 0 ≤ x ≤ 1, and −(1 − ) log(1 − p) ≥ 0 for all 0 ≤ , p ≤ 1. Substituting this in Eq. (34) yields the result. Table 1 : 1Comparison of computational complexity for robust weighted filtrations. 3 that V [d n,Q ] and V [R d , d n,Q ] are (id, α)−interleaved for α : t → 2 Furthermore, consider the intermediate filtrations V [X * n-m , d n,Q ]. From Proposition 3.2 and Lemma 3.5 we have that V [R d , d n,Q ] and V [X * n-m , d n,Q ] are (η, id)− interleaved for η : t → 2p−1 p t. p−1 p t + sup x∈X * n-m d n,Q (x). ii)≤ 1 an Q W 0 n Q exp 4Q log(2/δ) Q − 2m 1/b + 1 an W 0 (n exp {4 log(2/δ)}) 1/b (iii) ≤ k a 1/b 1 n Q W 0 n Q exp 4Q log(2/δ) Q − 2m + 1 n W 0 (n exp {4 log(2/δ)}) 1/b (iv) ≤ k a 1/b log n Q n Q + log n n + 4 log(2/δ) Q n Q (Q − 2m) + 1 n 1/b (v) ≤ k2 1/b a 1/b H ρ (X, Y) : Hausdorff distance between X ⊆ M and Y ⊆ M measured w.r.t. metric ρ.V t [f ] : Sublevel set of f at level t given by x ∈ R d : f (x) ≤ t V [f ] and V[f ] : Sublevel filtration V t [f ]: t ∈ R and its persistence module The inclusion map ι t : U t → U β(t) can be decomposed as ι t = j t • κ t where j t : U t → W t and κ t : W t → U β(t) . Since W t is star-shaped and contractible, i.e., W t ∼ {x 0 }, the linear map between the homology groups induced by κ t , i.e., v t : W t → U β(t) will be trivial. AcknowledgementsBKS is supported by National Science Foundation (NSF) CAREER Award DMS-1945396. SK is partially supported by JSPS KAKENHI Grant Number 21H03403. SV, KF, and SK were supported by JST, CREST Grant Number JPMJCR15D3, Japan.The point y satisfies the following three properties: (PI) ρ(x * y , y ) = ρ(x * y , y), since y ∈ ∂B f,ρ x * y , r ; (PII) ρ(z, y ) ≤ ρ(z, y) by definition of y ; and (PIII) ρ(x * y , y ) + ρ(y , z) ≥ ρ(x * y , z) from the triangle inequality.Since z ∈ B f,ρ (y, t), when ρ(x * y , y) ≤ a we may use the triangle inequality to obtainAlternatively, when ρ(x * y , y) > a we obtain the following inequality,where (iii) holds from the assumption on f , (iv-vi) follow from (PI-PIII) respectively, and (vii) uses Lemma B.1 (i). Rearranging the terms of Eq. (25) we get ρ(x * y , z) ≤ a + 2 1− 1 p t. Therefore, from Eq. (24) and Eq. (25), in case (II) we have thatwhere (viii) uses Lemma B.1 (vi). Similar to case (I), we obtain z ∈ B f,ρ (x, α(t)) ⊆ V α(t) .Proof of Lemma 3.6Let t(X) = inf t > 0 : x∈X B f,ρ (x, t) = ∅ , and let x 0 ∈ x∈X B f,ρ (x, t). To ease the notation, letdenote the usual f -weighted filtration, and let W t be defined asWith this background, the proof closely follows that ofAnai et al. (2019, Proposition 4.8). Specifically, the proof is based on the following outline:1 We first establish that for any y ∈ Y \ X, there existsis star-shaped around x 0 . Since this holds for all y ∈ Y \ X, it also holds for y∈Y\X B f,ρ (y, t), and, therefore, W t is star-shaped and contractible to x 0 .To this end, let A = {q ∈ [Q] : S q ∩ Y m = ∅} be the blocks containing no outliers. For y ∈ Y m and every q ∈ A, we have that S q ⊆ X * n-m , and thereforeSince this holds for every q ∈ A, taking the infimum on the right hand side over Q yieldsd n,q (y).Since 2m < Q by assumption, using the pigeonhole principle we further have thatwhich implies that inf x∈X * n-m x − y ≤ d n,Q (y) for every y ∈ Y m . Therefore, taking a = 0 in Lemma 3.5 and Lemma 3.6 we obtainTurning our attention to the quantity appearing in the statement of the theorem, note that an application of the triangle inequality yieldswhere the first term in ( ) follows from Eq. (28) and the last term follows from Lemma 2.2 (i). This gives us the desired result. Furthermore, when p = 1 note that 1 − 1/p = 0, giving us the tighter bound in this case.Proof of Theorem 3.3We begin by noting that V[X] = V[X, d X ]. Indeed, the distance function d X (x) = 0 for all x ∈ X. We may further conclude thatThe bottleneck distance between V[X * n-m ∪ Y m , d n,Q ] and V[X] may be bounded above asB Supplementary ResultsThe following lemma is a collection of well-known inequalities (and their slight variants). We state them here for reference, as they are used frequently in the proofs.Lemma B.1. For 0 < y ≤ x and p ≥ 1, the following inequalities hold:for all 0 < y ≤ x and p ≥ 1. Therefore f (y) is strictly non-decreasing, and f (y) ≥ f (0) = 0. This gives us the first inequality. For the second inequality, note that g(z) = z p is convex for z ≥ 0. This follows from the fact that g (z) = p(p − 1)z p−2 ≥ 0 for all z ≥ 0 and p ≥ 1. By convexity, we obtainwhich leads to the second inequality.Part (ii). Let z = (x − y). Applying the first inequality from the preceding part to z and y we get z p ≤ (y + z) p − y p , i.e., (x − y) p ≤ x p − y p . Similarly, from the second inequality, (z + y) p ≤ 2 p−1 (z p + y p ), which is the same as 2 1−p x p − y p ≤ (x − y) p .Part(iii). Note that f (z) = z 1 p is concave for all z ≥ 0 and p ≥ 1, sincefor all z ≥ 0, p ≥ 1. Therefore, by concavity,which leads to the right hand side inequality, i.e., x 1 p + y Part (iv). The proof is identical to the proof in Part (ii). The inequalities are obtained by taking z = (x − y), and applying the results of Part (iii).Part(v). Since y ≤ x, it follows that 1 ≤ (x/y) DTM-based filtrations. Hirokazu Anai, Frédéric Chazal, Marc Glisse, Yuichi Ike, Hiroya Inakoshi, Raphaël Tinarrage, Yuhei Umeda, SoCG 2019-35th International Symposium on Computational Geometry. Hirokazu Anai, Frédéric Chazal, Marc Glisse, Yuichi Ike, Hiroya Inakoshi, Raphaël Tinarrage, and Yuhei Umeda. DTM-based filtrations. In SoCG 2019-35th International Symposium on Computational Geometry, 2019. Robust linear least squares regression. Jean- , Yves Audibert, Olivier Catoni, The Annals of Statistics. 395Jean-Yves Audibert and Olivier Catoni. Robust linear least squares regression. The Annals of Statistics, 39 (5):2766-2794, 2011. An alternative point of view on Lepski's method. State of the Art in Probability and Statistics: Festschrift for Willem R. Lucien Birgé, Van Zwet36114Lucien Birgé. An alternative point of view on Lepski's method. State of the Art in Probability and Statistics: Festschrift for Willem R. Van Zwet, 36:114, 2001. Computing the interleaving distance is NP-hard. Magnus Bakke Håvard Bakke Bjerkevik, Michael Botnan, Kerber, Foundations of Computational Mathematics. 205Håvard Bakke Bjerkevik, Magnus Bakke Botnan, and Michael Kerber. Computing the interleaving distance is NP-hard. Foundations of Computational Mathematics, 20(5):1237-1271, 2020. Geometric and Topological Inference. Jean-Daniel Boissonnat, Frédéric Chazal, Mariette Yvinec, Cambridge University Press57Jean-Daniel Boissonnat, Frédéric Chazal, and Mariette Yvinec. Geometric and Topological Inference, volume 57. Cambridge University Press, 2018. Metrics for generalized persistence modules. Peter Bubenik, Vin De Silva, Jonathan Scott, Foundations of Computational Mathematics. 156Peter Bubenik, Vin De Silva, and Jonathan Scott. Metrics for generalized persistence modules. Foundations of Computational Mathematics, 15(6):1501-1531, 2015. Efficient and robust persistent homology for measures. Mickaël Buchet, Frédéric Chazal, Y Steve, Donald R Oudot, Sheehy, Computational Geometry. 58Mickaël Buchet, Frédéric Chazal, Steve Y Oudot, and Donald R Sheehy. Efficient and robust persistent homology for measures. Computational Geometry, 58:70-96, 2016. The theory of multidimensional persistence. Discrete & Computational Geometry. Gunnar Carlsson, Afra Zomorodian, 42Gunnar Carlsson and Afra Zomorodian. The theory of multidimensional persistence. Discrete & Computa- tional Geometry, 42(1):71-93, 2009. An introduction to topological data analysis: Fundamental and practical aspects for data scientists. Frédéric Chazal, Bertrand Michel, arXiv:1710.04019arXiv preprintFrédéric Chazal and Bertrand Michel. An introduction to topological data analysis: Fundamental and practical aspects for data scientists. arXiv preprint arXiv:1710.04019, 2017. Geometric inference for probability measures. Frédéric Chazal, David Cohen-Steiner, Quentin Mérigot, Foundations of Computational Mathematics. 116Frédéric Chazal, David Cohen-Steiner, and Quentin Mérigot. Geometric inference for probability measures. Foundations of Computational Mathematics, 11(6):733-751, 2011. Subsampling methods for persistent homology. Frédéric Chazal, Brittany Fasy, Fabrizio Lecci, Bertrand Michel, Alessandro Rinaldo, Larry Wasserman, International Conference on Machine Learning. PMLRFrédéric Chazal, Brittany Fasy, Fabrizio Lecci, Bertrand Michel, Alessandro Rinaldo, and Larry Wasserman. Subsampling methods for persistent homology. In International Conference on Machine Learning, pages 2143-2151. PMLR, 2015a. Convergence rates for persistence diagram estimation in topological data analysis. Frédéric Chazal, Marc Glisse, Catherine Labruere, Bertrand Michel, Journal of Machine Learning Research. 16Frédéric Chazal, Marc Glisse, Catherine Labruere, and Bertrand Michel. Convergence rates for persistence diagram estimation in topological data analysis. Journal of Machine Learning Research, 16:3603-3635, 2015b. The Structure and Stability of Persistence Modules. Frédéric Chazal, Vin De Silva, Marc Glisse, Steve Oudot, SpringerFrédéric Chazal, Vin De Silva, Marc Glisse, and Steve Oudot. The Structure and Stability of Persistence Modules. Springer, 2016. Robust topological inference: Distance to a measure and kernel distance. Frédéric Chazal, Brittany Fasy, Fabrizio Lecci, Bertrand Michel, Alessandro Rinaldo, Larry Wasserman, Journal of Machine Learning Research. 181Frédéric Chazal, Brittany Fasy, Fabrizio Lecci, Bertrand Michel, Alessandro Rinaldo, and Larry Wasserman. Robust topological inference: Distance to a measure and kernel distance. Journal of Machine Learning Research, 18(1):5845-5884, 2017. Robust inference via multiplier bootstrap. Xi Chen, Wen-Xin Zhou, The Annals of Statistics. 483Xi Chen and Wen-Xin Zhou. Robust inference via multiplier bootstrap. The Annals of Statistics, 48(3): 1665-1691, 2020. Stability of persistence diagrams. David Cohen, - Steiner, Herbert Edelsbrunner, John Harer, Discrete & Computational Geometry. 371David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. Stability of persistence diagrams. Discrete & Computational Geometry, 37(1):103-120, 2007. Introduction to Algorithms. Charles E Thomas H Cormen, Ronald L Leiserson, Clifford Rivest, Stein, MIT pressThomas H Cormen, Charles E Leiserson, Ronald L Rivest, and Clifford Stein. Introduction to Algorithms. MIT press, 2009. On boundary estimation. Antonio Cuevas, Alberto Rodríguez-Casal, Advances in Applied Probability. Antonio Cuevas and Alberto Rodríguez-Casal. On boundary estimation. Advances in Applied Probability, pages 340-354, 2004. The MNIST database of handwritten digit images for machine learning research. Li Deng, IEEE Signal Processing Magazine. 296Li Deng. The MNIST database of handwritten digit images for machine learning research. IEEE Signal Processing Magazine, 29(6):141-142, 2012. Sub-Gaussian mean estimators. Luc Devroye, Matthieu Lerasle, Gabor Lugosi, Roberto I Oliveira, The Annals of Statistics. 446Luc Devroye, Matthieu Lerasle, Gabor Lugosi, and Roberto I Oliveira. Sub-Gaussian mean estimators. The Annals of Statistics, 44(6):2695-2725, 2016. Being robust (in high dimensions) can be practical. Ilias Diakonikolas, Gautam Kamath, M Daniel, Jerry Kane, Ankur Li, Alistair Moitra, Stewart, International Conference on Machine Learning. PMLRIlias Diakonikolas, Gautam Kamath, Daniel M Kane, Jerry Li, Ankur Moitra, and Alistair Stewart. Being robust (in high dimensions) can be practical. In International Conference on Machine Learning, pages 999-1008. PMLR, 2017. Computational Topology: An Introduction. Herbert Edelsbrunner, John Harer, American Mathematical SocietyHerbert Edelsbrunner and John Harer. Computational Topology: An Introduction. American Mathematical Society, 2010. Confidence sets for persistence diagrams. Brittany Terese Fasy, Fabrizio Lecci, Alessandro Rinaldo, Larry Wasserman, Sivaraman Balakrishnan, Aarti Singh, The Annals of Statistics. 426Brittany Terese Fasy, Fabrizio Lecci, Alessandro Rinaldo, Larry Wasserman, Sivaraman Balakrishnan, and Aarti Singh. Confidence sets for persistence diagrams. The Annals of Statistics, 42(6):2301-2339, 2014. Curvature sets over persistence diagrams. Mario Gómez, Facundo Mémoli, arXiv:2103.04470arXiv preprintMario Gómez and Facundo Mémoli. Curvature sets over persistence diagrams. arXiv preprint arXiv:2103.04470, 2021. A guided tour of Chernoff bounds. Torben Hagerup, Christine Rüb, Inf. Process. Lett. 33Torben Hagerup and Christine Rüb. A guided tour of Chernoff bounds. Inf. Process. Lett., 33:305-308, 1990. Robust Statistics: The Approach Based on Influence Functions. Frank R Hampel, M Elvezio, Ronchetti, J Peter, Werner A Rousseeuw, Stahel, John Wiley & Sons196Frank R Hampel, Elvezio M Ronchetti, Peter J Rousseeuw, and Werner A Stahel. Robust Statistics: The Approach Based on Influence Functions, volume 196. John Wiley & Sons, 2011. Probability inequalities for sums of bounded random variables. Wassily Hoeffding, Journal of the American Statistical Association. 58301Wassily Hoeffding. Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association, 58(301):13-30, 1963. Inequalities on the Lambert W function and hyperpower function. Abdolhossein Hoorfar, Mehdi Hassani, J. Inequal. Pure and Appl. Math. 92Abdolhossein Hoorfar and Mehdi Hassani. Inequalities on the Lambert W function and hyperpower function. J. Inequal. Pure and Appl. Math, 9(2):5-9, 2008. Robust estimation of a location parameter. J Peter, Huber, The Annals of Mathematical Statistics. Peter J Huber. Robust estimation of a location parameter. The Annals of Mathematical Statistics, pages 73-101, 1964. Robust Statistics. J Peter, Huber, Wiley Series in Probability and Statistics -Applied Probability and Statistics Section Series. WileyPeter J. Huber. Robust Statistics. Wiley Series in Probability and Statistics -Applied Probability and Statistics Section Series. Wiley, 2004. Robust kernel density estimation with median-of-means principle. Pierre Humbert, Le Batiste, Ludovic Bars, Nicolas Minvielle, Vayatis, arXiv:2006.16590arXiv preprintPierre Humbert, Batiste Le Bars, Ludovic Minvielle, and Nicolas Vayatis. Robust kernel density estimation with median-of-means principle. arXiv preprint arXiv:2006.16590, 2020. Robust estimation of U-statistics. Emilien Joly, Gábor Lugosi, Stochastic Processes and their Applications. 126Emilien Joly and Gábor Lugosi. Robust estimation of U-statistics. Stochastic Processes and their Applications, 126(12):3760-3773, 2016. Robust machine learning by median-of-means: Theory and practice. Guillaume Lecué, Matthieu Lerasle, Annals of Statistics. 482Guillaume Lecué and Matthieu Lerasle. Robust machine learning by median-of-means: Theory and practice. Annals of Statistics, 48(2):906-931, 2020. On a problem of adaptive estimation in Gaussian white noise. Oleg V Lepski, Theory of Probability & Its Applications. 35Oleg V. Lepski. On a problem of adaptive estimation in Gaussian white noise. Theory of Probability & Its Applications, 35(3):454-466, 1991. MONK: Outlier-robust mean embedding estimation by median-of-means. Matthieu Lerasle, Zoltán Szabó, Timothée Mathieu, Guillaume Lecué, International Conference on Machine Learning. PMLRMatthieu Lerasle, Zoltán Szabó, Timothée Mathieu, and Guillaume Lecué. MONK: Outlier-robust mean embedding estimation by median-of-means. In International Conference on Machine Learning, pages 3782-3793. PMLR, 2019. Mean estimation and regression under heavy-tailed distributions: A survey. Gábor Lugosi, Shahar Mendelson, Foundations of Computational Mathematics. 195Gábor Lugosi and Shahar Mendelson. Mean estimation and regression under heavy-tailed distributions: A survey. Foundations of Computational Mathematics, 19(5):1145-1190, 2019a. Risk minimization by median-of-means tournaments. Gabor Lugosi, Shahar Mendelson, Journal of the European Mathematical Society. 223Gabor Lugosi and Shahar Mendelson. Risk minimization by median-of-means tournaments. Journal of the European Mathematical Society, 22(3):925-965, 2019b. Probability measures on the space of persistence diagrams. Yuriy Mileyko, Sayan Mukherjee, John Harer, Inverse Problems. 2712124007Yuriy Mileyko, Sayan Mukherjee, and John Harer. Probability measures on the space of persistence diagrams. Inverse Problems, 27(12):124007, 2011. Geometric median and robust estimation in Banach spaces. Stanislav Minsker, Bernoulli. 214Stanislav Minsker. Geometric median and robust estimation in Banach spaces. Bernoulli, 21(4):2308-2335, 2015. Sub-Gaussian estimators of the mean of a random matrix with heavy-tailed entries. Stanislav Minsker, The Annals of Statistics. 466AStanislav Minsker. Sub-Gaussian estimators of the mean of a random matrix with heavy-tailed entries. The Annals of Statistics, 46(6A):2871-2903, 2018. Problem Complexity and Method Efficiency in Optimization. Arkadij Semenovič Nemirovskij, David Borisovich Yudin, Wiley-InterscienceArkadij Semenovič Nemirovskij and David Borisovich Yudin. Problem Complexity and Method Efficiency in Optimization. Wiley-Interscience, 1983. A roadmap for the computation of persistent homology. Nina Otter, A Mason, Ulrike Porter, Peter Tillmann, Heather A Grindrod, Harrington, EPJ Data Science. 6Nina Otter, Mason A Porter, Ulrike Tillmann, Peter Grindrod, and Heather A Harrington. A roadmap for the computation of persistent homology. EPJ Data Science, 6:1-38, 2017. Geometric inference on kernel density estimates. M Jeff, Bei Phillips, Yan Wang, Zheng, 31st International Symposium on Computational Geometry. 34Jeff M Phillips, Bei Wang, and Yan Zheng. Geometric inference on kernel density estimates. In 31st International Symposium on Computational Geometry, volume 34, pages 857-871, 2015. Elchanan Solomon, Alexander Wagner, Paul Bendich, arXiv:2101.12288From geometry to topology: Inverse theorems for distributed persistence. arXiv preprintElchanan Solomon, Alexander Wagner, and Paul Bendich. From geometry to topology: Inverse theorems for distributed persistence. arXiv preprint arXiv:2101.12288, 2021. A survey of sampling from contaminated distributions. Contributions to probability and statistics. W John, Tukey, John W Tukey. A survey of sampling from contaminated distributions. Contributions to probability and statistics, pages 448-485, 1960. Fréchet means for distributions of persistence diagrams. Katharine Turner, Yuriy Mileyko, Sayan Mukherjee, John Harer, Discrete & Computational Geometry. 521Katharine Turner, Yuriy Mileyko, Sayan Mukherjee, and John Harer. Fréchet means for distributions of persistence diagrams. Discrete & Computational Geometry, 52(1):44-70, 2014. Multiparameter persistent homology landscapes identify immune cell spatial patterns in tumors. Oliver Vipond, Joshua A Bull, S Philip, Ulrike Macklin, Tillmann, W Christopher, Helen M Pugh, Heather A Byrne, Harrington, Proceedings of the National Academy of Sciences. 118412021Oliver Vipond, Joshua A Bull, Philip S Macklin, Ulrike Tillmann, Christopher W Pugh, Helen M Byrne, and Heather A Harrington. Multiparameter persistent homology landscapes identify immune cell spatial patterns in tumors. Proceedings of the National Academy of Sciences, 118(41), 2021. Robust persistence diagrams using reproducing kernels. Siddharth Vishwanath, Kenji Fukumizu, Satoshi Kuriki, K Bharath, Sriperumbudur, Advances in Neural Information Processing Systems. 33Siddharth Vishwanath, Kenji Fukumizu, Satoshi Kuriki, and Bharath K Sriperumbudur. Robust persistence diagrams using reproducing kernels. Advances in Neural Information Processing Systems, 33:21900-21911, 2020. On building fast k−d trees for ray tracing, and on doing that in O(N log N ). Ingo Wald, Vlastimil Havran, 2006 IEEE Symposium on Interactive Ray Tracing. IEEEIngo Wald and Vlastimil Havran. On building fast k−d trees for ray tracing, and on doing that in O(N log N ). In 2006 IEEE Symposium on Interactive Ray Tracing, pages 61-69. IEEE, 2006. jl: Flexible and efficient persistent homology computation in Julia. Matija Čufar, Ripserer, Journal of Open Source Software. 5542614Matija Čufar. Ripserer.jl: Flexible and efficient persistent homology computation in Julia. Journal of Open Source Software, 5(54):2614, 2020.
{'fraction_non_alphanumeric': 0.08662821709561275, 'fraction_numerical': 0.02249466595546073, 'mean_word_length': 3.5746911697422603, 'pattern_counts': {'":': 0, '<': 62, '<?xml version=': 0, '>': 86, 'https://': 1, 'lorem ipsum': 0, 'www.': 1, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 81, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'The distance function to a compact set plays a crucial role in the paradigm of topological data analysis. In particular, the sublevel sets of the distance function are used in the computation of persistent homologya backbone of the topological data analysis pipeline. Despite its stability to perturbations in the Hausdorff distance, persistent homology is highly sensitive to outliers. In this work, we develop a framework of statistical inference for persistent homology in the presence of outliers. Drawing inspiration from recent developments in robust statistics, we propose a median-of-means variant of the distance function (MoM Dist), and establish its statistical properties. In particular, we show that, even in the presence of outliers, the sublevel filtrations and weighted filtrations induced by MoM Dist are both consistent estimators of the true underlying population counterpart, and their rates of convergence in the bottleneck metric are controlled by the fraction of outliers in the data. Finally, we demonstrate the advantages of the proposed methodology through simulations and applications. arXiv:2206.01795v1 [math.ST] 3 Jun 2022 computation of persistence diagrams comes from the celebrated stability of persistence diagrams . In a nutshell, the stability result for persistence diagrams guarantees that (i) the persistence diagrams resulting from two compact sets X and Y are close whenever the sets themselves are close in the Hausdorff distance, and, (ii) the functional persistence diagrams resulting from two filter functions f and g are close whenever f and g are close w.r.t. the · ∞ metric.In the statistical setting, one has access to X only through samples X n = {X 1 , X 2 , . . . , X n } obtained using a probability distribution P which is supported on the (unknown) set X. The objective, in a statistical inference framework, is to use the samples X n to infer the true population persistence diagram Dgm(V X ). The offset X n (r) and filter function f n , constructed using the sample points, are themselves random quantities associated with their population counterparts X(r) and f X , respectively, and these may be used to construct a sample estimator Dgm(V Xn ). To this end, several existing works have studied the statistical properties of Dgm(V Xn ), e.g., constructing confidence bands and characterizing the convergence rate of Dgm(V Xn ) to Dgm(V X ) in the space of persistence diagrams(Fasy et al., 2014;Chazal et al., 2015aChazal et al., ,b, 2017Vishwanath et al., 2020).ContributionsIn practical settings, real-world data is likely subject to measurement errors and the presence of outliers. While some assumptions may be imposed on the noise and the outliers, in the most baneful settings, the given data may be subject to adversarial contamination. In this setting, for m < n/2, we assume that the samples X n , which we have access to, contain only n − m points obtained from the probability distribution P with supp(P) = X, and make no further assumptions on the remaining m points. In principle, the m outliers may be carefully chosen by an adversary after examining the remaining n − m points. The overarching objective of this paper is to construct an estimator of the (unknown) population quantity Dgm(V[X]) using the corrupted sample points X n which is, both, statistically consistent and computationally efficient.While the stability of persistence diagrams guarantees that small perturbations in the sample points induce only small changes in the resulting persistence diagrams, even a few outliers in the samples can lead to deleterious effects. This issue is further exacerbated in the adversarial setting, where the adversary is free to place the m points where it may drastically impact the resulting topological inference.In this paper, we introduce MoM Dist, denoted by d n,Q , as an outlier-robust variant of the empirical distance function which is constructed using the median-of-means principle, and we establish its theoretical properties. Notably the MoM Dist relies on a tuning parameter Q which is easy to interpret. While the persistence diagram resulting from the sublevel filtration of d n,Q is a valid candidate for statistical inference, it can be expensive to compute in practice. To overcome this, we use the weighted filtrations introduced by Buchet et al.(2016)and Anai et al. (2019) to construct d n,Q -weighted filtrations, V [X n , d n,Q ], as computationally efficient estimators of Dgm(V [X]). Our main contributions are the following: (I) We show that sublevel set persistence diagrams of d n,Q are consistent estimators of the sublevel set persistence diagram of the true population counterpart d X even in the presence of outliers (Theorem 3.1).(II) We establish a stability result for the the d n,Q -weighted filtrations, V [X n , d n,Q ], and we show that they are stable w.r.t. adversarial contamination (Theorem 3.2).(III) Furthermore, we show that the persistence diagram Dgm(V[X n , d n,Q ]) is both a computationally efficient and statistically consistent estimator of Dgm(V[X]), and we establish its convergence rate (Theorem 3.3).(IV) Next, in a sensitivity analysis framework, we quantify the gain in robustness achieved when using the d n,Q -weighted filtrations vis-à-vis its non-robust d n -weighted counterpart (Theorem 3.4).', 'arxivid': '2206.01795', 'author': ['Siddharth Vishwanath ', 'Bharath K Sriperumbudur ', 'Kenji Fukumizu \nThe Institute of Statistical Mathematics\nTokyoJapan\n', 'Satoshi Kuriki \nThe Institute of Statistical Mathematics\nTokyoJapan\n', '\nDepartment of Statistics\nThe Pennsylvania State University\nUniversity ParkPAUSA\n'], 'authoraffiliation': ['The Institute of Statistical Mathematics\nTokyoJapan', 'The Institute of Statistical Mathematics\nTokyoJapan', 'Department of Statistics\nThe Pennsylvania State University\nUniversity ParkPAUSA'], 'corpusid': 249394535, 'doi': None, 'github_urls': ['https://www.github.com/sidv23/RobustTDA.jl/'], 'n_tokens_mistral': 41981, 'n_tokens_neox': 37113, 'n_words': 21920, 'pdfsha': '85be6c64442dba024f67565e72eede9e01e6cc3e', 'pdfurls': ['https://arxiv.org/pdf/2206.01795v1.pdf'], 'title': ['Robust Topological Inference in the Presence of Outliers', 'Robust Topological Inference in the Presence of Outliers'], 'venue': []}
arxiv
NVIDIA FLARE: Federated Learning from Simulation to Real-World Holger R Roth NVIDIA Corporation * Shanghai Munich, Bethesda, Santa ClaraChina, Germany, USA Yan Cheng NVIDIA Corporation * Shanghai Munich, Bethesda, Santa ClaraChina, Germany, USA Yuhong Wen NVIDIA Corporation * Shanghai Munich, Bethesda, Santa ClaraChina, Germany, USA Isaac Yang NVIDIA Corporation * Shanghai Munich, Bethesda, Santa ClaraChina, Germany, USA Ziyue Xu NVIDIA Corporation * Shanghai Munich, Bethesda, Santa ClaraChina, Germany, USA Yuan-Ting Hsieh NVIDIA Corporation * Shanghai Munich, Bethesda, Santa ClaraChina, Germany, USA Kristopher Kersten NVIDIA Corporation * Shanghai Munich, Bethesda, Santa ClaraChina, Germany, USA Ahmed Harouni NVIDIA Corporation * Shanghai Munich, Bethesda, Santa ClaraChina, Germany, USA Can Zhao NVIDIA Corporation * Shanghai Munich, Bethesda, Santa ClaraChina, Germany, USA Kevin Lu NVIDIA Corporation * Shanghai Munich, Bethesda, Santa ClaraChina, Germany, USA Zhihong Zhang NVIDIA Corporation * Shanghai Munich, Bethesda, Santa ClaraChina, Germany, USA Wenqi Li NVIDIA Corporation * Shanghai Munich, Bethesda, Santa ClaraChina, Germany, USA Andriy Myronenko NVIDIA Corporation * Shanghai Munich, Bethesda, Santa ClaraChina, Germany, USA Dong Yang NVIDIA Corporation * Shanghai Munich, Bethesda, Santa ClaraChina, Germany, USA Sean Yang NVIDIA Corporation * Shanghai Munich, Bethesda, Santa ClaraChina, Germany, USA Nicola Rieke NVIDIA Corporation * Shanghai Munich, Bethesda, Santa ClaraChina, Germany, USA Abood Quraini NVIDIA Corporation * Shanghai Munich, Bethesda, Santa ClaraChina, Germany, USA Chester Chen NVIDIA Corporation * Shanghai Munich, Bethesda, Santa ClaraChina, Germany, USA Daguang Xu NVIDIA Corporation * Shanghai Munich, Bethesda, Santa ClaraChina, Germany, USA Nic Ma NVIDIA Corporation * Shanghai Munich, Bethesda, Santa ClaraChina, Germany, USA Prerna Dogra NVIDIA Corporation * Shanghai Munich, Bethesda, Santa ClaraChina, Germany, USA Mona Flores NVIDIA Corporation * Shanghai Munich, Bethesda, Santa ClaraChina, Germany, USA Andrew Feng NVIDIA Corporation * Shanghai Munich, Bethesda, Santa ClaraChina, Germany, USA NVIDIA FLARE: Federated Learning from Simulation to Real-World Federated learning (FL) enables building robust and generalizable AI models by leveraging diverse datasets from multiple collaborators without centralizing the data. We created NVIDIA FLARE 1 as an open-source software development kit (SDK) to make it easier for data scientists to use FL in their research and realworld applications. The SDK includes solutions for state-of-the-art FL algorithms and federated machine learning approaches, which facilitate building workflows for distributed learning across enterprises and enable platform developers to create a secure, privacy-preserving offering for multiparty collaboration utilizing homomorphic encryption or differential privacy. The SDK is a lightweight, flexible, and scalable Python package. It allows researchers to apply their data science workflows in any training libraries (PyTorch, TensorFlow, XGBoost, or even NumPy) in real-world FL settings. This paper introduces the key design principles of NVFlare and illustrates some use cases (e.g., COVID analysis) with customizable FL workflows that implement different privacy-preserving algorithms. Introduction Federated learning (FL) has become a reality for many real-world applications [31]. It enables multinational collaborations on a global scale to build more robust and generalizable machine learning and AI models. In this paper, we introduce NVIDIA FLARE (NVFlare), an open-source software development kit (SDK) that makes it easier for data scientists to collaborate to develop more generalizable and robust AI models by sharing model weights rather than private data. While FL is attractive in many industries, it is particularly beneficial for healthcare applications where patient data needs to be protected. For example, FL has been used for predicting clinical outcomes in patients with COVID-19 [6] or to segment brain lesions in magnetic resonance imaging [35,34]. NVFlare is not limited to applications in healthcare and is designed to allow cross-silo FL [15] across enterprises for different industries and researchers. In recent years, several efforts (both open-source and commercial) have been made to bring FL technology into the healthcare sector and other industries, like TensorFlow Federated [1], PySyft [44], FedML [11], FATE [23], Flower [2], OpenFL [30], Fed-BioMed [36], IBM Federated Learning [24], HP Swarm Learning [38], Federat-edScope [40], FLUTE [7], and more. Some focus on simulated FL settings for researchers, while others prioritize production settings. NVFlare aims to be useful for both scenarios: 1) for researchers by providing efficient and extensible simulation tools and 2) by providing an easy path to transfer research into real-world production settings, supporting high availability and server failover, and by providing additional productivity tools such as multi-tasking and admin commands. NVIDIA FLARE Overview NVIDIA FLARE -or short NVFlare -stands for "NVIDIA Federated Learning Application Runtime Environment". The SDK enables researchers and data scientists to adapt their machine learning and deep learning workflows to a federated paradigm. It enables platform developers to build a secure, privacy-preserving offering for distributed multiparty collaboration. NVFlare is a lightweight, flexible, and scalable FL framework implemented in Python that is agnostic to the underlying training library. Developers can bring their own data science workflows implemented in PyTorch, TensorFlow, or even in pure NumPy, and apply them in a federated setting. A typical FL workflow such as the popular federated averaging (FedAvg) algorithm [25], can be implemented in NVFlare using the following main steps. Starting from an initial global model, each FL client trains the model on their local data for a while and sends model updates to the server for aggregation. The server then uses the aggregated updates to update the global model for the next round of training. This process is iterated many times until the model converges. Though used heavily for federated deep learning, NVFlare is a generic approach for supporting collaborative computing across multiple clients. NVFlare provides the Controller programming API for researchers to create workflows for coordinating clients for collaboration. FedAvg is one such workflow. Another example is cyclic weight transfer [4]. The central concept of collaboration is the notion of "task". An FL controller assigns tasks (e.g., deep-learning training with model weights) to one or more FL clients and processes results returned from clients (e.g., model weight updates). The controller may assign additional tasks to clients based on the processed results and other factors (e.g., a pre-configured number of training rounds). This task-based interaction continues until the objectives of the study are achieved. The API supports typical controller-client interaction patterns like Figure 1: NVFlare job execution. The Controller is a Python object that controls or coordinates the Workers to get a job done. The controller is run on the FL server. A Worker is capable of performing tasks. Workers run on FL clients. broadcasting a task to multiple clients, sending a task to one or more specified clients, or relaying a task to multiple clients sequentially. Each interaction pattern has two flavors: wait (block until client results are received) or no-wait. A workflow developer can use these interaction patterns to create innovative workflows. For example, the Scat-terAndGather controller (typically used for FedAvg-like algorithms) is implemented with the broadcast_and_wait pattern, and the CyclicController is implemented with the relay_and_wait pattern. The controller API allows the researcher to focus on the control logic without needing to deal with underlying communication issues. Figure 1 shows the principle. Each FL client acts as a worker that simply executes tasks assigned to it (e.g., model training) and returns execution results to the controller. At each task interaction, there can be optional filters that process the task data or results before passing it to the Controller (on the server side) or task executor (client side). The filter mechanism can be used for data privacy protection (e.g., homomorphic encryption/decryption or differential privacy) without having to alter the training algorithms. Key Components NVFlare is built on a componentized architecture that allows FL workloads to move from research and simulation to real-world production deployment. Some of the key components of this SDK include: • FL Simulator for rapid development and prototyping. • NVFlare Dashboard for simplified project management, secure provisioning, and deployment, orchestration. • Reference FL algorithms (e.g., FedAvg, FedProx, SCAFFOLD) and workflows, like scatter and gather, cyclic, etc. • Privacy preservation with differential privacy, homomorphic encryption, and more. • Specification-based API for extensibility, allowing customization with plug-able components. • Tight integration with other learning frameworks like MONAI [3], XGBoost [5], and more. High-Level Architecture NVFlare is designed with the idea that less is more, using a specification-based design principle to focus on what is essential. This allows other people to be able to do what they want to do in real-world applications by following clear API definitions. FL is an open-ended space. The API-based design allows others to bring their implementations and solutions for various components. Controllers, task executors, and filters are just examples of such extensible components. NVFlare provides an end-to-end operation environment for different personas. It provides a comprehensive provisioning system that creates security credentials for secure communications to enable the easy and secure deployment of FL applications in the real world. It also provides an FL Simulator for running proof-of-concept studies locally. In production mode, the researcher conducts an FL study by submitting jobs using admin commands using Notebooks or the NVFlare Console -an interactive command tool. NVFlare provides many commands for system operation and job management. With these commands, one can start and stop a specific client or the entire system, submit new jobs, check the status of jobs, create a job by cloning from an existing one, and much more. With NVFlare's component-based design, a job is just a configuration of components needed for the study. For the control logic, the job specifies the controller component to be used and any components required by the controller. System Concepts A NVFlare system is a typical client-server communication system that comprises one or more FL server(s), one or more FL client(s), and one or more admin clients. The FL Servers open two ports for communication with FL clients and admin clients. FL clients and admin clients connect to the opened ports. FL clients and admin clients do not open any ports and do not directly communicate with each other. The following is an overview of the key concepts and objects available in NVFlare and the information that can be passed between them. Workers and Controller NVFlare's collaborative computing is achieved through the Controller/Worker interactions. Shareable Object that represents a communication between server and client. Technically, the Shareable is implemented as a Python dictionary that could contain different information, e.g., model weights. Data Exchange Object (DXO) Standardizes the data passed between the communicating parties. One can think of the Shareable as the envelope and the DXO as the letter. Together, they comprise a message to be shared between communicating parties. FLComponent The base class of all the FL components. Executors, controllers, filters, aggregators, and their subtypes are all FLComponents. FLComponent comes with some useful built-in methods for logging, event handling, auditing, and error handling. Executors Type of FLComponent for FL clients that has an execute method that produces a Shareable from an input Shareable. NVFlare provides both single-and multi-process executors to implement different computing workloads. FLContext One of the most important features of NVFlare is to pass data between the FL components. FLContext is available to every method of all common FLComponent types. Through FLContext, the component developer can get services provided by the underlying infrastructure and share data with other components of the FL system. Communication Drivers NVFlare abstracts the communication layers out so that different deployment scenarios can implement customizable communication drivers. By default, we use GRPC for data communication in taskbased communication. However, the driver can be replaced to run other communication protocols, for example, TCP. The customizable nature of communication in NVFlare allows for both server-centric and peer-to-peer communication patterns. This enables the user to utilize both scatter and gather-type workflows like FedAvg [25], decentralized training patterns like swarm learning [38], or direct peer-to-peer communication as in split learning [9]. Fig. 2 compares the times for model upload and download from the client's perspective using different communication protocols available in NVFlare using a model of ∼18MB in size. The experiment runs in a multi-cloud environment with the server and eight clients running on Azure, while two clients run on AWS. One can observe that the global model download is slower as all clients are trying to download the global model at the same time, and hence the server is more busy. In contrast, the clients' model uploads happen at slightly different times and therefore are faster. One can also see how this multi-cloud setup causes the clients on AWS to take slightly longer during model download due to communication across different cloud infrastructures. Filters Filters in NVFlare are a type of FLComponent that have a process method to transform the Shareable object between the communicating parties. A Filter can provide additional processing to shareable data before sending or after receiving from a peer. Filters can convert data formats and a lot more and are NVFlare's primary mechanism for data privacy protection [21,10]: • ExcludeVars to exclude variables from shareable. • PercentilePrivacy for truncation of weights by percentile. • SVTPrivacy for differential privacy through sparse vector techniques. • Homomorphic encryption filters used for secure aggregation. The clients are distributed between Azure and AWS. The message size is ∼18MB. Communication times were measured over 100 rounds of FedAvg. Error bars indicate the 95% confidence intervals. As an example, we show the average encryption, decryption, and upload times when using homomorphic encryption for secure aggregation 2 . We compare raw data to encrypted model gradients uploaded in Table 1 when hosting the server on AWS 3 and connecting 30 client instances using an on-premise GPU cluster. One can see the longer upload times due to the larger message sizes needed by homomorphic encryption. Event Mechanism NVFlare comes with a powerful event mechanism that allows dynamic notifications to be sent to all event handlers. This mechanism enables data-based communication among decoupled components: one component fires an event when a certain condition occurs, and other components can listen to that event and processes the event data. Each FLComponent is automatically an event handler. To listen to and process an event, one can simply implement the handle_event() method and process desired event types. Events represent some important moments during the execution of the system logic. For example, before and after aggregation or when important data becomes available, e.g., a new "best" model was selected. Productivity Features NVFlare contains features that enable efficient, collaborative, and robust computing workflows. 2 https://developer.nvidia.com/blog/federated-learning-with-homomorphic-encryption 3 For reference, we used an m5a.2xlarge instance with eight vCPUs, 32-GB memory, and up to 2,880 Gbps network bandwidth. Multi-tasking For systems with a large capacity, computing resources could be idle most of the time. NVFlare implements a resource-based multi-tasking solution, where multiple jobs can be run concurrently when overall system resources are available. Multi-tasking is made possible by a job scheduler on the server side that constantly tries to schedule a new job. For each job to be scheduled, the scheduler asks each client whether they can satisfy the required resources of the job (e.g., number of GPU devices) by querying the client's resource manager. If all clients can meet the requirement, the job will be scheduled and deployed to the clients. High Availability and Server Failover To avoid the FL server as a single point of failure, a solution has been implemented to support multiple FL servers with automatic cut-over when the currently active server becomes unavailable. Therefore, a component called Overseer is added to facilitate automatic cut-over. The Overseer provides the authoritative endpoint info of the active FL server. All other system entities (FL servers, FL clients, admin clients) constantly communicate (i.e., every 5 seconds) with the Overseer to obtain and act on such information. If the server cutover happens during the execution of a job, then the job will continue to run on the new server. Depending on how the controller is written, the job may or may not need to restart from the beginning but can continue from a previously saved snapshot. Simulator NVFlare provides a simulator to allow data scientists and system developers to easily write new FLComponents and novel workflows. The simulator is a command line tool to run a NVFlare job. To allow simple experimentation and debugging, the FL server and multiple clients run in the same process during simulation. A multi-process option allows efficient use of resources, e.g., training multiple clients on different GPUs. The simulator follows the same job execution as in real-world NVFlare deployment. Therefore, components developed in simulation can be directly deployed in real-world federated scenarios. Secure Provisioning in NVFlare Security is an important requirement for FL systems. NVFlare provides security solutions in the following areas: authentication, communication confidentiality, user authorization, data privacy protection, auditing, and local client policies. Authentication NVFlare ensures the identities of communicating peers using mutual Transport Layer Security (TLS). Each participating party (FL Servers, Overseer, FL Clients, Admin Clients) must be properly provisioned. Once provisioned, each party receives a startup kit containing TLS credentials (public cert of the root, the party's own private key and certificate) and system endpoint information, see Fig. 3. Each party can only connect to the NVFlare system with the startup kit. Communication confidentiality is also achieved with the use of TLS-based messaging. Federated Authorization NVFlare's admin command system is very rich and powerful. Not every command is for everyone. NVFlare implements a role-based user authorization system that controls what a user can or cannot do. At the time of provision, each user is assigned a role. Authorization policies specify which commands are permitted for which roles. Each FL client can define its authorization policy that specifies what a role can or cannot do to the client. For example, one client could allow a role to run jobs from any researchers. In contrast, another client may only allow jobs submitted by its researchers (i.e., the client and the job submitter belong to the same organization). NVFlare automatically records all user commands and job events in system audit files on both the server and client sides. In addition, the audit API can be used by application developers to record additional events in the audit files. Client-Privacy NVFlare enhances the overall system security by allowing each client to define its policies for authorization, data privacy (filters), and computing resource management. The client can change its policies at any time after the system is up and running without having to be re-provisioned. For example, the client could require all jobs running on it to be subject to a set of filters. The client could also change the number of computing resources (e.g., GPU devices) to be used by the FL client. Federated Data Science As a general distributed computing platform, NVFlare can be used for various applications in different industries. Here we describe some of the most common use cases where NVFlare was deployed. Federated Deep Learning A go-to example dataset for benchmarking different FL algorithms is CIFAR-10 [17]. NVFlare allows users to experiment with different algorithms and data splits using different levels of heterogeneity based on a Dirichlet sampling strategy [37]. Figure 4a shows the impact of varying alpha values, where lower values cause higher heterogeneity on the performance of the FedAvg. Apart from FedAvg, currently available in NVFlare include FedProx [20], FedOpt [29], and SCAFFOLD [16]. Figure 4b compares an α setting of 0.1, causing a high data heterogeneity across clients and its impact on more advanced FL algorithms, namely FedProx, FedOpt, and SCAFFOLD. FedOpt and SCAFFOLD show markedly better convergence rates and achieve better performance than FedAvg and FedProx with the same alpha setting. SCAFFOLD achieves this by adding a correction term when updating the client models, while FedOpt utilizes SGD with momentum to update the global model on the server. Therefore, both perform better with the same number of training steps as FedAvg and FedProx. Other algorithms available in or coming soon to NVFlare include federated XGBoost [5], Ditto [19], FedSM [41], Auto-FedRL [8], and more. Federated Machine Learning Traditional machine learning methods, such as linear models, support vector machine (SVM), and k-means clustering, can be formulated under a federated setting. With certain libraries, the federated machine learning algorithms need to be designed considering two factors: algorithm-wise, each of these models has distinct training schemes and model representations; and implementationwise, popular libraries providing these functionalities (e.g., scikit-learn, XGBoost) have different APIs and inner logics. Hence, when developing an FL variant of a particular traditional machine learning method, several questions need to be answered at these two levels: First, at the algorithm level, we need to break down the optimization process into individual steps/rounds (if possible) and have answers to three major questions: 1. What information should clients share with the server? 2. How should the server aggregate the collected information from clients? 3. What should clients do with the global aggregated information received from the server? Second, at the implementation level, we need to know what APIs are available and how to utilize them in a federated pipeline to implement a distributed version of the algorithm. A major difference between federated traditional machine learning and federated deep learning is that, for traditional machine learning methods, the boundary between "federated" and "distributed", or even "ensemble", can be much more vague than for deep learning. Due to the characteristics of a given algorithm and its API design, the concepts can be equivalent. Take XGBoost and SVM, for example: Algorithm-wise, XGBoost can distribute the training samples to several workers and construct trees based on the collected histograms from each worker. Such a process can be directly adopted under a federated setting because the communication cost is affordable. In this case, "federated" is equivalent to "distributed" learning. API-wise, some algorithms can be constrained by their implementation. Take scikit-learn's SVM for instance. Although theoretically SVM can be formulated as an iterative optimization process, the API only supports one-shot "fitting" without the capability of separately calling the optimization steps. Hence a federated SVM algorithm using the scikit-learn library can only be implemented as a two-step process. In this case, "federated" is equivalent to "ensemble". For clarification, we provide the full formulation for tree-based federated XGBoost, illustrated in Fig. 5: 1. XGBoost, by definition, is a sequential optimization process: each step adds one extra tree to the model to reduce the residual error. Hence, federated XGBoost can be formulated as follows: each round of FL corresponds to one boosting step at the local level. Clients share the newly added tree trained on local data with the server at the end of local boosting. 2. The model representation is a decision/regression tree. To aggregate the information from all clients, the server will bag all received trees to form a "forest" to be added to the global boosting model. 3. With the updated global model from the server, each client will continue the boosting process by learning a new tree starting from the global model of the boosted forest. Boosting Communication: Client to Server Server to Client Split learning Split learning assumes a vertical data partitioning [42] that can be useful in many distributed learning scenarios involving neural network architectures [9]. As an introductory example, we can assume that one client holds the images, and the other holds the labels to compute losses and accuracy metrics. Activations and corresponding gradients are being exchanged between the clients using NVFlare, as illustrated in Fig. 6. We use a cryptographic technique called private set intersection (PSI) [39] to compute the alignment between images and labels on both clients. NVFlare's implementation of PSI can be extended to multiple parties and applied to other use cases than split learning, e.g., requiring a secure and privacy-preserving alignment of different databases. Using NVFlare's capability to implement different communication patterns, we can investigate the communication speed-ups one can achieve by implementing split learning using direct peer-to-peer communication as opposed to routing the messages between the two clients through a central server. The table in Fig. 6 compares the training speeds of split learning on the CIFAR-10 dataset in a local simulation scenario. First, we use the same PyTorch script to simulate split learning. Then, we implement two distributed solutions using NVFlare. One that routes the messages through the server and one using a direct peer-to-peer connection between the clients. As expected, the direct peer-to-peer connection is more efficient, achieving only a slight overhead in total training time compared to the standalone PyTorch script, which could not be translated to real-world scenarios. Federated Statistics NVFlare provides built-in federated statistics operators (Controller and Executors) that will generate global statistics based on local client statistics. Each client could have one or more datasets, such as "train" and "test" datasets. Each dataset may have many features. NVFlare will calculate and combine the statistics for each feature in the dataset to produce global statistics for all the numeric features. The output gathered on the server will be the complete statistics for all datasets in clients and global, as illustrated in Fig. 7. Real-world Use Cases NVFlare and its predecessors have been used in several real-world studies exploring FL for healthcare scenarios. The collaborations between multinational institutions tested and validated the utility of federated learning, pushing the envelope for training robust, generalizable AI models. These initiatives included FL for breast mammography classification [32], prostate segmentation [33], pancreas segmentation [37], and most recently, chest X-ray (CXR) and electronic health record (EHR) analysis to predict the oxygen requirement for patients arriving in the emergency department with symptoms of COVID-19 [6]. Summary & Conclusion We described NVFlare, an open-source SDK to make it easier for data scientists to use FL in their research and to allow an easy transition from research to real-world deployment. As discussed above, NVFlare's Controller programming API supports various interaction patterns between the server and clients over internet connections, which could be unstable. Therefore, the API design mitigates various failure conditions and unexpected crashes of the client machines, such as allowing developers to process timeout conditions properly. NVFLare's unique flexibility and agnostic approach towards the deployed training libraries make it the perfect solution for integrating with different deep learning frameworks, including popular ones used for training large language models (LLM). With our dedication to addressing the current limitations of communication protocols, we are working towards supporting the communication of large message sizes, enabling the federated fine-tuning of AI models with billions of parameters, such as those used for ChatGPT [28] and GPT-4 [27]. Moreover, our team is implementing parameter-efficient federated methods to adapt LLM models to downstream tasks [43], utilizing techniques such as prompt tuning [18] and p-tuning [22], adapters [13,12], LoRA [14], showing promising performance. Our commitment to innovation and excellence in this field ensures that we continue to push the boundaries of what is possible with federated learning. We did not go into all details of exciting features available in NVFlare, like homomorphic encryption, Ten-sorBoard streaming, provisioning web dashboard, integration with MONAI 4 [26,3], etc. However, we hope that this overview of NVFlare gives a good starting point for developers and researchers on their journey to using FL and federated data science in simulation and the real world. NVFlare is an open-source project. We invite the community to contribute and grow NVFlare. For more information, please visit the code repository at https://github.com/NVIDIA/NVFlare. Figure 2 : 2Comparison of GRPC and TCP communication drivers in NVFlare. The server is running on Azure. Figure 3 : 3High-level steps for running a real-world study with secure provisioning with NVFlare. Figure 4 : 4Federated learning experiments with NVFlare. Figure 5 : 5Tree-based federated XGBoost: a "boosting of forests." Figure 6 : 6Simple split learning scenario using CIFAR-10. The table compares multiple communication patterns. Using 50,000 training samples and 15,625 rounds of communication with a batch size of 64. statistics. Note the data of "site-4" violates the client's privacy policy and therefore does not share its statistics with the server.(b) Histogram visualization. Figure 7 : 7Federated statistics with NVFlare. (a) Mammography. (b) Prostate. (c) Pancreas. (d) CXR & EHR. Figure 8 : 8Real-world use cases of NVFlare. Table 1 : 1Federated learning exchanging homomorphic encrypted vs. raw model updates.Time in seconds Mean Std. Dev. Encryption 5.01 1.18 Decryption 0.95 0.04 Enc. upload 38.00 71.17 Raw upload 21.57 74.23 https://monai.io {TensorFlow}: a system for {Large-Scale} machine learning. M Abadi, P Barham, J Chen, Z Chen, A Davis, J Dean, M Devin, S Ghemawat, G Irving, M Isard, 12th USENIX symposium on operating systems design and implementation (OSDI 16). M. Abadi, P. Barham, J. Chen, Z. Chen, A. Davis, J. Dean, M. Devin, S. Ghemawat, G. Irving, M. Isard, et al. {TensorFlow}: a system for {Large-Scale} machine learning. In 12th USENIX symposium on operating systems design and implementation (OSDI 16), pages 265-283, 2016. D J Beutel, T Topal, A Mathur, X Qiu, T Parcollet, P P De Gusmão, N D Lane, arXiv:2007.14390Flower: A friendly federated learning research framework. arXiv preprintD. J. Beutel, T. Topal, A. Mathur, X. Qiu, T. Parcollet, P. P. de Gusmão, and N. D. Lane. Flower: A friendly federated learning research framework. arXiv preprint arXiv:2007.14390, 2020. M J Cardoso, W Li, R Brown, N Ma, E Kerfoot, Y Wang, B Murrey, A Myronenko, C Zhao, D Yang, arXiv:2211.02701An open-source framework for deep learning in healthcare. arXiv preprintM. J. Cardoso, W. Li, R. Brown, N. Ma, E. Kerfoot, Y. Wang, B. Murrey, A. Myronenko, C. Zhao, D. Yang, et al. Monai: An open-source framework for deep learning in healthcare. arXiv preprint arXiv:2211.02701, 2022. Distributed deep learning networks among institutions for medical imaging. K Chang, N Balachandar, C Lam, D Yi, J Brown, A Beers, B Rosen, D L Rubin, J Kalpathy-Cramer, Journal of the American Medical Informatics Association. 258K. Chang, N. Balachandar, C. Lam, D. Yi, J. Brown, A. Beers, B. Rosen, D. L. Rubin, and J. Kalpathy-Cramer. Distributed deep learning networks among institutions for medical imaging. Journal of the American Medical Informatics Association, 25(8):945-954, 2018. XGBoost: A scalable tree boosting system. T Chen, C Guestrin, Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD '16. the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD '16New York, NY, USAACMT. Chen and C. Guestrin. XGBoost: A scalable tree boosting system. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD '16, pages 785-794, New York, NY, USA, 2016. ACM. Federated learning for predicting clinical outcomes in patients with covid-19. I Dayan, H R Roth, A Zhong, A Harouni, A Gentili, A Z Abidin, A Liu, A B Costa, B J Wood, C.-S Tsai, Nature medicine. 2710I. Dayan, H. R. Roth, A. Zhong, A. Harouni, A. Gentili, A. Z. Abidin, A. Liu, A. B. Costa, B. J. Wood, C.-S. Tsai, et al. Federated learning for predicting clinical outcomes in patients with covid-19. Nature medicine, 27(10):1735-1743, 2021. Flute: A scalable, extensible framework for high-performance federated learning simulations. D Dimitriadis, M H Garcia, D M Diaz, A Manoel, R Sim, arXiv:2203.13789arXiv preprintD. Dimitriadis, M. H. Garcia, D. M. Diaz, A. Manoel, and R. Sim. Flute: A scalable, extensible framework for high-performance federated learning simulations. arXiv preprint arXiv:2203.13789, 2022. Auto-fedrl: Federated hyperparameter optimization for multi-institutional medical image segmentation. P Guo, D Yang, A Hatamizadeh, A Xu, Z Xu, W Li, C Zhao, D Xu, S Harmon, E Turkbey, arXiv:2203.06338arXiv preprintP. Guo, D. Yang, A. Hatamizadeh, A. Xu, Z. Xu, W. Li, C. Zhao, D. Xu, S. Harmon, E. Turkbey, et al. Auto-fedrl: Feder- ated hyperparameter optimization for multi-institutional medical image segmentation. arXiv preprint arXiv:2203.06338, 2022. Distributed learning of deep neural network over multiple agents. O Gupta, R Raskar, Journal of Network and Computer Applications. 116O. Gupta and R. Raskar. Distributed learning of deep neural network over multiple agents. Journal of Network and Computer Applications, 116:1-8, 2018. Do gradient inversion attacks make federated learning unsafe?. A Hatamizadeh, H Yin, P Molchanov, A Myronenko, W Li, P Dogra, A Feng, M G Flores, J Kautz, D Xu, arXiv:2202.06924arXiv preprintA. Hatamizadeh, H. Yin, P. Molchanov, A. Myronenko, W. Li, P. Dogra, A. Feng, M. G. Flores, J. Kautz, D. Xu, et al. Do gradient inversion attacks make federated learning unsafe? arXiv preprint arXiv:2202.06924, 2022. C He, S Li, J So, X Zeng, M Zhang, H Wang, X Wang, P Vepakomma, A Singh, H Qiu, arXiv:2007.13518A research library and benchmark for federated machine learning. arXiv preprintC. He, S. Li, J. So, X. Zeng, M. Zhang, H. Wang, X. Wang, P. Vepakomma, A. Singh, H. Qiu, et al. FedML: A research library and benchmark for federated machine learning. arXiv preprint arXiv:2007.13518, 2020. Towards a unified view of parameter-efficient transfer learning. J He, C Zhou, X Ma, T Berg-Kirkpatrick, G Neubig, arXiv:2110.04366arXiv preprintJ. He, C. Zhou, X. Ma, T. Berg-Kirkpatrick, and G. Neubig. Towards a unified view of parameter-efficient transfer learning. arXiv preprint arXiv:2110.04366, 2021. Parameter-efficient transfer learning for nlp. N Houlsby, A Giurgiu, S Jastrzebski, B Morrone, Q De Laroussilhe, A Gesmundo, M Attariyan, S Gelly, International Conference on Machine Learning. PMLRN. Houlsby, A. Giurgiu, S. Jastrzebski, B. Morrone, Q. De Laroussilhe, A. Gesmundo, M. Attariyan, and S. Gelly. Parameter-efficient transfer learning for nlp. In International Conference on Machine Learning, pages 2790-2799. PMLR, 2019. E J Hu, Y Shen, P Wallis, Z Allen-Zhu, Y Li, S Wang, L Wang, W Chen, arXiv:2106.09685Lora: Low-rank adaptation of large language models. arXiv preprintE. J. Hu, Y. Shen, P. Wallis, Z. Allen-Zhu, Y. Li, S. Wang, L. Wang, and W. Chen. Lora: Low-rank adaptation of large language models. arXiv preprint arXiv:2106.09685, 2021. P Kairouz, H B Mcmahan, B Avent, A Bellet, M Bennis, A N Bhagoji, K Bonawitz, Z Charles, G Cormode, R Cummings, arXiv:1912.04977Advances and open problems in federated learning. arXiv preprintP. Kairouz, H. B. McMahan, B. Avent, A. Bellet, M. Bennis, A. N. Bhagoji, K. Bonawitz, Z. Charles, G. Cormode, R. Cummings, et al. Advances and open problems in federated learning. arXiv preprint arXiv:1912.04977, 2019. Scaffold: Stochastic controlled averaging for federated learning. S P Karimireddy, S Kale, M Mohri, S Reddi, S Stich, A T Suresh, International Conference on Machine Learning. PMLRS. P. Karimireddy, S. Kale, M. Mohri, S. Reddi, S. Stich, and A. T. Suresh. Scaffold: Stochastic controlled averaging for federated learning. In International Conference on Machine Learning, pages 5132-5143. PMLR, 2020. Learning multiple layers of features from tiny images. A Krizhevsky, G Hinton, A. Krizhevsky, G. Hinton, et al. Learning multiple layers of features from tiny images. 2009. The power of scale for parameter-efficient prompt tuning. B Lester, R Al-Rfou, N Constant, arXiv:2104.08691arXiv preprintB. Lester, R. Al-Rfou, and N. Constant. The power of scale for parameter-efficient prompt tuning. arXiv preprint arXiv:2104.08691, 2021. Ditto: Fair and robust federated learning through personalization. T Li, S Hu, A Beirami, V Smith, International Conference on Machine Learning. PMLRT. Li, S. Hu, A. Beirami, and V. Smith. Ditto: Fair and robust federated learning through personalization. In International Conference on Machine Learning, pages 6357-6368. PMLR, 2021. Federated optimization in heterogeneous networks. T Li, A K Sahu, M Zaheer, M Sanjabi, A Talwalkar, V Smith, Proceedings of Machine Learning and Systems. Machine Learning and Systems2T. Li, A. K. Sahu, M. Zaheer, M. Sanjabi, A. Talwalkar, and V. Smith. Federated optimization in heterogeneous networks. Proceedings of Machine Learning and Systems, 2:429-450, 2020. Privacy-preserving federated brain tumour segmentation. W Li, F Milletarì, D Xu, N Rieke, J Hancox, W Zhu, M Baust, Y Cheng, S Ourselin, M J Cardoso, International workshop on machine learning in medical imaging. SpringerW. Li, F. Milletarì, D. Xu, N. Rieke, J. Hancox, W. Zhu, M. Baust, Y. Cheng, S. Ourselin, M. J. Cardoso, et al. Privacy-preserving federated brain tumour segmentation. In International workshop on machine learning in medical imaging, pages 133-141. Springer, 2019. . X Liu, Y Zheng, Z Du, M Ding, Y Qian, Z Yang, J Tang, arXiv:2103.10385Gpt understands, too. arXiv preprintX. Liu, Y. Zheng, Z. Du, M. Ding, Y. Qian, Z. Yang, and J. Tang. Gpt understands, too. arXiv preprint arXiv:2103.10385, 2021. Fate: An industrial grade platform for collaborative learning with data protection. Y Liu, T Fan, T Chen, Q Xu, Q Yang, J. Mach. Learn. Res. 22226Y. Liu, T. Fan, T. Chen, Q. Xu, and Q. Yang. Fate: An industrial grade platform for collaborative learning with data protection. J. Mach. Learn. Res., 22(226):1-6, 2021. Ibm federated learning: an enterprise framework white paper v0. H Ludwig, N Baracaldo, G Thomas, Y Zhou, A Anwar, S Rajamoni, Y Ong, J Radhakrishnan, A Verma, M Sinn, arXiv:2007.109871arXiv preprintH. Ludwig, N. Baracaldo, G. Thomas, Y. Zhou, A. Anwar, S. Rajamoni, Y. Ong, J. Radhakrishnan, A. Verma, M. Sinn, et al. Ibm federated learning: an enterprise framework white paper v0. 1. arXiv preprint arXiv:2007.10987, 2020. Communication-efficient learning of deep networks from decentralized data. B Mcmahan, E Moore, D Ramage, S Hampson, B A Arcas, Artificial intelligence and statistics. PMLRB. McMahan, E. Moore, D. Ramage, S. Hampson, and B. A. y Arcas. Communication-efficient learning of deep networks from decentralized data. In Artificial intelligence and statistics, pages 1273-1282. PMLR, 2017. MONAI: Medical Open Network for AI. Monai Consortium, 92022MONAI Consortium. MONAI: Medical Open Network for AI, 9 2022. . Openai, Gpt-4 technical reportOpenAI. Gpt-4 technical report, 2023. Training language models to follow instructions with human feedback. L Ouyang, J Wu, X Jiang, D Almeida, C Wainwright, P Mishkin, C Zhang, S Agarwal, K Slama, A Ray, Advances in Neural Information Processing Systems. 35L. Ouyang, J. Wu, X. Jiang, D. Almeida, C. Wainwright, P. Mishkin, C. Zhang, S. Agarwal, K. Slama, A. Ray, et al. Training language models to follow instructions with human feedback. Advances in Neural Information Processing Systems, 35:27730-27744, 2022. S Reddi, Z Charles, M Zaheer, Z Garrett, K Rush, J Konečnỳ, S Kumar, H B Mcmahan, arXiv:2003.00295Adaptive federated optimization. arXiv preprintS. Reddi, Z. Charles, M. Zaheer, Z. Garrett, K. Rush, J. Konečnỳ, S. Kumar, and H. B. McMahan. Adaptive federated optimization. arXiv preprint arXiv:2003.00295, 2020. G A Reina, A Gruzdev, P Foley, O Perepelkina, M Sharma, I Davidyuk, I Trushkin, M Radionov, A Mokrov, D Agapov, arXiv:2105.06413An open-source framework for federated learning. arXiv preprintG. A. Reina, A. Gruzdev, P. Foley, O. Perepelkina, M. Sharma, I. Davidyuk, I. Trushkin, M. Radionov, A. Mokrov, D. Agapov, et al. Openfl: An open-source framework for federated learning. arXiv preprint arXiv:2105.06413, 2021. The future of digital health with federated learning. N Rieke, J Hancox, W Li, F Milletari, H R Roth, S Albarqouni, S Bakas, M N Galtier, B A Landman, K Maier-Hein, NPJ digital medicine. 31N. Rieke, J. Hancox, W. Li, F. Milletari, H. R. Roth, S. Albarqouni, S. Bakas, M. N. Galtier, B. A. Landman, K. Maier-Hein, et al. The future of digital health with federated learning. NPJ digital medicine, 3(1):1-7, 2020. Federated learning for breast density classification: A real-world implementation. H R Roth, K Chang, P Singh, N Neumark, W Li, V Gupta, S Gupta, L Qu, A Ihsani, B C Bizzo, Domain Adaptation and Representation Transfer, and Distributed and Collaborative Learning. SpringerH. R. Roth, K. Chang, P. Singh, N. Neumark, W. Li, V. Gupta, S. Gupta, L. Qu, A. Ihsani, B. C. Bizzo, et al. Federated learning for breast density classification: A real-world implementation. In Domain Adaptation and Representation Transfer, and Distributed and Collaborative Learning, pages 181-191. Springer, 2020. Federated learning improves site performance in multicenter deep learning without data sharing. K V Sarma, S Harmon, T Sanford, H R Roth, Z Xu, J Tetreault, D Xu, M G Flores, A G Raman, R Kulkarni, Journal of the American Medical Informatics Association. 286K. V. Sarma, S. Harmon, T. Sanford, H. R. Roth, Z. Xu, J. Tetreault, D. Xu, M. G. Flores, A. G. Raman, R. Kulkarni, et al. Federated learning improves site performance in multicenter deep learning without data sharing. Journal of the American Medical Informatics Association, 28(6):1259-1264, 2021. Federated learning in medicine: facilitating multi-institutional collaborations without sharing patient data. M J Sheller, B Edwards, G A Reina, J Martin, S Pati, A Kotrotsou, M Milchenko, W Xu, D Marcus, R R Colen, Scientific reports. 101M. J. Sheller, B. Edwards, G. A. Reina, J. Martin, S. Pati, A. Kotrotsou, M. Milchenko, W. Xu, D. Marcus, R. R. Colen, et al. Federated learning in medicine: facilitating multi-institutional collaborations without sharing patient data. Scientific reports, 10(1):1-12, 2020. Multi-institutional deep learning modeling without sharing patient data: A feasibility study on brain tumor segmentation. M J Sheller, G A Reina, B Edwards, J Martin, S Bakas, International MICCAI Brainlesion Workshop. SpringerM. J. Sheller, G. A. Reina, B. Edwards, J. Martin, and S. Bakas. Multi-institutional deep learning modeling without sharing patient data: A feasibility study on brain tumor segmentation. In International MICCAI Brainlesion Workshop, pages 92-104. Springer, 2018. Fed-biomed: A general open-source frontend framework for federated learning in healthcare. S Silva, A Altmann, B Gutman, M Lorenzi, Domain Adaptation and Representation Transfer, and Distributed and Collaborative Learning. SpringerS. Silva, A. Altmann, B. Gutman, and M. Lorenzi. Fed-biomed: A general open-source frontend framework for federated learning in healthcare. In Domain Adaptation and Representation Transfer, and Distributed and Collaborative Learning, pages 201-210. Springer, 2020. H Wang, M Yurochkin, Y Sun, D Papailiopoulos, Y Khazaeni, arXiv:2002.06440Federated learning with matched averaging. arXiv preprintH. Wang, M. Yurochkin, Y. Sun, D. Papailiopoulos, and Y. Khazaeni. Federated learning with matched averaging. arXiv preprint arXiv:2002.06440, 2020. Swarm learning for decentralized and confidential clinical machine learning. S Warnat-Herresthal, H Schultze, K L Shastry, S Manamohan, S Mukherjee, V Garg, R Sarveswara, K Händler, P Pickkers, N A Aziz, Nature. 5947862S. Warnat-Herresthal, H. Schultze, K. L. Shastry, S. Manamohan, S. Mukherjee, V. Garg, R. Sarveswara, K. Händler, P. Pickkers, N. A. Aziz, et al. Swarm learning for decentralized and confidential clinical machine learning. Nature, 594(7862):265-270, 2021. Private set intersection -Wikipedia, the free encyclopedia. Wikipedia contributors. 27Wikipedia contributors. Private set intersection -Wikipedia, the free encyclopedia, 2023. [Online; accessed 27-April-2023]. Federatedscope: A comprehensive and flexible federated learning platform via message passing. Y Xie, Z Wang, D Chen, D Gao, L Yao, W Kuang, Y Li, B Ding, J Zhou, arXiv:2204.05011arXiv preprintY. Xie, Z. Wang, D. Chen, D. Gao, L. Yao, W. Kuang, Y. Li, B. Ding, and J. Zhou. Federatedscope: A comprehensive and flexible federated learning platform via message passing. arXiv preprint arXiv:2204.05011, 2022. Closing the generalization gap of cross-silo federated medical image segmentation. A Xu, W Li, P Guo, D Yang, H R Roth, A Hatamizadeh, C Zhao, D Xu, H Huang, Z Xu, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionA. Xu, W. Li, P. Guo, D. Yang, H. R. Roth, A. Hatamizadeh, C. Zhao, D. Xu, H. Huang, and Z. Xu. Closing the generalization gap of cross-silo federated medical image segmentation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 20866-20875, 2022. Federated machine learning: Concept and applications. Q Yang, Y Liu, T Chen, Y Tong, ACM Transactions on Intelligent Systems and Technology (TIST). 102Q. Yang, Y. Liu, T. Chen, and Y. Tong. Federated machine learning: Concept and applications. ACM Transactions on Intelligent Systems and Technology (TIST), 10(2):1-19, 2019. Reduce communication costs and preserve privacy: Prompt tuning method in federated learning. H Zhao, W Du, F Li, P Li, G Liu, arXiv:2208.12268arXiv preprintH. Zhao, W. Du, F. Li, P. Li, and G. Liu. Reduce communication costs and preserve privacy: Prompt tuning method in federated learning. arXiv preprint arXiv:2208.12268, 2022. Pysyft: A library for easy federated learning. A Ziller, A Trask, A Lopardo, B Szymkow, B Wagner, E Bluemke, J.-M Nounahon, J Passerat-Palmbach, K Prakash, N Rose, Federated Learning Systems. SpringerA. Ziller, A. Trask, A. Lopardo, B. Szymkow, B. Wagner, E. Bluemke, J.-M. Nounahon, J. Passerat-Palmbach, K. Prakash, N. Rose, et al. Pysyft: A library for easy federated learning. In Federated Learning Systems, pages 111-139. Springer, 2021.
{'fraction_non_alphanumeric': 0.05208781747739991, 'fraction_numerical': 0.019289506590410593, 'mean_word_length': 4.704396632366698, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 3, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 5, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'Federated learning (FL) enables building robust and generalizable AI models by leveraging diverse datasets from multiple collaborators without centralizing the data. We created NVIDIA FLARE 1 as an open-source software development kit (SDK) to make it easier for data scientists to use FL in their research and realworld applications. The SDK includes solutions for state-of-the-art FL algorithms and federated machine learning approaches, which facilitate building workflows for distributed learning across enterprises and enable platform developers to create a secure, privacy-preserving offering for multiparty collaboration utilizing homomorphic encryption or differential privacy. The SDK is a lightweight, flexible, and scalable Python package. It allows researchers to apply their data science workflows in any training libraries (PyTorch, TensorFlow, XGBoost, or even NumPy) in real-world FL settings. This paper introduces the key design principles of NVFlare and illustrates some use cases (e.g., COVID analysis) with customizable FL workflows that implement different privacy-preserving algorithms.', 'arxivid': '2210.13291', 'author': ['Holger R Roth \nNVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA\n', 'Yan Cheng \nNVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA\n', 'Yuhong Wen \nNVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA\n', 'Isaac Yang \nNVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA\n', 'Ziyue Xu \nNVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA\n', 'Yuan-Ting Hsieh \nNVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA\n', 'Kristopher Kersten \nNVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA\n', 'Ahmed Harouni \nNVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA\n', 'Can Zhao \nNVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA\n', 'Kevin Lu \nNVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA\n', 'Zhihong Zhang \nNVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA\n', 'Wenqi Li \nNVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA\n', 'Andriy Myronenko \nNVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA\n', 'Dong Yang \nNVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA\n', 'Sean Yang \nNVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA\n', 'Nicola Rieke \nNVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA\n', 'Abood Quraini \nNVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA\n', 'Chester Chen \nNVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA\n', 'Daguang Xu \nNVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA\n', 'Nic Ma \nNVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA\n', 'Prerna Dogra \nNVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA\n', 'Mona Flores \nNVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA\n', 'Andrew Feng \nNVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA\n'], 'authoraffiliation': ['NVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA', 'NVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA', 'NVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA', 'NVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA', 'NVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA', 'NVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA', 'NVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA', 'NVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA', 'NVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA', 'NVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA', 'NVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA', 'NVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA', 'NVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA', 'NVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA', 'NVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA', 'NVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA', 'NVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA', 'NVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA', 'NVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA', 'NVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA', 'NVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA', 'NVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA', 'NVIDIA Corporation * Shanghai\nMunich, Bethesda, Santa ClaraChina, Germany, USA'], 'corpusid': 253098004, 'doi': '10.48550/arxiv.2210.13291', 'github_urls': ['https://github.com/NVIDIA/NVFlare.'], 'n_tokens_mistral': 13620, 'n_tokens_neox': 12194, 'n_words': 7186, 'pdfsha': '04315fbe6554edc9b5d76c99f64b8d1a6f26f7af', 'pdfurls': ['https://export.arxiv.org/pdf/2210.13291v3.pdf'], 'title': ['NVIDIA FLARE: Federated Learning from Simulation to Real-World', 'NVIDIA FLARE: Federated Learning from Simulation to Real-World'], 'venue': []}
arxiv
Chern-Weil theory for Haefliger-singular foliations 12 Mar 2022 March 2022 Lachlan E Macdonald Australian Institute for Machine Learning School of Mathematical Sciences The University of Adelaide Adelaide The University of Adelaide Adelaide 5000, 5000SA, SA Benjamin Mcmillan Australian Institute for Machine Learning School of Mathematical Sciences The University of Adelaide Adelaide The University of Adelaide Adelaide 5000, 5000SA, SA Chern-Weil theory for Haefliger-singular foliations 12 Mar 2022 March 2022 We give a Chern-Weil map for the Gel'fand-Fuks characteristic classes of Haefligersingular foliations, those foliations defined by smooth Haefliger structures with dense regular set. Our characteristic map constructs, out of singular geometric structures adapted to singularities, explicit forms representing characteristic classes in de Rham cohomology. The forms are functorial under foliation morphisms. We prove that the theory applies, up to homotopy, to general smooth Haefliger structures: subject only to obvious necessary dimension constraints, every smooth Haefliger structure is homotopic to a Haefliger-singular foliation, and any morphism of Haefliger structures is homotopic to a morphism of Haefligersingular foliations. As an application, we provide a generalisation to the singular setting of the classical construction of forms representing the Godbillon-Vey invariant. Introduction In this paper, we give a Chern-Weil construction of the Gel'fand-Fuks characteristic classes of certain singular foliations in terms of singular metrics and connections. Our methods apply specifically to singular foliations arising from Haefliger structures [24], which are the closest singular cousins of regular foliations, as their leaves are still locally determined by complete families of first integrals. We show that this class is sufficiently general for our theory to apply (up to homotopy) to all smooth Haefliger structures on sufficiently high-dimensional manifolds. Our work opens the way for further study of the topology of Haefliger's classifying space with all the flexibility afforded by singularities (cf. [47]), and opens up the potential advancement of the study of singular foliations via noncommutative geometry and index theory, which has made great strides in recent years [2,3,4,5]. Let us briefly recall some of the history of foliation characteristic classes most relevant to our work. A regular foliation of codimension q on an n-manifold M is given by an involutive subbundle TF ⊂ TM of rank n − q. By the Frobenius theorem, the data TF is equivalent to a decomposition F of M as a union of non-intersecting, immersed submanifolds of dimension n−q. Assuming the foliation is transversely orientable (i.e. that the normal bundle νF := TM/TF is an orientable vector bundle), the subbundle TF can alternatively be regarded as the kernel of a nowhere vanishing, decomposable q-form ω = ω 1 ∧ . . . ∧ω q on M that is integrable, in the sense that there exists a 1-form η for which dω = η∧ω. The starting point for the tremendous advances made in understanding regular foliations via their characteristic classes was the discovery of Godbillon and Vey [22] that, for codimension 1 foliations, the associated 3-form η∧dη is closed, and its class in de Rham cohomology (now called the Godbillon-Vey class) depends only on F, and not on the choices of η and ω. Nontriviality of the Godbillon-Vey class was first shown by Roussarie (also in [22]), while in [47] Thurston gave a construction exhibiting continuous variation of the Godbillon-Vey class for a family of foliations of the 3-sphere. It was discovered by Bott [9] that the Godbillon-Vey class of codimension 1 foliations is the simplest example of a large family of so-called secondary characteristic classes associated to regular foliations of arbitrary codimension, transversely orientable or not. More precisely, Bott showed in [8] that the normal bundle of any regular codimension q foliation admits connections that are flat along leaves (now called Bott connections), and proved as an easy consequence that any Pontryagin polynomial of degree more than 2q vanishes when evaluated on the curvature of any Bott connection. Bott deduced that certain Chern-Simons [14] transgression forms of such polynomials-in-curvature therefore define de Rham cohomology classes; in particular, for a codimension 1 foliation, the Godbillon-Vey class arises as a transgression of the square of the first Pontryagin form. These characteristic classes, being associated to the normal bundle of the foliation, can be regarded as characteristic classes for the leaf space of the foliation. Bott's Chern-Weil account of the characteristic classes of foliations coincided with deep work by Gel'fand and Fuks studying the (continuous) cohomology of infinite-dimensional Lie algebras of vector fields [18,19,20,21]. Gel'fand and Fuks in particular computed the O(q)-basic cohomology H * (A(a q ), O(q)) of the Lie algebra a q of ∞-jets at zero of vector fields on R q , which is now frequently referred to simply as the "Gel'fand-Fuks cohomology". They discovered that, in addition to encoding the usual Pontryagin classes of real vector bundles, this Lie algebra cohomology automatically encodes Bott's vanishing theorem and the consequent secondary characteristic classes, and is isomorphic to the cohomology of a well-understood finite-dimensional subalgebra W O q ⊂ A * (a q ). The subalgebra W O q furthermore factors through the 2-jets of vector fields. The relationship between Gel'fand-Fuks cohomology and the characteristic classes of foliations was formalised by Bott and Haefliger [11,10], drawing on work of Kobayashi regarding higher order frame bundles [33]. One of the key properties of vector bundle characteristic classes is functorialiality under pullbacks. One must expect the same to hold for the characteristic classes of foliations, but immediately the problem arises that a smooth map φ : N → M into a regularly foliated manifold M does not in general pull back a regular foliation. More precisely, by the Frobenius theorem, a regular foliation of M is determined by an open covering U := {U α } α∈A together with submersive first integrals f α : U α → R q whose level sets define the leaves of the foliation in U α . The "pullback foliation" of N , defined by the open covering {φ −1 (U α )} α∈A and functions f α • φ : φ −1 (U α ) → R q , admits singularities wherever φ is not transverse to leaves. So, that the characteristic classes of foliations be functorial necessitates their definition for singular foliations of this nature. In an abstract sense, this lack of functoriality was solved by Haefliger, whose brilliant insight in [24] was to categorify codimension q foliations, regular or not, by considering the transition functions between local first integrals as fundamental. Since Haefliger's insight is essential to our work, we record his definition here. Definition 1.1. Let X be a topological space. A Haefliger cocycle of codimension q on X consists of an open covering {U α } α∈A of X together with continuous functions f α : U α → R q , called Haefliger charts, and for each x ∈ U α ∩ U β , the transition function h x αβ , a local diffeomorphism of R q defined on a neighbourhood of f β (x), such that for all indices α, β and x ∈ U α ∩ U β , 1. the assignment x → h x αβ is continuous, in the sense that the map (y, s) → h y αβ ( s) is continuous for (y, s) near (x, f β (x)) ∈ X × R q , Of particular concern to Haefliger were the numerable Haefliger structures, those which may be representated by a cocycle defined over an open cover admitting a subordinate partition of unity. This class includes all Haefliger structures on smooth manifolds and, in particular, all regular foliations. Haefliger showed that the category H q , whose objects are numerable Haefliger structures of codimension q and whose morphisms are continuous maps pulling back the structure on the codomain to that on the domain, admits a terminal object (BΓ q , γ) [25,Theorem 7], the classifying space of codimension q Haefliger structures (here Γ q is the groupoid of germs of local diffeomorphisms of R q ). As an application of this fact, he showed that there exists a universal characteristic map H * (A(a q ), O(q)) → H * (BΓ q ) from Gel'fand-Fuks cohomology to the cohomology of (BΓ q , γ), thereby furnishing a characteristic map for all numerable Haefliger structures of codimension q [26, p. 53], which is fully functorial. 2. f α = h x αβ • f β Haefliger's categorical lens enables us to systematically expose the limits of the existing characteristic class theory of foliations as follows. Let T denote the category whose objects are pairs (X, c), where X is a topological space and c is a cohomology class on X, and whose morphisms are continuous maps pulling back the class associated to the codomain to that associated to the domain. Then Haefliger's characteristic map furnishes an assignment A * (a q ) O(q) × H q → T , which is functorial in the second argument and homomorphic (with respect to the ring structures of A * (a q ) O(q) and of cohomology) in the first. Let us denote by F q reg the category of regular foliations of codimension q on smooth manifolds, whose morphisms are transverse maps pulling back the foliation of the codomain to that of the domain. Then the Frobenius theorem induces an inclusion F q reg ֒→ H q of categories, and Bott's Chern-Weil homomorphism amounts to a similarly homomorphic-functorial assignment W O q × F q reg → T . Bott proves in [9,Theorem 10.16] that his Chern-Weil characteristic map is compatible with that of Haefliger, in the sense that it recovers the characteristic classes coming from Haefliger's classifying space. Put another way, Bott's Chern-Weil characteristic map makes the diagram W O q × F q reg A * (a q ) O(q) × H q T (1) commute. We note that, by the homotopy-invariance of cohomology, both characteristic maps factor through the category H q htpy , whose objects and morphisms are the homotopy classes of those of H q . With this diagram in mind, the limits of the existing theory are clear. On the one hand, the category F q reg is not sufficiently large to accurately model the full subcategory of smooth manifolds in H q , even up to homotopy, since there exist manifolds with Haefliger structure not homotopic to a regular foliation. This phenomenon can be seen already by choosing a codimension 1 Haefliger structure on S 2 , which cannot be homotoped to a regular structure, because S 2 admits no regular codimension 1 foliation. For such examples, Bott's Chern-Weil theory offers no way to probe topology. On the other hand, while Haefliger's characteristic map works for all Haefliger structures, and the abstract definition allows quick proofs of general properties, this very abstraction makes it difficult to use differential geometric methods for the concrete study of topology, as is typically necessary for finer-grained results [22,47]. A middle ground is needed, one permitting the construction of characteristic classes for all Haefliger structures on manifolds, yet in terms of differential geometry. This gap in the literature has stood for almost five decades. Our primary result is the provision of such a middle ground. Our solution is an extension of Bott's Chern-Weil theory to a natural class of singular foliations, which we now define, restricting ourselves to smooth Haefliger cocycles on smooth manifolds (those whose Haefliger charts and transition functions are smooth). Definition 1.2. The singular set Σ of a smooth Haefliger cocycle of codimension q on a manifold M of dimension at least q is the closed set of all critical points of each f α ; the singular set depends only on the associated Haefliger structure. The regular set is the complement of Σ, which we often denote asM := M − Σ. IfM is dense in M , we say that the Haefliger cocycle determines a Haefliger-singular foliation (M, F ) of codimension q, whose regular subfoliation is the regular foliation ofM determined by the restriction of the Haefliger cocycle toM . We note that the term "singular foliation" is often used to refer to Stefan-Sussmann singular foliations, namely those defined by integrable families of vector fields [45,46]. We show in the appendix that every Haefliger-singular foliation (in fact, every smooth Haefliger structure) does indeed define a Stefan-Sussmann singular foliation. However, the converse is not true; for instance, the foliation of R 2 by the integrals of x∂ x + y∂ y is Stefan-Sussmann, but admits no nontrivial first integrals in any neighbourhood of the origin, so cannot be associated to a Haefliger structure. We show that the category F q sing of codimension q Haefliger-singular foliations, whose morphisms are smooth maps pulling back the singular foliation of the codomain to that of the domain, fits snugly in between F q reg and H q . In addition, we describe a Chern-Weil map extending that of Bott to Haefliger-singular foliations, expanding the commuting diagram (1) to W O q × F q reg W O q × F q sing A * (a q ) O(q) × H q T .(2) Importantly, we prove that F q sing is large enough so that every object in H q whose underlying space is a manifold of dimension at least q is homotopic to a Haefliger-singular foliation, and that every morphism in H q between two such objects is homotopic to a morphism in F q sing . In this way, our theory provides a complete, Chern-Weil solution to the problem of accessing the Gel'fand-Fuks characteristic classes of Haefliger structures on manifolds in terms of geometric data. As a further, geometric application of our theory, we describe how the classical algorithm for the geometric construction of the Godbillon-Vey invariant, used extensively for the study of regular foliations [22,47], extends cleanly to the singular setting. Section 2 consists of a review of the necessary prerequisites, on Gel'fand-Fuks cohomology and on the Chern-Weil approach to characteristic classes for regular foliations. In Section 3 we prove a smooth (diffeological) generalisation of Haefliger's classifying theorem, and an associated de Rham-theoretic universal characteristic map. This is the most complete and explicit account of the universal characteristic map that we are aware of, enabling a clean proof that our Chern-Weil construction recovers the correct characteristic classes. Section 4 gives a new, de Rham-theoretic proof that the Chern-Weil approach to regular foliations recovers the characteristic classes coming from Haefliger's classifying space. Although the result, originally due to Bott [9], has been known for some decades, our presentation is novel in that it invokes an under-utilised family of fibre bundles, that we call the Haefliger bundles, which are defined over both regular and singular foliations. The Haefliger bundles act as an intermediary between the universal characteristic map and the Chern-Weil characteristic map for regular foliations. Section 5 consists of an extension of these ideas to singular foliations. We characterise those metric-Bott-connection pairs on the normal bundle of the regular subfoliation of a Haefligersingular foliation which are adapted to the singularities via the jets of their exponential maps. Such pairs we term adapted geometries. We then prove our Chern-Weil homomorphism for singular foliations. Background We work under the convention that the set of natural numbers N includes zero. All manifolds are assumed to be connected, paracompact, Hausdorff, and without boundary, unless otherwise stated. Gel'fand-Fuks cohomology Given a Lie group G with Lie algebra g, a G-differential graded algebra, or G-DGA, is a differential graded R-algebra (A • , d), where A • carries a smooth, degree 0 action of G, with associated Lie derivative defined by L ξ (a) := d dt 0 exp(tξ) · a, ξ ∈ g, a ∈ A, and a contraction operator ι ξ : A • → A •−1 for each ξ ∈ g such that d and ι anticommute up to the Lie derivative, L ξ = dι ξ + ι ξ d. Given any Lie subgroup K of G, with Lie algebra k, the K-basic subalgebra of A • is the differential graded subalgebra A • K := a ∈ A • : k · a = a for all k ∈ K, ι ξ a = 0 for all ξ ∈ k . We denote the cohomology of A • K by H * (A, K). Our primary example of such will be the Gel'fand-Fuks cohomology, which we turn to presently. For 0 ≤ q, k < ∞, fix the Lie group G k q , the k th -order jet group of R q , comprising the k-jets at zero of diffeomorphisms of R q that fix zero. Let g k q denote the Lie algebra of G k q . As k ranges from 0 to ∞, the natural maps between the G k q form a projective system of Lie groups. The projective limit G ∞ q of this system is the infinite order jet group of R q , which inherits a natural smooth structure [43,Section 7.1]. With this smooth structure, G ∞ q is an infinite-dimensional Lie group, with Lie algebra g ∞ q equal to the projective limit of the projective system of Lie algebras determined by the g k q . For q ≥ 1, denote by a q the Lie algebra of ∞-jets at zero of smooth vector fields on R q . Endow a q with a projective limit topology by identifying it as the projective limit of the system {a k q , π k } of finite-dimensional manifolds of k-jets a k q at zero of vector fields on R q , with π k : a k q → a k−1 q the canonical projection. This, plus the Lie bracket on a q determined by that of vector fields, determines a topological Lie algebra structure on a q [21]. Any X ∈ a q can be realised as the time derivative of a path of ∞-jets, X = d dt 0 j ∞ 0 (ϕ t )(3) where t → ϕ t , ϕ 0 = id is the flow of any vector field representing X. (There is no issue of completeness of flows here, because the flow is only required in an infinitesimal neighbourhood of 0). The construction also exhibits a natural inclusion g ∞ q ֒→ a q of Lie algebras, with image characterised as the jets of those vector fields that vanish at 0. For k ≥ 1, denote by A k (a q ) the space of continuous, alternating, multi-linear functionals ∧ k a q → R. The usual Chevalley-Eilenberg formula defines a differential d : A k (a q ) → A k+1 (a q ), given for c ∈ A k (a q ) and X 0 , . . . , X k ∈ a q by dc(X 0 , . . . , X k ) := i<j (−1) i+j c([X i , X j ], X 0 , . . . ,X i , . . . ,X j , . . . , X k+1 ). Now, A * (a q ) is a G ∞ q -differential graded algebra. Indeed, the inclusion g ∞ q ֒→ a q defines a contraction operator ι ξ : A * (a q ) → A * −1 (a q ) for all ξ ∈ g ∞ q , and the right action of G ∞ q on a q defined by d dt 0 j ∞ 0 (ϕ t ) · j ∞ 0 (g) = d dt 0 j ∞ 0 g −1 • ϕ t • g , j ∞ 0 (g) ∈ G ∞ q , d dt 0 j ∞ 0 (ϕ t ) ∈ a q , induces an action of G ∞ q on A * (a q ) compatible with the contraction operator. The O(q)-basic cohomology H * (A(a q ), O(q)) of A * (a q ) is frequently referred to as simply the Gel'fand-Fuks cohomology, and was computed by Gel'fand and Fuks in [21]. We now review the method outlined by Bott [10] for the computation of Gel'fand-Fuks cohomology H * (A(a q ), O(q)), and in the next subsection, how the O(q)-basic A * (a q )-cocycles define characteristic classes. Fix the standard linear coordinates s i on R q . For any multi-index α ∈ N q and for 1 ≤ i ≤ q, the Dirac-derivative functional on a q defined by δ i α (X) := (−1) |α| ∂ |α| X i ∂s α (0), X ∈ a q is an element of A 1 (a q ), and by elementary distribution theory, the collection of all such functionals generates A 1 (a q ) linearly, hence generates A * (a q ) as an algebra. The following structure equations then follow from a routine calculation: dδ i + δ i j ∧ δ j = 0, dδ i j + δ i jk ∧ δ k + δ i k ∧ δ k j = 0,(4) with the Einstein summation convention assumed. Notice that the first of these equations resembles the structure equation for torsion-free affine connections (with the δ i playing the role of the components of the solder form, and the δ i j the components of the connection form), while the second resembles the equation defining the curvature ∆ i j := −δ i jk ∧δ k = dδ i j + δ i k ∧δ k j of the connection in terms of the components of the connection form. We will see in the next subsection that this is not a coincidence-these structure equations are the universal structure equations for torsion-free affine connections on manifolds. Define the q × q matrices of 1-and 2-forms δ := δ i j , ∆ := ∆ i j , i, j = 1, . . . , q(5) corresponding to the functionals δ i j and ∆ i j respectively, and denote by δ := δ s + δ o and ∆ := ∆ s + ∆ o their respective decompositions into symmetric and antisymmetric components. Let R[c 1 , . . . , c q ] q denote the quotient of the polynomial algebra in symbols c i of degree 2i by the ideal of elements consisting of total degree greater than 2q, and let ∧(h 1 , h 3 , . . . , h l ) denote the exterior algebra generated by symbols h i of degree 2i − 1, with l the largest odd integer that is less than or equal to q. Equip the graded-commutative algebra W O q := R[c 1 , . . . , c q ] q ⊗ ∧(h 1 , h 3 , . . . , h l ) with the differential d defined by dc i = 0 for all i and dh j = c j for all j odd. Then by the results of [23], W O q embeds as a differential graded subalgebra of A * (a q ) O(q) according to the formulae c i := Tr(∆ ∧i ), 1 ≤ i ≤ q,(6)h j := j Tr 1 0 δ s (t∆ s + ∆ o + (t 2 − 1)δ ∧2 s ) ∧(j−1) dt , 1 ≤ j ≤ q, j odd.(7) In fact, the c i are GL(R q )-basic, and correspond to the Pontryagin classes of tangent bundles, as we will see in the next subsection. One has the following theorem, which computes the cohomology of the infinite-dimensional algebra A * (a q ) O(q) in terms of the finite-dimensional subalgebra W O q . Frame bundles and tautological forms Let (M, F ) be a regular foliation of codimension q. For x ∈ M , a transverse embedding through x is an embedding u : R q → M such that u(0) = x and for each s ∈ R q one has T u( s) M = du s (T s R q )⊕ T u( s) F. Two k-jets j k 0 (u 1 ) and j k 0 (u 2 ) of transverse embeddings through x are said to be leaf space equivalent if for some (hence any) Haefliger chart f : U → R q defined around x one has j k 0 (f • u 1 ) = j k 0 (f • u 2 ). The leaf space equivalence class of a k-jet j k 0 (u) will be denoted j k 0,t (u) (the t being used to denote transverse). Definition 2.2. [41] Let (M, F ) be a regular foliation of codimension q, and let 0 ≤ k ≤ ∞. The transverse k-frame bundle of (M, F ) is the principal G k q -bundle Fr k (M/ F ) → M whose fibre over x ∈ M is the set of leaf space equivalence classes of k-jets at 0 of transverse embeddings through x. ω k j k 0 (u) d dt 0 j k 0 (u t ) := d dt 0 j k−1 0 (u −1 • u t ) for d dt 0 j k 0 (u t ) ∈ T j k 0 (u) Fr k (M ) is easily seen to be an element of Ω 1 (Fr k (M )) Diff (M ) ⊗ a k−1 q . The tautological 1-forms ω k were introduced by Kobayashi [33]. The a q -valued 1-form ω := ω ∞ on Fr ∞ (M ) satisfies the Maurer-Cartan identity dω ∞ + of G ∞ q -differential graded algebras (cf. [32,Proposition 3.5]). It is known [34, p. 131, Proposition] that for each finite k ≥ 1, the natural projection G k q → GL(R q ) is a principal fibre bundle, with typical fibre a contractible nilpotent Lie group whose Lie exponential map is a global diffeomorphism. The same is therefore also true of the fibration G ∞ q → GL(R q ). It follows that Fr ∞ (M ) → Fr 1 (M ) has contractible fibres, and always admits sections. In particular, letting ∇ be an affine connection on M , with exponential map exp ∇ , the formula σ ∇ ( e x ) := j ∞ 0 s → exp ∇ x ( s · e x ) e x ∈ Fr 1 (M )(9) defines a G (9). Then (σ ∇ ) * δ(ω) ∈ Ω 1 (Fr 1 (M ); gl(R q )) is the connection form associated to ∇ and (σ ∇ ) * ∆(ω) ∈ Ω 2 (Fr 1 (M ); gl(R q )) is its curvature. It follows immediately from Proposition 2.4 that the forms (σ ∇ ) * c i (ω) ∈ Ω 2i (Fr 1 (M )) GL(R q ) from Equation (6) are the Pontryagin forms of M defined with respect to the curvature tensor of ∇, which displays the fundamental relationship between Chern-Weil theory and Gel'fand-Fuks cohomology. Moreover, since the c i and h i only depend on 2-jets, and W O q ֒→ A(a q ) is a quasiisomorphism, one sees that in cohomology it suffices to work with ω 2 ∈ Ω 1 (Fr 2 (M )) Diff(M ) ⊗ a 1 q . The facts elucidated in Example 2.3 generalise to the transverse frame bundles of foliations more generally. Specifically, the transverse frame bundles of a regular foliation admit tautological forms, which model geometric structures on the normal bundle of the foliation. This is folklore, but will be made precise after our introduction of Haefliger bundles in Section 3. Review of Chern-Weil for regular foliations In the early nineteen-seventies, R. Bott showed that characteristic classes for a regular foliation (M, F ) of codimension q could be obtained in the following manner [9]. Let the normal bundle νF := TM/TF of F be regarded as a subbundle of TM that is complementary to TF. (For instance, the orthogonal complement of TF with respect to some Riemannian metric on M .) For a vector field Z on M , denote by Z = Z F + Z ν its decomposition into leafwise and normal components respectively. Bott showed [8] that there exist connections ∇ on νF satisfying the equation ∇ X Y = ∇ Xν Y + [X F , Y ] ν , X ∈ X(M ), Y ∈ Γ(νF) ⊂ X(M ).c i → Tr(R ∧i ∇ ), vanishes on all monomials (in the c i ) of degree greater than 2q. Bott's vanishing theorem has the following consequence. Let ∇ 1 be a Bott connection on νF and let ∇ 0 be a connection on νF that is compatible with some Euclidean metric on νF. For t ∈ [0, 1], denote by ∇ t := t∇ 1 + (1 − t)∇ 0 the affine combination of ∇ 0 and ∇ 1 . λ ∇ 1 ,∇ 0 (c i ) := Tr(R ∧i ∇ 1 ) ∈ Ω 2i (M ),(10)λ ∇ 1 ,∇ 0 (h i ) := i 1 0 Tr (∇ 1 − ∇ 0 ) ∧ R ∧(i−1) ∇ t dt ∈ Ω 2i−1 (M )(11) is a homomorphism of differential graded algebras whose descent to cohomology does not depend on the Bott connection or metric connection chosen. Continuing with a regular foliation (M, F ), denote projection to the normal bundle νF by p : TM → νF. The torsion of a Bott connection ∇ on νF is the tensor T ∈ Γ ∞ (M ; T * M ⊗ T * M ⊗ νF) defined by T (X, Y ) := ∇ X (pY ) − ∇ Y (pX) − p[X, Y ] for all X, Y ∈ X(M ). The connection ∇ is said to be torsion-free if T (X, Y ) = 0 for all X, Y ∈ X(M ) . Torsion-free Bott connections will play an essential role in our theory. One way of constructing torsion-free Bott connections is via a choice of Riemannian metric g on M . Such a choice gives rise to three canonical objects: 1. an identification of νF with the orthogonal complement of T F, 2. a Euclidean structure ǫ induced by g on νF ⊂ TM , defining a section M → Fr(νF)/ O(q), and 3. a torsion-free Bott connection, the Bott Levi-Civita connection ∇, defined by ∇ X Y := [X F , Y ] ν + (∇ LC Xν Y ) ν , where ∇ LC is the Levi-Civita connection associated to g. These geometric structures are used by Guelorget in the definition of an alternative characteristic map to that of Bott [23], to which our own map bears substantial resemblance. Moreover, as will be clear in Section 5, the geometric structures arising from appropriately singular Riemannian metrics can be used in our Chern-Weil map for Haefliger-singular foliations. In Section 4, we will describe how Bott's Chern-Weil map for a regular foliation relates to the higher transverse frame bundles of the foliation. This will furnish a new proof of the fact that Bott's Chern-Weil map recovers the characteristic classes arising from the classifying space BΓ q of codimension q Haefliger structures. Following this, in Section 5, we will use the same techniques to give our Chern-Weil characteristic map for singular Haefliger foliations. Classifying spaces and the universal characteristic map In this section, following a review of some well-known definitions and constructions concerning classifying spaces, we prove a generalisation of Haefliger's classifying theorem [25,Theorem 7] to the smooth setting. Our proof factors through Mostow's smooth structure for classifying spaces [42], which we identify with a diffeology that we call the Mostow diffeology. It is via diffeological methods that we are able to view our classifying theorem as an honest generalisation of that of Haefliger, and by which we can give a neat, explicit, de Rham-theoretic, universal characteristic map from Gel'fand-Fuks cohomology to the cohomology of the smooth Haefliger classifying space. To the best of our knowledge, ours is the most explicit (and the only smooth) construction of a universal characteristic map available in the literature. Ultimately, this de Rham characteristic map is necessary to facilitate a clean identification of our Chern-Weil classes with those coming from the classifying space. Classifying spaces We begin by briefly recalling the framework of diffeology, referring to the book [31] for details. A diffeological space is a set X with a smooth structure, determined by a collection of plots U → X, plots P of X. Unless otherwise stated, diffeological spaces will always be assumed to carry the D-topology. Conversely, any topological space carries a natural diffeology, called the continuous diffeology, whose plots are the continuous maps. A map f : X 1 → X 2 of diffeological spaces is smooth if f • P is a plot of X 2 for each plot P of X 1 . By definition, smooth maps between diffeological spaces are continuous with respect to the underlying D-topologies. Given a subset U ⊂ X 1 , a map f : U → X 2 is locally smooth if P −1 (U ) is open and the composition P −1 (U ) → U → X 2 is a plot of X 2 , for all plots P of X 1 . To check that f : U → X 2 is locally smooth, it suffices to check first that U is D-open, and then that f sends the plots of X 1 with image in U to plots of X 2 . Subsets and quotients of diffeological spaces carry natural diffeologies (the subspace and quotient diffeologies), and the category of diffeological spaces and smooth maps is both complete and cocomplete with respect to limits, and contains the category of manifolds and smooth maps as a full, faithful subcategory. Finally, any diffeological space X has an associated de Rham complex (Ω * (X), d), which is contravariantly functorial in the manner familiar from manifold theory, and which coincides with the usual de Rham complex when X is a manifold. Having discussed diffeology, recall now that a groupoid is a small category with inverses. The range (or target) and source maps of a groupoid will always be denoted r and s respectively. A groupoid is said to be diffeological if its morphism set is a diffeological space, with object set equipped, via the unit map, with the subspace diffeology, for which the range, source, TheČech groupoidǓ of U is the groupoid whose morphism space is the disjoint unioň U (1) := (α,β)∈A 2 U α ∩ U β . An element ofǓ is an ordered triple (x, α, β) with x ∈ U α ∩ U β , so that the source and target maps are defined by the rules s(x, α, β) = (x, β, β) and r(x, α, β) = (x, α, α) respectively. In particular, the unit spaceǓ is the disjoint union of the degenerate intersections U α ∩ U α . Composition is given by (x, α, β) · (x, β, δ) := (x, α, δ). (Note the convention that morphisms act right to left.) As a disjoint union of diffeological spaces,Ǔ inherits a canonical diffeology (the sum diffeology [31, Section 1.39]), with respect to which it is anétale diffeological groupoid. Example 3.2 (Haefliger groupoid). The Haefliger groupoid Γ q is the groupoid of germs of local diffeomorphisms of R q [24]. For γ ∈ Γ (1) q with source x ∈ R q , representative local diffeomorphism g : dom(g) → R q and any open neighbourhood U of x contained in dom(g), define an open neighbourhood of γ by N (γ, g, U ) := {germ x ′ (g) : x ′ ∈ U }. The collection of these for all (γ, g, U ) determine the basis for a topology on Γ q . This topology induces the standard topology on the unit space R q , with respect to which the range and source are local homeomorphisms. As such, the manifold structure of R q induces a (non-Hausdorff) manifold structure on Γ q , which is thus anétale diffeological groupoid. The following definition is the diffeological analogue of Haefliger's [25, Definition 2]. For the rest of the paper, we will freely use the abbreviation h αβ for the restriction h| Ultimately we will show that certain groupoid structures on diffeological spaces are classified by smooth homotopy classes of maps into some diffeological classifying space associated to the groupoid. These classifying spaces will be constructed semi-simplicially. A semi-simplicial object in a category C is a sequence X • = {X (n) } n∈N of objects in C together with morphisms ∂ i : X (n+1) → X (n) defined for i = 0, . . . , n + 1, called face maps, satisfying the relations ∂ i • ∂ j = ∂ j−1 • ∂ i , for i < j. A morphism of semi-simplicial objects X • and Y • in C is a natural transformation, a family φ • = {φ (n) : X (n) → Y (n) } of morphisms that commute with respective face maps. Moreover, if C is a Cartesian category, and X • and Y • are two semi-simplicial objects in C, then their product X • × Y • is also a semi-simplicial object in C, with face maps ∂ X×Y i := ∂ X i × ∂ Y i . The following examples are key for the classifying space construction. Example 3.4. Any diffeological groupoid Γ has a natural associated semi-simplicial space Γ • , its nerve, with Γ (n) the subspace of composable n-tuples in (Γ (1) ) n and the face maps ∂ i : Γ (n+1) → Γ (n) defined by the formulae ∂ i (γ 1 , . . . , γ n+1 ) :=          (γ 2 , . . . , γ n+1 ) if i = 0, (γ 1 , . . . , γ i γ i+1 , . . . , γ n+1 ) if 1 ≤ i ≤ n (γ 1 , . . . , γ n ) if i = n + 1 for n ≥ 1. The zero-tuples Γ (0) are the objects (identity morphisms) of Γ, and the maps ∂ 0 , ∂ 1 : Γ (1) → Γ (0) are the source and range maps respectively. Any morphism of groupoids induces an obvious morphism of their nerves. In particular, the map h :Ǔ → Γ defining a smooth Γ-cocycle determines, by a mild abuse of notation, a map h : Ǔ • → Γ • q of semi-simplicial diffeological spaces. Example 3.5. Consider the natural numbers N (taken to include 0), equipped with their zerodimensional manifold structure. Define a semi-simplicial space N • by setting N (k) to be the set of all strictly increasing k + 1-tuples α 0 < · · · < α k of natural numbers. The face maps ∂ j : N (k) → N (k−1) are given by ommission: (α 0 , . . . , α k ) → (α 0 , . . . ,α j , . . . , α k ). Associated to any semi-simplicial diffeological space X • is a diffeological space, the fat realisation X • . For each n ∈ N, let ∆ n denote the standard n-simplex, thought of as a diffeological subspace of Euclidean space. Let (t 0 , . . . , t n ) denote the barycentric coordinates on ∆ n , and denote by d i : ∆ n → ∆ n+1 the i th face inclusion (which inserts a 0 at position i). The fat realisation is the quotient X • := n∈N ∆ n × X (n) / ∼ by the relation (d i ( t), x) ∼ ( t, ∂ i (x)) for ( t, x) ∈ ∆ n × X (n+1) , n ≥ 0 and i = 0, . . . , n + 1. The fat realisation may be equipped with the quotient diffeology, but for our purposes it is necessary to consider a coarser (that is, larger [31, Section 1.18]) diffeology, as we now describe, following Mostow [42]. Let Γ be a diffeological groupoid, and consider the fat realisation N • ×Γ (•) as a set. For each α ∈ N there is a natural barycentric coordinate map u α : N • ×Γ (•) → [0, 1], given by the (well-defined) quotient of the maps ∆ n × N n ×Γ (n) ∋ (t 0 , . . . , t n ; α 0 , . . . , α n ; γ 1 , . . . , γ n ) −−−→    t j if α = α j 0 otherwise. The sets U α := u −1 α (0, 1] define a cover of N • ×Γ (•) (which will be an open cover in the topology to follow). Given a composable tuple (γ 1 , . . . , γ n ) ∈ Γ (n) , define elements of Γ by g 0 := r(γ 1 ) and g i := γ 1 · · · γ i for i = 1, . . . n. Then there are functions γ αβ : U α ∩ U β → Γ given by γ αβ ([t 0 , . . . , t n ; α 0 , . . . , α n ; γ 1 , . . . , γ n ]) := g −1 i g j where α i = α and α j = β. Definition 3.6. Let Γ be a diffeological groupoid. The classifying space BΓ of Γ is the diffeological space whose underlying set is N • ×Γ (•) , equipped with the Mostow diffeology, which is the largest diffeology for which the maps u α : To be explicit, P : U → BΓ is a plot of BΓ whenever the u α • P are smooth, the preimages N • ×Γ (•) ) → [P −1 (U α ∩ U β ) are open, and the compositions P −1 (U α ∩ U β ) P − → BΓ γ αβ − − → Γ are plots of Γ. Note that while Mostow never invokes diffeology explicitly in [42], the Mostow diffeology defined here is implicit in his constructions, via the definition of differential forms [42, Section 2]. It follows immediately from the definitions that classifying spaces are functorial: for any morphism φ : Γ 1 → Γ 2 of diffeological groupoids, there is a smooth map Bφ : BΓ 1 → BΓ 2 , which furthermore preserves the canonical cocycles: u 2 α • Bφ = u 1 α and φ • γ 1 αβ = γ 2 αβ • Bφ, with u i α and γ i αβ the maps defining the canonical cocycle of BΓ i . Just as in the topological setting, Definition 3.6 is a special case of a more general construction, which assigns to any semi-simplicial diffeological X • its unwound geometric realisation µ(X • ) (cf. [44,49,42]). In [25], Haefliger uses Milnor's infinite join construction to construct the classifying space BΓ for any topological groupoid Γ, so that homotopy classes of continuous, numerable Γ-structures on a topological space X are in bijective correspondence with homotopy classes of continuous maps X → BΓ. Haefliger's construction can easily be identified with the topological version of Definition 3.6 [42, p. 278], provided one equips BΓ with the so-called strong topology, which is the weakest topology making the maps u α and γ αβ continuous. For a diffeological groupoid Γ, the strong topology on BΓ is coarser than the D-topology induced by the Mostow diffeology, since smooth maps are always continuous with respect to the D-topology. As such, any smooth map of a diffeological X into BΓ is automatically continuous with respect to the D-topology on X and the strong topology on BΓ. theorem. An alternative diffeological approach to classifying spaces, which is insufficient for our purposes, is presented in [38]. Theorem 3.7. Let Γ be a diffeological groupoid. 1. The canonical Γ-structure γ on BΓ is smoothly numerable. 2. For any smoothly numerable Γ-structure h on a diffeological space X, there is a smooth map η : X → BΓ q such that h = η * γ. gives rise to a well-defined map η : X → BΓ. To see this, let us denote (λ α 0 (y), . . . , λ αn (y)) by λ α (y), and (h α 0 α 1 (y), . . . , h α n−1 αn (y)) by h α (y). Suppose then that W ′ were some other open neighbourhood of x intersecting supp λ α only for α contained in some other finite list α ′ := (α ′ 0 , . . . , α ′ m ), and that y ∈ W ∩ W ′ . Since both W and W ′ contain x, the intersection α ∩ α ′ = (α i 0 , . . . , α i k ) is nonempty, and there are sequences ∂ and ∂ ′ of composites of face maps on N • such that ∂ α = α ∩ α ′ = ∂ ′ α ′ . Moreover, only those functions λ α for which α ∈ α ∩ α ′ will take nonzero values at y, so that λ α (y) = d( λ α∩ α ′ (y)) and λ α ′ (y) = d ′ ( λ α∩ α ′ (y)), where d and d ′ denote the composites of face maps for the standard simplices corresponding to ∂ and ∂ ′ respectively, with order of composition reversed. Therefore, by definition of the equivalence relation defining BΓ, we then have d( λ α∩ α ′ (y)); α; h α (y) ∼ λ α∩ α ′ (y); ∂( α; h α (y)) and λ α∩ α ′ (y); ∂ ′ ( α ′ ; h α ′ (y)) ∼ d ′ ( λ α∩ α ′ (y)); α ′ ; h α ′ (y) so that λ α (y); α; h α (y) ∼ λ α ′ (y); α ′ ; h α ′ (y) as claimed. To see that η is smooth, let P : U → X be a plot. For u ∈ U , choose an open neighbourhood W of P (u) in X such that W intersects only those sets supp λ α for α contained in some sequence α. Then W := P −1 (W ) is open, and for any α ∈ N the composite u α • η • P | W =    λ α i • P if α = α i ∈ α 0 otherwise is smooth. It follows that u α •η •P is smooth, hence that η is smooth with respect to the largest diffeology for which the u α are smooth. Therefore for any pair α, β ∈ N, η −1 (U α ∩ U β ) is open in X, and to complete the proof that η is smooth with respect to the Mostow diffeology on BΓ, it remains only to show that γ αβ • η • P | P −1 η −1 (Uα∩U β ) is a plot for Γ. If P −1 η −1 (U α ∩ U β ) is empty then we are done. Otherwise, for u ∈ P −1 η −1 (U α ∩ U β ), let us again take W to be an open neighbourhood of P (u) which intersects only those supp λ α ′ for which α ′ ∈ α. We then have α = α i ∈ α and β = α j ∈ α, and setting W : = P −1 (W ∩ η −1 (U α ∩ U β )), the composite γ αβ • η • P | W = (h α 0 α 1 • P ) · · · (h α i−1 α i • P ) −1 (h α 0 ,α 1 • P ) · · · (h α j−1 ,α j • P ) is smooth by the smoothness of h, and therefore γ αβ • η • P | P −1 η −1 (Uα∩U β ) is smooth. It follows that η : X → BΓ is smooth. To complete the proof of the second item, one need only show that h and η * γ are cocyclic. This follows from the definition of η. (3) Suppose that η 0 , η 1 : X → BΓ are smooth maps. It is clear that if η 0 and η 1 are smoothly homotopic then η * 0 γ and η * 1 γ are smoothly, numerably homotopic. Suppose conversely that η * 0 γ and η * 1 γ are smoothly, numerably homotopic through a Γ structure on X × [0, 1] which by item (2) may be assumed to be of the form η * γ for some smooth map η : X × [0, 1] → BΓ. We then need only prove that η i is smoothly homotopic to η| X×{i} . Thus it suffices to prove item (3) assuming that η * 0 γ = η * 1 γ. Our argument is inspired by those found in [30, p.57-58] and in [38,Proposition 3.16], however it is sufficiently different from both that we feel obliged to spell it out in some detail for the reader. Let t i (y, s) :=    (1 − s) t 0 j (y) if α i = α 0 j s t 1 j (y) if α i = α 1 j . Then the formula η(y, s) = [t 0 (y, s), . . . , t n (y, s); α 0 , . . . , α n ; h α 0 α 1 (y), . . . , h α n−1 αn (y)], y ∈ W. defines a homotopy η : X × [0, 1] → BΓ between η 0 and η 1 over W . Universal characteristic map via Haefliger bundles In this section we define the Haefliger bundles of a smoothly numerable Haefliger structure on a diffeological space, which are natural, locally trivial 1 principal G k q -bundles on the space. Although previously identified (at least for regular foliations [41, Section 3.4.1]), Haefliger bundles appear to have been underutilised in the literature. As we will see in the next section, the Haefliger bundles of a regular Haefliger structure are isomorphic to the transverse frame bundles of the associated regular foliation. However, while transverse frame bundles do not make sense over singularities, Haefliger bundles are defined globally even in the singular case. Consider a smoothly numerable Haefliger cocycle h :Ǔ → Γ q on a diffeological space X. For k ∈ N ∪{∞}, the Haefliger k-frame bundle of h, or simply the k-Haefliger bundle, is the smooth quotient Fr k (h) := α∈N (s • h αα ) * Fr k (R q ) ∼,(13) the equivalence relation ∼ given by (x, α, ϕ) ∼ (x, β, h βα (x) · ϕ) for all x ∈ U α ∩ U β . Note moreover that by the Diff(R q )-invariance of the tautological forms ω k on Fr k (R q ), the pullbacks (s • h αα ) * ω k glue to give an a k q -valued 1-form ω k h on Fr k (h). We have the following result. Proposition 3.8. Let X be a diffeological space and let k ∈ N ∪{∞}. If U is an open cover of X and h :Ǔ → Γ q represents a smoothly numerable Haefliger structure on X, then Fr k (h) is a smooth, principal G k q -bundle over X, locally trivial over U . 2. If η : X → BΓ q is a smooth map, then Fr k (η * γ) and η * Fr k (γ) are canonically isomorphic as principal G k q -bundles over X, and under this isomorphism η * ω k γ identifies with ω k η * γ . 3. If h 0 and h 1 are smoothly, numerably homotopic Haefliger structures on X, then Fr k (h 0 ) and Fr k (h 1 ) are isomorphic as principal G k q -bundles over X. As a consequence, isomorphism classes of Haefliger bundles are functorial for diagrams X BΓ q Y of smooth maps that commute up to homotopy. Proof. Item (1) follows from the triviality of Fr k (R q ) → R q , while item (2) The tautological form ω ∞ h on the Haefliger bundle Fr ∞ (h) → X associated to a smoothly numerable Haefliger structure on a diffeological space X induces a homomorphism A * (a q ) O(q) → Ω * (Fr ∞ (h)/ O(q)). Thus, given a smooth section X → Fr ∞ (h)/ O(q), one obtains a homomorphism A * (a q ) O(q) → Ω * (X) of DGAs. Our next theorem shows that, despite the potentially pathological topology of X, such sections are in abundance, and their induced maps on cohomology unambiguous. Proof. The theorem is essentially a consequence of the fact that the fibre G ∞ q / O(q) of the bundle is contractible. When taken with the strong topology, BΓ q has the homotopy type of a CW complex [24, Theorem 1.6], and the existence of sections is then classical. A full, constructive proof in our diffeological case is given in Appendix C. By the homotopy-invariance of de Rham cohomology, Theorem 3.9 enables the following definition of de Rham characteristic maps for smoothly numerable Haefliger structures. Definition 3.10. If h is a smoothly numerable Haefliger structure on a diffeological space X, the characteristic map associated to h is the homomorphism H * (A(a q ); O(q)) → H * (Ω(X)) induced by the composite A * (a q ) O(q) ω ∞ h − − → Ω * (Fr ∞ (h)/ O(q)) σ * − → Ω * (X), where σ : X → Fr ∞ (h)/ O(q) is any smooth section. In particular, the characteristic map u : H * (A(a q ); O(q)) → H * (Ω(BΓ q )) associated to the canonical Haefliger structure γ on BΓ q is called the universal characteristic map. It follows easily from Proposition 3.8 that characteristic maps for smoothly numerable Haefliger structures are functorial under smooth maps. In particular if h = η * γ is the smoothly numerable Haefliger structure defined by a smooth map η : X → BΓ q , then the characteristic map associated to h is equal to η * • u : H * (A(a q ); O(q)) → H * (Ω(BΓ q ))) → H * (Ω(X)). Our final result in this section is a corollary of Theorem 3.9, which describes the characteristic map in terms of the quasi-isomorphic, finite-dimensional subalgebra W O q of A * (a q ) O(q) , and will be the means by which we relate our Chern-Weil theory to classifying space theory. Corollary 3.11. Let X be a diffeological space and h = η * γ a smoothly numerable Haefliger structure on X. If σ : X → Fr 2 (h)/ O(q) is any smooth section, then the map induced on cohomology by the homomorphism W O q ω 2 h −→ Ω * (Fr 2 (h)/ O(q)) σ * − → Ω * (X) of DGAs is equal to the characteristic map η * • u : H * (W O q ) → H * (Ω(X)). We now turn to describing how sections of Fr 2 (h)/ O(q) → X may be induced via geometric structures (connections and curvatures). We first consider the regular case, and then the singular case. Regular foliations and Haefliger bundles We begin by establishing notation. Given a regular foliation (M, F ) of codimension q, we will assume it to be defined by a smooth map η F : M → BΓ q . Abusing notation, we will assume that the pullback h F := η * F γ of the canonical Haefliger cocycle γ on BΓ q is defined over an open cover {U α } α∈N of M , with submersive Haefliger charts f α := s • γ αα • η F : U α → R q . The following will allow us to give a new proof of Bott's result that the Chern-Weil map for (M, F ) recovers the characteristic classes arising from the map η F : M → BΓ q . Ultimately, it is this approach that will allow us to extend Chern-Weil theory to singular foliations. Proof. The local maps i k,α : Fr k (M/ F )| Uα → Fr k (h F )| Uα defined by i k,α j k 0,t (u) → (u(0), α, j k 0 (f α • u)) ∼ glue to give the isomorphism i k . Indeed, since each f α is a submersion and each u a frame, the composition f α • u is a local diffeomorphism of R q . The maps i k,α are manifestly equivariant for the respective right actions of G k q , and on each overlap U α ∩ U β , one has x, α, j k 0 (f α • u) = x, α, j k 0 (h x αβ • f β • u) ∼ x, β, j k 0 (f β • u) for each j k 0,t (u) ∈ Fr k (M/ F )| Uα∩U β , so that i k,α (x, j k 0,t (u)) = i k,β (x, j k 0,t (u)). Note that Fr k (h F ) has a natural bundle foliation: the trivial bundle Fr k (R q ) ∼ = R q ×G k q has horizontal foliation from constant sections, and each chart (U α , f α ) of h F pulls this foliation back to a bundle foliation of Fr k (h F ) over U α , compatible on overlaps. The well-known fact that 3] to show that T F 1 ⊂ ker(σ * ω 2 h ), with F 1 the natural bundle foliation on Fr 1 (M/ F ). Suppose therefore that ε : t → ε(t) ∈ Fr 1 (M/ F ) is a smooth path contained in a leaf of F 1 , whose projection to M is contained in the domain U α of a Haefliger chart map f α . It follows that Df α (ε(t)) is constant in t. Representing σ(ε(t)) by a transversal embedding u t : R q → M , we compute σ * i * 2 ω 2 h d dt 0 ε(t) =ω 2 h d dt 0 j 2 0 (f α • u t ) = d dt 0 D(f α • u 0 ) −1 fα(u 0 ( 0)) D(f α • u t ) 0 = d dt 0 D(f α • u 0 ) −1 fα(u 0 ( 0)) Df α (ε(t)) = 0, yielding T F 1 ⊂ ker(σ * i * 2 ω 2 h ) as claimed. Conversely, let ∇ ν be a torsion-free Bott connection on νF. The aim is to construct from ∇ ν an equivariant lift σ ∇ ν : Fr 1 (M/ F ) → Fr 2 (M/ F ), which will be done via local construction and then shown to be independent of choices made. To this end, fix index sets i, j, k, . . . We call any such local framing an adapted lift of ε, and remark that the vector fields e i have horizontally parallel projection, in the sense that To be precise, the connection ∇ is given by ∇ X Y = ∇ X ι(v) + w := ι∇ ν X v + ∇ F X w for any vectors X, Y ∈ X and the associated splitting Y = ι(v) + w into image-of-ι and T F components. All this granted, we define the (clearly equivariant) lift σ ∇ ν : Fr 1 (M/ F ) → Fr 2 (M/ F ) by the rule that to a frame ε of νF at x assigns the transverse 2-jet of the exponential through any adapted lift ( e, f ) of ε: σ ∇ ν (ε) := j 2 0,t s → exp ∇ ( s · e) with argument s ∈ R q .(14) It γ A (t) = −Γ A BC (γ(t))γ B (t)γ C (t) for t in the domain of γ. But for any geodesic through x and tangent to ι(νF) at x, this reduces toγ A (0) = −Γ A jk (x)γ j (0)γ k (0), and the normal component of this reduces further tö γ i (0) = −Γ i jk (x)γ j (0)γ k (0). The point here is that only 'normal' indices i, j, k appear, but these particular components of the Christoffel symbol depend only on the Bott connection. So, the normal component of the two-jet of the exponential map is independent of choice of ∇ F . Now, suppose given two different choices of foliation coordinates lifting a fixed frame ε of νF x , with associated adapted lifts ( e, f ) and ( e ′ , f ′ ) near x ∈ M . Denoting all data associated to the 'primed' lift with a prime, note that there exists a matrix function A a i so that e ′ i = e i +A a i f a . Furthermore, we have e i = ι(p(e i )) and e ′ i = ι ′ (p(e ′ i )) and that p( e i ) x = ǫ i = p(e ′ i ) x . Computing, at x we have ∇ ′ e ′ i e ′ j = ι ′ ∇ ν e ′ i p(e ′ j ) = ι ′ ∇ ν e i p(e j ) + ι ′ ∇ ν A a i fa p(e j ) = ι ′ ∇ ν e i p(e j ), while ∇ e i e j = ι∇ ν e i p(e j ). These have equal projection to the normal bundle, and it follows that the normal component of the 'bare' and 'primed' Christoffel symbols at x does not depend on the choice of foliations coordinates made. These determine the transverse two jet of the exponential map (14), which is seen to be independent of choices, as required. With of the form a(i * 2 ω 2 h F ) ∈ Ω * (Fr 2 (M/ F )/ O(q)). But this latter is defined by evaluation of the Gel'fand-Fuks cocycle associated to a ∈ W O q ⊂ A(a q ) O(q) on the a q -valued form i * ∞ ω ∞ h F on Fr ∞ (M/ F ). Since ω ∞ satisfies the Maurer-Cartan identity, a → a(i * ∞ ω ∞ h F ) is a homomorphism of DGAs (cf. Equation (8)). To see that the diagram commutes, recall that the universal characteristic map u is induced by the cochain map W O q A * (a q ) O(q) Ω * (Fr ∞ (γ)/ O(q)) Ω * (BΓ q ),Fr 2 (M/ F )/ O(q) Fr 2 (h F )/ O(q) η * F Fr 2 (γ)/ O(q) Fr 2 (γ)/ O(q) M M M BΓ q i 2 ∼ = ∼ = σ ∇ σǫ id id η F σ Chern-Weil for Haefliger-singular foliations In this section we extend the ideas developed in the previous section to yield a Chern-Weil map for Haefliger-singular foliations admitting adapted geometries, which are pairs consisting of a Bott connection and Euclidean structure over the normal bundle of the regular subfoliation. For such a pair to be adapted means essentially that the Euclidean structure blows up appropriately towards singularities. As an immediate geometric application, we describe how the classical algorithm for the construction of the Godbillon-Vey class, using differential forms, may be extended to the Haefliger-singular setting. As a final application of our theory, we describe how any smooth Haefliger structure on a manifold is homotopic to a Haefliger-singular foliation. This is fortuitous, because all Haefligersingular foliations admit adapted geometries. In this way, the theory realises a Chern-Weil map for all homotopy classes of smooth Haefliger structures. Furthermore, the theory is fully functorial (up to homotopy) for maps between Haefliger structures, because any map into a Haefliger-singular foliation may be slightly perturbed to a map pulling back a Haefliger-singular foliation. Extending the notational convention of the previous section, assume any Haefliger foliation (M, F ) of codimension q to be associated to a smooth map η F : M → BΓ q , with accompanying Haefliger cocycle h F := η * F γ. Adapted geometries and the Chern-Weil map Chern-Weil theory provides for the construction of explicit forms that represent the characteristic classes of regular foliations, taking as input certain geometric data-a Bott connection and a Euclidean structure on the normal bundle. The results of the previous section allow the same construction on Haefliger-singular foliations, provided one allows geometric data that are singular but appropriately adapted. It will follow from functoriality, to be discussed in the next subsection, that all Haefligersingular foliations admit adapted geometries (Corollary 5.7). We first describe how an adapted geometry allows for the construction of a Chern-Weil map. i ) = (i 2 σ ∇ σ ǫ ) * c i (ω 2 h F ) and λ ǫ,∇ (h i ) = (i 2 σ ∇ σ ǫ ) * h i (ω 2 h F ), where ω 2 h F is the tautological gl(q, R)-valued 1-form on Fr 2 (h F ). The extension of λ ǫ,∇ (c i ) and λ ǫ,∇ (h i ) to globally smooth forms then follows from the fact that (ǫ, ∇) is an adapted geometry. Commutativity of the diagram follows from essentially the same argument as in Theorem 4.3. Functoriality It is a well-established fact that regular foliations are functorial under transverse maps [12,13], and that the characteristic classes of regular foliations are similarly functorial. Here we extend this functoriality to smooth maps which are transverse only on a dense set. W O q Ω * (M ) f * λ ǫ,∇ λ f * ǫ,f * ∇(15) commutes. Proof. The pullback data f * ǫ and f * ∇ are associated to the pullbacks by f of the corresponding sections σ ǫ :Ñ → Fr 1 (Ñ / F )/ O(q) and σ ∇ : Fr 1 (Ñ / F ) → Fr 2 (Ñ / F ). Consider the commuting diagram, with σ the unique smooth section extending i 2 σ ∇ σ ǫ . N N Fr 2 (Ñ / F )/ O(q) Fr 2 (h F )/ O(q) Fr 2 (M / F )/ O(q) Fr 2 (h f * F )/ O(q) M M σ ∇ σǫ σ i 2 i 2 f σ f * ∇ σ f * ǫ f f * σ (16) Functoriality of the Haefliger bundle affords the identification Fr 2 (h f * F ) ∼ = f * Fr 2 (h F ), and thus a well defined, smooth pullback section f * σ. At any regular point x ∈M , one sees that i 2 σ ∇ σ ǫ (x) = f * σ(x) by chasing around the top of the diagram, andM is dense, so the section f * σ is the unique extension of i 2 σ f * ∇ σ f * ǫ . As such, (f * ǫ, f * ∇) is adapted. geometries. Moreover, in the primary application of our theory, we will show that the category of Haefliger-singular foliations with adapted geometries, whose morphisms are Haefligersingular maps pulling back the foliation and geometry of the codomain to that of the domain, is homotopy-equivalent to the category of all smooth Haefliger structures on manifolds. Before turning to this, we describe an immediate geometric application. Application: a singular Godbillon-Vey algorithm The classical Godbillon-Vey algorithm [22] for a regular, transversely orientable foliation (M, F ) of codimension 1 proceeds by choosing a nonvanishing 1-form ω defining F . Then by the classical Frobenius theorem, there exists a 1-form η such that dω = η ∧ ω. The 3-form η ∧ dη is closed, and its class in de Rham cohomology, which is independent of the choices of η and ω, is the Godbillon-Vey class of the foliation. This algorithm is a key tool in the constructions of Roussarie [22] and Thurston [47] of foliations with nonvanishing Godbillon-Vey invariant. In the recent paper [37], an attempt is made to employ the Godbillon-Vey algorithm in the context of codimension 1 singular foliations, in an application to fluid mechanics. However, its use in this singular context requires more care. For instance, if a 1-form ω is singular in that is allowed to have zeroes, then it is no longer true that the identity ω ∧ dω = 0 suffices to guarantee integrability in the sense that ω is locally of the form f dg for some functions f and g [39]. More generally, integrability of singular q-forms is an extremely subtle problem [40], and the relationship between integrable singular q-forms and codimension q Haefliger structures (and therefore to characteristic classes) is far from clear. Our next theorem supplies a singular generalisation of the Godbillon-Vey algorithm using our Chern-Weil theorem. We will say that a Haefliger-singular foliation is transversely orientable if each connected component of its regular subfoliation is transversely orientable. Proof. Let h denote the Haefliger structure defining the foliation. We begin with a computation using the tautological forms ω k h on the Haefliger bundles. For notational convenience, denote the tautological R q -valued form ω 1 h on the Haefliger bundle Fr 2 (h) associated to h by the tuple (ω i ) q i=1 , and similarly denote the tautological gl(q, R)-valued form ω 2 h by the tuple (ω i j ) q i,j=1 . Denote byω the q-form ω 1 ∧ · · · ∧ ω q . Using the first structure equation of Equation (4) one has dω = q k=1 (−1) k+1 ω 1 ∧ · · · ∧ ∧ω k−1 ∧ dω k ∧ ω k+1 ∧ · · · ∧ ω q = q k=1 (−1) k ω 1 ∧ · · · ∧ ω k−1 ∧ ω k j ∧ ω j ∧ ω k+1 ∧ · · · ∧ ω q = q k=1 −ω k k ∧ω = − Tr(ω 2 h ) ∧ω. Being O(q)-basic, the formω descends to a form on Fr 2 (h)/ O(q), hence so too does Tr(ω 2 h ). Now, the section σ ǫ σ ∇ pulls back ω 1 ∧ · · · ∧ ω q to a smooth q-form ω on M which, since (M, F ) is transversely orientable, defines the regular subfoliation on each connected component. The calculation above shows, moreover, taking η to be the pullback of − Tr(ω 2 h ) along σ ǫ σ ∇ , that one has dω = η ∧ ω. Finally, by Theorem 5.2, we see that the Godbillon-Vey invariant of (M, F ) is represented by λ ǫ,∇ (h 1 c q 1 ) = Tr(ω 2 h ) ∧ d Tr(ω 2 h ) = (−1) q+1 η ∧ (dη) q as claimed. In the next and final subsection, we prove one of the main results of the paper -that smooth Application to Haefliger structures This section is devoted to proving the general applicability of our theory; specifically that the class of Haefliger-singular foliations is homotopically identical to that of smooth Haefliger structures on manifolds. The first result pertains to the graph of a Haefliger structure, which allows embeddings of an arbitrary Haefliger foliation into a regular foliation. The construction is due to Haefliger, that pull back to zero the q-form dy 1 ∧ · · · ∧dy q on R n ; this condition determines polynomial equations in the matrix coefficients, which can be written out explicitly if required. One might also compare the proof of Theorem 3.2.6 in [29]. The singular set of a map g : M → N is contained in g −1 (Σ ′ ) ∪ j 1 (g) −1 (V), with Σ ′ the singular set of N . As such, any map in the intersection of the set of maps g : M → N such that g −1 (Σ ′ ) has dense complement, and the set of maps g so that j 1 (g) is transverse to V, is a Write V = ∪W Haefliger-singular map. Both sets are residual, so their intersection is residual, so is dense, as required. Proof That the set A is residual follows from the more general following claim: given topological spaces X, Y such that Y is a metric space admitting countable dense sequence y n , and a subset D ⊂ X × Y , denote by D x := π Y ({x} × Y ) ∩ D ⊂ Y the x-slice of D in Y , and by X the set of points x ∈ X so that D x is dense in Y . If D is open and dense in X × Y , then X is residual in X. For each n, let B n be the ball of radius 1/n about y n , and let D Bn := {x ∈ X : ({x} × B n ) ∩ D = ∅} = {x ∈ X : B n ∩ D x = ∅}. Each X ′ = n∈N D Bn is contained in X by construction, which is to say that for each x ∈ X ′ , we have D x dense in Y . Indeed, fix x ∈ X ′ , and for any y ∈ Y , take a subsequence so that y n → y. There is for each n some y ′ n ∈ B n ∩ D x , so that (x, y ′ n ) ∈ D. The new sequence y ′ n also converges to y, so D x contains a sequence converging to y for all y ∈ Y , i.e. D x is dense in Y . Putting together the results of this subsection, we obtain one of the primary theorems of our paper, enabling the application of our Chern-Weil theorem to all smooth Haefliger structures on manifolds of sufficiently high dimension. Let F q sing denote the category whose objects are Haefliger-singular foliations of codimension q with adapted geometries, and whose morphisms are Haefliger-singular maps pulling back the foliation and geometry on the codomain to that on the domain. Let H q man denote the category whose objects are smooth Haefliger structures on manifolds of dimension at least q, and whose morphisms are smooth functions pulling back the Haefliger structure on the codomain to that on the domain. Discussion Our work opens up a number of directions for future research, which we briefly discuss now. Noncommutative geometry: The study of singular foliations has gained substantial traction in recent years, following the seminal construction of the holonomy groupoid of a Stefan-Sussmann singular foliation by I. Androulidakis and G. Skandalis [2] (as we show in Appendix A, all foliations we consider fall into this class). As is well-known, the Gel'fand-Fuks classes of a regular foliation play an important role in noncommutative geometry, where they define cyclic cocycles that pair with K-theory elements of the convolution algebra associated to the holonomy groupoid to yield numerical invariants [15]. We anticipate that our theory will facilitate the construction of analogous cocycles to pair with groupoid algebras of Haefliger-singular foliations, allowing deeper insight into the structure of the noncommutative leaf spaces of such objects. Residue formulae: Recall Bott's celebrated residue formula [7], which expresses characteristic numbers of a Riemannian manifold in terms of quantities localised to the zeros of a Killing field. This may have an analogy for singular foliations. Similar residue formulae for Chern forms associated to the normal bundle of (not necessarily Haefliger-) singular foliations have already appeared in the literature and aided the study of characteristic classes [6,27]. Motivated by this, we ask whether there exists a Haefliger-singular foliation whose singular set is an embedded submanifold, which admits adapted geometries that are preserved by leafwise vector fields vanishing along the singular set. By similar arguments to those in [7], the Godbillon-Vey invariant of such a foliation (thought of as a current) could be localised to the singular set in a residue formula, which may shed new light on the topological dynamics of singular foliations. Any distribution that is so spanned by a tangent subsheaf is called a smooth distribution. A Haefliger structures and Stefan-Sussmann singular foliations An integral submanifold of a smooth distribution is an imbedded submanifold ι : Σ → M (not necessarily homeomorphic onto its image) that through each image point has tangent plane equal to the distribution, dι(T x Σ) = ∆ ι(x) . A smooth distribution is integrable if it has an integral submanifold through each point; in this case, the maximal integral submanifolds are unique through each point, and determine a partition of M into imbedded submanifolds. Stefan [45] and Sussmann [46] independently and near simultaneously discovered the conditions under which a smooth distribution is integrable, the statement as follows. Theorem A.1 ([45], [46]). On a smooth manifold M , a smooth distribution ∆ (spanned by D) is integrable if and only if ∆ is invariant under the local flows of (local) vector fields in D. Notice that the requirement that sections of ∆ be closed under Lie bracket is implicit in the condition that ∆ be invariant under the local flows of vector fields valued in ∆. Now, let h :Ǔ → Γ q be a Haefliger cocycle over an open cover U = {U α } α∈A of a manifold M (Definition 3.3). We will denote by f α := s • h αα : U α → R q the corresponding Haefliger charts. There is a naturally associated differential ideal I on M , which is generated locally, in each Haefliger chart f α : U α → R q , by ω i α := f * α dx i (where the x i are standard coordinates on R q ). This ideal is clearly formally Frobenius, in the sense that it is generated algebraically by the 1-forms ω i α ; equivalently, for each i = 1, . . . , q, dω i α ≡ 0 mod ω j α , j = 1, . . . q. (In fact, dω i α = 0 identically, but it is the weaker displayed condition that is basis independent, and crucial.) There is associated to this ideal a sheaf of vector fields; for U open in M , D(U ) = {X ∈ X(U ) : ω(X) = 0 all ω ∈ I(U )} . That this sheaf is involutive follows from the fact that I is formally Frobenius, viz. ω([X, Y ]) = dω(X, Y ) − Xω(Y ) + Y ω(X) = 0 for any X, Y ∈ D(U ) and any ω ∈ I(U ). The sheaf D spans a distribution ∆ ⊂ T M , and per the Stefan-Sussmann Theorem, ∆ is seen to be intregrable once it is shown to be invariant under flows of elements of D. This is true, as we show now for completeness. let ω ′ = Φ * −t ω ∈ I; we find, for y near x, that ω y (W ) = (Φ * t ω ′ ) y (W ) = ω ′ Φt(y) (dΦ t W ) = ω ′ Φt(y) (V ) = 0. So, W is in D, its evaluation at x is in ∆ x , and we see that ∆ Φt(x) ⊆ dΦ t (∆ x η : X → BΓ such that h = η * γ. 3. If η 0 , η 1 : X → BΓ are continuous maps, then η * 0 γ and η * 1 γ are numerably homotopic if and only if η 0 and η 1 are homotopic. To see that Theorem B.1 follows from Theorem 3.7, equip the topological groupoid Γ with the continuous diffeology. Then the canonical partition of unity on the diffeological space BΓ is smooth, hence continuous with respect to the D-topology, hence continuous with respect to the strong topology, which is contained in the D-topology. That the second and third items of Haefliger's classifying theorem follow from the corresponding items of Theorem 3.7 can be seen immediately by the application of the following lemma, whose proof is an elementary consequence of the definitions. Suppose now that h is a numerable Γ-structure on a topological space X. Equipping X with the continuous diffeology, h is smoothly numerable, hence by Theorem 3.7 is associated to a smooth (hence strongly continuous) map η : X → BΓ such that h = η * γ. Conversely, by Lemma B.2, any strongly continuous map η : X → BΓ is automatically smooth when X is equipped with the continuous diffeology and BΓ with the Mostow diffeology, and Theorem 3.7 then applies to yield a corresponding continuous Γ-structure η * γ on X. Finally suppose that η 0 , η 1 : X → BΓ are strongly continuous maps. By Lemma B.2 they are then smooth for the continuous diffeology on X and the Mostow diffeology on BΓ, and Theorem 3.7 then applies to show that η * 0 γ and η * 1 γ are numerably homotopic if and only if η 0 and η 1 are homotopic. C Sections of Haefliger bundles mod O(q) Here we describe an explicit construction, inspired by Dupont [17], of a smooth section of the bundle Fr ∞ (γ)/ O(q) → BΓ q , from which the existence of analogous sections for all smoothly numerable Haefliger structures follows. Note that the existence of continuous sections follows from the fact that BΓ q (with the strong topology) has the homotopy type of a CW-complex together with contractibility of the fibre, but the existence of smooth sections in the diffeological setting requires additional argumentation. To this end, we show in this appendix how to construct a section σ of Fr ∞ (γ)/ O(q) from any section σ 0 of Fr ∞ (R q ) → R q . In fact, we will identify Fr ∞ (γ) with the classifying space BΓ q,∞ of a smooth groupoid Γ q,∞ , and then σ will given functorially as Bσ 0 . The construction will furthermore show that any two smooth sections of a Haefliger bundle mod O(q) are smoothly homotopic, so that their induced characteristic maps (Definition 3.10) coincide. Proposition C.1. For k ∈ N ∪{∞}, define Γ q,k to be the action groupoid Γ q ⋉ Fr k (R q ). Then BΓ q,k → BΓ q is a principal G k q -bundle over BΓ q that is canonically isomorphic to Fr k (γ) → BΓ q . Proof. The groupoid Γ q,k = Γ q ⋉ Fr k (R q ) = Γ q × s,π k Fr k (R q ) is equipped with the subspace diffeology of the product Γ q × Fr k (R q ), meaning that a parameterisation P : U → Γ q ⋉ Fr k (R q ) is a plot if and only if each of its component maps U → Γ q and U → Fr k (R q ) are plots. A smooth action of a diffeological group G on a diffeological space X is by definition principal if and only if the action map a : X × G ∋ (x, g) → (x, x · g) ∈ X × X is a diffeological induction [31,Section 8.11], meaning that a is injective (the action is free) and each parameterisation P : U → X × G is a plot if and only if a • P is. Since Fr k (R q ) → R q is a principal G k q -bundle, the action map a : Fr k (R q ) × G k q → Fr k (R q ) × Fr k (R q ) is an induction. It induces an action map a Γ : Γ q,k × G k q → Γ q,k × Γ q,k by the rule a Γ (γ, φ, g) = (γ, φ), (γ, a(φ, g)) , which is inductive because a is and by definition of the diffeology on Γ q,k . The map a Γ induces in turn an action map Ba Γ : BΓ q,k × G k q → BΓ q,k × BΓ q,k , given by the formula Ba Γ t; α; − −− → (γ, φ) , g := t; α; − −− → (γ, φ) , t; α; − −−−−−− → a Γ (γ, φ, g) . Here the composable tuple − −− → (γ, φ) = (γ 1 , φ 1 ), . . . , (γ n , φ n ) ∈ Γ (n) q,k is mapped to the composable tuple − −−−−−− → a Γ (γ, φ, g) = (γ 1 , a(φ 1 , g)), . . . , (γ n , a(φ n , g)) . The map a Γ preserves each open U α of the canonical cover of BΓ q,k , and is such that for any α, β ∈ N, the diagram (U α ∩ U β ) × G k q (U α ∩ U β ) × (U α ∩ U β ) Γ q,k × G k q Γ q,k × Γ q,k Ba Γ γ k αβ ×id γ k αβ ×γ k αβ a Γ commutes. Indeed, this follows because the right action of G k q on Fr k (R q ) commutes with the left action of Γ q . That Ba Γ is an induction then follows from the inductivity of a Γ by a diagram chase. Finally, we come to identifying BΓ q,k with Fr k (γ). For this, observe that for any α ∈ N we have a canonical, G k q -equivariant identification of U α ⊂ BΓ q,k with (s • γ αα ) * Fr k (R q ), defined by [ t; α; − −− → (γ, φ)] → [ t; α; γ], φ j ,(17) where α = α j ∈ α. Furthermore, if [ t; α; − −− → (γ, φ)] ∈ U α ∩ U β , where β = α k ∈ α, then one has φ j = γ αβ ([ t; α; γ]) · φ k . It follows that the local identifications of Equation (17) patch together to a global identification of BΓ q,k with Fr k (γ). Recall the natural identification of G 1 q with GL(q, R). Recall too the inclusions O(q) ֒−→ GL(q, R) ֒−→ G ∞ q , defined by taking the infinite jet at 0 of diffeomorphisms that fix the metric and linear structures of R q respectively. The construction of a section of Fr ∞ (γ)/ O(q) → BΓ q requires the following lemma, which defines a canonical 'exponential' path in G ∞ q / O(q) to each element from the identity equivalence class. For this we make use of the fact that, as a projective limit of manifolds, G ∞ q admits a natural tangent structure [1, Chapter 1]. Lemma C.2. There exists a G 1 q -equivariant diffeomorphism e : T O(q) G ∞ q / O(q) → G ∞ q / O(q). Proof. The natural projection π : G ∞ q → G 1 q of groups is split, so, letting N = ker(π), there is an identification of G ∞ q as the semidirect product G 1 q ⋉ N . Explicitly, the identification is given by G ∞ q ∋ g −−−→ π(g), gπ(g) −1 ∈ G 1 q ⋉ N. The right action of O(q) only disturbs the first factor, so induces a diffeomorphism G ∞ q / O(q) ∼ = G 1 q / O(q) × N . This diffeomorphism is equivariant for the left action of G 1 q , in that A[g] → (A[π(g)], Agπ(g) −1 A −1 ) for A ∈ G 1 q and [g] ∈ G ∞ q / O(q). This gives also the identification T O(q) G ∞ q / O(q) ∼ = (g 1 q / so(q)) ⊕ n. There is a canonical bi-O(q)-invariant, left-G 1 q -invariant Riemannian structure on G 1 q , which descends to a Riemannian structure on the homogeneous space G 1 q / O(q). The Riemannian exponential map for this metric induces a diffeomorphism exp R : T O(q) (G 1 q / O(q)) → G 1 q / O(q), which is equivariant for the left multiplication actions of G 1 q [28, Chapter VI, Theorem 1.1]. From [34,Proposition 13.4], the exponential map exp N : n → N is a global diffeomorphism, equivariant for the adjoint actions of G 1 q on domain and codomain. Combining these two facts, we have the diffeomorphism e = exp R × exp N : (g 1 q / so(q)) ⊕ n −−−→ G 1 q / O(q) × N, which is equivariant in each factor, hence equivariant. The associated bundle construction works internal to the diffeological category [31,Section 8.16]. In particular, given a principal G ∞ q -bundle Y → X, the quotient Y / O(q) is the associated G ∞ q / O(q)-bundle to the action of G ∞ q on G ∞ q / O(q). Define its vertical tangent bundle V (Y / O(q)) as the associated bundle Y × G ∞ q T (G ∞ q / O(q)). We have a commutative diagram, V (Y / O(q)) Y / O(q) Y / O(q) X p σ where the lower three arrows are bundle projections, and the top arrow is given, using the associated bundle construction on both sides, by the rule y, [g, v] = y · g, [id, g −1 * v] −−−→ y · g, [e(g −1 * v)] for y ∈ Y and [g, v] ∈ T (G ∞ q / O(q)) ∼ = G ∞ q / O(q) × T O(q) (G ∞ q / O(q)). The top arrow defines fibrewise diffeomorphisms, in that it maps the fibre over any y ∈ Y / O(q) diffeomorphically to the fibre over p(y). The next lemma follows immediately. Using the equivalence between Fr ∞ (γ) and BΓ q,∞ , we construct a section of Fr ∞ (γ)/ O(q) → BΓ q . This is done semi-simplicially, using Lemma C.3 in the construction of sections σ k : ∆ k × N (k) ×Γ (k) q → ∆ k × N (k) ×(Γ (k) q,∞ / O(q)) that satisfy (id ×∂ j ) • σ k • (d j × id) = (d j × id) • σ k−1 • (id ×∂ j ) j = 0, . . . , k,(18) guaranteeing the sections glue to a global section. We take inspiration from Dupont [17]. Now letting (t 0 , . . . , t k ) denote the barycentric coordinates on the standard simplex ∆ k , write s i := t i + · · · + t k for i = 1, . . . , k, and define σ k : ∆ k × N (k) ×Γ (k) q → ∆ k × N (k) ×(Γ (k) q,∞ / O(q)) by the formula σ k ( t; α; γ) := ( t; α;σ 1 k ( t; γ), . . . ,σ k k ( t; γ)), wherẽ σ i k ( t; γ) := γ −1 k · · · γ −1 i · g s 1 ,r(γ 1 ) γ 1 · g s 2 s 1 ,r(γ 2 ) γ 2 · · · · · g s k s k−1 ,r(γ k ) γ k · σ(s(γ k )) · · · . As in [17, p. 241], σ k may be assumed to be smooth, and a routine calculation shows that they satisfy the identities of Equation (18). As a consequence, the σ k glue to a smooth section Bσ of Fr ∞ (γ)/ O(q) ∼ = BΓ q,∞ / O(q) → BΓ q . We can now easily prove Theorem 3.9. According to the above construction, smooth sections of Fr ∞ (γ)/ O(q) → BΓ q exist. Now let h be a smoothly numerable Haefliger structure on a diffeological space X, associated to a smooth map η : X → BΓ q . From the isomorphism on some open neighbourhood of x in U α ∩ U β , and 3. the transition functions satisfy the following cocycle condition: for allx ∈ U α ∩ U β ∩ U δ , one has h x αβ • h x βδ = h x αδ as germs.Two Haefliger cocycles are equivalent if there exists a third Haefliger cocycle refining them, and a Haefliger structure is an equivalence class of Haefliger cocycles. Theorem 2. 1 . 1[11, Theorem 2] The algebra inclusion W O q ֒→ A * (a q ) O(q) induces an isomorphism on cohomology, H * (W O q ) ∼ = H * (A(a q ), O(q)). Example 2 . 3 . 23If q = dim(M ), then F is a regular foliation of M by points, Haefliger charts are coordinate charts, and this definition recovers the standard definition of the k-frame bundle Fr k (M ) of M . In this case, the diffeomorphism group Diff(M ) acts from the left on Fr k (M ) by postcomposition, and this action commutes with the principal right action of G k q . Denote by Ω * (Fr k (M )) Diff(M ) the G k q -DGA of Diff(M )-invariant forms on Fr k (M ). Using Equation (3), the tautological a k−1 q -valued 1-form ω k defined by the formula 1 2 1[ω ∞ , ω ∞ ] = 0 [41, p. 113] (the identities in Equation (4) are low-order manifestations of this fact), has trivial kernel and defines a canonical trivialisation T Fr ∞ (M ) ∼ = Fr ∞ (M ) × a q of the tangent bundle of Fr ∞ (M ). As such, any c ∈ A * (a q ) naturally defines a form c(ω∧ · · · ∧ω) ∈ Ω * (Fr ∞ (M )) Diff(M ) , and this assignment gives rise to a canonical isomorphism A * (a q ) Ω * (Fr ∞ (M )) Diff(M ) ω 1 q -equivariant section σ ∇ of Fr ∞ (M ) → Fr 1 (M ). Recall now the functionals δ and ∆ introduced in Equation (5). Proposition 2.4. [16, Lemma 18] Let ∇ be an affine connection on a manifold M , and let σ ∇ : Fr 1 (M ) → Fr ∞ (M ) be the section given in Equation Such connections, now called Bott connections, are flat along leaves, in that their curvature forms R ∇ vanish on restriction to the tangent distribution of leaves. As a consequence, one has Bott's vanishing theorem: for any Bott connection ∇ on νF, the associated Chern-Weil characteristic map R[c 1 , . . . , c q ] → Ω * (M ) encoding the Pontryagin classes of νF, defined on generators by Theorem 2. 5 ( 5Bott-Chern-Weil characteristic map). [9, p. 67-69] The map λ ∇ 1 ,∇ 0 : W O q → Ω * (M ) defined on generators by the formulae as U ranges over the open subsets of finite dimensional Euclidean spaces. This collection of plots, called a diffeology, must satisfy three axioms: constant maps are plots, all maps that are locally given by plots are plots, and closure under precomposition by smooth functions between open subsets of Euclidean space. (The map from the empty set is vacuously locally a plot, so a plot). A diffeology on X determines a topology on X, the D-topology, which is the finest topology for which all plots are continuous-a subset U ⊂ X is D-open if and only if P −1 (U ) is open for all composition and inversion maps are all smooth maps. A diffeological groupoid isétale if the source and target maps are local homeomorphisms with respect to the D-topology. The most important examples for our purposes are the following. Example 3. 1 ( 1Čech groupoid). Let M be a diffeological space and U = {U α } α∈A an open cover of M . Definition 3. 3 . 3Let X be a diffeological space and Γ a diffeological groupoid. A Γ-cocycle on X defined over a D-open cover U of X is a smooth morphism h :Ǔ → Γ of diffeological groupoids. Two Γ-cocycles over open covers U and V are said to be equivalent if they are restrictions toǓ andV respectively of a smooth Γ-cocycle over U ∪ V. An equivalence class of Γ-cocycles is called a Γ-structure. Two Γ-structures are homotopic if there exists a Γ-structure on X × [0, 1] inducing the given structures on X ∼ = X × {0} and X ∼ = X × {1} respectively. In particular we call Γ q -structures Haefliger structures of codimension q. Uα∩U β of a Haefliger cocycle h to a single intersection U α ∩ U β . It is clear that Definition 1.1 can be reformulated as a special case of Definition 3.3 for the Haefliger groupoid Γ q . In particular, a Γ q -cocycle h :Ǔ → Γ q over an open cover U = {U α } α∈A of some manifold M induces Haefliger charts f α := s • h αα : U α → R q , and if these chart maps are submersions, one obtains a regular foliation of M . 0, 1 ] 1are smooth and the maps γ αβ : U α ∩ U β → Γ are locally smooth. With respect to the Mostow diffeology, U := {U α } α∈N is an open cover, the canonical open cover, and γ :Ǔ → Γ is a smooth Γ-cocycle, the canonical cocycle. The Haefliger structure determined by γ is the canonical Γ-structure on BΓ. 3 . 3If η 0 , η 1 : X → BΓ are smooth maps, then the Γ-structures η * 0 γ and η * 1 γ are smoothly, numerably homotopic if and only if η 0 and η 1 are smoothly homotopic. Proof. (1) We show first smooth numerability of the canonical cocycle on the classifying space B{e} of the trivial groupoid. Note that it follows from the definition that B{e} may be identified with the infinite simplex ∆ ∞ , which comprises the infinite sequences {t α } α∈N for which only finitely many elements are nonzero and for which α t α = 1. Under this identification, the barycentric coordinate maps t α : ∆ ∞ → [0, 1] are smooth maps, and the canonical open cover is given by {t −1 α (0, 1]} α∈N . The barycentric coordinates {t α } α∈N define a pointwise-finite, smooth partition of unity on ∆ ∞ . Mostow [42, p. 273] constructs from this a locally finite partition of unity {s α } α∈N that remains subordinate to the canonical open cover with respect to the strong topology (the weakest topology for which the t α are continuous). But the D-topology on ∆ ∞ is finer than the strong topology, so {s α } α∈N is also subordinate in the D-topology. To see the claim for general Γ, consider the smooth map u ∞ : BΓ → ∆ ∞ = B{e} obtained by applying the classifying space functor to Γ → {e}. It pulls back the canonical open cover of B{e} to the same on BΓ, so the pullback functions v α := s α • u ∞ define a smooth, locally finite partition of unity subordinate to the canonical open cover of BΓ. ( 2 ) 2Let h :V → Γ be a Γ-cocycle for X, defined over a countable open cover V = {V α } α∈N with smooth, subordinate, locally finite partition of unity {λ α } α∈N . For x ∈ X, choose a Dopen neighbourhood W of x such that W has nonempty intersection with supp λ α only for α contained in some finite list α := (α 0 , . . . , α n ) of indices. Then the formula η(y) := [(λ α 0 (y), . . . , λ αn (y); α 0 , . . . , α n ; h α 0 α 1 (y), . . . , h α n−1 αn (y))], y ∈ W BΓ 0 and BΓ 1 denote the subsets of BΓ consisting of tuples [ t, α; γ] for which α consists entirely of even or odd numbers respectively. Replacing the linear functions α n : I n := [1 − 2 −n , 1 − 2 −n−1 ] → [0, 1] of [30, p. 57] with smooth functions b n : I n → [0, 1] which are everywhere nondecreasing, and constant on a small neighbourhood of each endpoint, the arguments of [30, p.57-58] can be used to show that the maps h 1 : BΓ → BΓ and h 0 : BΓ → BΓ [t 0 , . . . , t k ; α 0 , . . . , α k ; γ 1 , . . . , γ k ] := [t 0 , . . . , t k ; 2α 0 + 1, . . . , 2α k + 1; γ 1 , . . . , γ k ], h 0 [t 0 , . . . , t k ; α 0 , . . . , α k ; γ 1 , . . . , γ k ] := [t 0 , . . . , t k ; 2α 0 , . . . , 2α k ; γ 1 , . . . , γ k ] are both smoothly homotopic to the identity. Hence we may replaceη i with h i • η i , which takes values in BΓ i . Now η * 0 γ is defined over the cover {V 2α := η −1 0 (U 2α )} α∈N while η * 1 γ is defined over the cover {V 2α+1 := η * 1 (U 2α+1 )} α∈N .By hypothesis η * 0 γ and η * 1 γ are cocyclic, and we denote by V = {V α } α∈N the corresponding cover and h :V → Γ the corresponding cocycle. Now fixx ∈ X. Since the maps η i are smooth, we can find an open neighbourhood W of x, indicesα 0 := (α 0 0 , . . . , α 0 n 0 ) and α 1 := (α 1 0 , . . . , α 1 n 1 ), and smooth functions {t i j i : W → (0, 1]} j i =0,...,n i such that η i (y) = [t i 0 (y), . . . , t i n i (y); α i 0 , . . . , α i n i ; y ∈ W , i = 0, 1. The indices α 0 and α 1 may be formed uniquely into a new list α ∈ N • of length n = n 0 + n 1 . For i = 0, . . . , n, define t i : W × [0, 1] → [0, 1] by Theorem 3. 9 . 9Let h be a smoothly numerable Haefliger structure on a diffeological space X. Then Fr ∞ (h)/ O(q) → X admits smooth sections, and any two such sections are smoothly homotopic. Proposition 4 . 1 . 41Let (M, F ) be a regular foliation of codimension q. For all 1 ≤ k ≤ ∞, there are canonical isomorphisms i k : Fr k (M/ F ) → Fr k (h F ) of principal G k q -bundles. These commute with the natural projection maps. the transverse frame bundles Fr k (M/ F ) → M of a regular foliation (M, F ) are foliated bundles then follows easily the isomorphism of Proposition 4.1. Moreover, Proposition 4.1 allows us to extend Proposition 2.4 to the transverse frame bundles of regular foliations. The next theorem is folklore, and elements of its proof appear in a less precise form in [36, Section 5.2]. We give a precise proof here, which makes the role of Haefliger bundles clear. Theorem 4 . 2 . 42Let (M, F ) be a regular foliation of codimension q. Torsion-free Bott connections on νF are in bijective correspondence with GL(q, R)-equivariant sections Fr 1 (M/ F ) → Fr 2 (M/ F ). Proof. The isomorphism i 2 : Fr 2 (M/ F ) → Fr 2 (h F ) pulls back the tautological matrix-valued form ω 2 h on Fr 2 (h F ) to a matrix-valued form i * 2 ω 2 h on Fr 2 (M/ F ). The pullback of this form by any GL(q, R)-equivariant section σ : Fr 1 (M/ F ) → Fr 2 (M/ F ) is then easily verified, by the properties of ω 2 h and the equivariance of σ, to define a torsion-free connection form on Fr 1 (M/ F ). To see that σ * i * 2 ω 2 h is a Bott connection, it suffices [35, Definition 2. = 1 , 1. . . , q and a, b, c, . . . = 1, . . . , dim(M ) − q, as well as A, B, C, . . . = 1, . . . , dim(M ). Recall the notation p : TM → νF for projection to the normal bundle. For a point of Fr 1 (M/ F ) over x-a frame ε = (ε 1 , . . . , ε q ) of νF x -there exist local foliation coordinates x i , y a about x so that p(∂/∂x i ) x = ε i . Denote by e i := ∂/∂x i , f a := ∂/∂y a the induced local framing of TM . i ) = [X, e i ] ν = 0 for all local vector fields X ∈ T F in the neighborhood on which they are defined. Now, choose a connection ∇ F on T F . Any adapted lift determines a local embedding ι : νF → TM that splits the projection TM → νF; equivalently, a choice of splitting TM ∼ = T F ⊕νF. As such, ∇ F plus an adapted lift determine an affine connection ∇ = ∇ ν ⊕ ∇ F on TM near x. remains only to check that the transverse component of this two-jet does not depend on the choice of adapted lift, nor the choice of tangential connection ∇ F . This follows by writing the geodesic equation in local foliation coordinates. First suppose that the foliation coordinates, and consequent adapted lift ( e, f ), are fixed. Relative e i , f a , the Christoffel symbols are given by ∇ e i e j = Γ k ij e k + Γ a ij f a , ∇ fa e j = Γ k aj e k + Γ b aj f b , . . . and the two other obvious permutations of the e i and f a . The exponential map is determined by geodesics; a path γ(t) through x ∈ M , with components γ = (γ A ) in the foliation coordinates, is geodesic if and only if it satisfies the equation Theorem 4.2 in hand, we can now reformulate the Chern-Weil theorem for regular foliations. Using this reformulation as a starting point, it will be relatively clear how to proceed in the singular case. To state the theorem, recall that if ǫ is a Euclidean structure on a vector bundle, then any connection ∇ on the bundle induces an ǫ-compatible connection ∇ ǫ , whose connection form on the orthogonal frame bundle is simply the antisymmetric component of the connection form of ∇. Theorem 4. 3 ( 3Chern-Weil theorem for regular foliations). Let (M, F ) be a regular foliation of codimension q, and suppose νF is equipped with a Euclidean structure ǫ and a torsion-free Bott connection ∇. Denote by ∇ ǫ the ǫ-compatible connection on νF induced by ∇, and fort ∈ [0, 1] denote by ∇ t := t∇ + (1 − t)∇ ǫ the affine combination on M × I. The map of algebras λ ǫ,∇ : W O q → Ω * (M ) defined on generators of W O q by λ ǫ,∇ (c i ) := Tr(R ∧i ∇ ) ∈ Ω 2i (M ) λ ǫ,∇ (h i ) ∈ Ω 2i−1 (M )is a DGA map. Descending to cohomology, the diagram H * (W O q ) H * (Ω(BΓ q )) Let σ ǫ : M → Fr 1 (M/ F )/ O(q) be the section defined by the Euclidean structure, locally determined by any choice of orthonormal framing. Per Theorem 4.2, the Bott connnection ∇ determines an equivariant section σ ∇ : Fr 1 (M/ F ) → Fr 2 (M/ F ), which thus descends to a section σ ∇ : Fr 1 (M/ F )/ O(q) → Fr 2 (M/ F )/ O(q). It is a routine calculation on generators to verify that for a ∈ W O q , the image λ ǫ,∇ (a) is the pullback by σ ∇ σ ǫ : M → Fr 2 (M/ F )/ O(q) some choice of smooth section σ of Fr ∞ (γ)/ O(q) → BΓ q . Using contractibility of the fibre of Fr 2 (M/ F )/ O(q) → M , commutativity of the diagram follows from the commutativity up to homotopy of the following diagram. Definition 5. 1 . 1Let (M, F ) be a Haefliger-singular foliation of codimension q. A pair (ǫ, ∇) consisting of a Euclidean metric ǫ and a torsion-free Bott connection ∇ on νF →M is an adapted geometry for (M, F ) if the composition i 2 σ ∇ σ ǫ :M → Fr 2 (h F )/ O(q) of the associated sections σ ǫ :M → Fr 1 (M / F )/ O(q) and σ ∇ : Fr 1 (M / F )/ O(q) → Fr 2 (M / F )/ O(q) with i 2 : Fr 2 (M / F )/ O(q) ֒→ Fr 2 (h F )/ O(q) extends smoothly to a section of Fr 2 (h F )/ O(q) → M . Theorem 5. 2 (F 2Chern-Weil for Haefliger-singular foliations). Let (M, F ) be a Haefliger-singular foliation of codimension q, equipped with an adapted geometry (ǫ, ∇). Denote by ∇ ǫ the ǫcompatible connection on νF induced by ∇, and by ∇ t := t∇ + (1 − t)∇ ǫ the affine-interpolating family of connections between ∇ and ∇ ǫ . Then the forms λ ǫ,∇ (c i ) := Tr(R ∧i ∇ ) ∈ Ω 2i (M ) λ ǫ,∇ (h i ) globally-defined smooth forms on M . Furthermore, the resulting homomorphism λ ǫ,∇ : W O q → Ω * (M ) of DGAs makes the following diagram commute. H * (W O q ) H * (Ω(BΓ q )) Proof. Let σ ǫ :M → Fr 1 (M / F )/ O(q) and σ ∇ : Fr 1 (M / F ) → Fr 2 (M / F ) be the sections associated to ǫ and ∇. Then λ ǫ,∇ (c Definition 5 . 3 . 53Let (N, F ′ ) be a Haefliger foliation, and f : M → N a smooth map. Say that f is regular at x ∈ M if for some (and so any) Haefliger chart h α : U α → R q with f (x) ∈ U α , it holds that h α f is a submersion at x. Say that f is Haefliger-singular if it is regular on a dense open set. It follows immediately from Definition 5.3 that Haefliger-singular foliations are functorial under pullbacks by Haefliger-singular maps. Similarly, one has functoriality of adapted geometries. Theorem 5.4. Let (N, F ) be a Haefliger-singular foliation of codimension q, and let f : M → (N, F ) be a Haefliger-singular map. Let f * F be the induced Haefliger-singular foliation of M . If (ǫ, ∇) is an adapted geometry for (N, F ′ ), then (f * ǫ, f * ∇) is an adapted geometry for (M, f * F), and the diagram Ω * (N ) The commutativity of Diagram(15) follows also from commutativity of Diagram(16), because the Chern-Weil morphism factors through the forms on the Haefliger bundle.Theorem 5.4 greatly enlarges the class of Haefliger-singular foliations admitting adapted geometries. Indeed, all geometries (ǫ, ∇) on a regular foliation (N, F ) are adapted, so every Haefliger-singular map f : M → (N, F ) induces an adapted geometry on (M, f * F). In the final subsection, we will show that in fact every Haefliger-singular foliation admits adapted Theorem 5 . 5 ( 55Singular Godbillon-Vey algorithm). Let (M, F ) be a transversely orientable, Haefliger-singular foliation of codimension q, and suppose that (ǫ, ∇) is an adapted geometry for (M, F ). Then there is a singular q-form ω on M for which: 1. ω|M is nonvanishing and defines the regular subfoliation, 2. dω = η ∧ ω for some smooth 1-form η on M , and 3. (−1) q+1 η ∧ (dη) q ∈ Ω 2q+1 (M ) is a closed form representing the Godbillon-Vey class of the foliation. Haefliger structures on manifolds of sufficiently high dimension are categorically homotopyequivalent to Haefliger-singular foliations. This result furnishes the primary application of our theory in giving a complete geometric solution to the problem of representing the characteristic classes of Haefliger structures. [ 24, p. 188], but we recall it for the convenience of the reader. Fix a representative Haefliger cocycle over some locally finite open cover {U α } α∈N of a manifold M , with Haefliger charts f α : U α → R q . Form an open manifold G(h) by gluing appropriate neighborhoods V α ofΓ(f α ) = {(x, f α (x)) : x ∈ U α } in U α × R qalong the change of chart maps associated to the cocycle. One obtains a regular foliation on G(h) by gluing the foliations given on each V α by the level sets of V α ֒→ U α × R q → R q . The graph of h is the map i : M → G(h) obtained by gluing the graphs U α → Γ(f α ), well-defined by construction of G(h). The following is easy from the construction.Proposition 5.6. Every Haefliger structure (M, h) on a manifold M admits a Haefliger embedding into a regular foliation, namely, G(h) with its regular foliation. In particular, h is the pullback of the regular foliation on G(h), and the embedding is regular wherever h is regular.Since regular foliations trivially admit adapted geometries, one has the following immediate corollary of Proposition 5.6 and Theorem 5.4.Corollary 5.7. Every Haefliger-singular foliation admits adapted geometries. Recall that a Haefliger-singular map M → (N, F ′ ) is one whose regular set is dense. A map M → N is Haefliger-singular if and only if the induced pullback foliation on M is Haefligersingular. Proposition 5.8. Given a codimension q Haefliger-singular foliation (N, F ′ ), every smooth map f : M → N from a manifold M of dimension at least q, is homotopic to a Haefligersingular map. Proof. Up to a small perturbation, we may assume f smooth. We will show the stronger statement that the subset of Haefliger-singular maps is dense (in fact, residual) in C ∞ (M, N ) equipped with the strong topology. Since C ∞ (M, N ) is locally path connected in this topology, there exists a continuous path to some Haefliger-singular map, which induces a homotopy by uncurrying. First suppose that the foliation on N is regular. Associated to (N, F ′ ) is a regular distribution D ⊂ T N , of codimension q. The argument proceeds through the Thom jet-transversality theorem, to show that the set of smooth maps M → N transverse to D on a dense subset of M is itself dense in C ∞ (M, N ). (Note that the integrability of D is not used in the following, and the argument works for any regular smooth distribution.) Fix m = dim M and n = dim N . In the 1-jet bundle J 1 (M, N ), let V denote the subset of J 1 (M, N ) comprising 1-jets that are not transverse to D. Although V is not generally a manifold, it is a finite union of manifolds, each of positive codimension. To see this, note that the bundle map J 1 (M, N ) → M × N restricts to a bundle map V → M × N , with fiber over each (x, y) ∈ M × N a certain variety V 0 , and V 0 is a finite union of manifolds, each of positive codimension in the fiber J 1 (M, N ) (x,y) ∼ = Hom(R m , R n ). More precisely, V 0 is, up to isomorphism, the variety of elements in Hom(R m , R n ) a for a finite collection of manifolds W a . Given a map g : M → N , the set of points where g is not transverse to D is exactly j 1 (g) −1 (V). But each W a has positive codimension, so if j 1 (g) is transverse to each W a , the set j 1 (g) −1 (V) is contained in a finite union of positive-codimension submanifolds, so has dense complement. On the other hand, byTheorem 3.2.8 of [29], the set of elements in C ∞ (M, N ) that have 1-jet lift transverse to W a is a residual set in C ∞ (M, N ). Since C ∞ (M, N ) is Baire ([29] Theorem 2.4.2), residual sets are dense. Now suppose (N, F ′ ) is Haefliger-singular, so that the regular setÑ is a dense open submanifold of N . Let D be the (regular) distribution overÑ , and V the set of 1-jets with target inÑ and not transverse to D. As in the regular case, V is a V 0 bundle overÑ , a finite union of positive-codimension submanifolds in J 1 (M, N ). Theorem 3.2.8 of [29] still applies, and the set of maps densely transverse to D remains residual in C ∞ (M, N ). On the other hand, the following holds, with proof deferred momentarily. Lemma 5.9. Given an open dense setÑ ⊂ N , let A be the set of smooth maps g : M → N such that g −1 (Ñ ) is dense in M . The set A is residual in C ∞ (M, N ). of Lemma 5.9. Consider the evaluation map ev : C ∞ (M, N ) × M → N . Since ev is continuous, open, and surjective, the pullback D = ev −1 (Ñ ) is open and dense in C ∞ (M, N ) × M . (The evaluation map is open because it is open on each product open U ×V , which is a union of U ×{x} over x in V . Each ev(U ×{x}) is already open, sufficing to take U basic in C ∞ (M, N ), say the set of all maps M → N whose graph lies in some open neighborhood W of the graph of a fixed g 0 : M → N . The projection of the set W ∩ ({x} × N ) is open in N , and equals ev(U ×{x}).) D Bn is dense in X: if not, there would be a nonempty open U ⊂ X so that U ∩ D Bn = ∅, and then U × B n would be an open nonempty set disjoint from D. The sets D Bn are open because D is open. The (residual) countable intersection Theorem 5 . 10 . 510The inclusion of categories F q sing ֒→ H q is a homotopy equivalence. Consequently, the characteristic map for all smooth Haefliger structures on manifolds of sufficiently high dimension is given functorially by the Chern-Weil map for Haefliger-singular foliations. Here we fill in a gap in the literature by showing that all smooth Haefliger structures on manifolds induce Stefan-Sussmann foliations. We begin by recalling the definition of a Stefan-Sussmann foliation. Start with a distribution on a smooth manifold M , here meaning some fiberwise linear subset ∆ ⊆ T M , so a choice of linear subspace ∆ x ⊆ T x M (of possibly varying dimension) for each point x ∈ M . Now, take the smooth functions C ∞ (M ) as a sheaf on M , and the vector fields X(M ) as a module over this sheaf. We consider sub-modules D of X(M ), subject to the condition that each point M be contained in some open set U so that D(U ) is not empty (the module covers M ; the zero vector field is valid). We will want D to be involutive, closed under the local application of Lie brackets, but D can always be replaced with its closure under Lie brackets if necessary. There is a map from such tangent subsheaves to distributions; the subsheaf D determines ∆ which for each x ∈ M is spanned by the germinal vector fields in the stalk of D at x. Lemma B. 2 . 2Let Γ be a topological groupoid, equipped with the continuous diffeology. Then the corresponding Mostow diffeology on BΓ coincides with the continuous diffeology on BΓ induced by the strong topology. Consequently, if X is a topological space with the continuous diffeology, a map η : X → BΓ is smooth if and only if it is strongly continuous. Lemma C. 3 . 3For any section σ : X → Y / O(q), there is a canonical diffeomorphism of fibre bundles e σ : σ * V (Y / O(q)) → Y / O(q). : ( * ; α; x) −−−→ * ; α; [σ( x)] , x ∈ R q .We now construct σ k for k > 0. The diffeomorphism e :σ * V (Fr ∞ (R q )/ O(q)) → Fr ∞ (R q )/ O(q)of Lemma C.3 allows the definition of a fibrewise contraction: for t ∈ [0, 1] and x ∈ R q , defineg t, x : Fr ∞ (R q )/ O(q) x → Fr ∞ (R q )/ O(q) x by the formula g t, x (b) := e σ (t e −1 σ (b)), b ∈ Fr ∞ (R q )/ O(q) x . Fr ∞ (h)/ O(q) ∼ = η * Fr ∞ (γ)/ O(q) (Proposition 3.8), any smooth section of Fr ∞ (γ)/ O(q) → BΓ q induces a corresponding smooth section of Fr ∞ (h)/ O(q) → X. Finally, any two smooth sections are smoothly homotopic via an exponentiated-linear homotopy using Lemma C.3. Theorem 1.3. Let M be a Haefliger-singular foliation of codimension q. Any adapted geometry on M specifies a unique Chern-Weil homomorphism from W O q ⊂ A * (a q ) to Ω * (M ). The Chern-Weil homomorphism descends to cohomology to agree with the Haefliger characteristic map from Gel'fand-Fuks cohomology. prove in addition that our Chern-Weil homomorphism is functorial under Haefligersingular maps, namely those smooth maps which pull back a Haefliger-singular foliation of the codomain to a Haefliger-singular foliation of the domain. As an application of our geometric theory, we generalise the classical algorithm for the construction of the Godbillon-Vey invariant, famously used by Thurston to study the topology of BΓ q [48], to Haefliger-singular foliations. Section 5 is concluded by showing that Haefliger-singular foliations with adapted geometries suffice to recover the characteristic classes of all Haefliger structures on manifolds of sufficiently high dimension. That is: Theorem 1.4. All Haefliger-singular foliations admit adapted geometries. Moreover, the category F q sing of codimension q Haefliger-singular foliations with adapted geometries is homotopyequivalent to the category H q man consisting of codimension q Haefliger structures on smooth manifolds of dimension at least q. LM was supported by the Australian Research Council Discovery Project grant DP200100729. BM was supported by the Australian Research Council Discovery Project grant DP190102360. LM wishes to thank Iakovos Androulidakis for encouraging him to think about the characteristic classes of singular foliations, and Adam Rennie, Alan Carey, Mathai Varghese and David Roberts for helpful discussions. BM wishes to thank Mike Eastwood and Thomas Leistner for helpful discussions.We 1.1 Acknowledgements The next theorem generalises Haefliger's[25, Theorem 7] to the diffeological setting, and justifies the nomenclature of "classifying space" for the diffeological space BΓ. Say that a smooth, countable partition of unity {λ α } α∈N on a diffeological space is locally finite if the covering λ −1 α (0, 1] of X by D-closures is locally finite, and subordinate to an open cover{U α } α∈N of X if λ −1 α (0, 1] ⊂ U α for all α.A countable open cover of a diffeological space X is smoothly numerable if it admits a subordinate, locally finite, smooth partition of unity. A smooth Γstructure on X is smoothly numerable if it admits a representative cocycle over a countable, smoothly numerable open cover, and two smooth, numerable Γ-structures on X are said to be smoothly, numerably homotopic if there exists a smoothly numerable homotopy between them.Our next theorem generalises Haefliger's classifying space theorem to the diffeological category (that our theorem really is a generalisation can be seen by equipping any topological groupoid, as considered by Haefliger, with the continuous diffeology; see Appendix B). Although the diffeology for BΓ is inspired by Mostow[42], Mostow does not prove a classifying follows from the naturality of pullbacks. The final item follows from [38, Lemma 3.11]. Note that the hypothesis there that X be Hausdorff, second-countable and smoothly paracompact is required only for the existence of a smoothly numerable open cover over which the bundle is locally trivial. In our setting, this follows from the hypothesis that h 0 and h 1 are smoothly numerable Haefliger structures. ). The reverse inclusion follows analogously, as required. B The diffeological generalisation of Haefliger's classifying the-In this appendix we detail our claim that our diffeological classifying space theorem (Theorem 3.7) generalises Haefliger's topological classifying theorem, which we now recall for the reader's convenience. Theorem B.1 (Haefliger's classifying theorem). [25, Theorem 7] Let Γ be a topological groupoid, and regard BΓ with the strong topology (the weakest topology making the canonical partition of unity and cocycle maps on BΓ continuous). 1. The canonical Γ-structure γ on BΓ is numerable.2. For any numerable Γ-structure h on a topological space X, there is a continuous maporem In the diffeological setting, principal does not imply locally trivial, but only locally trivial along plots[31, Section 8.13]. Proposition A.2. The distribution ∆ associated to a Haefliger cocycle on a manifold M is integrable.Proof. It suffices to show for each point x ∈ M and vector field X ∈ D defined near x, thatt the local flow of X. But this suffices to be shown for small enough t that a neighborhood of the curve t → Φ t (x) is contained in the domain of a single Haefliger chart f α : U α → R q . Now, the subspace ∆ y depends only on the stalk of D at y, which in turn depends only on the stalk (or germ) of I at y. The latter is invariant under local flows along D, so the former is too. To be explicit, we have first (Φ * t I) x = I x , as follows from the observation that f α • Φ t = f α and the implication thatplus the opposite inclusion deduced from an application of Φ * −t . Then, given v ∈ ∆ Φt(x) and V ∈ D extending v, let W = dΦ −t V be the vector field pullback of V . For any ω ∈ I near x, The Variational Bicomplex. I Anderson, Utah State UniversityI. Anderson, The Variational Bicomplex, Utah State University, 1989. The holonomy groupoid of a singular foliation. I Androulidakis, G Skandalis, J. Reine Angew. Math. 626I. Androulidakis and G. Skandalis, The holonomy groupoid of a singular foliation, J. Reine Angew. Math. 626 (2009), 1-37. The analytic index of elliptic pseudo differential operators on a singular foliation. J. K-theory. 8, The analytic index of elliptic pseudo differential operators on a singular foliation, J. K-theory 8 (2011), 363-385. Pseudodifferential calculus on a singular foliation. J. Noncommut. Geom. 5, Pseudodifferential calculus on a singular foliation, J. Noncommut. Geom. 5 (2011), 125-152. Connes conjecture for singular foliations. Baum, Ann. K-theory. 4, A Baum-Connes conjecture for singular foliations, Ann. K-theory 4 (2019), 561- 620. Singularities of holomorphic foliations. P Baum, R Bott, J. Differential Geom. 7P. Baum and R. Bott, Singularities of holomorphic foliations, J. Differential Geom. 7 (1972), 279-342. Vector fields and characteristic numbers. R Bott, Mich. Math. J. 14R. Bott, Vector fields and characteristic numbers, Mich. Math. J. 14 (1967), 231-244. On a topological obstruction to integrability. Proc. Sympos. Pure Math. 16, On a topological obstruction to integrability, Proc. Sympos. Pure Math. 16 (1970), 127-131. Lectures on characteristic classes and foliations, Lectures on Algebraic and Differential Topology. Springer279, Lectures on characteristic classes and foliations, Lectures on Algebraic and Dif- ferential Topology, vol. 279, Springer, 1972. On characteristic classes in the framework of Gelfand-Fuks cohomology. Astérique. 32, On characteristic classes in the framework of Gelfand-Fuks cohomology, Astérique 32-33 (1976), 113-139. On characteristic classes of Γ-foliations. R Bott, A Haefliger, Bull. Amer. Math. Soc. 78R. Bott and A. Haefliger, On characteristic classes of Γ-foliations, Bull. Amer. Math. Soc. 78 (1972), 1039-1044. A Candel, L Conlon, Foliations I (Graduate Studies in Mathematics. Amer. Math. SocA. Candel and L. Conlon, Foliations I (Graduate Studies in Mathematics), Amer. Math. Soc., 1999. . I I Foliations, Graduate Studies in Mathematics Series. 60Amer Mathematical Society, Foliations II (Graduate Studies in Mathematics Series Volume 60), Amer Mathe- matical Society, 2003. Characteristic forms and geometric invariants. S S Chern, J Simons, Ann. Math. 99S. S. Chern and J. Simons, Characteristic forms and geometric invariants, Ann. Math. 99 (1974), 48-69. Cyclic cohomology and the transverse fundamental class of a foliation, Geometric methods in operator algebras. A Connes, Longman Sci. Tech. H. Araki and E. G. EffrosA. Connes, Cyclic cohomology and the transverse fundamental class of a foliation, Geomet- ric methods in operator algebras (Kyoto, 1983) (H. Araki and E. G. Effros, eds.), Longman Sci. Tech., Harlow, 1986, pp. 52-144. Differentiable cyclic cohomology and Hopf algebraic structures in transverse geometry. A Connes, H Moscovici, Essays on geometry and related topics. Geneva1A. Connes and H. Moscovici, Differentiable cyclic cohomology and Hopf algebraic structures in transverse geometry, Essays on geometry and related topics, Vol. 1, 2, vol. 38, pp. 217- 255, Monogr. Enseign. Math., Geneva, 2001. Simplicial de Rham cohomology and characteristic classes of flat bundles. J Dupont, Topology. 15J. Dupont, Simplicial de Rham cohomology and characteristic classes of flat bundles, Topol- ogy 15 (1976), 233-245. Cohomologies of the Lie algebra of vector fields on a circle. I , D Fuks, Funct. Anal. Its Appl. 2I. Gel'fand and D. Fuks, Cohomologies of the Lie algebra of vector fields on a circle, Funct. Anal. Its Appl. 2 (1968), 342-343. Cohomologies of Lie algebra of tangential vector fields of a smooth manifold. I Ge&apos;lfand, D Fuks, Funct. Anal. Its Appl. 3I. Ge'lfand and D. Fuks, Cohomologies of Lie algebra of tangential vector fields of a smooth manifold, Funct. Anal. Its Appl. 3 (1969), 194-210. Cohomologies of Lie algebra of tangential vector fields. I , D Fuks, Funct. Anal. Its Appl. III. Gel'fand and D. Fuks, Cohomologies of Lie algebra of tangential vector fields. II, Funct. Anal. Its Appl. 4 (1970), 110-116. Cohomology of the Lie algebra of formal vector fields. Math. USSR-Izv. 4, Cohomology of the Lie algebra of formal vector fields, Math. USSR-Izv. 4 (1970), 327-342. Un invariant des feuilletages de codimension un. C Godbillon, J Vey, C.R. Acad. Sci. Paris. 273C. Godbillon and J. Vey, Un invariant des feuilletages de codimension un, C.R. Acad. Sci. Paris 273 (1971), 92-95. Algèbre de Weil du groupe linéaire, Application aux classes caractéristiques d'un feuilletage. S Guelorget, Springer Lecture Notes in Math. 484SpringerS. Guelorget, Algèbre de Weil du groupe linéaire, Application aux classes caractéristiques d'un feuilletage, Springer Lecture Notes in Math., vol. 484, pp. 179-191, Springer, Berlin, Heidelberg, 1975. Feuilletages sur les variétés ouvertes. A Haefliger, Topology. 9A. Haefliger, Feuilletages sur les variétés ouvertes, Topology 9 (1970), 183-194. . Homotopy, Springer, Lecture Notes in Math. 197Springer, Homotopy and integrability, Springer Lecture Notes in Math., vol. 197, pp. 133- 163, Springer, Berlin, Heidelberg, 1971. . Differentiable Cohomology, CIME. , Differentiable cohomology, pp. 19-70, CIME, Varenna 1976, Ligouri, Naples, 1979. Independent variation of secondary classes. J Heitsch, Ann. Math. 108J. Heitsch, Independent variation of secondary classes, Ann. Math. 108 (1978), 421-460. Differential geometry and symmetric spaces. S Helgason, Academic PressS. Helgason, Differential geometry and symmetric spaces, Academic Press, 1962. Morris W Hirsch, 1976. MR 0448362Differential topology. New York-HeidelbergSpringer-VerlagMorris W. Hirsch, Differential topology, Graduate Texts in Mathematics, No. 33, Springer- Verlag, New York-Heidelberg, 1976. MR 0448362 . D Husemoller, Fibre Bundles, Graduate Texts in Mathematics. Springer Science + Business MediaThird EditionD. Husemoller, Fibre Bundles, Third Edition, Graduate Texts in Mathematics, Springer Science + Business Media, New York, 1994. . P Iglesias-Zemmour, Diffeology, Mathematical Surveys and Monographs. 185American Mathematical SocietyP. Iglesias-Zemmour, Diffeology, Mathematical Surveys and Monographs, vol. 185, Ameri- can Mathematical Society, 2013. Foliated bundles and characteristic classes. F W Kamber, P Tondeur, Lecture Notes in Mathematics. 493SpringerF. W. Kamber and P. Tondeur, Foliated bundles and characteristic classes, Lecture Notes in Mathematics, no. 493, Springer, 1975. Canonical forms on frame bundles of higher order contact. S Kobayashi, Proc. Sympos. Pure Math. 3S. Kobayashi, Canonical forms on frame bundles of higher order contact, Proc. Sympos. Pure Math. 3 (1961), 186-193. I Kolář, P W Michor, J Slovák, Natural Operations in Differential Geometry. Springer-VerlagI. Kolář, P. W. Michor, and J. Slovák, Natural Operations in Differential Geometry, Springer-Verlag, 1993. A characteristic map for the holonomy groupoid of a foliation. L E Macdonald, Math. Z. 300L. E. MacDonald, A characteristic map for the holonomy groupoid of a foliation, Math. Z. 300 (2022), 1093-1115. The Godbillon-Vey invariant and equivariant KK-theory. L E Macdonald, A Rennie, Ann. K-theoryL. E. MacDonald and A. Rennie, The Godbillon-Vey invariant and equivariant KK-theory, Ann. K-theory (2020), 249-294. Godbillon-Vey invariant as topological vorticity compression and obstruction to steady flow in ideal fluids. T Machon, Proc. R. Soc. A. 476T. Machon, Godbillon-Vey invariant as topological vorticity compression and obstruction to steady flow in ideal fluids, Proc. R. Soc. A 476 (2020). The diffeology of Milnor's classifying space. J P Magnot, J Watts, Topology Appl. 232J. P. Magnot and J Watts, The diffeology of Milnor's classifying space, Topology Appl. 232 (2017), 189-213. Frobenius avec singularités. I. Codimension un. B Malgrange, Publ. Math. IHÉS. 46B. Malgrange, Frobenius avec singularités. I. Codimension un, Publ. Math. IHÉS 46 (1976), 163-173. Frobenius avec singularités. 2. Le cas général. Invent. Math. 39, Frobenius avec singularités. 2. Le cas général, Invent. Math. 39 (1977), 67-89. S Morita, Geometry of Characteristic Classes (Translations of Mathematical Monographs). American Mathematical SocietyS. Morita, Geometry of Characteristic Classes (Translations of Mathematical Monographs), American Mathematical Society, 2001. The differentiable space structures of Milnor classifying spaces, simplicial complexes, and geometric realizations. M A Mostow, J. Differential Geometry. 14M. A. Mostow, The differentiable space structures of Milnor classifying spaces, simplicial complexes, and geometric realizations, J. Differential Geometry 14 (1979), 255-293. The Geometry of Jet Bundles. D J Saunders, London Mathematical Society Lecture Note Series. 142Cambridge University PressD. J. Saunders, The Geometry of Jet Bundles, London Mathematical Society Lecture Note Series 142, Cambridge University Press, 1989. Classifying spaces and spectral sequences. G Segal, Inst. HautesÉtudes Sci. Publ. Math. 34G. Segal, Classifying spaces and spectral sequences, Inst. HautesÉtudes Sci. Publ. Math. 34 (1968), 105-112. Accessible sets, orbits, and foliations with singularities. P Stefan, Proc. London Math. Soc. 29P. Stefan, Accessible sets, orbits, and foliations with singularities, Proc. London Math. Soc. 29 (1974), 699-713. Orbits of families of vector fields and integrability of distributions. H J Sussmann, Trans. Amer. Math. Soc. 180H. J. Sussmann, Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc. 180 (1973), 171-188. Noncobordant Foliations of S 3. W Thurston, Bull. Amer. Math. Soc. 78W. Thurston, Noncobordant Foliations of S 3 , Bull. Amer. Math. Soc. 78 (1972), 511-514. Foliations and groups of diffeomorphisms. Bull. Amer. Math. Soc. 80, Foliations and groups of diffeomorphisms, Bull. Amer. Math. Soc. 80 (1974), 304- 307. On the homotopy type of classifying spaces. T Tom Dieck, Manuscripta Math. 11T. tom Dieck, On the homotopy type of classifying spaces, Manuscripta Math. 11 (1974), 41-49.
{'fraction_non_alphanumeric': 0.07197414040301348, 'fraction_numerical': 0.016257831304362938, 'mean_word_length': 3.5255210381439244, 'pattern_counts': {'":': 0, '<': 5, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 139, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': "We give a Chern-Weil map for the Gel'fand-Fuks characteristic classes of Haefligersingular foliations, those foliations defined by smooth Haefliger structures with dense regular set. Our characteristic map constructs, out of singular geometric structures adapted to singularities, explicit forms representing characteristic classes in de Rham cohomology. The forms are functorial under foliation morphisms. We prove that the theory applies, up to homotopy, to general smooth Haefliger structures: subject only to obvious necessary dimension constraints, every smooth Haefliger structure is homotopic to a Haefliger-singular foliation, and any morphism of Haefliger structures is homotopic to a morphism of Haefligersingular foliations. As an application, we provide a generalisation to the singular setting of the classical construction of forms representing the Godbillon-Vey invariant.", 'arxivid': '2106.10078', 'author': ['Lachlan E Macdonald \nAustralian Institute for Machine Learning\nSchool of Mathematical Sciences\nThe University of Adelaide Adelaide\nThe University of Adelaide Adelaide\n5000, 5000SA, SA\n', 'Benjamin Mcmillan \nAustralian Institute for Machine Learning\nSchool of Mathematical Sciences\nThe University of Adelaide Adelaide\nThe University of Adelaide Adelaide\n5000, 5000SA, SA\n'], 'authoraffiliation': ['Australian Institute for Machine Learning\nSchool of Mathematical Sciences\nThe University of Adelaide Adelaide\nThe University of Adelaide Adelaide\n5000, 5000SA, SA', 'Australian Institute for Machine Learning\nSchool of Mathematical Sciences\nThe University of Adelaide Adelaide\nThe University of Adelaide Adelaide\n5000, 5000SA, SA'], 'corpusid': 247447759, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 38485, 'n_tokens_neox': 33750, 'n_words': 21381, 'pdfsha': '10c0d481345be2957972f6ce548d7188e79acf55', 'pdfurls': ['https://arxiv.org/pdf/2106.10078v2.pdf'], 'title': ['Chern-Weil theory for Haefliger-singular foliations', 'Chern-Weil theory for Haefliger-singular foliations'], 'venue': []}
arxiv
A Resource Letter on Physical Eschatology Milan M Ćirković [email protected] Astronomical Observatory Belgrade Volgina 7 11160Belgrade SerbiaYugoslavia A Resource Letter on Physical Eschatology This Resource Letter treats the nascent discipline of physical eschatology, which deals with the future evolution of astrophysical objects, including the universe itself, and is thus both a counterpart and a complement to conventional cosmology. While sporadic interest in these topics has flared up from time to time during the entire history of humanity, a truly physical treatment of these issues has only become possible during the last quarter century. This Resource Letter deals with these recent developments. It offers a starting point for understanding what the physical sciences might say about the future of our universe and its constituents. Journal articles, books, and web sites are provided for the following topics: history and epistemology of physical eschatology, the future of the Solar system, the future of stars and stellar systems, the global future of the universe, information processing and intelligent communities, as well as some side issues, like the possible vacuum phase transition and the so-called Doomsday Argument. http://www.santafe.edu/~shalizi/Bernal/. A great inspiration of futurologists, prophets, and physical eschatologists since it appeared. The concluding section starts with a phrase appropriate for almost any PE study: "By now it should be possible to make a picture of the general scheme of development as a unified whole, and though each part may seem plausible in detail, yet in some obscure way the total result seems unbelievable." 5. " 5 B. The epistemological basis of physical eschatology and the philosophy of time We obviously do not think in the same way about past and future. We remember the past, but not the future. In other words, we claim to have secure knowledge (memories) of past events, but only vague hunches, at best, of future events. We seem to feel the passage of time, as the special moment we call "now" moves from past to future. Many tomes have been devoted to philosophical, psychological, artistic, and even social aspects of this sensation. These are beyond the scope of this Resource Letter, but they should be considered within a broader framework. We are concerned here with epistemological properties, as well as differences (if any) between prediction and retrodiction in physical science. This is a formidable topic that has been investigated many times in different contexts (especially in thermodynamics and in classical and quantum field theory), and I present here a point of entry into the literature that is likely to be of interest from the PE point of view. A general feature in the development of the sciences seems to be that philosophical considerations play a role mostly in their formative phases (or in periods of crisis or controversy). In the case of physical eschatology, we do seem to be in a rather early part of its formative phase. proper, with plenty of neat excursions into history of science and philosophy. F. Conference proceedings To date there has been only two conferences devoted mainly to PE. The first was a Symposium that was held in Budapest and Debrecen, Hungary, July 2-6, 1999. Its The most local and "practical" aspect of physical eschatology pertains to the future of our immediate cosmic neighborhood-the Earth and the Solar system. There are several reasons for investigating this issue. First, the Solar system is regarded traditionally as a good approximation to an isolated astrophysical system; exceptions dealing for instance with the influence of the Galactic tides on cometary orbits as a rule are regarded as highly controversial. Second, the timescales for the future evolution of the Solar system are driven essentially by the evolution of the Sun up and off the Main Sequence, which is regarded as a well-established part of stellar evolution. Third, the timescale for the demise of the Sun and Earth (at least as a viable habitat) is significantly shorter than the vast majority of timescales encountered in the works I survey later. Following the premise that a prediction is more precise and persuasive as its temporal locus approaches the present, this should be the "firmest" aspect of PE. These advantages should be weighed against the extraordinary high precision by astrophysical and cosmological standards that is required to decide meaningfully on the course of future events. Thus, the variation of the Solar radius of the order of 1% or less during the thermal pulsations on the asymptotic giant branch (AGB) will decide whether our Earth will be evaporated or not (see, e.g., Ref. 66 Confirms earlier rough conclusions that the radius of the AGB Sun is "surprisingly close" to 1 AU, and thus the fate of Earth hangs in the balance. However, in that case of ejection into interstellar space, Earth will cool with timescale of ∼10 6 years, and settle down in a quasi-equilibrium state; life in hydrothermal vents could "continue in largely unperturbed fashion" even then. descendants-if any-will certainly find strategies for survival more efficient than those presented in this paper. However, as a pioneering contribution in this respect, it certainly deserves attention today! "On the Final III. FATE OF STARS AND STELLAR SYSTEMS A. Fate of stars and the final mass distribution function Before the sun and the light, and the moon, and the stars are darkened and the clouds return after the rain. that the universe is not only homogeneous and istotropic in space, but also homogeneous in time. In such a universe, PE is reduced to the trivial statement that the universe on large scales will remain the same as it is today throughout the temporal limit t → +∞. The qualification "on large scales" is crucial here, since stars, for instance, live and die and their populations are slowly extinguished in essentially the same manner as in any evolutionary cosmology (as discussed in section III.A. above); thus, the "local" part of PE is still valid in the steady-state context. The difference is that on scales of galaxies and larger, things stay the Attempts to demonstrate the significance of early decision-making in the context of the entire history of an intelligent community; dependence on the realistic cosmological model is particularly emphasized. B. The Doomsday Argument One of the most intriguing side issues in discussing the future of humanity is the so- idea can be expressed through the following urn-ball experiment. Place two large urns in front of you, one of which you know contains ten balls, the other a million, but you do not know which is which. The balls in each urn are numbered 1, 2, 3, 4,... . Now take one ball at random from the left urn; it shows the number 7. This clearly is a strong indication that the left urn contains only ten balls. If the odds originally were fifty-fifty (identically-looking urns), an application of Bayes' theorem gives the posterior probability that the left urn is the one with only ten balls as P post (n=10) = 0.99999. Now consider the case where instead of two urns you have two possible models of humanity, and instead of balls you have human individuals, ranked according to birth order. One model suggests that the human race will soon become extinct (or at least that the number of individuals will be greatly reduced), and as a consequence the total number of humans that ever will have existed is about 100 billion. The other model indicates that humans will colonize other planets, spread through the Galaxy, and continue to exist for many future millennia; we consequently can take the number of humans in this model to be of the order of, say, 10 18 . As a matter of fact, you happen to find that your rank is about sixty billion. According to Carter and Leslie, we should reason in the same way as we did with the urn balls. That you should have a rank of sixty billion is much more likely if only 100 billion humans ever will have lived than if the number was 10 18 . Therefore, by Bayes' theorem, you should update your beliefs about mankind's prospects and realize that an impending doomsday is much more probable than you thought previously. The following references show clearly that the Doomsday Argument continues to be a highly controversial topic. In addition, it is one that obviously requires a truly crossdisciplinary approach to explore, considering that the authors are both physicists and philosophers. See also the references describing the possible hazards owing to the vacuum phase transition and similar calamities (Refs. 157-166), which, one suspects, motivated some of the interest of physicists in the Doomsday Argument. The End of the World: from the Standpoint of Mathematical Physics," A. S. Eddington, Nature 127, 447-453 (1931). (I) 6. The Beginning and the End of the World, E. T. Whittaker (Oxford University Press, Oxford, 1942). (I) A fine essay describing the genesis, contents and evolution of Haldane's eschatological writings is 7. "Last Judgment: The Visionary Biology of J. B. S. Haldane," M. B. Adams, Journal of the History of Biology 33, 457-491 (2000). (E) Some thoughts about the influence of early futurology on modern PE can be found in Dyson's Jerusalim lectures, which he published as 8. Imagined Worlds, F. J. Dyson (Harvard University Press, Cambridge, Mass., 1998). (E) A book manifestly not belonging to PE, but exercising a continuous influence on more than one modern physical eschatologist is 9. The Phenomenon of Man, Teilhard de Chardin, translated by B. Wall (Collins, London, 1959). (I,A) This posthumously published study of the great and controversial paleontologist and theologian can be recognized as a strong inspiration not only for Tipler's Omega Point theory (Refs. 52, 168, and 174), but also for the entire twentiethcentury future-oriented thinking (e.g. Refs. 47, 48, 54, 55, and Sec. V.A.). 6 ( 6Some additional philosophical background can be found in Refs. 47, 49, 98, 135, 144-153, and in Sec. V.B. 10. The Poverty of Historicism, K. R. Popper (Routledge and Kegan Paul, London, 1957; originally published in Economica, 1944/45). (I) The relevant part of this famous bookdeals with the severe limitations of prediction in both "hard" and "soft" sciences. Even if we were perfect Laplacian calculators, we would need to know not just what the true laws of physics were, but that our knowledge of these laws was accurate and complete, which could never be determined on empirical grounds. This argument lies at the core of our necessary assumptions in PE.27. "Will the Universe Expand Forever?" J. R. Gott III, J. E. Gunn, D. N. Schramm, and B. M. Tinsley, Scientific American 234 (March), 62-79 (1976). (E) Explains the "basic dilemma" of the large-scale PE: will the universe expand forever or recollapse? Argues strongly for the ever-expanding case, and some arguments are still very relevant. 28. "The Future History of the Universe," J. K. Lawrence, Mercury VII, no. November/December), 132-138 (1978). (E) 29. "The Ultimate Fate of the Universe," J. N. Islam, Sky & Telescope 57 (January), 13-18 (1979). (E) 30. "The Future of the Universe," D. N. Page, and M. R. McKee, Mercury 12 (January-February), 17-23 (1983). (E) 31. "Not the end of the world," J. Silk, Nature 304, 191-192 (1983). (E) A review of the book by Jamal N. Islam (Ref. 46) from the pen of one of the most distinguished contemporary astrophysicists. 32. "[Review of] The Ultimate Fate of the Universe," A. Lawrence, Observatory 103, 268-269 (1983). (E) Another review of Islam's book (Ref. 46). Contains an interesting philosophical conclusion: "Theories of the once-only future can be tested only by waiting. And then-tested by whom? If a scientific test requires a conscious scientist who understands the result, then the only possible scientific theory on the future of life is that it survives!" 33. "The Future of the Universe," D. A. Dicus, J. R. Letaw, D. C. Teplitz, and V. L. Teplitz, Scientific American 248 (March), 74-85 (1983). (E) "A forecast for the expanding universe through the year 10 100 ." Based partially on the research by the same authors in Ref. 102. 34. "The far future of the Universe," J. N. Islam, Endevoar 8(1), 32-34 (1984). (E) Based on the same material as Refs. 29, 46, and 97 of the same author. Investigates the far future of an open universe. 35. "L'Avenir de l'univers," N. Prantzos, and M. Cassé, La Recherche 15, 839-847 (1984) (in French). (E) 36. "After the Sun Dies," T. A. Heppenheimer, Omni (August), 37-40 (1986). (E) 37. "The Future History of the Solar System," J. Maddox, Nature 372, 611 (1994). (E) News-and-views column by the long-standing editor of Nature, devoted to uncertainties about the fate of the Earth faced with the post-Main Sequence Solar evolution; compare Sec. II. 38. "This Too Shall Pass," M. Szpir, American Scientist 85 (May-June), 223-225 (1997). proceedings have been published as 56. The Future of the Universe and the Future of Our Civilization, edited by V. Burdyuzha and G. Khozin (World Scientific, Singapore, 2000). (A) The second was a conference on the "Far-Future Universe: Eschatology from a Cosmic Perspective," which was held in Rome, Italy, November 7-9, 2000. Its proceedings have been published as 57. Far-Future Universe: Eschatology from a Cosmic Perspective, edited by G. F. R. Ellis (Templeton Press, Radnor, 2002). (A) Another partially relevant meeting was a Symposium that was held in conjunction with the 160th Annual Meeting of the Astronomical Society of the Pacific, at the University of Maryland, College Park, June 26-28, 1995. Its proceedings have been published as: 58. Clusters, lensing, and the future of the universe, edited by V. Trimble and A. Reisenegger (ASP, San Francisco, 1996). (A) II. FATE OF THE EARTH, THE SUN, AND THE SOLAR SYSTEM So I travelled, stopping ever and again, in great strides of a thousand years or more, drawn on by the mystery of the earth's fate, watching with a strange fascination the sun grow larger and duller in the westward sky, and the life of the old earth ebb away. H. G. Wells, The Time Machine (1895) 59. "Survival of the Earth and the Future Evolution of the Sun," S. C. Vila, Earth, Moon and Planets 31, 313-315 (1984). (I) On the basis of rather simplistic assumptions, Vila argues that Earth will be destroyed in the solar red giant's envelope. Calculates the amount of mass accreted by Earth from the solar wind. 60. "The fate of the Earth in the red giant envelope of the Sun," J. Goldstein, Astronomy and Astrophysics 178, 283-285 (1987). (A) 61. "Advanced stages in the evolution of the Sun," U. G. Jorgensen, Astronomy and Astrophysics 246, 118-136 (1991). (A) First detailed study of "the solar evolution all the way from the ZAMS [zero-age Main Sequence] to the end of its life as a red giant." 62. "Our Sun. III. Present and Future," I.-J. Sackmann, A. I. Boothroyd, and K. E. Kraemer, Astrophysical Journal 418, 457-468 (1993). (A) The crucial detailed astrophysical study of the future solar evolution.63. "The Expected Morphology of the Solar System Planetary Nebula," N. Soker, Publications of the Astronomical Society of the Pacific 106, 59-62 (1994). (A) Deals mainly with the possible influence of Jupiter on the planetary nebula created by Sun in its AGB phase. 64. "The Effects of Post-Main-Sequence Solar Mass Loss on the Stability of Our Planetary System," M. J. Duncan and J. J. Lissauer, Icarus 134, 303-310 (1998). (A) Discusses long-term stability of planetary orbits taking into account both Solar mass loss and accretion drag exerted on planets.65. "The Frozen Earth: Binary Scattering Events and the Fate of the Solar System," G. Laughlin and F. C. Adams, Icarus 145, 614-627 (2000). (A) Discusses the fate of Earth in view of random binary scatterings of passing stars; concludes that chances of serious disruption of Earth's orbit prior to effects of Solar evolution are very small, ∼10 -5 . Destiny of the Earth and the Solar System," K. R. Rybicki and C. Denis, Icarus 151, 130-137 (2001). (A) Investigates thermal pulses during the Solar red giant and AGB phases. Conclusion: "Mercury will evaporate... and Venus will most probably be destroyed as well. The Earth's fate still remains controversial, but according to the existing evolution sequences for solar models, it is likely that our planet will evaporate during the giant stage of the Sun." 67. "Astronomical engineering: a strategy for modifying planetary orbits," D. G. Korycansky, G. Laughlin, and F. C. Adams, Astrophysics and Space Science 275, 349-366 (2001). (A) When faced with the Solar ascent up the Main Sequence, our remote Ecclesiastes, 12: 2 2Stellar evolution is the pride and glory of theoretical astrophysics. The detail and accuracy of stellar models over a range of three orders of magnitude in masses, and over almost seven orders of magnitude in evolutionary timescales, have become something of a yardstick for the quality of the modelling endeavor. Thus, it is somewhat surprising that, apart from a pile of potentially applicable results, there is relatively modest interest in PErelated problems in this area. In a nice expression used in Ref. 69, we live in the stelliferous era in the history of the universe. This era is characterized by active star formation from interstellar matter throughout the disks of spiral galaxies, and possibly in some other, more exotic environments, like cluster cooling flows and galaxy-merger events. Since the recycling of matter through stellar mass-loss and supernovae obviously is not perfect (since the matter is continually being locked in inert remnants at the rate of a few Solar masses per year in a galaxy like the Milky Way), this process necessarily will come to an end. But how long into the future this era will last is still very, very uncertain. Other unsolved problems abound. The concept of the final mass function of stars, introduced in Ref. 69, can be defined as precisely as the concept of an initial mass function, and is potentially of similar interest, but so far it has not been investigated much. A lot of work has to be done in this PE subfield.68. "The coldest neutron star," G. Feinberg, Physical Review D 23, 3075 (1981). (I) "So the temperature of a neutron star cannot drop below 100 K. The prospects of observing such cold stars do not seem very bright until interstellar travel becomes commonplace, if then." 69. "A dying universe: the long-term fate and evolution of astrophysical objects," F. C. Adams and G. Laughlin, Reviews of Modern Physics 69, 337-372 (1997). (A) Together with Refs. 98, 101, and 114, constitutes a set of landmark papers in physical eschatology. Contains a fascinating treasure of results in various aspects of PE, notably the one linked with the fate of stars and galaxies.70. "The End of the Main Sequence," G. Laughlin, P. Bodenheimer, and F. C. Adams, Astrophysical Journal 482, 420-432 (1997). (A) First detailed modeling of the complete evolution of the lowest-mass stars over their stupendously long timescales. 71. "Gravitational demise of cold degenerate stars," F. C. Adams, G. Laughlin, M. Mbonye, and M. J. Perry, Physical Review D 58, 083003-1/7 (1998). (A) Notices that "the wavefunction of the star will contain a small admixture of the black hole states" that will emit Hawking's radiation. 72. "Future of Galaxies and the Fate of Intelligent Beings," M. M. Ćirković, Serbian Astronomical Journal 159, 79-86 (1999). (I) Considers the duration of the stelliferous era in several simple models with infall, which are consistent with the usual chemical evolution constraints.73. "The Galactic Millenium," P. Hodge, Publications of the Astronomical Society of the Pacific 112, 1005-1007 (2000) (E) Supposing that one "galactic year" lasts about 100 million years-period of the revolution of the Solar system around the Galactic center-Hodge reviews predictions for the state of our environment in 100 billion years (see also Ref. 42). B. The fate of the larger gravitating systems Surprisingly little has been written about the future evolution of large-scale density perturbations, in particular those that manifest themselves today as clusters and superclusters of galaxies. The future of the large-scale structure itself is connected tightly to the realistic cosmological model and the exact form of the density perturbation power spectrum. Both issues are still controversial, although we have made great progress on both during the past decade. In particular, after the results from cosmological supernovae surveys began to be published in 1998, cosmology began to converge on the flat, Ω m ≈ 0.3, Ω Λ ≈ 0.7, dark-energy dominated model. 74. "Orbits of the nearby galaxies," P. J. E. Peebles, Astrophysical Journal 429, 43-65 (1994). (A) Peebles analyzes dynamics of the Local Group galaxies and discusses a controversial possibility of future collision between our Galaxy and M31. 75. "Future Evolution of Nearby Large Scale Structure in a Universe Dominated by a Cosmological Constant," K. Nagamine and A. Loeb, New Astronomy, in press (2002) (preprint astro-ph/0204249). (A) C. The fate of black holes The bright sun was extinguish'd, and the stars Did wander darkling in the eternal space, Rayless, and pathless... Lord Byron, Manfred (1816) Among astrophysical objects of interest to PE, the ones that occupy the most elevated place are black holes. The reason is obvious: their longevity surpasses by far that of any other known astrophysical object. Before the "black hole revolution" of the 1970s, black holes were believed to be eternal, and that once formed they cannot be undone. However, after the discovery of the black-hole evaporation process by Stephen Hawking in 1974, and the elaboration of the new field of black-hole thermodynamics by Hawking, Jacob Bekenstein, Roger Penrose, Robert Geroch, Robert M. Wald, William G. Unruh and others, the finiteness of their lifetimes became known. But their exact fate (especially in light of the informationloss puzzle) is still not completely clear. In any case, their lifetimes are enormous: a black hole of 1 Solar mass will evaporate (at least until it reaches a mass on the order of a Planck mass) in about 10 65 years, and supermassive black holes of galactic mass likely will live about 10 98 years! Eventually, in the ever-expanding universe, as was shown by Fred C. Adams and Gregory Laughlin (Ref. 69), the incredibly weak Hawking radiation will come to dominate the radiation energy density of the universe. Even stranger problems are posed by the conjecture of Hawking (Ref. 76) that black holes may pose a fundamental obstacle to any kind of long-term prediction. Let us consider a pure quantum state corresponding to a distribution of matter of mass M » M Pl (Planck mass),which collapses under its own weight. The density matrix of such a state is given by ρ = |ψ〉〈ψ|, with vanishing entropy S = -Tr (ρ ln ρ). If M is high enough, the matter will inevitably form a black hole. Subsequently, the black hole will slowly evaporate by the Hawking process, emitting blackbody radiation (which by definition carries out no information). The semiclassical treatment used by Hawking in his discovery of the black-hole evaporation and in all subsequent discussions will certainly break down when the mass of black hole approaches M Pl , but what will happen with the information from the initial state still locked in the black hole? This is the puzzle of black-hole information loss. As is wellknown, the possibility Hawking himself proposed is that the black hole simply evaporates completely and the information is irreversibly lost. Although this idea remains the simplest and the least problematic answer to the puzzle, it has provoked a lot of controversy, since it implies that the evolution of the complete system (universe plus black hole) is fundamentally non-unitary, and leads to evolution of pure into mixed quantum states.Among piles of literature on the long-term behavior of black holes, some of the useful points of entry are the following: 76. "Black hole explosions?" S. W. Hawking, Nature 248, 30-31 (1974). (A) The discovery of finite black-hole lifetimes. "The black hole would therefore have a finite time of the order of 10 71 [M Solar /M] -3 s. For a black hole of solar mass this is much longer than the age of the Universe..."-a strong understatement, indeed. 77. "Breakdown of predictability in gravitational collapse," S. W. Hawking, Physical Review D 14, 2460-2473 (1976). (A) The celebrated paper exposing the possible nonunitarity of the evolution of evaporating black holes. 78. "Is Black-Hole Evaporation Predictable?" D. N. Page, Physical Review Letters 44, 301-304 (1980). (A) 79. "Resource Letter BH-1: Black Holes," S. Detweiler, American Journal of Physics 49, 394-400 (1981). (E,I,A) 80. "The unpredictability of quantum gravity," S. W. Hawking, Communications in Mathematical Physics 87, 395-415 (1982). (A) 81. "Lectures on Black Holes and Information Loss," T. Banks, Nuclear Physics B (Proc. Suppl.) 41, 21-65 (1995). (A) 82. "Black holes and massive remnants," S. B. Giddings, Physical Review D 46, 1347-1352 (1992). (A) One of the best exposition of the massive remnant hypothesis: Hawking evaporation must end at M ~ several M Pl , and a stable remnant (sometimes called "cornucopion") remains. These are bound to be important in what Adams and Laughlin dubbed the black-hole era (Refs. 39, 53, and 69). 83. "The Hawking information loss paradox: the anatomy of a controversy," G. Belot, J. Earman, and L. Ruetsche, British Journal for the Philosophy of Science 50, 189-229 (1999). (A) Gives an overview of the nonunitarity puzzle. 84. "Gravitation, thermodynamics and quantum theory," R. M. Wald, Classical and Quantum Gravity 16, A177-A190 (1999). (A) Another interesting issue is the interaction between the dark energy (usually exemplified by the cosmological constant) and black holes, which has been the topic of a lively (and so far unresolved) debate, as indicated in the references below.85. A cosmological constant limits the size of black holes," S. A. Hayward, T. Shiromizu, and K. Nakao, Physical Review D 49, 5080-5085 (1994). (A) 86. "Possible effects of a cosmological constant on black hole evolution," F. C. Adams, M. Mbonye, and G. Laughlin, Physics Letters B 450, 339-342 (1999). (A) 87. "Black holes must die," N. Dalal and K. Griest, Physics Letters B 490, 1-5 (2000). (A) 88. "The Life and Times of Extremal Black Holes," F. C. Adams, General Relativity and Gravitation 32, 2229-2234 (2000). (A) Though we do not expect to encounter them in nature, they may still play an important role in far future, especially in conjuction with the intelligent influences (cf. Refs. 104 and 149). 89. "Proton decay, black holes and large extra dimensions," F. C. Adams, G. L. Kane, M. Mbonye, and M. J. Perry, International Journal of Modern Physics A 16, 2399-2410 (2001). (A) For additional discussions of relevance to black holes and their future evolution see also Refs. 127, 128, 131, 149 and 151. IV. GLOBAL COSMOLOGICAL FUTURE Time ends. That is the lesson of the Big Bang. It is also the lesson of the black hole. John A. Wheeler, The Lesson of the Black Hole (1981) This is the "true" eschatological topic. Modeling the future of the entire universe depends on our choice of the cosmological model. Obviously, there are cosmological models in which PE is trivial. Historically, the most important of these has been the steady-state model of Herman Bondi and Thomas Gold, and Fred Hoyle. In Bondi and Gold's version, the main premise of steady-state cosmology is the Perfect Cosmological Principle, which states same owing to the creation of low-entropy matter out of nothing (or out of the universal field of negative energy density, as in Hoyle's and William McCrea's subsequent elucidations of 21 the steady-state concept). However, during the "great controversy" (Ref. 90) this view has been rejected by almost all cosmologists in favor of-fortunately from the PE point of view-an evolutionary picture of the universe. Thus, we may expect that events different from those already seen will occur in the cosmological future.90. Cosmology and Controversy, H. Kragh (Princeton University Press,Princeton, 1996).(A) By far the best and most comprehensive reference for the formative period of modern cosmology(ca. 1930-1970). Contains an excellent discussion of the motivation behind the steady-state cosmology, some of which (e.g. uniformity of the laws of nature) is relevant to PE.The basic duality presented by the "standard model" of evolutionary cosmology is whether the universe will expand forever or the gravitational pull of matter fields will be strong enough to halt the expansion and turn it into contraction towards the "Big Crunch." In the older literature, one can find equality between eternal expansion and topological openness and, conversely, between recollapse and topological closeness. However, as elaborated by Lawrence Krauss and Michael Turner, dark energy introduces a degeneracy into the cosmological future, which indicates that even a topologically closed universe (Ω > 1) can expand forever in the presence of, say, a positive cosmological constant. Conversely, a topologically open universe can recollapse into the Big Crunch if the dark energy is attractive (e.g. a negative cosmological constant). However, since all observations suggest a repulsive form of dark energy, this option is of rather academic interest.91. "Geometry and Destiny," L. M. Krauss and M. S. Turner, General Relativity and Gravitation 31,1453-1459 (1999).(I) The same degeneracy has been studied from somewhat different perspective by LawrenceFord: 92. "Unstable fields and the recollapse of an open universe," L. H. Ford, Physics Letters A 110, 21-23 (1985). (A) 93. "Does Ω < 1 imply that the Universe will expand forever?" L. H. Ford, General Relativity and Gravitation 19, 325-329 (1987). (A) A. The future of the standard cosmological model: the ever-expanding universeWe almost certainly live in an ever-expanding cosmological domain ("universe"). This has followed from the discovery of large dark-energy density (interpreted as either the cosmological constant or quintessence) in 1998. Of course, observational cosmology long ago suggested similar conclusions on the long-term future of the universe, since all surveys of gravitating matter fell short of the critical density for recollapse. I list some references dealing with the future of ever-expanding universes, either topologically open or dominated by dark energy.94. "The Thermal Future of the Universe," P. C. W. Davies, Monthly Notices of the Royal Astronomical Society 161, 1-5 (1973). (A) 95. "Possible Ultimate Fate of the Universe," J. N. Islam, Quarterly Journal of the Royal Astronomical Society 18, 3-8 (1977). (I) 96. "Eternity is unstable," J. D. Barrow and F. J. Tipler, Nature 276, 453-459 (1978). (A)The first comprehensive survey of PE in the ever-expanding universe, with particular emphasis on the late metric distortions. "In our view the space-time geometry is becoming more and more irregular at very late times. Both pictures envision an asymptotic approach to a state of maximum entropy; our version of the heat death is different because we have included the gravitational entropy."97. "The long-term future of the universe," J. N. Islam, Vistas in Astronomy 23, 265-277 (1979). (I) Another of Islam's pioneering contributions to our present-day understanding of large-scale PE. 98. "Time without end: Physics and biology in an open universe," F. Dyson, Reviews of Modern Physics 51, 447-460 (1979). (A) This paper is crucial for our present understanding of PE. It describes evolution of an open or flat (matter-dominated) universe with a host of philosophical, epistemological, and information-theoretic asides. Notable is Dyson's analogy of our position in physics and astronomy with that in mathemathics; for him, PE is a physical analogue of Gödel's theorem on the incompleteness of mathematics.99. "Matter annihilation in the late universe," D. N. Page and M. R. McKee, Physical Review D 24, 1458-1469 (1981). (A) 100. "Eternity matters," D. N. Page and M. R. McKee, Nature 291, 44-45 (1981). (A) A companion paper toRef. 99. Concludes that for the flat Friedman universe "radiation will never completely dominate the density... matter will always be important." The asymptotic ratio of matter-to-radiation density has been calculated to be about 0.60.101. "Entropy in an Expanding Universe," S. Frautschi, Science 217,593-599 (1982).(A)Refutes the more than century-old idea of the "heat death" of the universe, confirming the early intuition of Pierre Duhem that entropy in the cosmological context only can approach its maximum value asymptotically. Thus, there will always be a thermodynamical arrow of time, although the number and intensity of relevant processes will decrease without limit. However, the conclusion does not apply to themodels with event horizons (cf. Ref. 104). 102. "Effects of proton decay on the cosmological future," D. A. Dicus, J. R. Letaw, D. C. Teplitz, and V. L. Teplitz, Astrophysical Journal 252, 1-9 (1982). (A) 103. "Future and Origin of our Universe: Modern View," A. A. Starobinsky, invited talk at the Symposium "The Future of the Universe and the Future of our Civilization" (Ref. 56). (I) "In any branch of science, sure forecasts exist for finite periods of time only, ranging from days in meteorology to millions of years in the Solar system astronomy. So, how can cosmology be an exception from this general rule? Evidently, it can't." 104. "Life, The Universe, and Nothing: Life and Death in an Ever-Expanding Universe," L. M. Krauss and G. D. Starkman, Astrophysical Journal 531, 22-30 (2000). (A)Concludes, contrary to Dyson, that "assuming that consciousness has a physical computational basis, and therefore is ultimately governed by quantum mechanics, life cannot be eternal."105. "Can the Universe escape eternal acceleration?" J. D. Barrow, R. Bean, and J. Magueijo, Monthly Notices of the Royal Astronomical Society 316, L41-L44 (2000). (A) 106. "Dark Energy and the Observable Universe," E. H. Gudmundsson and G. Björnsson, Astrophysical Journal 565, 1-16 (2001). (A) Future of Λand quintessence-dominated models from an observational point of view; complementary to Ref. 129 by the same authors.107. "Can we predict the fate of the Universe?" P. P. Avelino, J. P. M. de Carvalho, and C. J. A. P. Martins, Physics Letters B 501, 257-263 (2001). (A) 108. "The Fate of the Accelerating Universe," J.-A. Gu and W.-Y. P. Hwang, Physical Review D, in press (2002) (preprint astro-ph/0106387). (A) 109. "The Long-Term Future of Extragalactic Astronomy," A. Loeb, Physical Review D 65, 047301-1/4 (2002). (A) Considers the sky in dark-energy dominated cosmological future; compare with Ref. 106. "In contrast to a matter-dominated universe... the statistics of visible sources in a Λ-dominated universe are getting worse with the advance of cosmic time." 110. "Vacuum Decay Constraints on a Cosmological Scalar Field," J. S. Heyl and A. Loeb, Physical Review Letters 88, 121302-1/3 (2002). (A) Shows that lack of bubbles of collapsing space-time at present constrains the nature of dark energy and makes untenable the cyclic or ekpyrotic models-our Big Bang preceded by Big Crunch of the previous cycle with minimal value of the scalar potential equaling zero. 111. "Future Island Universes in a Background Universe Accelerated by a Cosmological Constant and by Quintessence," T. Chiueh and X.-G. He, Physical Review D 66, 123518-1/8 (2002). (A) 112. "Is the Universe Inflating? Dark Energy and the Future of the Universe," D. Huterer, G. D. Starkman, and M. Trodden, Physical Review D 66, 043511-1/6 (2002). (A) 113. "Accelerating Universe and Event Horizon," X.-G. He, Physical Review D, submitted (2002) (preprint astro-ph/0105005). (A) B. The future of the standard cosmological model: the recollapsing universeRecollapsing-universe models have been associated traditionally with topologicallyclosed models containing a finite amount of matter (those with Ω m > 1). The inadequacy of this formulation in the general case has been explained above; nonetheless, I list here references treating such recollapsing world-models. Of course, nowadays it seems highly unlikely that a recollapse will occur. Recent observations of both cosmological supernovae and the CMB anisotropies speak strongly against the possibility of recollapse. This is corroborated by estimates of the age of the universe (coupled with recent data on the Hubble constant) and by the general failure to find anything even remotely close to the amount of gravitating matter necessary for recollapse. The references below show that-in sharp contradistinction to the ever-expanding universe-interest in recollapsing models obviously has declined during the past decade (with an exception of the ekpyrotic model of Steinhardt and Turok, admittedly a "special case").Recollapsing universes are distinguished by possessing only finite physical time in the future, which may obviate other eschatological results. For instance, if the universe is topologically closed by a large margin (say Ω m ∼ 2), the maximal future time is of the order of 10 11 years, so that processes like Hawking's evaporation of black holes of stellar mass will never occur.A special case of recollapsing universes which has been quite popular during the 20th century are oscillating models in which the universe passes through a series (allegedlyinfinite, but see Ref. 130) of expansion and contraction cycles. These models, like the classical steady-state model, blur the difference between past and future, and thus are of only limited interest from the physical eschatological point of view. However, I include here some of the literature dealing with them, both for the sake of completeness and because of the great historical role they played in generating interest in cosmology. 114. "The collapse of the universe: an eschatological study," M. J. Rees, Observatory 89, 193-198. (1969). (I) The pioneering PE study, starting the entire field and coining a new meaning for the old word.115. "Singularities in Cosmology," R. Penrose, in M. S. Longair, ed., Confrontation of Cosmological Theory with Observational Data (IAU, D. Reidel Publishing Co., Boston, 1974), 263-272. (I) 116. "Speculation on cosmological bounce," M. Bailyn, Physical Review D 15, 957-964 (1977). (A) 117. "General relativity, thermodynamics, and the Poincaré cycle," F. J. Tipler, Nature 280, 203-205 (1979). (A) Shows the impossibility of "eternal return," i.e., Poincaré recurrence in the cycles of the closed universe governed by general relativity.118. "Gravitational bounce," K. Lake and L. A. Nelson, Physical Review D 22, 1266-1269 (1980). (A) 119. "Phase transitions and dynamics of the universe," V. Petrosian, Nature 298, 805-808 (1982). (A) "The restoration of symmetry at grand unification in a closed contracting 26 Robertson-Walker universe could slow down and halt the contraction, causing the universe to bounce and avoid the singular state or the big crunch." 120. "The impossibility of a bouncing universe," A. H. Guth and M. Sher, Nature 302, 505-506 (1982). (A) A criticism of Petrosian (Ref. 119) with respect to the possibility of a bounce.121. "Reply by Vahé Petrosian," V. Petrosian, Nature 302, 806-807 (1982). (A) Reply to Guth and Sher, Ref. 120. 122. "Acceleration and dissolution of stars in the antibang," E. R. Harrison, in G. O. Abell and G. Chincarini, eds., Early Evolution of the Universe and Its Present Structure (IAU, D. Reidel Publishing Co., Boston, 1983), 453-455. (A) "Antibang" is Harrison's preferred term for the Big Crunch. 123. "Black holes and the fate of a closed universe," D. Kazanas, in G. O. Abell and G. Chincarini, eds., Early Evolution of the Universe and Its Present Structure (IAU, D. Reidel Publishing Co., Boston, 1983), 331. (A) 124. "Thermodynamics and the end of a closed Universe," S. A. Bludman, Nature 308, 319-322 (1984). (A) 125. "A place for teleology?" W. H. Press, Nature 320, 315-316 (1986). (I) Criticism of Barrow and Tipler's book (Ref. 47), including its PE aspects. 126. "Achieved spacetime infinity," F. J. Tipler, Nature 325, 201-202 (1987). (I) Reply to the criticism of Press, dealing explicitly with the history-laden issue of whether it is meaningful to state that an actual infinity of events occur before the final singularity. 127. "Black holes and structure in an oscillating universe," W. C. Saslaw, Nature 350, 43-45 (1991). (A) "If black holes exist in the contracting phase of a closed universe, they will give rise to a pressure and entropy catastrophe. First, the black holes absorb all radiation; then their apparent horizons merge, and coalesce with the cosmological apparent horizon. ...in these oscillating universes containing black holes, the formation of structure, as well as the existence of life, always gets another chance." 128. "Black-hole mergers and mass inflation in a bouncing universe," A. E. Sikkema and W. Israel, Nature 349, 45-47 (1991). (A) Argues that, contrary to usual considerations, black holes may be states of very low entropy. This would circumvent most of the problems with the bouncing closed universe given since Tolman's time. 129. "Cosmological observations in a closed universe," G. Björnsson and E. H. Gudmundsson, Monthly Notices of the Royal Astronomical Society 274, 793-807 27 (1995). (A) "Practical" study of observations in the recollapsing universe. "To an observer in a contracting universe, the night sky would present a colourful zoo of cosmological objects, a vast collection of primaries and ghosts, some blueshifted, others redshifted, where apparent brightness, or size, by itself would not be a reliable indicator of distance, even if all objects were intrinsically the same and not evolving with time." 130. "Oscillating universes," J. D. Barrow and M. P. Dąbrowski, Monthly Notices of the Royal Astronomical Society 275, 850-862 (1995). (A) "If we live in a closed Friedmann universe that has undergone an infinite number of past oscillations, and if there is a positive cosmological constant, then, no matter how small its value, we might expect most likely to be living in the first phase after the oscillations have ceased, which will eventually become dominated by the cosmological constant." 131. "The Ultimate Future of the Universe, Black Hole Event Horizon Topologies, Holography and The Values of the Cosmological Constant," F. J. Tipler in Relativistic Astrophysics: 20th Texas Symposium, AIP Conference Proceedings, volume 586 (AIP, Melville, New York, 2001), 769-772. (A) 132. "A Cyclic Model of the Universe," P. J. Steinhardt and N. Turok, Science 296, 1436-1439 (2002). (A) 133. "Cosmic Evolution in a Cyclic Universe," P. J. Steinhardt and N. Turok, Physical Review D 66, 126003-1/20 (2002). (A) A colorful astrophysical process clearly relevant for late stages of a recollapsing universe is modelled in: 134. "The evolution of irradiated stars," C. A. Tout, P. P. Eggleton, A. C. Fabian, and J. E. Pringle, Monthly Notices of the Royal Astronomical Society 238, 427-438 (1989). (A) C. The future of exotic or nonstandard cosmological models Models different from standard Friedmann models also have been considered from the point of view of their future evolution. A somewhat peculiar example, which I list here for 28 the sake of completeness, is the famous recollapsing model of Thomas Gold, in which the arrow of time reverses with the reversal of expansion: 135. "The Arrow of Time," T. Gold, American Journal of Physics 30, 403-410 (1962). (I) 136. "Will entropy decrease if the Universe recollapses?" D. N. Page, Physical Review D 32, 2496-2499 (1985). (A) Criticizes the Gold universe, as well as Hawking's support for it; ends with: "Actually, it would not be surprising if the relative probability of our being in the expanding phase is much closer to unity, because this phase is predicted to last an arbitrarily long time, and hence during the subsequent recollapse all stars may have burned out and there may not be much around except for large black holes continually coalescing." 137. "Time-symmetric cosmology and the opacity of the future light cone," P. C. W. Davies and J. Twamley, Classical and Quantum Gravity 10, 931-945 (1993). (A) 138. "Observation of the Final Boundary Condition: Extragalactic Background Radiation and the Time Symmetry of the Universe," D. A. Craig, Annals of Physics 251, 384-425 (1996). (A) The most detailed analysis so far of the time-symmetric cosmological models. "On the dual grounds of theory and experiment, it therefore appears unlikely that we live in a time symmetric universe. (A definitive expurgation must await more thorough investigation of at least some of the aforementioned difficulties.)" Craig finds that "[t]his is therefore a demonstration by example that physics today can be sensitive to the presence of a boundary condition in the arbitrarily distant future." 139. "Causality in time-neutral cosmologies," A. Kent, Physical Review D 59, 043505-1/5 (1998). (A) This model is arguably closer to the steady-state theory from the PE point of view, since it does not tell us anything particularly interesting or new about the future except, of course, the bizarre and superficially counterintuitive situations encountered in the "counter-clock world"-bizarre, that is, from our perspective but completely normal from the perspective of hypothetical contemporary intelligent beings. Some of the other non-standard models with some PE aspects are: 140. "An Isothermal Universe," W. C. Saslaw, S. D. Maharaj, and N. Dadhich, Astrophysical Journal 471, 571-574 (1996). (A) Derives a class of inhomogeneous cosmologies that "may represent the ultimate state of an Einstein-de Sitter universe that undergoes a phase transition caused by gravitational clustering." 141. "Structure and future of the 'new' universe," Ya. B. Zeldovich and L. P. Grishchuk, Monthly Notices of the Royal Astronomical Society 207, 23P-28P (1984). (A) 142. "Optimistic cosmological model," N. S. Kardashev, Monthly Notices of the Royal Astronomical Society 243, 252-256 (1990). (A) "It is demonstrated that, for a certain type of hidden mass… a positive curvature cosmological model can realize a regime of periodic oscillations of the Universe without approaching singularity or even a steadystate regime... Finally, note that the model mentioned is most optimistic because it does not lead to the extermination of life as a result of the unlimited expansion of the Universe and of a density decrease or collapse to singularity. This statement also may be accepted as part of the Anthropic Cosmological Principle." 143. "Effects on the Structure of the Universe of an Accelerating Expansion," G. A. Baker, Jr., General Relativity and Gravitation 34, 767-791 (2002). (A) This paper investigates inhomogeneous mass-distributions in the background universe dominated by cosmological constant. "[I]t appears that for larger scale structures composed of galaxies and inter-galactic space, the observed increase in the rate of expansion may be an important feature in determining the size of self-bound gravitating systems. For smaller structures like galaxies, globular clusters, etc. other mechanisms are presumably dominant." D. Information processing, intelligent beings, and the cosmological future Dyson taught us in his seminal paper (Ref. 98) that, "It is impossible to calculate in detail the long-range future of the universe without including the effects of life and intelligence. It is impossible to calculate the capabilities of life and intelligence without touching, at least peripherally, philosophical questions. If we are to examine how intelligent life may be able to guide the physical development of the universe for its own purposes, we cannot altogether avoid considering what the values and purposes of intelligent life may be. But as soon as we mention the words value and purpose, we run into one of the most firmly entrenched taboos of twentieth-century science." The authors listed below have tried to undermine this taboo. Nonetheless, it should be noted that discussions of life and information 30 processing are still on a different footing than predictions of the future evolution of stars, stellar systems, and the physical universe. The laws of physics are relatively well-known, and even battle-tested. We are still trying to figure out the basic definitions of life, and are far from having a deep, predictive theory of life and intelligence. In spite of this limitation, however, progress can be made and the battle is still raging. 144. "Cosmological limits on computation," F. J. Tipler, International Journal of Theoretical Physics 25, 617-661 (1986). (A) The basic paper on the crucial link among astrophysical evolution, information theory, and intelligent communities.145. "Life after inflation," A. D. Linde, Physics Letters B 211, 29-31 (1988). (A) In the very first sentence the author states that "one of the main purposes of science is to investigate the future evolution of life in the universe"; concludes that our cosmological domain probably will evolve into an exponential black hole containing inflationary regions inside on huge timescales of ∼10 10000 years! Suggests a "moving" strategy for indefinite survival of intelligent species. 146. "World as system self-synthesized by quantum networking," J. A. Wheeler, IBM Journal of Research and Development 32, 4-15 (1988). (I) This beautifully written paper expounds Wheeler's celebrated notion of the participatory universe; there are several passages of relevance to PE, for instance: "Minuscule though the part is today that such acts of observer-participancy play in the scheme of things, there are billions of years to come. There are billions upon billions of living places yet to be inhabited. The coming explosion of life opens the door to an all-encompassing role for observerparticipancy: to build, in time to come, no minor part of what we call its past-our past, present and future-but this whole vast world." 147. "The ultimate fate of life in universes which undergo inflation," F. J. Tipler, Physics Letters B 286, 36-43 (1992). (A) Criticizes Linde's optimism (cf. Ref. 145) as far as survival of life and intelligence in (chaotic) inflationary universes. 148. "Life at the End of the Universe?" G. F. R. Ellis and D. H. Coule, General Relativity and Gravitation 26, 731-739 (1994). (I) A critical comment on Ref. 147. 149. "Possible Implications of the Quantum Theory of Gravity," L. Crane (1994), preprint hep-th/9402104. (E) Expounds what the author calls the meduso-anthropic principleadvanced civilizations creating black holes as a way of proliferating universes in Smolin's manner! "Although it has been generally believed by people with a scientific frame of mind that human life and history take place within the rule of physical law, it has generally been assumed that the relationship between the specific laws of physics and human events was complex and accidental. This has, in fact, placed science in conflict with the otherwise dominant current of Western (and by no means only Western) thought." 150. "Can the Universe create itself?" J. R. Gott, III and L.-X. Li, Physical Review D 58, 023501-1/43 (1998). (A) Opening sections of this remarkable paper briefly consider fates of various cosmological models from the point of view of quantum cosmologies. Section X deals with "baby universes" and possible role of advanced intelligent communities in creating them. Contains one of the best relevant bibliographies. 151. "Eternal inflation, black holes, and the future of civilizations," J. Garriga, V. F. Mukhanov, K. D. Olum, and A. Vilenkin, International Journal of Theoretical Physics 39, 1887-1900 (2000). (A) Considers in detail the problem of information transmission from one inflating region to another; concludes that obstacles (mainly in the form of quantum-energy conditions) are formidable, but that there still is room for the total number of civilized regions in the branching tree of universes to be infinite. 152. "The Physics of Information Processing Superobjects: Daily Life Among the Jupiter Brains," A. Sandberg, Journal of Transhumanism 5 (now Journal of Evolution and Technology, at http://www.jetpress.org/volume5/Brains2.pdf), 1-34 (2000). (A) Analyses specific information technologies available to far-future human or advanced extraterrestrial civilizations; many issues are related to PE, which is explicitly considered in §8.4. 153. "Cosmological Constant and the Final Anthropic Hypothesis," M. M. Ćirković and N. Bostrom, Astrophysics and Space Science 274, 675-687 (2000). (I) Reformulates the Final Anthropic Principle of Barrow and Tipler (Ref. 47) into a serious hypothesis about the physical universe. The authors investigate the chances of such a Final Anthropic Hypothesis being true in the realistic cosmological model, dominated by cosmological constant.154. "Ultimate physical limits to computation," S. Lloyd, Nature 406, 1047-1054 (2000). (A) Although it does not explicitly address PE issues, this paper is important for Lloyd's bold speculations on the future computing technologies, as well as on the computing capacities of black holes. Compare Refs. 144, 152, and 155.155. "On The Maximal Quantity Of Processed Information In The Physical Eschatological Context," M. M. Ćirković and M. Radujkov, Serbian Astronomical Journal 163, 53-56 (2001). (I) 156. "The Ultimate Fate of Life in an Accelerating Universe," K. Freese and W. H. Kinney, Astrophysical Journal, in press (2002) (preprint astro-ph/0205279). (A) Compare to Refs. 98 and 104. Attempts to salvage some of the optimism of the former, arguing that in models going beyond the simplest accelerating expansion, the Dysonian hybernation method might be feasible, in spite of the conclusions of Ref. 104.E. Vacuum decay in the future and other quantum-field apocalypsesA small industry has grown up around the notion of a possible future vacuum phase transition. This is not only an eschatological issue in the most literal sense, but it also is connected with the topic of technological development and the capacities of intelligent communities, since the basic idea is that such communities may trigger the phase transition (presumably unwittingly) by conducting very high-energy physical experiments. Although admittedly smacking of science fiction, this idea has been taken seriously even by high-level administrators of modern particle-accelerator laboratories (Ref. 165)! The reason is easy to understand: even if the chance of such an occurence is exceedingly small, its catastrophical ecological impact is incomparably greater than any other conceivable threat, so it deserves close scrutiny. Topics usually investigated together with the vacuum phase-transition threat are the accidental production of strangelets or even mini black-holes in high-energy experiments.157. "Gravitational Effects on and of Vacuum Decay," S. Coleman and F. De Luccia, Physical Review D 21, 3305-3315 (1980). (A) Classical paper, always quoted in connection with vacuum phase transition at late cosmological times.158. "Is our vacuum metastable?" M. S. Turner and F. Wilczek, Nature 298, 633-634 (1982). (A) 159. "How stable is our vacuum?" P. Hut and M. J. Rees, Nature 302, 508-509 (1983). (A) First mention of the possibility that the vacuum phase transition may be induced by high-energy physics experiments; rejects the idea for foreseeable human technologies on the basis of comparison with natural cosmic-ray interactions. 160. "Cosmic-ray induced vacuum decay in the Standard model," M. Sher and H. W. Zaglauer, Physics Letters B 206, 527-532 (1988). (A) 161. "Comment on 'Slightly massive photon,'" M. Sher, Physical Review D 39, 3513-3514 (1989). (A) Contains a brief discussion of possible phase transition at late cosmological times. 162. "The environmental impact of vacuum decay," M. M. Crone and M. Sher, American Journal of Physics 59, 25-32 (1991). (I) 163. "Will relativistic heavy-ion colliders destroy our planet?" A. Dar, A. De Rújula, and U. Heinz, Physics Letters B 470, 142-148 (1999). (A) 164. "Problems with empirical bounds for strangelet production at RHIC," A. Kent (2000), preprint hep-ph/0009130. (A) 165. "Review of speculative 'disaster scenarios' at RHIC," R. L. Jaffe, W. Busza, F. Wilczek, and J. Sandweiss, Reviews of Modern Physics 72, 1125-1140 (2000). (A) An officially commissioned study of possible hazardous scenarios dealing with inducing vacuum phase transitions or strangelet production at energies available to the new Brookhaven heavy ion collider. 166. "A critical look at catastrophe risk assessments," A. Kent, Risk, in press (2002) (preprint hep-ph/0009204). (A) A criticism of the conclusions of Ref. 165 from the "devil's advocate" point of view. V. PHILOSOPHY, THEOLOGY, SOCIOLOGY OF THE FUTURE A. Theological, philosophical, sociological inferences As mentioned above, eschatological issues have been understood traditionally as part of the religious, rather than the scientific domain. The transition that occurred mainly in the 1920s (Refs. 1-4) led to the realization that the physical sciences and, ultimately, technology may be used to predict and influence the future on a large scale. This should not be construed, however, as severing all of the links between religious and physical eschatology. The most obvious (although probably not the most instructive) example of the persisting interaction 177. "Cosmological Forecast and Its Practical Significance," M. M. Ćirković, Journal of Evolution called Doomsday Argument, which was conceived (but not published) by the astrophysicist Brandon Carter in the early 1980s, and first expounded in print by John Leslie in 1989 (Ref. 178) and by Richard Gott in 1993 (Ref. 181). The most comprehensive discussion of the issues involved is Leslie's monograph of 1996, The End of The World (Ref. 188). The core (E) A review-somewhat jovial-of the seminal paper by Adams and Laughlin (Ref.39. "The Future of the Universe," F. C. Adams, and G. Laughlin, Sky & Telescope 96The conventional approach to eschatological issues is exemplified by the cursory (though not unsympathetic) treatments in general cosmological reviews, such as the following three.43. "Our Universe and Others," M. J. Rees, Quarterly Journal of the Royal Astronomical The Halley Lecture for 1985). (I) It devotes precious little space to the questions of the future, exemplifying the prevailing (misguided) notion that the cosmological future is somehow less interesting than the past, but it does contain a Topics pertaining to PE have gained a disproportionate amount of attention in popular or semi-popular books, in comparison to the volume of the research literature in the field. This is unusual, since in scientific fields a large number of research papers usually appear in print before the first popular expositions. Consider, for instance, research on the cosmic microwave background (CMB) or even on extrasolar planetology, disciplines that bear some similarity to PE. Although we may speculate why these fields are the reverse of PE (I provide a close-to-exhaustive list of research publications on PE in the following sections), one reason may be a cultural bias towards the future in many strands of Western life during the last quarter century (and particularly after the end of the Cold War). Many popular books, however, do devote much space to "classical" cosmological issues; this is natural in light of the relative scarcity of results in PE proper. 47. The Anthropic Cosmological Principle, J. D. Barrow and F. J. Tipler, (Oxford University Press, New York, 1986). (A) Chapter X of this famous-but fairly controversial-book is devoted to physical-eschatological issues. For a detailed bibliography of reviews and reactions to this book up to 1991, see Ref. 26. 48. The Omega Point: The Search for the Missing Mass and the Ultimate Fate of the 50. End: Cosmic Catastrophes and the Fate of the Universe, F. Close (Simon & Schuster, New York, 1988). (E) 51. The Last Three Minutes, P. C. W. Davies (Basic Books, New York, 1994). (E)69). (August), 32-39 (1998). (E) A popular exposition of the research in PE from the pen of its two distinguished protagonists; compare Refs. 53 and 69. 40. "The Great Cosmic Battle," F. C. Adams and G. Laughlin, Mercury 29 (January/February), 10-15 (2000). (E) 41. "Embracing the End: When the Stars Burn Out," F. C. Adams and G. Laughlin, Astronomy 28 (October), 48-53 (2000). (E) 42. "The Galactic Millenium," G. Laughlin and F. C. Adams, Astronomy 29 (November), 38-45 (2001). (E) Compare with Ref. 73. Society 22, 109-124 (Fourth Milne Lecture) (1981). (E) Part of this breathtaking essay is devoted to reviewing Rees's own (closed-cosmologies) and Dyson's (open/flat- cosmologies) PE results. 44. "The Universe-Present, Past and Future," M. S. Longair, Observatory 105, 171-188 (1985) (wonderful remark: "The future of our Universe is a splendid topic for after-dinner speculation." 45. "The Epoch of Observational Cosmology," T. Rothman, and G. F. R. Ellis, Observatory 107, 24-29 (1987). (I) E. Books 11 46. The Ultimate Fate of the Universe, J. N. Islam (Cambridge University Press, Cambridge, 1983). (E) See Refs. 29, 31, 32, 34, and 97. Universe, J. Gribbin (Bantam, New York and Heinemann, London, 1987). (E) 49. Infinite in all Directions, F. Dyson (Harper & Row, New York, 1988). (E) 52. The Physics of Immortality, F. J. Tipler (Doubleday, New York, 1994). (E) Although the single most controversial reference here, this book is otherwise very hard to classify. It expounds a particular cosmological model-of the topologically-closed and recollapsing-universe type-and interprets it in quasireligious terms, which sometimes seem appropriate, but mostly just funny or absurd. For severe criticisms of Tipler's approach, see Refs. 148, 167, 171, 173 and 175. 53. The Five Ages of the Universe, F. C. Adams and G. Laughlin (The Free Press, New York, 1999). (E,I) This is a beautiful popular exposition of the crucial specialized article by the same authors on the topic (Ref. 69), enriched with much of the "classical" cosmological lore, in particular, inflationary models and the primordial nucleosynthesis. 54. The Future of the Universe: Chance, Chaos, God?, A. Benz (Continuum, New York, 2000). (E) Contemporary astrophysics reviewed from an openly theist viewpoint; part IV deals with PE. See also Refs. 167-176. 55. Our Cosmic Future: Humanity's Fate in the Universe, N. Prantzos, translated by Stephen Lyle (Cambridge University Press, Cambridge, 2000) (E) Rather technologically and optimistically oriented survey of the future; chapter 4 deals with PE the two is Tipler's book (Ref. 52), which left a lasting impression on its scientific and philosophical readers, as seen in the references below. the two is Tipler's book (Ref. 52), which left a lasting impression on its scientific and philosophical readers, as seen in the references below. Is Religion Refuted by Physics or Astronomy. Vistas in Astronomy. 10Herman ZanstraI) A companion paper to Ref. 20. Contrasts, among other things. Teilhard de Chardin's eschatological theory to our knowledge about the expanding universe"Is Religion Refuted by Physics or Astronomy," Herman Zanstra, Vistas in Astronomy 10, 1-22 (1968). (I) A companion paper to Ref. 20. Contrasts, among other things, Teilhard de Chardin's eschatological theory to our knowledge about the expanding universe. The Omega Point as Eschaton: Answers to Pannenberg's Questions for Scientists. F. J. Tipler, Zygon. 24I)"The Omega Point as Eschaton: Answers to Pannenberg's Questions for Scientists," F. J. Tipler, Zygon 24, 217-253 (1989). (I) London, 1992). (I) Gifford Lectures containing an over-skeptical and often-rhetorical critique of Tipler's Omega-point theory. Science as Salvation: A Modern Myth and its Meaning, M. Midgley (Routledge. Science as Salvation: A Modern Myth and its Meaning, M. Midgley (Routledge, London, 1992). (I) Gifford Lectures containing an over-skeptical and often-rhetorical critique of Tipler's Omega-point theory. The Metaethical Alternative to the Idea of Eternal Life in Modern Cosmology. A. V. Nesteruk, Diotima. 21E)"The Metaethical Alternative to the Idea of Eternal Life in Modern Cosmology," A. V. Nesteruk, Diotima 21, 70-74 (1993). (E) The Idea of Eternal Life in Modern Cosmology: Its Ultimate Reality and Metaethical Meaning. A. V. Nesteruk, Ultimate Reality and Meaning. 17115Nature. E) A very strong and sometimes unwarranted criticism of Tipler's theory"The Idea of Eternal Life in Modern Cosmology: Its Ultimate Reality and Metaethical Meaning," A. V. Nesteruk, Ultimate Reality and Meaning 17, 222-231 (1994). (I) 172. "Piety in the Sky," G. F. R. Ellis, Nature 371, 115 (1994). (E) A very strong and sometimes unwarranted criticism of Tipler's theory. The Final Anthropic Cosmology as Seen by Transcedental Philosophy: Its Underlying Theology and Ethical Contradiction. The Interplay Between Scientific and Theological Worldviews, Part I. 5"The Final Anthropic Cosmology as Seen by Transcedental Philosophy: Its Underlying Theology and Ethical Contradiction," A. V. Nesteruk in Studies in Science and Theology, Volume 5: The Interplay Between Scientific and Theological Worldviews, Part I, 43-54 (1997). There are no limits to the open society. F. J. Tipler, Critical Rationalist. 302E) Puts the Omega-point theory in a Popperian context"There are no limits to the open society," F. J. Tipler, Critical Rationalist 3, no. 02 (available at http://www.eeng.dcu.ie/~tkpw/tcr/volume-03/), 1-20 (1998). (E) Puts the Omega-point theory in a Popperian context. (E) Another harsh criticism of Tipler's Omega Point theory from a philosophical viewpoint. Uses-rather superficially and unfairly-Tipler's theory as a yardstick for all of physical eschatology. 39Colonising the Galaxies"Colonising the Galaxies," G. Oppy, Sophia 39, 117-141 (2000). (E) Another harsh criticism of Tipler's Omega Point theory from a philosophical viewpoint. Uses-rather superficially and unfairly-Tipler's theory as a yardstick for all of physical eschatology. ). (I) Gives arguments to the effect that emotional involvement is inappropriate when dealing with bleak eschatological perspectives of life and intelligence. G. Oppy, Philo. 42Physical Eschatology"Physical Eschatology," G. Oppy, Philo 4, (available at http://www.philoonline.org/) no. 2 (2001). (I) Gives arguments to the effect that emotional involvement is inappropriate when dealing with bleak eschatological perspectives of life and intelligence. E) The very first exposition of the Doomsday Argument in print. J. Leslie, Bulletin of the Canadian Nuclear Society. 21Risking the World's End"Risking the World's End," J. Leslie, Bulletin of the Canadian Nuclear Society 21 (May 1989), 10-15 (1989). (E) The very first exposition of the Doomsday Argument in print. (I) First suggestion of what came to be called the "Self-Indication Assumption" as an answer to the Doomsday Argument conundrum. J Leslie, D. Dieks, Philosophical Quarterly. 40Doomsday -Or: The Dangers of Statistics. roughly suggests that your existence favors the existence of many observers in the universe"Is the end of the world nigh?" J. Leslie, Philosophical Quarterly 40, 65-72 (1990). (I) 180. "Doomsday -Or: The Dangers of Statistics," D. Dieks, Philosophical Quarterly 42, 78-84 (1992). (I) First suggestion of what came to be called the "Self-Indication Assumption" as an answer to the Doomsday Argument conundrum; roughly suggests that your existence favors the existence of many observers in the universe. Implications of the Copernican principle for our future prospects. J R N Gott ; S, Goodman, P. Buch. A. L. Mackay363Nature. I). "Implications of the Copernican principle for our future prospects," J. R. Gott, Nature 363, 315-319 (1993). (I) Gott's-rather fragile-version of the Doomsday Argument. 182. "Future prospects discussed," S. N. Goodman, Nature 368, 106-107 (1994). (I) 183. "Future prospects discussed," A. L. Mackay, Nature 368, 107 (1994). (I) 184. "Future prospects discussed," P. Buch, 1994, Nature 368, 107-108. (I) 108. (I) Gott's reply to criticisms published in Nature (Refs. 182-184) of his version of the Doomsday Argument. J. R. Gott. 368Future Prospects Discussed: Gott Replies"Future Prospects Discussed: Gott Replies," J. R. Gott, 1994, Nature 368, 108. (I) Gott's reply to criticisms published in Nature (Refs. 182-184) of his version of the Doomsday Argument. Too Soon for the Doom Gloom. T Kopf, P Krtous, D N Page, gr- qc/9407002A) Proves that the Self-Indication Assumption exactly cancels the Doomsday Argument probability shift"Too Soon for the Doom Gloom?" T. Kopf, P. Krtous, and D. N. Page, preprint gr- qc/9407002 (1994). (A) Proves that the Self-Indication Assumption exactly cancels the Doomsday Argument probability shift. E) Elaboration of Gott's views of Refs. J. R. Gott, in Ref. 58185Our future in the universe"Our future in the universe," J. R. Gott, in Ref. 58, pp. 140-151 (1996). (E) Elaboration of Gott's views of Refs. 181 and 185. (I) Monograph largely inspired by the Doomsday Argument. LondonThe End of the World: The Ethics and Science of Human Extinction, J. Leslie (Routledgebut containing a lot of interesting empirical material on possible threats to humanityThe End of the World: The Ethics and Science of Human Extinction, J. Leslie (Routledge, London, 1996). (I) Monograph largely inspired by the Doomsday Argument, but containing a lot of interesting empirical material on possible threats to humanity. Doom Soon. T. Tännsjö, Inquiry. 40I)"Doom Soon?" T. Tännsjö, Inquiry 40, 243-252 (1997). (I) A Refutation of the Doomsday Argument. K K Korb, J J Oliver, 107I) Lists several-rather intuitive-arguments against the conclusion of the Doomsday Argument"A Refutation of the Doomsday Argument," K. K. Korb and J. J. Oliver, Mind 107, 403-410 (1998). (I) Lists several-rather intuitive-arguments against the conclusion of the Doomsday Argument. How to predict everything: Has the physicist J. Richard Gott really found a way. "How to predict everything: Has the physicist J. Richard Gott really found a way?" T. E) A review of Gott's version of the Doomsday Argument. Ferris, The New Yorker. 75Ferris, The New Yorker 75 (July 12 1999), 35-39 (1999). (E) A review of Gott's version of the Doomsday Argument. The Doomsday Argument is Alive and Kicking. N. Bostrom, Mind. 108I) A successful reply to Korb and Oliver"The Doomsday Argument is Alive and Kicking," N. Bostrom, Mind 108, 539-550 (1999). (I) A successful reply to Korb and Oliver. Comment on Nick Bostrom's 'The Doomsday Argument is Alive and Kicking. "Comment on Nick Bostrom's 'The Doomsday Argument is Alive and Kicking'," K. . K Korb, J J Oliver, 108I)K. Korb and J. J. Oliver, Mind 108, 551-553 (1999). (I) No one knows the date or the hour: an unorthodox application of Rev. Bayes' Theorem. C Hitchcock, Philosophy of Science. 66"No one knows the date or the hour: an unorthodox application of Rev. Bayes' Theorem," P. Bartha and C. Hitchcock, Philosophy of Science 66, S339-S353 (1999). The Shooting-Room Paradox and Conditionalizing on 'Measurably Challenged' Sets. C Hitchcock, Synthese. 118A)195. "The Shooting-Room Paradox and Conditionalizing on 'Measurably Challenged' Sets," P. Bartha and C. Hitchcock, Synthese 118, 403-437 (1999). (A) French). (A) Develops the analogy between Hempel's raven paradox and the Doomsday Argument. P. Franceschi, The Canadian Journal of Philosophy. 29Comment l'Urne de Carter et Leslie se Déverse dans celle de Hempelin"Comment l'Urne de Carter et Leslie se Déverse dans celle de Hempel," P. Franceschi, The Canadian Journal of Philosophy 29, 139-156 (1999) (in French). (A) Develops the analogy between Hempel's raven paradox and the Doomsday Argument. A) Summarizes causal problems inherent in the underlying assumption of the Doomsday Argument. U N Adam &amp; Eve, Quantum ++, Joe, C. Caves, Contemporary Physics. 41(I) Attempts to refute Gott's version (Refs. 181, 185, 187) of the Doomsday Argument. 198. christened by Bostrom as the Self-Sampling Assumption"Predicting Future Duration from Present Age: A Critical Assessment," C. Caves, Contemporary Physics 41, 143-153 (2000). (I) Attempts to refute Gott's version (Refs. 181, 185, 187) of the Doomsday Argument. 198. "The Doomsday Argument, Adam & Eve, UN ++ and Quantum Joe," N. Bostrom, Synthese 127, 359-387 (2001). (A) Summarizes causal problems inherent in the underlying assumption of the Doomsday Argument, christened by Bostrom as the Self- Sampling Assumption. A) Argues for acceptance of the Self-Indication Assumption in anthropic reasoning. K D Olum, Philosophical Quarterly. 52The doomsday argument and the number of possible observers"The doomsday argument and the number of possible observers," K. D. Olum, Philosophical Quarterly 52, 164-184 (2002). (A) Argues for acceptance of the Self- Indication Assumption in anthropic reasoning. A) Argues that the Self-Indication Assumption is a poor guideline in dealing with the Doomsday Argument. M M Ćirković, Philosophical Quarterly. The Doomsday Argument and the Self-Indication Assumption: Reply to Olum. criticizes Ref. 199"The Doomsday Argument and the Self-Indication Assumption: Reply to Olum," N. Bostrom and M. M. Ćirković, Philosophical Quarterly, in press (scheduled for January 2003). (A) Argues that the Self-Indication Assumption is a poor guideline in dealing with the Doomsday Argument; criticizes Ref. 199. A) A wonderfully detailed treatment of many facets of anthropic reasoning, including both the Doomsday Argument and the issue of statistical prediction in cosmology (and PE). Anthropic Bias: Observation Selection Effects, N. Bostrom, (Routledge. New YorkAnthropic Bias: Observation Selection Effects, N. Bostrom, (Routledge, New York, 2002). (A) A wonderfully detailed treatment of many facets of anthropic reasoning, including both the Doomsday Argument and the issue of statistical prediction in cosmology (and PE). A Critique of Two Versions of the Doomsday Argument -Gott's Line and Leslie's Wedge. E. Sober, Synthese. in press (scheduled for early 2003). (A)"A Critique of Two Versions of the Doomsday Argument -Gott's Line and Leslie's Wedge," E. Sober, Synthese, in press (scheduled for early 2003). (A) . De Rerum Lucretius, Natura, ca. 50 BCLucretius, De Rerum Natura (ca. 50 BC) The Rubáiyát (ca. 1100). Omar Khayyám, Omar Khayyám, The Rubáiyát (ca. 1100) Robert Frost, Fire and Ice. Robert Frost, Fire and Ice (1920) No field is more pregnant with the future than gravitational collapse. No more revolutionary views of man and the universe has one ever been driven to consider seriously than those that come out of pondering the paradox of collapse. No predictions subject to early test are more entrancing than those on the formation and properties of a black hole. laboratory model" for some of what is predicted for the universe itself. the greatest crisis of physics of all timeNo predictions subject to early test are more entrancing than those on the formation and properties of a black hole, "laboratory model" for some of what is predicted for the universe itself. No field is more pregnant with the future than gravitational collapse. No more revolutionary views of man and the universe has one ever been driven to consider seriously than those that come out of pondering the paradox of collapse, the greatest crisis of physics of all time. Charles Misner, Kip Thorne, John A Wheeler, Gravitation. Charles Misner, Kip Thorne, and John A. Wheeler, Gravitation (1973) The world of brute matter offers room for great but limited growth. The world of mind and pattern, though, holds room for endless evolution and change. The possible seems room enough. The world of brute matter offers room for great but limited growth. The world of mind and pattern, though, holds room for endless evolution and change. The possible seems room enough. . K Eric Drexler, Engines of CreationK. Eric Drexler, Engines of Creation (1987) One of the main purposes of science is to investigate the future evolution of life in the universe. One of the main purposes of science is to investigate the future evolution of life in the universe. Andrei Linde, Inflation and Quantum Cosmology. Andrei Linde, Inflation and Quantum Cosmology (1990) In my view, the future of the universe is as interesting as its past and so I do not understand why there are not many more papers on this topic. In my view, the future of the universe is as interesting as its past and so I do not understand why there are not many more papers on this topic. . Abraham Loeb, private communicationAbraham Loeb, private communication (2001) What in the world is physical eschatology? Anonymous referee. rejecting a previous manuscript of the authorWhat in the world is physical eschatology? Anonymous referee, rejecting a previous manuscript of the author (2001) My foremost thanks go to Roger H. Stuewer, for his kind help, encouragement, and careful editorial work on a previous version of this manuscript. I am also happy to express special gratitude to Vesna Milošević-Zdjelar, Branislav K. Acknowledgements, Nikolić, OlgaAcknowledgements. My foremost thanks go to Roger H. Stuewer, for his kind help, encouragement, and careful editorial work on a previous version of this manuscript. I am also happy to express special gratitude to Vesna Milošević-Zdjelar, Branislav K. Nikolić, Olga Nedeljković for their invaluable help in locating several hard-to-find references, as well as for their useful comments. Vladan Čelebonović and the Dutch Embassy in Belgrade kindly enabled obtaining several electronic subscriptions wherein important references were found. The useful suggestions and encouragement of. Milan Latinović, Saša Bogosavljević, Latinović, Milan Bogosavljević, and Saša Nedeljković for their invaluable help in locating several hard-to-find references, as well as for their useful comments. Vladan Čelebonović and the Dutch Embassy in Belgrade kindly enabled obtaining several electronic subscriptions wherein important references were found. The useful suggestions and encouragement of . Freeman J Dyson, Zoran Živković, Fred C. Adams, Nick Bostrom, Sir Martin J. Rees, PetarFreeman J. Dyson, Zoran Živković, Fred C. Adams, Nick Bostrom, Sir Martin J. Rees, Petar . Robert Grujić, Yuri Bradbury, George Balashov, Marina Musser, Radujkov, Srdjan Kacper Rafal Rybicki, Ken D Samurović, Jelena Olum, Ivana Milogradov-Turin, John D Dragićević, Srdjan Barrow, Mark Keča, Zoran Walker, Knežević, and Abraham Loeb are also hereby acknowledgedGrujić, Robert Bradbury, Yuri Balashov, George Musser, Marina Radujkov, Kacper Rafal Rybicki, Srdjan Samurović, Ken D. Olum, Jelena Milogradov-Turin, Ivana Dragićević, John D. Barrow, Srdjan Keča, Mark Walker, Zoran Knežević, and Abraham Loeb are also hereby acknowledged.
{'fraction_non_alphanumeric': 0.058607413372253954, 'fraction_numerical': 0.03694104028787841, 'mean_word_length': 4.528973913043478, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 4, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'This Resource Letter treats the nascent discipline of physical eschatology, which deals with the future evolution of astrophysical objects, including the universe itself, and is thus both a counterpart and a complement to conventional cosmology. While sporadic interest in these topics has flared up from time to time during the entire history of humanity, a truly physical treatment of these issues has only become possible during the last quarter century. This Resource Letter deals with these recent developments. It offers a starting point for understanding what the physical sciences might say about the future of our universe and its constituents. Journal articles, books, and web sites are provided for the following topics: history and epistemology of physical eschatology, the future of the Solar system, the future of stars and stellar systems, the global future of the universe, information processing and intelligent communities, as well as some side issues, like the possible vacuum phase transition and the so-called Doomsday Argument.', 'arxivid': 'astro-ph/0211413', 'author': ['Milan M Ćirković [email protected] \nAstronomical Observatory Belgrade Volgina 7\n11160Belgrade SerbiaYugoslavia\n'], 'authoraffiliation': ['Astronomical Observatory Belgrade Volgina 7\n11160Belgrade SerbiaYugoslavia'], 'corpusid': 119506821, 'doi': '10.1119/1.1528470', 'github_urls': [], 'n_tokens_mistral': 23398, 'n_tokens_neox': 19504, 'n_words': 12350, 'pdfsha': '38ebbf52921f1b4bae8cd1dfcb0593f95b61fd47', 'pdfurls': ['https://arxiv.org/pdf/astro-ph/0211413v1.pdf'], 'title': ['A Resource Letter on Physical Eschatology', 'A Resource Letter on Physical Eschatology'], 'venue': []}
arxiv
29 Mar 2018 Alessandro Carotenuto Scuola Internazionale Superiore di Studi Avanzati (SISSA) Laboratoire de Physique Théorique CNRS Bonomea 265I-34136TriesteItaly D ֒ Ludwik Scuola Internazionale Superiore di Studi Avanzati (SISSA) Laboratoire de Physique Théorique CNRS Bonomea 265I-34136TriesteItaly Abrowski [email protected] Scuola Internazionale Superiore di Studi Avanzati (SISSA) Laboratoire de Physique Théorique CNRS Bonomea 265I-34136TriesteItaly Michel Dubois-Violette [email protected] Université Paris-Sud Université Paris-Saclay Bâtiment 210F-91405OrsayFrance 29 Mar 2018arXiv:1803.08373v3 [math.QA] DIFFERENTIAL CALCULUS ON JORDAN ALGEBRAS AND JORDAN MODULES Having in mind applications to particle physics we develop the differential calculus over Jordan algebras and the theory of connections on Jordan modules. In particular we focus on differential calculus over the exceptional Jordan algebra and provide a complete characterization of the theory of connections for free Jordan modules. Introduction It is quite legitimate to expect that the finite spectrum of fundamental particles of matter (fundamental fermions) corresponds to representations of some finite quantum space. Such a virtual space should be described by its observables, i.e. a quantum analogue of some class of real functions over it. This is of course at the core of noncommutative geometry, where C*-algebras correspond to the noncommutative analogues of algebras of continuous complex functions. This formalism, extended also to real C*algebras and enriched with additional structure, has been in particular applied to the Standard Model ( [6], [5], [4], [3], [7], [25]) as well as to other Higgs gauge models ( [9], [10], [8]). Moreover quaternionic C*-algebras, seen as generalizations of the algebra of continuous quaternionic functions on a classical space, have been studied within this formalism. However already at the early beginning of quantum theory it was pointed out that the appropriate algebraic structures for finite quantum systems are the finite-dimensional formally real Jordan algebras, since this is the right framework in which one has spectral theory and the physical interpretation in terms of observables and states ( [19], [13]). The real vector space of self-adjoint elements of a C* algebra is a formally real Jordan algebra (it is in fact a JB-algebra which implies the formal reality and is equivalent to it in finite dimension, see e.g. [23] or [16]). In fact Jordan subalgebras of this kind of algebras cover almost all the possible cases, with the only exception of the real Albert factor, which is the 27-dimensional algebra of three by three hermitian matrices with octonionic entries [1]. In a recent work [11] it has been suggested that this exceptional algebra may play a key role in the description of the internal space of fundamental fermions in the Standard Model of particle physics and in particular, that the implicit triality underlining the exceptional algebra may be related to the three generations of fundamental fermions. The aim of this work is to outset the representation theory of formally real (also called Euclidean) finite-dimensional finite-dimensional Jordan algebras. We investigate Jordan modules over Jordan algebras and elaborate differential calculus and the theory of connections on Jordan modules. From a physical point of view, this corresponds to develop gauge theories for a quantum theory in which one allows for Jordan algebras as algebras of observables. It is needless to say that the groups of automorphisms of the Jordan algebras play a fundamental role in this theory (for instance acting as gauge group). We also provide certain more general constructions in the setting of Jordan algebras, and also in a broader setting of noncommutative and nonassociative algebras. Jordan algebras and finite quantum spaces Here and in the following a Jordan algebra is meant to be unital and finitedimensional if not differently specified. We will provide some definitions for the category of all algebras, thus the term algebra without further specification will generally denote a noncommutative and nonassociative algebra. Also every vector space is meant as vector space over the field of real numbers if not differently specified. We use the Einstein summation for repeated up-down indices. Definition 2.1. A Jordan algebra is a vector space J together with a bilinear product • : J × J → J such that x • y = y • x(1) and x • (y • x 2 ) = (x • y) • x 2(2) for any x, y ∈ J. Condition (2) is called Jordan identity and it is equivalent to [L x•y , L z ] + [L z•x , L y ] + [L y•z , L x ] = 0(3) where L x denotes the (left) multiplication by x ∈ J. We provide two examples of Jordan algebras. Example 2.2. Let (A, ) be an associative algebra, we define the Jordan algebra A + = (A, •) to be the vector space A equipped with the product given by x • y = 1 2 (xy + yx)(4) for all x, y ∈ A. One verifies by direct check of properties (1) and (2) that A + is a Jordan algebra. Any Jordan algebra which is isomorphic to a Jordan subalgebra of a Jordan algebra of this kind is called a special Jordan algebra. In particular if the associative algebra A is endowed with an involution * : A → A that is (x * ) * = x (xy) * = y * x * then the subspace A sa = {a ∈ A | a * = a} of self-adjoint elements in A is not a subalgebra of (A, ) but it is a Jordan subalgebra of the special Jordan algebra A + and it is therefore special ⋄ Example 2.3. The exceptional Jordan algebra (J 8 3 , •) is defined as follows: its elements are 3 × 3 hermitian matrices with octonionic entries J 8 3 = {x ∈ M 3 (O) | x = x * } and the product • is given by the anticommmutator x • y = 1 2 (xy + yx)(5) for any x, y ∈ J 8 3 . It is a classical result ( [1]) that J 8 3 is a Jordan algebra which is not a special one. ⋄ In the following we shall write xy for the product of two elements in Jordan algebras (and in other kind of algebras) when no confusion arises. Definition 2.4. An Euclidean (or formally real) Jordan algebra is a real Jordan algebra J satisfying the formal reality condition x 2 + y 2 = 0 ⇔ x = y = 0(6) for any x, y ∈ J. Any Euclidean finite-dimensional Jordan algebra J has a unit, moreover if x ∈ J there is a spectral resolution of x with real eigenvalues (see [11] for more details). The above examples are quite exhaustive in view of the following classical theorem ( [19]). Theorem 2.5. Any finite-dimensional Euclidean Jordan algebra is a finite direct sum of simple Euclidean finite-dimensional Jordan algebras. Any finite-dimensional simple Euclidean Jordan algebra is isomorphic to one of the following: R, JSpin n+2 = J n+1 2 = R ⊕ R n+2 , J 1 n+3 , J 2 n+3 , J 4 n+3 , J 8 3 for n ∈ N. In the above statement J 8 3 is the only non special Jordan algebra, while J 1 n , J 2 n , J 4 n denote n × n hermitian matrices with real, complex and quaternionic entries respectively, with product given by the anticommutator. JSpin n = R ⊕ R n are the spin factors equipped with the product (s ⊕ v)(s ′ ⊕ v ′ ) = (ss ′ + v, v ′ ) ⊕ (sv ′ + s ′ v)(7) where ·, · denotes the Euclidean scalar product on R n . The spin factor JSpin 1 is absent from this list since it is isomorphic to the commutative and associative algebra R 2 which is not simple. The following isomorphisms hold: J 1 1 = J 2 1 = J 4 1 = J 8 1 = R and J 1 2 = JSpin 2 , J 2 2 = JSpin 3 , J 4 2 = JSpin 5 , J 8 2 = JSpin 9 while J 8 n is not a Jordan algebra for n ≥ 4. This list gives all finitedimensional Jordan algebras corresponding to finite quantum spaces. Center and derivations Definition 3.1. Let A be an algebra, define the associator by [x, y, z] = (xy)z − x(yz)(8) for any x, y, z ∈ A. The center of A, denoted by Z(A), is the associative and commutative subalgebra of elements z ∈ A satisfying [x, z] = 0, [x, y, z] = [x, z, y] = [z, x, y] = 0(9) for any x, y ∈ A. One has the following result. for all x, y ∈ A. Proof. The condition [x, z] = 0 is trivial for any x, z ∈ A since we have taken A commutative. If the condition [x, y, z] = 0 holds, then for every x, y ∈ A one has: 0 = [x, y, z] − [y, x, z] = [y, z, x](11) and 0 = −[x, y, z] = [z, x, y](12) for any x, y ∈ A, in view of the commutativity. In particular, the proposition above is valid for all Jordan algebras. Definition 3.3. A derivation of an algebra A is a linear endomorphism X of A, such that one has X(xy) = X(x)y + xX(y) for all x, y ∈ A. 3. The center of A is stable with respect to derivations, that is X(z) ∈ Z(A) for all X ∈ Der(A) and for any z ∈ Z(A). 4. The following formula holds: [X 1 , zX 2 ] = X 1 (z)X 2 + z[X 1 , X 2 ](14) for all X 1 , X 2 ∈ A and z ∈ Z(A). Proof. (1), (2) and (4) are trivial, we have only to prove stability of the center. Let z ∈ Z(A) and X ∈ Der(A), we have: [x, y, X(z)] = (xy)X(z) − x(yX(z)) = = X ((xy)z) − X(xy)z − (xX(yz) − x (X(y)z)) = = X ((xy)z) − X(xy)z − X(x(yz)) + X(x)(yz) + x (X(y)z) = = X ([x, y, z]) − [X(x), y, z] − [x, X(y), z] = 0(15) for any x, y ∈ A. Similarly one proves that [x, X(z), y] = [X(z), x, y] = 0 and [x, X(z)] = 0. Thus the pair (Z(A), Der(A)) form a Lie-Rinehart algebra ( [24], [15]). For Jordan algebras, the list of derivations for the finite-dimensional nonexceptional simple Euclidean Jordan algebras covers the list of the non exceptional simple Lie algebra, i.e. the Lie algebras denoted by a n , b n , c n and d n in the Cartan classification, while for the exceptional Jordan algebra J 8 3 the algebra of derivations is given by the exceptional Lie algebra f 4 as shown in the following example. Example 3.5. As just mentioned, the Lie algebra of derivations of the exceptional Jordan algebra J 3 8 is the exceptional Lie algebra f 4 (see e.g. [28]). Introduce the standard basis of the exceptional Jordan algebra E 1 =   1 0 0 0 0 0 0 0 0   , E 2 =   0 0 0 0 1 0 0 0 0   , E 3 =   0 0 0 0 0 0 0 0 1   F j 1 =   0 0 0 0 0 ǫ j 0 ǫ j 0   , F j 2 =   0 0 ǫ j 0 0 0 ǫ j 0 0   , F j 3 =   0 ǫ j 0 ǫ j 0 0 0 0 0   where ǫ j j ∈ {0, . .., 7} are a basis of the octonions, so ǫ 0 = 1, ǫ 2 j = −1 for j = 0 and the multiplication table of octonions holds (see e.g. on [2]). As vector space, f 4 admits a decomposition f 4 = D 4 ⊕ M − given as follows. D 4 is the subspace of derivations which annihilates the diagonal of any element in J 8 3 , that is δE i = 0 i ∈ {1, 2, 3} for any δ ∈ D 4 . An interesting and concrete characterization of D 4 is given by the following theorem (see e.g. chapter 2 of [28]). Theorem 3.6. The algebra D 4 is isomorphic to so(8) = d 4 . The isomorphism is given via the equality: δ   ξ 1 x 3 x 2 x 3 ξ 2 x 1 x 2 x 1 ξ 3   =   0 D 3 x 3 D 2 x 2 D 3 x 3 0 D 1 x 1 D 2 x 2 D 1 x 1 0   where δ ∈ D 4 and D 1 , D 2 , D 3 ∈ so(8). D 2 , D 3 are determined by D 1 from the principle of infinitesimal triality (D 1 x)y + x(D 2 )y = D 3 (xy)(16) for any x, y ∈ O. Elements of the vector space M − are 3 × 3 antihermitian octonion matrices with every element on the diagonal equal to zero. Every M ∈ M − defines a linear endomorphismM : J 8 3 → J 8 3 , via the commutator M (x) = M x − xM where in the expression above juxtaposition is understood as the usual raw by column matrix product. ⋄ The following classical result about derivations of Jordan algebras, due to Jacobson ( [17]) and Harris [14], is the equivalent of Witehead's first lemma for Lie algebras. Theorem 3.7. Let J be a finite-dimensional semi-simple Jordan algebra, let X ∈ Der(J). There exists a finite number of couples of elements x i , y i ∈ J such that one has X(z) = [x i , z, y i ](17) for any z ∈ J. Jordan modules The familiar definition of bimodules over associative algebras is not suitable for nonassociative algebras such as Jordan algebras. Indeed, due to nonassociativity, such a definition would imply that a Jordan algebra is not a module over itself if one takes the multiplication as action of the algebra. A more correct definition is the following ( [12], [18], see also [10], [21]): Definition 4.1. Let J be a Jordan algebra, a Jordan bimodule over J is a vector space M together with two bilinear maps J ⊗ M → M x ⊗ m → xm M ⊗ J → M m ⊗ x → mx such that J ⊕ M, endowed with the product (x, m)(x ′ , m ′ ) = xx ′ , xm ′ + mx ′(18) is a Jordan algebra by itself. This definition is equivalent to require the following properties of the action of the Jordan algebra J on its module M :            mx = xm x(x 2 m) = x 2 (mx) (x 2 y)m − x 2 (ym) = 2((xy)(xm) − x(yxm)) 1 J m = m(19) far any x, y ∈ J and m ∈ M. Notice that from the first of relations above one has not to specify if using left or right multiplication so we shall call Jordan module any bimodule over a Jordan algebra. The second relation can also be written as [L x , L x 2 ] = 0(20) while the third is written as L x 2 y − L x 2 L y − 2L xy L x + 2L x L y L x = 0(21) which is equivalent to the conditions L x 3 − 3L x 2 L x + 2L 3 x = 0 [[L x , L y ] , L z ] + L [x,y,z] = 0(22) for every x, y, z ∈ J, where here L x denote the multiplication by x ∈ J in M. Example 4.2. It follows from definition (4.1) that any Jordan algebra J is a module over itself. More generally, let J be a finite-dimensional Jordan algebra, a free J-module M is of the form M = J ⊗ E where E is a finite-dimensional vector space and the action of J on M is given by multiplication on the first component of M. It turns out that, when J is the exceptional Jordan algebra, any finite module over J is a free module [18]. ⋄ Example 4.3. Let A be an associative algebra, let J ⊆ A + be a special Jordan algebra as in example (2.2). Any element x ∈ J is also an element of A and A is endowed with J-module structure by setting L x a = x • a = 1 2 (ax + xa)(23) for any x ∈ J and a ∈ A. In the two following examples the same construction is explicitely given for the antihermitian real, complex and quaternionic matrices as module over hermitian matrices and for the Clifford algebras Cl(R n ) as modules over the spin factors JSpin n . Example 4.4. Denote by A i n (i = 1, 2, 4) the vector space of antihermitian matrices with real, complex and quaternionic entries respectively. A i n is a module over the special Jordan algebra J i n with action given by the matrix anticommutator: L x a = x • a = 1 2 (ax + xa)(24) for any x ∈ J i n and a ∈ A i n . Moreover, taking J i n as free module over itself we have: J i n ⊕ A i n = M i n which is the J-module of n × n real, complex or quaternionic matrices with action of J defined as above by (24). ⋄ Example 4.5. The Clifford algebra Cl (R n ) = T (R n ) ({x ⊗ x = ||x|| 2 , ∀x ∈ R n }) is a module over the Jordan algebra JSpin n = R ⊕ R n with action given by L x [y] = 1 2 ([x ⊗ y] + [y ⊗ x])(25) for any x ∈ R n and [y] ∈ Cl(R n ). for all m ∈ M and x ∈ J. For homomorphisms between free modules over a fixed Jordan algebra, one has the following results. ϕ(x ⊗ v) = x ⊗ Av x ∈ J, v ∈ E(27) where A : M → N is a linear map. Proof. For sake of simplicity, start by taking M = N = J, then a module homomorphism is a linear map f : J → J such that: xϕ(y) = ϕ(xy)(28) for any x, y ∈ J. In particular: ϕ(x) = xϕ(1) = xA(29) for some A ∈ J such that A = ϕ(1). Now, from definition of module homomorphism, we have: ϕ(xy) = (xy)A = xφ(y) = x (yA) ⇒ [x, y, A] = 0(30) for all x, y ∈ J, hence A ∈ Z(J). Thus A ∈ R, in view of simplicity of J. More generally let M = J ⊗ E and N = J ⊗ F, denote as e α and f α a basis of E and F respectively. We have ϕ(1 ⊗ e α ) = A λ α ⊗ f λ(31) for some A λ α ∈ J. With the same argument as above, we get: ϕ(xy ⊗ e α ) = (xy)ϕ(1 ⊗ e α ) = (xy)A λ α ⊗ f λ (32) ϕ(xy ⊗ e α ) = (xϕ(y ⊗ e α )) = x(yϕ(1 ⊗ e α )) = x(yA λ α ) ⊗ f λ(33) and so every A λ α ∈ Z(J) and it is a real number. Using properties of tensor product we have A λ α ⊗ f λ = 1 ⊗ A λ α f λ(34) and the statement follows by taking as map A from E into F the linear transformation defined by A(e α ) = A λ α f λ . If the Jordan algebra J is not simple the above theorem is generalized as follows: Lemma 4.8. Let J be a finite-dimensional unital Jordan algebra, let M = J ⊗ E and N = J ⊗ F be free modules over J, with E, F finite-dimensional vector space of dimension m and n respectively. Then if f : M → N is homomorphism of J modules, there exist α k ∈ Z(J) and f k ∈ M m×n such that: f (1 ⊗ e) = k α k ⊗ f k (e)(35) for any e ∈ E. From the above lemma we deduce the following result. Theorem 4.9. Let J be a finite-dimensional unital Jordan algebra with center Z(J). Denote as F M od J the category of free Jordan modules over J with homomorphisms of Jordan modules and as F M od Z(J) the category of free modules over the associative algebra Z(J) with homomorphisms of modules over associative algebras. Then the following functor is an isomorphism of categories: F : J ⊗ E → Z(J) ⊗ E (ϕ : J ⊗ E → J ⊗ F ) → (ϕ Z(J) : Z(J) ⊗ E → Z(J) ⊗ F )(36) where ϕ Z(J) is the restriction of ϕ to Z(J) ⊗ E. Proof. We begin by checking functoriality of F. Of course the image of the identity of F M od J is the identity of F M od Z(J) . Let ϕ : J ⊗ E → J ⊗ F and φ : J ⊗ F → J ⊗ H be two homomorphisms of free modules over J, then we have: F(φ • ϕ) = (φ • ϕ) Z(J)(37) From theorem (4.7) we know that ϕ(Z(J) ⊗ E) ⊆ Z(J) ⊗ F, and so: F(φ • ϕ) = (φ • ϕ) Z(J) = φ Z(J) • ϕ Z(J) = F(φ) • F(ϕ)(38) which proves that F is a functor. Define F −1 as: F −1 : Z(J) ⊗ E → J ⊗ E (ϕ : Z(J) ⊗ E → Z(J) ⊗ F ) → (ϕ J : J ⊗ E → J ⊗ F )(39) where ϕ J is defined by regarding the elements of Z(J) as elements of J and setting: ϕ J (x ⊗ e) := xϕ(1 ⊗ e)(40) for any x ∈ J and e ∈ E. Finally let us introduce the following notion that will be useful in the next section. d(xy) = d(x)y + xd(y)(41) for any x, y ∈ J. Differential calculi Let us recall the following standard "super version" of Jordan algebras (see e.g. in [20]). Definition 5.1. A Jordan superalgebra Ω = Ω 0 ⊕ Ω 1 is a Z 2 -graded vector space with a graded commutative product, meaning: xy = (−1) |x||y| yx for all x, y ∈ Ω and such that this product respects the Jordan identity. For a Jordan superalgebra it holds: [x, y, z] = (−1) |y||z| [z, y, x](42) for all x, y, z ∈ Ω. If we introduce the graded commutator of x and y as [x, y] gr = xy + (−1) |x||y| yx (43) the Jordan identity is equivalent to: (−1) |x||z| [L xy , L z ] gr + (−1) |z||y| [L zx , L y ] gr + + (−1) |y||x| [L yz , L x ] gr = 0(44) for all x, y, z ∈ Ω. In what follows, we will deal with N-graded Jordan superalgebras, that means we are going to consider N-graded algebras Ω = ⊕ N Ω n that are also Jordan superalgebras with respect to the Z 2 -grading induced by the decomposition in even and odd parts, that we shall denote respectively as Ω + and Ω − . Definition 5.2. A differential graded Jordan algebra is an N-graded Jordan superalgebra Ω equipped with a differential, which is a antiderivation d of degree 1 and with square zero, that is one has dΩ n ⊂ Ω n+1 for all x, y ∈ Ω. Such differential graded Jordan algebras are our models for generalizing differential forms, in particular when Ω 0 = J we say that (Ω, d) is a differential calculus over the Jordan algebra J (this terminology is inspired from [26]). A model of differential calculus over a Jordan algebra is the derivation-based differential calculus which has been introduced in [11] and generalizes differential forms as defined in [22]. Let us denote as Ω 1 Der (J) the J-module of Z(J)-homomorphisms from Der(J) into J. We define a derivation d Der : J → Ω 1 Der (J) by setting: (d Der x) (X) := X(x)(45) for any x ∈ J and X ∈ Der(J). We refer to the pair Ω 1 Der (J), d Der as the derivation-based first order differential calculus over J. Let Ω n Der (J) be the J-module of n-Z(J)-linear antisymmetric mapping of Der(J) into J, that is any ω ∈ Ω n Der (J) is a Z(J)-linear mapping ω : ∧ n Z(J) Der(J) → J. Then Ω Der (J) = ⊕ n≥0 Ω n Der (J), is an N-graded Jordan superalgebra with respect to wedge product of linear maps. One extends d to a linear endomorphism of Ω Der (J) by setting (d Der ω)(X 0 , ..., X n ) = 0≤k≤n (−1) k X k ω X 0 , ..., X k , ...X n + + 0≤r<s≤n (−1) r+s ω [X r , X s ], X 0 , ..., X r , ..., X s , ...X n for any ω ∈ Ω n (J). This extension of d Der is an antiderivation and d 2 Der = 0. Thus Ω Der (J) endowed with d Der is a differential graded Jordan superalgebra with Ω 0 = J. We refer to (Ω Der (J), d Der ) as the derivation-based differential calculus over J. In general, the derivation-based differential calculus does not play any privileged role in the theory of differential calculus over a given Jordan algebra. Howewer in the case of exceptional Jordan algebra J 8 3 , the derivation-based differential calculus is characterized up to isomorphism by the following universal property ( [11]). Theorem 5.3. Let (Ω, d) be a differential graded Jordan algebra and let φ : J 8 3 → Ω 0 be an homomorphism of unital Jordan algebras. Then φ has a unique extensionφ : Ω Der J 8 3 → Ω as homomorphism of differential graded Jordan algebras. To prove this theorem we shall need the following result. This result is a consequence of the following lemmas proved in [29] (Lemma 2 and Lemma 3 in [29]). Lemma 5.5. Let Γ = Γ + ⊕ Γ − be a unital Jordan superalgebra whose even component Γ + is associative, then either one of the two equalities [Γ − , Γ + , Γ + ] = 0 (46) or [Γ − , Γ + , Γ + ] = Γ − (47) holds. Lemma 5.6. Let Γ be as above and such that [Γ − , Γ + , Γ + ] = 0, then either one of the two equalities [Γ + , Γ − , Γ − ] = 0 (48) or [Γ + , Γ − , Γ − ] = Γ +(49) holds. Proof of theorem (5.4). Let ξ = i a i ⊗ b i and η = x ⊗ y be elements in J 8 3 ⊗ Γ we have to find whenever ξ 2 , η, ξ = 0, that is   i a i ⊗ b i 2 , x ⊗ y, j a j ⊗ b j   =   i,j<i a i a j ⊗ b i b j + (−1) |b i ||b j | a i a j ⊗ b i b j , x ⊗ y, k a k ⊗ b k   + +   i a 2 i ⊗ b 2 i , x ⊗ y, k =i a k ⊗ b k   = 0(50) Γ = Γ + ⊕ Γ − is a Jordan superalgebra and in particular Γ + is a graded subalgebra of Γ and one knows ( [27]) that the algebra J 8 3 ⊗ Γ + is a Jordan graded algebra if and only if Γ + is associative. We must then assume Γ + associative. In expression (50) let us take ξ = a −1 ⊗ 1 + a 0 ⊗ e + i a i ⊗ o i , ξ 2 = a 2 −1 ⊗ 1 + a 2 0 ⊗ e 2 + a −1 ⊗ e+ + 2 i a −1 a i ⊗ o i + 2 i a o a i ⊗õ i , η = x 0 ⊗ y e + x 1 ⊗ y o (51) where a i and x i ∈ J 8 3 , e and y e ∈ Γ + , o i and y o ∈ Γ − and finally we set o i = eo i ∈ Γ − . Then one has ξ 2 , η, ξ = a 2 0 ⊗ e 2 , x ⊗ y, a j ⊗ o j + [a −1 a 0 ⊗ e, x ⊗ y, a 0 ⊗ e + a j ⊗ o j ] + + [a −1 a i ⊗ o i , x ⊗ y, a 0 ⊗ e] + [a 0 a i ⊗õ i , x ⊗ y, a 0 ⊗ e] + + [a 0 a i ⊗õ i , x ⊗ y, a 0 a j ⊗õ j ] = 0(52) for all a i ∈ J 8 3 . We can choose elements a i 's and x in J 8 3 in such a way that this condition is equivalent to and varying elements in Γ we see that condition above implies Γ − , Γ + , Γ + + Γ − , Γ − , Γ + + Γ − , Γ − , Γ − = 0(54) then, combining lemma (5.5) with lemma (5.6), we see that the equality above can hold only if all the summands above are identically zero, hence Γ = Γ + ⊕ Γ − must be an associative superalgebra. Now the proof of theorem (5.3) is the same as in proposition 4 of [11], as we shall recall for sake of completeness. Proof of theorem (5.3). For all n ∈ N, Ω n is a Jordan module over J 8 3 and from general theory of J 8 3 modules we know that every module over J 8 3 is a free module, hence we have Ω n = J 8 3 ⊗ Γ n(55) where Γ n is a vector space. Any differential graded Jordan superalgebra over J 8 3 is then written as Ω = ⊕ n∈N J 8 3 ⊗ Γ n = J 8 3 ⊗ Γ where Γ = ⊕ n∈N Γ n is a Jordan superalgebra. Consider the J 8 3 -module Ω 1 = J 8 3 ⊗ Γ 1 , and let {e α } ⊂ Γ 1 be a basis of Γ 1 . Let {∂ k } be a basis of Der J 8 3 with dual basis {θ k } such that θ k (∂ j ) = δ kj . We have dx = ∂ k x ⊗ c k α e α(56) for all x ∈ J and for some real constants c k α 's. Define the linear mapφ from Ω 1 Der into Ω 1 byφ x ⊗ θ k = x ⊗ c k α e α .(57) and extend it as homomorphism of superalgebras. We haveφ •d Der = d•φ, and uniqueness ofφ follows from d 2 = 0 and the Leibniz rule. It is important to remark that this statement holds true only for the exceptional Jordan algebra and it is a direct consequence of the fact that the only irreducible module over J 8 3 is J 8 3 itself. Connections and curvature for Jordan modules There are two equivalent definitions of derivation-based connections for modules of Jordan algebras and correspondingly two definitions of curvature. Definition 6.1. Let J be a Jordan algebra, a derivation-based connection on a module M over J is a linear mapping ∇ from Der(J) into the space of linear endomorphisms of the module End(M ), ∇ : X → ∇ X such that ∇ X (xm) = X(x)m + x∇ X m(58) and ∇ zX (m) = z∇ X (m)(59) for any x ∈ J, m ∈ M and z ∈ Z(J). From the first property it follows that if ∇ and ∇ ′ are two connections on the Jordan module M , then ∇ X − ∇ ′ X is a J-module endomorphism. Definition 6.2. Let ∇ be a derivation-based connection on a Jordan module M. The curvature of ∇ is defined as R X,Y = [∇ X , ∇ Y ] − ∇ [X,Y ](60) for all X, Y ∈ Der(J). It follows that R X,Y is a J-module endomorphism. A connection will be called flat if its curvature is identically zero that is R X,Y (m) = 0 (61) for all X, Y ∈ Der(J) and m ∈ M. Remark 6.3. In view of applications to particle physics, and in particular to Yang-Mills models, we are interested in classifying flat connections for Jordan modules. In fact, according to a standard heuristic argument ( see e.g. [9], [8]), any flat connection corresponds to a different ground state of the theory and the specification of the latter leads to different physical situations. The second definition of derivation-based connections is more suitable to be generalized to connections not based on derivations. Let J be a Jordan algebra, let M be a module over J and denote as Ω n Der (M ) the J-module of all n-Z(J)-linear antisymmetric mapping of Der(J) into M, then Ω Der (M ) = ⊕Ω n Der (M ) is a module over Ω Der (J) in the following way: for ω ∈ Ω n Der (J) and Φ ∈ Ω l Der (M ), the action of ω on Φ is given by (ωΦ) (X 1 , ..., X n+l ) = 1 (n + l)! i (−1) |i| ω (X i 1 , ..., X in ) Φ(X i n+1 , ..., X i n+l ) where i denotes a permutation of (1, ..., n + l) and | i | denotes the parity of the permutation i. and ∇ (ωφ) = d(ω)Φ + (−1) n ω∇Φ.(63) for all ω ∈ Ω n Der (J) and Φ ∈ Ω l Der (M ). From (63) we see that if ∇ and ∇ ′ are two different connections, then their difference is an endomorphism of Ω Der (M ) as module over Ω Der (J). In this case the curvature of a connection is defined as R = ∇ 2 . Definitions (6.1) and (6.4) are equivalent, in fact if ∇ is a connection as in the second definition, one defines a map from Der(J) into End(M ) by setting ∇ X (m) = (∇(m))(X)(64) and the map X → ∇ X is a connection in the sense of (6.1). On the other hand, if ∇ : X → ∇ X is a connection according to the first definition, one sets ∇(Φ) (X 0 , ..., X n ) = 0≤k≤n (−1) k ∇ Xp Φ X 0 , ..., X k , . ..X n + + 0≤r<s≤n (−1) r+s Φ [X r , X s ], X 0 , ..., X r , ..., X s , ...X n for all Φ ∈ Ω n Der (M ) and X p ∈ Der(J) and ∇ is now a connection according to definition (6.4). In the following examples the term "connection" will stand for derivationbased connection. Example 6.5. Let J be a finite-dimensional and unital Jordan algebra, let M = J ⊗ E be a free J-module. On M we have a base connection ∇ 0 = d ⊗ Id E : J ⊗ E → Ω 1 Der ⊗ E. As map from Der(J) into End(M ), ∇ 0 is the lift of the differential on J, that is ∇ 0 X (x ⊗ e) = (dx) (X) ⊗ e(66) for any X ∈ Der(J) and x ⊗ e ∈ M. It is easy to check that ∇ 0 respects properties (58) and (59). Moreover, this connection is gauge invariant whenever the center of J is trivial. ⋄ Proposition 6.6. Let J be a finite-dimensional Jordan algebra, let M = J ⊗ E be a free module over J, where E is a real vector space. Then any connection on M is of the form ∇ = ∇ 0 + A (67) where A is a linear map A : Der(J) → Z(J) ⊗ End(E) and A(X) (x ⊗ e) = x ⊗ A(X)e(68) for all X ∈ Der(J) and x ∈ J. Proof. From the definition of connection, it has to be ∇ − ∇ 0 = A ∈ End J (M )(69) and from theorem (4.7), it follows A(X) ∈ Z(J) ⊗ End(E). For what concerns flat connections, the following result, very similar to its counterpart in the context of Lie algebras, holds. Proposition 6.7. Let M = J ⊗ E be a free module over a simple Jordan algebra J, then flat connections on M are in one to one correspondence with Lie algebra homomorphisms A : Der(J) → End(E). That is, for a basis {X µ } ⊂ Der(J) with structure constants c τ µν one has [A(X µ ), A(X ν )] = c τ µν A(X τ ).(70) where [X µ , X ν ] = c τ µν X τ . Proof. By direct computation one can check that if a given connection ∇ = ∇ 0 + A is flat then (70) must hold. On the converse, if A : Der(J) → End(E) is such that (70) holds on a basis {X µ } ⊂ Der(J), then ∇ = ∇ 0 + A is a flat connection on M. Summarizing all the derivation-based differential calculus for free modules over Jordan algebras is resumed by the following proposition. for all z ∈ J i n and a ∈ A i n . Moreover this base connection is flat, indeed: [∇ x , ∇ Y ] − ∇ [ for all x, y, z ∈ J. Hence the commutator [L x , L y ] defines an inner derivation for J. In fact, formula (73) is a consequence of this in the particular case of special Jordan algebras. The example above can be generalized to the case of a module M over any finite-dimensional, semisimple Jordan algebra. In fact, in view of theorem (3.7) all the derivations of such algebras are inner. Remark 6.12. Connection (80) can be defined for every Jordan module over a Jordan algebra for which all derivations are inner in the sense of theorem (3.7) and such that the extension of derivations of the algebra to derivations on the split null extension is unique. The set of Jordan algebras for which all derivations are inner contains all finite-dimensional semi-simple Jordan algebra over a field of characteristic zero but it is in fact much wider, for example from theorem 2 of [14] we see that this request holds true for finite-dimensional and separable Jordan algebras on any field of characteristic different from 2. More generally we can give the following definition for a connection. Definition 6.13. Let Ω = ⊕ N Ω n be a differential graded Jordan algebra and let Γ = ⊕ N Γ n be a graded Jordan module over Ω, a connection on Γ is a linear endomorphism ∇ : Γ → Γ such that (∇Φ) ∈ Γ l+1 (82) ∇(ωΦ) = d(ω)Φ + (−1) n ω∇(Φ)(83) for all ω ∈ Ω n and Φ ∈ Γ l . In particular when Ω 0 = J and Γ 0 = M one obtains the definition of Ω-connection over the J-module M. Proposition 3. 2 . 2Let A be a commutative algebra and let z ∈ A, then z ∈ Z(A) if and only if [x, y, z] = 0 (10) Proposition 3. 4 . 4The vector space Der(A) of derivations of an algebra A has the following properties:1. Der(A) is a Lie algebra with respect to the commutator of endomorphisms.2. Der(A) is a module over the center Z(A). Definition 4. 6 . 6Let J be a Jordan algebra, let M and N be two modules over J, then a module homomorphism between M and N is a linear map ϕ : M → N such that xϕ(m) = ϕ(xm) Theorem 4. 7 . 7Let J be a finite-dimensional unital simple Jordan algebra, let M = J ⊗ E and N = J ⊗ F, where E and F are two finite-dimensional vector spaces, be free modules over J. Then every module homomorphism ϕ : M → N is of the form Definition 4 . 10 . 410Let J be a Jordan algebra, let M be a module over J. A derivation d of J into M is a linear map d : J → M such that: xy) = (dx)y + (−1) |x| xd(y) Lemma 5 . 4 . 54Let Γ be a Jordan superalgebra, then J 8 3 ⊗ Γ = ⊕ n∈N J 8 3 ⊗ Γ n is a Jordan superalgebra if and only if Γ is an associative superalgebra. e 2 , y e , o j + e 2 , y o , o j + [e, y o , e] + [e, y e , o j ] + [e, y o , o j ] + [o i , y e , e] + + [o i , y o , e] + [õ i , y e , e] + [õ i , y o , e] + [õ i , y e ,õ j ] + [õ i , y o ,õ j ] = 0 (53) Definition 6. 4 . 4Let J be a Jordan algebra, let M be a module over J. A derivation-based connection on M is a linear endomorphism ∇ of Ω Der (M ) such that ∇(Φ) ∈ Ω l+1 Der (M ) Proposition 6. 8 . 8Let J be a unital Jordan algebra, let M = J ⊗ E be a free module over J then1. ∇ 0 = d ⊗ I E : J ⊗ E → Ω 1 (J) ⊗ E defines a flat connection on Mwhich is gauge-invariant whenever the center of J is trivial. X,Y ] (a) = [X [Y, a]] − [Y [X, a]] − [[X, Y ] , a] = [[a, Y ] , X] + [[X, a] , Y ] + [[Y, X] , a] = 0(77)in view of the Jacopi identity in the Lie algebra M n (R). ⋄ Remark 6.10. Due to commutativity, for any Jordan algebra J it holds [x, z, y] = − [L x , L y ] z Proposition 6 . 11 . 611Let J be a finite-dimensional semisimple Jordan algebra so that ∀X ∈ Der(J) there exist a finite number of couples of elementsx i , y i ∈ J such that X(z) = [x i , z, y i ](79)for every z ∈ J. Then the map∇ :Der(J) → End(M ) X → ∇ X = [x i , ·, y i ](80)is a connection on M.Proof. Let X = [x i , ·, y i ] ∈ Der(J), it extends to a derivationX on the split null extension J ⊕ M given bỹX(z, m) = [(x i , 0), (z, m), (y i , 0)](81)for all (z, m) ∈ J ⊕ M.If we identify M with elements of the form (0, m) in J ⊕ M, we see thatX restricts to a linear endomorphism on M. Then ∇ is a Z(J) linear map from Der(J) into End(M ) and from Leibniz rule applied toX ∈ Der(J ⊕ M ) we have ∇ X (zm) = X(z)m + z∇ X m. Any other connection ∇ on M is defined by3. For a derivation-based connection ∇ the curvature is given by Example 6.9. Consider again A i n as module over J i n . We can provide a base connection for this module. From (3.7) we know that for any X ∈ Der(J i n ) there exists a finite number of couples of x i , y i ∈ J i n such thatfor any z ∈ J i n and where we have explicitly written • to design matrix anticommutator. Let X i = [x i , y i ] , where the commutator is taken with respect to the standard row by column product, then the expression above can also be written as:for any z ∈ J i n . Recall that the commutator of two hermitian matrices is an antihermitian matrix, then a good base connection on the Jordan module A i n is given byfor all a ∈ A i n , indeed: On a Certain Algebra of Quantum Mechanic. A A Albert, Annals of Mathematics. Second Series 35A.A.Albert, On a Certain Algebra of Quantum Mechanic, Annals of Mathematics, Second Series 35, pp. 65 − 72, 1934. J Baez, arXiv:math/0105155The octonions. J. Baez, The octonions, arXiv:math/0105155, 2002 The spectral action principle. A Chamseddine, A Connes, Commun. Math. Phys. 186A.Chamseddine, A.Connes, The spectral action principle Commun. Math. Phys. 186, pp 731 − 750, 1997. Noncommutative differential geometry Pub. A Connes, IHES. 62A. Connes, Noncommutative differential geometry Pub. IHES 62, pp. 257 − 360, 1986. Lott Particles models and noncommutative geometry Nucl. Phys B18 Suppl. A Connes, J , A. Connes, J. Lott Particles models and noncommutative geometry Nucl. Phys B18 Suppl., pp. 29 − 47, 1990. Noncommutative geometry. A Connes, Accademic PressA. Connes,Noncommutative geometry, Accademic Press, 1994. Marcolli Noncommutative geometry, quantum fields and motives. A Connes, M , American Mathematical SocietyA. Connes, M. Marcolli Noncommutative geometry, quantum fields and motives American Mathematical Society, 2008. Gauge bosons in a noncommutative geometry. M Dubois-Violette, R Kerner, J Madore, Phys. Lett. 217M. Dubois-Violette, R. Kerner, J. Madore, Gauge bosons in a non- commutative geometry, Phys. Lett. B217, pp. 485 − 488, 1989. Noncommutative differential geometry, quantum mechanics and gauge theory. M Dubois-Violette, Differential Geometric Methods in Theoretical Physics. Bartocci C., Bruzzo U., Cianci R.Springer375M. Dubois-Violette Noncommutative differential geometry, quantum mechanics and gauge theory. In: Bartocci C., Bruzzo U., Cianci R. (eds) Differential Geometric Methods in Theoretical Physics. Lecture Notes in Physics 375 Springer, 1991. M Dubois-Violette, Lectures on graded differential geometry and noncommutative geometry in: Y. Maeda et al. Noncommutative differential geometry and its applications to physics. Shonan, JapanKluwer accademic pubblishersM. Dubois-Violette Lectures on graded differential geometry and non- commutative geometry in: Y. Maeda et al. Noncommutative differential geometry and its applications to physics, Shonan, Japan, pp. 245 − 306, Kluwer accademic pubblishers, 2001. M Dubois, Violette Exceptional quantum geometry and particle physics Nuclear Physics B912. M. Dubois-Violette Exceptional quantum geometry and particle physics Nuclear Physics B912, pp. 426 − 449, 2016. Extensions of general algebras Ann. S Eilenberg, Soc. Math. Pol. 21S. Eilenberg Extensions of general algebras Ann. Soc. Math. Pol. 21, pp. 125 − 134, 1948. Moufang plane and octionic quantum mechanics. M Gunaydin, C Piron, H Ruegg, Commun. Math. Phys. 61M. Gunaydin, C. Piron, H. Ruegg, Moufang plane and octionic quan- tum mechanics, Commun. Math. Phys. 61, pp. 69 − 85, 1978. . B Harris, Derivations of Jordan algebras Pacific J. Math. 9B. Harris,Derivations of Jordan algebras Pacific J. Math. 9, pp. 495 − 512, 1959. Lie-Rineheart algebras. J Huebschmann, Gerstenhaber algebras and Batalin-Vilkovisky algebras Ann. Inst. Fourier Grenoble. 48J. Huebschmann Lie-Rineheart algebras. Gerstenhaber algebras and Batalin-Vilkovisky algebras Ann. Inst. Fourier Grenoble 48, pp. 425 − 440, 1998. Jordan structures in analysis, geometry and physics. R Iordanescu, R. Iordanescu Jordan structures in analysis, geometry and physics, Ed. Acad. Romane, 2009. N Jacobson, General representation theory of Jordan algebras. 70N. Jacobson General representation theory of Jordan algebras, Trans. Amer. Math. Soc. 70, pp. 509 − 530, 1951. N Jacobson, Structure and Representations of Jordan Algebras. American Mathematical SocietyN. Jacobson, Structure and Representations of Jordan Algebras, Amer- ican Mathematical Society, 1968.. On an Algebraic Generalization of the Quantum Mechanical Formalism. P Jordan, J Neumann, E Wigner, Annals of Mathematics. 35P. Jordan , J. von Neumann, E. Wigner, On an Algebraic Generaliza- tion of the Quantum Mechanical Formalism, Annals of Mathematics, Princeton, 35, pp. 29 − 64, 1934. Kac Classification of simple Z-graded Lie superalgebras and simple Jordan superalgebras. V G , Commun. Algebra. 5V.G. Kac Classification of simple Z-graded Lie superalgebras and sim- ple Jordan superalgebras, Commun. Algebra 5, pp. 1375 − 1400 1977. Representation type of Jordan algebras. I Kashuba, S Osvienko, I Shestakov, Adv. Math. 226I. Kashuba, S. Osvienko, I. Shestakov, Representation type of Jordan algebras, Adv. Math. 226, pp. 1 − 35, 2011. Fibre bundles and diferential geometry, Tata Institute of fundamental research. J L Koszul, BombayJ.L. Koszul, Fibre bundles and diferential geometry, Tata Institute of fundamental research, Bombay 1960. K Mccrimmon, A taste of Jordan algebras. SpringerK. McCrimmon, A taste of Jordan algebras, Springer, 2004 Differential forms for general commutative algebras. G Rinehart, Trans. Amer. Math. Soc. 108G. Rinehart, Differential forms for general commutative algebras Trans. Amer. Math. Soc. 108, pp. 195 − 222, 1963. W Van Suijlekom, Noncommutative Geometry and Particle Physics. SpringerW. van Suijlekom, Noncommutative Geometry and Particle Physics, Springer, 2014. Differential calculus on compact matrix pseudogroups (quantum groups). S L Woronowicz, Commun. Math. Phys. 122S. L. Woronowicz, Differential calculus on compact matrix pseu- dogroups (quantum groups), Commun. Math. Phys. 122, pp. 125− 170, 1989. Tensor product of Jordan algebras. A Wulfsohn, Can. J. Math. XXVIIIA. Wulfsohn, Tensor product of Jordan algebras. Can. J. Math. Vol. XXVIII, 1975. . I Yokota, arXiv:0902.0431Exceptional Lie GroupsI. Yokota, Exceptional Lie Groups arXiv:0902.0431, 2009. Simple special Jordan superalgebras with associative even part. V N Zhelyabin, I P Shestakov, Siberian Mathematical Journal. 45V. N. Zhelyabin and I. P. Shestakov, Simple special Jordan superalge- bras with associative even part. Siberian Mathematical Journal 45, pp. 860 − 882, 2004.
{'fraction_non_alphanumeric': 0.0863556996757296, 'fraction_numerical': 0.029084559740583688, 'mean_word_length': 3.3164298018949183, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 98, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'Having in mind applications to particle physics we develop the differential calculus over Jordan algebras and the theory of connections on Jordan modules. In particular we focus on differential calculus over the exceptional Jordan algebra and provide a complete characterization of the theory of connections for free Jordan modules.', 'arxivid': '1803.08373', 'author': ['Alessandro Carotenuto \nScuola Internazionale Superiore di Studi Avanzati (SISSA)\nLaboratoire de Physique Théorique\nCNRS\nBonomea 265I-34136TriesteItaly\n', 'D ֒ Ludwik \nScuola Internazionale Superiore di Studi Avanzati (SISSA)\nLaboratoire de Physique Théorique\nCNRS\nBonomea 265I-34136TriesteItaly\n', 'Abrowski [email protected] \nScuola Internazionale Superiore di Studi Avanzati (SISSA)\nLaboratoire de Physique Théorique\nCNRS\nBonomea 265I-34136TriesteItaly\n', 'Michel Dubois-Violette [email protected] \nUniversité Paris-Sud\nUniversité Paris-Saclay\nBâtiment 210F-91405OrsayFrance\n'], 'authoraffiliation': ['Scuola Internazionale Superiore di Studi Avanzati (SISSA)\nLaboratoire de Physique Théorique\nCNRS\nBonomea 265I-34136TriesteItaly', 'Scuola Internazionale Superiore di Studi Avanzati (SISSA)\nLaboratoire de Physique Théorique\nCNRS\nBonomea 265I-34136TriesteItaly', 'Scuola Internazionale Superiore di Studi Avanzati (SISSA)\nLaboratoire de Physique Théorique\nCNRS\nBonomea 265I-34136TriesteItaly', 'Université Paris-Sud\nUniversité Paris-Saclay\nBâtiment 210F-91405OrsayFrance'], 'corpusid': 119151893, 'doi': '10.1007/s11005-018-1102-z', 'github_urls': [], 'n_tokens_mistral': 15064, 'n_tokens_neox': 13017, 'n_words': 7824, 'pdfsha': '1d6e0140e2e7f78cceedc03da4f12b31b717306f', 'pdfurls': ['https://arxiv.org/pdf/1803.08373v3.pdf'], 'title': [], 'venue': []}
arxiv
Heterogeneous Graph Transformer Ziniu Hu Yuxiao Dong [email protected] Kuansan Wang [email protected] Yizhou Sun [email protected] University of California Los Angeles Microsoft Research Redmond Microsoft Research Redmond University of California Los Angeles Heterogeneous Graph Transformer 10.1145/3366423.3380027ACM Reference Format: Ziniu Hu, Yuxiao Dong, Kuansan Wang, and Yizhou Sun. 2020. Hetero-geneous Graph Transformer. In Proceedings of The Web Conference 2020 (WWW '20), April 20-24, 2020, Taipei, Taiwan. ACM, New York, NY, USA, 11 pages. https://Graph Neural NetworksHeterogeneous Information NetworksRepresentation LearningGraph EmbeddingGraph Attention Recent years have witnessed the emerging success of graph neural networks (GNNs) for modeling structured data. However, most GNNs are designed for homogeneous graphs, in which all nodes and edges belong to the same types, making them infeasible to represent heterogeneous structures. In this paper, we present the Heterogeneous Graph Transformer (HGT) architecture for modeling Web-scale heterogeneous graphs. To model heterogeneity, we design node-and edge-type dependent parameters to characterize the heterogeneous attention over each edge, empowering HGT to maintain dedicated representations for different types of nodes and edges. To handle dynamic heterogeneous graphs, we introduce the relative temporal encoding technique into HGT, which is able to capture the dynamic structural dependency with arbitrary durations. To handle Web-scale graph data, we design the heterogeneous mini-batch graph sampling algorithm-HGSampling-for efficient and scalable training. Extensive experiments on the Open Academic Graph of 179 million nodes and 2 billion edges show that the proposed HGT model consistently outperforms all the state-of-the-art GNN baselines by 9%-21% on various downstream tasks. The dataset and source code of HGT are publicly available at https://github.com/acbull/pyHGT. INTRODUCTION Heterogeneous graphs have been commonly used for abstracting and modeling complex systems, in which objects of different types Figure 1: The schema and meta relations of Open Academic Graph (OAG). Given a Web-scale heterogeneous graph, e.g., an academic network, HGT takes only its one-hop edges as input without manually designing meta paths. interact with each other in various ways. Some prevalent instances of such systems include academic graphs, Facebook entity graph, LinkedIn economic graph, and broadly the Internet of Things network. For example, the Open Academic Graph (OAG) [28] in Figure 1 contains five types of nodes: papers, authors, institutions, venues (journal, conference, or preprint), and fields, as well as different types of relationships between them. Over the past decade, a significant line of research has been explored for mining heterogeneous graphs [17]. One of the classical paradigms is to define and use meta paths to model heterogeneous structures, such as PathSim [18] and metapath2vec [3]. Recently, in view of graph neural networks' (GNNs) success [7,9,22], there are several attempts to adopt GNNs to learn with heterogeneous networks [14,23,26,27]. However, these works face several issues: First, most of them involve the design of meta paths for each type of heterogeneous graphs, requiring specific domain knowledge; Second, they either simply assume that different types of nodes/edges share the same feature and representation space or keep distinct non-sharing weights for either node type or edge type alone, making them insufficient to capture heterogeneous graphs' properties; Third, most of them ignore the dynamic nature of every (heterogeneous) graph; Finally, their intrinsic design and implementation make them incapable of modeling Web-scale heterogeneous graphs. Take OAG for example: First, the nodes and edges in OAG could have different feature distributions, e.g., papers have text features whereas institutions may have features from affiliated scholars, and coauthorships obviously differ from citation links; Second, OAG has been consistently evolving, e.g., 1) the volume of publications doubles every 12 years [4], and 2) the KDD conference was more related to database in the 1990s whereas more to machine learning in recent years; Finally, OAG contains hundreds of millions of nodes and billions of relationships, leaving existing heterogeneous GNNs not scalable for handling it. In light of these limitations and challenges, we propose to study heterogeneous graph neural networks with the goal of maintaining node-and edge-type dependent representations, capturing network dynamics, avoiding customized meta paths, and being scalable to Web-scale graphs. In this work, we present the Heterogeneous Graph Transformer (HGT) architecture to deal with all these issues. To handle graph heterogeneity, we introduce the node-and edgetype dependent attention mechanism. Instead of parameterizing each type of edges, the heterogeneous mutual attention in HGT is defined by breaking down each edge e = (s, t) based on its meta relation triplet, i.e., ⟨ node type of s, edge type of e between s & t, node type of t⟩. Figure 1 illustrates the meta relations of heterogeneous academic graphs. In specific, we use these meta relations to parameterize the weight matrices for calculating attention over each edge. As a result, nodes and edges of different types are allowed to maintain their specific representation spaces. Meanwhile, connected nodes in different types can still interact, pass, and aggregate messages without being restricted by their distribution gaps. Due to the nature of its architecture, HGT can incorporate information from high-order neighbors of different types through message passing across layers, which can be regarded as "soft" meta paths. That said, even if HGT take only its one-hop edges as input without manually designing meta paths, the proposed attention mechanism can automatically and implicitly learn and extract "meta paths" that are important for different downstream tasks. To handle graph dynamics, we enhance HGT by proposing the relative temporal encoding (RTE) strategy. Instead of slicing the input graph into different timestamps, we propose to maintain all the edges happening in different times as a whole, and design the RTE strategy to model structural temporal dependencies with any duration length, and even with unseen and future timestamps. By end-to-end training, RTE enables HGT to automatically learn the temporal dependency and evolution of heterogeneous graphs. To handle Web-scale graph data, we design the first heterogeneous sub-graph sampling algorithm-HGSampling-for minibatch GNN training. Its main idea is to sample heterogeneous subgraphs in which different types of nodes are with similar proportions, since the direct usage of existing (homogeneous) GNN sampling methods, such as GraphSage [7], FastGCN [1], and LADIES [29], results in highly imbalanced ones regarding to both node and edge types. In addition, it is also designed to keep the sampled sub-graphs dense for minimizing the loss of information. With HGSampling, all the GNN models, including our proposed HGT, can train and infer on arbitrary-size heterogeneous graphs. We demonstrate the effectiveness and efficiency of the proposed Heterogeneous Graph Transformer on the Web-scale Open Academic Graph comprised of 179 million nodes and 2 billion edges spanning from 1900 to 2019, making this the largest-scale and longest-spanning representation learning yet performed on heterogeneous graphs. Additionally, we also examine it on domain-specific graphs: the computer science and medicine academic graphs. Experimental results suggest that HGT can significantly improve various downstream tasks over state-of-the-art GNNs as well as dedicated heterogeneous models by 9-21%. We further conduct case studies to show the proposed method can indeed automatically capture the importance of implicit meta paths for different tasks. PRELIMINARIES AND RELATED WORK In this section, we introduce the basic definition of heterogeneous graphs with network dynamics and review the recent development on graph neural networks (GNNs) and their heterogeneous variants. We also highlight the difference between HGT and existing attempts on heterogeneous graph neural networks. Heterogeneous Graph Mining Heterogeneous graphs [17] (a.k.a., heterogeneous information networks) are an important abstraction for modeling relational data for many real-world complex systems. Formally, it is defined as: Definition 1. Heterogeneous Graph: A heterogeneous graph is defined as a directed graph G = (V, E, A, R) where each node v ∈ V and each edge e ∈ E are associated with their type mapping functions τ (v) : V → A and ϕ(e) : E → R, respectively. Meta Relation. For an edge e = (s, t) linked from source node s to target node t, its meta relation is denoted as ⟨τ (s), ϕ(e), τ (t)⟩. Naturally, ϕ(e) −1 represents the inverse of ϕ(e). The classical meta path paradigm [17][18][19] is defined as a sequence of such meta relation. Notice that, to better model real-world heterogeneous networks, we assume that there may exist multiple types of relations between different types of nodes. For example, in OAG there are different types of relations between the author and paper nodes by considering the authorship order, i.e., "the first author of", "the second author of", and so on. Dynamic Heterogeneous Graph. To model the dynamic nature of real-world (heterogeneous) graphs, we assign an edge e = (s, t) a timestamp T , when node s connects to node t at T . If s appears for the first time, T is also assigned to s. s can be associated with multiple timestamps if it builds connections over time. In other words, we assume that the timestamp of an edge is unchanged, denoting the time it is created. For example, when a paper published on a conference at time T , T will be assigned to the edge between the paper and conference nodes. On the contrary, different timestamps can be assigned to a node accordingly. For example, the conference node "WWW" can be assigned any year. WWW @1994 means that we are considering the first edition of WWW, which focuses more on internet protocol and Web infrastructure, while WWW @2020 means the upcoming WWW, which expands its research topics to social analysis, ubiquitous computing, search & IR, privacy and society, etc. There have been significant lines of research on mining heterogenous graphs, such as node classification, clustering, ranking and representation learning [3,[17][18][19], while the dynamic perspective of HGs has not been extensively explored and studied. Graph Neural Networks Recent years have witnessed the success of graph neural networks for relational data [7,9,22]. Generally, a GNN can be regarded as using the input graph structure as the computation graph for message passing [6], during which the local neighborhood information is aggregated to get a more contextual representation. Formally, it has the following form: Definition 2. General GNN Framework: Suppose H l [t] is the node representation of node t at the (l)-th GNN layer, the update procedure from the (l-1)-th layer to the (l)-th layer is: H l [t] ← Aggregate ∀s ∈N (t ),∀e ∈E(s,t ) Extract H l −1 [s]; H l −1 [t], e(1) where N (t) denotes all the source nodes of node t and E(s, t) denotes all the edges from node s to t. The most important GNN operators are Extract(·) and Aggregate(·). Extract(·) represents the neighbor information extractor. It extract useful information from source node's representation H l −1 [s], with the target node's representation H l −1 [t] and the edge e between the two nodes as query. Aggregate(·) gather the neighborhood information of souce nodes via some aggregation operators, such as mean, sum, and max, while more sophisticated pooling and normalization functions can be also designed. Various (homogeneous) GNN architectures have been proposed following this framework. Kipf et al. [9] propose graph convolutional network (GCN), which averages the one-hop neighbor of each node in the graph, followed by a linear projection and non-linear activation operations. Hamilton et al. propose GraphSAGE that generalizes GCN's aggregation operation from average to sum, max and a RNN unit. Velickovi et al. propose graph attention network (GAT) [22] by introducing the attention mechanism into GNNs, which allows GAT to assign different importance to nodes within the same neighborhood. Heterogeneous GNNs Recently, studies have attempted to extend GNNs for modeling heterogeneous graphs. Schlichtkrull et al. [14] propose the relational graph convolutional networks (RGCN) to model knowledge graphs. RGCN keeps a distinct linear projection weight for each edge type. Zhang et al. [27] present the heterogeneous graph neural networks (HetGNN) that adopts different RNNs for different node types to integrate multi-modal features. Wang et al. [23] extend graph attention networks by maintaining different weights for different meta-path-defined edges. They also use high-level semantic attention to differentiate and aggregate information from different meta paths. Though these methods have shown to be empirically better than the vanilla GCN and GAT models, they have not fully utilized the heterogeneous graphs' properties. All of them use either node type or edge type alone to determine GNN weight matrices. However, the node or edge counts of different types can vary greatly. For relations that don't have sufficient occurrences, it's hard to learn accurate relation-specific weights. To address this, we propose to consider parameter sharing for a better generalization. Given an edge e = (s, t) with its meta relation as ⟨τ (s), ϕ(e), τ (t)⟩, if we use three interaction matrices to model the three corresponding elements τ (s), ϕ(e), and τ (t) in the meta relation, then the majority of weights could be shared. For example, in "the first author of" and "the second author of" relationships, their source and target node types are both author to paper, respectively. In other words, the knowledge about author and paper learned from one relation could be quickly transferred and adapted to the other one. Therefore, we integrate this idea with the powerful Transformer-like attention architecture, and propose Heterogeneous Graph Transformer. To summarize, the key differences between HGT and existing attempts include: (1) Instead of attending on node or edge type alone, we use the meta relation ⟨τ (s), ϕ(e), τ (t)⟩ to decompose the interaction and transform matrices, enabling HGT to capture both the common and specific patterns of different relationships using equal or even fewer parameters. (2) Different from most of the existing works that are based on customized meta paths, we rely on the nature of the neural architecture to incorporate high-order heterogeneous neighbor information, which automatically learns the importance of implicit meta paths. (3) Most previous works don't take the dynamic nature of (heterogeneous) graphs into consideration, while we propose the relative temporal encoding technique to incorporate temporal information by using limited computational resources. (4) None of the existing heterogeneous GNNs are designed for and experimented with Web-scale graphs, we therefore propose the heterogeneous Mini-Batch graph sampling algorithm designed for Web-scale graph training, enabling experiments on the billion-scale Open Academic Graph. HETEROGENEOUS GRAPH TRANSFORMER In this section, we present the Heterogeneous Graph Transformer (HGT). Its idea is to use the meta relations of heterogeneous graphs to parameterize weight matrices for the heterogeneous mutual attention, message passing, and propagation steps. To further incorporate network dynamics, we introduce a relative temporal encoding mechanism into the model. Figure 2 shows the overall architecture of Heterogeneous Graph Transformer. Given a sampled heterogeneous sub-graph (Cf. Section 4), HGT extracts all linked node pairs, where target node t is linked by source node s via edge e. The goal of HGT is to aggregate information from source nodes to get a contextualized representation for target node t. Such process can be decomposed into three components: Heterogeneous Mutual Attention, Heterogeneous Message Passing and Target-Specific Aggregation. Overall HGT Architecture We denote the output of the (l)-th HGT layer as H (l ) , which is also the input of the (l+1)-th layer. By stacking L layers, we can get the node representations of the whole graph H (L) , which can be used for end-to-end training or fed into downstream tasks. Heterogeneous Mutual Attention The first step is to calculate the mutual attention between source node s and target node t. We first give a brief introduction to the general attention-based GNNs as follows: H l [t] ← Aggregate ∀s ∈N (t ),∀e ∈E(s,t ) Attention(s, t) · Message(s)(2)< τ (s 1 ), ϕ(e 1 ), τ (t) > & < τ (s 2 ), ϕ(e 2 ) , τ (t) > as input to learn a contextualized representation H (L) for each node, which can be used for downstream tasks. Color decodes the node type. HGT includes three components: (1) meta relation-aware heterogeneous mutual attention, (2) heterogeneous message passing from source nodes, and (3) target-specific heterogeneous message aggregation. where there are three basic operators: Attention, which estimates the importance of each source node; Message, which extracts the message by using only the source node s; and Aggregate, which aggregates the neighborhood message by the attention weight. For example, the Graph Attention Network (GAT) [22] adopts an additive mechanism as Attention, uses the same weight for calculating Message, and leverages the simple average followed by a nonlinear activation for the Aggregate step. Formally, GAT has Attention GAT (s, t) = Softmax ∀s ∈N (t ) ì a W H l −1 [t] ∥ W H l −1 [s] Message GAT (s) = W H l −1 [s] Aggregate GAT (·) = σ Mean(·) Though GAT is effective to give high attention values to important nodes, it assumes that s and t have the same feature distributions by using one weight matrixW . Such an assumption, as we've discussed in Section 1, is usually incorrect for heterogeneous graphs, where each type of nodes can have its own feature distribution. In view of this limitation, we design the Heterogeneous Mutual Attention mechanism. Given a target node t, and all its neighbors s ∈ N (t), which might belong to different distributions, we want to calculate their mutual attention grounded by their meta relations, i.e., the ⟨τ (s), ϕ(e), τ (t)⟩ triplets. Inspired by the architecture design of Transformer [21], we map target node t into a Query vector, and source node s into a Key vector, and calculate their dot product as attention. The key difference is that the vanilla Transformer uses a single set of projections for all words, while in our case each meta relation should have a distinct set of projection weights. To maximize parameter sharing while still maintaining the specific characteristics of different relations, we propose to parameterize the weight matrices of the interaction operators into a source node projection, an edge projection, and a target node projection. Specifically, we calculate the h-head attention for each edge e = (s, t) (See Figure 2 (1)) by: Attention H GT (s, e, t) = Softmax ∀s ∈N (t ) ∥ i ∈[1,h] ATT -head i (s, e, t) (3) ATT -head i (s, e, t) = K i (s) W AT T ϕ(e) Q i (t) T · µ ⟨τ (s),ϕ(e),τ (t )⟩ √ d K i (s) = K-Linear i τ (s) H (l −1) [s] Q i (t) = Q-Linear i τ (t ) H (l −1) [t] First, for the i-th attention head ATT -head i (s, e, t), we project the τ (s)-type source node s into the i-th Key vector K i (s) with a linear projection K-Linear i τ (s) : R d → R d h , where h is the number of attention heads and d h is the vector dimension per head. Note that K-Linear i τ (s) is indexed by the source node s's type τ (s), meaning that each type of nodes has a unique linear projection to maximally model the distribution differences. Similarly, we also project the target node t with a linear projection Q-Linear i τ (t ) into the i−th Query vector. Next, we need to calculate the similarity between the Query vector Q i (t) and Key vector K i (s). One unique characteristic of heterogeneous graphs is that there may exist different edge types (relations) between a node type pair, e.g., τ (s) and τ (t). Therefore, unlike the vanilla Transformer that directly calculates the dot product between the Query and Key vectors, we keep a distinct edge- based matrix W AT T ϕ(e) ∈ R d h × d h for each edge type ϕ(e). In doing so, the model can capture different semantic relations even between the same node type pairs. Moreover, since not all the relationships contribute equally to the target nodes, we add a prior tensor µ ∈ R | A |× | R |× | A | to denote the general significance of each meta relation triplet, serving as an adaptive scaling to the attention. Finally, we concatenate h attention heads together to get the attention vector for each node pair. Then, for each target node t, we gather all attention vectors from its neighbors N (t) and conduct softmax, making it fulfill ∀s ∈N (t ) Attention H GT (s, e, t) = 1 h×1 . Heterogeneous Message Passing Parallel to the calculation of mutual attention, we pass information from source nodes to target nodes (See Figure 2 (2)). Similar to the attention process, we would like to incorporate the meta relations of edges into the message passing process to alleviate the distribution differences of nodes and edges of different types. For a pair of nodes e = (s, t), we calculate its multi-head Message by: Message H GT (s, e, t) = ∥ i ∈[1,h] MSG-head i (s, e, t)(4) Target-Specific Aggregation With the heterogeneous multi-head attention and message calculated, we need to aggregate them from the source nodes to the target node (See Figure 2 (3)). Note that the softmax procedure in Eq. 3 has made the sum of each target node t's attention vectors to one, we can thus simply use the attention vector as the weight to average the corresponding messages from the source nodes and get the updated vector H (l ) [t] as: H (l ) [t] = ⊕ ∀s ∈N (t ) Attention H GT (s, e, t) · Message H GT (s, e, t) . This aggregates information to the target node t from all its neighbors (source nodes) of different feature distributions. The final step is to map target node t's vector back to its typespecific distribution, indexed by its node type τ (t). To do so, we apply a linear projection A-Linear τ (t ) to the updated vector H (l ) [t], followed by residual connection [8] as: H (l ) [t] = A-Linear τ (t ) σ H (l ) [t] + H (l −1) [t].(5) In this way, we get the l-th HGT layer's output H (l ) [t] for the target node t. Due to the "small-world" property of real-world graphs, stacking the HGT blocks for L layers (L being a small value) can enable each node reaching a large proportion of nodes-with different types and relations-in the full graph. That is, HGT generates a highly contextualized representation H (L) for each node, which can be fed into any models to conduct downstream heterogeneous network tasks, such as node classification and link prediction. Through the whole model architecture, we highly rely on using the meta relation-⟨τ (s), ϕ(e), τ (t)⟩-to parameterize the weight matrices separately. This can be interpreted as a trade-off between the model capacity and efficiency. Compared with the vanilla Transformer, our model distinguishes the operators for different relations and thus is more capable to handle the distribution differences in heterogeneous graphs. Compared with existing models that keep a distinct matrix for each meta relation as a whole, HGT's triplet parameterization can better leverage the heterogeneous graph schema to achieve parameter sharing. On one hand, relations with few occurrences can benefit from such parameter sharing for fast adaptation and generalization. On the other hand, different relationships' operators can still maintain their specific characteristics by using a much smaller parameter set. Relative Temporal Encoding By far, we present HGT-a graph neural network for modeling heterogeneous graphs. Next, we introduce the Relative Temporal Encoding (RTE) technique for HGT to handle graph dynamic. The traditional way to incorporate temporal information is to construct a separate graph for each time slot. However, such a procedure may lose a large portion of structural dependencies across different time slots. Meanwhile, the representation of a node at time t might rely on edges that happen at other time slots. Therefore, a proper way to model dynamic graphs is to maintain all the edges happening at different times and allow nodes and edges with different timestamps to interact with each other. In light of this, we propose the Relative Temporal Encoding (RTE) mechanism to model the dynamic dependencies in heterogeneous graphs. RTE is inspired by Transformer's positional encoding method [15,21], which has been shown successful to capture the sequential dependencies of words in long texts. Specifically, given a source node s and a target node t, along with their corresponding timestamps T (s) and T (t), we denote the relative time gap ∆T (t, s) = T (t) − T (s) as an index to get a relative temporal encoding RT E(∆T (t, s)). Noted that the training dataset will not cover all possible time gaps, and thus RT E should be capable of generalizing to unseen times and time gaps. Therefore, we adopt a fixed set of sinusoid functions as basis, with a tunable linear projection T-Linear * : R d → R d as RT E: Base ∆T (t, s), 2i = sin ∆T t,s /10000 2i d (6) Base ∆T (t, s), 2i + 1 = cos ∆T t,s /10000 2i +1 d(7) RT E ∆T (t, s) = T-Linear Base(∆T t,s ) Finally, the temporal encoding relative to the target node t is added to the source node s' representation as follows: H (l −1) [s] = H (l −1) [s] + RT E ∆T (t, s)(9) In this way, the temporal augmented representation H (l −1) will capture the relative temporal information of source node s and target node t. The RTE procedure is illustrated in the Figure 3. WEB-SCALE HGT TRAINING In this section, we present HGT's strategies for training Webscale heterogeneous graphs with dynamic information, including an efficient Heterogeneous Mini-Batch Graph Sampling algorithm-HGSampling-and an inductive timestamp assignment method. HGSampling The full-batch GNN [9] training requires the calculation of all node representations per layer, making it not scalable for Web-scale graphs. To address this issue, different sampling-based methods [1,2,7,29] have been proposed to train GNNs on a subset of nodes. However, directly using them for heterogeneous graphs is prone to get sub-graphs that are extremely imbalanced regarding different node types, due to that the degree distribution and the total number of nodes for each type can vary dramatically. To address this issue, we propose an efficient Heterogeneous Mini-Batch Graph Sampling algorithm-HGSampling-to enable both HGT and traditional GNNs to handle Web-scale heterogeneous graphs. HGSampling is able to 1) keep a similar number of nodes and edges for each type and 2) keep the sampled sub-graph dense to minimize the information loss and reduce the sample variance. Algorithm 1 outlines the HGSampling algorithm. Its basic idea is to keep a separate node budget B[τ ] for each node type τ and to sample an equal number of nodes per type with an importance sampling strategy to reduce variance. Given node t already sampled, we add all its direct neighbors into the corresponding budget with Algorithm 2, and add t's normalized degree to these neighbors in line 8, which will then be used to calculate the sampling probability. Such normalization is equivalent to accumulate the random walk probability of each sampled node to its neighborhood, avoiding the sampling being dominated by high-degree nodes. Intuitively, the higher such value is, the more a candidate node is correlated with the currently sampled nodes, and thus should be given a higher probability to be sampled. Algorithm 1 Heterogeneous Mini-Batch Graph Sampling Require: Adjacency matrix A for each ⟨τ (s), ϕ(e), τ (t)⟩ relation pair; Output node Set OS; Sample number n per node type; Sample depth L. Ensure: Sampled node set N S; Sampled adjacency matrixÂ. 1: N S ← OS // Initialize sampled node set as output node set. 2: Initialize an empty Budget B storing nodes for each node type with normalized degree. 3: for t ∈ N S do 4: Add-In-Budget(B, t, A, N S) // Add neighbors of t to B. 5: end for 6: for l ← 1 to L do 7: for source node type τ ∈ B do Sample n nodes {t i } n i=1 from B[τ ] using prob (l −1) [τ ]. 12: for t ∈ {t i } n i=1 do 13: OS[τ ] .add(t) // Add node t into Output node set. 14: Add-In-Budget(B, t, A, N S) // Add neighbors of t to B. After the budget is updated, we then calculate the sampling probability in Algorithm 1 line 9, where we calculate the square of the cumulative normalized degree of each node s in each budget. As proved in [29], using such sampling probability can reduce the sampling variance. Then, we sample n nodes in type τ by using the calculated probability, add them into the output node set, update its neighborhood to the budget, and remove it out of the budget in lines 12-15. Repeating such procedure for L times, we get a sampled sub-graph with L depth from the initial nodes. Finally, we reconstruct the adjacency matrix among the sampled nodes. By using the above algorithm, the sampled sub-graph contains a similar number of nodes per type (based on the separate node budget), and is sufficiently dense to reduce the sampling variance (based on the normalized degree and importance sampling), making it suitable for training GNNs on Web-scale heterogeneous graphs. Inductive Timestamp Assignment Till now we have assumed that each node t is assigned with a timestamp T (t). However, in real-world heterogeneous graphs, many nodes are not associated with a fixed time. Therefore, we need to assign different timestamps to it. We denote these nodes as plain nodes. For example, the WWW conference is held in both 1974 and 2019, and the WWW node in these two years has dramatically different research topics. Consequently, we need to decide which timestamp(s) to attach to the WWW node. if s has no timestamp then 6: s.time = t .time // Inductively inherit timestamp. There also exist event nodes in heterogeneous graphs that have an explicit timestamp associated with them. For example, the paper node should be associated with its publication behavior and therefore attached to its publication date. We propose an inductive timestamp assignment algorithm to assign plain nodes timestamps based on event nodes that they are linked with. The algorithm is shown in Algorithm 2 line 6. The idea is that plan nodes inherit the timestamps from event nodes. We examine whether the candidate source node is an event node. If yes, like a paper published at a specific year, we keep its timestamp for capturing temporal dependency. If no, like a conference that can be associated with any timestamp, we inductively assign the associated node's timestamp, such as the published year of its paper, to this plain node. In this way, we can adaptively assign timestamps during the sub-graph sampling procedure. EVALUATION In this section, we evaluate the proposed Heterogeneous Graph Transformer on three heterogeneous academic graph datasets. We conduct the Paper-Field prediction, Paper-Venue prediction, and Author Disambiguation tasks. We also take case studies to demonstrate how HGT can automatically learn and extract meta paths that are important for downstream tasks † . Web-Scale Datasets To examine the performance of the proposed model and its realworld applications, we use the Open Academic Graph (OAG) [16,20,28] as our experimental basis. OAG consists of more than 178 million nodes and 2.236 billion edges-the largest publicly available heterogeneous academic dataset. In addition, all papers in OAG are associated with their publication dates, spanning from 1900 to 2019. To test the generalization of the proposed model, we also construct two domain-specific subgraphs from OAG: the Computer Science (CS) and Medicine (Med) academic graphs. The graph statistics are listed in Table 1, in which P-A, P-F, P-V, A-I, and P-P denote the edges between paper and author, paper and field, paper and venue, author and institute, and the citation links between two papers. Both the CS and Med graphs contain tens of millions of nodes and hundreds of millions of edges, making them at least one magnitude larger than the other CS (e.g., DBLP) and Med (e.g., Pubmed) academic datasets that are commonly used in existing heterogeneous GNN and heterogeneous graph mining studies. Moreover, the three datasets used are far more distinguishable than previously wide-adopted small citation graphs used in GNN studies, such as Dataset #nodes #edges #papers #authors #fields #venues #institutes #P-A #P-F #P-V #A-I #P-P CS 11,732,027 107,263,811 5,597,605 5,985,759 119,537 27,433 16,931 15,571,614 47,462,559 5,597,606 7,190,480 31,441,552 Med 51,044,324 451,468,375 21,931,587 28,779,507 289,930 25,044 18,256 85,620,479 149,728,483 21,931,588 28,779,507 165,408,318 OAG 178,663,927 2,236,196,802 89,606,257 88,364,081 615,228 53,073 25,288 300,853,688 657,049,405 89,606,258 167,449,933 1,021,237,518 Table 1: Open Academic Graph (OAG) Statistics. Cora, Citeseer, and Pubmed [9,22], which only contain thousands of nodes. There are totally five node types: 'Paper', 'Author', 'Field', 'Venue', and 'Institute'. The 'Field' nodes in OAG are categorized into six levels from L 0 to L 5 , which are organized with a hierarchical tree. Therefore, we differentiate the 'Paper-Field' edges corresponding to the field level. In addition, we differentiate the different author orders (i.e., the first author, the last one, and others) and venue types (i.e., journal, conference, and preprint) as well. Finally, the 'Self' type corresponds to the self-loop connection, which is widely added in GNN architectures. Except the 'Self' relationship, which are symmetric, all other relation types ϕ have a reverse relation type ϕ −1 . Experimental Setup Tasks and Evaluation. We evaluate the HGT model on four different real-world downstream tasks: the prediction of Paper-Field (L 1 ), Paper-Field (L 2 ), and Paper-Venue, and Author Disambiguation. The goal of the first three node classification tasks is to predict the correct L 1 and L 2 fields that each paper belongs to or the venue it is published at, respectively. We use different GNNs to get the contextual node representation of the paper and use a softmax output layer to get its classification label. For author disambiguation, we select all the authors with the same name and their associated papers. The task is to conduct link prediction between these papers and candidate authors. After getting the paper and author node representations from GNNs, we use a Neural Tensor Network to get the probability of each author-paper pair to be linked. For all tasks, we use papers published before the year 2015 as the training set, papers between 2015 and 2016 for validation, and papers between 2016 and 2019 as testing. We choose NDCG and MRR, which are two widely adopted ranking metrics [10,11], as the evaluation metrics. All models are trained for 5 times and, the mean and standard variance of test performance are reported. Baselines. We compare HGT with two classes of state-of-art graph neural networks. All baselines as well as our own model, are implemented via the PyTorch Geometric (PyG) package [5]. The first class of GNN baselines is designed for homogeneous graphs, including: • Graph Convolutional Networks (GCN) [9], which simply averages the neighbor's embedding followed by linear projection. We use the implementation provided in PyG. • Graph Attention Networks (GAT) [22], which adopts multihead additive attention on neighbors. We use the implementation provided in PyG. The second class considered is several dedicated heterogeneous GNNs as baselines, including: • Relational Graph Convolutional Networks (RGCN) [14], which keeps a different weight for each relationship, i.e., a relation triplet. We use the implementation provided in PyG. • Heterogeneous Graph Neural Networks (HetGNN) [27], which adopts different Bi-LSTMs for different node type for aggregating neighbor information. We re-implement this model in PyG following the authors' official code. • Heterogeneous Graph Attention Networks (HAN) [23] design hierarchical attentions to aggregate neighbor information via different meta paths. We re-implement this model in PyG following the authors' official code. In addition, to systematically analyze the effectiveness of the two major components of HGT, i.e., Heterogeneous weight parameterization (Heter) and Relative Temporal Encoding (RTE), we conduct an ablation study, but comparing with models that remove these components. Specifically, we use −Heter to denote models that uses the same set of weights for all meta relations, and use −RT E to denote models that doesn't include relative temporal encoding. By considering all the permutations, we have: HGT −RT E −H et er , HGT +RT E −H et er , HGT −RT E +H et er and HGT +RT E +H et er ‡ . We use our HGSampling algorithm proposed in Section 4 for all baseline GNNs to handle the large-scale OAG graph. To avoid data leakage, we remove out the links we aim to predict (e.g., the Paper-Field link as the label) from the sub-graph. Input Features. As we don't assume the feature of each node type belongs to the same distribution, we are free to use the most appropriate features to represent each type of nodes. For each paper, we use a pre-trained XLNet [24,25] to get the representation of each word in its title. We then average them weighted by each word's attention to get the title representation for each paper. The initial feature of each author is then simply an average of his/her published papers' representations. For the field, venue, and institute nodes, we use the metapath2vec model [3] to train their node embeddings by reflecting the heterogeneous network structures. The homogeneous GNN baselines assume the node features belong to the same distribution, while our feature extraction doesn't fulfill this assumption. To make a fair comparison, we add an adaptation layer between the input features and all used GNNs. This module simply conducts different linear projections for nodes of different types. Such a procedure can be regarded to map heterogeneous data into the same distribution, which is also adopted in literature [23,27]. Implementation Details. We use 256 as the hidden dimension throughout the neural networks for all baselines. For all multi-head attention-based methods, we set the head number as 8. All GNNs keep 3 layers so that the receptive fields of each network are exactly ‡ Unless other stated, HGT refers to HGT +RT E +H e t e r . Experimental Results We summarize the experimental results of the proposed model and baselines on three datasets in Table 2. All experiments for the four tasks are evaluated in terms of NDCG and MRR. The results show that in terms of both metrics, the proposed HGT significantly and consistently outperforms all baselines for all tasks on all datasets. Take, for example, the Paper-Field (L 1 ) classification task on OAG, HGT achieves relative performance gains over baselines by 15-19% in terms of NDCG and 18-21% in terms of MRR (i.e., the performance gap divided by the baseline performance). When compared to HAN-the best baseline for most of the cases, the average relative NDCG improvements of HGT on the CS, Med and OAG datasets are 11%, 10% and 8%, respectively. Overall, we observe that on average, HGT outperforms GCN, GAT, RGCN, HetGNN, and HAN by 20% for the four tasks on all three large-scale datasets. Moreover, HGT has fewer parameters and comparable batch time than all the heterogeneous graph neural network baselines, including RGCN, HetGNN, and HAN. This suggests that by modeling heterogeneous edges according to their meta relation schema, we are able to have better generalization with fewer resource consumption. Ablation Study. The core component in HGT are the heterogeneous weight parameterization (Heter) and Relative Temporal Encoding (RTE). To further analyze their effects, we conduct an ablation study by removing them from HGT. Specifically, the model that removes heterogeneous weight parameterization, i.e., HGT +RT E −H et er , drops 4% of performance compared with the full model HGT +RT E +H et er . By removing RTE (i.e., HGT −RT E +H et er ), the performance has a 2% drop. The ablation study shows the significance of parameterizing with meta relations and using Relative Temporal Encoding. In addition, we also try to implement a baseline that keeps a unique weight matrix for each relation. However, such a baseline contains too many parameters so that our experimental setting doesn't have enough GPU memory to optimize it. This also indicates that using the meta relation to parameterize weight matrices can achieve competitive performance with fewer resources. Case Study To further evaluate how Relative Temporal Encoding (RTE) can help HGT to capture graph dynamics, we conduct a case study showing the evolution of conference topic. We select 100 conferences in computer science with the highest citations, assign them three different timestamps, i.e., 2000, 2010 and 2020, and construct sub-graphs initialized by them. Using a trained HGT, we can get the representations for these conferences, with which we can calculate the euclidean distances between them. We select WWW, KDD, and NeurIPS as illustration. For each of them, we pick the top-5 most similar conferences (i.e., the one with the smallest euclidean distance) to show how the conference's topics evolve over time. As shown in Table 3, these venues' relationships have changed from 2000 to 2020. For example, WWW in 2000 was more related to some database conferences, i.e., SIGMOD and VLDB, and some networking conferences, i.e., NSDI and GLOBECOM. However, WWW in 2020 would become more related to some data mining and information retrieval conferences (KDD, SIGIR, and WSDM), in addition to SIGMOD and GLOBECOM. Also, KDD in 2000 was more related to traditional database and data mining venues, while in 2020 it will tend to correlate with a variety of topics, i.e. machine learning (NeurIPS), database (SIGMOD), Web (WWW), AI (AAAI), and NLP (EMNLP). Additionally, our HGT model can capture the difference brought by new conferences. For example, NeurIPS in 2020 would relate with ICLR, which is a newly organized deep learning conference. This case study shows that the relative temporal encoding can help capture the temporal evolution of the heterogeneous academic graphs. Visualize Meta Relation Attention To illustrate how the incorporated meta relation schema can benefit the heterogeneous message passing process, we pick the schema that has the largest attention value in each of the first two HGT layers and plot the meta relation attention hierarchy tree in Figure 5. For example, to calculate a paper's representation, ⟨Paper, is_published_at, Venue, is_published_at −1 , Paper⟩, ⟨Paper, has_L 2 _f ield_o f , Field, has_L 5 _f ield_o f −1 , Paper⟩, and ⟨Institute, is_a f f iliated_with −1 , Author, is_f irst_author _o f , Paper⟩ are the three most important meta relation sequences, which can be regarded as meta paths PVP, PFP, and IAP, respectively. Note that Venue Time Top−5 Most Similar Venues these meta paths and their importance are automatically learned from the data without manual design. Another example of calculating an author node's representation is shown on the right. Such visualization demonstrates that Heterogeneous Graph Transformer is capable of implicitly learning to construct important meta paths for specific downstream tasks, without manual customization. Figure 2 : 2The Overall Architecture of Heterogeneous Graph Transformer. Given a sampled heterogeneous sub-graph with t as the target node, s 1 & s 2 as source nodes, the HGT model takes its edges e 1 = (s 1 , t) & e 2 = (s 2 , t) and their corresponding meta relations MSG-head i (s, e, t) = M-Linear i τ (s) H (l −1) [s] W MSG ϕ(e) To get the i-th message head MSG-head i (s, e, t), we first project the τ (s)-type source node s into the i-th message vector with a linear projection M-Linear i τ (s) : R d → R d h . It is then followed by a matrix W M SG ϕ(e) ∈ R d h × d h for incorporating the edge dependency. The final step is to concat all h message heads to get the Message H GT (s, e, t) for each node pair. Figure 3 : 3Relative Temporal Encoding (RTE) to model graph dynamic. Nodes are associated with timestamps T (·). After the RTE process, the temporal augmented representations are fed to the HGT model. Figure 4 : 4HGSampling with Inductive Timestamp Assignment.Algorithm 2 Add-In-BudgetRequire: Budget B storing nodes for each type with normalized degree; Added node t; Adjacency matrix A for each ⟨τ (s), ϕ(e), τ (t)⟩ relation pair; Sampled node set N S. Ensure: Updated Budget B.1: for each possible source node type τ and edge type ϕ do 2:D t ← 1 / len A ⟨τ ,ϕ,τ (t )⟩ [t] // get normalized degree of added node t regarding to ⟨τ , ϕ, τ (t)⟩.3:for source node s in A ⟨τ ,ϕ,τ(t )⟩ [t] do 4:if s has not been sampled (s N S) then 5: Figure 5 : 5Hierarchy of the learned meta relation attention. end for 18: end for 19: Reconstruct the sampled adjacency matrix among the sampled nodes OS from A. 20: return OS andÂ;15: B[τ ].pop(t) // Remove sampled node t from Budget. 16: end for 17: B[τ ][s] ← B[τ ][s] +D t // Add candidate node s to budget B with target node t's normalized degree.7: end if 8: 9: end if 10: end for 11: end for 12: return Updated Budget B Table 2 : 2Experimental results of different methods over the three datasets.the same. All baselines are optimized via the AdamW optimizer[13] with the Cosine Annealing Learning Rate Scheduler[12]. For each model, we train it for 200 epochs and select the one with the lowest validation loss as the reported model. We use the default parameters used in GNN literature and donot tune hyper-parameters. Table 3 : 3Temporal Evolution of Conference Similarity. * For simplicity, we denote a linear projection L : R a → R b as a function to conduct linear transformation to vector x ∈ R a as: L(x ) = W x + b, where matrix W ∈ R a+b and bias b ∈ R b . W and b are learnable parameters for L. † The dataset and code are publicly available at https://github.com/acbull/pyHGT. CONCLUSIONIn this paper, we propose the Heterogeneous Graph Transformer (HGT) architecture for modeling Web-scale heterogeneous and dynamic graphs. To model heterogeneity, we use the meta relation ⟨τ (s), ϕ(e), τ (t)⟩ to decompose the interaction and transform matrices, enabling the model to have the similar modeling capacity with fewer resources. To capture graph dynamics, we present the relative temporal encoding (RTE) technique to incorporate temporal information using limited computational resources. To conduct efficient and scalable training of HGT on Web-scale data, we design the heterogeneous Mini-Batch graph sampling algorithm-HGSampling. We conduct comprehensive experiments on the Open Academic Graph, and show that the proposed HGT model can capture both heterogeneity and outperforms all the state-of-the-art GNN baselines on various downstream tasks.In the future, we will explore whether HGT is able to generate heterogeneous graphs, e.g., predict new papers and their titles, and whether we can pre-train HGT to benefit tasks with scarce labels.Acknowledgements. We would like to thank Xiaodong Liu for helpful discussions. This work is partially supported by NSF III-1705169, NSF CAREER Award 1741634, NSF#1937599, Okawa Foundation Grant, and Amazon Research Award. . Vldb Sigmod, Globecom Nsdi, Cikm Kdd, Sigir, Sigmod, SIGMOD, VLDB, NSDI, GLOBECOM, SIGIR 2010 GLOBECOM, KDD, CIKM, SIGIR, SIGMOD Globecom Kdd, Wsdm Sigir, Sigmod, Icde, Cikm Icdm, Icde, Www, Neurips, Sigmod, Neurips, Sigmod, Aaai Www, Neurips, ICML, ECCV, AAAI, CVPR 2010 ICML, CVPR, ACL, KDD, AAAI 2020 ICML, CVPR, ICLR, ICCV, ACL. KDD, GLOBECOM, SIGIR, WSDM, SIGMOD KDD 2000 SIGMOD, ICDE, ICDM, CIKM, VLDB 2010 ICDE, WWW, NeurIPS, SIGMOD, ICML 2020 NeurIPS, SIGMOD, WWW, AAAI, EMNLP NeurIPS 2000 ICCV, ICML, ECCV, AAAI, CVPR 2010 ICML, CVPR, ACL, KDD, AAAI 2020 ICML, CVPR, ICLR, ICCV, ACL FastGCN: Fast Learning with Graph Convolutional Networks via Importance Sampling. Jie Chen, Tengfei Ma, Cao Xiao, ICLR'18. Jie Chen, Tengfei Ma, and Cao Xiao. 2018. FastGCN: Fast Learning with Graph Convolutional Networks via Importance Sampling. In ICLR'18. Stochastic Training of Graph Convolutional Networks with Variance Reduction. Jianfei Chen, Jun Zhu, Le Song, ICML. Jianfei Chen, Jun Zhu, and Le Song. 2018. Stochastic Training of Graph Convolu- tional Networks with Variance Reduction. In ICML. 941-949. 2017. metapath2vec: Scalable Representation Learning for Heterogeneous Networks. Yuxiao Dong, V Nitesh, Ananthram Chawla, Swami, 17Yuxiao Dong, Nitesh V Chawla, and Ananthram Swami. 2017. metapath2vec: Scalable Representation Learning for Heterogeneous Networks. In KDD '17. A Century of Science: Globalization of Scientific Collaborations, Citations, and Innovations. Yuxiao Dong, Hao Ma, Zhihong Shen, Kuansan Wang, KDD '17. ACM. Yuxiao Dong, Hao Ma, Zhihong Shen, and Kuansan Wang. 2017. A Century of Science: Globalization of Scientific Collaborations, Citations, and Innovations. In KDD '17. ACM, 1437-1446. Fast Graph Representation Learning with PyTorch Geometric. Matthias Fey, Jan Eric Lenssen, ICLR 2019 Workshop: Representation Learning on Graphs and Manifolds. Matthias Fey and Jan Eric Lenssen. 2019. Fast Graph Representation Learning with PyTorch Geometric. ICLR 2019 Workshop: Representation Learning on Graphs and Manifolds (2019). Neural Message Passing for Quantum Chemistry. Justin Gilmer, Samuel S Schoenholz, Patrick F Riley, Oriol Vinyals, George E Dahl, Justin Gilmer, Samuel S. Schoenholz, Patrick F. Riley, Oriol Vinyals, and George E. Dahl. 2017. Neural Message Passing for Quantum Chemistry. In ICML. 1263-1272. Inductive Representation Learning on Large Graphs. William L Hamilton, Zhitao Ying, Jure Leskovec, NeurIPS'17. William L. Hamilton, Zhitao Ying, and Jure Leskovec. 2017. Inductive Represen- tation Learning on Large Graphs. In NeurIPS'17. Deep Residual Learning for Image Recognition. Kaiming He, Xiangyu Zhang, Shaoqing Ren, Jian Sun, CVPR'16. Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. 2016. Deep Residual Learning for Image Recognition. In CVPR'16. Semi-Supervised Classification with Graph Convolutional Networks. N Thomas, Max Kipf, Welling, 17Thomas N. Kipf and Max Welling. 2017. Semi-Supervised Classification with Graph Convolutional Networks. In ICLR'17. Learning to Rank for Information Retrieval and Natural Language Processing. Hang Li, 10.2200/S00607ED2V01Y201410HLT026Morgan & Claypool PublishersSecond EditionHang Li. 2014. Learning to Rank for Information Retrieval and Natural Language Processing, Second Edition. Morgan & Claypool Publishers. https://doi.org/10. 2200/S00607ED2V01Y201410HLT026 Learning to Rank for Information Retrieval. Tie-Yan Liu, SpringerTie-Yan Liu. 2011. Learning to Rank for Information Retrieval. Springer. SGDR: Stochastic Gradient Descent with Warm Restarts. Ilya Loshchilov, Frank Hutter, 17Ilya Loshchilov and Frank Hutter. 2017. SGDR: Stochastic Gradient Descent with Warm Restarts. In ICLR'17. Decoupled Weight Decay Regularization. Ilya Loshchilov, Frank Hutter, ICLR'19. Ilya Loshchilov and Frank Hutter. 2019. Decoupled Weight Decay Regularization. In ICLR'19. Modeling Relational Data with Graph Convolutional Networks. Michael Sejr, Thomas N Schlichtkrull, Peter Kipf, Rianne Bloem, Van Den, Ivan Berg, Max Titov, Welling, In ESWC. Michael Sejr Schlichtkrull, Thomas N. Kipf, Peter Bloem, Rianne van den Berg, Ivan Titov, and Max Welling. 2018. Modeling Relational Data with Graph Convo- lutional Networks. In ESWC'2018. Self-Attention with Relative Position Representations. Peter Shaw, Jakob Uszkoreit, Ashish Vaswani, NAACL-HLT. Peter Shaw, Jakob Uszkoreit, and Ashish Vaswani. 2018. Self-Attention with Relative Position Representations. In NAACL-HLT. 464-468. . Arnab Sinha, Zhihong Shen, Yang Song, Hao Ma, Darrin Eide, Bo-June Paul Hsu, Kuansan Wang, An Overview of Microsoft Academic Service (MAS) and Applications. In WWW Companion. Arnab Sinha, Zhihong Shen, Yang Song, Hao Ma, Darrin Eide, Bo-June Paul Hsu, and Kuansan Wang. 2015. An Overview of Microsoft Academic Service (MAS) and Applications. In WWW Companion 2015. Mining Heterogeneous Information Networks: Principles and Methodologies. Yizhou Sun, Jiawei Han, Morgan & Claypool PublishersYizhou Sun and Jiawei Han. 2012. Mining Heterogeneous Information Networks: Principles and Methodologies. Morgan & Claypool Publishers. Pathsim: Meta path-based top-k similarity search in heterogeneous information networks. Yizhou Sun, Jiawei Han, Xifeng Yan, Philip S Yu, Tianyi Wu, 11Yizhou Sun, Jiawei Han, Xifeng Yan, Philip S. Yu, and Tianyi Wu. 2011. Pathsim: Meta path-based top-k similarity search in heterogeneous information networks. In VLDB '11. Integrating meta-path selection with user-guided object clustering in heterogeneous information networks. Yizhou Sun, Brandon Norick, Jiawei Han, Xifeng Yan, Philip S Yu, Xiao Yu, KDD'12. Yizhou Sun, Brandon Norick, Jiawei Han, Xifeng Yan, Philip S. Yu, and Xiao Yu. 2012. Integrating meta-path selection with user-guided object clustering in heterogeneous information networks. In KDD'12. Arnetminer: extraction and mining of academic social networks. Jie Tang, Jing Zhang, Limin Yao, Juanzi Li, Li Zhang, Zhong Su, KDD. Jie Tang, Jing Zhang, Limin Yao, Juanzi Li, Li Zhang, and Zhong Su. 2008. Arnet- miner: extraction and mining of academic social networks. In KDD. Attention is All you Need. Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, Illia Polosukhin, NeurIPS'17. Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. 2017. Attention is All you Need. In NeurIPS'17. Petar Velickovic, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Liò, Yoshua Bengio, Graph Attention Networks. In ICLR'18. Petar Velickovic, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Liò, and Yoshua Bengio. 2018. Graph Attention Networks. In ICLR'18. Heterogeneous Graph Attention Network. Xiao Wang, Houye Ji, Chuan Shi, Bai Wang, Yanfang Ye, Peng Cui, Philip S Yu, KDD'19. Xiao Wang, Houye Ji, Chuan Shi, Bai Wang, Yanfang Ye, Peng Cui, and Philip S. Yu. 2019. Heterogeneous Graph Attention Network. In KDD'19. 2022-2032. Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, arXiv:cs.CL/1910.03771RÃľmi Louf, Morgan Funtowicz, and Jamie Brew. 2019. Transformers: State-of-the-art Natural Language Processing. Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, RÃľmi Louf, Morgan Funtowicz, and Jamie Brew. 2019. Transformers: State-of-the-art Natural Language Processing. arXiv:cs.CL/1910.03771 XLNet: Generalized Autoregressive Pretraining for Language Understanding. Zhilin Yang, Zihang Dai, Yiming Yang, Jaime G Carbonell, Ruslan Salakhutdinov, V Quoc, Le, In NeurIPS'19Zhilin Yang, Zihang Dai, Yiming Yang, Jaime G. Carbonell, Ruslan Salakhutdi- nov, and Quoc V. Le. 2019. XLNet: Generalized Autoregressive Pretraining for Language Understanding. In NeurIPS'19. Graph Transformer Networks. Seongjun Yun, Minbyul Jeong, Raehyun Kim, Jaewoo Kang, Hyunwoo J Kim, NeurIPS'19. Seongjun Yun, Minbyul Jeong, Raehyun Kim, Jaewoo Kang, and Hyunwoo J. Kim. 2019. Graph Transformer Networks. In NeurIPS'19. Heterogeneous Graph Neural Network. Chuxu Zhang, Dongjin Song, Chao Huang, Ananthram Swami, Nitesh V Chawla, WWW'19. Chuxu Zhang, Dongjin Song, Chao Huang, Ananthram Swami, and Nitesh V. Chawla. 2019. Heterogeneous Graph Neural Network. In WWW'19. OAG: Toward Linking Large-scale Heterogeneous Entity Graphs. Fanjin Zhang, Xiao Liu, Jie Tang, Yuxiao Dong, Peiran Yao, Jie Zhang, Xiaotao Gu, Yan Wang, Bin Shao, Rui Li, Kuansan Wang, KDD'19. Fanjin Zhang, Xiao Liu, Jie Tang, Yuxiao Dong, Peiran Yao, Jie Zhang, Xiaotao Gu, Yan Wang, Bin Shao, Rui Li, and Kuansan Wang. 2019. OAG: Toward Linking Large-scale Heterogeneous Entity Graphs. In KDD'19. Layer-Dependent Importance Sampling for Training Deep and Large Graph Convolutional Networks. Difan Zou, Ziniu Hu, Yewen Wang, Song Jiang, Yizhou Sun, Quanquan Gu, In NeurIPS'19Difan Zou, Ziniu Hu, Yewen Wang, Song Jiang, Yizhou Sun, and Quanquan Gu. 2019. Layer-Dependent Importance Sampling for Training Deep and Large Graph Convolutional Networks. In NeurIPS'19.
{'fraction_non_alphanumeric': 0.05116141286354052, 'fraction_numerical': 0.01944691306393434, 'mean_word_length': 4.652876280535855, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 2, 'https://': 4, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 3, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'Recent years have witnessed the emerging success of graph neural networks (GNNs) for modeling structured data. However, most GNNs are designed for homogeneous graphs, in which all nodes and edges belong to the same types, making them infeasible to represent heterogeneous structures. In this paper, we present the Heterogeneous Graph Transformer (HGT) architecture for modeling Web-scale heterogeneous graphs. To model heterogeneity, we design node-and edge-type dependent parameters to characterize the heterogeneous attention over each edge, empowering HGT to maintain dedicated representations for different types of nodes and edges. To handle dynamic heterogeneous graphs, we introduce the relative temporal encoding technique into HGT, which is able to capture the dynamic structural dependency with arbitrary durations. To handle Web-scale graph data, we design the heterogeneous mini-batch graph sampling algorithm-HGSampling-for efficient and scalable training. Extensive experiments on the Open Academic Graph of 179 million nodes and 2 billion edges show that the proposed HGT model consistently outperforms all the state-of-the-art GNN baselines by 9%-21% on various downstream tasks. The dataset and source code of HGT are publicly available at https://github.com/acbull/pyHGT.', 'arxivid': '2003.01332', 'author': ['Ziniu Hu ', 'Yuxiao Dong [email protected] ', 'Kuansan Wang [email protected] ', 'Yizhou Sun [email protected] ', '\nUniversity of California\nLos Angeles\n', '\nMicrosoft Research\nRedmond\n', '\nMicrosoft Research\nRedmond\n', '\nUniversity of California\nLos Angeles\n'], 'authoraffiliation': ['University of California\nLos Angeles', 'Microsoft Research\nRedmond', 'Microsoft Research\nRedmond', 'University of California\nLos Angeles'], 'corpusid': 211818229, 'doi': '10.1145/3366423.3380027', 'github_urls': ['https://github.com/acbull/pyHGT.', 'https://github.com/acbull/pyHGT.'], 'n_tokens_mistral': 16004, 'n_tokens_neox': 14395, 'n_words': 8921, 'pdfsha': '0ca7d8c3250d43d14fdde46bf6fc299654d861ef', 'pdfurls': ['https://arxiv.org/pdf/2003.01332v1.pdf'], 'title': ['Heterogeneous Graph Transformer', 'Heterogeneous Graph Transformer'], 'venue': []}
arxiv
Data-Dependent Early Completion of Dose Finding Trials for Drug-Combination Masahiro Kojima [email protected] Biometrics Department, R&D Division Kyowa Kirin Co., Ltd TokyoJapan Department of Statistical Science School of Multidisciplinary Sciences Address: Biometrics Department, R&D Division, Kyowa Kirin Co., Ltd. Otemachi Financial City Grand Cube The Graduate University for Advanced Studies 1-9-2 Otemachi, Chiyoda-ku, 100-004Tokyo, TokyoJapan., Japan Masahiro Kojima Data-Dependent Early Completion of Dose Finding Trials for Drug-Combination 1 Acknowledgments: The author thanks Professor Masahiko Gosho and Associate Professor Hisashi Noma for their encouragement and helpful suggestions. Corresponding author Name: Running title: Early Completion of Drug-Combination trials List of where and when the study has been presented in part elsewhere: NoneModel-assisted designsEarly completion of drug combination finding trialsBOIN design, Keyboard design Financial support: NoneConflict of interest disclosure statement: NoneContext Summaries:We propose a data-dependent early completion of dose finding trials for drug-combination.The early completion is determined when a beta-binomial probability for dose retainment with the trial data and the number of remaining patients is high. This paper also proposes an 3 early completion method that a dose retainment probability is adjusted by a bivariate isotonic regression. We demonstrate the early completion for a virtual trial. We evaluate the performance of early completion method through simulation studies with 12 scenarios. We confirmed the superior performance for our proposed early completion methods. We show the number of patients for determining early completion before a trial starts and a program code for calculating dose retainment probability in this paper.Word count: 2998The number of figures: 3The number of tables: 3 4 Abstract Purpose: Model-assisted designs for drug combination trials have been proposed as novel designs with simple and superior. The model-assisted designs require to set a sample size in advance, and the trial cannot complete until the number of patients treated reaches the sample size. We propose a data-dependent early completion of dose finding trials for drugcombination.Methods:The early completion is determined when a beta-binomial probability for dose retainment with the trial data and the number of remaining patients is high. This paper also proposes an early completion method that a dose retainment probability is adjusted by a bivariate isotonic regression. We demonstrate the early completion for a virtual trial. We evaluate the performance of early completion method through simulation studies with 12 scenarios.Results: From simulation studies, the percentage of early completion was averagely determined about 70% and the number of patients treated reduced by 25% from planned sample size. The percentage of a correct maximum tolerated dose-combination selection for early completion designs showed little change from the non-early completion designs by an average of about 3%. 5 Conclusion: We confirmed the superior performance for our proposed early completion methods. We showed the number of patients for determining early completion before the trial starts and a program code for calculating dose retainment probability in this paper. correct MTD Key: Keyboard, EC: Early completion based on normal dose retainment probability, ECI: Early completion based on dose retainment probability using bivariate isotonic regression Figure 3. Percentage of early completion Key: Keyboard, EC: Early completion based on normal dose retainment probability, ECI: Early completion based on dose retainment probability using bivariate isotonic regression Introduction We consider a dose-finding for drug-combination trial that aims to identify a maximum tolerated dose-combination (MTD). The MTD is a dose-combination whose dose limiting toxicity (DLT) rate is closest to a target toxicity level (TTL). To identify the MTD, dose adjustments are repeatedly needed for experience multiple doses. A design for the dose adjustments is classically a 3+3 design 1 . The 3+3 design is simple to operate for the dose adjustment based on a simple algorithm, but the performance of correct MTD selection is poor. Hence, designs based on a statistical model have been proposed 2,3,4,5,6,7,8,9,10,11,12,13,14,15 . However, the model-based designs are rarely used in actual clinical trials because the designs require complex assumptions 16 . Therefore, a Bayesian optimal interval (BOIN) 17,18,19 ,Keyboard 20,21 , and extended model-assisted designs 22,23,24,25,26,27,28,29,30,31,32,33 did not discuss an applicability of early completion for dose-combination trials. In this paper, we propose a data-dependent early completion of dose finding trials for drug-combination. The early completion is determined when a beta-binomial probability for dose retainment with the trial data and the number of remaining patients is high. This paper also proposes an early completion method that a dose retainment probability is adjusted by a bivariate isotonic regression. We demonstrate the early completion for a virtual trial. We evaluate the performance of early completion method through simulation studies with 12 scenarios. The paper is structured as follows: Section 2 presents early completion methods for dose-combination trials. In addition, we demonstrate the early completion for a virtual trial and present a simulation configuration to evaluate a performance for the early completion method. Section 3 presents the simulation results. Section 4 discusses our proposed designs and the results. Methods We consider a drug-combination trial with the following: a sample size of , dose levels of drug A and dose levels of drug B, a current combined-dose level ( , ) of , , the total number of patients treated at , is , , the total number of DLTs at the current dose , is , , the observed DLT rate at the current dose is ̂, = , / , , the TTL is , and a prior distribution of each dose-combination is Beta(1,1)(=Uniform(0,1)). Bayesian optimal interval (BOIN) design conducts a dose-assignment compared For drug-combination trials, if the dose for next cohort also retains the current dose, the dose adjustment is simple. On the other hand, the problems with combination trials are that there are many ways to the dose escalation and de-escalation. It would be nice to be able to try all combinations without safety issues, but a strategic dose selection is necessary to identify MTD in a limited sample size. Figure 1 shows the strategy for dose-assignment. The two candidates for dose escalation from dose combination 1,1 were the blue dotted line for 1,2 and the blue solid line for 2,1 , and the dose was escalated to dose 2,1 whose DLT rate was closer to the TTL. Subsequent dose escalations were selected in the same manner. If a dose de-escalation is determined after the administration of dose combination 3,2 , the dosecombination de-escalates to 3,1 or 2,3 whose DLT rate is closer to the TTL. Pan et al. 19 proposed algorithms to escalate or de-escalate the doses of two drugs simultaneously, but this paper deals with the method to increase or decrease only one dose of two drugs. We summarize the algorithm for dose escalation/de-escalation in the dose-combination trial below. [Algorithm of dose escalation/de-escalation] • When the dose combination for next cohort is determined as an escalation after , is administered, escalate to a dose combination which has the highest interval posterior probability of the interval for either +1, or , +1 . If the candidate combination for dose escalation has never been administered, the interval probability is calculated by using the prior distribution. If the interval probabilities are equal, choose a drug combination randomly. • When the dose combination for next cohort is determined as a de-escalation after , is administered, de-escalate to a dose combination which has the highest interval posterior probability of the interval for either +1, or , +1 . If the candidate combination for dose escalation has never been administered, the interval probability is calculated using the prior distribution. If the interval probabilities are equal, choose a drug combination randomly. Early completion method We propose a data-dependent early completion of dose finding trials for drug-combination. The early completion is determined when a beta-binomial probability for dose retainment with the trial data and the number of remaining patients is high. Because the ℛ includes not only the number of patients for dose retainment but also the number of patients for dose escalation, the dose retainment probability ○ 1 is divided by 2. The maximum dose-combination means the maximum dose of both two drugs, and if the planned maximum doses are changed by the stopping rule, the maximum doses are also changed. The minimum dose is calculated in the same way. We also propose the dose retainment probability which is calculated by replacing , in the equation ○ 1 with , × adjusted toxicity rate. We demonstrate the early completion for a virtual trial. From here, we call the normal dose retainment probability DRP and the dose retainment probability based on the bivariate isotonic regression DRP-I. Example trial We assume the dose-finding trial with the 3 dose levels of drug A and the 3 dose levels of ). The adjusted Discussion We proposed the early completion methods for dose finding trials for drug-combination. The early completion is determined when a beta-binomial probability for dose retainment with the trial data and the number of remaining patients is high. We also proposed an early completion method that a dose retainment probability is adjusted by a bivariate isotonic regression. We demonstrated the early completion for the virtual trial with 3 × 3 drug combinations and sample size of 36. By completing the trial early, we reduced 6 patients from the sample size. The six fewer patients mean a six-month reduction in trial duration, for example, if the safety evaluation period is one month and the enrollment of patient takes one month. For the simulation study, the percentages for the correct MTDs selection (PCMSs) in all early completion designs were almost the same as compared non-early completion designs. The Keyboard designs with early completion methods were smaller the change in PMCSs from the keyboard design. The results of the sensitivity analyses with changed planned sample size or TTL were observed same trend. In comparison with the stopping rule proposed by Trialdesign.org, the overall PCMSs of the designs with the stopping rule were lower than our proposed method, and the PCMSs reduced by about 10% compared with nonearly completion designs in scenarios 9-11 in Supplemental Table 9. We consider that the number of remaining patients should be considered for the early completion. By the application of early completion method, we confirmed the non-higher percentage of lower/higher doses selection than correct MTD compared with non-early completion designs. We confirmed that the early completion was averagely determined about 70% and the number of patients treated reduced averagely about 20-30%. We confirmed the superior performance for our proposed early completion methods. In specially, the keyboard with early completion was superior. Among the two early completion methods, ECI was slightly superior. However, we recommend EC because it is easier to implement. Our proposed method can be applied to dose-finding trial with more two drugs in the same way. We can show the number of patients for determining early completion before the trial starts and show in Table 3. This table can be also applied to the single drug and more two drugs dose-finding trial. We described the sample program code of early completion in supplemental material. Author's Contributions M. Kojima: Conception and design; development of methodology; acquisition of data (provided animals, acquired and managed patients, provided facilities, etc.): analysis and interpretation of data (e.g., statistical analysis, biostatistics, computational analysis); writing, review, and revision of the manuscript; administrative, technical, and material support (i.e., reporting and organizing data, constructing databases); and study supervision. If the current dose-combination is the maximum dose-combination, the dose-combination retained even if it is below the number of patients treated above. If the current dose-combination is the minimum dose-combination, the dose-combination retained even if it is over the number of patients treated above. have been proposed as simple and good performing. The BOIN and Keyboard designs can prepare a dose-assignment decision table based on a simple statistical model before the trial starts. The BOIN and Keyboard designs require to set a sample size in advance, and the trial cannot complete until the number of patients treated reaches the sample size. Although the BOIN and Keyboard designs have a stopping rule to terminate the trial if the number of patients treated exceeds the predetermined number, there is no statistical evidence for the number. ( ), ( )) for the DLT rate around a target DLT level. ( ) is the maximum value at which a dose escalation is determined and ( ) is the minimum value at which a dose de-escalation is determined. The boundaries of the proper dosing interval are calculated by minimizing the misjudgment of dose adjustment. For example, when = 0.3 , , = (0.236, 0.358) . Keyboard design conducts a doseassignment compared an interval probability of a proper dosing interval , = ( − 0.05, + 0.05) called a target key and other interval probabilities. When we do not distinguish , and , , we write just . If the drug combination administered is too toxic ( ( , > ) > 0.95) , stop the administration of the current dose combination and all higher dose combinations in the dosefinding trial. After completion of the trial, the MTD is selected as the dose combination whose observed toxicity rate is closest to the TTL. Because of the small sample size of the dosefinding trial, the observed toxicity rates are adjusted by a bivariate isotonic regression 34 to give a monotonically increase for the toxicity rates, and then the MTD is selected. Whenthere are multiple dose combinations that are closest to the TTL, if the observed toxicity rates of the multiple dose combinations are below the TTL, we select the highest dose combination as MTD. If the observed toxicity rates of the multiple dose combinations are over the TTL, we select the lowest dose combination as MTD. If there are simultaneously the multiple dose combination with higher and lower observed toxicity rates than the TTL, we select randomly the minimum dose combination in the dose combinations with higher observed toxicity and the maximum dose combination in the dose combinations with lower observed toxicity. The BOIN and Keyboard designs require to set a sample size in advance, and the trial cannot complete until the number of patients treated reaches the sample size. Although the BOIN and Keyboard designs have a stopping rule to terminate the trial if the number of patients treated exceeds the predetermined number, there is no statistical evidence for the number. 2, 2 2is 3.015. DRP-I is 0.491. The DRP-I is over the threshold of 0.4. Hence, this trial halts early. This trial can complete without dosing six patients by the early completion method. The six fewer patients mean a six-month reduction in trial duration, for example, if the safety evaluation period is one month and the enrollment of patient takes one month. We described the sample R program code to calculate the dose retainment probabilities in Supplemental material.Simulation studySimulation settings. We demonstrated the performance of the early completion methods via a Monte Carlo simulation. We prepared 6 designs: BOIN, BOIN using the early completion method based on DRP (BOIN-EC), BOIN using the early completion method based onDRP-I (BOIN-ECI), Keyboard (Key), Keyboard using the early completion method based on DRP (Key-EC), Keyboard using the early completion method based on DRP-I (Key-ECI). The dose-combinations were assumed 12 scenarios including two sizes of dose matrices (3 × 4 and 5 × 6). The detailed true DLT rates for each scenario were provided in SupplementalTable 1. The sample sizes were 45 for 3 × 4 drug combinations and 90 for 5 × 6 drug combinations. The cohort size was 3. The target toxicity level (TTL) was 30%. The threshold for early completion was 0.4. Sensitivity analyses also were performed by changing the sample size to 36 for 3 × 4 and 75 for 5 × 6 and the TTL to 20%, respectively. In addition, we conduct sensitivity analyses with changed to the threshold for early completion to 0.35 and 0.45. The simulation study was conducted in R. We evaluated each method using the following criteria.Evaluation criteria. 1. Percentage for correct MTD selection (PCMS). 2. Percentage for doses selection lower correct MTD 3. Percentage for doses selection higher correct MTD 4. The average number of patients treated 5. Percent change of patients treated from planned sample size Results Percentage for the correct MTDs selection (PCMS). Figure 2 shows the results of PCMSs for 12 scenarios. The change in PCMSs of BOIN-EC from BOIN averaged -3.9%, with a minimum -0.9% at scenario 6 and a maximum of -9.9 at scenario 9. The change in PCMSs of BOIN-ECI from BOIN averaged -4.6%, with an increasing maximum of 4.0% at scenario 6 and a decreasing maximum of -8.6 at scenario 10. The change in PCMSs of Key-EC from Key averaged -2.3%, with an increasing maximum of 0.8% at scenario 1 and a decreasing maximum of -4.8 at scenario 2. The change in PCMSs of Key-ECI from Key averaged -1.7%, with an increasing maximum 1.4% at scenario 6 and a decreasing maximum of -3.8% at scenario 9. The change in keyboard design using the early completion method was smaller than the BOIN design. Percentage for doses selection lower correct MTD. We show the difference (the result of early completion designs − the result of non-early completion design). For BOIN-EC, the average was 3.5%, with a minimum of 0.9% at scenario 1 and a maximum of 6.8% at scenario 9. For BOIN-ECI, the average was 7.0%, with a minimum 3.6% at scenario 12 and a maximum 13.4% at scenario 7. For Key-EC, the average was 1.2%, with a minimum of -0.7% at scenario 12 and a maximum of 2.5% at scenario 10. For Key-ECI, the average was 2.2%, with a minimum of 0.4% at scenario 12 and a maximum of 3.1% at scenario 3. The change of Key-EC was the smallest. Percentage for doses selection higher correct MTD. We show the difference (the result of early completion designs − the result of non-early completion design). For BOIN-EC, the average was 0.7%, with a minimum of -2.3% at scenario 7 and a maximum of 3.7% at scenario 2. For BOIN-ECI, the average was -2.0%, with a minimum -6.5% at scenario 7 and a maximum 1.3% at scenario 9. For Key-EC, the average was 1.2%, with a minimum of -1.3% at scenario 7 and a maximum of 4.7% at scenario 2. For Key-ECI, the average was -0.3%, with a minimum of -2.3% at scenario 7 and a maximum of 1.9% at scenario 12. The change of BOIN-EC was the smallest. The average number of patients treated/ Percent change of patients treated from planned sample size. For 3 × 4 drug combinations with the sample size of 45, the percent changes of patients treated from planned sample size for BOIN-EC and BOIN-ECI were averagely -24.6% and -25.1% (reduced about 11 patients), with a minimum of -14.2% and -14.0% (reduced about 6 patients) at scenario 8 and a maximum of -54.7% and -60.0% (reduced about 25 patients) at scenario 6. For Key-EC and Key-ECI, the percent changes were averagely -20.6% and -17.4% (reduced about 9 patients), with a minimum of -11.3% and -8.4% (reduced about 4 patients) at scenario 8 and a maximum of -48.2% and -47.6% (reduced about 23 patients) at scenario 6. For 5 × 6 drug combinations with the sample size of 90, the percent changes of patients treated from planned sample size for BOIN-EC and BOIN-ECI were averagely -31.6% and -29.8% (reduced about 27 patients), with a minimum of -28.1% and -26.2% (reduced about 24 patients) at scenario 11 and a maximum of -34.0% and -32.7% (reduced about 30 patients) at scenario 10. For Key-EC and Key-ECI, the percent changes were averagely -21.2% and -20.1% (reduced about 19 patients), with a minimum of -17.4% and -16.2% (reduced about 15 patients) at scenario 11 and a maximum of -24.0% and -22.9% (reduced about 21 patients) at scenario 10. On average, all changes were similar. normal dose retainment probability adjusted by pool-adjacent-violators algorithm. Figure 1 . 1Example of strategy for dose-assignmentThe solid line means that the drug was administered, and the dotted line means that the drug was a candidate for doses escalation/de-escalation. The blue means the dose escalation and the red means the dose de-escalation. Figure 2 . 2Percent change relative to the non-early completion version for selection of the The BOIN and keyboard designs can summarize the number of patients treated for which a dose retainment is determined in a table and the dose retainment probability uses this number. We assume that ℛ is the set of the number of patients for dose retainment for , + . For example, ℛ includes 3 and 4 patients, when the number of patients is 12 patients for the BOIN design.The dose retainment probability is calculated by the equation below,BetaBinom is the beta-binomial probability function that − , DLTs occur in the remaining patients when there are , DLTs for , patients. The trial is completed early when the dose retainment probability over a threshold. Kojima 29 recommends a threshold of 0.4 because the dose retainment interval for the DLT rate for BOIN design is a maximum of 0.4. In addition, because the MTD is selected by using the toxicity rate adjusted by a bivariate isotonic regression, we consider the early completion method using the toxicity rate adjusted by the bivariate isotonic regression. If the current dose-combination is the maximum dosecombination, ℛ includes the number of patients for which a dose escalation is determined.dose retainment probability = ∑ ( − , ; , , + 1, , − , + 1) ∈ℛ . ○ 1 drug B, the sample size of 30, the cohort size of 3, the TTL of 30, and the threshold for early and the safety evaluation has been completed. ℛ for 15 patients (9 patients treated and 6 remaining patients) for the BOIN design is {4,5}. Because ℛ of the Keyboard design is the same as one of the BOIN design, we consider an early completion without distinguishing between the designs.completion of 0.4. We show the result of 24 patients treated to Table 2. Dose adjustments for up to 24 patients are shown in Supplemental Example. The current dose combination is 2,2 The DRP is calculated below, (4 − 3; 6, 3 + 1, 9 − 3 + 1) + (5 − 3; 6, 3 + 1, 9 − 3 + 1) = 0.493. The DRP is over the threshold of 0.4. Hence, this trial halts early. Next, we calculate the DRP-I. The observed DLT rates adjusted by the bivariate isotonic regression are ( 0.000 0.000 0.000 0.167 0.335 0.664 0.000 0.664 0.000 ), the dose corresponding to each element is ( 1,1 1,2 1,3 2,1 2,2 2,3 3,1 3,2 3,3 Table 1 . 1The number of patients treated for dose retainment (TTL=0.3)Design Number of patients treated at current dose-combination 3 6 9 12 15 18 Keyboard 1 2 3 4-5 4-5 5-6 BOIN 1 2 3 3-4 4-5 5-6 Table 2 . 2The number of patients treated and DLTs for each dose combination ( , , , ) Dark gray indicates not administered. Bold indicates that the safety evaluation has completed at the current dose.1,1 (3,0) 1,2 1,3 2,1 (6,1) , (9,3) 2,3 (3,2) 3,1 3,2 (3,2) 3,3 Table 3 . 3Decision table for early completion on BOIN and Keyboard design with sample size of 36[Keyboard] #Patients at current dose #DLTs at current dose #Remining patients 3 1 ≤2 6 2 ≤2 9 3 ≤3 Accelerated titration designs for phase I clinical trials in oncology. R Simon, L Rubinstein, S G Arbuck, M C Christian, B Freidlin, J. Natl. Cancer Inst. 89Simon R, Rubinstein L, Arbuck SG, Christian MC, Freidlin B, et al, Accelerated titration designs for phase I clinical trials in oncology, J. Natl. Cancer Inst. 1997;89:1138-1147. Continual reassessment method: a practical design for phase 1 clinical trials in cancer. J O&apos;quigley, Pepe M Fisher, L , Biometrics. 46O'Quigley J, Pepe M and Fisher L. Continual reassessment method: a practical design for phase 1 clinical trials in cancer. Biometrics 1990; 46: 33-48. Dose-finding with two agents in phase I oncology trials. P F Thall, R E Millikan, P Muller, S J Lee, Biometrics. 59Thall PF, Millikan RE, Muller P, Lee SJ. Dose-finding with two agents in phase I oncology trials. Biometrics 2003; 59: 487-496. Designs for single-or multiple-agent phase I trials. M R Conaway, S Dunbar, S D Peddada, Biometrics. 60Conaway MR, Dunbar S and Peddada SD. Designs for single-or multiple-agent phase I trials. Biometrics 2004; 60: 661-669. A nonparametric approach to the design and analysis of twodimensional dose-finding trials. A Ivanova, K Wang, Stat Med. 23Ivanova A and Wang K. A nonparametric approach to the design and analysis of two- dimensional dose-finding trials. Stat Med 2004; 23: 1861-1870. A parallel phase I/II clinical trial design for combination therapies. X Huang, S Biswas, Y Oki, Biometrics. 63Huang X, Biswas S, Oki Y, et al. A parallel phase I/II clinical trial design for combination therapies. Biometrics 2007; 63: 429-436. Design issues in dose-finding phase I trials for combinations of two agents. S K Fan, A P Venook, Y Lu, J Biopharm Stat. 19Fan SK, Venook AP and Lu Y. Design issues in dose-finding phase I trials for combinations of two agents. J Biopharm Stat 2009; 19: 509-523. Bayesian dose finding in oncology for drug combinations by copula regression. G Yin, Y Yuan, J R Stat Soc Ser C Appl Stat. 61Yin G and Yuan Y. Bayesian dose finding in oncology for drug combinations by copula regression. J R Stat Soc Ser C Appl Stat 2009; 61: 211-224. Dose-finding design for multi-drug combinations. N A Wages, M R Conaway, O Quigley, J , Clin Trials. 8Wages NA, Conaway MR and O'Quigley J. Dose-finding design for multi-drug combinations. Clin Trials 2011; 8: 380-389. A two-dimensional search algorithm for dose-finding trials of two agents. Lee Bl, Fan Sk, J Biopharm Stat. 22Lee BL and Fan SK. A two-dimensional search algorithm for dose-finding trials of two agents. J Biopharm Stat 2012; 22: 802-818. Adaptive designs for dual-agent phase I dose-escalation studies. J A Harrington, G M Wheeler, M J Sweeting, Nat Rev Clin Oncol. 10Harrington JA, Wheeler GM, Sweeting MJ, et al. Adaptive designs for dual-agent phase I dose-escalation studies. Nat Rev Clin Oncol 2013; 10: 277-288. A dose-finding approach based on shrunken predictive probability for combinations of two agents in phase I trials. A Hirakawa, C Hamada, S Matsui, Stat Med. 32Hirakawa A, Hamada C and Matsui S. A dose-finding approach based on shrunken predictive probability for combinations of two agents in phase I trials. Stat Med 2013; 32: 4515-4525. A Bayesian dose-finding design for drug combination trials with delayed toxicities. S Liu, J Ning, Bayesian Anal. 8Liu S and Ning J. A Bayesian dose-finding design for drug combination trials with delayed toxicities. Bayesian Anal 2013;8: 703-722. Non-parametric overdose control for dose finding in drug combination trials. K C Lam, R Lin, G Yin, JRSS Appl Statist. 68Lam KC, Lin R, Yin G. Non-parametric overdose control for dose finding in drug combination trials. JRSS Appl Statist 2019;68: 1111-1130. A nonparametric Bayesian method for dose finding in drug combinations cancer trials. S Z Razaee, G Cook-Wines, M Tighiouart, Stat Med. 2022Razaee SZ, Cook-Wines G, Tighiouart M. A nonparametric Bayesian method for dose finding in drug combinations cancer trials. Stat Med 2022:1-22 Designs of drug-combination phase I trials in oncology: a systematic review of the literature. K M Riviere, Le Tourneau, C Paoletti, X Dubois, F Zohar, S , Annals of Oncology. 26Riviere KM, Le Tourneau C, Paoletti X, Dubois F, Zohar S. Designs of drug-combination phase I trials in oncology: a systematic review of the literature. Annals of Oncology 2015;26:669-674 Bayesian optimal interval designs for phase I clinical trials. S Liu, Y Yuan, Appl Statist. 64Liu S, Yuan Y. Bayesian optimal interval designs for phase I clinical trials. Appl Statist 2015;64:507-523. Model-Assisted Designs for Early-Phase Clinical Trials: Simplicity Meets Superiority. Y Yuan, J J Lee, G S Hilsenbeck, JCO PO. 3Yuan Y, Lee JJ, Hilsenbeck GS, Model-Assisted Designs for Early-Phase Clinical Trials: Simplicity Meets Superiority. JCO PO 2019;3;1-12 Keyboard design for phase I drug-combination trials. H Pan, R Lin, Y Zhou, Y Yuan, Contemporary Clinical Trials. 92Pan H, Lin R, Zhou Y, Yuan Y. Keyboard design for phase I drug-combination trials. Contemporary Clinical Trials 2020;92:1-11 Keyboard: A Novel Bayesian Toxicity Probability Interval Design for Phase I Clinical Trials. F Yan, J S Mandrekar, Y Yuan, Clin Cancer Res. 2315Yan F, Mandrekar JS, Yuan Y. Keyboard: A Novel Bayesian Toxicity Probability Interval Design for Phase I Clinical Trials. Clin Cancer Res 2017;23(15):3994-4003. Bayesian optimal interval design for dose finding in drug-combination trials. R Lin, G Yin, Statistical Methods in Medical Research. 265Lin R, Yin G. Bayesian optimal interval design for dose finding in drug-combination trials. Statistical Methods in Medical Research 2017;26(5):2155-2167 Bayesian optimal interval design for dose finding based on both efficacy and toxicity outcomes. K Takeda, M Taguri, S Morita, Boin-Et, Pharma Stat. 17Takeda K, Taguri M, Morita S. BOIN-ET: Bayesian optimal interval design for dose finding based on both efficacy and toxicity outcomes. Pharma Stat 2018,17:383-395. Time-to-Event Bayesian Optimal Interval Design to Accelerate Phase I Trials. Y Yuan, R Lin, D Li, L Nie, K E Warren, Clin Cancer Res. 24Yuan Y, Lin R, Li D, Nie L, Warren KE. Time-to-Event Bayesian Optimal Interval Design to Accelerate Phase I Trials. Clin Cancer Res 2018;24:4921-4930. gBOIN: a unified model-assisted phase I trial design accounting for toxicity grades, and binary or continuous end points. R Mu, Y Yuan, J Xu, S J Mandrekar, J Yin, JRSS Appl Statist. 68Mu R, Yuan Y, Xu J, Mandrekar SJ, Yin J. gBOIN: a unified model-assisted phase I trial design accounting for toxicity grades, and binary or continuous end points. JRSS Appl Statist 2019;68: 289-308. Time-to-event model-assisted designs for dose-finding trials with delayed toxicity. R Lin, Y Yuan, Biostatistics. 21Lin R, Yuan Y. Time-to-event model-assisted designs for dose-finding trials with delayed toxicity. Biostatistics 2020;21:807-824. BOIN12: Bayesian Optimal Interval Phase I/II Trial Design for Utility-Based Dose Finding in Immunotherapy and Targeted Therapies. R Lin, Y Zhou, F Yan, D Li, Y Yuan, JCO PO. 4Lin R, Zhou Y, Yan F, Li D, Yuan Y. BOIN12: Bayesian Optimal Interval Phase I/II Trial Design for Utility-Based Dose Finding in Immunotherapy and Targeted Therapies. JCO PO 2020;4:1393-1402 TITE-BOIN-ET: time-to-event Bayesian optimal interval design to accelerate dose-finding based on both efficacy and toxicity outcomes. K Takeda, S Morita, M Taguri, Pharma Stat. 19Takeda, K, Morita S, Taguri M. TITE-BOIN-ET: time-to-event Bayesian optimal interval design to accelerate dose-finding based on both efficacy and toxicity outcomes. Pharma Stat 2020;19:335-349. Early completion of phase I cancer clinical trials with Bayesian optimal interval design. M Kojima, Stat Med. 40Kojima M. Early completion of phase I cancer clinical trials with Bayesian optimal interval design. Stat Med 2021;40:3215-3226 Early completion of model-assisted designs for dose-finding trials. M Kojima, JCO Precis Oncol. 20215Kojima M. Early completion of model-assisted designs for dose-finding trials. JCO Precis Oncol 2021: 5, 1449-1457 Adaptive design for identifying maximum tolerated dose early to accelerate dose-finding trial. M Kojima, ID: 2110.02413Arxiv. Kojima M. Adaptive design for identifying maximum tolerated dose early to accelerate dose-finding trial. Arxiv. 2021;Arxiv ID: 2110.02413. Early completion based on multiple dosages to accelerate maximum tolerated dose-finding. M Kojima, ID: 2110.00563Arxiv. Kojima M. Early completion based on multiple dosages to accelerate maximum tolerated dose-finding. Arxiv. 2021;Arxiv ID: 2110.00563. Application of Multi-Armed Bandits to Model-assisted designs for Dose-Finding Clinical Trials. M Kojima, ID: 2201.05268Arxiv. Kojima M. Application of Multi-Armed Bandits to Model-assisted designs for Dose- Finding Clinical Trials. Arxiv. 2021;Arxiv ID: 2201.05268. Bayesian adaptive model selection design for optimal biological dose finding in phase I/II clinical trials. R Lin, G Yin, H Shi, Biostatistics. Lin R, Yin G, Shi H. Bayesian adaptive model selection design for optimal biological dose finding in phase I/II clinical trials. Biostatistics 2021. An algorithm for isotonic regression for two or more independent variable. L R Dykstra, T Robertson, Ann Stat. 103Dykstra LR, Robertson T. An algorithm for isotonic regression for two or more independent variable. Ann Stat;10(3):708-716
{'fraction_non_alphanumeric': 0.045335942596216566, 'fraction_numerical': 0.03335705390499911, 'mean_word_length': 4.364561794178464, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'Financial support: NoneConflict of interest disclosure statement: NoneContext Summaries:We propose a data-dependent early completion of dose finding trials for drug-combination.The early completion is determined when a beta-binomial probability for dose retainment with the trial data and the number of remaining patients is high. This paper also proposes an 3 early completion method that a dose retainment probability is adjusted by a bivariate isotonic regression. We demonstrate the early completion for a virtual trial. We evaluate the performance of early completion method through simulation studies with 12 scenarios. We confirmed the superior performance for our proposed early completion methods. We show the number of patients for determining early completion before a trial starts and a program code for calculating dose retainment probability in this paper.Word count: 2998The number of figures: 3The number of tables: 3 4 Abstract Purpose: Model-assisted designs for drug combination trials have been proposed as novel designs with simple and superior. The model-assisted designs require to set a sample size in advance, and the trial cannot complete until the number of patients treated reaches the sample size. We propose a data-dependent early completion of dose finding trials for drugcombination.Methods:The early completion is determined when a beta-binomial probability for dose retainment with the trial data and the number of remaining patients is high. This paper also proposes an early completion method that a dose retainment probability is adjusted by a bivariate isotonic regression. We demonstrate the early completion for a virtual trial. We evaluate the performance of early completion method through simulation studies with 12 scenarios.Results: From simulation studies, the percentage of early completion was averagely determined about 70% and the number of patients treated reduced by 25% from planned sample size. The percentage of a correct maximum tolerated dose-combination selection for early completion designs showed little change from the non-early completion designs by an average of about 3%. 5 Conclusion: We confirmed the superior performance for our proposed early completion methods. We showed the number of patients for determining early completion before the trial starts and a program code for calculating dose retainment probability in this paper. correct MTD Key: Keyboard, EC: Early completion based on normal dose retainment probability, ECI: Early completion based on dose retainment probability using bivariate isotonic regression Figure 3. Percentage of early completion Key: Keyboard, EC: Early completion based on normal dose retainment probability, ECI: Early completion based on dose retainment probability using bivariate isotonic regression', 'arxivid': '2202.04252', 'author': ['Masahiro Kojima [email protected] \nBiometrics Department, R&D Division\nKyowa Kirin Co., Ltd\nTokyoJapan\n\nDepartment of Statistical Science\nSchool of Multidisciplinary Sciences\nAddress: Biometrics Department, R&D Division, Kyowa Kirin Co., Ltd. Otemachi Financial City Grand Cube\nThe Graduate University for Advanced Studies\n1-9-2 Otemachi, Chiyoda-ku, 100-004Tokyo, TokyoJapan., Japan\n', 'Masahiro Kojima '], 'authoraffiliation': ['Biometrics Department, R&D Division\nKyowa Kirin Co., Ltd\nTokyoJapan', 'Department of Statistical Science\nSchool of Multidisciplinary Sciences\nAddress: Biometrics Department, R&D Division, Kyowa Kirin Co., Ltd. Otemachi Financial City Grand Cube\nThe Graduate University for Advanced Studies\n1-9-2 Otemachi, Chiyoda-ku, 100-004Tokyo, TokyoJapan., Japan'], 'corpusid': 246680011, 'doi': '10.1177/09622802231155094', 'github_urls': [], 'n_tokens_mistral': 9637, 'n_tokens_neox': 8328, 'n_words': 5219, 'pdfsha': '01dd3cf69020e9b9c63984cf02627dadaeb52a56', 'pdfurls': ['https://arxiv.org/pdf/2202.04252v1.pdf'], 'title': ['Data-Dependent Early Completion of Dose Finding Trials for Drug-Combination', 'Data-Dependent Early Completion of Dose Finding Trials for Drug-Combination'], 'venue': []}
arxiv
Understanding Individual and Team-based Human Factors in Detecting Deepfake Texts Adaku Uchendu Jooyoung Lee Hua Shen Thai Le Ting-Hao &apos;kenneth&apos; Huang Dongwon Lee Adaku Uchendu Jooyoung Lee Hua Shen Thai Le Ting-Hao Kenneth &apos; Huang Pennsylvania State University USA Pennsylvania State University USA Pennsylvania State University USA University of Mississippi USA Pennsylvania State University USA Pennsylvania State University USA Understanding Individual and Team-based Human Factors in Detecting Deepfake Texts ACM Reference Format:Additional Key Words and Phrases: deepfakeindividualcollaborationexpertnon-expert In recent years, Natural Language Generation (NLG) techniques in AI (e.g., T5, GPT-3, ChatGPT) have shown a massive improvement and are now capable of generating human-like long coherent texts at scale, yielding so-called deepfake texts. This advancement, despite their benefits, can also cause security and privacy issues (e.g., plagiarism, identity obfuscation, disinformation attack). As such, it has become critically important to develop effective, practical, and scalable solutions to differentiate deepfake texts from human-written texts. Toward this challenge, in this work, we investigate how factors such as skill levels and collaborations impact how humans identify deepfake texts, studying three research questions: (1) do collaborative teams detect deepfake texts better than individuals? (2) do expert humans detect deepfake texts better than non-expert humans?(3)what are the factors that maximize the detection performance of humans? We implement these questions on two platforms: (1) non-expert humans or asynchronous teams on Amazon Mechanical Turk (AMT) and (2) expert humans or synchronous teams on the Upwork. By analyzing the detection performance and the factors that affected performance, some of our key findings are: (1) expert humans detect deepfake texts significantly better than non-expert humans, (2) synchronous teams on the Upwork detect deepfake texts significantly better than individuals, while asynchronous teams on the AMT detect deepfake texts weakly better than individuals, and (3) among various error categories, examining coherence and consistency in texts is useful in detecting deepfake texts. In conclusion, our work could inform the design of future tools/framework to improve collaborative human detection of deepfake texts. INTRODUCTION In recent years, AI technologies have drastically advanced, enabling the generation of human-quality artifacts in various modalities, including texts, images, and videos [12,31,40]. Collectively, these AI-generated artifacts are known as Deepfakes. In particular, the advanced Natural Language Generation (NLG) techniques, especially large language models (e.g., GPT-3, T5, ChatGPT), are now able to generate long coherent texts without human intervention. In this work, we refer to such machine-generated texts as Deepfake Texts 1 and generative language models as Neural Text Generator (NTG). Such deepfake texts have many obvious benefits in diverse applications. For instance, individuals can use deepfake texts in writing draft blog profiles and program codes for learning while business can use them for writing bulk emails or product descriptions at scale. However, as it is true for any technology, deepfake texts also can be misused in many applications. For instance, students may abuse deepfake texts in writing essay homework, scammers may generate sophisticated phishing messages from deepfake texts, and state-backed operators may use deepfake texts as part of disinformation attack. Therefore, a great need to differentiate deepfake texts from human-written texts has naturally risen. In essence, this task (i.e., given a text T , determine if T is a deepfake text or human-written text) resembles so-called Turing Test 2 or binary classification in Machine Learning. In this work, we investigate this task of determining if a given text is a deepfake text or not. While the field of open-ended text generation is still relatively new, both computational and noncomputational detection of deepfake texts have been extensively studied in recent years and well surveyed in Uchendu et al. [35]. In particular, what we are interested in answering is how "humans" are able to detect deepfake texts better. Clearly understanding human capacity and their limitations in detecting deepfake texts would help the development of both computational and non-computational (and even hybrid) tools for detecting deepfake text better. Recent literature (e.g., [5,6,10,37]) has shown that, by and large, humans are not good at detecting deepfake texts, performing only slightly better than the level of random guessing. Even if humans are trained to detect deepfake texts, the performance has not improved significantly (e.g., [6,9,34]). Therefore, in this work, we aim to find ways to improve humans in detecting deepfake texts better and understand human factors at play. Especially, we wonder if individual humans, trained or not, are not good at detecting deepfake texts, does their collaboration or expertise matter? As such, we pose the following three research questions (RQs): RQ1 Do collaborative teams/groups perform better than individuals in deepfake text detection? RQ2 Do experts perform better than non-experts in deepfake text detection? RQ3 What are the factors that maximize the performance gain? RQ1 aims to investigate what improves human performance from the baseline -team collaboration or individuals, and if collaboration improves human performance significantly, which collaboration technique matter -synchronous or asynchronous collaboration? The hypothesis here is that synchronous collaboration will improve human performance in deepfake text detection because humans perform better when there is informal discussion and sharing of ideas,as shown in prior literature (e.g., [26], [18], [28]). Given the benefits of collaboration, we hypothesize that collaboration will improve human performance. RQ2 aims to investigate how English experts vs. English non-experts detect deepfake texts differently. English experts are defined as individuals with at least a Bachelor's degree in English (and related programs). We aim to investigate the characteristics that make one or more settings significantly outperform others. An example here is that we hypothesize that experts will focus more on high-level errors such as logical fallacies and non-experts will focus more on low-level errors such as grammar issues. RQ3 aims to investigate the human factors that may help improve human performance in deepfake text detection. See Figure 1(C) for a visual representation of the research questions. Finally, our main contribution is investigating human participants' ability to detect deepfake texts with different settings -non-expert vs. expert and individual vs. collaborative. Our key findings are summarized as follows: (1) both expert and non-expert in the individual settings outperform the baseline significantly; (2) experts improve significantly from individual to collaborative settings; and (3) experts more frequently use strong indicators of deepfake texts as justification (which explains their superior performance). RELATED WORK Automatic Evaluation of Deepfake Texts As Neural Text Generators (NTGs) such as GPT-2 can be maliciously used to generate misinformation at scale, several techniques have been employed to detect deepfake texts. Using stylometric 3 classifiers, researchers adopted stylometry from traditional authorship attribution solutions to achieve automatic deepfake text detection [14,36]. However, due to the flaws of stylometric classifiers, deeplearning techniques have been proposed [1,3,[19][20][21]39]. While these deep-learning techniques achieved high performance and significantly improved from stylometric classifiers, they are not interpretable. To mitigate this issue, statistical-based classifiers are proposed [15,16,29,30]. Lastly, to combine the benefits of each of the 3 types of classifiers for deepfake text detection, 2 or more of these classifier types are combined to build a more robust classifier. Uchendu et al. [35] defines these classifiers as hybrid classifiers and they achieve superior performance [23,24,41]. Lastly, using automatic deepfake text detectors, deepfake detection has been achieved with reasonable performance. However, in the real world, as humans cannot solely depend on these models to detect deepfakes, they need to be equipped at performing the task themselves. A common theme in most of the detectors are that newer NTGs are harder to detect, which can sometimes make the older detectors obsolete. Thus, it is imperative that humans are also able to perform the task of deepfake text detection. For this reason, a few researchers have evaluated human performance in this task under several settings. See below. Human Evaluation of Deepfake Texts The quality of deepfake texts has always been compared to human-written texts. Thus, since humans still remain the gold standard when evaluating machine-generated texts, several works have investigated human performance in distinguishing between human-written and machine-generated texts. This has been studied in clever and nuanced ways which include training and not training. Human Evaluation Without Training. GROVER [39], a NTG trained to generate news articles can easily be used maliciously. To evaluate the quality of GROVER-generated news (fake) articles, they are compared to human-written news articles. Humans are asked to pick which articles are more believable and GROVER-generated fake news was found to be more trustworthy [39]. Donahue et al. [8] recruits human participants from Amazon Mechanical Turk (AMT) to detect machine-generated words in a sentence. Uchendu et al. [37] also recruits human participants from AMT and asks them to detect which one of two articles is machine-generated and given one article, decide if it is machine-generated or not. Ippolito et al. [20] evaluates the human ability to perform comparably given 2 different generation strategies. Brown et al. [5] evaluates human performance in distinguishing human-written texts from GPT-3-generated texts. Finally, in all these works, the themes remain the same -humans perform poorly at detecting machine-generated texts, achieving about or below chance-level during evaluation. Human Evaluation with Training. Since human performance in deepfake text detection is very poor, researchers proposed improving there experimental framework by training human participants before the task. Humans without training achieve a 54% accuracy in distinguishing GPT-2-generated texts from human-written texts [16]. To improve performance, GLTR (Giant Language Model Test Room), a color-coded tool is proposed. GLTR color codes words based on the distribution level which improves human performance from 54% to 72% [16]. Dugan et al. [11] gamifies machinegenerated text detection by training humans to detect the boundary at which a document becomes deepfake to earn points. Humans are given the option to select one of many reasons or include their own reasons for which a sentence could be machine-generated [10]. Our framework is modeled more closely after Dugan et al. [11]'s work. Next, Clark et al. [6] proposes 3 training techniques -Instruction-based, Example-based, and Comparison-based. Example-based training improved the accuracy from 50% to 55% [6]. Lastly, Dou et al. [9] recruits human participants to annotate the error types of machine-generated texts. Participants were evaluated on an extensive qualification task which trains them [9]. A score ≥ 90 out of 100 is considered a pass so the participant can move to the next round. Finally, all methods except for GLTR did not yield significant improvements in human performance. However, GLTR achieved an average of 56% F1 score on 19 pairs of human vs. NTGs [37]. This means that while GLTR outperformed all other methods, it was built in 2019, and the results from Uchendu et al. [37] suggest that newer NTG render older deepfake text detectors inferior/obsolete. We hypothesize that previous techniques to improve human performance failed because they did not consider that collaboration and skill levels could affect performance. Based on the skill levels of humans, they will understand hints and potentially use them differently. For instance, if given a task to highlight the grammar issues in a piece of text, a person with college or post-college level of reading & writing will find higher-level grammar errors (e.g., re-worded repetition, run-on sentences) than a person with 11th-grade level. Furthermore, the point of collaboration is to encourage the exchange of ideas which could de-mystify the task for humans and improve performance. Therefore, while we implement the example-based training technique, we also improve human performance by incorporating collaboration. METHODOLOGY To improve human performance in deepfake text detection, we first, define a realistic problemdetecting deepfake texts in an article authored by both human and an AI (i.e., . This is done by randomly replacing 1 out 3 human-written paragraphs with a GPT-2-generated paragraph. Next, we define 2 variables for our studyindividual vs. collaboration and non-expert vs. expert. After using these variables to facilitate crowdsoursing recruitment, we ask the human participants to select 1/3 paragraphs that is deepfake and provide justification for selection. Data Generation Our task is modeled similarly to the Turing Test problem defined by Alan Turing in the 1950s. The Turing Test is a test administered by a human as the judge who has a conversation with an unknown entity and decides if they are speaking with a human or machine/AI model. If the machine is labeled as human, then the machine has passed the test. However, for our task, due to the security risks deepfake texts pose, it is imperative that the NTG does not pass the test. To that end, we propose a framework that increases the probability of GPT-2 failing the Turing Test. Therefore, we first define the goal of distinguishing deepfake texts from human-written texts. As this problem has been studied by several researchers [6,11,16,20,36] and shown to be very non-trivial and difficult to solve, we develop a novel way of administering the Turing Test. Thus, we implement a realistic setting of the problemdetecting deepfake texts in an article authored by both humans and an NTG. This setting is motivated by the fact that while NTGs currently have very impressive generations, humans still produce more natural speech than NTGs. This means that it will be natural for humans to edit machine-generated articles to make them sound more authentic. We study this non-trivial problem by asking human evaluators to: (1) Of 3 paragraphs, two written by a human, and one machine-generated, select the machine-generated paragraph. (2) Please check all explanations that satisfy the reason(s) for your choice. See Figure 1(B) for a visual representation of our task. We provide seven pre-defined rationales that correspond to flaws typically observed in deepfake text [10] -grammatical issues, repetition, lacks common sense, contains logical errors, contradicts previous sentences, lack of creativity or boring to read, writing is erratic (i.e. does not have a good flow) or choose to write on their own. To build this dataset, we collected 200 human-written news articles (mostly politics since this work is motivated by mitigating the risk of mis/disinformation or fake news dissemination) from reputable news sources such as CNN and Washington Post. Next, of the 200 articles, we took the first suitable 50 articles with at least 3 paragraphs. Then, we removed all paragraphs after the 3rd paragraph. Since the goal is to have a multi-authored article (human and AI), we randomly select 1/3 paragraphs to be replaced by GPT-2's [32] generated texts. We used only GPT-2 to generate the deepfake texts because: (1) GPT-2 and GPT-3 are similar. Based on [6,36], human performance on detecting GPT-2 and GPT-3 texts have similar accuracies; and (2) GPT-2 is cheaper to generate texts with than GPT-3 since GPT-2 is open-source and GPT-3 is not. For generation, we used GPT-2 XL Table 1. Data labels of deepfake texts which has 1.5 billion parameters and aitextgen 4 , a robust implementation of GPT-2 to generate texts with the default parameters. We followed the following replacement process: (1) If paragraph 1 is to be replaced: Use Title as a prompt to generate GPT-2 replacement (2) If paragraph 2 is to be replaced: Use Paragraph 1 as a prompt to generate GPT-2 replacement (3) If paragraph 3 is to be replaced: Use Paragraph 2 as a prompt to generate GPT-2 replacement Since, we are unable to control the number of paragraphs GPT-2 generates given a prompt, we use a Masked Language model to choose the best GPT-2 replacement that fits well with the article. We use a BERT-base [7] as the Masked Language model to get the probability of the next sentence. Let us call this model G(.), it takes 2 inputs -the first and probable second sentence/paragraph (G(T ext_1, T ext_2)) and outputs a score. The lower the score, the more probable T ext_2 is the next sentence. For instance, say GPT-2 texts is to replace Paragraph 2 (P2) of an article • We use P1 as prompt to generate P2 with GPT-2 • GPT-2 generates another 3-paragraph article with P1 as the prompt • To find the suitable P2 replacement, we do G(P1, each GPT-2 generated paragraph) • Since low scores with G(.) is considered most probable, the P2 replacement is the GPT-2 paragraph that yielded the lowest score with G(.) We use a random number generator to select which paragraphs are to be replaced and got the following deepfake text replacement for the paragraphs in Table 1. After we created these multiauthored articles, we manually did a quality check of a few of these articles by checking for consistency and coherence. See Figures 2 for the data generation process and 1(A) for an example of the final multi-authored article. Next, as we have defined this realistic scenario, we hypothesize that collaboration will improve human detection of deepfake texts. Thus, we define 2 variables for this experiment -Individual vs. Collaboration and English expert vs. English non-expert. We investigate how collaboration (both synchronous and asynchronous) improves from individual-based detection of deepfake texts. The hypothesis here is that when humans come together to solve a task, collaborative effort will be a significant improvement from average individual efforts. Additionally, as human detection of deepfake texts is non-trivial, we want to investigate if the task is non-trivial because English non-experts focus on misleading cues as opposed to English experts. Finally, we observe that based on the replacement algorithm, some bias in detection may be introduced. Replacing paragraph 3 may be seen as easier because there is no other paragraph after it to judge the coherency. However, we keep the generation process fair by only using the text right before the paragraph as a prompt to generate the next paragraph. Thus, to replace paragraph 3, we only use paragraph 2 as a prompt, not the previous paragraphs and title. Participant Recruitment AMT. Inspired by Clark et al. [6], Dugan et al. [11], and Van Der Lee et al. [38], we used Amazon Mechanical Turk (AMT) to collect responses from non-expert evaluators. We deployed a two-stage process to conduct the non-expert human studies. First, we posted a Qualification Human Intelligence Task (HIT) that pays $0.50 per assignment on MTurk to recruit 240 qualified workers In terms of the qualification requirements, in addition to our custom qualification used for worker grouping, three built-in worker qualifications are used in all the HITS, including i) HIT Approval Rate (≤98%), Number of Approved HITs (≥3000), and Locale (US Only) Qualification. Next, we only enable the qualified workers to enter the large-scale labeling tasks. The approximate time to finish each labeling task is around 5 minutes (i.e., the average time of two authors on finishing a random HIT). Therefore, we aim for $7.25 per hour and set the final payment as $0.6 for each assignment. Further, we provide "double-payment" to workers who made correct submissions as the extra bonus. Upwork. We utilized Upwork to recruit expert evaluators, especially those with expertise in writing domains.Upwork is one of the leading freelance websites with a substantial network size: Upwork generates 40 million monthly visits on average, and its gross services volume reached 3.5 billion dollars in 2021. 5 It has facilitated the freelance industry by introducing skilled freelancers in diverse categories like writing, graphic design, and web development. With its automated recommendation system, Upwork is capable of effectively matching clients and workers based on their needs. Through Upwork, we first posted a task description as a client to gather participants. We mentioned in the description that this is for research and provided all necessary information such as research objectives and questions we anticipated that they will solve. Our recruitment advertisement also highlighted the mandatory requirements: (1) a participant should be at least 18 years old; and (2) a participant should be a native English speaker. Lastly, if they were willing to proceed, they were asked to submit a proposal answering the following questions: (1) what is the highest level of degree you have completed in school? (2) did you major in English or English Literature? and (3) describe your recent experience with similar projects. One useful feature for accelerating the recruitment process in Upwork is that not only workers can apply to the postings but also clients like us can invite prospective candidates that seem suitable for the task to submit proposals. We manually reviewed workers' profile descriptions who specified their skill sets as copywriting, editing/proofreading, content writing and then sent them invites. While making recruitment decisions, we verified participants' eligibility by checking their selfreported age, language, and education in the profile, in addition to evaluating their proposal responses. It resulted in a total of 18 finalists to officially begin the study. Next, we sent them the consent form via the platform's messaging function and activated Upwork contracts only after they returned the signed form. A primary purpose of the contracts was for clients to compensate workers based on submitted hours through the Upwork system. Participants' requested hourly wages ranged from $25-$35 per hour depending on their prior experiences and education levels. All 18 individuals successfully signed both documents and were compensated accordingly. Table 2 gives the self-reported demographic breakdown of recruited Upworkers. Experiment Design AMT. During the large-scale labeling task, we divide the recruited qualified workers into two groups to represent the individual vs. collaborative settings, respectively. We define the group1 as Individual Group, in which each worker was asked to select the machine-generated paragraph without any references. See Figure 3, for example, humans in Individual Group can only see the introduction with panel (A) (B) and (C). On the other hand, we design the group 2 to be Collaborative Group, where the workers were asked to conduct the same task after the Individual Group finishes all HITs (i.e., see panel (A), (B), (C) in Figure 3). In addition, workers from the Collaborative Group could also see the selection results from the group1 in an asynchronously manner, as the example shown in Figure 3(D), to support their own selection. In addition, we investigate the capability of Individual vs. Collaboration of non-expect human participants to improve human performance in deepfake text detection. To do this, we compare 2 ways the human participants can provide justification for their answer -Select & Write. For the select setting, the question type was inspired by RoFT [11], a gamification technique for improving human performance in deepfake text detection. In the RoFT framework, participants were asked to select from a pre-defined list one or more reasons such as repetition, grammar errors, etc. Participants were also given another option, where they can enter their own justification if they do not find any suitable selection from the provided list. However, as this may be limiting because we pre-define the justifications, we also investigated another question typewrite. In this setting, participants were asked to provide their reasoning. To help, we also share the list of justifications in the select setting to give the participants an idea of what justifications look like. See Figure 5 for select and write AMT interface. Furthermore, we take actions to incentivize workers to provide qualified results: i) in our instruction, we provide immediate feedback on the worker's selection to calibrate their accuracy. In specific, after reading the HIT instruction (i.e., Figure 4 (A)), workers can get a deeper understanding of "which paragraph is generated by AI machine" by trial and error on selecting one example (i.e., Figure 4 (B)). Participants were given unlimited chances to change their answers. This example-based training process was inspired by Clark et al. [6]'s human evaluation study and was found to be the most effective training technique. ii) We pay double compensations to the workers who provide correct answers as mentioned in Section 3.2.1. This aims to encourage workers to get high accuracy on selecting the correct machine-generated paragraphs. iii) We set the minimum time constraint (i.e., one minute) for workers to submit their HITs, so that the workers will concentrate on the task for at least one minute instead of randomly selecting one answer and submitting the HIT. Note that we also disabled the copy and paste functions in the user interface to prevent workers from searching for the paragraphs from online resources. Upwork. Given that we aim to compare experts' deepfake text detection accuracy with respect to individual vs. collaborative settings, our Upwork study consists of two sub-experiments. The first experiment asks Upwork participants to perform a given task on their own. The second experiment requires three individuals to solve the questions as one group in a synchronous manner. We used Qualtrics 6 service to generate and disseminate the study form. The user interface was equivalent to the select scenario in Figure 5. Upwork participants were given one week to complete the survey. Upon completion, we randomly grouped 3 participants per team, resulting 6 teams in total for synchronous collaboration ( Table 2). All discussions were conducted on the video communications software -Zoom 7 and we leveraged Zoom's built-in audio transcription feature, which is powered by Otter.ai 8 for discourse analyses. In addition to the written consent obtained during the recruitment procedure, verbal consent for participation in the discussion and for audio recording was obtained prior to the start of each session. One member of the study team served as a moderator for the meetings. Depending on the participant's schedule and level of commitment in their group, each meeting lasted 1.5 -3 hours. Analysis Methods To investigate RQ1 and RQ2, we first quantitatively compare the performance of human participants (Section 4.2). This is done by measuring the mean accuracy of participants at the article level. Next, we categorize their provided rationales and compare their distributions based on correct and incorrect responses to support RQ3 (Section 4.3 & 4.4). Consistent with RQ1 and RQ2 analyses, for RQ3, we compare the frequency of justifications across individual vs. collaboration and non-experts (AMT) vs. experts (Upwork) settings. In addition to quantitative examination, we conduct qualitative studies on discussion transcripts from Upwork participants to gain insights into how Upwork groups benefited more from collaboration (Section 4.5). Finally, the implications of preceding investigation are outlined in Section 5. RESULTS Performance Measurement We measure how well participants perform the tasks and compared them across different experiment settings. To quantify the detection performance of each setting, we computed the proportion of people who got the answer correct given a set of 50 questions Q={q 1 , q 2 ,..., q 50 }. Suppose l n is the number of participants with correct answers, and m n is the total number of participants for the question q n , we calculated the accuracy using this formula: acc n =l n /m n * 100. This resulted in a list of accuracy scores ACC={acc 1 , acc 2 , ..., acc 50 }, representing the participants' performance of 50 articles. To further evaluate whether the means of two groups (individual vs. collaborative & non-experts vs. experts settings) are statistically different, we conducted an independent sample T-test. Since the T-test is grounded on the assumption of normality [17], we ran the Kolmogorov-Smirnov test on our data and confirmed that the requirement was satisfied. Following, we summarize the results of statistical testing. Deepfake Text Detection Performance (RQ1 & RQ2) 4.2.1 RQ1: Individual vs. Collaboration. The baseline (i.e., randomly guessing) accuracy is 32% (i.e., 1 out of 3 paragraphs) for AMT participants. AMT-Individual improved from randomly guessing by achieving 45% and 46% in the select and write settings, respectively. AMT-Collaboration achieved 51% and 52% accuracy in the select and write settings, respectively. In terms of AMT experiments, both individual and team-based problem-solving significantly outperformed random guessing with p-value < 0.05 (Table 3). Still, a mean difference between individual and collaborative environments was found to be insignificant. Additionally, Upwork participants had a baseline of 31%. They achieved an accuracy of 56% and 69% accuracy for Individual and Collaboration, respectively. All Upwork results were found to be significant. See Tables 3 and 4 for the average accuracy of AMT and Upwork participants, respectively. RQ2 : Non-Experts (AMT) vs. Experts (Upwork). AMT participants represent the laypeople/nonexperts while Upwork participants represent experts. For the Individual setting non-experts vs. experts achieved an accuracy of 45% vs. 56%. Also, for the Collaboration setting non-experts vs. experts achieved an accuracy of 51% vs. 69%. And since all p-value < 0.05 for baseline vs. individual, baseline vs. collaboration, and individual vs. collaboration, the results are statistically significant between non-experts and experts. See Table 5 for a detailed result of the T-test of human performance on deepfake text detection of non-experts vs. experts. Justification Patterns -select (RQ3) Using the findings of prior literature [6,10], we used 7 justifications for participants to select a paragraph as deepfake: grammar issues, repetition, lacks common sense, contains logical errors/fallacies, Figure 6 and 7 display the justification frequency based on correct and incorrect responses, respectively. Table 6 details the significance scores of individual vs. collaboration comparison. For the correct responses, the top-3 dominant categories that AMT mentioned were 'grammar', 'common sense', and 'coherence', and 'lack of creativity'. The top-3 least frequent justifications for correct responses that AMT participants used are 'other', 'repetition', and 'self-contradiction'. During the experiment, none of them selected 'other' to provide explanations for their choices that did not fall into the main 7 existing types. Three categories ('grammar', 'common sense' and 'coherence') had the most increase in use from Individual to Collaboration, but only a 9.11% increase for grammar error type and a 6.23% increase for common sense was found to be significant (p = 0.016, p = 0.036). Next, for incorrect responses, AMT participants frequently used 'grammar', 'logical errors', 'selfcontradiction', and 'lack of creativity'. The frequency of all justifications reduced from Individual to Collaboration, except for 'grammar'. Among those categories, a drop in 'logical errors', and 'self-contradiction' are statistically significant. The most common reasons Upwork participants gave for their correct responses, regardless of the presence of discussion, are 'coherence', 'grammar', and 'common sense'. They also chose 'other' considerably often. Our manual inspection of written justifications under the 'other' category revealed that off-topic and off-prompt were the most frequent ones. Similar to AMT workers, Upwork participants least frequently cited 'repetition' to justify their decisions. As opposed to AMT, we find that when they collectively solved the task the overall frequency of providing particular reasons surged. Specifically, the following justifications exhibited a significant spike in frequency: 'grammar' (+9.27%, p = 0.013), 'common sense' (+15%, p < 0.0001), 'logical errors' (+6.62%, p = 0.006), 'self-contradiction' (+7.13%, p = 0.011), 'coherence' (+11.44%, p = 0.019) types. 'Lack of Creativity' is the only category that was used less often in collaborative problem-solving than individual problem-solving, but it was statistically insignificant. Next, the top-3 most frequent justifications are 'grammar', 'coherence', and 'lack of creativity' for individual-based responses from Upwork participants regarding their incorrect answers. Also, all (Table 7). AMT participants are non-experts while Upwork participants are experts. Our analyses of correct responses reveal that 'grammar' and 'coherence' were the most commonly submitted rationales, regardless of individuals' expertise. The least frequent justification for both AMT-and Upwork-Individuals was 'repetition'. While Upwork participants cited 'grammar', 'common sense', 'coherence' and 'other' more often than AMT participants, those differences were found to be insignificant, except for 'other' (0% vs. 12.22%, p < 0.0001). This is because AMT participants Table 7. T-test Results of select Justification Frequency w.r.t. Correctness (Non-experts vs. Experts) never described their rationale in writing form. Meanwhile, the frequency of 'lack of creativity' (12.87% vs. 8.33%) was significantly lower for Upwork-Individuals than for AMT-Individual s (p = 0.0264). While 'grammar' and 'coherence' remain as popular categories both AMT and Upwork participants cited for their correct answers even in the collaborative environment, the 'common sense' justification was cited more often than 'grammar' in Upwork-Collaboration. Moreover, reasons such as 'contradicts previous sentences' (+9.23%, p = 0.002), 'lacks common sense' (+12.7%, p = 0.011), and 'coherence' (+12.51%, p = 0.004) were more strongly associated with correct responses in Upwork-Collaboration compared to AMT-Collaboration. For incorrect responses, 'logical errors', 'self-contradiction', and 'lack of creativity' were top-3 dominant justifications that AMT-Individual s provided, whereas Upwork-Individuals frequently chose 'grammar', 'coherence' and 'lack of creativity'. All justification categories, except for 'coherence' and 'other', decreased in frequency from AMT-Individual s to Upwork-Individuals. Yet, the observed drop in 'grammar' was found to be insignificant with p > 0.05. Also, despite an increase in the mentions of coherence, the gap was statistically insignificant. Grammatical errors were mentioned the most within erroneous answers in the collaborative context by both AMT and Upwork participants. Yet, 'repetition' (-3.63%, p = 0.029) and 'lack of creativity' (-10.53%, p < 0.0001) were the only categories that demonstrated a substantial drop in frequency from AMT-Collaboration to Upwork-Collaboration. Justification Patterns -write (RQ3) Unlike the select setting, the write setting does not offer any selection alternatives, allowing us to conduct more comprehensive studies that are not confined to the 7 error types (excluding 'other'). Motivated by Clark et al. [6]'s findings that failing participants concentrated on the form of the text rather than content, we manually inspected and categorized the justifications into three subsets: low-level, high-level, and hybrid. A low-level justification is related to the format, style, and tone of the text; high-level justification is an error type that can be determined based on the text's meaning; and hybrid justification represents a case where evaluators cited both low-level and highlevel justifications. For annotation, one researcher initially created a codebook (Table 8) using 7 pre-defined justification categories and coded written responses submitted by AMT workers. We integrated new justifications into the code book as we noticed new information in the data. Next, two additional researchers independently labeled them. Lastly, Fleiss' Kappa coefficient [13] was used to calculate the agreement amongst three annotators (0.924 for individuals & 0.94 for collaboration), supporting the reliability of the generated labels. As Upwork experiments do not have the writing setting, we relied on the justifications provided in the select style. This time we also annotated writings submitted in the 'Other' type. Following, we describe the T-test results which were used to test the distribution differences across three labels. Table 9, AMT participants' correct responses were more strongly associated with either low or high error levels rather than both when they solved the task individually. In the collaborative setting, their mentions of hybrid reasoning surged as well as low and high-level rationales. Yet, none of the increases was statistically significant. For incorrect answers, 25% of AMT individuals on average used low-level or high-level justifications to explain their judgments. Meanwhile, a relatively smaller percentage of participants (3%) provided both low and high-level justifications. When they collectively performed the task, the frequency of low-level rationale decreased to 20%. Still, the drop was found to be insignificant. Also, a change in high and hybrid-level errors was minimal. Individual vs. Collaboration. As shown in Next, Upwork participants, similar to AMT participants cited one error type more often than both for correct responses in individual problem-solving. The frequency of hybrid reasoning almost doubled and became the most frequent justification type after group discussions, with p < 0.05. A decrease in low or high levels, on the other hand, was not significant. For incorrect responses, low-level errors were most frequently mentioned when Upwork participants performed the task individually. Per question, 23.11% of participants on average utilized low-level errors to justify their decisions. In a collaborative environment, its percentage dropped to 12.8%, and this drop was statistically significant. Similarly, we observe that the shift in frequency of high-level justifications-from 15.11% to 8.4%-was significant. Hybrid-level errors, on the contrary, became more frequent after the collaboration but were not substantial. Table 10 includes T-test results of AMT vs. Upwork experiments. Non-Experts vs. Experts. In both individual and collaborative problem-solving environments, the proportion of people who stated low-or high-level reasoning for their correct responses did not differ that much across AMT and Upwork studies. Specifically, all percentages associated with low-or high-level ranged from 20%-23%, regardless of participants' linguistic abilities. Meanwhile, the frequency of hybrid reasoning was 8.11% greater for Upwork than for AMT when the task was performed independently, and the measured gap was statistically significant. When participants complete the task together, the difference between AMT and Upwork in the frequency of both low-and high-level reasons increased from 8.11% to 19%, yielding greater statistical significance. Now, for incorrect responses, AMT-Individual participants provided the low-level error as their justification slightly more often than Upwork-Individual participants, but the gap is negligible with p > 0.05. Likewise, the observed difference in hybrid-level errors was not substantial. Upwork had a much lower frequency of high-level explanations than AMT, and this result was statistically significant. We also discover that in the collaborative setting, AMT participants cited more low-and high-level experiments than on Upwork participants. Especially, the rate of high-level justifications was approximately 3 times greater in the AMT setting compared to the Upwork setting. Hybrid-level errors, on the contrary, were more common in Upwork, and the difference was significant. Qualitative Analysis of Upwork Transcripts To better understand why the synchronous collaboration of Upwork evaluators yielded better performance than other scenarios, we run a transcript analysis. Transcript analysis is a common methodology that has been applied in the HCI field to qualitatively examine discourse content or patterns. To begin, we first construct a coding scheme covering the mentions of syntactic or semantic elements of texts (Table 11). We primarily follow categories from [6,10]. The annotation process was conducted by one researcher (responsible for assigning the codes) and supported by another researcher (responsible for double-checking whether the code assignment made sense, i.e., if a code would fit to a given statement). Computed Fleiss's Kappa values are shown in Table 12. Following, we introduce three key strategies that were commonly adopted by Upwork-Collaboration participants: deductive reasoning, advanced linguistic skills, and leveraging their prior experiences. We further support them with selected quotes. Deductive reasoning. Deduction, by definition, produces valid conclusions, which must be true if the premises are true [22]. Based on the transcript analyses, Upwork participants preferred to use deductive reasoning to come to a conclusion; they first attempted to come up with reasons why 2 out of 3 paragraphs could not be deepfake, and then picked the remaining one as deepfake. Specifically, there was an instance where P10 from G4 chose the paragraph that lacked common sense the most: "Number two and three makes sense. Actually they all make sense, but it's just this first one that makes the least sense." -[P10] In this example, she hypothesized that human-written texts do not lack common sense. G2 members also sometimes used elimination strategies when the answer was not clear: "All three have grammatical issues. I think it's I feel the best about number three like actually being part of the article." -[P4] "yeah. let's say one or two. The way the last sentence and paragraph three, is written, I feel like that's that was just written by a human getting short shrift." -[P5] "I agree. Let's go with two then." -[P6] They relied on their intuitions to make a final judgment after confirming all 3 paragraphs contain grammar errors. Although they did not provide detailed explanations, they could come to a consensus depending on their beliefs about human-written articles. Interestingly, P16 from G6 cited the following sentences when he applied the deductive logic: "I have to admit, all three sound perfectly fine to me. I'd say one actually makes the most sense to me so I'm going to buy again process of elimination, I think one is. That could be the AI just because the other two people can't write very well." -[P16] He chose the paragraph that makes the most sense as deepfake due to a similarity in writing styles of the other two paragraphs. This could have been influenced by his focus on the consistency of texts. Advanced linguistic skills. The quantitative findings showed that Upwork participants cited grammar errors very often as part of their justifications. Congruent with the fact that Upwork participants are proficient English speakers, their discussion around linguistic properties was complex and comprehensive. For instance, several participants (e.g., P4, P13) talked about the length of sentences and how they are not joined with the proper conjunction or punctuation: "I mean, I feel like number two is just hard to read and long and confusing and if I were writing it or editing it I would want to trim it maybe turn it into more sentences." -[P4] "One of those calculations, paragraph one, has three of them I would assume that and they would be programmed to pump out the longer sentences." -[P13] Not limited to the identification of run-on sentences, they could capture more sophisticated grammatical errors resulting from tense shifts, dashes or capitalization. For example, P2 from G1 mentioned a missing dash, and P4 from G2 discussed inconsistent tense usage: "I was wondering, to the Washington at the beginning, there should be a dash after." -[P2] "Studies on historical facts and policy talking about the Capitol Hill security issue is in the present tense and this is taking us to 13 and 2012." -[P4] Our quantitative analysis on justification frequencies (Section 4.3) revealed that Upwork participants frequently selected the 'other' option to cite 'off-topic' and 'off-prompt' as their rationales. Transcript annotation results further confirm their recurring mentions of consistency issues within given paragraphs. See these examples below: "Why, I didn't get it up here well but yeah it is that is bad and nothing I didn't get about paragraph two is recently bringing the Russians in when the first paragraph, had nothing to do with Russia. " -[P2] "For this one, I thought it was paragraph two, just because that question seems a little random and irrelevant to the other paragraphs." -[P8] "Paragraph one is talking about North Korea, while paragraph two and three are talking about something else. So it seems like the AI was just gotten into that place without having information of what the rest of the article was talking about." -[P15] Another topic that Upwork participants actively discussed during the process of synchronous collaboration was text structure or awkward word selections. Specifically, P14 from G5 and P10 from G4 cited the following: "I think this one I choose, I chose number two just because 'jumping too much for joy' seems like an odd statement. The word orders are unusual." -[P14] "Number three says, this was some way to pay back and so love for the country like this was some way, I thought that sounded a little odd." -[P10] These cases overall indicate Upwork participants' abilities to analyze a piece of text in greater depth and contextualize its relationship to other paragraphs. 4.5.3 Leveraging one's prior experiences. As could be expected given the professional backgrounds of Upwork participants, some of them shared their experiences with AI-powered writing and advised a course of action. For example, P17 from G6 said that: "The thing is, I use AI writing to help me with my own writing and least the software that I use it's it does not make those mistakes when it comes to just like spelling (...). It seems like more human error is for it to be grammatically related because that's something we'd be making more mistakes compared to logic and fallacy and flow." -[P17] "If you give the AI prompt it'll start writing on that and I've had it happen, sometimes when it doesn't know what to write it just repeats back the crop." - [P17] "Since all of them have been clear grammatical errors we're supposed to be looking for something else that is Missing logically." -[P17] Another G2 member (P6) voiced his opinion on deepfake content: "I tried to generate machine-written content for the purpose of this research. And there's a grammatical errors that we find here, sometimes a I think a more human related than the machines, because (...)." -[P6] Following, P4 formulated a hypothesis and further reshaped her viewpoint accordingly: "The thing so interesting what P5 is saying is like we are assuming that the most clearly written material will be written by humans and you're saying well, maybe we should assume that the most clearly written material is written by a computer. I would say that I rely a lot less on grammatical errors and more on logical errors like if something has a grammatical error." -[P4] The Upwork participants also anticipated certain writing styles from human-written texts. Using, this knowledge, they concluded that the paragraphs are from news articles that incorporate formal writing. For instance, P1, P3, and P13 mentioned as follows: "yeah this was just filtering to me. Where would a machine get the details, but also this is so odd I feel like a human journalist when right this weird tangent about these people." -[P1] "I feel like there could be more like journalistic voice here, or it could have been presented in a more interesting way." -[P3] "Another reason I chose paragraph three is because of the 'so i'm sure' which usually isn't a way that any writer would begin." -[P13] In conclusion, the examples above suggest that Upwork participants took an advantage of their previous experiences and skills to complete the task. Further, the overall analysis supports the role of synchronous discussion in allowing participants to exchange stories and thoughts to appropriately reshape their perspectives. DISCUSSION Summary of Results This paper entails a holistic examination of the performance of AMT (laypeople) and Upwork (English professionals) participants in detecting deepfake texts, as well as providing insights into the influence of asynchronous and synchronous collaboration on their performance. It also attempts to elucidate the relationship between various textual components and detection performance. Our major findings from Section 4 can be summarized as below: Individual vs. Collaboration (1) Experts benefited from collaboration while non-experts did not: While there is no difference in the deepfake text detection performance between AMT-Individual and AMT-Collaboration, Upwork-Collaboration's (69%) performance is significantly higher than that of Upwork-Individual (56%). (2) Experts' mentions of coherence, logical fallacies, and self-contradiction errors as justifications for deepfake text detection were significantly higher in the collaborative setting than the individual setting: There are frequency differences between coherence, logical fallacies, and self-contradiction for Upwork-Individual and Upwork-Collaboration, with Collaboration more frequently citing these errors, whereas AMT experiments show no difference. (3) In contrast to experts, non-experts' usage of three error levels did not differ between individual and collaborative scenario: While the frequency of low, high, and hybrid error levels did not differ between AMT-Individual and AMT-Collaboration, Upwork-Collaboration's mentions of the hybrid level was significantly more common than Upwork-Individual -(Individual vs. Collaboration usage percentage -13.67 vs. 28.67 for correct responses). Experts vs. Non-experts (1) Experts were better at deepfake text detection than non-experts in terms of detection accuracy: Both AMT (45%) and Upwork (56%) Individual groups significantly outperformed the baseline (32% & 33%, respectively) performance. Still, Upwork participants outperformed AMT participants in both Independent and Collaborative environments. (2) Experts paid less attention to the creativity of articles than non-experts: Upwork participants mentioned a lack of creativity less frequently than AMT participants for correct responses. For analysis of non-experts vs. experts (12.87 vs. 8.33 for correct and 13.49 vs. 7.6 for incorrect responses) frequent usage of creative errors as justification. (3) Experts put more emphasis on consistency issues in the articles than non-experts: Upwork participants were capable of capturing consistency-related errors such as off-topic and off-prompt while AMT participants could not. (4) Experts relied on hybrid-level justifications more often than non-experts: Upwork participants in general cited the hybrid-level justification more frequently than AMT participants for correct responses. For non-experts vs. experts analysis frequency comparison of hybrid-level errors in the collaboration setting, we have 9.67 vs. 28.67 for correct responses. Implications We discuss the implications of these summarized results below to extrapolate the reasoning or phenomena of the findings. Individual vs. Collaboration. (1) Experts benefited from collaboration while non-experts did not: It has long been established that the performance of a group may surpass that of even the most knowledgeable person [27]. However, we discovered that there was no performance difference between independent and asynchronous collaboration regarding AMT experiments, whereas Upwork-Individual participants not only detected significantly better than the baseline but also benefited from the synchronous collaboration. A reason for this could be because synchronous collaboration context, for example, could have encouraged workers to be more involved, creative, and social. A body of CSCW literature (e.g., [4,25,33]) has also argued that the gains of synchronous collaboration outweigh the benefits of asynchronous collaboration. There is another possible reason to explain these mixed results. Since Upwork collaborators share the same background in English expertise, their familiarity with each other's fields and the advanced degree of individual intelligence may have positively impacted the group's intelligence. (2) Experts' mentions of coherence, logical fallacies, and self-contradiction errors as justifications for deepfake text detection were significantly higher in the collaborative setting than the individual setting: Non-experts showed no pattern differences in coherence, logical errors, and self-contradiction justifications between individuals and collaboration. However, expert participants used them, especially coherence and self-contradiction, more in collaboration when they detected the deepfake texts successfully and less in collaboration when they detected deepfake texts inaccurately. This result corroborates Dou et al. [10]'s finding that machines are prone to fall short of those categories. Moreover, coherence errors and self-contradiction are considered high-level errors (Table 8). This implies that it is more challenging to find them and thus, advanced linguistic skills might have shown to be helpful. Therefore, it is expected that English professionals citing these errors more frequently than non-experts. Taking into account experts' superior performance in deepfake text detection, we conclude that both coherence errors and self-contradiction errors are strong indicators of deepfake text. Regarding logical fallacy errors, expert participants used them more frequently in the collaborative setting for both correct and incorrect responses. That said, our findings imply that logical flaws may be a weak predictor of deepfake texts. (3) In contrast to experts, non-experts' usage of three error levels did not differ between individual and collaborative scenarios: We have three error levels -low (i.e., form of text) vs. high (i.e., content of text) vs. hybrid (i.e., form & content). We observe no patterns in non-experts' use of error levels between the independent and collaborative settings for both incorrect and correct answers. Given their similarities in error type usage, it is reasonable that non-experts' performance gain from the collaboration was marginal. The prevalence of hybridlevel errors, on the other hand, is more than doubled for correct responses between individual and collaborative settings of experts. This suggests that when experts collaborate, they are possibly more able to discuss more and find more errors than when they work independently. Furthermore, they could detect deepfake texts more effectively using these hybrid-level errors than they could individually. This hints that considering both form and content-wise errors is a useful strategy for deepfake text detection. Experts vs. Non-experts. (1) Experts were better at deepfake text detection than non-experts: As opposed to previous works where humans performed either at chance or below chance level [6,9,11,37], both experts and non-experts from our studies exceeded the baseline performance. For the nonexperts, we tested various interface designs to improve performance as well as the incorporation of example-based training. These experimental designs may have impacted the performance of non-experts. Given Upwork participants' profound experiences in writing domains, it is expected that they performed significantly better than laypeople like AMT workers. What is more interesting is that, despite additional training processes of non-experts, they were unable to outperform experts who had not received any training. Thus, these results indicate that existing training procedures are insufficient for non-experts, while experts may not need as much training as non-experts. Finally, more nuanced approaches that contain patterns that English professionals are likely to adopt should be explored further. (2) Experts paid less attention to the creativity of articles than non-experts: While the statistical significance of the difference was partially supported, experts, in general, mentioned a lack of creativity less frequently than non-experts for correct responses. Since experts significantly outperformed non-experts in both independent and collaboration scenarios, their lack of focus on the creativity error category suggests that it is not a strong indicator of deepfake texts. We also observe non-experts' frequent mentions of lack of creativity in both incorrect and incorrect responses, reinforcing our argument that lack of creativity is a misleading indicator of deepfake texts. Finally, our transcript analysis (Section 4.5 ) illustrates that UpWork participants were able to leverage their previous knowledge in writing domains and expected specific writing styles from the political news articles when making a decision. Since political news contain reporting of events/facts and are not fictional, their writing styles are expected to be formal and factual. In summary, creativity errors were misleading error types for our task possibly because the goal is for humans to detect deepfake texts in the politics-related news domain. Thus, future works are needed to draw further conclusions on other news topics. (3) Experts put more emphasis on consistency issues in the articles than non-experts: Unlike non-experts, expert participants were able to detect off-topic and off-prompt errors. According to Badaskar et al. [2], language models find it challenging to stick to a single topic but cover diverse, often unrelated topics in a single text. This explains experts' enhanced performance over non-experts. Moreover, off-prompt/off-topic error detection is more likely to require careful reading and thinking than the detection of other error types. From Section 4.5, we also observe that Upwork participants were able to identify one paragraph that introduce a new topic, causing it to be off-prompt. As a whole, this demonstrates that English experts, as opposed to laypeople like AMT participants, can detect errors that go beyond the provided categories, such as off-topic and off-prompt. This also implies that consistency errors are good markers of deepfake texts. (4) Experts tended to rely on hybrid-level justifications more often than non-experts: Analyzing non-experts vs. experts, the frequency of the hybrid-level error level was found to be statistically different. These results suggest that using a mix of both low-level and high-level errors rather than solely one yields more accurate detection performance. Hybrid-level errors require more careful analysis to cite compared to, for instance, repetition or grammar issues. Since low-level errors are easiest to detect but can lead to erroneous judgments, experts' abilities to look beyond obvious error types appear to be beneficial for deepfake detection tasks. Lastly, the results imply that experts' ability to reason deductively, their use of advanced linguistic skills, and leveraging their own prior experiences informed their use of hybrid-level errors. The discussions of experts in Section 4.5 showcased their ability to detect deepfake texts by relying on their professional skills. It also showed their ability to use linguistic cues that will have been otherwise missed by non-experts. ETHICS, LIMITATIONS, AND FUTURE WORK Ethical Statement Our research protocol was approved by the Institutional Review Board (IRB) at our institution. We only recruited human participants 18 years old or over. Participants did not have to complete the entire task to be paid. Using AMT, participants' identification was already anonymized, but for Upwork we anonymized participants by assigning them numerical values for the analysis. For performing the deepfake text detection task, all our human participants, from both AMT and Upwork, were paid over minimum wage rate. Next, the articles that we used for the experiments are the first 3-paragraphs of news articles. While we did not share the answer to the task, we clearly informed participants that the presented texts (and one of three paragraphs therein) contains deepfake texts. Therefore, we believe that participants are unlikely to be negatively influenced by their exposure to the test news articles with deepfake paragraphs. Limitations To implement design choices and run manageable experiments, we made a few simplifications that may limit our findings. First, since, we only use GPT-2 to generate deepfake texts, our findings may not be directly applicable to other NTGs. However, we believe that the choice of GPT-2 is reasonable because: (1) prior research reported that human detection performance of deepfake texts by the later GPT-3 and GPT-2 is similar [6,37], and (2) using the largest parameter size of GPT-2 enabled us to generate deepfake texts more effectively that closely resembles GPT-3 quality. Furthermore, as we use the default hyperparameters of GPT-2 to generate the texts, we believe that the results may be limited to that sampling technique. However, we mitigated this issue by manually checking the quality of a few of the articles and found the deepfake texts to be human-like. This preserved the integrity of the experiments as the task remained non-trivial. Next, for the non-expert vs. expert analysis, we compared AMT to Upwork, where AMT are the non-experts and Upwork, experts. However, since they are different forms of collaboration, AMT being asynchronous and Upwork being synchronous, the comparison may be different. Although a few prior works explored the space of synchronous collaboration between AMT workers, these systems tend to be engineering-heavy and are rarely used in real-world applications. We thus believe asynchronous collaboration is a more realistic way of using AMT. We understand that this design decision might compromise the comparability of the two settings but believe it is a reasonable trade-off. Furthermore, Upwork's interface supports the recruitment of English experts, as well as provides a framework for synchronous collaboration. To mitigate this limitation, we calculate the accuracy of AMT and Upwork participants in the same way (per-article accuracy) to have a fairer comparison. Future Work In the future, we aim to improve the AMT write setting framework by proposing a Turing Test framework that supports non-experts' creativity. Furthermore, using our framework, we will investigate and re-run all experimental studies on Upwork. This experiment will have the following variablesnon-experts vs. experts, Individual vs. Collaboration, and asynchronous vs. synchronous collaboration. Next, we will improve the performance of collaborating for deepfake text detection using a highlight tool to color code the error types. As indicated above, hybrid-level errors, self-contradiction, and coherence are good indicators of deepfake texts. Therefore, our highlighting tool can be used to color codes these errors to reduce their cognitive load (especially for non-experts). Next, using this highlighting tool, we will train human participants before testing their ability to use such a tool to improve detection. Finally, most human evaluation research on deepfake text detection setup the articles to be either fully human-written or fully deepfake. However, in the real world, deepfake articles will be edited by humans, creating a multi-authored article. Therefore, there are several ways to generate a more realistic dataset for this study. These include: (1) given a 3-paragraph article, 2 deepfake, and 1 human-written, ask human participants to select the human-written paragraph; (2) given a 3-paragraph article, each paragraph is generated by a unique NTG, ask humans select which paragraph is deepfake. The goal here is to see which of the state-of-the-art NTGs are easier for humans to detect; (3) given a 3-paragraph article, all written by humans, ask humans to select which is deepfake. The goal here is to see if humans have an affinity for selecting a particular paragraph (e.g. mostly the second paragraph). CONCLUSION In this paper, we studied human performance in deepfake text detection. To be more realistic, we built a 3-paragraph article with 1/3 paragraphs, machine-generated (deepfake) and 2/3 paragraphs, human-written. We ask human participants to select which paragraph is deepfake and to provide justification for their selection out of 7 error types. Specifically, we studied human performance with two variablesindividual vs. collaboration and English non-expert vs. English expert. To achieve this, we recruit non-expert human participants from AMT and experts from Upwork. Furthermore, we run asynchronous collaboration with AMT and compared it to synchronous collaboration with Upwork. Finally, our results suggest that synchronous collaboration of expert human participants significantly improves human performance in deepfake text detection. We further identify several factors (such as coherence and consistency) that deepfake texts excel at or fail by analyzing their justification patterns. Lastly, the enhanced performance of participants (particularly non-experts) from baseline in the individual setting indicates that our Turing Test framework facilitates the improvement of humans' deepfake text detection performance. Fig. 1 . 1(A) Example of a multi-authored (Human & Deepfake) 3-paragraph article; (B) Task: Detecting DeepFake texts; (C) Description of three research questions. Fig. 3 . 3User Interface for the AMT Collaborative Group workers to choose the machine-generated paragraph. A HIT Introduction B Example Trial and Error Fig. 4. The instructions to train users by providing prompt feedback. Fig. 5 . 5Select (A) vs. Write (B) justification question type Fig. 6 . 6Justification Category Distribution w.r.t. Correct Responses justification types, excluding 'repetition', 'common sense', 'logical errors', and 'other', reduced in frequency from Individual to Collaboration. The surge in the frequency of four justification types was not statistically supported. Finally, 'lack of creativity' and 'coherence' categories were less prevalent during the team-based approach in comparison to the individual-based approach. Fig. 7 . 7Justification Category Distribution w.r.t. Incorrect Responses 4.3.2 Non-Experts vs. Experts. We now report the results of comparative analyses in terms of AMT-Individual vs. Upwork-Individual and AMT-Collaboration vs. Upwork-Collaboration, respectively Fig. 8 . 8The distribution of eight justification categories. Table 2 . 2Upwork participant demographics Table 3 . 3T-test Results for AMT Experiments (RQ1)SETTING Select Mean Accuracy p-value Baseline vs. Individual 31% vs. 56.11% 9.1e-12 Baseline vs. Collaboration 31% vs. 68.87% 7.3e-15 Individual vs. Collaboration 56.11% vs. 68.87% 0.008 Table 4 . 4T-test Results for Upwork Experiments (RQ1)SETTING Select Write Mean Accuracy p-value MeanAccuracy p-value AMT-Individual vs. Upwork-Individual 44.99% vs. 56.11% 0.005 45.92% vs. 56.11% 0.028 AMT-Collaboration vs. Upwork-Collaboration 51.35% vs. 68.87% 0.002 51.97% vs. 68.87% 0.003 Table 5 . 5T-test Results for AMT vs. Upwork (RQ2) Accuracy p-value Mean Accuracy p-value Mean Accuracy p-value Mean Accuracy p-valueJustification Type Correct Incorrect AMT Upwork AMT Upwork Mean Grammar 13.97 vs. 23.08 0.016 15.33 vs. 24.6 0.013 15.65 vs. 16.89 0.675 14.22 v.s 12.07 0.469 Repetition 6.73 vs. 6.69 0.986 4 vs. 6.4 0.266 8.53 vs. 5.62 0.113 1.67 vs. 2 0.725 Common Sense 9.25 vs. 15.48 0.036 13 vs. 28 9.67e-05 13.02 vs. 9.94 0.206 3.33 vs. 5.56 0.171 Logical Errors 11.64 vs. 10.24 0.589 7.78 vs. 14.4 0.006 18.54 vs. 7.7 4.66e-05 3.89 vs. 4 0.942 Self-Contradiction 9.35 vs. 5.57 0.092 7.67 vs. 14.8 0.011 18.01 vs. 6.7 1.53e-06 6.56 vs. 3.6 0.054 Lack of Creativity 12.87 vs. 13.49 0.843 8.33 vs. 7.6 0.734 16.9 vs. 14.13 0.322 8.11 v.s 3.6 0.004 Coherence 14.64 vs. 19.29 0.174 20.56 vs. 32 0.019 11.65 vs. 10.06 0.513 13.78 vs. 9.2 0.05 Other 0 vs. 0 N/A 12.22 vs.18.4 0.053 0 vs. 0 N/A 6.78 vs. 8.4 0.519 Table 6 . 6T-test Results of select Justification Frequency w.r.t. Correctness (Individual vs. Collaboration) groups cited a justification category. We calculate the overall frequency of justification and the frequency of justification used for incorrect and correct responses. Lastly, we calculate the statistical significance test with the independent sample T-test for these error types.4.3.1 Individual vs. Collaboration.contradicts previous sentences, lack of creativity or boring to read, writing is erratic/incoherent. If none of 7 error types are found suitable, participants choose an additional justification, other, and write their own justification. To compare the frequency of justifications regarding two variable pairs (individual vs. collaboration & non-experts vs. experts), we first compute the frequency at which each of 4 Accuracy p-value Mean Accuracy p-value Mean Accuracy p-value Mean Accuracy p-valueJustification Type Correct Incorrect Individual Collaboration Individual Collaboration Mean Grammar 13.98 vs. 15.33 0.57 23.08 vs. 24.6 0.747 15.65 vs. 14.22 0.53 16.89 vs. 12.07 0.175 Repetition 6.73 vs. 4 0.107 6.69 vs. 6.4 0.919 8.54 vs. 1.67 3.46e-07 5.62 vs. 2 0.029 Common Sense 9.25 vs. 13 0.086 15.48 vs. 28 0.004 13.02 vs. 3.33 6.49e-07 9.95 vs. 5.6 0.06 Logical Errors 11.64 vs. 7.77 0.056 10.24 vs. 14.4 0.151 18.54 vs. 3.89 1.14e-10 7.7 vs. 4 0.089 Self-Contradiction 9.35 vs. 7.67 0.398 5.57 vs. 14.8 0.002 18.01 vs. 6.56 4.7e-08 6.7 vs. 3.6 0.097 Lack of Creativity 12.87 vs. 8.33 0.026 13.49 vs. 7.6 0.071 16.9 vs. 8.11 1.17e-05 14.13 vs. 3.6 7.26e-05 Coherence 14.64 v.s 20.56 0.066 19.29 vs. 32 0.011 11.65 vs. 13.78 0.283 10.06 vs. 9.2 0.75 Other 0 vs. 12.22 1.1e-11 0 vs. 18.4 1.11e-09 0 vs. 6.78 6.5e-08 0 vs. 8.4 0.0003 Table 8 . 8Code book for error level annotation Accuracy p-value Mean Accuracy p-value Mean Accuracy p-value Mean Accuracy p-valueJustification Level Correct Incorrect AMT Upwork AMT Upwork Mean Low 19.54 vs. 21.08 0.675 21.89 vs. 20.4 0.668 24.95 vs. 19.78 0.121 23.11 vs. 12.8 0.0009 High 20.94 vs. 21.88 0.812 20.33 vs. 19.13 0.744 25.75 vs. 24 0.68 15.11 vs. 8.4 0.006 Hybrid 5.56 vs. 9.67 0.103 13.67 vs. 28.67 0.0003 3.27 vs. 3.6 0.825 5.67 vs. 9.6 0.067 Table 9 . 9T-test Results of Justification Level Frequency w.r.t. Correctness (Individual v.s. Collaboration) Accuracy p-value Mean Accuracy p-value Mean Accuracy p-value Mean Accuracy p-valueJustification Level Correct Incorrect Individual Collaboration Individual Collaboration Mean Low 19.54 vs. 21.89 0.482 21.08 vs. 20.4 0.858 24.95 vs. 23.11 0.558 19.78 vs. 12.8 0.032 High 20.94 vs. 20.33 0.838 21.88 vs. 19.13 0.545 25.75 vs. 15.11 0.003 24 vs. 8.4 8.9e-06 Hybrid 5.56 vs. 13.67 0.0007 9.67 vs. 28.67 1.3e-05 3.27 vs. 5.67 0.083 3.59 vs. 9.6 0.008 Table 10. T-test Results of Justification Level Frequency w.r.t. Correctness (Non-Experts v.s. Experts) Code Description Grammar mentions of the spelling and grammar of the text Repetition mentions of words/phrases/content being repetitive Factuality mentions of whether the text describes things that are true Consistency mentions of how the text relates to the context and other pieces of the text Common sense mentions of whether the text makes sense within the world that it is written Coherence mentions of the structure, wording, or coherence of the text Self-Contradiction mentions of whether the text contradicts itself Creativity mentions of whether the text seems boring to read Writers' capabilities mentions of writer's intent or capabilities Table 11 . 11Annotation categories for Upwork transcript Table 12 . 12The Fleiss' Kappa score of human annotation on the categories This is also known as neural texts.2 Turing Test measures how human-like a model is. If a model shows intelligent behavior usually attributed to a human and is thus, labeled a human, the model is said to have passed the Turing Test. stylometry is the statistical analysis of an author's writing style/signature https://github.com/minimaxir/aitextgen https://sellcoursesonline.com/Upwork-statistics https://www.qualtrics.com 7 https://zoom.us 8 https://otter.ai Whodunit? Learning to Contrast for Authorship Attribution. Bo Ai, Yuchen Wang, Yugin Tan, Samson Tan, Proceedings of the 2nd Conference of the Asia-Pacific Chapter of the Association for Computational Linguistics and the 12th International Joint Conference on Natural Language Processing. the 2nd Conference of the Asia-Pacific Chapter of the Association for Computational Linguistics and the 12th International Joint Conference on Natural Language ProcessingBo Ai, Yuchen Wang, Yugin Tan, and Samson Tan. 2022. Whodunit? Learning to Contrast for Authorship Attribution. In Proceedings of the 2nd Conference of the Asia-Pacific Chapter of the Association for Computational Linguistics and the 12th International Joint Conference on Natural Language Processing. 1142-1157. Identifying real or fake articles: Towards better language modeling. Sameer Badaskar, Sachin Agarwal, Shilpa Arora, Proceedings of the Third International Joint Conference on Natural Language Processing. the Third International Joint Conference on Natural Language ProcessingIISameer Badaskar, Sachin Agarwal, and Shilpa Arora. 2008. Identifying real or fake articles: Towards better language modeling. In Proceedings of the Third International Joint Conference on Natural Language Processing: Volume-II. Real or fake? learning to discriminate machine from human generated text. Anton Bakhtin, Sam Gross, Myle Ott, Yuntian Deng, Marc&apos;aurelio Ranzato, Arthur Szlam, arXiv:1906.03351arXiv preprintAnton Bakhtin, Sam Gross, Myle Ott, Yuntian Deng, Marc'Aurelio Ranzato, and Arthur Szlam. 2019. Real or fake? learning to discriminate machine from human generated text. arXiv preprint arXiv:1906.03351 (2019). Tracking changes in collaborative writing: edits, visibility and group maintenance. Jeremy Birnholtz, Steven Ibara, Proceedings of the ACM 2012 conference on Computer Supported Cooperative Work. the ACM 2012 conference on Computer Supported Cooperative WorkJeremy Birnholtz and Steven Ibara. 2012. Tracking changes in collaborative writing: edits, visibility and group maintenance. In Proceedings of the ACM 2012 conference on Computer Supported Cooperative Work. 809-818. Language models are few-shot learners. Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, Advances in neural information processing systems. 33Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. 2020. Language models are few-shot learners. Advances in neural information processing systems 33 (2020), 1877-1901. All That's 'Human' Is Not Gold: Evaluating Human Evaluation of Generated Text. Elizabeth Clark, Sofia August, Nikita Serrano, Suchin Haduong, Noah A Gururangan, Smith, 10.18653/v1/2021.acl-long.565Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing. the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language ProcessingAssociation for Computational Linguistics1Long Papers)Elizabeth Clark, Tal August, Sofia Serrano, Nikita Haduong, Suchin Gururangan, and Noah A. Smith. 2021. All That's 'Human' Is Not Gold: Evaluating Human Evaluation of Generated Text. In Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers). Association for Computational Linguistics, Online, 7282-7296. https: //doi.org/10.18653/v1/2021.acl-long.565 Jacob Devlin, Ming-Wei Chang, Kenton Lee, Kristina Toutanova, arXiv:1810.04805Bert: Pre-training of deep bidirectional transformers for language understanding. arXiv preprintJacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2018. Bert: Pre-training of deep bidirectional transformers for language understanding. arXiv preprint arXiv:1810.04805 (2018). Enabling Language Models to Fill in the Blanks. Chris Donahue, Mina Lee, Percy Liang, Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics. the 58th Annual Meeting of the Association for Computational LinguisticsChris Donahue, Mina Lee, and Percy Liang. 2020. Enabling Language Models to Fill in the Blanks. In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics. 2492-2501. Scarecrow: A framework for scrutinizing machine text. Yao Dou, Maxwell Forbes, Rik Koncel-Kedziorski, A Noah, Yejin Smith, Choi, arXiv:2107.01294arXiv preprintYao Dou, Maxwell Forbes, Rik Koncel-Kedziorski, Noah A Smith, and Yejin Choi. 2021. Scarecrow: A framework for scrutinizing machine text. arXiv preprint arXiv:2107.01294 (2021). Is GPT-3 Text Indistinguishable from Human Text? Scarecrow: A Framework for Scrutinizing Machine Text. Yao Dou, Maxwell Forbes, Rik Koncel-Kedziorski, A Noah, Yejin Smith, Choi, Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics. the 60th Annual Meeting of the Association for Computational LinguisticsLong Papers1Yao Dou, Maxwell Forbes, Rik Koncel-Kedziorski, Noah A Smith, and Yejin Choi. 2022. Is GPT-3 Text Indistinguishable from Human Text? Scarecrow: A Framework for Scrutinizing Machine Text. In Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers). 7250-7274. RoFT: A Tool for Evaluating Human Detection of Machine-Generated Text. Liam Dugan, Daphne Ippolito, Arun Kirubarajan, Chris Callison-Burch, Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing: System Demonstrations. the 2020 Conference on Empirical Methods in Natural Language Processing: System DemonstrationsLiam Dugan, Daphne Ippolito, Arun Kirubarajan, and Chris Callison-Burch. 2020. RoFT: A Tool for Evaluating Human Detection of Machine-Generated Text. In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing: System Demonstrations. 189-196. TweepFake: About detecting deepfake tweets. Tiziano Fagni, Fabrizio Falchi, Margherita Gambini, Antonio Martella, Maurizio Tesconi, Plos one. 16251415Tiziano Fagni, Fabrizio Falchi, Margherita Gambini, Antonio Martella, and Maurizio Tesconi. 2021. TweepFake: About detecting deepfake tweets. Plos one 16, 5 (2021), e0251415. Measuring nominal scale agreement among many raters. L Joseph, Fleiss, Psychological bulletin. 76378Joseph L Fleiss. 1971. Measuring nominal scale agreement among many raters. Psychological bulletin 76, 5 (1971), 378. Feature-based detection of automated language models: tackling GPT-2, GPT-3 and Grover. Leon Fröhling, Arkaitz Zubiaga, PeerJ Computer Science. 7443Leon Fröhling and Arkaitz Zubiaga. 2021. Feature-based detection of automated language models: tackling GPT-2, GPT-3 and Grover. PeerJ Computer Science 7 (2021), e443. Unsupervised and Distributional Detection of Machine-Generated Text. Matthias Gallé, Jos Rozen, Germán Kruszewski, Hady Elsahar, arXiv:2111.02878arXiv preprintMatthias Gallé, Jos Rozen, Germán Kruszewski, and Hady Elsahar. 2021. Unsupervised and Distributional Detection of Machine-Generated Text. arXiv preprint arXiv:2111.02878 (2021). GLTR: Statistical Detection and Visualization of Generated Text. Sebastian Gehrmann, Hendrik Strobelt, Alexander M Rush, Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics: System Demonstrations. the 57th Annual Meeting of the Association for Computational Linguistics: System DemonstrationsSebastian Gehrmann, Hendrik Strobelt, and Alexander M Rush. 2019. GLTR: Statistical Detection and Visualization of Generated Text. In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics: System Demonstrations. 111-116. A brief review of independent, dependent and one sample t-test. Banda Gerald, International Journal of Applied Mathematics and Theoretical Physics. 4Banda Gerald. 2018. A brief review of independent, dependent and one sample t-test. International Journal of Applied Mathematics and Theoretical Physics 4, 2 (2018), 50-54. The potential of synchronous communication to enhance participation in online discussions: A case study of two e-learning courses. Stefan Hrastinski, Information & Management. 45Stefan Hrastinski. 2008. The potential of synchronous communication to enhance participation in online discussions: A case study of two e-learning courses. Information & Management 45, 7 (2008), 499-506. GPT-2 Output Detector Demo. Huggingface, Huggingface. 2019. GPT-2 Output Detector Demo. https://huggingface.co/openai-detector/ (2019). Automatic Detection of Generated Text is Easiest when Humans are Fooled. Daphne Ippolito, Daniel Duckworth, Chris Callison-Burch, Douglas Eck, 10.18653/v1/2020.acl-main.164Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics. the 58th Annual Meeting of the Association for Computational LinguisticsAssociation for Computational LinguisticsDaphne Ippolito, Daniel Duckworth, Chris Callison-Burch, and Douglas Eck. 2020. Automatic Detection of Generated Text is Easiest when Humans are Fooled. In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics. Association for Computational Linguistics, Online, 1808-1822. https://doi.org/10.18653/v1/2020.acl- main.164 Automatic Detection of Entity-Manipulated Text using Factual Knowledge. Ganesh Jawahar, Muhammad Abdul-Mageed, Laks Lakshmanan, Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics. the 60th Annual Meeting of the Association for Computational LinguisticsShort Papers2Ganesh Jawahar, Muhammad Abdul-Mageed, and Laks Lakshmanan. 2022. Automatic Detection of Entity-Manipulated Text using Factual Knowledge. In Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers). 86-93. Deductive reasoning. Annual review of psychology. Philip N Johnson-Laird , 50Philip N Johnson-Laird. 1999. Deductive reasoning. Annual review of psychology 50, 1 (1999), 109-135. Artificial Text Detection via Examining the Topology of Attention Maps. Laida Kushnareva, Daniil Cherniavskii, Vladislav Mikhailov, Ekaterina Artemova, Serguei Barannikov, Alexander Bernstein, Irina Piontkovskaya, Dmitri Piontkovski, Evgeny Burnaev, Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing. the 2021 Conference on Empirical Methods in Natural Language ProcessingLaida Kushnareva, Daniil Cherniavskii, Vladislav Mikhailov, Ekaterina Artemova, Serguei Barannikov, Alexander Bernstein, Irina Piontkovskaya, Dmitri Piontkovski, and Evgeny Burnaev. 2021. Artificial Text Detection via Examining the Topology of Attention Maps. In Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing. 635-649. CoCo: Coherence-Enhanced Machine-Generated Text Detection Under Data Limitation With Contrastive Learning. Xiaoming Liu, Zhaohan Zhang, Yichen Wang, Yu Lan, Chao Shen, arXiv:2212.10341arXiv preprintXiaoming Liu, Zhaohan Zhang, Yichen Wang, Yu Lan, and Chao Shen. 2022. CoCo: Coherence-Enhanced Machine- Generated Text Detection Under Data Limitation With Contrastive Learning. arXiv preprint arXiv:2212.10341 (2022). A study of synchronous versus asynchronous collaboration in an online business writing class. Mark Mabrito, The American Journal of Distance Education. 202Mark Mabrito. 2006. A study of synchronous versus asynchronous collaboration in an online business writing class. The American Journal of Distance Education 20, 2 (2006), 93-107. A meta-analysis on the effects of synchronous online learning on cognitive and affective educational outcomes. Florence Martin, Ting Sun, Murat Turk, Albert D Ritzhaupt, International Review of Research in Open and Distributed Learning. 22Florence Martin, Ting Sun, Murat Turk, and Albert D Ritzhaupt. 2021. A meta-analysis on the effects of synchronous online learning on cognitive and affective educational outcomes. International Review of Research in Open and Distributed Learning 22, 3 (2021), 205-242. Why do humans reason? Arguments for an argumentative theory. Hugo Mercier, Dan Sperber, Behavioral and brain sciences. 342Hugo Mercier and Dan Sperber. 2011. Why do humans reason? Arguments for an argumentative theory. Behavioral and brain sciences 34, 2 (2011), 57-74. Synchronous and asynchronous discussions: Effects on cooperation, belonging, and affect. Patrick N Amy T Peterson, Ralph T Beymer, Putnam, Online Learning. 22Amy T Peterson, Patrick N Beymer, and Ralph T Putnam. 2018. Synchronous and asynchronous discussions: Effects on cooperation, belonging, and affect. Online Learning 22, 4 (2018), 7-25. Krishna Pillutla, Lang Liu, John Thickstun, Sean Welleck, Swabha Swayamdipta, Rowan Zellers, Sewoong Oh, Yejin Choi, Zaid Harchaoui, arXiv:2212.14578MAUVE Scores for Generative Models: Theory and Practice. arXiv preprintKrishna Pillutla, Lang Liu, John Thickstun, Sean Welleck, Swabha Swayamdipta, Rowan Zellers, Sewoong Oh, Yejin Choi, and Zaid Harchaoui. 2022. MAUVE Scores for Generative Models: Theory and Practice. arXiv preprint arXiv:2212.14578 (2022). Krishna Pillutla, Swabha Swayamdipta, Rowan Zellers, John Thickstun, Sean Welleck, Yejin Choi, Zaid Harchaoui, arXiv:2102.01454An information divergence measure between neural text and human text. arXiv preprintKrishna Pillutla, Swabha Swayamdipta, Rowan Zellers, John Thickstun, Sean Welleck, Yejin Choi, and Zaid Harchaoui. 2021. An information divergence measure between neural text and human text. arXiv preprint arXiv:2102.01454 (2021). Jiameng Pu, Zain Sarwar, Muhammad Sifat, Abdullah Abdullah, Yoonjin Rehman, Parantapa Kim, Mobin Bhattacharya, Javed, Bimal Viswanath, Virginia Tech, and LUMS Pakistan. 2023. Deepfake Text Detection: Limitations and Opportunities. 44th IEEE Symposium on Security and Privacy. Jiameng Pu, Zain Sarwar, Sifat Muhammad Abdullah, Abdullah Rehman, Yoonjin Kim, Parantapa Bhattacharya, Mobin Javed, Bimal Viswanath, Virginia Tech, and LUMS Pakistan. 2023. Deepfake Text Detection: Limitations and Opportunities. 44th IEEE Symposium on Security and Privacy (2023). Language models are unsupervised multitask learners. Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, Ilya Sutskever, OpenAI blog. 19Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, and Ilya Sutskever. 2019. Language models are unsupervised multitask learners. OpenAI blog 1, 8 (2019), 9. Task and technology fit: a comparison of two technologies for synchronous and asynchronous group communication. I Ashraf, Shirani, H A Mohammed, John F Tafti, Affisco, Information & management. 36Ashraf I Shirani, Mohammed HA Tafti, and John F Affisco. 1999. Task and technology fit: a comparison of two technologies for synchronous and asynchronous group communication. Information & management 36, 3 (1999), 139-150. Detecting Cross-Modal Inconsistency to Defend Against Neural Fake News. Reuben Tan, Bryan Plummer, Kate Saenko, Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP. the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLPReuben Tan, Bryan Plummer, and Kate Saenko. 2020. Detecting Cross-Modal Inconsistency to Defend Against Neural Fake News. In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP). 2081-2106. Attribution and Obfuscation of Neural Text Authorship: A Data Mining Perspective. Adaku Uchendu, Thai Le, Dongwon Lee, SIGKDD Explorations (2023). 25Adaku Uchendu, Thai Le, and Dongwon Lee. 2023. Attribution and Obfuscation of Neural Text Authorship: A Data Mining Perspective. SIGKDD Explorations (2023), vol. 25. Authorship attribution for neural text generation. Adaku Uchendu, Thai Le, Kai Shu, Dongwon Lee, Conf. on Empirical Methods in Natural Language Processing (EMNLP). Adaku Uchendu, Thai Le, Kai Shu, and Dongwon Lee. 2020. Authorship attribution for neural text generation. In Conf. on Empirical Methods in Natural Language Processing (EMNLP). TURINGBENCH: A Benchmark Environment for Turing Test in the Age of Neural Text Generation. Adaku Uchendu, Zeyu Ma, Thai Le, Rui Zhang, Dongwon Lee, Findings of the Association for Computational Linguistics: EMNLP 2021. Adaku Uchendu, Zeyu Ma, Thai Le, Rui Zhang, and Dongwon Lee. 2021. TURINGBENCH: A Benchmark Environment for Turing Test in the Age of Neural Text Generation. In Findings of the Association for Computational Linguistics: EMNLP 2021. 2001-2016. Best practices for the human evaluation of automatically generated text. Chris Van Der, Albert Lee, Emiel Gatt, Sander Van Miltenburg, Emiel Wubben, Krahmer, Proceedings of the 12th International Conference on Natural Language Generation. the 12th International Conference on Natural Language GenerationChris Van Der Lee, Albert Gatt, Emiel Van Miltenburg, Sander Wubben, and Emiel Krahmer. 2019. Best practices for the human evaluation of automatically generated text. In Proceedings of the 12th International Conference on Natural Language Generation. 355-368. Defending against neural fake news. Rowan Zellers, Ari Holtzman, Hannah Rashkin, Yonatan Bisk, Ali Farhadi, Franziska Roesner, Yejin Choi, Advances in neural information processing systems. 32Rowan Zellers, Ari Holtzman, Hannah Rashkin, Yonatan Bisk, Ali Farhadi, Franziska Roesner, and Yejin Choi. 2019. Defending against neural fake news. Advances in neural information processing systems 32 (2019). Deepfake generation and detection, a survey. Tao Zhang, Multimedia Tools and Applications. 81Tao Zhang. 2022. Deepfake generation and detection, a survey. Multimedia Tools and Applications 81, 5 (2022), 6259-6276. Neural Deepfake Detection with Factual Structure of Text. Wanjun Zhong, Duyu Tang, Zenan Xu, Ruize Wang, Nan Duan, Ming Zhou, Jiahai Wang, Jian Yin, Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP. the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLPWanjun Zhong, Duyu Tang, Zenan Xu, Ruize Wang, Nan Duan, Ming Zhou, Jiahai Wang, and Jian Yin. 2020. Neural Deepfake Detection with Factual Structure of Text. In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP). 2461-2470.
{'fraction_non_alphanumeric': 0.04397569594070499, 'fraction_numerical': 0.02849478583125054, 'mean_word_length': 4.813552952965491, 'pattern_counts': {'":': 0, '<': 6, '<?xml version=': 0, '>': 2, 'https://': 7, 'lorem ipsum': 0, 'www.': 1, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'In recent years, Natural Language Generation (NLG) techniques in AI (e.g., T5, GPT-3, ChatGPT) have shown a massive improvement and are now capable of generating human-like long coherent texts at scale, yielding so-called deepfake texts. This advancement, despite their benefits, can also cause security and privacy issues (e.g., plagiarism, identity obfuscation, disinformation attack). As such, it has become critically important to develop effective, practical, and scalable solutions to differentiate deepfake texts from human-written texts. Toward this challenge, in this work, we investigate how factors such as skill levels and collaborations impact how humans identify deepfake texts, studying three research questions: (1) do collaborative teams detect deepfake texts better than individuals? (2) do expert humans detect deepfake texts better than non-expert humans?(3)what are the factors that maximize the detection performance of humans? We implement these questions on two platforms: (1) non-expert humans or asynchronous teams on Amazon Mechanical Turk (AMT) and (2) expert humans or synchronous teams on the Upwork. By analyzing the detection performance and the factors that affected performance, some of our key findings are: (1) expert humans detect deepfake texts significantly better than non-expert humans, (2) synchronous teams on the Upwork detect deepfake texts significantly better than individuals, while asynchronous teams on the AMT detect deepfake texts weakly better than individuals, and (3) among various error categories, examining coherence and consistency in texts is useful in detecting deepfake texts. In conclusion, our work could inform the design of future tools/framework to improve collaborative human detection of deepfake texts.', 'arxivid': '2304.01002', 'author': ['Adaku Uchendu ', 'Jooyoung Lee ', 'Hua Shen ', 'Thai Le ', 'Ting-Hao &apos;kenneth&apos; Huang ', 'Dongwon Lee ', 'Adaku Uchendu ', 'Jooyoung Lee ', 'Hua Shen ', 'Thai Le ', 'Ting-Hao ', 'Kenneth &apos; Huang ', '\nPennsylvania State University\nUSA\n', '\nPennsylvania State University\nUSA\n', '\nPennsylvania State University\nUSA\n', '\nUniversity of Mississippi\nUSA\n', '\nPennsylvania State University\nUSA\n', '\nPennsylvania State University\nUSA\n'], 'authoraffiliation': ['Pennsylvania State University\nUSA', 'Pennsylvania State University\nUSA', 'Pennsylvania State University\nUSA', 'University of Mississippi\nUSA', 'Pennsylvania State University\nUSA', 'Pennsylvania State University\nUSA'], 'corpusid': 257913864, 'doi': '10.48550/arxiv.2304.01002', 'github_urls': ['https://github.com/minimaxir/aitextgen'], 'n_tokens_mistral': 24856, 'n_tokens_neox': 21682, 'n_words': 13378, 'pdfsha': 'bec51f0fcdc6cc5856f8484ebb3735eac0c814ff', 'pdfurls': ['https://export.arxiv.org/pdf/2304.01002v1.pdf'], 'title': ['Understanding Individual and Team-based Human Factors in Detecting Deepfake Texts', 'Understanding Individual and Team-based Human Factors in Detecting Deepfake Texts'], 'venue': []}
arxiv
Thermal annealing of sputtered Nb 3 Sn and V 3 Si thin films for superconducting radio-frequency cavities December 2022 2 Jan 2023 Katrina Howard [email protected]. Cornell Laboratory for Accelerator-Based Sciences and Education Cornell University 14853IthacaNYUSA Department of Physics University of Chicago 60637ChicagoILUSA Zeming Sun Cornell Laboratory for Accelerator-Based Sciences and Education Cornell University 14853IthacaNYUSA Matthias U Liepe Cornell Laboratory for Accelerator-Based Sciences and Education Cornell University 14853IthacaNYUSA Thermal annealing of sputtered Nb 3 Sn and V 3 Si thin films for superconducting radio-frequency cavities December 2022 2 Jan 2023Submitted to: Superconductor Science and Technologythermal annealingA15 superconductorssputteringthin filmSRF Nb 3 Sn and V 3 Si thin films are promising candidates as thin films for the next generation of superconducting radio-frequency (SRF) cavities. However, sputtered films often suffer from stoichiometry and strain issues during deposition and post annealing. In this study, we explore the structural and chemical effects of thermal annealing, both in-situ and post-sputtering, on DC-sputtered Nb 3 Sn and V 3 Si films of varying thickness on Nb or Cu substrates, extending from our initial studies[1]. Through annealing at 950°C, we successfully enabled recrystallization of 100 nm thin Nb 3 Sn films on Nb substrate with stoichiometric and strain-free grains. For 2 µm thick films, we observed the removal of strain and a slight increase in grain size with increasing temperature. Annealing enabled a phase transformation from unstable to stable structure on V 3 Si films, while we observed significant Sn loss in 2 µm thick Nb 3 Sn films after high temperature anneals. We observed similar Sn and Si loss on films atop Cu substrates during annealing, likely due to Cu-Sn and Cu-Si phase generation and subsequent Sn and Si evaporation. These results encourage us to refine our process to obtain high-quality sputtered films for SRF use. Introduction Nb 3 Sn and V 3 Si thin films are of increasing interest to the superconducting radiofrequency community owing to the quest of achieving high accelerating gradient and efficiency. As niobium-based superconducting radio-frequency (SRF) cavities are reaching the theoretical limits [2], alternative materials are of great interest to continue the quest of increasing quality factors, accelerating gradients, and efficiency [3]. A15 superconductors Nb 3 Sn and V 3 Si are promising candidates for this role, used as thin films inside either Nb or Cu cavities [3,4]. Both candidates have relatively high critical temperatures (T c,N b 3 Sn = 18.3 K and T c,V 3 Si = 17.1 K), and Nb 3 Sn is predicted to yield a superheating field of ∼ 440 mT that doubles the Nb limit of ∼ 240 mT [3,5,6,7,8]. These properties could allow cavity operation at an elevated temperature of ∼ 4 K and the potential for increased accelerating gradients [9]. This higher operating temperature allows for reduced cryogenic costs and simpler infrastructure for particle accelerators and their applications [3]. Due to their brittle nature and low thermal conductivity, Nb 3 Sn and V 3 Si are best suited for use as a thin film inside a host cavity with better thermal conductivity, such as Nb or Cu [3,10,11]. Nb 3 Sn thin films have been achieved through vapor diffusion, sputtering, electroplating, and chemical vapor deposition [12,13,14,15,16,17]. In the state-of-the-art vapor diffusion, a niobium cavity is placed in a hightemperature vacuum furnace, and then tin or tin chloride sources are vaporized and allowed to diffuse into the niobium surface for alloying [3,9,13,18,19,20]. In contrast, sputtering utilizes high-energy plasma to directly eject target materials onto a substrate at low temperatures [4,6,12,21,22]. Alternatively, Nb 3 Sn films are fabricated via electroplating in aqueous solutions working at near-room temperatures and atmospheric pressure followed by heat treatment [14,15,16,23], or via chemical vapor deposition that takes advantage of reactions between volatile precursors [24]. Nb 3 Sn has been successfully vapor-diffused inside cavities, where a single-cell reached gradients of 24 MV/m, while Nb 3 Sn 9-and 5-cells reached 15 MV/m, both with Q 0 's on the order of 10 10 at operating temperature 4.4 K [19,20]. In cavity tests, maximum surface fields of 120 mT (pulsed operation) and 80 -100 mT (CW) have been achieved, showing that Nb 3 Sn cavities can be operated reliably in a flux-free metastable state above the lower critical field of this material (around 40 mT) [7,25]. In the sputtering process, the film properties are tailored by controlling the Ar/Kr plasma pressure, substrate temperature, sputtering voltage, sputtering current, rate of deposition, and post-sputtering anneal temperature/duration. In literature [4,6,9,12,26], sputtered Nb 3 Sn films have been demonstrated on Nb and Cu surfaces by using a stoichiometric Nb 3 Sn target, by co-sputtering with Nb and Sn targets, or through annealing a sputtered Nb/Sn multilayer. A stoichiometric target allows for a design where only a single target is used [4,6,26,27]. Co-sputtering involves the use of separate Nb and Sn targets that are sputtering at the same time, allowing for tuning of the power applied to each target [21,28]. Multilayer sputtering also uses separate Nb and Sn targets but alternates the use of each target to create many ultrathin layers of each material [11,12,22]. T c 's above 17.8 K have been observed for single-target and multilayer sputtering [6,11,12]. V 3 Si films have been attempted by thermal diffusion, magnetron sputtering, and highpower impulse magnetron sputtering (HiP-IMS) [10,11,27,28]. In thermal diffusion, a vanadium layer on a silicon-on-insulator substrate is annealed at high temperature to produce V 3 Si [28]. In the HiPIMS method, power is applied as a set of discrete high-energy pulses at a low-duty cycle, which can be used to ion bombard the substrate, recrystallizing films at a low temperature and allowing more control of the stoichiometry; this method of depositing V 3 Si films on Cu substrates produced T c up to 10 K [10]. CERN's magnetron sputtered V 3 Si films on a silver buffer layer upon a Cu substrate have reached T c of 11.2K [27]. Thermal annealing of the sputtered films, either in situ or post-deposition, is required to minimize the internal stress induced by the sputtering process and improve the stoichiometry and grain structures, which are important for their critical temperature and cavity RF performance [4,6,12]. However, during annealing of sputtered Nb 3 Sn or Nb/Sn multilayers, the films suffer from issues such as Sn loss, Cu incorporation into the film from Cu substrates, high strain, and interface issues at the substrate-film boundary [4,6,12]. Sn loss is a critical issue because of the dependence of T c on Sn concentration [3]. While annealing is frequently performed on Nb 3 Sn films, these high temperatures have led to Sn loss in the furnace and Nb-rich films with reduced T c [6,12], which motivates us to mechanistically understand the phase transformation associated with annealing. Cu incorporation can occur during annealing, which lowers the T c [4]. This issue can be addressed by using a barrier layer such as tantalum to reduce the interdiffusion [27]. The interface between Nb 3 Sn and Cu also suffers from strain because of their different thermal expansion coefficients and lattice mismatch, which can cause cracking in the film [4]. Cracking can release high initial strain in the lattice, but does not relieve microstrain and increases surface roughness while decreasing the uniformity of the film [4,29]. Currently, no sputtered Nb 3 Sn cavity test has been reported. Moreover, V 3 Si is much less studied than Nb 3 Sn, and there has been no RF test to date [10,11,27]. One goal of this work is to optimize the sputtering capability of these alternative SRF materials at Cornell and compare our results with existing efforts in the SRF field. Most importantly, we aim to systematically investigate the effect of thermal annealing on the sputtered Nb 3 Sn and V 3 Si thin films in order to better understand these observed issues and design an optimal process for SRF use. By understanding the impacts of deposition and annealing parameters, our goal is to find the root of the issues in stoichiometry and strain of thin films. With such knowledge, we hope to provide insights for the development of sputtered Nb 3 Sn and V 3 Si cavities. In this study, we investigate Nb 3 Sn and V 3 Si films of different thicknesses on both Nb and Cu substrates to optimize the best conditions that minimize strain while producing required stoichiometry and superconducting properties. Methods Nb 3 Sn and V 3 Si thin films were deposited using a DC-sputtering system at the Cornell Center for Materials Research. A high vacuum of 10 −6 torr base pressure was achieved using a cryo-pumped system. All depositions were performed at 5 mTorr Ar pressure. A rotating stage was used, when possible, to ensure uniformity during deposition. As summarized in Table 1, the sputtering parameters varied were the film material (Nb 3 Sn vs. V 3 Si), substrate material (Nb Cu), deposition temperature (room temperature vs. 550°C in situ heating), and film thickness (100 nm, 300 nm, and 2 µm). Bulk Nb 3 Sn and V 3 Si targets were used, and they were purchased from ACI alloy, Inc. The impurity concentrations as received were 0.01 at.%. Nb and Cu squared substrates of 1 cm 2 area were used in order to provide insights for applications in Nb and Cu substrate cavities. Before deposition, Nb substrates were electropolished, and Cu substrates were chemically polished to ensure a smooth surface. The Nb 3 Sn and V 3 Si films were designed to have thicknesses of 100 nm and 2 µm on Nb substrates and 300 nm on Cu substrates. The deposition rate was 2.5Å/s for all samples except for the V 3 Si film on Cu substrate which was 1.8Å/s (as there was difficulty lighting the plasma). The deposition temperature for the thick 2 µm samples is subject to error because the temperature is uncontrolled upon the rotating stage and increased through the 133-minute deposition. Subsequently, a 550 ℃ heating stage was applied to investigate the effect of in situ heating during deposition. After the sputtering process, films were annealed in a series of elevated temperatures at 600 ℃, 700 ℃, 800 ℃, and 950 ℃, each for 6 hours, in a Lindberg high-vacuum (3 × 10 −7 Torr) furnace. The heating rate was 10 ℃ per minute, and the annealing was followed by furnace cooling. Structural and chemical analyses were conducted between anneals to characterize the films. These analysis methods included scanning electron microscope (SEM) to observe the grain structure and size, energy dispersive X-ray (EDS) and X-ray photoelectron (XPS) spectroscopies to determine the atomic composition, and X-ray diffraction (XRD) to gain insight into the crystal structure of the film and calculate the strain. In this analysis, the key features are the quality of the film surfaces (smoothness, uniformity, grain shape/size), the stoichiometry of the films, and the existence and strain of Nb 3 Sn and V 3 Si diffraction planes. Note that EDS results were calibrated with regard to the electron penetration depth in each material and the film thickness. Finally, on the 100 nm thin Nb 3 Sn film that yields the best performance, we verified its critical temperature using a quantum design physical property measurement system (PPMS) and quantified the surface roughness using atomic force microscopy (AFM). Results and Discussion In this section, we first analyze the recrystallization behavior observed in the 100 nm thin Nb 3 Sn films annealed and discuss the superconducting, composition, and surface properties of these films. Next, we show the composition and strain evolutions as a function of annealing temperature in the 2 µm thick Nb 3 Sn and V 3 Si films (on Nb) and attempt to understand the Sn loss and strain relief mechanisms. Finally, we show the ternary phase transformation upon annealing in the 300 nm thick Nb 3 Sn and V 3 Si films that were deposited on Cu substrates. Representative surface morphologies of samples upon deposition and after 700°C and 950°C anneals are shown in figure S1. Thin Nb 3 Sn film: Recrystallization 3.1.1. Recrystallization behavior Figure 1 shows the evolution of surface morphology with increasing temperature for the 100 nm thin Nb 3 Sn film on Nb substrate. We observed evident grain recrystallization at 950 ℃ anneals. The grain size increased from a few nanometers as deposited (figure 1a) to approximately 300 nm after annealing (figure 1d). Recrystallization occurs through the release of strain energy during annealing and the subsequent migration of grain boundaries [30]. Here, we discuss the driving force and boundary mobility for thermodynamic considerations of this recrystallization annealing. The stored energy per unit volume (E s ) at a strain level ( ) of 0.2, the maximum strain measured from our sputtered films, is 2.7 × 10 9 J/m 3 , based on a 1-dimensional elastic assumption, E s = 1/2 E 2 , where E is Young's modulus and the value for Nb 3 Sn at 300 K is 13.7 × 10 11 dyn/cm 2 [31]. Indeed, this value is dramatically larger than the typical lightly- deformed energy of 10 5 J/m 3 for driving recrystallization in metals [32]. This suggests a sufficient driving force from the sputtering-induced strain within the film to enable the recrystallization annealing. Our X-ray diffraction data (figure 2) shows the Nb 3 Sn phase is consistent in terms of grain orientation at all annealing temperatures including 950 ℃ recrystallizations. We find Nb 3 Sn peaks near the known powder diffraction peaks at 2θ = 33.6°, 37.7°, 41.5°, 62.8°, 65.6°, 70.6°, and 82.9° [34]. Due to the large penetration depth of the X-ray probe, strong Nb substrate diffractions are seen at 2θ = 38.4°, 53.3°, and 69.3°. A complete list of known peak locations is shown in table S1. As the annealing temperature was increased to 950 ℃, the grain orientations of Nb 3 Sn remained while the growth of grain size was significant. We assume this recrystallization follows a boundary migration mechanism and evaluate the boundary mobility by the Arrhenius law, d = A × exp (-E a / RT), where d is the equilibrium grain size, A is the pre-exponential factor, E a is the activation energy, and T is annealing temperature. The apparent values of the pre-exponential factor and activation energy were determined to be 2.59 × 10 5 and 63 ± 2 kJ/mol, respectively, by Schelb [33]. At an annealing temperature of 950 ℃, the maximum attainable grain size is in the range of 434 -643 nm. The observed ∼ 300 nm grain size in our work is reasonable considering the influence from annealing time (6 h in our work versus up to 200 h in Schelb's work). In summary, recrystallization anneal above 800 ℃ is effective in relieving the built-in strain from sputtering and thus forming stoichiometric Nb 3 Sn along with grain coarsening. Film properties Superconducting properties, atomic composition, and surface roughness were investigated on the 100 nm Nb 3 Sn sample after the 950 ℃ annealing for 6 hours. As shown in figure 3, the critical temperature is determined to be 17.5 K, while the Nb/Sn stoichiometry is 3/1 after sputtering away the surface oxides. Similarly, Sayeed et al. [6] reported T c values of 17.68 -17.83 K for 350 nm Nb 3 Sn sputtered films that were annealed at 800 ℃ for 24 hours and 1000 ℃ for 1 hour with low Sn loss. They observed significant degradation of T c down to 10.95 K as a consequence of the dramatic Sn loss down to 4% after annealing for 24 h at 1000 ℃. In contrast, we did not observe the Sn loss in the 100 nm thin films after annealing. We infer recrystallization plays a major role in retaining the Sn ratio as well as maintaining T c ∼ 17.5 K in the 100 nm thin films, with the relatively short annealing time helping to retain the film properties. However, our 2 µm thick films as detailed in Section 3.2 showed similar Sn loss behavior at increasing annealing temperatures as compared to the 350 nm thick films in Sayeed's work [6], which indicates the importance of a recrystallization process to obtain stoichiometric Nb 3 Sn films with T c 17.5 K. Additionally, the atomic force microscopy (AFM) result is Thick Nb 3 Sn and V 3 Si films: relation of strain and composition change Thermal annealing was performed on 2 µm thick Nb 3 Sn and V 3 Si films on the Nb substrate. The initial thickness of the films greatly altered the annealing behaviors as compared to results from 100 nm thin films. 3 Sn films The Nb 3 Sn grains nucleated in a triangular shape and remained in that shape at all annealing temperatures studied ( figure 4a and 4b). We speculate that these small triangular-shaped grains with 100 -200 nm in size were induced by the high built-in stress and subsequent plastic deformation during deposition. Inplane stress is typical in physical vapor sputtering, and small grains with angular shapes are favored under the stress [35]. This argument is supported by the in situ stress versus grain size relationship in Leib's work [36]. 2 µm thick Nb Upon annealing, as shown in figure 5a, the 2 µm thick Nb 3 Sn films experienced significant Sn loss from the as-deposited ∼ 24% down to 21% after the initial anneal at 600 ℃ and further down to nearly 2% after the 950 ℃ anneal. Conversely, Nb 3 Sn phases were barely observed in the X-ray diffraction until Nb 3 Sn peaks appeared at 800 ℃ and 950 ℃ anneals. The strain ( ) for a given plane was calculated by = (a T -a 0 ) / a 0 , where a T is the measured lattice constant from Nb 3 Sn plane diffraction and a 0 is the lattice parameter from database [34]. (Internal strains calculated for all samples are summarized in Table S2.) The relative strain (∆ ), shown in figure 6a, was obtained by normalizing strain to the hightemperature anneal limit where we observe negligible strains. Here, we analyze the effect of film thickness on the strain. The internal strain ( ) in a biaxial thin film system where inplane stresses are equal (δ = δ 11 = δ 22 ) can be described by a linear relationship as = (2 S 1 + 1/2 S 2 sin 2 φ) δ, where φ is the angle from the film normal to the diffraction plane normal, and S 1 and S 2 are the Xray elastic constants that are determined, in an elastic isotropic scenario, by Young's modulus (E) and Poisson's ratio (υ) and given by -υ / E and (1 + υ) / E, respectively [36]. The built-in stress increases with film thickness (or deposition time at a fixed deposition rate) in a polycrystalline film system when the deposition goes beyond the initial instantaneous stress stage (< 10 nm thickness) [35]. This positive correlation, although slightly affected by the growthinterrupt stress relaxation effect and the heating effect, suggests high strain in the 2 µm thick film; however, the high in-plane stress during thicker film sputtering results in plastic deformation as indicated by the observation of small grain sizes and high density of boundaries (figure 4a). Furthermore, we cannot fully explain the Sn loss in the course of annealing the sputtered Nb 3 Sn films, i.e., the decrease of Sn/(Nb+Sn) atomic ratios with the increasing annealing temperature (figure 5a). Note that this Sn loss behavior is repeatedly observed in previous Nb 3 Sn sputtering work [6,12]. At the annealing temperatures studied, pure Sn phases are not expected due to their low vaporization temperatures (e.g., 800 ℃ at 10 −6 Torr), so we primarily consider Nb-Sn alloy phases in the film. The as-deposited film showed a 23 -25% Sn atomic ratio which suggests minimal Sn-rich phases (Nb 6 Sn 5 and NbSn 2 ) based on the Nb-Sn phase diagram [3]; these Sn-rich phases were also not observed in the X-ray diffraction. Without Sn or Sn- rich phases, merely Nb 3 Sn is expected in this study as indicated by the 23 -25% Sn ratio, but XRD did not show any detectable Nb 3 Sn diffractions upon deposition; the crystalline Nb 3 Sn phase has an extremely high (> 2100 ℃) phase transformation temperature, making it unlikely to explain the Sn loss. We, therefore, suspect the generation of Figure 6. (a) Relative strain that is normalized to the high-temperature anneal limit, as a function of annealing temperature for the 100 nm and 2 µm Nb 3 Sn films on Nb substrates together with the 300 nm Nb 3 Sn film on Cu substrates. (b) Temperature-dependent strain diagram calculated from stable (s) and unstable (u) (220) diffraction peak shifting for 2 µm and 300 nm V 3 Si films on the Nb and Cu substrates, respectively. (c) Example of the XRD patterns taken on the 2 µm thick V 3 Si film showing the stable and unstable (220) diffraction peaks for generating the strain diagram. amorphous Nb 3 Sn phases in the film. Such amorphous phases were reported when using non-equilibrium processing techniques [38,39]. This could cause the loss of Sn alloys via the generation of α-Nb and also explain the appearance of Nb 3 Sn diffraction for anneals above 800 ℃, which likely corresponds to the crystallization temperature. This requires further investigation. 2 µm thick V 3 Si films Different from thick Nb 3 Sn films, the as-deposited 2 µm thick V 3 Si film (figure 6b) exhibits a high strain of 15%, which supports the positive relationship between strain and film thickness in an elastic scenario for thick (> 10 nm) polycrystalline films. The initial film shows a near-stoichiometric value of Si (∼ 23%) shown in figure 5b. Upon annealing, the V 3 Si film shows a constant Si concentration for all temperatures; see figure 5a. In contrast, the strain within the film is significantly relieved together with a transition from the unstable V 3 Si structure to the stable structure between 800 ℃ and 950 ℃ (figure 6b). The structural transformation is observed through the shifting of the (220) and (222) diffraction peaks in figure 6c. These behaviors demonstrate that thermal annealing contributes to strain reduction and structural stabilization in a thick sputtered film. However, as shown in figure 4d, large cracks begin to appear on the film after the first anneal at 600 ℃, coinciding with a shift toward a more angular grain shape with increasing temperature. The high strain induced by the sputtering deposition is responsible for the cracks although thermal relaxation has reduced a significant amount of lattice strain. Nb 3 Sn and V 3 Si films on Cu substrates: ternary alloy systems By studying the temperature-atomic percentage phase diagrams of Nb-Sn [3] and V-Si [45], as well as the three-element composition phase diagrams of Cu-Nb-Sn [40,41,42] and Cu-V-Si [43,44], we can gain insight into the phase transformations our films undergo during the annealing process. As shown in figures 7a and 7c, 300 nm thick Nb 3 Sn and V 3 Si films were sputtered on Cu substrates using a 550 ℃ in situ heating stage. Upon annealing, both films undergo dramatic grain structure changes due to the generation of Cu-Sn or Cu-Si phases. Nb 3 Sn grains start with rounded grains collecting in finger-like formations as deposited on a Cu surface (figure 7a) and they remelt into small angular grains collecting in regions of differing densities after 950 ℃ anneal (figure 7b). In contrast, V 3 Si grains begin with a finger-like pattern after deposition (figure 7c) and end with small angular grains and large artifacts scattered across the surface after 950 ℃ anneal (figure 7d). Overall, there is a trend of grain angularization and pattern restructuring with increasing temperature. The ternary phase transformation that includes Cu-alloy in the films primarily determines the film properties. As shown in figure 5a, the 300 nm Nb 3 Sn films on Cu substrates suffer from the Sn loss similar to 2 µm thick films on Nb substrates, but the mechanism is different. According to the Nb-Sn-Cu phase diagram [40,41,42], Cu-Sn and Nb-Sn phases generate at low temperatures (e.g., 675 ℃ [40]) and these Cu-Sn transform into liquid at 800 ℃ under atmospheric pressure [41], and at high temperatures (e.g., 1000 ℃ [42]), only Nb 3 Sn and Cu exist. In our study, high-vacuum annealing vaporized the Cu-Sn phases leading to a continuous loss of Sn with increasing temperature. Also, we observed low-intensity Nb 3 Sn diffraction at all temperatures whereas convoluted XRD peaks that are possibly from Cu-Sn and other Nb-Sn phases appeared at low temperatures. These observations match with the existence of Nb 3 Sn in the phase diagram at high temperatures although the majority of the film was evaporated. Different from Nb-Sn-Cu, the V-Si-Cu phase diagram [43,44] shows Cu-Si and V-Si phases at low temperatures (e.g., 700 ℃ [43]), but there is no liquid phase at high temperatures (e.g., 800 ℃ [44]). Instead, these phases transform into V-Si and Cu phases. In our study, as shown in figure 5a, the Si/(Si+V) ratio begins with a high value of 42% due to the presence of Cu-Si phases generated during the 550 ℃ in situ heated deposition; note that Cu signal is evident, but is not included in the calculation. After annealing, the Si/(Si+V) ratio drops to 20% at 950 ℃. This phenomenon strongly supports the disappearance of Cu-Si phases in the phase diagram, and only V-Si phases together with some Cu metallic inclusions are expected in the annealed films. Our diffraction data suggest these V-Si phases include the stable (220) and unstable (222) V 3 Si structures. Conclusions and Outlook In this study, we have demonstrated the capability of annealing the sputtered thin films to produce successful Nb 3 Sn and V 3 Si surfaces that have the potential for use inside SRF cavities. We observe that annealing is required to release the strain in the film and promote grain growth. For our Nb 3 Sn samples, the best results are found on the recrystallized 100 nm film, where large grains form at 950 ℃ anneals. These films are also smooth and have minimal surface defects. The 2 µm Nb 3 Sn films are not able to overcome the built-in stress and plastic deformation during sputtering, and likely form an amorphous Nb-Sn phase that leads to nearly complete Sn loss upon annealing. In contrast, the V 3 Si samples retain the stoichiometry at high temperatures, along with a transition in the grain shape to become more angular. Most interesting was the behavior of these films with respect to the unstable and stable phases of V 3 Si. In the 2 µm film, there was a complete transition from unstable to stable at 800 ℃ along with consistent stoichiometry. Because we observe this transition and the proper stoichiometry at high temperatures, we determine these are successful V 3 Si films. For the Cu substrate samples, 550°C in situ heated deposition and the subsequent lowtemperature anneals produce Cu-Si and Cu-Sn phases. These phases transform at high temperatures, extracting high concentrations of Cu inclusions in the film. The Cu impurities and Cu-related phases could adversely affect the SRF performance of Nb 3 Sn/V 3 Si films inside Cu cavities. In a future study, we would be interested in the use of an ultrathin buffer layer between the Cu and the superconducting layer to prevent this effect [27]. In our results, we observed a similar Sn loss as in previous studies [6]. We are interested in finding ways to prevent this loss such as minimizing strong undercooling and avoiding disordered Nb-Sn phases or using encapsulation during the annealing process. We would like to obtain the benefits of annealing such as recrystallization and strain removal while avoiding events such as Sn loss and cracking. Because the 100 nm Nb 3 Sn film was successful, it would be important in a future study to further investigate films of similar thickness to optimize grain growth and RF performance. Data Availability Statement The data that support the findings of this study are available upon reasonable request from the authors. Conflicts of Interest The authors declare no competing financial interests. Table S1. X-ray diffraction (XRD) peaks of Nb 3 Sn, V 3 Si (stable vs. unstable), substrates Nb and Cu, and other possibly relevant phases (NbSn 2 , Nb 6 Sn 5 , V 5 Si 3 , V 6 Si 5 , Nb 3 Cu, V 3 Cu Cu 15 Si 4 , Cu 3 Si 4 , and V (unstable) from reference [34]. For NbSn 2 , Nb 6 Sn 5 , V 5 Si 3 , and V 6 Si 5 peaks, we only listed the prominent points. Figure 1 . 1Surface SEM images for 100 nm thin Nb 3 Sn films on Nb substrates: (a) as-deposited (a), and (b-d) after annealing: (b) 600 ℃, (c) 800 ℃, and (d) 950 ℃. Figure 2 . 2XRD patterns taken from the 100 nm thin Nb 3 Sn film as a function of annealing temperature: (a) as-deposited, (b) 600 ℃, (c) 700 ℃, (d) 800 ℃, (e) 950 ℃. Observed Nb 3 Sn diffraction planes are labeled at the top. Figure 3 . 3Film properties for the 100 nm thin Nb 3 Sn film on a Nb substrate after 950 ℃ annealing. (a) Resistive transition and critical temperature, (b) XPS spectrum showing the atomic composition after sputtering away the surface 20 nm layer, and (c) AFM image showing low surface roughness. shown in figure 3c. The film shows low surface roughness, with an average roughness of 18.3 nm, RMS roughness of 25.3 nm, and a maximum height difference of 600 nm. In contrast, Nb substrates used have an average roughness of ∼ 70 nm. Figure 4 . 4Surface SEM images for 2 µm thick Nb 3 Sn (a, b) and V 3 Si (c, d) films on Nb substrates: (a, c) as-deposited and (b, d) after 950 ℃ annealing. Figure 5 . 5(a)Sn/[Sn+Nb] or Si/[Si+V] ratios in the Nb 3 Sn and V 3 Si films, respectively, as a function of annealing temperature for the 2 µm thick films sputtered on Nb substrates and the 300 nm thick films on Cu substrates (discussed in Section 3.3). Note that the high Si ratios for Cu substrate samples are due to exclusion of Cu signals for calculation. As-deposited Sn/Si composition on 2 µm thick films are 23 -25 %. (b) Example of the EDS spectrum taken on the 2 µm thick V 3 Si film for generating the composition dataset. Figure 7 . 7Surface SEM images for 300 nm Nb 3 Sn (a, b) and V 3 Si (c, d) films on Cu substrates: (a, c) as-deposited and (b, d) after 950 ℃ annealing. Table 1 . 1Sputtering parameters for Nb 3 Sn and V 3 Si film deposition.Film Substrate Substrate holder Temperature (°C) Voltage (V) Current (A) Nominal thickness Nb 3 Sn Nb Rotating 25 596 0.15 100 nm Nb 3 Sn Nb Rotating > 25 589 0.26 2 µm Nb 3 Sn Cu Heated 550 466 0.214 300 nm V 3 Si Nb Rotating > 25 811 0.196 2 µm V 3 Si Cu Heated 550 819 0.222 300 nm vs. AcknowledgementsThis work was supported by the U.S. National Science Foundation under Award PHY-1549132, the Center for Bright Beams. This work made use of the Cornell Center for Materials Research Shared Facilities which are supported through the NSF MRSEC program (DMR-1719875).AppendixFigure S1. SEM map of all samples upon deposition, after 700°C anneal, and after 950°C anneal. Thermal annealing of sputtered Nb 3 Sn and V 3 Si thin films for superconducting RF cavities. K Howard, M Liepe, Z Sun, 10.18429/JACoW-SRF2021-SUPFDV009Proc. SRF'21. SRF'21East Lansing, MI, USAHoward K, Liepe M and Sun Z 2021 Thermal annealing of sputtered Nb 3 Sn and V 3 Si thin films for superconducting RF cavities Proc. SRF'21 (East Lansing, MI, USA) 82-85 doi: 10.18429/JACoW-SRF2021-SUPFDV009 Unprecedented quality factors at accelerating gradients up to 45. A Grassellino, Grassellino A et al. 2017 Unprecedented qual- ity factors at accelerating gradients up to 45 MVm −1 in niobium superconducting resonators via low temperature nitrogen infusion Supercond. 10.1088/1361-6668/aa7afeSci. Technol. 309MVm −1 in niobium superconducting resonators via low temperature nitrogen infusion Super- cond. Sci. Technol. 30 9 doi: 10.1088/1361- 6668/aa7afe Nb 3 Sn superconducting radiofrequency cavities: fabrication, results, properties, and prospects Supercond. S Posen, D L Hall, 10.1088/1361-6668/30/3/033004Sci. Technol. 3033004Posen S and Hall D L 2017 Nb 3 Sn supercon- ducting radiofrequency cavities: fabrication, results, properties, and prospects Supercond. Sci. Technol. 30 033004 doi: 10.1088/1361- 6668/30/3/033004 Development of sputtered Nb 3 Sn films on copper substrates for superconducting radiofrequency applications Supercond. E Ilyina, Ilyina E A et al. 2019 Development of sputtered Nb 3 Sn films on copper substrates for supercon- ducting radiofrequency applications Supercond. . 10.1088/1361-6668/aaf61fSci. Technol. 3235002Sci. Technol. 32 035002 doi: 10.1088/1361- 6668/aaf61f N Valles, M Liepe, The Superheating Field of Niobium: Theory and Experiment Proc. SRF'2011. Chicago, IL, USAValles N and Liepe M 2011 The Superheating Field of Niobium: Theory and Experiment Proc. SRF'2011 (Chicago, IL, USA) 293-301 Properties of Nb 3 Sn films fabricated by magnetron sputtering from a single target. M Sayeed, 10.1016/j.apsusc.2020.148528Appl. Surf. Sci. 541148528Sayeed M N et al. 2021 Properties of Nb 3 Sn films fabricated by magnetron sputtering from a single target Appl. Surf. Sci. 541 148528 doi: 10.1016/j.apsusc.2020.148528 Critical fields of Nb 3 Sn prepared for superconducting cavities. S Keckert, 10.1088/1361-6668/ab119eSupercond. Sci. Technol. 3275004Keckert S et al. 2019 Critical fields of Nb 3 Sn pre- pared for superconducting cavities Supercond. Sci. Technol. 32 075004 doi: 10.1088/1361- 6668/ab119e Temperature dependence of the superheating field for superconductors in the high-κ London limit. G Catelani, J P Sethna, 10.1103/PhysRevB.78.224509Phys. Rev. B. 78224509Catelani G and Sethna J P 2008 Temperature de- pendence of the superheating field for supercon- ductors in the high-κ London limit Phys. Rev. B 78 224509 doi: 10.1103/PhysRevB.78.224509 R Porter, 10.18429/JACoW-NAPAC2019-MOYBB3Progress in Nb 3 Sn SRF cavities at Cornell University Proc. NAPAC'. 19Porter R D et al. 2019 Progress in Nb 3 Sn SRF cavities at Cornell University Proc. NAPAC'19 37-40 doi: 10.18429/JACoW-NAPAC2019- MOYBB3 Using HiPIMS for V 3 Si superconducting thin films presented at 9th Int. F Estrin, Workshop on Thin Films and New Ideas for Pushing the Limits of RF Superconductivity. Estrin F L et al. 2021 Using HiPIMS for V 3 Si superconducting thin films presented at 9th Int. Workshop on Thin Films and New Ideas for Pushing the Limits of RF Superconductivity https://indico.jlab.org/event/405/contributions/ 8111/ S Deambrosis, The progress on Nb 3 Sn and V 3 Si Proc. SRF'2007. Beijing, ChinaDeambrosis S M et al. 2007 The progress on Nb 3 Sn and V 3 Si Proc. SRF'2007 (Beijing, China) 392-399 Structural and superconducting properties of Nb 3 Sn films grown by multilayer sequential magnetron sputtering. M Sayeed, 10.1016/j.jallcom.2019.06.017J. Alloys Compd. 800Sayeed M N et al. 2019 Structural and super- conducting properties of Nb 3 Sn films grown by multilayer sequential magnetron sputter- ing J. Alloys Compd. 800 272-278 doi: 10.1016/j.jallcom.2019.06.017 Atomic-scale analyses of Nb 3 Sn on Nb prepared by vapor diffusion for superconducting radiofrequency cavity applications: a correlative study. J Lee, 10.1088/1361-6668/aaf268Supercond. Sci. Technol. 3224001Lee J et al. 2019 Atomic-scale analyses of Nb 3 Sn on Nb prepared by vapor diffusion for super- conducting radiofrequency cavity applications: a correlative study Supercond. Sci. Technol. 32 024001 doi: 10.1088/1361-6668/aaf268 Synthesis of superconducting Nb 3 Sn coatings on Nb substrates. E Barzi, 10.1088/0953-2048/29/1/015009Supercond. Sci. Technol. 2915009Barzi E et al. 2016 Synthesis of superconducting Nb 3 Sn coatings on Nb substrates Supercond. Sci. Technol. 29 015009 doi: 10.1088/0953- 2048/29/1/015009 Toward stoichiometric and low-surface-roughness Nb 3 Sn thin films via direct electrochemical deposition Proc. Z Sun, Sun Z et al. 2021 Toward stoichiometric and low-surface-roughness Nb 3 Sn thin films via direct electrochemical deposition Proc. 10.18429/JACoW-SRF2021-WEOTEV03SRF'21. East Lansing, MI, USASRF'21 (East Lansing, MI, USA) 710-713 doi: 10.18429/JACoW-SRF2021-WEOTEV03 Electroplating of Sn film on Nb substrate for generating Nb 3 Sn thin films and post laser annealing Proc. Z Sun, 10.18429/JACoW-SRF2019-MOP014GermanySRF'19 (DresdenSun Z et al. 2019 Electroplating of Sn film on Nb substrate for generating Nb 3 Sn thin films and post laser annealing Proc. SRF'19 (Dres- den, Germany) 51-54 doi: 10.18429/JACoW- SRF2019-MOP014 L Xiao, 10.18429/JACoW-SRF2019-THP008The technical study of Nb 3 Sn film deposition on copper by HiPIMS Proc. SRF'19. Dresden, GermanyXiao L et al. 2019 The technical study of Nb 3 Sn film deposition on copper by HiPIMS Proc. SRF'19 (Dresden, Germany) 846-847 doi: 10.18429/JACoW-SRF2019-THP008 Analysis of Nb 3 Sn surface layers for superconducting radio frequency cavity applications. C Becker, 10.1063/1.4913617Appl. Phys. Lett. 10682602Becker C et al. 2015 Analysis of Nb 3 Sn surface layers for superconducting radio frequency cavity applications Appl. Phys. Lett. 106 082602 doi: 10.1063/1.4913617 Advances in Nb 3 Sn superconducting radiofrequency cavities towards first practical accelerator applications. S Posen, 10.1088/1361-6668/abc7f7Supercond. Sci. Technol. 3425007Posen S et al. 2021 Advances in Nb 3 Sn super- conducting radiofrequency cavities towards first practical accelerator applications Supercond. Sci. Technol. 34 025007 doi: 10.1088/1361- 6668/abc7f7 Nb 3 Sn multicell cavity coating system at. G Eremeev, 10.1063/1.5144490Jefferson Lab Rev. Sci. Instrum. 9173911Eremeev G et al. 2020 Nb 3 Sn multicell cavity coating system at Jefferson Lab Rev. Sci. Instrum. 91 073911 doi: 10.1063/1.5144490 Kinetically induced lowtemperature synthesis of Nb 3 Sn thin films. N Schäfer, 10.1063/5.0015376J. Appl. Phys. 128133902Schäfer N et al. 2020 Kinetically induced low- temperature synthesis of Nb 3 Sn thin films J. Appl. Phys. 128 133902 doi: 10.1063/5.0015376 Nb 3 Sn thin film coating method for superconducting multilayered structure. R Ito, 10.18429/JACoW-SRF2019-TUP077Proc. SRF'19. SRF'19Dresden, GermanyIto R et al. 2019 Nb 3 Sn thin film coating method for superconducting multilayered struc- ture Proc. SRF'19 (Dresden, Germany) 628- 631 doi: 10.18429/JACoW-SRF2019-TUP077 M Lu, 10.18429/JACoW-SRF2019-TUP076Electrochemical deposition of Nb 3 Sn on the surface of copper substrates Proc. SRF'19. Dresden, GermanyLu M et al. 2019 Electrochemical deposition of Nb 3 Sn on the surface of copper substrates Proc. SRF'19 (Dresden, Germany) 624-627 doi: 10.18429/JACoW-SRF2019-TUP076 M Ge, 10.18429/JACoW-SRF2019-TUFUB8CVD coated copper substrate SRF cavity research at Cornell University Proc. SRF'19. Dresden, GermanyGe M et al. 2019 CVD coated copper substrate SRF cavity research at Cornell University Proc. SRF'19 (Dresden, Germany) 381-385 doi: 10.18429/JACoW-SRF2019-TUFUB8 Nb 3 Sn for radio frequency cavities Lawrence Berkeley National Laboratory. A Godeke, Godeke A et al. 2006 Nb 3 Sn for radio frequency cavities Lawrence Berkeley National Laboratory https://escholarship.org/uc/item/6d3753q7 Development of Nb 3 Sn coatings by magnetron sputtering for SRF cavities. G Rosaz, 10.18429/JACoW-SRF2015-TUPB051Proc. SRF'15. SRF'15Whistler, BC, CanadaRosaz G et al. 2015 Development of Nb 3 Sn coat- ings by magnetron sputtering for SRF cavities Proc. SRF'15 (Whistler, BC, Canada) 691-694 doi: 10.18429/JACoW-SRF2015-TUPB051 Magnetron sputtered Nb 3 Sn and V 3 Si thin films on copper substrates for SRF application presented at Nb 3 SnSRF. S Fernandez, 20Fernandez S 2020 Magnetron sputtered Nb 3 Sn and V 3 Si thin films on copper substrates for SRF application presented at Nb 3 SnSRF'20 Thin film synthesis of superconductor on insulator A15 vanadium silicide Sci. W Zhang, 10.1038/s41598-021-82046-1Rep. 112358Zhang W et al. 2021 Thin film synthesis of superconductor on insulator A15 vanadium silicide Sci. Rep. 11 2358 doi: 10.1038/s41598- 021-82046-1 Nb 3 Sn thin film deposition on copper by DC magnetron sputtering. W Tan, 10.18429/JACoW-SRF2017-TUPB055Proc. SRF'. 17Tan W et al. 2017 Nb 3 Sn thin film deposi- tion on copper by DC magnetron sputtering Proc. SRF'17 512-515 doi: 10.18429/JACoW- SRF2017-TUPB055 . R Doherty, 10.1016/S0921-5093(97)00424-3Current issues in recrystallization: a review Mater. Sci. Eng. A. 238Doherty R D et al. 1997 Current issues in recrys- tallization: a review Mater. Sci. Eng. A 238 219-274 doi: 10.1016/S0921-5093(97)00424-3 Young's modulus of polycrystalline Nb 3 Sn between 4.2 and 300 K. J Bussiere, 10.1063/1.327730J. Appl. Phys. 511024Bussiere J F et al. 1980 Young's modulus of polycrystalline Nb 3 Sn between 4.2 and 300 K J. Appl. Phys. 51 1024 doi: 10.1063/1.327730 . F J Humphreys, M Hatherly, Recrystallization and Related Annealing Phenomena. Elsevier Ltd2nd EditionHumphreys F J and Hatherly M Recrystallization and Related Annealing Phenomena, 2nd Edi- tion, Elsevier Ltd., Amsterdam, 2004. Electron microscopic examination of multifilamentary bronze-processed Nb 3 Sn. W Schelb, 10.1007/BF01113599J. Mater. Sci. 16Schelb W 1981 Electron microscopic exami- nation of multifilamentary bronze-processed Nb 3 Sn J. Mater. Sci. 16 2575-2582 doi: 10.1007/BF01113599 The Materials Project: A materials genome approach to accelerating materials innovation APL Mater. A Jain, 10.1063/1.4812323111002Jain A et al. 2013 The Materials Project: A materials genome approach to accelerating materials innovation APL Mater. 1 011002 doi: 10.1063/1.4812323 Stress in thin films and coatings: Current status, challenges and prospects. G Abadias, 10.1116/1.5011790J. Vac. Sci. Technol. A. 3620801Abadias G et al. 2018 Stress in thin films and coatings: Current status, challenges and prospects J. Vac. Sci. Technol. A 36 020801 doi: 10.1116/1.5011790 Relationships between grain structure and stress in thin Volmer-Weber metallic films Massachusetts Institute of Technology. J Leib, Leib J S Relationships between grain structure and stress in thin Volmer-Weber metallic films Massachusetts Institute of Technology, 2018. Stress, strain, and microstructure in thin tungsten films deposited by dc magnetron sputtering. T Vink, 10.1063/1.354842J. Appl. Phys. 7415Vink T J et al. 1993 Stress, strain, and microstructure in thin tungsten films deposited by dc magnetron sputtering J. Appl. Phys. 74 15 doi: 10.1063/1.354842 New amorphous Cu-Nb-(Si, Ge or Sn) alloys prepared by mechanical alloying of elemental powders Mater. K Matsuki, 10.1016/0025-5416(88)90010-9Sci. Eng. 97Matsuki K et al. 1988 New amorphous Cu-Nb- (Si, Ge or Sn) alloys prepared by mechanical alloying of elemental powders Mater. Sci. Eng. 97 47-51 doi: 10.1016/0025-5416(88)90010-9 Superconductivity of ductile Nb-Based amorphous alloys. T Masumoto, 10.2320/mater-trans1960.21.115Trans. Jpn. Inst. Met. 21Masumoto T et al. 1980 Superconductivity of duc- tile Nb-Based amorphous alloys Trans. Jpn. Inst. Met. 21 115-122 doi: 10.2320/mater- trans1960.21.115 The ternary system Nb-Sn-Cu at 675°C. W L Neijmeijer, B H Kolster, 10.1515/ijmr-1987-781009Int. J. Mater. Res. 78Neijmeijer W L and Kolster B H 1987 The ternary system Nb-Sn-Cu at 675°C Int. J. Mater. Res. 78 730-737 doi: 10.1515/ijmr-1987-781009 Phase equilibria and superconducting properties in niobium-tincopper alloys USSR. V Pan, Pan V M et al. 1980 Phase equilibria and superconducting properties in niobium-tin- copper alloys USSR R Hopkins, Phase relations and diffusion layer formation in the systems Cu-Nb. Hopkins R H et al. 1977 Phase relations and diffusion layer formation in the systems Cu-Nb- . Cu-Nb-Ge Sn, Metall, 10.1515/ijmr-1987-781009Trans. A. 8Sn and Cu-Nb-Ge Metall. Trans. A 8 91-97 doi: 10.1515/ijmr-1987-781009 Si-Cu ternary systems. J Reid, 10.1557/JMR.1992.2424Thermodynamics of. 7Reid J S et al. 1992 Thermodynamics of (Cr, Mo, Nb, Ta, V, or W)-Si-Cu ternary systems J. Mater. Res. 7 2424-2428 doi: 10.1557/JMR.1992.2424 Influence of copper on the structure and superconducting properties of transition metals. E Savitskii, Inorg. Mater. 15Savitskii E M et al. 1979 Influence of copper on the structure and superconducting properties of transition metals Inorg. Mater. 15 512-515 Si-V (Silicon-Vanadium). H Okamoto, 10.1007/s11669-010-9733-5J. Phase Equilib. Diffus. 31Okamoto H et al. 2010 Si-V (Silicon-Vanadium) J. Phase Equilib. Diffus. 31 409-410 doi: 10.1007/s11669-010-9733-5
{'fraction_non_alphanumeric': 0.04693181308075594, 'fraction_numerical': 0.06594245185617453, 'mean_word_length': 4.304166666666666, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 4, 'https://': 2, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 4, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'Nb 3 Sn and V 3 Si thin films are promising candidates as thin films for the next generation of superconducting radio-frequency (SRF) cavities. However, sputtered films often suffer from stoichiometry and strain issues during deposition and post annealing. In this study, we explore the structural and chemical effects of thermal annealing, both in-situ and post-sputtering, on DC-sputtered Nb 3 Sn and V 3 Si films of varying thickness on Nb or Cu substrates, extending from our initial studies[1]. Through annealing at 950°C, we successfully enabled recrystallization of 100 nm thin Nb 3 Sn films on Nb substrate with stoichiometric and strain-free grains. For 2 µm thick films, we observed the removal of strain and a slight increase in grain size with increasing temperature. Annealing enabled a phase transformation from unstable to stable structure on V 3 Si films, while we observed significant Sn loss in 2 µm thick Nb 3 Sn films after high temperature anneals. We observed similar Sn and Si loss on films atop Cu substrates during annealing, likely due to Cu-Sn and Cu-Si phase generation and subsequent Sn and Si evaporation. These results encourage us to refine our process to obtain high-quality sputtered films for SRF use.', 'arxivid': '2301.00756', 'author': ['Katrina Howard [email protected]. \nCornell Laboratory for Accelerator-Based Sciences and Education\nCornell University\n14853IthacaNYUSA\n\nDepartment of Physics\nUniversity of Chicago\n60637ChicagoILUSA\n', 'Zeming Sun \nCornell Laboratory for Accelerator-Based Sciences and Education\nCornell University\n14853IthacaNYUSA\n', 'Matthias U Liepe \nCornell Laboratory for Accelerator-Based Sciences and Education\nCornell University\n14853IthacaNYUSA\n', 'Katrina Howard [email protected]. \nCornell Laboratory for Accelerator-Based Sciences and Education\nCornell University\n14853IthacaNYUSA\n\nDepartment of Physics\nUniversity of Chicago\n60637ChicagoILUSA\n', 'Zeming Sun \nCornell Laboratory for Accelerator-Based Sciences and Education\nCornell University\n14853IthacaNYUSA\n', 'Matthias U Liepe \nCornell Laboratory for Accelerator-Based Sciences and Education\nCornell University\n14853IthacaNYUSA\n'], 'authoraffiliation': ['Cornell Laboratory for Accelerator-Based Sciences and Education\nCornell University\n14853IthacaNYUSA', 'Department of Physics\nUniversity of Chicago\n60637ChicagoILUSA', 'Cornell Laboratory for Accelerator-Based Sciences and Education\nCornell University\n14853IthacaNYUSA', 'Cornell Laboratory for Accelerator-Based Sciences and Education\nCornell University\n14853IthacaNYUSA', 'Cornell Laboratory for Accelerator-Based Sciences and Education\nCornell University\n14853IthacaNYUSA', 'Department of Physics\nUniversity of Chicago\n60637ChicagoILUSA', 'Cornell Laboratory for Accelerator-Based Sciences and Education\nCornell University\n14853IthacaNYUSA', 'Cornell Laboratory for Accelerator-Based Sciences and Education\nCornell University\n14853IthacaNYUSA'], 'corpusid': 255372987, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 15102, 'n_tokens_neox': 11763, 'n_words': 7088, 'pdfsha': 'cd69a70ad5e9b9945be4a24e3a08b7c16901b0af', 'pdfurls': ['https://export.arxiv.org/pdf/2301.00756v1.pdf'], 'title': ['Thermal annealing of sputtered Nb 3 Sn and V 3 Si thin films for superconducting radio-frequency cavities', 'Thermal annealing of sputtered Nb 3 Sn and V 3 Si thin films for superconducting radio-frequency cavities', 'Thermal annealing of sputtered Nb 3 Sn and V 3 Si thin films for superconducting radio-frequency cavities', 'Thermal annealing of sputtered Nb 3 Sn and V 3 Si thin films for superconducting radio-frequency cavities'], 'venue': []}
arxiv
Quantum work relations and response theory 24 May 2008 David Andrieux Center for Nonlinear Phenomena and Complex Systems Université Libre de Bruxelles Code Postal 231, Campus PlaineB-1050BrusselsBelgium Pierre Gaspard Center for Nonlinear Phenomena and Complex Systems Université Libre de Bruxelles Code Postal 231, Campus PlaineB-1050BrusselsBelgium Quantum work relations and response theory 24 May 2008 A universal quantum work relation is proved for isolated time-dependent Hamiltonian systems in a magnetic field as the consequence of microreversibility. This relation involves a functional of an arbitrary observable. The quantum Jarzynski equality is recovered in the case this observable vanishes. The Green-Kubo formula and the Casimir-Onsager reciprocity relations are deduced thereof in the linear response regime. Nonequilibrium work relations have recently attracted much interest [1,2]. They provide relations for the work dissipated in time-dependent driven systems, independently of the form of the driving. They are of great interest to evaluate free energies under general nonequilibrium conditions and they provide new methods to study nanosystems. In the nanoscopic world, the extension of these classical relations to quantum systems is of particular importance and different approaches have been proposed. A first scheme was introduced by Kurchan [3]. In this framework, a measurement of the system state is performed at the initial time. In the sequel, the system is perturbed by a time-dependent Hamiltonian before performing another measurement at the final time. The random work performed on the system is associated with the energy difference between the final and initial eigenstates. This setup leads to the quantum extension of Jarzynski equality and Crooks fluctuation theorem [4,5,6,7]. Another possibility is to introduce a quantum work operator which measures the energy difference [8], in which cases quantum corrections to the fluctuation theorem must be taken into account. On the other hand, quantum fluctuation theorems have been obtained in suitable limits where the dynamics admits a Markovian description, allowing in particular the applications to nonequilibrium steady states [9,10,11,12,13,14]. Yet, the connection between the quantum work relations and response theory is still an open question even in the linear regime. The purpose of the present paper is to derive a new type of work relations which involves a functional of an arbitrary observable. This generating functional can be related to another functional but averaged over the timereversed process. This new work relation turns out to be of great generality since we can recover known results such as Jarzynski equality as special cases. Furthermore, this universal work relation allows us to formulate the response theory, to derive the quantum linear response functions, the quantum Green-Kubo relations [15,16], as well as the Casimir-Onsager reciprocity relations [17,18] in the regime close to the thermodynamic equilibrium. Functional symmetry relations. We suppose that the system is described by a Hamiltonian operator H(t; B) which depends on the time t and the magnetic field B. The time-reversal operator Θ is an antilinear operator such that Θ 2 = I and which has the effect of changing the sign of all odd parameters such as magnetic fields: ΘH(t; B)Θ = H(t; −B) .(1) We first introduce the forward process. The system is initially in thermal equilibrium at the inverse temperature β = 1/k B T . The initial state of the system is described by the canonical density matrix ρ(0) = e −βH(0;B) Z(0) ,(2) where the partition function is given in terms of the corresponding free energy F (0) by Z(0) = tr e −βH(0;B) = e −βF (0) . Starting from this equilibrium situation at the initial time t = 0, the system evolves until some final time t = T under the Hamiltonian dynamics. The corresponding forward time evolution is defined as i ∂ ∂t U F (t; B) = H(t; B)U F (t; B) ,(3) with the initial condition U F (0; B) = I [19]. In the Heisenberg representation, the observables evolve according to A F (t) = U † F (t) A U F (t)(4) which also concerns the time-dependent Hamiltonian H F (t) = U † F (t)H(t; B)U F (t) .(5) The average of an observable is thus obtained from A F (t) = tr ρ(0)A F (t) .(6) We note that the dependence on the magnetic field is implicit in these expressions. The backward process is introduced similarly but in the magnetic field reversed. The system is perturbed according to the time-reversed protocol H(T − t; −B), starting at the initial time t = 0 from the density matrix ρ(T ) = e −βH(T ;−B) Z(T ) ,(7) where the free energy F (T ) is given in terms of the partition function according to Z(T ) = tr e −βH(T ;−B) = e −βF (T ) . The system ends at time t = T with the Hamiltonian H(0; −B). The evolution operator of the backward process is defined as i ∂ ∂t U R (t; B) = H(T − t; B)U R (t; B) ,(8) with the initial condition U R (0; B) = I [19], and is related to the one of the forward process by the following Lemma: The forward and backward time evolution operators are related to each other according to ΘU F (T − t; B)U † F (T ; B)Θ = U R (t; −B) ,(9) where t is an arbitrary time 0 ≤ t ≤ T . This lemma is proved by first substituting T − t for t in Eq. (3) to get − i ∂ ∂t U F (T − t; B) = H(T − t; B)U F (T − t; B) .(10) Multiplying this equation by U † F (T ; B)Θ from the right and by Θ from the left, we find i ∂ ∂t ΘU F (T − t; B)U † F (T ; B)Θ = H(T − t; −B)ΘU F (T − t; B)U † F (T ; B)Θ ,(11) where we used the antilinearity Θi = −iΘ of the time-reversal operator and its further property (1). This shows that the expression ΘU F (T − t; B)U † F (T ; B)Θ obeys the same evolution equation (8) With this lemma, we can now demonstrate the Theorem: Let us consider an arbitrary time-independent observable A with a definite parity under time reversal: ΘAΘ = ǫ A A, with ǫ A = ±1. It satisfies the following functional relation: e R T 0 dtλ(t)AF(t) e −βHF(T ) e βH(0) F,B = e −β∆F e ǫA R T 0 dtλ(T −t)AR(t) R,−B ,(12) where λ(t) is an arbitrary function, while the subscripts F and R stand for the forward or backward protocol, respectively. ∆F = F (T ) − F (0) is the difference of the free energies of the initial equilibrium states (7) and (2) of the backward and forward processes. In order to prove Eq. (12), we first consider the quantity A F (t), which can be written as A F (t) = U † F (t) A U F (t) = U † F (T )U F (T ) U † F (t) A U F (t) U † F (T )U F (T ) = ǫ A U † F (T ) Θ A R (T − t) Θ U F (T ) ,(13) where we have inserted the identity U † F (T )U F (T ) = I to go at the second equality. At the third equality, we inserted Θ 2 = I between the evolution operators and we used ΘAΘ = ǫ A A along with Eq. (9). The connection is thus established with the backward process. Integrating over time with an arbitrary function λ(t) and taking the exponential of both sides, the previous expression becomes exp T 0 dt λ(t) A F (t) = U † F (T ) Θ exp ǫ A T 0 dt λ(T − t) A R (t) Θ U F (T ) ,(14) after the change of integration variables t → T − t in the right-hand side. Starting from the left-hand side of Eq. (12), we get tr ρ(0) exp T 0 dt λ(t)A F (t) exp[−βH F (T )] exp[βH(0)] = 1 Z(0) tr exp ǫ A T 0 dt λ(T − t)A R (t) Θ exp[−βH(T ; B)]Θ = Z(T ) Z(0) tr exp ǫ A T 0 dt λ(T − t)A R (t) ρ(T ) = e −β∆F exp ǫ A T 0 dt λ(T − t)A R (t) R,−B .(15) We used the invariance of the trace over cyclic permutations as well as the exponential of Eq. (13) at the first equality. In the second equality, we introduced the equilibrium density matrix (7) which is precisely the initial condition of the backward process. To obtain the last equality, we used that the partition functions have been expressed in terms of the corresponding free energies. This completes the proof of the theorem. QED. We notice that related results have previously been considered in the restricted case where there is no change in free energy ∆F = 0 [20,21]. The present theorem allows us to recover in particular the quantum Jarzynski equality as a special case of Eq. (12) if λ = 0: e −βHF(T ) e βH(0) F,B = e −β∆F .(16) The factor inside the bracket can indeed be interpreted in the quantum setting in terms of the work performed on the system during the forward process [3,4,6,9] in spite of the non-commutativity of the energy operators H F (T ) and H(0) and thanks to the protocol with von Neumann quantum measurements of the energy at the initial and final times. It is only in the classical limit that both energies commute and the classical work can be formed as W cl = [H F (T ) − H(0)] cl . In this case, both exponentials in the left-hand side of the relation (12) becomes exp(−βW cl ) which is the classical version of this relation. Response theory. We can obtain different correlation functions by taking functional derivatives of the relation (12) with respect to the arbitrary function λ(t). In this way, we can obtain the expression of linear response theory from the generalized symmetry relation (12). For this purpose, we consider a perturbation of the form H(t) = H 0 − X(t)B ,(17) where the perturbation X(t) is such that X(t) = 0 for t ≤ 0 and X(t) = 0 for T ≤ t. The observable B is here arbitrary and should not be confused with the magnetic field B. In order to obtain the linear response of an observable A with respect to the perturbation −X(t)B, we take the functional derivative of Eq. (12) with respect to λ(T ), around λ = 0. This yields A F (T )e −βHF(T ) e βH0 F,B = ǫ A A R (0) R,−B = ǫ A A eq,−B ,(18) where we used that ∆F = 0 since X(0) = X(T ) = 0. Since the reversed process also starts at equilibrium, the average in the right-hand side is an equilibrium average, albeit with a reversed magnetic field. Nevertheless, we have that ǫ A A eq,−B = A eq,B by using time reversal. We now have to calculate the exponentials of the initial and final Hamiltonians. Since, in the Heisenberg representation, the total time derivative of the Hamiltonian equals its partial derivative, dH F /dt = (∂H/∂t) F , we can write exp[−βH F (T )] = exp[−β(H 0 + E)](19) with E = T 0 dt ∂H ∂t F = − T 0 dtẊ(t) B F (t) = T 0 dt X(t)Ḃ F (t) ,(20) where the last equality follows from an integration by parts. We now use the expression exp[β(P + Q)] exp(−βP ) = 1 + β 0 du exp[u(P + Q)] Q exp(−uP ) ,(21) which can be proved by differentiating with respect to β. To first order in Q, we may neglect Q in the last exponential function, exp[u(P + Q)]. Taking P = −H 0 and Q = −E and developing to first order in X, we get e −βHF(T ) e βH0 = 1 − T 0 dt X(t) β 0 du e −uH0Ḃ (t)e uH0 + O(X 2 ) = 1 − T 0 dt X(t) β 0 duḂ(t + i u) + O(X 2 ) , where B(t) = exp(iH 0 t/ )B exp(−iH 0 t/ ) since, at first order in the driving force, the time evolution proceeds under the unperturbed Hamiltonian H 0 . Inserting this expansion into Eq. (18) and after some manipulations using the time invariance of correlation function as well as the KMS-like property ρA = A(i β)ρ [22], we finally find A F (T ) B = A eq,B + T 0 dt X(T − t)φ AB (t) + O(X 2 ) ,(22) with the response function φ AB (t) = β 0 du Ḃ (−i u)A(t) eq,B .(23) Equations (22) and (23) are the well-known expressions of linear response theory in the canonical ensemble, also known as the Green-Kubo formula [15,16]. The Casimir-Onsager reciprocity relations for the conductivities [17,18] are obtained by taking A = J µ /V andḂ = J ν in terms of the current J µ = n e nẋnµ and the volume V , in which case the time-reversal symmetry implies φ µν (t; B) = φ νµ (t; −B) and σ µν (ω; B) = σ νµ (ω; −B) for the tensor of conductivities σ µν (ω; B) = ∞ 0 dt e iωt φ µν (t; B). Higher-order terms in the expansion can be obtained as well. Conclusions. In this paper, we have obtained a universal quantum work relation which involves arbitrary observables at arbitrary times. This result relates an average over the forward process ponderated by the quantum analogue of the work to an average over the reversed process. By taking functional derivatives, we can obtain relations for arbitrary correlation functions, which are the consequence of microreversibility. In the simplest case, it can be used to recover the well-known Jarzynski equality. On the other hand, we can also straightforwardly derive from the universal relation the linear response theory of an arbitrary observable. In this regard, this relation unifies in a common framework the work relations and the response theory, thereby opening the possibility to obtain further general relations which are valid not only close to equilibrium but also in the far-from-equilibrium regime. as U R (t; −B). Since they also satisfy the same initial condition, ΘU F (T ; B)U † F (T ; B)Θ = U R (0; −B) = I, we have proven Eq. (9). QED. Acknowledgments. D. Andrieux thanks the F.R.S.-FNRS Belgium for financial support. This research is financially supported by the Belgian Federal Government (IAP project "NOSY") and the "Communauté française de Belgique" (contract "Actions de Recherche Concertées" No. 04/09-312). . C Jarzynski, Phys. Rev. Lett. 782690C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997). . G E Crooks, Phys. Rev. E. 602721G. E. Crooks, Phys. Rev. E 60, 2721 (1999). . J Kurchan, cond-mat/0007360J. Kurchan, cond-mat/0007360. . H Tasaki, cond-mat/0009244H. Tasaki, cond-mat/0009244. . T Monnai, Phys. Rev. E. 7227102T. Monnai, Phys. Rev. E 72, 027102 (2005). . P Talkner, E Lutz, P Hänggi, Phys. Rev. E. 7550102P. Talkner, E. Lutz, and P. Hänggi, Phys. Rev. E 75, 050102 (2007). . P Talkner, P Hänggi, J. Phys. A. 40569P. Talkner and P. Hänggi, J. Phys. A 40, F569 (2007). . T Monnai, S Tasaki, cond-mat/0308337T. Monnai and S. Tasaki, cond-mat/0308337. . S Mukamel, Phys. Rev. Lett. 90170604S. Mukamel, Phys. Rev. Lett. 90, 170604 (2003). . W De Roeck, C Maes, Phys. Rev. E. 6926115W. De Roeck and C. Maes, Phys. Rev. E 69, 026115 (2004). . M Esposito, S Mukamel, Phys. Rev. E. 7346129M. Esposito and S. Mukamel, Phys. Rev. E 73, 046129 (2006). . U Harbola, M Esposito, S Mukamel, Phys. Rev. B. 7685408U. Harbola, M. Esposito, and S. Mukamel, Phys. Rev. B 76, 085408 (2007). . M Esposito, U Harbola, S Mukamel, Phys. Rev. E. 7631132M. Esposito, U. Harbola, and S. Mukamel, Phys. Rev. E 76, 031132 (2007). . M Esposito, U Harbola, S Mukamel, Phys. Rev. B. 75155316M. Esposito, U. Harbola, and S. Mukamel, Phys. Rev. B 75, 155316 (2007). . M S Green, J. Chem. Phys. 20398M. S. Green, J. Chem. Phys. 20, 1281 (1952); 22, 398 (1954). . R Kubo, J. Phys. Soc. Jpn. 12570R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957). . L Onsager, Phys. Rev. 37405L. Onsager, Phys. Rev. 37, 405 (1931). . H B G Casimir, Rev. Mod. Phys. 17343H. B. G. Casimir, Rev. Mod. Phys. 17, 343 (1945). . D Andrieux, P Gaspard, T Monnai, S Tasaki, in preparationD. Andrieux, P. Gaspard, T. Monnai, and S. Tasaki, in preparation (2008). . G N Bochkov, Yu E Kuzovlev, Zh. Eksp. Teor. Fiz. 72238G. N. Bochkov and Yu. E. Kuzovlev, Zh. Eksp. Teor. Fiz. 72, 238 (1977); . Sov. Phys. JETP. 45125Sov. Phys. JETP 45, 125 (1977). R L Stratonovich, Nonlinear nonequilibrium thermodynamics II. Advanced theory. BerlinSpringerR. L. Stratonovich, Nonlinear nonequilibrium thermodynamics II. Advanced theory (Springer, Berlin, 1994). Indeed, the equilibrium density matrix ρ is related to the unperturbed evolution U (t) = exp(−iH0t/ ) according to Zρ = U † (i β). Indeed, the equilibrium density matrix ρ is related to the unperturbed evolution U (t) = exp(−iH0t/ ) according to Zρ = U † (i β).
{'fraction_non_alphanumeric': 0.08296858908341916, 'fraction_numerical': 0.03662461380020597, 'mean_word_length': 3.7484718826405867, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'A universal quantum work relation is proved for isolated time-dependent Hamiltonian systems in a magnetic field as the consequence of microreversibility. This relation involves a functional of an arbitrary observable. The quantum Jarzynski equality is recovered in the case this observable vanishes. The Green-Kubo formula and the Casimir-Onsager reciprocity relations are deduced thereof in the linear response regime.', 'arxivid': '0805.3770', 'author': ['David Andrieux \nCenter for Nonlinear Phenomena and Complex Systems\nUniversité Libre de Bruxelles\nCode Postal 231, Campus PlaineB-1050BrusselsBelgium\n', 'Pierre Gaspard \nCenter for Nonlinear Phenomena and Complex Systems\nUniversité Libre de Bruxelles\nCode Postal 231, Campus PlaineB-1050BrusselsBelgium\n'], 'authoraffiliation': ['Center for Nonlinear Phenomena and Complex Systems\nUniversité Libre de Bruxelles\nCode Postal 231, Campus PlaineB-1050BrusselsBelgium', 'Center for Nonlinear Phenomena and Complex Systems\nUniversité Libre de Bruxelles\nCode Postal 231, Campus PlaineB-1050BrusselsBelgium'], 'corpusid': 6491830, 'doi': '10.1103/physrevlett.100.230404', 'github_urls': [], 'n_tokens_mistral': 5323, 'n_tokens_neox': 4753, 'n_words': 2735, 'pdfsha': 'd2b92741dd13b13efe30f2290b6035d6cc16c1cc', 'pdfurls': ['https://arxiv.org/pdf/0805.3770v1.pdf'], 'title': ['Quantum work relations and response theory', 'Quantum work relations and response theory'], 'venue': []}
arxiv
The Morley-type virtual element method for the Navier-Stokes equations in stream-function form on general meshes D Adak D Mora A Silgado The Morley-type virtual element method for the Navier-Stokes equations in stream-function form on general meshes Nonconforming virtual elementsStokes complexstream-function formenriching operatordiscrete Sobolev embeddingsoptimal error estimatesvelocity-vorticity-pressure recoverypolygonal meshes Mathematics subject classifications (2000): 65N3065N1276D0565N15 The nonconforming Morley-type virtual element method for the incompressible Navier-Stokes equations formulated in terms of the stream-function on simply connected polygonal domains (not necessarily convex) is designed. A rigorous analysis by using a new enriching operator is developed. More precisely, by employing such operator, we provide novel discrete Sobolev embeddings, which allow to establish the well-posedness of the discrete scheme and obtain optimal error estimates in broken H 2 -, H 1 -and L 2 -norms under minimal regularity condition on the weak solution. The velocity and vorticity fields are recovered via a postprocessing formulas. Furthermore, a new algorithm for pressure recovery based on a Stokes complex sequence is presented. Optimal error estimates are obtained for all the postprocessed variables. Finally, the theoretical error bounds and the good performance of the method are validated through several benchmark tests. Introduction The two dimensional steady Navier-Stokes equations in its standard velocity-pressure form reads as: given a sufficiently smooth force density f : Ω → R 2 , find (u, p) such that −ν∆u + (∇u)u + ∇p = f , div u = 0 in Ω, u = 0 on Γ := ∂Ω, (p, 1) 0,Ω = 0, (1.1) where u : Ω → R 2 is the velocity field, p : Ω → R is the pressure fields and ν > 0 represents the fluid viscosity. This system model the behaviour of a viscous incompressible fluid in the domain Ω. The first and second equations in (1.1) dictates the momentum and mass conservation of the fluid, while the third identity indicates non-slip boundary conditions for the velocity field and the last equation represents the mean value of p over Ω vanishing, which is used for the uniqueness of the pressure solution. Due to the important role it plays in the study of viscous incompressible flows, several numerical schemes have been developed to efficiently approximate the Navier-Stokes system. In particular, we are interested in discretizing this system by using general polygonal decompositions and introducing the stream-function of the velocity field. In the last years, numerical methods for PDEs on polytopal meshes have received substantial attention. Different approaches have been proposed (see for instance [13] and the references therein), offering significant flexibility in terms of dealing with complicated domains. Among them, we can find the Virtual Element Method (VEM), which was presented for first time in [11], as an evolution of mimetic finite differences and a generalization of the Finite Element Method (FEM). The approach of VEM allows avoid an explicit construction of the discrete shape functions and this fact implies a high flexibility of the method, which is reflected, for instance in the ability to construct numerical schemes of high-order on general polygonal meshes (including "hanging vertexes" and nonconvex shapes). Moreover, in the construction of discrete spaces with high-regularity and of schemes with the divergence-free property (in the context of fluid problems). In virtue of these features, the VEM technology has enjoyed extensive success in numerical modeling and engineering applications, both in its conforming and nonconforming approaches (see for instance [19,21,12,8,15,10,22]). In particular, many works have been devoted to solving problems in fluid mechanics by using the VEM. Below are two representative list works in the conforming and nonconforming cases; [7,33,16,14,2] and [20,39,50,38], respectively. For a current state of the art on VEM, we refer to book [6]. In [9,49] the authors have introduced fully-nonconforming VEMs of high-order, independently and by using different approaches to solve biharmonic problems. In particular, the lowest-order configuration (i.e., k = 2) of these VEMs, can be consider as the extension of the popular Morley FE [43] to general polygonal meshes. Since then, several schemes and analysis based on these VEMs have been developed for linear problems; see for instance [28,37,47,30,35,23,3]. In the present work we are interesting to extend the Morley-type VEM to solve the nonlinear fourth-order Navier-Stokes equations in stream-function form on simply connected domains (not necessarily convex) by using general polygonal decompositions. Typically, the velocity-pressure formulation (2.1) is the most used to discretize the Navier-Stokes problem. However, the stream-function formulation has shown to be a competitive alternative to discretize flow problems, which has been the focus of study in the last decades. In particular, we can highlight the following features: the system is reduced in a singular scalar weak formulation, with automatic satisfaction of the incompressibility constrain (the velocity field is equal to the curl of the stream-function), the possibility to recover further variables of interest such as the velocity, vorticity and pressure fields by postprocessing from the stream-function. Besides, for nonlinear problems, the resulting trilinear form is naturally skew-symmetric, allowing more direct stability and convergence arguments. On the other hand, the stream-function approach avoid the difficulties related with the boundary values for the vorticity field, which are present in stream-function-vorticity formulation. Due to the attractive features discussed above, over past decades the stream-function formulation has received great attention from many researchers. In particular, in the area of Numerical Analysis several works have been devoted to the development and study of efficient numerical schemes to approximate this system. For instance; conforming and nonconforming FEMs in [25,26,32,24], bivariate spline [41], hp-version discontinuous FE [44], NURBS-based Isogeometric Analysis in [46]. Moreover, in [36] the nonconforming Morley FEM have been used to solve the steady Quasi-Geostrophic equations, which can be seen as an extension (in form) of the two dimensional Navier-Stokes equations in stream-function formulation. In the present contribution we configure the Stokes complex structure of the nonconforming VEM introduced in [50] to solve the fourth-order nonlinear Navier-Stokes equations in stream-function form on domains not necessarily convex and employing general polygonal partitions of the domain, allowing additionally the reconstruction of the pressure field. By using the enhancement technique, we introduce a discrete Stokes complex structure associate to the Morley-and Crouzeix-Raviart-type VE spaces. Then, we construct suitable projections useful to build the discrete trilinear form, which mimics the interesting and naturally skew-symmetry property of the continuous version. In order to establish the well-posedness of the discrete nonconforming formulation, is necessary to prove the continuity of the resulting discrete trilinear form respect to the natural norm in the Morley-type VE space M h . However, this fact does not follow directly, since it involves a discrete Sobolev inclusion (namely, M h ⊂ W 1,4 (Ω)). The derivation of the Sobolev embeddings require particular attention for the nonconforming approach, which is usually considered a challenging task. To the best of our knowledge, this is the first work where Sobolev embeddings for the Morley-type VE space are established. More precisely, with the aim of achieving such purpose, we introduce a novel enriching operator, which is a special kind of quasi-interpolation operator that maps the elements of the sum space between the continuous and nonconforming spaces (namely, Φ + M h ) to the conforming counterpart of the nonconforming space. Then, by using this operator and its approximation properties we provide new discrete Sobolev embeddings for the sum space Φ+M h and we prove the well-posedness of the discrete problem by using the fixed point Banach Theorem. It well know that due to nonconformity of the space increases the technicalities in the demonstrations of error estimates in the nonconforming approach, implying in some cases high-regularity of the solution, which are not realistic. Furthermore, for nonlinear problems these difficulties increase remarkably. In the present work, by employing the naturally skew-symmetry property of the discrete trilinear form and the discrete Sobolev inclusion, we write elegantly an abstract convergence result for the nonlinear VE scheme. Then, by exploiting again the enriching operator, we establish key approximation properties involving the bilinear and trilinear forms, together with the consistency errors, allowing the derivation of an optimal error estimate in broken H 2 -norm under the minimal regularity condition on the weak stream-function solution (see below Theorem 2.2). In addition, by using duality arguments and the enriching operator we also provided new optimal error estimates in the H 1 -and L 2 -norm under the same regularity condition on the stream-function and the density force. On the another hand, by exploiting the stream-function approach, we present techniques to recover further variables of physical interest, such as, the primitive velocity and pressure variables, along with the important vorticity field. More precisely, we recover the velocity and vorticity fields through a postprocess of the discrete stream-function by using adequate polynomial projections, which are directly computable from the degrees of freedom. The pressure recovery procedure require a special attention . Indeed, we approximate the fluid pressure by exploiting the Stokes complex sequence associate to the Morley-and Crouzeix-Raviart-type VE spaces, and solving an additional Stokes-like system with right hand side coming from the virtual stream-function solution and the force density f . For all the postprocessed variables, we provide optimal a priori error estimates. Furthermore the numerical method is tested with several benchmark tests, including the Kovasznay and cavity problems, where the theoretical accuracy and the good performance of the scheme are corroborated. Finally, we expect that the results reported here constitute a stepping-stone towards for the development and analysis of new numerical schemes based on the Morley-type VEM, for solving fourth-order related problems, allowing the derivation of optimal order error estimates in different broken norms, under less regularity assumptions of the solution in more complicated situations, such as, nonlinear coupled and/or time dependent systems. The outline of the remaining parts of this paper reads as follows: in Section 2 we introduce some preliminaries notations and the stream-function weak formulation of the Navier-Stokes problem (1.1). Moreover, we recall its well-posedness and regularity property. The Morley-type VE discretization, together with the Crouzeix-Raviart VE space are described in Section 3. In Section 4 we introduce the enriching operator, provide the discrete Sobolev embeddings and the well-posedness of the discrete problem by using a fixed-point strategy. In Section 5 we develop the error analysis of the scheme under minimal regularity condition on the weak solution. In Section 6 we describe the recovery techniques for the velocity, vorticity and pressure fields by using the discrete streamfunction solution. Finally, several numerical tests on different polygonal meshes are reported in Section 7. Preliminaries and continuous weak form 2.1 Notations In this subsection we introduce notations that we will use along the paper, including those already employed above. We will follow the standard notations of Sobolev spaces and their respective seminorms and norms according [4]. Hence, for every open bounded domain D, the seminorms and norms in the spaces L q (D) and W ,q (D) (with ≥ 0 and q ∈ [1, +∞)), are denoted by | · | ,q,D and · ,q,D , respectively. We adopt the usual convention W 0,q (D) := L q (D). In particular when q = 2, we write H (D) instead to W ,2 (D) and the corresponding convention for the seminorms and norms is also adopted, i.e., | · | ,D and · ,D , respectively. For any tensor fields τ = (τ ij ) i,j=1,2 and σ = (σ ij ) i,j=1,2 , we consider the standard scalar product of 2 × 2matrices: τ : σ = 2 i=1 τ ij σ ij and for simplicity the scalar, vectorial and tensorial L 2 -inner products will be denoted by (ϕ, φ) 0,D = D ϕφ (v, w) 0,D = D v · w (τ , σ) 0,D = D τ : σ. Moreover, with the usual notations, for scalar functions the symbols ∇, ∆, ∆ 2 and D 2 denote the gradient, Laplacian, Bilaplacian operators and the Hessian matrix, respectively, while the bold symbols ∇ and ∆ denote the gradient and Laplacian operators for vector fields, respectively. In addition, for smooth scalar and vectorial functions φ and v = (v 1 , v 2 ), we define the curl, divergence and rotational operators, as follow: curl φ := ∂ y ϕ −∂ x ϕ , div v := ∂ x v 1 + ∂ y v 2 , and rot v := ∂ x v 2 − ∂ y v 1 . Henceforth, Ω will denote a simply connected bounded domain of R 2 with polygonal Lipschitz boundary Γ := ∂Ω. The symbol n = (n i ) 1≤i≤2 is the outward unit normal vector to the boundary Γ, while the vector t = (t i ) i=1,2 is the unit tangent to Γ oriented such that t 1 = −n 2 , t 2 = n 1 . Moreover, ∂ n φ = ∇φ · n and ∂ t φ = ∇φ · t denote the normal and tangential derivatives, respectively. In addition, c or C, with or without subscripts, will represent a generic constant, which is independent of the mesh parameter h that might have distinct values at different places. The Navier-Stokes in velocity-pressure weak form. The standard variational formulation of problem (1.1) reads as: find (u, p) ∈ H × Q, such that ν(∇u, ∇v) 0,Ω + ((∇u)u, v) 0,Ω − (p, div v) 0,Ω = (f , v) 0,Ω ∀v ∈ H, −(g, div u) 0,Ω = 0 ∀g ∈ Q, (2.1) where the Hilbert spaces H and Q are defined by: H := v ∈ H 1 (Ω) 2 : v = 0 on Γ and Q := g ∈ L 2 (Ω) : (g, 1) 0,Ω = 0 . (2.2) It is well known that problem (2.1) admits a unique solution (see [34]) under smallness assumption on the data. Moreover, several works have been devoted to develop numerical schemes to approximate this formulation. For instance, see [14,40,39,48] in the VEM context. In this work, we will study the Navier-Stokes equations with a different approach. More precisely, under assumption that the domain is simply connected and by using the incompressibility condition of the velocity field (i.e., div u = 0), we write an equivalent variational formulation in terms of the stream-function of the velocity field. The stream-function weak form Since Ω ⊂ R 2 is simply connected, is well known that a vector function v ∈ Z := {v ∈ H : div v = 0} if and only if there exists a function ϕ ∈ H 2 (Ω) (called stream-function), such that v = curl ϕ. Let us consider the following Hilbert space Φ := ϕ ∈ H 2 (Ω) : ϕ = 0, ∂ n ϕ = 0 on Γ , and we endow this space with the norm ϕ 2,Ω := D 2 ϕ, D 2 ϕ 1/2 0,Ω ∀ϕ ∈ Φ. Then, we have that a variational formulation of problem (1.1), formulated in terms of stream-function, read as (see for instance [45,Section 10.4]): From the definition of the bilinear form A(·, ·) and equivalence of norms, we obtain its Φ-ellipticity. Moreover, by using the Cauchy-Schwarz inequality is easily obtain: given f ∈ L 2 (Ω) 2 , find ψ ∈ Φ, such that νA(ψ, φ) + B(ψ; ψ, φ) = F (φ) ∀φ ∈ Φ, (2.3) where the multilineal forms A : Φ × Φ → R, B : Φ × Φ × Φ → R and F : Φ → R are defined by: A(ψ, φ) := (D 2 ψ, D 2 φ) 0,Ω ,(2.|A(ϕ, φ)| ≤ ϕ 2,Ω φ 2,Ω ∀ϕ, φ ∈ Φ, |F (φ)| ≤ C F f 0,Ω φ 2,Ω ∀φ ∈ Φ, where C F is a positive constant. Now, we recall the following continuous Sobolev inclusion: for all v ∈ H 1 (Ω) 2 , there exists C sob > 0 such that v L 4 (Ω) ≤ C sob v 1,Ω . (2.7) Then, by using the Hölder inequality and the above inclusion, there exists C B := C 2 sob > 0, such that |B(ζ; ϕ, φ)| ≤ C B ζ 2,Ω ϕ 2,Ω φ 2,Ω ∀ζ, ϕ, φ ∈ Φ. From the above properties and the fixed-point Banach Theorem, we can prove that problem (2.3) is well-posed. More precisely, we have the following existence and uniqueness result (see for instance, [34, Chapter IV, Section 2.2]). Theorem 2.1 If C B C F ν −2 f 0,Ω < 1, then there exists a unique ψ ∈ Φ solution to problem (2.3), which satisfies the following continuous dependence on the data ψ 2,Ω ≤ C F ν −1 f 0,Ω . Now, we state an additional regularity result for the solution of problem (2.3) (see for instance [18]). Theorem 2.2 Let ψ ∈ Φ be the unique solution of problem (2.3). Then, there exist γ ∈ (1/2, 1] and C reg > 0, such that ψ ∈ H 2+γ (Ω) and ψ 2+γ,Ω ≤ C reg f 0,Ω . Morley-type virtual element approximation This section is devoted to the construction of a VEM to solve problem (2.3). We will introduce a Morley-type VE space by using some auxiliaries local virtual spaces and the enhancement technique. More precisely, the present framework is based on the discrete Stokes complex sequence for the Morley-and Crouzeix-Raviart-type VE spaces presented in [50]. This Stokes complex structure will allow us to approximate the main unknown in problem (2.3) and as an important topic, also it will allow to compute the pressure variable of the Navier-Stokes system (1.1) as a postprocess, by solving a Stokes-like problem with right hand side coming form the discrete-stream function solution and force density f (cf. subsection 6.3). We start with a subsection introducing the polygonal decompositions and some useful notations, these preliminaries are following by a subsection on the local and global nonconforming virtual spaces, their degrees of freedom and the classical VEM local projectors. Later on, we introduce other polynomial projections useful to build the discrete trilinear form. The polygonal decompositions and basic setting Let {T h } h>0 be a sequence of decompositions of Ω into general non-overlapping simple polygons K, where h := max K∈T h h K and h K is the diameter of K. We will denote by ∂K, N K and |K| the boundary, the number of vertices and area of each polygon K, respectively. For each element K we denote by E K h the set of its edges, while the set of all the edges in T h will be denote by E h . We decompose this set as the following union: E h := E int h ∪ E bdry h , where E int h and E bdry h are the set of interior and boundary edges, respectively. For the set of all the vertices we have an analogous notation. More precisely, we will denote by V h := V int h ∪ V bdry h the set of vertices in T h , where V int h and V bdry h are the set of interior and boundary vertices, respectively. In addition, we denote by e a generic edge of E h and by h e its length. Besides, for each K ∈ T h , we denote by n K its unit outward normal vector and by t K its tangential vector along the boundary ∂K. Moreover, we will adopt the notation n e and t e for a unit normal and tangential vector of an edge e ∈ E h , respectively. For every > 0 and q ∈ [1, +∞), we define the following broken Sobolev spaces W ,q (T h ) := {φ ∈ L 2 (Ω) : φ| K ∈ W ,q (K) ∀K ∈ T h }, and we endow these spaces with the following broken seminorm: |φ| ,q,h := K∈T h |φ| q ,q,K 1/q , where | · | ,q,K is the usual seminorm in W ,q (K). When q = 2, we omit q and write H (T h ) instead W ,2 (T h ), with the corresponding seminorm denoted by | · | ,h . Next, we will define the jump operator. First, for each φ h ∈ H 2 (T h ), we denote by φ ± h the trace of φ h | K ± , with e ⊂ ∂K + ∩ ∂K − . Then, the jump operator [[·]] is defined as follows: [[φ h ]] := φ + h − φ − h for every e ∈ E int h , φ h | e for every e ∈ E bdry h . The same notation is adopted for vectorial fields. Let us define a subspace of H 2 (T h ) with certain continuity: H 2,NC (T h ) := φ h ∈ H 2 (T h ) : φ h ∈ C 0 (V int h ), φ h (v i ) = 0 ∀v i ∈ V bdry h , ([[∂ ne φ h ]], 1) 0,e = 0 ∀e ∈ E h , where C 0 (V int h ) is the set of functions continuous at internal vertexes. Finally, for each subset D ⊂ R 2 and every integer ≥ 0, P (D) is the space of polynomials of degree up to defined on D. Furthermore, the piecewise -order polynomial space is defined by: P (T h ) := {χ ∈ L 2 (Ω) : χ| K ∈ P (K) ∀K ∈ T h }. In what follows, we will introduce some preliminary spaces, which are useful to construct the Morley-type VE space to approximate the solution of problem (2.3). Some auxiliary spaces For every polygon K ∈ T h , first we consider the following auxiliary finite dimensional space [9,49,37]: M h (K) := φ h ∈ H 2 (K) : ∆ 2 φ h ∈ P 2 (K), φ h | e ∈ P 2 (e), ∆φ h | e ∈ P 0 (e) ∀e ∈ ∂K . Next, for a given φ h ∈ M h (K), we introduce the following sets: • D M 1: the values of φ h (v i ) for all vertex v i of the polygon K; • D M 2: the edge moments (∂ ne φ h , 1) 0,e ∀ edge e ∈ E K h . For each polygon K, we define the following projector Π D K : M h (K) → P 2 (K) ⊆ M h (K), as the solution of the local problems: A K (Π D K φ h , χ) = A K (φ h , χ) ∀χ ∈ P 2 (K), Π D K φ h , χ K = φ h , χ K ∀χ ∈ P 1 (K), where ϕ h , φ h K is defined as follows: ϕ h , φ h K := N K i=1 ϕ h (v i )φ h (v i ), with v i , 1 ≤ i ≤ N K , being the vertices of K and A K (·, ·) is the restriction of the continuous form A(·, ·) (cf. (2.4)) on the element K. The operator Π D K : M h (K) → P 2 (K) is explicitly computable for every φ h ∈ M h (K), using only the information of the linear operators D M 1 − D M 2 (for further details, we refer to [49]). Now, we will introduce another auxiliary local spaces. Indeed, following [50] we define the spaces: U (K) := v h ∈ H 1 (K) 2 : div v h ∈ P 0 (K), rot v h ∈ P 0 (K), v h · n e ∈ P 1 (e) ∀e ∈ E K h , and Z(K) := φ ∈ H 2 (K) : ∆ 2 φ h = 0, φ h | e = 0, ∆φ h | e ∈ P 0 (e) ∀e ∈ E K h . By adding U (K) and curl of the functions belongs to Z(K), we define the space U 0 (K) := U (K) + curl ( Z(K)). Then, for each v h ∈ U 0 (K) we introduce the set of vector-valued, bounded linear functional • D U : the edge moments h −1 e (v h , 1) 0,e ∀e ∈ E K h . We observe that P 1 (K) 2 ⊂ U 0 (K), and we introduce the grad projection operator Π ∇ K : U 0 (K) → P 1 (K) 2 as the solution of the following problem: (∇(Π ∇ K v h − v h ), ∇χ) 0,K = 0 ∀χ ∈ P 1 (K) 2 , (Π ∇ K v h − v h , 1) 0,∂K = 0. (3.1) By using an integration by parts, we can deduce that the polynomial [50]). Π ∇ K v h is computable for all v h ∈ U 0 (K) from the set of values D U (see Next, by employing the grad projection operator Π ∇ K , we define the local Crouzeix-Raviart-like VE space U h (K) := v h ∈ U 0 (K) : (v h · n e − Π ∇ K v h · n e , χ) 0,e ∀χ ∈ P 1 (e) \ P 0 (e), ∀e ∈ E K h . Further, from [50] we have that the set D U characterize uniquely the functions of U h (K). Moreover, for each φ h ∈ M h (K), the function Π ∇ K curl φ h is computable using the sets D M 1 and D M 2. The global Crouzeix-Raviart-like space is defined as follows [50]: U h := v h ∈ L 2 (Ω) 2 : v h | K ∈ U h (K) ∀K ∈ T h , ([[v h ]], 1) 0,e = 0 ∀e ∈ E h . (3.2) We have that the dimension of the space U h is equal to 2N E h , where N E h is the total number of mesh edges of the discretization T h . This space will be useful in subsection 6.3 to present the pressure recovery technique. Remark 3.1 The nonconforming VE space defined in (3.2) coincides with the Crouzeix-Raviart finite element space when the polygon K is a triangle. Therefore, this space can be seen as an extension of the classical Crouzeix-Raviart space from triangle to polygonal element in the nonconforming VEM context. For further details of this discussion, see [50,Remark 8]. The Morley-type nonconforming virtual element space By using the auxiliary spaces defined in the above subsection, for each K ∈ T h we introduce the local Morley-type VE space [50]: M h (K) := φ h ∈ M h (K) : (curl φ h · n e − Π ∇ K (curl φ h · n e ), χ) 0,e = 0 ∀χ ∈ P 1 (e) \ P 0 (e) ∀e ∈ E K h , (φ h − Π D K φ h , χ) 0,K = 0 ∀χ ∈ P 2 (K) . (3.3) In the next result we summarize the main properties of the local Morley-type VE space. Lemma 3.1 For each polygons K, the space M h (K) defined in (3.3), we have P 2 (K) ⊆ M h (K) . Moreover, we can deduce the following properties: • The linear operators D M 1 − D M 2 constitutes a set of degrees of freedom for M h (K); • The operator Π D K : M h (K) → P 2 (K) is computable using the sets D M 1 − D M 2; • For each φ h ∈ M h (K), the function Π ∇ K curl φ h is computable using the degrees of freedom D M 1 − D M 2. With the above preliminaries we can introduce the global Morley-type VE space to the numerical approximation of the problem (2.3). Indeed, for every decomposition T h of Ω into polygons K, the global nonconforming VE space is given by: M h := φ h ∈ H 2,NC (T h ) : φ h | K ∈ M h (K), ∀K ∈ T h . (3.4) We have that M h ⊂ H 2,NC (T h ), but M h Φ. Moreover, we observe that the nonconforming VE does not require that the C 0 -continuity over Ω. This space can be seen as an extension of the popular Morley FE [43] to general polygonal meshes. For further details about this discussion, we refer to [50,Remark 20] and [49,Remark 4.1]. For the continuous bilinear form A(·, ·), we adopt the following notation: A(ϕ h , φ h ) := K∈T h A K (ϕ h , φ h ) ∀ϕ h , φ h ∈ Φ + M h . We also adopt the same notation by the continuous forms B(·; ·, ·) and F (·). Polynomial projection operators and discrete multilinear forms This subsection is dedicated to the presentation of other important polynomial projections, along with the construction of the trilinear form and the load term, by using such projections. Moreover, we build the bilinear discrete form. For each m ∈ N ∪ {0}, we consider the usual L 2 -projection, Π m K : L 2 (K) → P m (K), defined by the function such that (φ − Π m K φ, χ) 0,K = 0 ∀χ ∈ P m (K). (3.5) Moreover, we define its vectorial Π m K version in an analogous way. For the projection previously defined we have the following result. We recall that there exists C bd > 0 such that (see [14]): Π m K φ L 4 (K) ≤ C bd φ L 4 (K) and Π m K φ 0,K ≤ φ 0,K ∀φ ∈ L 2 (K). (3.6) Lemma 3.2 Let Π 2 K , Π 0 K and Π 1 K be the operators defined by relation (3.5) and by its vectorial version. Then, for each φ h ∈ M h (K), the polynomial functions Π 2 K φ h , Π 0 K ∆φ h , Π 1 K curl φ h and Π 1 K ∇φ h are computable using only the information of the degrees freedom D M 1 − D M 2. Proof. Let φ h ∈ M h (K), the proof of the function Π 2 K φ h follows from the definition of the space M h (K) (cf. (3. 3)). Moreover, using integration by parts we obtain (curl φ h , χ) 0,K = rot χ(Π 2 K φ h , 1) 0,K − (φ h , χ · t K )) 0,e ∀χ ∈ P 1 (K) 2 , then we also conclude that the Π 1 K curl φ h is fully computable from the degrees of freedom. Similarly, we prove that function Π 1 K ∇φ h is computable from the degrees of freedom D M 1 − D M 2. Next, we will prove that the polynomial function Π 0 K ∆φ h is also computable. Indeed, using integration by parts, we have Π 0 K ∆φ h = |K| −1 (∂ n K φ h , 1) 0,∂K = |K| −1 e∈∂K (∂ ne φ h , 1) 0,e , and note that the above integral is computable using the output values of the set D M 2. In this part, we will build the discrete version of the continuous forms defined in (2.4), (2.5) and (2.6) using the operators introduced previously. First, we consider the following discrete local bilinear form, A K h : M h (K) × M h (K) → R approximating the continuous form A(·, ·): A K h (ϕ h , φ h ) := A K Π D K ϕ h , Π D K φ h + S K D (I − Π D K )ϕ h , (I − Π D K )φ h ∀ϕ h , φ h ∈ M h (K). (3.7) where S K D (·, ·) is any symmetric positive definite bilinear form to be chosen as to satisfy: c * A K (φ h , φ h ) ≤ S K D (φ h , φ h ) ≤ c * A K (φ h , φ h ) ∀φ h ∈ Ker(Π D K ), (3.8) with c * and c * positive constants independent of K. More precisely, we choose the following computable representation satisfying property (3.8) (see [23, Lemma 5.1]): S K D (ϕ h , φ h ) := h −2 K N K dof i=1 dof i (ϕ h )dof i (φ h ) ∀ϕ h , φ h ∈ M h (K), where N K dof denote the number of degrees freedom of M h (K) and dof i (·) is the operator that to each smooth enough function φ associates the ith local degree of freedom dof i (φ), with 1 ≤ i ≤ N K dof . To approximate the local trilinear form B K (·; ·, ·), we consider the following expression: B K h (ζ h ; ϕ h , φ h ) := Π 0 K ∆ζ h Π 1 K curl ϕ h , Π 1 K ∇φ h 0,K ∀ζ h , ϕ h , φ h ∈ M h (K). (3.9) Finally, for the functional (2.6) we consider the following local approximation: F K h (φ h ) := (Π 1 K f , curl φ h ) 0,K ≡ (f , Π 1 K curl φ h ) 0,K ∀φ h ∈ M h (K). Thus, for all ζ h , ϕ h , φ h ∈ M h , we define the global multilineal forms, as follows: A h : M h × M h → R, A h (ϕ h , φ h ) := K∈T h A K h (ϕ h , φ h ), (3.10) B h : M h × M h × M h → R, B h (ζ h ; ϕ h , φ h ) := K∈T h B K h (ζ h ; ϕ h , φ h ), (3.11) F h : M h → R, F h (φ h ) := K∈T h F K h (φ h ). (3.12) We recall that all the forms defined above are computable using the degrees freedom and the trilinear form B h (·; ·, ·) is extendable to the whole Φ. Now, we establish the classical consistency and stability VEM properties (see [11,10,21,50]). Lemma 3.3 The local bilinear forms A K (·, ·) and A K h (·, ·) satisfy the following properties: • consistency: for all h > 0 and for all K ∈ T h , we have that A K h (χ, φ h ) = A K (χ, φ h ) ∀χ ∈ P 2 (K), ∀φ h ∈ M h (K),(3. 13) • stability and boundedness: there exist positive constants α 1 and α 2 , independent of h and K, such that: α 1 A K (φ h , φ h ) ≤ A K h (φ h , φ h ) ≤ α 2 A K (φ h , φ h ) ∀φ h ∈ M h (K). (3.14) 4 Discrete formulation and its well-posedness In this section we write the nonconforming discrete VE formulation and we provide its well-posedness by using a fixed-point strategy. The nonconforming VE problem reads as: find ψ h ∈ M h , such that νA h (ψ h , φ h ) + B h (ψ h ; ψ h , φ h ) = F h (φ h ) ∀φ h ∈ M h , (4.1) where the multilineal forms A h (·, ·), B h (·; ·, ·) and F h (·) are defined in (3.10), (3.11) and (3.12), respectively. In order to prove that problem (4.1) is well-posed, in next section, we will introduce an enriching operator E h , from the sum space Φ + M h into the conforming counterpart of the space M h . Moreover, we establish some approximation properties for this operator, and by using such estimates we provide novel embedding results for the sum space Φ + M h , which will be useful to establish the well-posedness of discrete problem and the error estimates. We remark that the operator E h constructed here can be seen as an extension of the enriching operator defined in [35] and the quasi-interpolation operator constructed in [29]. A new enriching operator With the aim of introducing the aforementioned operator and establish its approximation properties, we start by assuming the classical assumptions on the polygonal decomposition. There exists a uniform number ρ > 0 independent of T h , such that for every K ∈ T h it holds [11]: A 1 : K is star-shaped with respect to every point of a ball of radius ≥ ρh K ; A2 : the length h e of every edge e ∈ ∂K, satisfies h e ≥ ρh K . From reference [27] we have that if the mesh T h fulfilling the assumptions A 1 and A 2 , then the mesh also satisfy the following property: P 1 : For each K ∈ T h , there exists a virtual triangulation T K h of K such that T K h is uniformly shape regular and quasi-uniform. The corresponding mesh size h T of T K h is proportional to h K . Every edge of K is a side of a certain triangle in T K h . Remark 4.1 From property P 1 , we have that the number of triangles of each virtual triangulation T K h is uniformly bounded by a number L and the size of each triangle is comparable to that of the polygon (for further details, see [27]). Now, for the sake of completeness, we will recall the construction of the H 2 -conforming virtual space [8]. Conforming virtual local and global space. For every polygon K ∈ T h , we introduce the following preliminary finite dimensional space [8]: W C h (K) := φ h ∈ H 2 (K) : ∆ 2 φ h ∈ P 2 (K), φ h | ∂K ∈ C 0 (∂K), φ h | e ∈ P 3 (e) ∀e ⊆ ∂K, ∇φ h | ∂K ∈ C 0 (∂K) 2 , ∂ ne φ h | e ∈ P 1 (e) ∀e ⊆ ∂K , Next, for a given φ h ∈ W C h (K), we introduce two sets D v 1 and D ∇ 2 of linear operators from the local virtual space W C h (K) into R: • D C v : the values of φ h (v) for all vertex v of the polygon K; • D C ∇ : the values of h vi ∇φ h (v) for all vertex v of the polygon K, where h v is a characteristic length attached to each vertex v, for instance to the average of the diameters of the elements with v as a vertex. Now, we consider the operator Π D,C K : W C h (K) −→ P 2 (K) ⊆ W C h (K) associated to the conforming approach, which is computable using the sets D C v and D C ∇ (for further details see [8, Lemma 2.1]). Next, for each K ∈ T h , we consider the conforming local virtual space given by: W C h (K) := φ h ∈ W C h (K) : (φ h − Π D,C K φ h , χ) 0,K = 0 ∀χ ∈ P 2 (K) . For every decomposition T h of Ω into polygons K, we define the conforming virtual spaces W C h : W C h := φ h ∈ Φ : φ h | K ∈ W C h (K) ∀K ∈ T h . We recall that the global DOFs are defined by D C v and D C ∇ excluding the DOFs on the boundary Γ. Construction of the Enriching operator. We will extend the ideas of [35,29]. First, we will introduce some additional notations. Indeed, for each vertex v ∈ V h and for all e ∈ E h we define the following sets (patches): ω(v) := K ∈ T h : v ∈ K and ω(e) := K ∈ T h : e ∈ ∂K . Moreover, for each K ∈ T h we define ω(K) := K ∈ T h : K ∩ K = ∅ , and for a function φ h ∈ H 2 (T h ), we defined the following broken seminorm |φ h | 2 2,ω(K),h := K∈ω(K) |φ h | 2 2, K 1/2 . We will denote by N (v) and by N (e) the number of elements in ω(v) and ω(e), respectively. In addition, for (3.5). any ϕ h ∈ Φ + M h , we introduce the piecewise L 2 -projection Π 2 , as Π 2 ϕ h | K = Π 2 K (ϕ h | K ), where Π 2 K is the usual L 2 -projection onto P 2 (K) defined inLet N C dof := dim(W C h ) , then as in [35,29] we can relabel the degrees of freedom using a single subindex j = 1, . . . , N C dof and will denote the degrees of freedom by {D C j } N C dof j=1 , which are associated with the shape basis functions {ζ j } N C dof j=1 of the space W C h . Employing this notation the enriching operator E h : Φ + M h → W C h is defined by: E h ϕ h (x) = N C dof j=1 D C j ( E h ϕ h )ζ j (x), where the degrees of freedom for E h ϕ h are determined by: 1. D C 1,v ( E h ϕ h ) = E h ϕ h (v) := ϕ h (v) ∀v ∈ V int h ; 2. D C 2,v ( E h ϕ h ) := 1 N (v) K∈ω(v) h v ∇(Π 2 ϕ h | K (v)) ∀v ∈ V int h . The following result establishes approximation properties of the enriching operator E h . Proposition 4.1 For all φ h ∈ Φ + M h , there exists C > 0 independent of h, such that 2 j=0 h 2j K |φ h − E h φ h | 2 j,K ≤ Ch 4 K |φ h | 2 2,ω(K),h ∀K ∈ T h . Proof. First, we note that using the same arguments used in [35,Lemma 4.2] and [29, Lemma 4.1] (see also [3]), for all φ h ∈ Φ + M h , we have that φ h − E h φ h 0,K ≤ Ch 2 K |φ h | 2,ω(K),h and |φ h − E h φ h | 2,K ≤ C|φ h | 2,ω(K),h . (4.2) Now, by using standard inequality (see [35, equation (3.3)]) and (4.2), there exists a constant C > 0, independent to h K , such that |φ h − E h φ h | 1,K ≤ C(h K |φ h − E h φ h | 2,K + h −1 K φ h − E h φ h 0,K ) ≤ C(h K |φ h | 2,ω(K),h + h 2 K h −1 K |φ h | 2,ω(K),h ) ≤ Ch K |φ h | 2,ω(K),h . (4.3) The desired result follows from (4.2) and (4.3). Discrete Sobolev embeddings and properties of the discrete forms In this subsection we establish two important estimates, which are useful to prove the continuity of the discrete multilineal forms. We start presenting the main result of this section, which establishes discrete Sobolev embeddings for the space Φ + M h . |φ h | 1,q,h ≤ C sob |φ h | 2,h ∀φ h ∈ Φ + M h . Proof. Let 2 ≤ q < ∞, φ h ∈ Φ + M h and E h : Φ + M h → W C h be the enriching operator defined in the above subsection. Then, by using the triangle inequality, the embedding of H 2 (Ω) into W 1,q (Ω) and stability property in Proposition 4.1, we have that |φ h | 1,q,h ≤ |φ h − E h φ h | 1,q,h + | E h φ h | 1,q,Ω ≤ |φ h − E h φ h | 1,q,h + C| E h φ h | 2,Ω ≤ |φ h − E h φ h | 1,q,h + C|φ h | 2,h . (4.4) In what follows we will estimate the term |φ h − E h φ h | 1,q,h in the right-hand side of (4.4). To do that, for each K ∈ T h , we consider the sub-triangulation T K h of property P 1 . Next, let ϕ := ∇(φ h − E h φ h )| K and ϕ be the image of ϕ under the affine transformation from T to the reference triangle T . Then, by using scaling arguments and the embedding of H 1 ( T ) into L q ( T ), there is C > 0 independent of K, such that |φ h − E h φ h | 1,q,T = ϕ L q (T ) ≤ C|T | 1/q ϕ L q ( T ) ≤ C|T | 1/q ϕ 1, T ≤ C|T | (2−q)/2q ( ϕ 2 0,T + h 2 T |ϕ| 2 1,T ) 1/2 ≤ C(h 2 T ) (2−q)/2q (|φ h − E h φ h | 2 1,T + h 2 T |φ h − E h φ h | 2 2,T ) 1/2 ≤ Ch (2−q)/q K (|φ h − E h φ h | 2 1,K + h 2 K |φ h − E h φ h | 2 2,K ) 1/2 , where we have used the relation |T | ≈ h 2 T and that the size of each triangle in T K h is comparable with the polygon mesh size h K (see Remark 4.1). Now, from the above estimate and Proposition 4.1 it holds |φ h − E h φ h | 1,q,T ≤ Ch (2−q)/q K h K |φ h | 2,ω(K),h ≤ Ch 2/q K |φ h | 2,ω(K),h . (4.5) From bound (4.5) and since the number of triangles of each virtual triangulation T K h is uniformly bounded by a number L (see again Remark 4.1), we obtain |φ h − E h φ h | q 1,q,K = T ∈K |φ h − E h φ h | q 1,q,T ≤ C T ∈K h 2 K |φ h | q 2,ω(K),h ≤ CLh 2 K |φ h | q 2,ω(K),h . Summing over each K ∈ T h , using the fact that q ≥ 2 and a q -norms inequality, along with 0 < h ≤ C < 1, we obtain |φ h − E h φ h | 1,q,h = K∈T h |φ h − E h φ h | q 1,q,K 1/q ≤ Ch 2/q K∈T h |φ h | q 2,ω(K),h 1/q ≤ Ch 2/q K∈T h |φ h | 2 2,ω(K),h 1/2 ≤ Ch 2/q |φ h | 2,h ≤ C|φ h | 2,h ,(4.6) where the constant C > 0 is independent of h. Finally, combining the estimates (4.4) and (4.6) we conclude the proof. φ h 0,Ω + |φ h | 1,h ≤ C|φ h | 2,h , where C > 0 is a constant independent of h. The following lemma summarize other properties of the discrete forms defined in (3.10)-(3.12), which will be used to establish the well-posedness of the discrete problem. Lemma 4.2 There exist positive constants C A h , α, C B h , C F h , independent of h, such that for all ζ h , ϕ h , φ h ∈ M h the forms defined in (3.10)-(3.12) satisfies the following properties: |A h (ϕ h , φ h )| ≤ C A h |ϕ h | 2,h |φ h | 2,h and A h (φ h , φ h ) ≥ α|φ h | 2 2,h , (4.7) B h (ζ h ; ϕ h , φ h ) ≤ C B h |ζ h | 2,h |ϕ h | 2,h |φ h | 2,h , (4.8) B h (ζ h ; φ h , φ h ) = 0, and B h (ζ h ; ϕ h , φ h ) = −B h (ζ h ; φ h , ϕ h ), (4.9) |F h (φ h )| ≤ C F h f 0,Ω |φ h | 2,h . (4.10) Proof. Properties in (4.7) are obtained from the definition of bilinear form A h (·, ·) and the stability (3.14). To prove property (4.8), we use the definition of trilinear form B h (·; ·, ·) and Hölder inequality to obtain that B h (ζ h ; ϕ h , φ h ) ≤ C 2 bd K∈T h ∆ζ h 2 0,K 1/2 K∈T h curl ϕ h 4 L 4 (K) 1/4 K∈T h ∇φ h 4 L 4 (K) 1/4 ≤ C 2 bd |ζ h | 2,h |ϕ h | 1,4,h |φ h | 1,4,h ≤ C B h |ζ h | 2,h |ϕ h | 2,h |φ h | 2,h , where C B h := (C bd C sob ) 2 > 0, and C bd , C sob are the constants in (3.6) and Theorem 4.1, respectively. Finally, the proof of properties (4.9) and (4.10) are obtained from the definition of forms B h (·; ·, ·) and F h (·). A fixed-point strategy In this subsection we will develop a fixed-point strategy to establish the well-posedness of discrete problem (4.1). Indeed, for a given ξ h ∈ M h , we define the operator T h : M h −→ M h ξ h −→ T h (ξ h ) = ϕ h , where ϕ h is the solution of the following linear problem: find ϕ h ∈ M h , such that νA h (ϕ h , φ h ) + B h (ξ h ; ϕ h , φ h ) = F h (φ h ) ∀φ h ∈ M h . Next, we consider the ball Y h := φ h ∈ M h : φ h 2,Ω ≤ C F h ( αν) −1 f 0,λ h := C B h C F h (α 1 ν) −2 f 0,Ω < 1. (4.11) Then, T h : Y h → Y h is a contraction mapping. Proof. The demonstration follows from the definition of operator T h , and Lemma 4.2 the Lax-Milgram Theorem. We finish this section with the following result, which establishes that the discrete problem is well-posed. Theorem 4.2 If condition (4.11) is satisfied, then there exists a unique ψ h ∈ M h solution to problem (4.1) satisfying the following dependence of the data |ψ h | 2,h ≤ C F h ( αν) −1 f 0,Ω . (4.12) Proof. The proof follows from Lemma 4.3 and the Banach point-fixed Theorem. Error analysis In this section we will develop an error analysis for the VEM proposed in (4.1). By exploiting the naturally skewsymmetry property of the discrete trilinear form, and the consistence and boundedness properties of discrete bilinear form we write an abstract convergence result for the nonlinear VE scheme. Then, by using the enriching operator, we establish key approximation properties involving the bilinear and trilinear forms, together with the consistency errors, which allow the derivation of an optimal error estimate in broken H 2 -norm under the minimal regularity condition on the weak solution (cf. Theorem 2.2). Moreover, by using duality arguments and the enriching operator we also establish optimal error estimates in the H 1 -and L 2 -norm under the same regularity condition on the stream-function ψ and the density force f . An abstract convergence result We start with two technical lemmas involving the continuous and discrete forms B(·, ·, ·) and B h (·, ·, ·) defined in (2.5) and (3.11), respectively. Lemma 5.1 Let B(·; ·, ·) be the trilinear form defined in (2.5). Then, for all ζ ∈ H 2+t (Ω), with t ∈ (1/2, 1], and for all ϕ ∈ H 2 (Ω) and φ h ∈ H 1 (T h ), it holds: B(ζ; ϕ, φ h ) ≤ C ζ 2+t,Ω ϕ 2,Ω |φ h | 1,h . Proof. By using the Hölder inequality, for each ζ ∈ H 2+t (Ω), with t ∈ (1/2, 1], for all ϕ ∈ H 2 (Ω) and for all φ h ∈ H 1 (T h ), we have B(ζ; ϕ, φ h ) ≤ K∈T h ∆ζ 4 L 4 (K) 1/4 K∈T h ∇ϕ 4 L 4 (K) 1/4 K∈T h ∇φ h 2 0,K 1/2 ≤ |ζ| 2,4,Ω |ϕ| 1,4,Ω |φ h | 1,h . Then, by using the Sobolev embeddings H 2 (Ω) → W 1,4 (Ω) and H 2+t (Ω) → W 2,4 (Ω), with t ∈ (1/2, 1], we obtain B(ζ; ϕ, φ h ) ≤ C ζ 2+t,Ω ϕ 2,Ω |φ h | 1,h , where C depends only on Ω. The proof is complete. Remark 5.1 Following the above arguments, we can also prove that for all ζ ∈ H 2+t (Ω), with t ∈ (1/2, 1], and for all ϕ h ∈ H 1 (T h ) and φ ∈ H 2 (Ω), it holds B(ζ; ϕ h , φ) ≤ C ζ 2+t,Ω |ϕ h | 1,h |φ| 2,Ω . Lemma 5.2 Let ϕ ∈ Φ and ϕ h ∈ M h . Then, for each φ h ∈ M h , it holds |B h (ϕ; ϕ, φ) − B h (ϕ h ; ϕ h , φ h )| ≤ C B h (|ϕ h | 2,h |φ h | 2,h + |ϕ − ϕ h + φ h | 2,h ( ϕ 2,Ω + |ϕ h | 2,h )) |φ h | 2,h . Proof. The proof follows by adding and subtracting adequate terms together with property (4.9) and Theorem 4.1. In order to derive the abstract error estimate for the nonlinear VE scheme, we will introduce the following consistence errors. Let ψ ∈ Φ be the solution of continuous problem (2.3), then we define: N h (ψ; φ h ) := νA(ψ, φ h ) + B(ψ; ψ, φ h ) − F (φ h ) ∀φ h ∈ M h , (5.1) C h (ψ; φ h ) := B(ψ; ψ, φ h ) − B h (ψ; ψ, φ h ) ∀φ h ∈ M h . (5.2) The first term above measures to what extent the continuous solution ψ does not satisfy the nonconforming virtual element formulation (4.1) and the second term measure of the variational crime perpetrated in the discretization of the trilinear form B(·; ·, ·). In addition, we define the following quantity: F − F h := sup φ h ∈M h φ h = 0 |F (φ h ) − F h (φ h )| |φ h | 2,h . (5.3) In subsection 5.2 we will establish approximation properties for the above terms. Next, we provide the following Strang-type result for our nonlinear VE scheme. |ψ − ψ h | 2,h ≤ C inf φ h ∈M h |ψ − φ h | 2,h + inf χ∈P2(T h ) |ψ − χ| 2,h + F − F h + sup φ h ∈M h φ h = 0 |N h (ψ; φ h )| |φ h | 2,h + |C h (ψ; φ h )| |φ h | 2,h , where N h (ψ; ·) and C h (ψ; ·) are the consistency errors defined in (5.1) and (5.2). Proof. Let φ h ∈ M h and set δ h := φ h − ψ h . Then, by using triangle inequality we obtain |ψ − ψ h | 2,h ≤ |ψ − φ h | 2,h + |δ h | 2,h . (5.4) Now, by using the property (4.7), the consistence of bilinear forms A K h (·, ·) (cf. (3.13)), we have ν α|δ h | 2 2,h ≤ νA h (δ h , δ h ) = νA h (φ h , δ h ) − νA h (ψ h , δ h ) = νA h (φ h , δ h ) − F h (δ h ) + B h (ψ h ; ψ h , δ h ) = ν K∈T h A K h (φ h − χ, δ h ) + A K (χ − ψ, δ h ) + ν K∈T h A K (ψ, δ h ) − F h (δ h ) + B h (ψ h ; ψ h , δ h ) = ν K∈T h A K h (φ h − χ, δ h ) + A K (χ − ψ, δ h ) + (νA(ψ, δ h ) − F h (δ h ) + B h (ψ h ; ψ h , δ h )) = ν K∈T h A K h (φ h − χ, δ h ) + A K (χ − ψ, δ h ) + N h (ψ; ψ, δ h ) + [F (δ h ) − F h (δ h )] + [B h (ψ h ; ψ h , δ h ) − B(ψ; ψ, δ h )],(5.5) where we have added and subtracted adequate terms and χ is an arbitrary element of P 2 (T h ). From the continuity of bilinear forms A K (·, ·), A K h (·, ·), and by using the triangular inequality, we have K∈T h A K h (φ h − χ, δ h ) + A K (χ − ψ, δ h ) ≤ C(|φ h − ψ| 2,h + |ψ − χ| 2,h )|δ h | 2,h . Now, we add and subtract the term B h (ψ; ψ, δ h ), then applying Lemma 5.2, we obtain |B h (ψ h ; ψ h , δ h ) − B(ψ; ψ, δ h )| ≤ |B h (ψ h ; ψ h , δ h ) − B h (ψ; ψ, δ h )| + |B h (ψ; ψ, δ h ) − B(ψ; ψ, δ h )| ≤ C B h (|ψ h | 2,h |δ h | 2,h + |ψ − φ h | 2,h ψ 2+s,Ω ( ψ 2,Ω + |ψ h | 2,h )) |δ h | 2,h + |C h (ψ; δ h )|. (5.6) Therefore, combining (5.5)-(5.6), we get ν α|δ h | 2,h ≤ C(|ψ − φ h | 2,h + |ψ − χ| 2,h ) + C B h |ψ h | 2,h + F − F h + |N h (ψ; δ h )| + |C h (ψ; δ h )|. From the inequality above, we obtain ν α(1 − C B h (ν α) −1 |ψ h | 2,h )|δ h | 2,h ≤ C |ψ − φ h | 2,h + |ψ − χ| 2,h + F − F h + |N h (ψ; δ h )| + |C h (ψ; δ h )| . By using (4.12) and condition (4.11) we have that ( 1 − C B h (ν α) −1 |ψ h | 2,h ) ≥ 1 − λ h > 0. Therefore, from above inequality, we have |δ h | 2,h ≤ C |ψ − φ h | 2,h + |ψ − χ| 2,h + F − F h + |N h (ψ; δ h )| + |C h (ψ; ψ, δ h )| . Finally, the desired result follows from (5.4) and the above estimate. The next step is to provide approximation properties that can be used in Theorem 5.1. In next subsection we will establish such properties. Approximation results and a priori error estimate We have the following approximation result for polynomials on star-shaped domains. Proposition 5.1 For every φ ∈ H 2+t (K), with t ∈ [0, 1], there exist φ π ∈ P 2 (K) and C > 0, independent of h, such that φ − φ π ,K ≤ Ch 2+t− K |φ| 2+t,K , = 0, 1, 2. For the virtual space M h we have the following approximation result (see [9,49,37,23]). Proposition 5.2 For each φ ∈ H 2+t (Ω), with t ∈ [0, 1], there exist φ I ∈ M h and C > 0, independent of h, such that φ − φ I ,K ≤ Ch 2+t− K |φ| 2+t,K , = 0, 1, 2. Let E h : M h → W C h be the restriction of the operator E h to the space M h , i.e., E h := E h | M h . We note that this operator satisfies the approximation properties in Proposition 4.1. Then, by using the operator E h , we will establish an error estimate involving the bilinear form A(·, ·), which will be useful to obtain an error estimate in broken H 2 -norm under minimal regularity condition on the exact stream-function ψ (cf. Theorem 2.2). Lemma 5.3 Let ϕ ∈ H 2+t (Ω), with t ∈ [0, 1]. Then, for all φ h ∈ M h there exists a positive constant C, independent of h, such that A(ϕ, φ h − E h φ h ) ≤ Ch t ϕ 2+t,Ω |φ h | 2,h . Proof. The proof has been established in [3, Lemma 4.10]. The following result establishes error estimates for the consistence errors N h (ψ; ·) and C h (ψ; ·) defined in (5.1) and (5.2), respectively. Lemma 5.4 Let ψ ∈ H 2+γ (Ω) ∩ Φ be the solution of problem (2.3). Then, for all φ h ∈ M h , there exists a constant C > 0, independent to h, such that |N h (ψ; φ h )| ≤ Ch γ ( ψ 2+γ,Ω + f 0,Ω )|φ h | 2,h , |C h (ψ; φ h )| ≤ Ch γ ( ψ 1+γ,Ω + ψ 2,Ω ) ψ 2+γ,Ω |φ h | 2,h . Proof. Let φ h ∈ M h . Then, we can take E h φ h ∈ W C h ⊂ Φ as test function in (2.3) to obtain νA(ψ, E h φ h ) + B(ψ; ψ, E h φ h ) = F (E h φ h ). (5.7) Thus, from (5.1) and (5.7), we get N h (ψ; φ h ) = νA(ψ, φ h ) + B(ψ; ψ, φ h ) − F (φ h − E h φ h ) − F (E h φ h ) = νA(ψ, φ h − E h φ h ) + B(ψ; ψ, φ h − E h φ h ) − F (φ h − E h φ h ). (5.8) By using, identity (5.8), the Cauchy-Schwarz inequality, Lemmas 5.1 and 5.3, we get |N h (ψ; φ h )| ≤ Cνh γ ψ 2+γ,Ω |φ h | 2,h + C|ψ| 2+γ,Ω |ψ| 2,Ω |φ h − E h φ h | 1,h + C F f 0,Ω |φ h − E h φ h | 1,h ≤ Ch γ ( ψ 2+γ,Ω + f 0,Ω )|φ h | 2,h , where C > 0 is independent of h. The proof of second property follows by adapting the arguments used in [42,Lemma 4.2] to the nonconforming case and using Theorem 4.1. For the consistence error in the approximation defined in (5.3), we have the following result. Lemma 5.5 Let f ∈ L 2 (Ω) 2 , F (·) and F h (·) be the functionals defined in (2.6) and (3.12), respectively. Then, we have the following estimate: F − F h ≤ Ch f 0,Ω . Proof. The proof follows from the definition of the functionals F (·) and F h (·), together with approximation properties of the projector Π 1 K . The following result provides the rate of convergence of our virtual element scheme in broken H 2 -norm. Error estimates in H 1 and L 2 In this section we provide new optimal error estimates in broken H 1 -and L 2 -norms for the stream-function by using duality arguments and employing the enriching operator E h , under same regularity of the weak solution ψ and of the density force f , considered in Theorem 5.2. We start establishing the following key preliminary result involving the forms B(·; ·, ·) and B h (·; ·, ·), which will useful to provide the error estimates in the weak norms. This term will take care of the consistency error associate to the trilinear form present in the VEM approach and as we will observe, its manipulation is not direct, so it will require special attention due to the nonlinearity involved. Lemma 5.6 Let ψ ∈ Φ∩H 2+γ (Ω) and ψ h ∈ M h be the unique solutions of problems (2.3) and (4.1), respectively. Assuming that f ∈ L 2 (Ω) 2 and let ϕ ∈ H 2+t (Ω), with t ∈ (1/2, 1]. Then, it holds T B (ϕ) := B h (ψ h ; ψ h , ϕ) − B(ψ h ; ψ h , ϕ) ≤ C h γ+t + h 2γ ( f 0,Ω + ψ 2+γ,Ω ) ϕ 2+t,Ω + 2C reg C 2 sob C 2 bd f 0,Ω |ψ − ψ h | 1,h ϕ 1+t,Ω , where C > 0 is a constant independent of h, and C sob , C reg and C bd are the constants in (2.7), Theorem 2.2 and (3.6), respectively. Proof. By using the definition of trilinear forms B(·; ·, ·) and B h (·; ·, ·), adding and subtracting suitable terms and using the orthogonality property of the L 2 -projections, we have the following identity In what follows, we will establish estimates for each terms on the right hand side of the previous identity. For the term T 1 we use the Hölder and triangle inequalities, along with approximations properties of Π 0 K , to obtain T 1 ≤ K∈T h ∆ψ h − Π 0 K ∆ψ h 0,K curl ψ h − curl ψ L 4 (K) ∇ϕ L 4 (K) ≤ K∈T h (2 ∆ψ h − ∆ψ 0,K + ∆ψ − Π 0 K ∆ψ 0,K ) curl (ψ h − ψ) L 4 (K) ∇ϕ L 4 (K) ≤ C(|ψ − ψ h | 2,h + h γ ψ 2+γ,Ω )|ψ − ψ h | 1,4,h ∇ϕ L 4 (Ω) ≤ Ch 2γ ( f 0,Ω + ψ 2+γ,Ω ) ϕ 2+t,Ω , where we have used the Hölder inequality (for sequences), continuous Sobolev inclusion, along with Theorems 4.1 and 5.2. Now, for T 2 we follow similar arguments to obtain T 2 ≤ Ch 2γ ( f 0,Ω + ψ 2+γ,Ω ) ϕ 2+t,Ω . For the term T 3 we employ again the Hölder inequality, the continuity of the projector Π 1 K , along with Theorems 4.1 and 5.2, to obtain: T 3 ≤ K∈T h Π 0 K (∆ψ h − ∆ψ) 0,K Π 1 K curl ψ h L 4 (K) ∇ϕ − Π 1 K ∇ϕ L 4 (K) ≤ C|ψ − ψ h | 2,h curl ψ h 1,4,h h t |∇ϕ| W t 4 (Ω) ≤ Ch γ+t ( f 0,Ω + ψ 2+γ,Ω ) ϕ 2+t,Ω . For the term T 4 , we follow similar steps to those used above, to get T 4 ≤ K∈T h Π 0 K ∆ψ 0,K Π 1 K curl (ψ h − ψ) L 4 (K) ∇ϕ − Π 1 K ∇ϕ L 4 (K) ≤ Ch γ+t ( f 0,Ω + ψ 2+γ,Ω ) ϕ 2+t,Ω . Now, for the term T 5 , we add and subtract suitable terms, use the Hölder inequality, properties of the L 2projections Π 1 K and Π 0 K , together with continuous Sobolev embeddings to obtain T 5 ≤ K∈T h Π 0 K ∆ψ L 4 (K) curl ψ h − Π 1 K curl ψ h 0,K ∇ϕ L 4 (K) ≤ 2|ψ − ψ h | 1,h + Ch 1+γ ψ 2+γ,Ω (C bd ∆ψ L 4 (Ω) C bd ∇ϕ L 4 (Ω) ) ≤ 2C reg C 2 sob C 2 bd f 0,Ω |ψ − ψ h | 1,h ϕ 1+t,Ω + Ch γ+t ψ 2+γ,Ω ϕ 2+t,Ω , Repeating the same arguments, we obtain the following bounds for the terms T 6 and T 7 : T 6 + T 7 ≤ Ch γ+t ( f 0,Ω + ψ 2+γ,Ω ) ϕ 2+t,Ω . (5.9) Finally, by combining the above bounds we obtain the desired result. Moreover, for the bilinear form A(·, ·) we have the following auxiliary result [3,Lemma 4.11]. Lemma 5.7 For ϕ ∈ H 2+t (Ω) and φ ∈ Φ ∩ H 2+t (Ω), with t ∈ [0, 1], it holds: A(ϕ, φ − φ I ) ≤ Ch 2t ϕ 2+t,Ω φ 2+t,Ω , where φ I ∈ M h is the interpolant of φ in the virtual space M h (cf. Proposition 5.2). In order to establish the desired error estimates we consider the following assumption: 2C reg C 2 sob C 2 bd f 0,Ω < 1,(5.ψ − ψ h 0,Ω + |ψ − ψ h | 1,h ≤ Ch 2γ ( ψ 2+γ ,Ω + f 0,Ω ). (5.11) Proof. First we will prove the H 1 estimate in (5.11). To this propose, let ψ I ∈ M h be the interpolant of ψ such that Proposition 5.2 holds true. We set δ h := (ψ h − ψ I ) ∈ M h . Then, we write ψ h − ψ = (ψ h − ψ I ) + (ψ I − ψ) = (ψ I − ψ) + (δ h − E h δ h ) + E h δ h . Thus, by using the triangle inequality together with Remark 5.1, Proposition 5.2, Lemma 4.1 and Theorem 5.2, we obtain |ψ − ψ h | 1,h ≤ |ψ − ψ I | 1,h + |δ h − E h δ h | 1,h + |E h δ h | 1,h ≤ Ch 2γ ψ 2+s,Ω + ∇E h δ h 0,Ω . (5.12) Now, the goal is to estimate the term ∇E h δ h 0,Ω . To do that, we consider the following dual problem: given ψ ∈ Φ (the unique solution of the formulation (2.3)), find φ ∈ Φ, such that A DP (ψ; ϕ, φ) := νA(ϕ, φ) + B(ψ; ϕ, φ) + B(ϕ; ψ, φ) = (∇(E h δ h ), ∇ϕ) 0,Ω ∀ϕ ∈ Φ, (5.13) where A(·, ·) and B(·; ·, ·) are the continuous forms defined in (2.4) and (2.5), respectively. Following the same arguments in [36] we have that problem (5.13) is well-posed and from Theorem 2.2, we obtain that φ ∈ Φ∩H 2+γ (Ω) and φ 2+γ,Ω ≤ C ∇E h δ h 0,Ω , (5.14) where C > 0 is a constant independent of h. Taking ϕ = E h δ h ∈ W C h ⊂ Φ as test function, adding and subtracting δ h in problem (5.13), we get ∇E h δ h 2 0,Ω = A DP (ψ; E h δ h , φ) = A DP (ψ; E h δ h − δ h , φ) + A DP (ψ; δ h , φ) =: I 1 + I 2 . (5.15) Now, we will obtain bounds for the terms I 1 and I 2 in the above identity. For I 1 , we apply Lemma 5.3 and Proposition 5.2 to obtain I 1 := A DP (ψ; E h δ h − δ h , φ) = νA(E h δ h − δ h , φ) + B(ψ; E h δ h − δ h , φ) + B(E h δ h − δ h ; ψ, φ) ≤ Cνh γ |δ h | 2,h φ 2+γ,Ω + C ψ 2+γ,Ω |E h δ h − δ h | 1,h φ 2,Ω + B(E h δ h − δ h ; ψ, φ) ≤ Cνh 2γ ψ 2+γ,Ω φ 2+s,Ω + Ch 2γ ψ 2+γ,Ω φ 2,Ω + B(E h δ h − δ h ; ψ, φ). (5.16) To estimate the term B(E h δ h − δ h ; ψ, φ) we start recalling that ψ, φ ∈ H 2+γ (Ω), with γ ∈ (1/2, 1], then by using the Sobolev inclusion H 2+γ (Ω) → W 1,4 (Ω), we have |curl ψ · ∇φ| 1,Ω ≤ curl ψ 1,4,Ω ∇φ 1,4,Ω ≤ C 2 sob ψ 2+γ,Ω φ 2+γ,Ω < +∞. Therefore, curl ψ · ∇φ ∈ H 1 (Ω) (hence belongs to H 1 (K) for each K ∈ T h ). Thus, by using the definition of B(·; ·, ·) we have B(E h δ h − δ h ; ψ, φ) = K∈T h (∆(E h δ h − δ h ), curl ψ · ∇φ) 0,K ≤ K∈T h ∆(E h δ h − δ h ) −1,K curl ψ · ∇φ 1,K . Now, by using the definition of the dual norm and an integration by part, we obtain ∆(E h δ h − δ h ) −1,K = sup ϕ∈H 1 0 (K) (∆(E h δ h − δ h ), ϕ) 0,K |ϕ| 1,K = sup ϕ∈H 1 0 (K) (∇(E h δ h − δ h ), ∇ϕ) 0,K |ϕ| 1,K ≤ |E h δ h − δ h | 1,K . From the two estimates above, the Hölder inequality for sequences and (5.14), we have B(E h δ h − δ h ; ψ, φ) ≤ K∈T h |E h δ h − δ h | 1,K curl ψ · ∇φ 1,K ≤ |E h δ h − δ h | 1,h curl ψ · ∇φ 1,Ω ≤ Ch 2γ ψ 2+γ,Ω φ 2+γ,Ω ≤ Ch 2γ ψ 2+γ,Ω ∇E h δ h 0,Ω . Consequently, inserting the above inequality in (5.16), we arrive to I 1 ≤ Ch 2γ ψ 2+γ,Ω ∇E h δ h 0,Ω . (5.17) Now, we will estimate the remaining term I 2 . Indeed, we split again δ h := (ψ h − ψ) + (ψ − ψ I ), then I 2 = −A DP (ψ; ψ − ψ h , φ) + A DP (ψ; ψ − ψ I , φ) =: −I 21 + I 22 . (5.18) By using analogous arguments those employed to bound the term I 1 and applying Proposition 5.2 and Lemma 5.7, we can obtain I 22 ≤ Ch 2γ ψ 2+γ,Ω ∇E h δ h 0,Ω . (5.19) Next, adding and subtracting φ I , B(ψ; ψ, φ I ) and other suitable terms together with the definition of the continuous and discrete problems (cf. (2.3) and (4.1), respectively), we obtain I 21 = νA(ψ − ψ h , φ) + B(ψ; ψ − ψ h , φ) + B(ψ − ψ h ; ψ, φ) = νA(ψ − ψ h , φ − φ I ) + νA(ψ − ψ h , φ I ) + B(ψ; ψ − ψ h , φ) + B(ψ − ψ h ; ψ, φ) = νA(ψ − ψ h , φ − φ I ) + F (φ I ) − F h (φ I ) + νA h (ψ h , φ I ) + B h (ψ h ; ψ h , φ I ) − B(ψ; ψ, φ I ) − νA(ψ h , φ I ) + B(ψ; ψ − ψ h , φ) + B(ψ − ψ h ; ψ, φ) = νA(ψ − ψ h , φ − φ I ) + ν[A h (ψ h , φ I ) − A(ψ h , φ I )] + [F (φ I ) − F h (φ I )] + [B h (ψ h ; ψ h , φ I − φ) − B(ψ; ψ, φ I − φ)] + B(ψ − ψ h ; ψ − ψ h , φ) + [B h (ψ h ; ψ h , φ) − B(ψ h ; ψ h , φ)] =: T A1 + T A2 + T F + T B1 + T B2 + T B3 ,(5.20) where also we have used also the identity B(ψ; ψ − ψ h , φ) + B(ψ − ψ h ; ψ, φ) + B h (ψ h ; ψ h , φ) − B(ψ; ψ, φ) = B(ψ − ψ h ; ψ − ψ h , φ) + [B h (ψ h ; ψ h , φ) − B(ψ h ; ψ h , φ)]. By using standard arguments and (5.14) we obtain that T A1 + T A2 + T F + T B2 ≤ Ch 2γ ( f 0,Ω + ψ 2+γ,Ω ) ∇E h δ h 0,Ω . (5.21) For the remaining term T B1 , we employ Lemmas 5.2 and 5.4, to obtain |T B1 | ≤ |B(ψ; ψ, φ I − φ) − B(ψ; ψ, φ I − φ)| + |B h (ψ; ψ, φ I − φ) − B h (ψ h ; ψ h , φ I − φ)| ≤ Ch γ ( ψ 1+γ,Ω + ψ 2,Ω ) ψ 2+γ,Ω |φ I − φ| 2,h + C B h (|ψ h | 2,h |φ I − φ| 2,h + |(ψ − ψ h ) + (φ I − φ)| 2,h (|ψ| 2,h + ψ h 2,Ω )) |φ I − φ| 2,h ≤ Ch 2γ ( ψ 1+γ,Ω + ψ 2,Ω ) ψ 2+γ,Ω ∇E h δ h 0,Ω + Ch 2γ ( ψ 2+γ,Ω + f 0,Ω )( ψ 2,Ω + |ψ h | 2,h ) ∇E h δ h 0,Ω + Ch 2γ (|ψ h | 2,h + ψ 2,Ω ) ∇E h δ h 0,Ω ,(5.22) where we have used Theorem 5.2 and (5.14). For the term T B3 , we observe that T B3 = T B (φ), then by using Lemma 5.6 and (5.14) we get T B3 ≤ Ch 2γ ( ψ 2+γ,Ω + f 0,Ω ) φ 2+γ,Ω + 2C reg C 2 sob C 2 bd f 0,Ω |ψ − ψ h | 1,h φ 2+γ,Ω ≤ Ch 2γ ( ψ 2+γ,Ω + f 0,Ω ) ∇E h δ h 0,Ω + 2C reg C 2 sob C 2 bd f 0,Ω |ψ − ψ h | 1,h ∇E h δ h 0,Ω .|I 2 | ≤ C( ψ 2+s,Ω + f 0,Ω ) ∇E h δ h 0,Ω + 2C reg C 2 sob C 2 bd f 0,Ω |ψ − ψ h | 1,h ∇E h δ h 0,Ω . (5.24) The desired result follows by combining the estimates (5.12), (5.15), (5.17) and (5.24) together with the fact that (1 − 2C reg C 2 sob C 2 bd f 0,Ω ) > 0 (see assumption (5.10)). Finally, the L 2 estimate in (5.11) is obtained from the triangle inequality, Proposition 5.2, Lemma 5.7 and Theorem 5.2 as follow: ψ − ψ h 0,Ω ≤ ψ − ψ I 0,Ω + δ h − E h δ h 0,Ω + E h δ h 0,Ω ≤ Ch 2+γ ψ 2+γ,Ω + Ch 2 (|ψ h − ψ| 2,h + |ψ − ψ I | 2,h ) + C|E h δ h | 1,Ω ≤ Ch 2γ ( ψ 2+γ,Ω + f 0,Ω ), where we have used norm equivalence in Φ. The proof is complete. We finish this section establishing the following remark. Remark 5.2 If f is a smooth function, then applying an integration by parts and the boundary conditions in (2.6), we have that (f , curlφ) 0,Ω = (rot f , φ) 0,Ω ∀φ ∈ Φ. Inspired by this identity, we can consider an alternative right hand side as follows: F h (φ h ) := K∈T h (rot f , Π 2 K φ h ) 0,K ∀φ h ∈ M h . (5.25) We note that F h (·) is fully computable using the degrees of freedom D M 1 − D M 2, since Π 2 K is computable (cf. Lemma (3.2)). For the VE scheme (4.1) considering the alternative load term (5.25), we can provide an analogous analysis as the one develop in the above sections. Therefore, we can obtain the rate of convergences as in Theorems 5.2 and 5.3. We will present a numerical test to confirm the error estimates in this case (cf. Subsection 7.4). Moreover, we observe that if the density force is irrotational, i.e., f = ∇ϕ (for some ϕ), it is possible improve the error estimate in Theorem 5.2 by removing the dependence of the error by the load term f . Postprocessing of further fields of interest In this section we propose post-processing techniques that allow obtain approximations of the velocity, vorticity and pressure fields from the discrete stream-function ψ h . Postprocessing the velocity field In order to propose an approximation for the velocity field, we recall that if ψ ∈ Φ the unique solution of continuous problem (2.3), then u = curl ψ. (6.1) At the discrete level, we define a piecewise linear approximation of the velocity field u as u h | K := Π 1 K curl ψ h ,(6.2) where ψ h ∈ M h is discrete virtual solution delivered by solving problem (4.1) and the operator Π 1 K is defined by the vectorial version of (3.5). We have the following result for velocity vector u h . Postprocessing the vorticity field Due its importance and applications in fluid mechanics, different works have been devoted to approximate the vorticity field of the incompressible Navier-Stokes equations; see for instance [17,31,5] and the references therein. By solving the nonconforming discrete problem (4.1), we only obtain an approximation for the stream-function. Nevertheless, in this subsection we propose an approximation for the vorticity field ω via postprocessing formula through the discrete stream-function ψ h and the projection Π 0 K defined by the relation (3.5). First, we recall that ω = rot u, then using the identity u = curl ψ, we have obtain ω = rot u = rot(curl ψ) = −∆ψ. Then, at discrete level we define the following approximation for the vorticity: ω h | K := −Π 0 K (∆ψ h ), (6.3) where ψ h ∈ M h is the unique solution of problem (4.1) and Π 0 K is defined in (3.5). We have the following result for the discrete vorticity. Proof. The proof follows by using the same arguments in Theorem 6.1. Postprocessing the pressure field This subsection is devoted to developing a strategy to recover the pressure variable form the discrete streamfunction solution ψ h of problem (4.1), which is based on the algorithm presented in [26] and extended to the nonconforming VEM approach. We start by recalling that if ψ ∈ Φ is the unique solution of the weak formulation (2.3), then the velocity field is given by u = curl ψ. Thus, we can write b(v, p) := (p, div v) 0,Ω = ν(∇u, ∇v) 0,Ω + ((∇u)u, v) 0,Ω − (f , v) 0,Ω = ν(∇curl ψ, ∇v) 0,Ω + ((∇curl ψ)curl ψ, v) 0,Ω − (f , v) 0,Ω ∀v ∈ H. (6.4) Now, we consider the functional F(ψ, f )(·) : H → R given by F(ψ, f )(v) := ν(∇curl ψ, ∇v) 0,Ω + ((∇curl ψ)curl ψ, v) 0,Ω − (f , v) 0,Ω ∀v ∈ H. (6.5) By using (6.4) and (6.5), we reformulate (2.1) as a variational problem for the pressure variable: given ψ ∈ Φ the unique solution of problem (2.3) and f ∈ L 2 (Ω) 2 , find p ∈ Q such that b(v, p) = F(ψ, f )(v) ∀v ∈ H,(6.6) where H and Q are the spaces defined in (2.2). From an equivalence of problems and the LBB theory we have that problem (6.6) has a unique solution p ∈ Q (see [34]). The difficulties to discretize directly problem (6.6) have been discussed in [25,Section 9]. Thus, inspired in this work we consider the following equivalent problem: find (w, p) ∈ H × Q, such that a(w, v) + b(v, p) = F(ψ, f )(v) ∀v ∈ H b(w, q) = 0 ∀q ∈ Q, (6.7) where a( v, v) := (∇ v, ∇v) 0,Ω ∀ v, v ∈ H. We have that this Stokes-like problem is well-posed. Moreover, w = 0. Now the goal is to discretize the problem (6.7). Nonconforming Crouzeix-Raviart-type VE discretization In this subsection we will present a VE scheme to solve problem (6.7). First, we recall that the Morley-type VE space M h is in a Stokes-complex relation with the Crouzeix-Raviart type VE space U h , defined in (3.4) and (3.2), respectively. Apart from the previously mentioned spaces, we introduce the space for pressure approximation as Q h := {q h ∈ Q : q h | K ∈ P 0 (K) ∀K ∈ T h }. (6.8) At last, we introduce the auxiliary space U h := v h ∈ U h : K∈T h (q h , div v h ) 0,K = 0 ∀q h ∈ Q h ,(6.9) where U h is the Crouzeix-Raviart-type VE space defined in (3.2). Now, we present the virtual element discretization of the Stokes problem (6.7) that reads as: find (w h , p h ) ∈ U h × Q h such that a h (w h , v h ) + b h (v h , p h ) = F h (ψ h , f )(v h ) ∀v h ∈ U h , b h (w h , q h ) = 0 ∀q h ∈ Q h ,(6.12) where Q h is the space defined in (6.8). The scheme (6.12) is well-posed since a h (·, ·) is coercive and continuous, the bilinear form b h (·, ·) is continuous and satisfies a discrete inf-sup condition on the pair of functional spaces U h -Q h (see [50]) and curl M h = U h . We summarize this fact in the following result. Theorem 6.3 Let b h (·, ·) be the discrete bilinear form defined in (6.10). Then, there exists a strictly positive, real constant C b > 0 such that sup v h ∈ U h \{0} b h (v h , q h ) |v h | 1,h ≥ β q h 0,Ω ∀q h ∈ Q h . Moreover, there exist a unique (w h , p h ) ∈ U h × Q h , solution of problem (6.12). Error estimate for the pressure scheme In this subsection we develop an abstract error result for the virtual scheme presented above. Moreover, we provide error estimates involving some consistent errors. Finally, by combining these results we derive an optimal error estimate for the pressure field. First, we focus on deriving a bound on the difference between the functional (6.11) applied to the stream-function ψ solving the continuous variational formulation (2.3) and its virtual element approximation solving (4.1). Lemma 6.2 Let ψ ∈ Φ and ψ h ∈ M h be the solution to problems (2.3) and (4.1), respectively. Moreover, let F h (ψ, f )(·) and F h (ψ h , f )(·) be the functionals defined in (6.11) (applied to ψ and ψ h , respectively). Then, there exists a real, positive constant C F h > 0, independent of h, such that ≤ C|ψ − ψ h | 2,h |ψ| 2,Ω |v h | 1,h + C|ψ h | 2,h |ψ − ψ h | 2,h |v h | 1,h . The result follows by combining the above estimates. In continuation, we define the consistency error Θ h (·, ·) as follows: given ψ ∈ Φ the solution of problem (2.3), we consider Θ h (ψ, v h ) := F h (ψ, f )(v h ) − b h (v h , p) ∀v h ∈ U h . (6.13) We have the following abstract error estimate for the pressure recovery scheme. Then, there exists a strictly positive, real constant C > 0, independent of h, such that p − p h 0,Ω ≤ C inf q h ∈Q h p − q h 0,Ω + sup v h ∈ U h v h = 0 |Θ(ψ, v h )| |v h | 1,h + |ψ − ψ h | 2,h ,(6. 14) where Θ(ψ, ·) is the consistency error defined in (6.13). Proof. Adding and subtracting adequate terms in (6.12), for each v h ∈ U h we have a h (w h , v h ) = F h (ψ h , f )(v h ) − b h (v h , p h ) = F h (ψ h , f )(v h ) − F h (ψ, f )(v h ) + F h (ψ, f )(v h ) − b h (v h , p) + b h (v h , p − p h ) = (F h (ψ h , f )(v h ) − F h (ψ, f )(v h )) + Θ(ψ, v h ) + b h (v h , p − p h ). (6.15) Taking v h = w h in (6.15), then by using the fact that b h (w h , q h ) = b h (w h , p h ) = 0 ∀q h ∈ Q h , the continuity of b h (·, ·) and Lemma 6.2, we get |w h | 1,h ≤ C |ψ − ψ h | 2,h + p − q h 0,Ω + sup v h ∈ U h v h = 0 |Θ(ψ, v h )| |v h | 1,h . (6.16) By using again (6.15) and the linearity of b h (·, ·), for all q h ∈ Q h we have b h (v h , q h − p h ) = b h (v h , q h − p) + b h (v h , p − p h ) = b h (v h , q h − p) + a h (w h , v h ) − (F h (ψ h , f )(v h ) − F h (ψ, f )(v h )) − Θ(ψ, v h ). Thus, by using the two last estimate above, Lemma 6.2, the inf-sup condition in Lemma 6.3, we obtain β q h − p h 0,Ω ≤ C p − q h 0,Ω + |w h | 1,h + sup v h ∈ U h v h = 0 |Θ(ψ, v h )| |v h | 1,h . The desired result follows from the triangle inequality, the above estimate and (6.16). Lemma 6.3 Let ψ ∈ Φ ∩ H 2+γ (Ω), γ ∈ (1/2, 1] , be the solution of problem (2.3). Then, there exists a strictly positive, real constant C > 0, independent of h, such that |Θ h (ψ, v h )| ≤ Ch γ ( p γ,Ω + ψ 2+γ,Ω + f 0,Ω )|v h | 1,h ∀v h ∈ U h . Proof. By using the definition of the consistency term Θ h (ψ, ·) (cf. (6.13)), the weak continuity of the discrete function of the Crouzeix-Raviart space on edges, and employing standard arguments as [50,Theorem 13], together with the real method of interpolation, we can obtain the required result. Finally, the next result provide the rate of convergent for our pressure VE scheme. Theorem 6.5 Under same assumptions of Theorem 6.4, for p ∈ Q ∩ H γ (Ω), there exists C > 0, independent of h, such that p − p h 0,Ω ≤ Ch γ ( p γ,Ω + ψ 2+γ,Ω + f 0,Ω ). Proof. The demonstration follows from (6.14), taking q h = Π 0 K p in Theorem 6.4, Lemma 6.3 and Theorem 5.2. , avoiding the construction of the Stokes complex sequence. Moreover, we are able to recover the velocity and vorticity fields by using the postprocessing of subsections 6.1 and 6.2, and obtain the theoretical analysis presented here. However, the pressure recovery would not be available. Thus, we point out that the main advantage to used Stokes complex sequence associated to M h and U h is that we can additionally compute the pressure field from the discrete stream-function, with optimal rate of convergence, making the suitable setting. Numerical result In this section, we present four numerical experiments to test the practical performance of the proposed virtual element discretization (4.1) and assess the theoretical predictions as estimated in Sections 5 and 6. We first approximate the discrete stream-function ψ by employing Morley-type VE space (3.4), and then we recovered other fields of interest such as velocity, and vorticity by employing suitable projection operators. Further, we recover the pressure variable by solving a saddle point problem, where the velocity space are in Stokes complex relationship with the stream-function space (cf. Section 6.3). In each test to solve the nonlinear system resulting from (4.1), we apply the Newton method, with a fixed tolerance of Tol = 10 −8 and the initial guess is given by ψ in h = 0. We have tested the method by using different families of meshes such as: which are posted in Figure 1. We quantify the errors by employing the projection operators: Π D K , Π 1 K , and Π 0 K . The following formulations are used for the computation of experimental errors: , and Re = ν −1 . We have computed the discrete stream-function for different choice of viscosity coefficients, e.g., ν = 1, 0.01, and errors for the stream-function (cf. (7.1)) are posted in Figure 2, and Figure 4, respectively. Further, by employing the formulas (6.2) and (6.3), we have recovered discrete velocity and vorticity fields for ν = 1, 0.01. The error curves of the velocity and vorticity are posted in Figure 3, and Figure 5, while the error curves for the pressure are posted in Figure 6 for both values of ν. Besides, for all the meshes the maximum number of iterations that are required to achieve the tolerance in the Newton method is 4 for ν = 1 and 6 for ν = 0.01. •E i (ψ) := K∈T h |ψ − Π D K ψ h | 2 i,K 1/2 ∀i ∈ {0, 1, 2}, E j (u) := K∈T h |u − Π 1 K curl ψ h | 2 j,K 1/2 ∀j ∈ {0, 1}; E 0 (ω) := K∈T h ω − Π 0 K ∆ψ h 2 0,K 1/2 , E 0 (p) := K∈T h p − p h 2 0, In Figure 7, we have posted the discrete stream-function and pressure fields for ν = 1, using the mesh T 1 h , with h = 1/32. Test 2. L-shaped domain In this example, we would like to focus to examine the rate of convergences of the discrete stream-function, velocity and vorticity fields on a nonconvex L-shaped domain, where the exact solution ψ has less regularity. For the computational domain, we considered Ω = [−1, 1] × [−1, 1] \ (0, 1) × (−1, 0). The exact solution is given by ψ(r, θ) = r 5/3 sin( 5θ 3 ), where r = (x 2 +y 2 ) 1/2 , and θ is the angle with the vertical axis. Since ∂ψ ∂r is unbounded near the origin, then the solution ψ has weak regularity near the origin. The rate of convergence of stream-function velocity and vorticity solutions are posted in Table 1 for viscosity ν = 1, and using the mesh T 2 h . From the posted results, we observed that the rates of convergence are in accordance with the theoretical prediction for all the variables. Further, we have chosen exact pressure as p := sin(x) − sin(y) − p, where p is a constant that is set to satisfy zero mean condition, i.e., (p, 1) 0,Ω = 0. The convergence behavior of the pressure field is posted in Table 2. It is observed that initially the rate of convergence is slightly higher than the predicted order as in Theorem 6.3. However, for finer mesh we observe expected order of convergence, i.e., O(h 2/3 ). Further, we report that the presence of singularity of the stream-function at re-entrant corner affects the convergence order of pressure field as proven in Theorem 6.3. In this example, the number of iterations that are required for the Newton method is 4. Test 3. The lid-driven cavity problem In the third example, we assess the nature of the fluid for the lid-driven cavity flow. This is a benchmark test to validate the numerical schemes for different values of viscosity ν. The computational domain is unit square with upper horizontal lid is moving with uniform velocity u := (1, 0), and fixed boundary condition, i.e., u := (0, 0) is applied to other static walls. In stream-function formulation, we have imposed the following Dirichlet boundary conditions: ψ = ψ x = 0, and ψ y = 1 on moving lid, and ψ = ∂ψ ∂n = 0 on all other static walls. In Figure 8, we posted the discrete stream-function and pressure field for ν = 0.01 and using the mesh T 3 h , with h = 1/64. The small values of ν exhibits singularities near x = 0, and x = 1 [31,44], which increases for smaller values of ν. Such behaviors are noticed in other methods [31], and persists also for finer grid. Further, we observed that the vortex center has moved towards the direction of velocity for small values of ν. Such characteristic of fluids with small viscosity coefficient are well observed in literature. Additionally, we report that our scheme preserves the property of the fluids with low viscosity coefficients on general shaped polygonal meshes. For this numerical experiment, the number of iterations that are required for the Newton method is 5. Test 4. Performance of the scheme for small viscosity In this example, we mainly focus to discuss the performance of the scheme for small values of viscosity coefficients. We consider the exact stream-function, velocity and pressure solutions as ψ(x, y) = x 2 y 2 (1 − x) 2 (1 − y) 2 , u(x, y) := x 2 (1 − x) 2 (2y − 6y 2 + 4y 3 ) −y 2 (1 − y) 2 (2x − 6x 2 + 4x 3 ) , p(x, y) := x 3 y 3 − 1 6 . The numerical approximations of the stream-functions are computed by employing the scheme (4.1), with the alternative load term given by (5.25). The computational domain is considered as Ω := [0, 1] × [0, 1]. Further, we discretize the domain with square elements with different mesh sizes, and computed the errors for stream-function in broken H 2 -norm for different values of ν, which are posted in Figure 9. We observed that the errors are accurate when the parameter ν within the range ν ∈ [10 −3 , 10 0 ] and the errors increase for ν = 10 −4 . We claim that these results are in accordance with the general behaviour of the exactly divergence-free Galerkin schemes are more robust with respect to small viscosity parameters, see for instance [14] in the VEM approach. Finally, we report that the maximum number of iterations that are required to achieve the tolerance in the Newton method is 7. Figure 9: Test 4. Errors of the stream-function E 2 (ψ), using the VE scheme (4.1) with the alternative load term (5.25), for different values of ν and the mesh T 1 h . ζ; ψ, φ) := (∆ζ curl ψ, ∇φ) 0,Ω , (2.5) F (φ) := (f , curl φ) 0,Ω .(2.6) Theorem 4. 1 1For any 2 ≤ q < ∞ there exists a positive constant C, independent of h, such that Theorem 5 . 1 ( 51Abstract convergence result) Let ψ and ψ h be the unique solutions to problems (2.3) and (4.1), respectively. There exists a positive constant C, independent of h, such that Theorem 5. 2 2Let ψ ∈ Φ ∩ H 2+γ (Ω) and ψ h ∈ M h be the unique solutions of problem (2.3) and problem (4.1), respectively. Then, there exists a positive constant C, independent of h, such that|ψ − ψ h | 2,h ≤ Ch γ ( ψ 2+γ,Ω + f 0,Ω ).Proof. The demonstration follows from Theorem 5.1, Propositions 5.1 and 5.2, together with Lemmas 5.4 and 5.5. 10) where C sob , and C reg and C bd are the constants in (2.7), Theorem 2.2 and (3.6), respectively. The next theorem establish the main result of this subsection. Theorem 5.3 Let ψ ∈ Φ ∩ H 2+γ (Ω) and ψ h ∈ M h be the unique solutions of problems (2.3) and (4.1), respectively. Then, under assumption (5.10) there exists a positive constant C, independent of h, such that Theorem 6. 1 1The discrete velocity field u h defined by the relation (6.2) is computable from the degrees of freedom D M 1 − D M 2. Moreover, under the hypotheses of Theorem 5.2, there exists a positive constant C, independent of h, such thatu − u h 0,Ω + h γ |u − u h | 1,h ≤ Ch 2γ ( ψ 2+γ,Ω + f 0,Ω ).Proof. From Lemma 3.2 we have immediately the computability of u h by using D M 1 − D M 2. On the other hand, the error estimate, follow from (6.1), (6.2), the triangular inequality, stability property of Π 1 K , together with Theorems 5.2 and 5.3. Theorem 6. 2 2The discrete vorticity field ω h defined by the relation (6.3) is computable from the degrees of freedom D M 1 − D M 2. Moreover, under the hypotheses of Theorem 5.2, there exists a positive constant C, independent of h, such that ω − ω h 0,Ω ≤ Ch γ ( ψ 2+γ,Ω + f 0,Ω ). Theorem 6. 4 4Let ψ ∈ Φ ∩ H 2+γ (Ω), with γ ∈ (1/2, 1] and ψ h ∈ M h be the solutions of problems (2.3) and (4.1), respectively. Moreover, let (w, p) ∈ H × Q and (w h , p h ) ∈ U h × Q h be the solutions of problems (6.7) and (6.12). Figure 2 :Figure 3 : 23Test 1: Convergence of the stream-function ψ in broken H 2 -, H 1 -and L 2 -norms with mesh refinement for different types of meshes, using ν Test 1: Convergence of the velocity field u in broken H 1 -and L 2 -norms, and vorticity field ω in L 2 -norm with mesh refinement for different types of meshes, using ν = 1. Left panel shows errors curve for velocity in broken H 1 -norm, and middle panel shows error curve for velocity in L 2 -norms, and right panel shows error curves for vorticity in L 2 -norm. Figure 4 : 0 Figure 5 : 405Test 1: Convergence of the stream-function in broken H 2 , H 1 -L 2 -norms with mesh refinement for different types of meshes, using ν = Test 1: Convergence of the velocity field u in broken H 1 -and L 2 -norms, and vorticity field ω in L 2 -norm with mesh refinement for different types of meshes, using ν = 0.01. Left panel shows error curves for velocity in discrete H 1 -norm, and middle panel shows error curves for velocity in L 2 -norms, and right panel shows error curves for vorticity in L 2 -norm. Figure 6 :Figure 7 : 67Test 1: Convergence of the pressure (p) in L 2 -norm with mesh refinement for different types of meshes, using ν = 1 and ν = 0.01. Left panel shows the errors curve of p for ν = 1, and right panel shows the errors curve of p for ν = 0.01. Test 1: "Snapshots" of the approximate stream-function and pressure, using ν = 1 and the mesh T 1 h , Figure 8 : 8Test 3: "Snapshots" of the approximate stream-function and pressure for ν = 0.01, using the mesh T 3 h , with h = 1/64. The next result has been established in[49, Lemma 5.1] and allows show that the application | · | 2,h is a norm in M h . Lemma 4.1 For all φ h ∈ M h , is holds: Ω . Then, we have the following result for the operator T h .Lemma 4.3 The operator T h is well defined. Moreover, if Remark 6.1 We recall that if we are interesting to approximate only the main unknown of problem (2.3), we can consider the Morley-type VE introduced in [3, Subsection 3.2] Furthermore, we let R i (χ), where χ ∈ {u, ψ, ω}, and i ∈ {0, 1, 2} denotes the rates of convergence of the approximate solutions in H 2 -, H 1 -and L 2 -norms.In this numerical test, we solve the Navier-Stokes problem (1.1) on the square domain Ω := [0, 1] 2 . We take the load term f and boundary conditions in such a way that the analytical solution is given by the Kovasznay solution:K 1/2 . (7.1) Figure 1: Sample meshes. T 1 h , T 2 h , T 3 h and T 4 h (from left to right). 7.1 Test 1. Kovasznay flow u(x, y) = 1 − exp(λx) cos(2πy) λ 2π exp(λx) sin(2πy) , ψ(x, y) = y − 1 2π exp(λx) sin(2πy), p(x, y) = −(1/2) exp(2λx) +p, ω = λ 2 − 4π 2 2π exp(λx) sin(2πy), where λ = Re 2 − Re 2 4 + 4π 2 1/2 Table 1 : 1Test 2. Errors for the stream-function, and the post-processed velocity, vorticity fields in broken H 2 -, H 1 -and L 2 -norms for ν = 1, using the mesh T 2 h . Table 2 : 2Test 2. Errors for the pressure variable in L 2 -norm for ν = 1, using the mesh T 2 h . AcknowledgementsThe first author was partially supported by the National Agency for Research and Development, ANID-Chile through Let M h and U h be the spaces defined in (3.4) and in (6.9), respectively. Then. 0it holds that h E2(ψ) R2(ψ) E1(ψ) R1(ψ) E0(ψ) R0(ψ) E1(u) R1(uLet M h and U h be the spaces defined in (3.4) and in (6.9), respectively. Then, it holds that h E2(ψ) R2(ψ) E1(ψ) R1(ψ) E0(ψ) R0(ψ) E1(u) R1(u) E0 The nonconforming virtual element method for Oseen's equation using a stream-function formulation. D Adak, G Manzini, Submitted for publicationD. Adak, G. Manzini, The nonconforming virtual element method for Oseen's equation using a stream-function formulation, Submitted for publication, 2022. A virtual element discretization for the time dependent Navier-Stokes equations in stream-function formulation. D Adak, D Mora, S Natarajan, A Silgado, ESAIM Math. Model. Numer. Anal. 555D. Adak, D. Mora, S. Natarajan and A. Silgado, A virtual element discretization for the time dependent Navier- Stokes equations in stream-function formulation, ESAIM Math. Model. Numer. Anal., 55(5) (2021), pp. 2535-2566. A Morley-type virtual element approximation for a wind-driven ocean circulation model on polygonal meshes. D Adak, D Mora, A Silgado, Preprint: CI2MA:2022-15, CI2MA-UdeCD. Adak, D. Mora and A. Silgado, A Morley-type virtual element approximation for a wind-driven ocean circu- lation model on polygonal meshes, Preprint: CI2MA:2022-15, CI2MA-UdeC. R A Adams, J J F Fournier, Sobolev Spaces. AmsterdamAcademic Press2nd ed.R.A. Adams and J.J.F. Fournier, Sobolev Spaces, 2nd ed., Academic Press, Amsterdam, 2003. Analysis and approximation of a vorticity-velocity-pressure formulation for the Oseen equations. V Anaya, A Bouharguane, D Mora, C Reales, R Ruiz-Baier, N Seloula, H Torres, J. Sci. Comput. 883V. Anaya, A. Bouharguane, D. Mora, C. Reales, R. Ruiz-Baier, N. Seloula and H. Torres, Analysis and approximation of a vorticity-velocity-pressure formulation for the Oseen equations, J. Sci. Comput., 88(3) (2019), pp. 1577-1606. The Virtual Element Method and its Applications. P F Antonietti, L Beirão Da Veiga, G Manzini, SEMA SIMAI Springer Series. 31SpringerP.F. Antonietti, L. Beirão da Veiga and G. Manzini, The Virtual Element Method and its Applications, SEMA SIMAI Springer Series, Springer, Cham, Vol. 31, 2022. A stream virtual element formulation of the Stokes problem on polygonal meshes. P F Antonietti, L Beirão Da Veiga, D Mora, M Verani, SIAM J. Numer. Anal. 52P.F. Antonietti, L. Beirão da Veiga, D. Mora and M. Verani, A stream virtual element formulation of the Stokes problem on polygonal meshes, SIAM J. Numer. Anal., 52 (2014), pp. 386-404. A C 1 virtual element method for the Cahn-Hilliard equation with polygonal meshes. P F Antonietti, L Beirão Da Veiga, S Scacchi, M Verani, SIAM J. Numer. Anal. 54P.F. Antonietti, L. Beirão da Veiga, S. Scacchi and M. Verani, A C 1 virtual element method for the Cahn- Hilliard equation with polygonal meshes, SIAM J. Numer. Anal., 54 (2016), pp. 36-56. The fully nonconforming virtual element method for biharmonic problems. P F Antonietti, G Manzini, M Verani, Math. Models Methods Appl. Sci. 282P. F. Antonietti, G. Manzini, and M. Verani, The fully nonconforming virtual element method for biharmonic problems, Math. Models Methods Appl. Sci., 28(2) (2018), pp. 387-407. The nonconforming virtual element method. B Ayuso De Dios, K Lipnikov, G Manzini, ESAIM Math. Model. Numer. Anal. 503B. Ayuso de Dios, K. Lipnikov and G. Manzini, The nonconforming virtual element method. ESAIM Math. Model. Numer. Anal., 50(3) (2016), pp. 879-904. Basic principles of virtual element methods. L Beirão Da Veiga, F Brezzi, A Cangiani, G Manzini, L D Marini, A Russo, Math. Models Methods Appl. Sci. 23L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L.D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci., 23 (2013), pp. 199-214. H(div) and H(curl)-conforming virtual element methods. L Beirão Da Veiga, F Brezzi, L D Marini, A Russo, Numer. Math. 1332L. Beirão da Veiga, F. Brezzi, L.D. Marini and A. Russo, H(div) and H(curl)-conforming virtual element methods, Numer. Math., 133(2) (2016), pp. 303-332. Preface Special issue -Polyhedral discretization for PDE. L Beirão Da Veiga, A Ern, ESAIM Math. Model. Numer. Anal. 503L. Beirão da Veiga and A. Ern, Preface Special issue -Polyhedral discretization for PDE, ESAIM Math. Model. Numer. Anal., 50(3) (2016), pp. 633-634. Virtual elements for the Navier-Stokes problem on polygonal meshes. L Beirão Da Veiga, C Lovadina, G Vacca, SIAM J. Numer. Anal. 563L. Beirão da Veiga, C. Lovadina and G. Vacca, Virtual elements for the Navier-Stokes problem on polygonal meshes, SIAM J. Numer. Anal., 56(3) (2018), pp. 1210-1242. A virtual element method for the acoustic vibration problem. L Beirão Da Veiga, D Mora, G Rivera, R Rodríguez, Numer. Math. 1363L. Beirão da Veiga, D. Mora, G. Rivera and R. Rodríguez, A virtual element method for the acoustic vibration problem, Numer. Math., 136(3) (2017), pp. 725-763. The Stokes complex for virtual elements with application to Navier-Stokes flows. L Beirão Da Veiga, D Mora, G Vacca, J. Sci. Comput. 812L. Beirão da Veiga, D. Mora and G. Vacca, The Stokes complex for virtual elements with application to Navier- Stokes flows, J. Sci. Comput., 81(2) (2019), pp. 990-1018. Convergence of a finite element discretization of the Navier-Stokes equations in vorticity and stream function formulation. C Bernardi, N Chorfi, ESAIM Math. Model. Numer. Anal. 335C. Bernardi and N. Chorfi, Convergence of a finite element discretization of the Navier-Stokes equations in vorticity and stream function formulation, ESAIM Math. Model. Numer. Anal., 33(5) (2006), pp. 1033-1056. On the boundary value problem of the biharmonic operator on domains with angular corners. H Blum, R Rannacher, Math. Methods Appl. Sci. 24H. Blum and R. Rannacher, On the boundary value problem of the biharmonic operator on domains with angular corners, Math. Methods Appl. Sci., 2(4) (1980), pp. 556-581. Virtual elements for plate bending problems. F Brezzi, L D Marini, Comput. Methods Appl. Mech. Engrg. 253F. Brezzi and L.D. Marini, Virtual elements for plate bending problems, Comput. Methods Appl. Mech. Engrg., 253 (2013), pp. 455-462. The nonconforming virtual element method for the Stokes equations. A Cangiani, V Gyrya, G Manzini, SIAM J. Numer. Anal. 546A. Cangiani, V. Gyrya and G. Manzini, The nonconforming virtual element method for the Stokes equations, SIAM J. Numer. Anal., 54(6) (2016), pp. 3411-3435. Conforming and nonconforming virtual element methods for elliptic problems. A Cangiani, G Manzini, O J Sutton, IMA J. Numer. Anal. 37A. Cangiani, G. Manzini and O.J. Sutton, Conforming and nonconforming virtual element methods for elliptic problems, IMA J. Numer. Anal., 37 (2017), pp. 1317-1354. A priori and a posteriori error analysis of the lowest-order NCVEM for second-order linear indefinite elliptic problems. C Carstensen, R Khot, A K Pani, Numer. Math. 1513C. Carstensen, R. Khot and A.K. Pani, A priori and a posteriori error analysis of the lowest-order NCVEM for second-order linear indefinite elliptic problems, Numer. Math., 151(3) (2022), pp. 551-600. C Carstensen, R Khot, A K Pani, arXiv:2205.08764Nonconforming virtual elements for the biharmonic equation with Morley degrees of freedom on polygonal meshes. math.NAC. Carstensen, R. Khot and A.K. Pani, Nonconforming virtual elements for the biharmonic equation with Morley degrees of freedom on polygonal meshes, arXiv:2205.08764 [math.NA] (2022). Nonconforming finite element discretization for semilinear problems with trilinear nonlinearity. C Carstensen, G Mallik, N Nataraj, IMA J. Numer. Anal. 41C. Carstensen, G. Mallik and N. Nataraj, Nonconforming finite element discretization for semilinear problems with trilinear nonlinearity, IMA J. Numer. Anal., 41 (2021), pp. 164-205. Finite element technique for optimal pressure recovery from stream function formulation of viscous flows. M Cayco, R A Nicolaides, Math. Comp. 46174M. Cayco and R.A. Nicolaides, Finite element technique for optimal pressure recovery from stream function formulation of viscous flows, Math. Comp., 46(174) (1986), pp. 371-377. Analysis of nonconforming stream function and pressure finite element spaces for the Navier-Stokes equations. M Cayco, R A Nicolaides, Comput. Math. Appl. 188M. Cayco and R.A. Nicolaides, Analysis of nonconforming stream function and pressure finite element spaces for the Navier-Stokes equations, Comput. Math. Appl., 18(8) (1989), pp. 745-760. Some error analysis on virtual element methods. L Chen, J Huang, Calcolo. 551L. Chen and J. Huang, Some error analysis on virtual element methods, Calcolo, 55(1) (2018), pp. 5-23. Nonconforming virtual element method for 2mth order partial differential equations in R n. L Chen, X Huang, Math. Comp. 89324L. Chen and X. Huang, Nonconforming virtual element method for 2mth order partial differential equations in R n , Math. Comp., 89(324) (2020), pp. 1711-1744. A posteriori error estimation for a C 1 virtual element method of Kirchhoff plates. M Chen, J Huang, S Lin, Comput. Math. Appl. 120M. Chen, J. Huang and S. Lin, A posteriori error estimation for a C 1 virtual element method of Kirchhoff plates, Comput. Math. Appl., 120 (2022), 132-150. Robust nonconforming virtual element methods for general fourth-order problems with varying coefficients. A Dedner, A Hodson, IMA J. Numer. Anal. 422A. Dedner and A. Hodson, Robust nonconforming virtual element methods for general fourth-order problems with varying coefficients, IMA J. Numer. Anal., 42(2) (2022), pp. 1364-1399. High-Re solutions for incompressible flow using the Navier-Stokes equation and a multigrid method. U Ghia, K N Ghia, C T Shin, J. Comput. Phys. 48U. Ghia, K.N. Ghia and C.T. Shin, High-Re solutions for incompressible flow using the Navier-Stokes equation and a multigrid method. J. Comput. Phys., 48 (1982), pp. 387-411. Numerical computations of viscous, incompressible flow problems using a two-level finite element method, SIAM. F Fairag, J. Sci. Comput. 24F. Fairag, Numerical computations of viscous, incompressible flow problems using a two-level finite element method, SIAM J. Sci. Comput., 24 (2003), pp. 1919-1929. A mixed virtual element method for the Navier-Stokes equations. G N Gatica, M Munar, F Sequeira, Math. Models Methods Appl. Sci. 2814G.N. Gatica, M. Munar and F. Sequeira, A mixed virtual element method for the Navier-Stokes equations, Math. Models Methods Appl. Sci., 28(14) (2018), pp. 2719-2762. V Girault, P A Raviart, Finite Element Methods for Navier-Stokes Equations. BerlinSpringer-VerlagV. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin, 1986. A medius error analysis for nonconforming virtual element methods for Poisson and biharmonic equations. J Huang, Y Yu, J. Comput. Appl. Math. 386ppPaper No. 113229J. Huang and Y. Yu, A medius error analysis for nonconforming virtual element methods for Poisson and biharmonic equations, J. Comput. Appl. Math., 386 (2021), Paper No. 113229, 20 pp. Morley finite element methods for the stationary quasi-geostrophic equation. D Kim, A K Pani, E J Park, Comput. Methods Appl. Mech. Engrg. 375ppPaper No. 113639D. Kim, A.K. Pani and E.J. Park, Morley finite element methods for the stationary quasi-geostrophic equation, Comput. Methods Appl. Mech. Engrg., 375 (2021), Paper No. 113639, 21 pp. Conforming and nonconforming VEMs for the fourth-order reactionsubdiffusion equation: a unified framework. M Li, J Zhao, S Huang, Chen, IMA J. Numer. Anal. 423M. Li, J. Zhao, C Huang and S. Chen, Conforming and nonconforming VEMs for the fourth-order reaction- subdiffusion equation: a unified framework, IMA J. Numer. Anal., 42(3) (2022), pp. 2238-2300. A divergence-free reconstruction of the nonconforming virtual element method for the Stokes problem. X Liu, R Li, Y Nie, Comput. Methods Appl. Mech. Engrg. 37221X. Liu, R. Li and Y. Nie, A divergence-free reconstruction of the nonconforming virtual element method for the Stokes problem, Comput. Methods Appl. Mech. Engrg., 372 (2020), 113351, 21 pp. The nonconforming virtual element method for the Navier-Stokes equations. X Liu, Z Chen, Adv. Comput. Math. 451X. Liu and Z. Chen, The nonconforming virtual element method for the Navier-Stokes equations, Adv. Comput. Math., 45(1) (2019), pp. 51-74. Divergence-preserving reconstructions on polygons and a really pressure-robust virtual element method for the Stokes problem. D Frerichs, C Merdon, IMA J. Numer. Anal. 421D. Frerichs and C. Merdon, Divergence-preserving reconstructions on polygons and a really pressure-robust virtual element method for the Stokes problem, IMA J. Numer. Anal., 42(1) (2022), pp. 597-619. Bivariate spline method for numerical solution of steady state Navier-Stokes equations over polygons in stream function formulation Numer. M J Lai, P Wenston, Methods Partial Differential Equations. 162M.J. Lai and P. Wenston, Bivariate spline method for numerical solution of steady state Navier-Stokes equations over polygons in stream function formulation Numer. Methods Partial Differential Equations, 16(2) (2000), pp. 147- 183. A C 1 virtual element method for the stationary quasi-geostrophic equations of the ocean. D Mora, A Silgado, Comput. Math. Appl. 116D. Mora and A. Silgado, A C 1 virtual element method for the stationary quasi-geostrophic equations of the ocean, Comput. Math. Appl., 116 (2022), pp. 212-228. The triangular equilibrium element in the solution of plate bending problems. L S D Morley, Aero. Quart. 19L.S.D. Morley, The triangular equilibrium element in the solution of plate bending problems, Aero. Quart., 19 (1968), pp. 149-169. Discontinuous Galerkin finite element approximation of the twodimensional Navier-Stokes equations in stream-function formulation. I Mozolevski, E Süli, P R Bösing, Comm. Numer. Methods Engrg. 236I. Mozolevski, E. Süli and P.R. Bösing, Discontinuous Galerkin finite element approximation of the two- dimensional Navier-Stokes equations in stream-function formulation, Comm. Numer. Methods Engrg., 23(6) (2007), pp. 447-459. A Quarteroni, A Valli, Numerical Aproximation of the Partial Differential Equation. Berlin HeidelbergSpringer-VerlagA. Quarteroni and A. Valli, Numerical Aproximation of the Partial Differential Equation, Springer-Verlag Berlin Heidelberg, 1994. Isogeometric analysis and error estimates for high order partial differential equations in fluid dynamics. A Tagliabue, A Dedé, Quarteroni, Comput. & Fluids. 102A. Tagliabue, L Dedé and A. Quarteroni, Isogeometric analysis and error estimates for high order partial differential equations in fluid dynamics, Comput. & Fluids, 102 (2014), pp. 277-303. Conforming and nonconforming virtual element methods for a Kirchhoff plate contact problem. F Wang, J Zhao, IMA J. Numer. Anal. 412F. Wang and J. Zhao, Conforming and nonconforming virtual element methods for a Kirchhoff plate contact problem, IMA J. Numer. Anal., 41(2) (2021), pp. 1496-1521. The divergence-free nonconforming virtual element for the Navier-Stokes problem. B Zhang, J Zhao, M Li, 10.1002/num.22812Numer Methods Partial Differential Equations. Article in pressB. Zhang, J. Zhao and M. Li, The divergence-free nonconforming virtual element for the Navier-Stokes problem, Numer Methods Partial Differential Equations (2021). Article in press, DOI: https://doi.org/10.1002/num.22812. The Morley-type virtual element for plate bending problems. J Zhao, B Zhang, S Chen, S Mao, J. Sci. Comput. 761J. Zhao, B. Zhang, S. Chen and S. Mao, The Morley-type virtual element for plate bending problems, J. Sci. Comput., 76(1) (2018), pp. 610-629. The divergence-free nonconforming virtual element for the Stokes problem. J Zhao, B Zhang, S Mao, S Chen, SIAM J. Numer. Anal. 576J. Zhao, B. Zhang, S. Mao and S. Chen, The divergence-free nonconforming virtual element for the Stokes problem, SIAM J. Numer. Anal., 57(6) (2019), pp. 2730-2759.
{'fraction_non_alphanumeric': 0.09400556270496907, 'fraction_numerical': 0.032846942587903195, 'mean_word_length': 3.285009116378352, 'pattern_counts': {'":': 0, '<': 8, '<?xml version=': 0, '>': 28, 'https://': 1, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 126, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'The nonconforming Morley-type virtual element method for the incompressible Navier-Stokes equations formulated in terms of the stream-function on simply connected polygonal domains (not necessarily convex) is designed. A rigorous analysis by using a new enriching operator is developed. More precisely, by employing such operator, we provide novel discrete Sobolev embeddings, which allow to establish the well-posedness of the discrete scheme and obtain optimal error estimates in broken H 2 -, H 1 -and L 2 -norms under minimal regularity condition on the weak solution. The velocity and vorticity fields are recovered via a postprocessing formulas. Furthermore, a new algorithm for pressure recovery based on a Stokes complex sequence is presented. Optimal error estimates are obtained for all the postprocessed variables. Finally, the theoretical error bounds and the good performance of the method are validated through several benchmark tests.', 'arxivid': '2212.02173', 'author': ['D Adak ', 'D Mora ', 'A Silgado '], 'authoraffiliation': [], 'corpusid': 254246674, 'doi': '10.48550/arxiv.2212.02173', 'github_urls': [], 'n_tokens_mistral': 36457, 'n_tokens_neox': 31810, 'n_words': 18834, 'pdfsha': '1bbb60db631b14cad6cb053f37fadb4dbb88c9ca', 'pdfurls': ['https://export.arxiv.org/pdf/2212.02173v1.pdf'], 'title': ['The Morley-type virtual element method for the Navier-Stokes equations in stream-function form on general meshes', 'The Morley-type virtual element method for the Navier-Stokes equations in stream-function form on general meshes'], 'venue': []}
arxiv
Interpreting Models of Infectious Diseases in Terms of Integral Input-to-State Stability 6 Apr 2020 Hiroshi Ito Interpreting Models of Infectious Diseases in Terms of Integral Input-to-State Stability 6 Apr 2020arXiv:2004.02552v1 [math.OC] 1 The notions of integral input-to-state stability (iISS) and input-to-state stability (ISS) have been effective in addressing nonlinearities globally without domain restrictions in analysis and design of control systems. In particular, they provide useful tools of module-based methods integrating characteristics of components. This paper applies the framework of module-based analysis to ordinary differential equations which deterministically describe dynamics of prevalence and the duration of epidemics. The objective is to express fundamental properties of models of infectious diseases and vaccination through the language of iISS and ISS. The systematic treatment is expected to facilitate development of effective schemes of controlling the spread of diseases via non-conventional Lyapunov functions.Index Terms-Epidemic models, integral input-to-state stability, Lyapunov functions, positive nonlinear network, small gain theorem. I. INTRODUCTION For many decades, mathematical models of infectious diseases have been recognized as useful tools for public health decision-making during epidemics [1]- [4]. Detailed models help predict the future course of outbreak, while simple models allow one to understand mechanisms whose interpretations can lead to ideas of control strategies, such as vaccination, isolation, digital contact tracing and culling, which slow or ultimately eradicate the infection from the population. The objective of this paper is to facilitate the development of the latter. This paper does not report any novel behavior of disease transmission. Instead, this paper is devoted to formulating a systematic understanding behavior of classical and simple models of infectious diseases in the language of integral inputto-state stability (iISS) and input-to-state stability (ISS). It aims to take a first step toward development of an iISS/ISStheoretic foundation for systematic control design to combat infectious diseases. This paper reports that popular models share essentially the same qualitative behavior which can be analyzed and explained systematically via the same tools of iISS/ISS. The notion of iISS and ISS have been accepted widely as mathematical tools to deal with and utilize nonlinearities effectively in the area of control [5], [6]. The notions offer a systematic framework of module-based design of control systems. Once a system or a network is divided into 'stable" components, aggregating characteristics of components gives answers to control design problems systematically. The answers are global, and they are not restricted to small domains of variables. Without relying linearity, ISS allows one to H. Ito is with the Department of Intelligent and Control Systems, Kyushu Institute of Technology, 68-4 Kawazu, Iizuka, 820-8502, Japan (e-mail: [email protected]). handle systems based on boundedness of states with respect to bounded inputs. Importantly, the boundedness does not require finite operator gain, so that replacing linearity with ISS, we can handle a large class of nonlinear systems. However, nonlinearity such as bilinearity and saturation often prevent systems from being ISS. They are cases where nonlinearities retain convergence of state variables in the absence of inputs, but prevent state variables from being bounded in the presence of inputs. Such nonlinear systems are covered by iISS. Systems whose state is bounded for small inputs are grouped into the class of Strong iISS systems [7]. ISS and (Strong) iISS characterize both internal and external stability properties. The weak "stability" of (Strong) iISS components can be compensated by ISS components. This fact is one of useful and powerful tools of the iISS/ISS framework. Some of main ideas of iISS/ISS module-based arguments are packed in the iISS small-gain theorem [8]- [11] which is an extension of the ISS small-gain theorem [12]- [16]. One of the important features of the small-gain methodology is that for interconnected systems and networks, it gives formula to explicitly construct nonconventional Lyapunov functions which not only establish stability properties of equilibria, but also properties with respect to external variables and parameters. For understanding behavior of diseases transmission, construction of Lyapunov functions has been one of major directions in mathematical epidemiology, and Lyapunov functions are known to be useful for analyzing global properties of stability of each given equilibrium (see [17]- [24] and references therein). However, the Lyapunov functions are classical, so that they do not unite characterizations of stability properties which vary with parameters and external elements. In other words, bifurcation analysis need to be performed separately to divide the Lyapunov function analysis into cases. This paper gives interpretations to behavior of typical models of infectious diseases in terms of basic characterizations of iISS and ISS [6], [25], [26] with the help of two tools provided by iISS and ISS. One tool is a criterion of the smallgain type. The other is a fusion of global and local nonlinear gain of components. The two tools formulated in this paper are not completely novel ideas, but they extend standard concepts in the iISS/ISS methodology by specializing in the setup of diseases models. In addition to their usefulness, this paper illustrates how the same set of the basic characterizations and tools of iISS/ISS can be applied to popular models of infectious diseases uniformly to help explain and capture their fundamental behavior globally without dividing the analysis into cases a prior. II. PRELIMINARIES Let the set of real numbers be denoted by R := (−∞, ∞). This paper uses the symbols R + := [0, ∞) and R n + := [0, ∞) n . used. A function η : R + → R + is said to be of class P and written as η ∈ P if η is continuous and satisfies η(0) = 0 and η(s) > 0 for all s ∈ R + \ {0}. A class P function is said to be of class K if it is strictly increasing, A class K function is said to be of class K ∞ if it is unbounded. A continuous function β : R + × R + → R + is said to be of class KL if, for each fixed t ≥ 0, β(·, t) is of class K and, for each fixed s > 0, β(s, ·) is decreasing and lim t→∞ β(s, t) = 0. The zero function of appropriate dimension is denoted by 0. Composition of η 1 , η 2 : R + → R + is expressed as η 1 • η 2 . For a continuous function η : R + → R + , the function η ⊖ : R + → R + := [0, ∞] is defined as η ⊖ (s) = sup{τ ∈ R + : s ≥ η(τ )}. By definition, for η ∈ K, η ⊖ (s) = ∞ holds for all s ≥ lim τ →∞ η(τ ), and η ⊖ (s) = η −1 (s) elsewhere. The set {1, 2, 3, . . . , n} is denoted by n. For a set U , its cardinality is denoted by |U |. For U ⊂ n and x ∈ R n , the vector x U ∈ R |U| is the collection of |U | components x i , i ∈ U . A system of the forṁ x(t) = f (x(t), u(t))(1) is said to be integral input-to-state stable (iISS) with respect to the input u [6] if there exist β ∈ KL, χ, µ ∈ K ∪ {0} such that, for all measurable locally essentially bounded functions u : R + → R p , all x(0) ∈ R n and all t ≥ 0, its solution x(t) exists and satisfies |x(t)| ≤ β(|x(0)|, t) + χ t 0 µ(|u(τ )|)dτ ,(2) where the symbol | · | denotes the Euclidean norm. System (1) is said to be strongly integral input-to-state stable (Strongly iISS) with respect to the input u [7] if there exist R > 0 and γ ∈ K ∪ {0} such that ess sup t∈R+ |u(t)| < R ⇒ |x(t)| ≤ β(|x(0)|, t) + γ(ess sup t∈R+ |u(t)|)(3) holds in addition to the above requirement of (2). The constant R is called an input threshold. If the requirement (3) is met with R = ∞, system (1) is said to be input-to-state stable (ISS) [5]. The function γ is called an ISS-gain function. An ISS system is Strongly iISS. A Strongly iISS system is iISS. Their converses do not hold true. iISS of (1) implies globally asymptotic stability of the equilibrium x = 0 for u = 0. If a radially unbounded and continuously differentiable function V : R n → R + satisfies ∀x ∈ R n ∀u ∈ R p ∂V ∂x (x)f (x, u) ≤ −α(V (x)) + σ(|u|)(4) for some σ ∈ K and some α ∈ P (resp., α ∈ K and lim s→∞ α(s) ≥ lim s→∞ σ(s)), the function V (x) is said to be an iISS (resp., ISS) Lyapunov function. The existence of an iISS (resp., ISS) Lyapunov function guarantees iISS (resp., ISS) of system (1) [25], [26]. If an iISS Lyapunov function admits α ∈ K, the system is guaranteed to be Strong iISS [7]. iISS and ISS Lyapunov functions are conventional Lyapunov functions when the input u is zero in system (1). The Lyapunov-type characterization (4) yields lim sup t→∞ V (x(t)) ≤ γ(ess sup t∈R+ |u(t)|),(5) where γ := α ⊖ • σ. If u(t) ≡ u 0 is a constant and (4) holds with an equality sign, we have lim t→∞ V (x(t)) = γ(|u 0 |). The inequality (5) is called the asymptotic gain property [25]. If system (1) is not ISS, there exists no γ ∈ K satisfying the asymptotic gain property (5). This is because (3) never holds with R = ∞. As in (2), an iISS system which is not ISS accumulates its input, and the solution x(t) exists for all t ∈ R + , but unbounded if the input is persistent. All the above are standard definitions given for sign-indefinite system (1). When the vector field f generates only non-negative x(t) in (1) defined with x(0) ∈ R n + and u(t) ∈ R p + , all the above definitions and facts are valid by replacing R with R + . III. A SMALL-GAIN THEOREM FOR GENERALIZED BALANCING KINETICS This section shows a small-gain type method for establishing stability of dynamical networks in the framework of iISS. To propose a generalized formulation which includes a previously-developed criterion as a special case, consider x(t) = [x 1 (t), x 2 (t), . . . , x n (t)] T : R + → R n + governed bẏ x i = − η i−1,i (x) + σ i,i−1 (x) − η i+1,i (x) + σ i,i+1 (x) − β i (x i ) + κ i (w i ), i ∈ n (6) for any x(0) ∈ R n + and any measurable and locally essentially bounded function w(t) = [w 1 (t), w 2 (t), . . . , w n (t)] T ∈ R n + . In (6), the subscripts of η and σ are integers which are circular of length n. If a subscript k by itself does not belong to n, it stands for ((k − 1) mod n) + 1, where i mod n denotes the non-negative reminder of the division of an integer i by n. All subscripts in this section are circular. Assume that the functions η i−1,i , η i+1,i , σ i,i−1 , σ i,i+1 : R n + → R + and β i ∈ P are locally Lipschitz and satisfy 1 x i = 0 ⇒ η i−1,i (0) = η i+1,i (0) = 0(7) for all i ∈ n. For all i ∈ n, κ i ∈ K ∪ {0} is assumed. Network (6) is made of balancing mechanisms between state variables. The component x i is consumed at the rate −η i+1,i (x), and the consumption leads to the production of the downstream component x i+1 at the rate +σ i+1,i (x). In the same way, the consumption −η i−1,i (x) of x i produces the downstream component x i−1 at the rate +σ i−1,i (x) . The rates are allowed to be functions of x instead of the local variable x i . The extra rate β i of consumption in either the upstream or the downstream direction is a function of the local variable x i . It is also important that the balancing between neighbors forms not only cycles of length 1, but also cycles of length n. Assume that the rate functions in (6) satisfy 1 Under the assumption (8), the implication (7) is necessary and sufficient for guaranteeing ∀i ∈ n ∀j ∈ {i − 1, i + 1} ∃ℓ i,j ≥ 0 ∀x ∈ R n + σ i,j (x) ≤ ℓ i,j η i,j (x).(8)x(t) ∈ R n + . The following theorem can be proved. Theorem 1: Assume that ∀i ∈ n ℓ i,i+1 ℓ i+1,i ≤ 1 (9) ∃k ∈ n ℓ k,k+1 ℓ k+1,k ≤ n i=1 ℓ i,i+1 ℓ i+1,i ≤ 1 ℓ k,k+1 ℓ k+1,k(10) are satisfied. Then network (6) is iISS with respect the input w. If β i ∈ K (resp., β i ∈ K ∞ ) holds for all i ∈ n in addition, network (6) is Strongly iISS (resp., ISS) with respect the input w. Furthermore, an iISS/ISS Lyapunov function V (x) is V (x) = i∈n λ i x i (11) ∀i ∈ {k+1, k+2, . . . , k+n} λ i = i j=k+2 ℓ j−1,j ℓ j,j−1 .(12) Proof: Due to (9), from (8) and (12) it follows that λ i+1 σ i+1,i (x) = ℓ i,i+1 ℓ i+1,i λ i σ i+1,i (x) ≤ ℓ i,i+1 ℓ i+1,i λ i η i+1,i (x i ) ≤ λ i η i+1,i (x) λ i σ i,i+1 (x) = ℓ i+1,i ℓ i,i+1 λ i+1 σ i,i+1 (x) ≤ λ i+1 η i,i+1 (x) for all x ∈ R n + at each i ∈ {k + 1, k + 2, . . . , k + n − 1}. By virtue of the second inequality in (10), properties (8) and (12) yield λ k σ k,k+1 (x) = k+n−1 i=k+1 ℓ i,i+1 ℓ i+1,i λ k+1 σ k,k+1 (x) ≤ ℓ k+1,k ℓ k,k+1 · 1 ℓ k,k+1 ℓ k+1,k λ k+1 σ k,k+1 (x) ≤ λ k+1 η k,k+1 (x). In the same way, the first inequality in (10) gives λ k+1 σ k+1,k (x) ≤ λ k η k+1,k (x). Therefore, the function V given in (11) satisfieṡ V ≤ i∈n λ i (−β i (x i ) + k i (w i )) along the trajectory x(t) of network (6). Defining α(s) := min {x∈R n + :V (x)=s} i∈n β i (x i ) σ(s) := i∈n κ i (s) satisfies α ∈ P, κ i ∈ K ∪ {0} and (4). Therefore, the function V is an iISS Lyapunov function establishing iISS of network (6). We have α ∈ K establishing Strong iISS of network (6) if β i ∈ K for all i ∈ n. In the case of β i ∈ K ∞ for all i ∈ n, we have α ∈ K ∞ , and the function V is an ISS Lyapunov function. Recall that circulating subscripts of length n are used for ℓ i,j and λ i . The above theorem extends a development of [27] in which η j,i and σ j,i were restricted to be functions of x i . The restriction disallows bilinearities and multiplicative nonlinearities to appear in η j,i and σ j,i . Removal of the restrictions by Theorem 1 is the theoretical key in this paper. Remark 1: If there exists k ∈ n such that ℓ k,k+1 = 0 achieving (8), the first inequality in (10) is satisfied automatically with a sufficiently small ℓ k+1,k > 0. The same applies to the case of ℓ k,k+1 = 0 with the second inequality in (10). Remark 2: Theorem 1 reduces to a special case of the iISS small-gain theorem for networks proposed in [10]. When the functions in (6) are restricted to ∃b i−i,i , b i+i,i > 0 ∀x ∈ R n + b i−i,i η i−i,i (x) = b i+i,i η i+i,i (x) = β i (x i ) (13) β i ∈ K (14) ∀x ∈ R n + ∀x ∈ R n + ∩ {x i = x i } σ i,i−i (x) = σ i,i−i (x), σ i,i+i (x) = σ i,i+i (x)(15) for all i ∈ n, network (6) fit in the setup of [10], and the conditions (9) and (10) coincide with the cyclic small-gain condition presented in [10], If all the functions in (6) are nonzero, the network consists of n simple directed cycles of length 1, and 2 simple directed cycles of length n. the inequality (9) is the small-gain requirement for the cycles of length 1, while the two inequalities in (10) are for length n. IV. CONVERGENCE VIA ZERO LOCAL ISS-GAIN This section focuses on an extra property which iISS systems often possess due to bilinear or multiplicative nonlinearities. To formulate it, consider x(t) ∈ R + governed bẏ x = f (x, w) :=      f 1 (x, w) f 2 (x, w) . . . f n (x, w)     (16) for any x(0) ∈ R n + and any measurable and locally essentially bounded function w(t) = [w 1 (t), w 2 (t), . . . , w p (t)] T ∈ R p + . It is assumed that f : R n + × R p + → R is locally Lipschitz functions satisfying f (0, 0) = 0 and x i = 0 ⇒ ∀x ∈ R n + ∀w ∈ R p + f i (x, w) ≥ 0.(17) Property (17) is necessary and sufficient for guaranteeing x(t) ∈ R + with respect to all x(0) and w. The following proposition describes a property of iISS systems which completely reject the effect of small inputs on some state variables. Proposition 1: Suppose that system (16) is iISS with respect the input w, and admits a non-empty set U ⊂ n, a radially unbounded and continuously differentiable function V U : R |U| → R + , and class K functions α U , σ U such hat ∀x ∈ R n + ∀w ∈ R p + ∂V U ∂x U (x U )f U (x, w) ≤ −α U (V U (x U )) + σ U (w). (18) If there exist a non-empty set L ⊂ n, a real number H > 0, and a radially unbounded and continuously differentiable function V L : R |L| → R + such that for each k ∈ (0, 1), ∀x ∈ R n + ∩ {V U (x U ) ≤ kH} ∀w ∈ R p + ∂V L ∂x L (x L )f L (x, w) ≤ −α L (V L (x L ))(19) holds for some α L ∈ P, then the solution x(t) of (16) satisfies sup t∈R+ w(t) < Q ⇒ ∀x(0) ∈ R n + lim t→∞ x L (t) = 0(20) for Q = lim s→H− σ ⊖ U • α U (s). Proof: Since system (16) is iISS, a solution x(t) < ∞ exists and unique for all t ∈ R + . Assume that a real number H ∈ (0, ∞) satisfies (19). Suppose that sup t∈R+ w(t) < Q for Q = lim s→H− σ ⊖ U • α U (s). In the case of Q = ∞, for each w, assumption (18) implies the existence of T ∈ R + and ǫ ∈ (0, H) such that ∀t ∈ [T, ∞) V U (x U (t)) ≤ H − ǫ.(21) In the case of Q < ∞, since for each w, there exists δ > 0 satisfying sup t∈R+ w(t) ≤ Q − δ , evaluating (18) allows one to verify the existence of T ∈ R + and ǫ ∈ (0, H) fulfilling (21) again. Therefore, the existence of α L ∈ P satisfying (19) for each k ∈ (0, 1) ensures (20). The above proposition makes use of property (19) which is the existence of a partial system completely rejecting the effect of its interconnecting inputs whose magnitude is smaller than a threshold. Let L C = n \ L. For x L -system defined bẏ x L = f L (x, w), x L C (t) and w(t) are exogenous. Property (19) implies that the ISS-gain from the input [x T L C , w T ] T to the state x L is zero as long as V U (x U ) < H. Hence, in Proposition 1, x L -system is required to admit zero ISS-gain locally, although it is not required to be ISS globally. In the case of L = n, property (19) implies Strong iISS of x L -system since iISS is assumed in Proposition 1. This property (19) is very common, although it may not have been focused very often in the literature of systematic control methodology. For example, any of scalar non-negative systemṡ ξ = −ξ + ξẇ ξ = − ξ 1 + ξ + ξw 1 + ξ ξ = − ξ 1 + ξ 2 + ξw 1 + ξ 2 has the threshold at H = 1 for x L = ξ and V (x L ) = x L . The state ξ converges to zero for w < 1, while ξ increases as long as w > 1, although the increase is within the property of iISS . The above three systems are Strong iISS. The zero local ISS-gain they actually have is "stronger" than Strong iISS. In this way, the threshold is a bifurcation point bringing in a superior stability property. Bilinearity and multiplicative nonlinearity give rise to the bifurcation, and Proposition 1 aims at highlighting such systems. Remark 3: Since Proposition 1 assumes the entire system (16) to be only iISS, the variable x L (t) can increase until V U (x U (t)) < H is achieved. Then an outbreak of x L occurs, and x L (t) exhibits a peak. Property (18) allows one to estimate that the smaller w is, the shorter the time when x L starts decreasing (the duration of the growth phase) is. The larger Q or H is, the shorter the growth phase is. The increasing rate of x L (t) until V U (x U (t)) < H can be estimated by the iISS property of the entire system, or x L -system, which is more specific than the entire system. For example, an upper bound of (∂V L /∂x L )f L for V U (x U ) > H gives such information, which is not stated explicitly in Proposition 1. V. SIR MODEL Considers the solution x(t) := [S(t), I(t), R(t)] T ∈ R 3 + of the ordinary differential equatioṅ S =B − µS − βIS (22a) I =βIS − γI − µI (22b) R =γI − µR (22c) defined for any [S(0), I(0), R(0)] T ∈ R 3 + and any measurable and locally essentially bounded function B : R + → R + . The variable S(t) describes the (continuum) number of susceptible, I(t) is the number of infectious, while R(t) is of people recovered with immunity. B(t) is the newborn rate. The positive number β, γ and µ are parameters describing the contact rate, the recovery rate and the death late, respectively. The equation (22) is referred to as the (classic) SIR endemic model [3]. When B = 0 and µ = 0, the equation is referred to as the (classic) SIR epidemic model. S-system (22a) and R-system (22c) are ISS with respect to the input [B, I] T and the input I, respectively. In fact, the positive variables S and R by themselves are ISS Lyapunov functions, due to the definition (4). The ISS property is also clear since R-system (22c) is linear, and S-system (22a) is bounded from above by the solution of the linear systeṁ S = B−µS. The two linear systems are asymptotically stable. By contrast, due to the bilinear term βIS, I-system (22b) is not ISS with respect to the input S since (22b) generates unbounded I(t) for any constant input S > (γ + µ)/β. Isystem (22b) is Strong iISS since V I (I) = log(1 + I) yieldṡ V I ≤ − (γ + µ)I 1 + I + βS. Hence, the SIR model (22) is a cascade consisting of an ISS system, an Strong iISS system and an ISS system. The argument of iISS cascade analysis [8], [28]- [31] can show that the SIR model (22) is Strong iISS with respect to the input B. This stability assessment is not yet very informative, although it holds true. The rest this section demonstrates that the developments in Sections III and IV improve the stability assessment for capturing discriminative behavior of the spread of infectious diseases. Theorem 1 establishes that the SIR model (22) is ISS, which is "stronger" than Strong iISS, although the SIR model involves I-system which is not ISS. In fact, the model (22) satisfies (6) and (8) with ℓ i+1,i = ℓ i,i+1 = 1, β i (s) = µs, i = 1, 2, 3.(23) These parameters satisfy (9) and (10), and yield λ 1 = λ 2 = λ 3 = 1. Theorem 1 gives the total number N (t) := S(t) + I(t) + R(t) = V (x(t)) as an ISS Lyapunov function for the SIR model (22). Indeed,Ṅ = −µN + B(24) is satisfied along the solution x(t) of (22). The above inequality implies lim t→∞ N (t) = B µ(25) for any constant B ∈ R + . If B is not constant, we have the asymptotic gain property lim sup t→∞ N (t) ≤ 1 µ ess sup t∈R+ B(t).(26) SI-system consisting of (22a) and (22b) is ISS with respect to the input B since defining the sum V U (x U (t)) = S(t)+I(t) yieldsV U = −µV U − γI + B for x U = [S, I] T . This confirms (18) with α U (s) = µs and σ U (s) = s. I-system defined by (22b) fulfills (19) with x L = I and H := γ + µ β (27) since S ≤ V U (x U ). Recall that ISS of the entire model (22) implies iISS of (22). Hence, Proposition 1 with Q = µH guarantees ess sup t∈R+ B(t) < µH ⇒ lim t→∞ I(t) = 0.(28) The convergence of I to zero implies lim t→∞ R(t) = 0 since (22c) is ISS. Indeed, it is a stable linear system. Thus For any constant B ∈ R + , by virtue of (25) or (22a), we arrive at B < µH ⇒ lim t→∞ x(t) = B µ , 0, 0 T .(30) As Proposition 1 is applied in the above analysis, regarding S as a parameter, I-system has a bifurcation point at S = H. The origin I = 0 is an unstable equilibrium if S > H. As explained in Remark 3, I(t) increases until S(t) ≤ H. It is not a possibility any more since it is confirmed that I-system is not ISS. The growth phase occurs. In the case of B < µH and S(0) > H, if I(0) > 0, the infectious I(t) peaks before converging to zero. The smaller B and the larger (γ + µ)/β are, the shorter the time to the peak is. The larger µ and the smaller β are, the smaller the increase rate of I in the growth phase is. Typical responses of the SIR model (22) are shown in Fig. 1 and Fig. 2 for β = 0.0002, µ = 0.015 and γ = 0.032 with S(0) = 700, I(0) = 200 and R(0) = 70. For a general newborn rate B, the SIR model (22) is ISS. It means that the disease can remain as endemic. That is, the infectious population I(t) is bounded, but it can become very large, and lim inf t→∞ I(t) > 0 can hold. Since I-system is Strong iISS and it is not ISS, the infectious I(t) starts with an growth phase unless the initial susceptible population is below the threshold H = 235. The infectious population I(t) never decreases to zero if the newborn rate B is above the threshold µH = 3.525 (Fig. 1). If the newborn rate B does not exceed the threshold, the convergence of I(t) to zero is guaranteed, and the disease is eradicated (Fig. 2). This bifurcation is the central feature of the SIR model (22). In mathematical epidemiology, for constant B ∈ R + , the value R 0 := βB µ(γ + µ) = B µH(31) is called the basic reproduction number. Solving the simultaneous equation ofṠ = 0,İ = 0 andṘ = 0 in (22) with constant B ∈ R + gives the steady-state value in (30) and R 0 ≥ 1 ⇒ x e = H, µ(R 0 − 1) β , γ(R 0 − 1) β T .(32) The steady-state value in (30) is called the disease-free equilibrium, while x e in (32) is called the endemic equilibrium. Since x(t) ∈ R + , the endemic equilibrium exists only if R 0 ≥ 1. For R 0 = 1, the endemic equilibrium is identical with the diseasefree equilibrium. The endemic equilibrium is consistent with (25) since the definition of R 0 yields H + µ(R 0 − 1) β + γ(R 0 − 1) β = (γ + µ)R 0 β = B µ .(33) Recall that the entire SIR model (22) and SI-systems are ISS. Since iISS of I-system which is not ISS accumulates the amount S(t) − H, there exists a unbounded increasing sequence {t i } i∈{0,1,2,...} in R + such that ∀x(0) ∈ R + \{I(0) > 0} lim i→∞ x 2 (t i ) = x e,2(34) if R 0 > 1 for constant B ∈ R + , where x e = [x e,1 , x e,2 , x e,3 ] T . with [S(0), E(0), I(0)] T ∈ R 3 + and B : R + → R + . The equation (35) is referred to as the SEIS model [18]. The SEIS model is known to be useful for describing diseases which have non-negligible incubation periods. The variable E represents the (continuum) number of infected individuals who are not yet infectious. The SEIS model also consider infections which do not give long lasting immunity, and recovered individuals become susceptible again. As seen in (35), the short immunity forms a circle of length 3, which the SIR model does not have. VI. SEIS MODEL Considers x(t) := [S(t), E(t), I(t)] T ∈ R 3 + governed bẏ S =B − µS − βIS + γI (35a) E =βIS − ǫE − µE (35b) I =ǫE − γI − µI (35c) The SEIS model (35) satisfies (6) and (8) with (23). Conditions (9) and (10) are satisfied. Thus Theorem 1 establishes ISS of the SEIS model (35) with respect to the input B, and an ISS Lyapunov function is obtained as (11) with λ 1 = λ 2 = λ 3 = 1, i.e., V (x) = S + E + I =: N . In fact, properties (24), (25) and (26) (18) with α U (s) = µs and σ U (s) = s. EI-system consisting of (35b) and (35c) is iISS with respect to the input S since the choice V EI = log(1 + E + I) satisfieṡ V EI ≤ − µ(E + I) 1 + E + I + βS(36) along the solution [E(t), I(t)] T of EI-system. EI-system is, however, not ISS. EI-system is a Strongly iISS system admitting a zero local ISS-gain. To see this, one can make use of the Lyapunov function proposed in Theorem 1 by regarding S as a constant S ♯ for EI-system. EI-system satisfies (6) and (8) with ℓ 1,2 = βS ♯ a(γ + µ) , ℓ 2,1 = ǫ a(ǫ + µ)(37)β 1 (s) = (1−a)(ǫ+µ)s, β 2 (s) = (1−a)(γ +µ)s(38) for an arbitrarily given a ∈ (0, 1). Defining V L (x L ) for x L = [E, I] T as in (11) and (12) gives Then along the solution of (35b) and (35c) we obtain V L (x L ) = E + λ 2 I, λ 2 = βS ♯ (ǫ + µ) (γ + µ)ǫ .(39)d dt V L (x L ) = βǫ(ǫ + µ) γ + µ √ S ♯ − (γ + µ)(ǫ + µ) βǫ E + β 2 S ♯ S √ S ♯ − (γ + µ)(ǫ + µ) βǫ I.(40) Define H := (γ + µ)(ǫ + µ) βǫ .(41) Equation (40) allows one to see that S = H is a bifurcation point of EI-system. To this end, pick S ♯ = a 2 H. Due to S ≤ V U (x U ), EI-system satisfies (19) with (41) since 2 such a parameter a ∈ (0, 1) can be taken for each k ∈ (0, 1). On the other hand, if S is a constant satisfying S > H, equation (40) with S ♯ = S yields dV L (x L )/dt > 0 unless E = I = 0. Therefore, EI-system is not ISS, but the convergence property (19) is met. Hence, Proposition 1 can be invoked for Q = µH, and concludes that sup s∈R+ B(t) < µH ⇒ lim t→∞ E(t) = lim t→∞ I(t) = 0.(42) From (25) VII. MSIR AND SEIR MODELS Considers x(t) := [M (t), S(t), I(t), R(t)] T ∈ R 4 + foṙ M =B − δM − µM (43a) S =δM − µS − βIS (43b) I =βIS − γI − µI (43c) R =γI − µR (43d) which is called the MSIR model [3], [32]. The variable M represents delay in becoming susceptible due to the maternally derived immunity. The analysis of the MSIR model is almost the same as that of the SIR model. With ℓ i+1,i = ℓ i,i+1 = 1, β i (s) = µs, i = 1, 2, 3, 4,(44) Theorem 1 assures that the function N (t) := M (t) + S(t) + I(t) + R(t) proves ISS of (43), and the ultimate bounds (25) and (26) via (24). Thus, the choices V U (x) = N and x = x U give (18) with α U (s) = µs and σ U (s) = s. Because of the bilinear term βIS, I-system in (43) is not ISS, but Strongly iISS. Thus, the variable I(t) increases until S(t) ≤ H, where the bifurcation point H is defined as (27). The ISS gain of I-system is zero for the input sup s∈R+ S(t) < H. In fact, I-system satisfies (19). Thus, Proposition 1 establishes B < µH ⇒ lim t→∞ x(t) = B µ , 0, 0, 0 T(45) for the MSIR model (43) in the case of constant B ∈ R + . Finally, the SEIR model consists oḟ S =B − µS − βIS (46a) E =βIS − ǫE − µE (46b) I =ǫE − γI − µI (46c) R =γI − µR. (46d) Its state vector is x(t) := [S(t), E(t), I(t), R(t)] T ∈ R 4 + [33]. The SEIR model can be analyzed as done for the SIES model. Theorem 1 qualifies N (t) := S(t) + E(t) + I(t) + R(t) as an ISS Lyapunov function proving ISS of (46), and provides the ultimate bounds (25) and (26) via (24). Property (18) holds with α U (s) = µs and σ U (s) = s for V U (x) = N and x = x U . EIR-system consisting of (46b), (46c) and (46d) is Strongly iISS with respect to the input S, although the bilinear term βIS prevent it from being ISS. Using V L (x L ) = λ E E + λ I I + λ R R for appropriate λ E , λ I , λ R > 0 given by Theorem 1, one can show that EIR-system is not ISS. The function V L (x L ) also shows that EIR-system admits the zero ISS gain for the input sup s∈R+ S(t) < H, where the bifurcation point H is defined as (41). EIR-system satisfies (19). Therefore, Proposition 1 concludes that the SEIR model (46) satisfies (45) for constant B ∈ R + . VIII. VACCINATION MODELS One way of eradicating infectious diseases is to vaccinate newborns. Let the constant P ∈ (0, 1) denote the vaccination fraction. Considering a vaccine giving lifelong immunity [3], the SIR model can be modified aṡ S =B(1 − P ) − µS − βIS (47a) I =βIS − γI − µI (47b) R =γI − µR (47c) A =BP − µA,(47d) where A is the number of vaccinated individuals. Since (8) is satisfied with (44), Theorem 1 assures that N (t) := S(t) + I(t) + R(t) + A(t) is an ISS Lyapunov function for the model (47), and establishes the ultimate bounds (25) and (26) via (24). S-system is ISS with respect to the input B. In fact, property (18) is met with V U (x U ) = S, x U = S, α U (s) = µs and σ U (s) = (1−P )s. IR system is the same as that of the SIR model. Hence, a bifurcation point H is obtained as (27). Asystem is ISS. The convergence of I to zero does not imply lim t→∞ R(t) = 0. The variable R reaches its steady state since R-system is ISS. Indeed, it is a stable linear system. Hence, Proposition 1 guarantees B(1−P ) < µH ⇒ lim t→∞ x(t) = B(1−P ) µ , 0, 0, BP µ T(48) for any constant B ∈ R + and P ∈ (0, 1). Hence, a vaccination fraction P which is sufficiently close to 1 can eradicate the disease. Another way to model the newborn vaccination within the SIR model isṠ =B(1 − P ) − µS − βIS (49a) I =βIS − γI − µI (49b) R =γI − µR + BP.(49c) Assumption (8) is satisfied with (23), Theorem 1 proves ISS of (49) with N (t) := S(t) + I(t) + R(t), which establishes (25) and (26) via (24). The reminder of the analysis is the same as that of the model (47) except that the convergence of I to zero does not imply lim t→∞ R(t) = 0. Since the scalar R-system is ISS, it is clear that B(1 − P ) < µH ⇒ lim t→∞ x(t) = B(1 − P ) µ , 0, BP µ T(50) for any constant B ∈ R + and P ∈ (0, 1). The same modifications to other disease models in the previous sections can be possible for modeling the newborn vaccination [34]. Their analysis goes in essentially the same as to the one described above for the SIR model. If non-newborns are vaccinated instead of the newborn [35], [36], a way to modify the SIR model iṡ S =B − ρS − µS − βIS (51a) I =βIS − γI − µI (51b) R =γI − µR (51c) A =ρS − µA,(51d) where the constant ρ ∈ R + is the vaccination rate. The analysis is the same as that of (47) except that ISS of S-system with respect to the input B yields property (18) with V U (x U ) = S, x U = S for α U (s) = (ρ + µ)s and σ U (s) = s. Hence, B < (ρ + µ)H ⇒ lim t→∞ x(t) = B(1−P ) µ , 0, 0, BP µ T(52) for any constant B ∈ R + and ρ ∈ R + . Thus, the disease can be eradicated by a sufficiently high vaccination rate ρ. Irrespective of B < (ρ + µ)H, the ultimate bounds (25) and (26) hold, and the model (51) also has the bifurcation point at S = H given in (27). IX. CONCLUDING REMARKS This paper has investigated popular models of infectious diseases from the viewpoint of iISS and ISS. It has been shown that behavior of all the models can be analyzed uniformly in terms of a asymptotic gain property of ISS, and a Strongly iISS component which is not globally ISS, but admits a zero local ISS-gain function. The outbreak is caused by the Strongly iISS component which is not ISS. However, the disease is eradicable since the Strongly iISS component possesses zero local ISS-gain which takes effect if a characteristic value is below a threshold. The notions of (i)ISS absorb changes of equilibria, and provide a module-based framework. The analysis of global properties does not require direct and heuristic construction of different Lyapunov functions of the entire network depending on equilibria. This demonstration is the main contribution of this paper. The same procedure and explanation are valid even in the presence of an outer-loop caused by sort-time immunity. The source of the particular iISS component is bilinearity. Indeed, scalar linear systems can never exhibit peaks, the outbreak. Although the bilinearity is the only nonlinearity in the popular simplest models, the theoretical tools presented in this paper accommodate a broad class of nonlinearities, such as saturation, non-monotone nonlinearities [21], [37]- [41] and others, as long as component models retain appropriate iISS and ISS properties. In fact, the arguments in this paper rely on neither linearity nor particular nonlinearities. Only ISS, iISS and ISS-gain characterizations are utilized. This paper has not reported any new epidemiologic discoveries. Nevertheless, the systematic treatment is expected to be superior to heuristic approaches in finding control strategies for eradicating or containing the spread of diseases. The new option provided by this paper aims to facilitate the research on control design with global guarantees. It is worth noticing that Lyapunov functions constructed in this paper are weighted sum of populations, which are simpler than logarithmic functions that have been popular in the field of mathematical epidemiology [17]. More importantly, (i)ISS Lyapunov functions constructed in this paper are different from Lyapunov functions in the conventional concept, and the construction of Lyapunov functions does not need preprocessing of equilibria. The vaccination discussed in this paper is openloop. Interesting future research includes introduction of the (i)ISS framework to closed-loop control design (see, e.g., [34], [42], [43] and references therein). Fig. 1 . 1Populations of the SIR model(22) with B(t) ≡ 12, which is R 0 = B/(µH) = 3.4043 > 1. Fig. 2 . 2Populations of the SIR model(22) with B(t) ≡ 3, which is R 0 = B/(µH) = 0.8511 < 1. are verified. For x U := x = [S, E, I] T and V U = V , property (24) achieves Fig. 3 . 3Populations of the SEIS model(35) with B(t) ≡ 12, which is R 0 = B/(µH) = 1.757 > 1. , the solution x(t) = [S(t), E(t), I(t)] T of (35) satisfies (30) for any constant B. Accordingly, all the observations made for the SIR model apply to the SEIS model except that the increase of the infectious individuals I is replaced by the increase of the weighted sum of infected individuals E + λ 2 I. Simulations of the SEIS model (35) are shown in Figs. 3 and 4 for β = 0.0002, µ = 0.015 and γ = 0.032 with S(0) = 700, I(0) = 200 and R(0) = 70. The thresholds are H = 455.3 and µH = 6.83. The disease is removed in Fig. 4 plotted for B = 3 < µH. The infected and infectious populations remain high in Fig. 3 computed for B = 12 > µH. Fig. 4 . 4Populations of the SEIS model(35) with B(t) ≡ 3, which is R 0 = B/(µH) = 0.4393 < 1. This is consistent with (9) and (10) evaluated with the existence of a ∈ (0, 1) in Theorem 1. A contribution to the mathematical theory of epidemics. W O Kermack, A G Mckendrick, Proc. R. Soc. Lond. 115W.O. Kermack and A.G. McKendrick, "A contribution to the mathemat- ical theory of epidemics." Proc. R. Soc. Lond., vol. A115, pp. 700-721, 1927. Population biology of infectious diseases: Part I. R M Anderson, R M May, Nature. 280R.M. Anderson and R.M. May, "Population biology of infectious dis- eases: Part I," Nature, vol. 280, pp. 361-367, 1979. The mathematics of infectious diseases. H W Hethcote, SIAM Rev. 424H.W. Hethcote, "The mathematics of infectious diseases," SIAM Rev., vol. 42, no. 4, pp. 599-653, 2000. M J Keeling, P Rohani, Modeling infectious diseases in humans and animals. Princeton Univ. PressM.J. Keeling and P. Rohani, Modeling infectious diseases in humans and animals, Princeton Univ. Press, 2008. Smooth stabilization implies coprime factorization. E D Sontag, IEEE Trans. Autom. Control. 344E.D. Sontag, "Smooth stabilization implies coprime factorization," IEEE Trans. Autom. Control, vol. 34, no. 4, pp. 435-443, 1989. Comments on integral variants of ISS. E D Sontag, Syst. Control Lett. 341-2E.D. Sontag, "Comments on integral variants of ISS," Syst. Control Lett., vol. 34, no. 1-2, pp. 93-100, 1998. Combining iISS and ISS with respect to small inputs: the Strong iISS property. A Chaillet, D Angeli, H Ito, IEEE Trans. Automat. Contr. 599A. Chaillet, D. Angeli and H. Ito, "Combining iISS and ISS with respect to small inputs: the Strong iISS property," IEEE Trans. Automat. Contr., vol.59, no.9, pp. 2518-2524, 2014. State-dependent scaling problems and stability of interconnected iISS and ISS systems. H Ito, IEEE Trans. Autom. Control. 5110H. Ito, "State-dependent scaling problems and stability of interconnected iISS and ISS systems," IEEE Trans. Autom. Control, vol. 51, no. 10, pp. 1626-1643, 2006. Necessary and sufficient small gain conditions for integral input-to-state stable systems: A Lyapunov perspective. H Ito, Z P Jiang, IEEE Trans. Autom. Contorl. 54H. Ito and Z.P. Jiang, "Necessary and sufficient small gain conditions for integral input-to-state stable systems: A Lyapunov perspective," IEEE Trans. Autom. Contorl, vol. 54, pp. 2389-2404, 2009. Robust stability of networks of iISS systems: construction of sum-type Lyapunov functions. H Ito, Z P Jiang, S Dashkovskiy, B S Rüffer, IEEE Trans. Autom. Control. 58H. Ito, Z.P. Jiang, S. Dashkovskiy and B.S. Rüffer, "Robust stability of networks of iISS systems: construction of sum-type Lyapunov func- tions," IEEE Trans. Autom. Control, vol. 58, pp. 1192-1207, 2013. A tight small gain theorem for not necessarily ISS systems. D Angeli, A Astolfi, Syst. Control Lett. 56D. Angeli and A. Astolfi. "A tight small gain theorem for not necessarily ISS systems," Syst. Control Lett., vol. 56, pp. 87-91, 2007. A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems. Z P Jiang, I Mareels, Y Wang, Automatica. 32Z.P. Jiang, I. Mareels and Y. Wang, "A Lyapunov formulation of the non- linear small-gain theorem for interconnected ISS systems," Automatica, vol. 32, pp. 1211-1215, 1996. A vector small-gain theorem for general nonlinear control systems. I Karafyllis, Z P Jiang, IMA J. Math. Control Info. 283I. Karafyllis and Z.P. Jiang, "A vector small-gain theorem for general nonlinear control systems," IMA J. Math. Control Info., vol. 28, no. 3, pp. 309-344, 2011 Lyapunov formulation of ISS smallgain in continuous-time dynamical networks. T Liu, D J Hill, Z P Jiang, Automatica. 47T. Liu, D.J. Hill and Z.P. Jiang, "Lyapunov formulation of ISS small- gain in continuous-time dynamical networks," Automatica, vol. 47, pp. 2088-2093, 2011. Small gain theorems for large scale systems and construction of ISS Lyapunov functions. S Dashkovskiy, B S Rüffer, F R Wirth, SIAM J. Control Optim. 48S. Dashkovskiy, B.S. Rüffer and F.R. Wirth, "Small gain theorems for large scale systems and construction of ISS Lyapunov functions," SIAM J. Control Optim., vol. 48, pp. 4089-4118, 2010. On a small-gain theorem for ISS networks in dissipative Lyapunov form. S Dashkovskiy, H Ito, F Wirth, European J. Contr. 17S. Dashkovskiy, H. Ito and F. Wirth, "On a small-gain theorem for ISS networks in dissipative Lyapunov form," European J. Contr., vol. 17, 357-365, 2011. Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models. A Korobeinikov, G C Wake, Appl. Math. Lett. 15A. Korobeinikov and G.C. Wake, "Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models, Appl. Math. Lett., vol. 15, pp. 955-960, 2002, Lyapunov functions and global properties for SEIR and SEIS epidemic models. A Korobeinikov, Math. Med. Biol. 21A. Korobeinikov, "Lyapunov functions and global properties for SEIR and SEIS epidemic models," Math. Med. Biol., vol. 21, pp. 75-83, 2004. Epidemiological models and Lyapunov functions. A Fall, A Iggidr, G Sallet, J J Tewa, Math. Model. Nat. Phenom. 21A. Fall, A. Iggidr, G. Sallet and J. J. Tewa, "Epidemiological models and Lyapunov functions," Math. Model. Nat. Phenom., vol. 2, no. 1, pp. 62-83, 2007. Lyapunov functions for SIR and SIRS epidemic models. S M O&apos;regan, T C Kelly, A Korobeinikov, M J A O&apos;callaghan, A V Pokrovskii, Appl. Math. Lett. 234S.M. O'Regan, T.C. Kelly, A. Korobeinikov, M.J.A. O'Callaghan and A.V. Pokrovskii, "Lyapunov functions for SIR and SIRS epidemic models," Appl. Math. Lett. vol. 23, no. 4, pp. 446-448, 2010. Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays. Y Enatsu, Y Nakata, Y Muroya, Disc. Cont. Dynam. Sys. B. 15Y. Enatsu, Y. Nakata and Y. Muroya, "Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays," Disc. Cont. Dynam. Sys. B, vol. 15, pp. 61-74, 2011. Stability of epidemic models with waning immunity. Y Nakata, Y Enatsu, H Inaba, T Kuniya, Y Muroya, Y Takeuchi, SUT J. Mathematics. 502Y. Nakata, Y. Enatsu, H. Inaba, T. Kuniya, Y. Muroya, and Y. Takeuchi, "Stability of epidemic models with waning immunity," SUT J. Mathe- matics, vol. 50, no. 2, pp. 205-245, 2014. Global stability of infectious disease models using Lyapunov functions. Z Shuai, P Van Den Driessche, SIAM J. Appl. Math. 734Z. Shuai and P. van den Driessche, "Global stability of infectious disease models using Lyapunov functions," SIAM J. Appl. Math., vol. 73, no. 4, pp. 1513-1532, 2013. The global stability of an SIRS model with infection age. Y Chen, J Yang, F Zhang, Math. Biosci. Eng. 11Y. Chen, J. Yang and F. Zhang, "The global stability of an SIRS model with infection age," Math. Biosci. Eng., vol. 11, pp. 449-469, 2014. On characterizations of input-to-state stability property. E D Sontag, Y Wang, Syst. Control Lett. 245E.D. Sontag and Y. Wang, "On characterizations of input-to-state stability property," Syst. Control Lett., vol. 24, no. 5, pp. 351-359, 1995. A characterization of integral input-to-state stability. D Angeli, E D Sontag, Y Wang, IEEE Trans. Autom. Control. 456D. Angeli, E.D. Sontag and Y. Wang, A characterization of integral input-to-state stability," IEEE Trans. Autom. Control, vol. 45, no. 6, pp. 1082-1097, 2000. Strong integral input-to-state stability of nonlinear networks through balancing kinetics. H Ito, submitted to 59th IEEE Conf. Decision ControlH. Ito, "Strong integral input-to-state stability of nonlinear networks through balancing kinetics," submitted to 59th IEEE Conf. Decision Control, 2020. Strong iISS is preserved under cascade interconnection. A Chaillet, D Angeli, H Ito, Automatica. 509A. Chaillet, D. Angeli and H. Ito, "Strong iISS is preserved under cascade interconnection," Automatica, vol. 50 no. 9, pp. 2424-2427, 2014. Integral input to state stable systems in cascade. A Chaillet, D Angeli, Syst. Control Lett. 57A. Chaillet and D. Angeli, "Integral input to state stable systems in cascade," Syst. Control Lett., vol. 57, pp. 519-527, 2008. A Lyapunov approach to cascade interconnection of integral input-to-state stable systems. H Ito, IEEE Trans. Autom. Control. 553H. Ito. "A Lyapunov approach to cascade interconnection of integral input-to-state stable systems," IEEE Trans. Autom. Control, vol. 55, no. 3, pp. 702-708, 2010. Relaxing growth rate assumption for integral input-to-state stability of cascade systems. H Ito, H. Ito, "Relaxing growth rate assumption for integral input-to-state stability of cascade systems," SICE Annual Conference 2017, pp. 689- 694, 2017. Measles in developing countries Part I. Epidemiological parameters and patterns. A R Mclean, R M Anderson, Epidemiology and Infection. 100A.R. McLean and R.M. Anderson, "Measles in developing countries Part I. Epidemiological parameters and patterns," Epidemiology and Infection, vol. 100, pp. 111-133, 1988. Global stability for the SEIR model in epidemiology. M Y Li, J S Muldowney, Math. Biosci. 125M.Y. Li and J.S. Muldowney, "Global stability for the SEIR model in epidemiology," Math. Biosci., vol. 125, pp. 155-164, 1995. On vaccination controls for the SEIR epidemic model. M De La Sen, A Ibeas, S Alonso-Quesada, Commun. Nonlinear Sci. Numer. Simul. 176M. De la Sen, A. Ibeas and S. Alonso-Quesada, "On vaccination controls for the SEIR epidemic model," Commun. Nonlinear Sci. Numer. Simul., vol. 17, no. 6, pp. 2637-2658, 2012. Vaccination strategies for epidemics in highly mobile populations. P Ogren, C F Martin, Appl. Math. Comput. 127P. Ogren and C.F. Martin, "Vaccination strategies for epidemics in highly mobile populations," Appl. Math. Comput., vol. 127, pp. 261-276, 2002. Stability analysis and optimal vaccination of an SIR epidemic model. G Zaman, Y H Kang, I H Jung, Biosystems. 933G. Zaman, Y.H. Kang and I.H. Jung, "Stability analysis and optimal vaccination of an SIR epidemic model," Biosystems, vol. 93, no. 3, pp. 240-249, 2008. A generalization of the Kermack-McKendrick deterministic epidemic model. V Capasso, G Serio, Math. Biosci. 42V. Capasso and G. Serio, "A generalization of the Kermack-McKendrick deterministic epidemic model," Math. Biosci., vol. 42, pp. 43-61, 1978. Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models. W M Liu, S A Levin, Y Iwasa, J. Math. Biol. 23W.M. Liu, S.A. Levin and Y. Iwasa, "Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models," J. Math. Biol., vol. 23, pp. 187-204, 1986. Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission. A Korobeinikov, Bulletin Math. Biol. 30A. Korobeinikov, "Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission," Bulletin Math. Biol., vol. 30, pp. 615-626, 2006. Global analysis of an epidemic model with nonmonotone incidence rate. D Xiao, S Ruan, Math. Biosci. 208D. Xiao and S. Ruan, "Global analysis of an epidemic model with nonmonotone incidence rate," Math. Biosci., vol. 208, pp. 419-429, 2007. Stability and bifurcation analysis of epidemic models with saturated incidence rates: an application to a nonmonotone incidence rate. Y Enatsu, Y Nakata, Math. Biosci. Eng. 11Y. Enatsu and Y. Nakata, "Stability and bifurcation analysis of epidemic models with saturated incidence rates: an application to a nonmonotone incidence rate," Math. Biosci. Eng., vol. 11, pp. 78-5-805, 2014. An observer-based vaccination control law for an SEIR epidemic model based on feedback linearization techniques for nonlinear systems. S Alonso-Quesada, M De La Sen, R P Agarwal, A Ibeas, Adv. Differ. Equ. 2012161S. Alonso-Quesada, M. De la Sen, RP. Agarwal and A. Ibeas, "An observer-based vaccination control law for an SEIR epidemic model based on feedback linearization techniques for nonlinear systems," Adv. Differ. Equ., vol. 2012, 161, 2012. Dynamic analysis of an SIR epidemic model with state dependent pulse vaccination. L F Nie, Z D Teng, A Torres, Nonlinear Anal., Real World Appl. 134L.F. Nie, Z.D. Teng and A. Torres, "Dynamic analysis of an SIR epidemic model with state dependent pulse vaccination," Nonlinear Anal., Real World Appl., vol. 13, no. 4, pp. 1621-1629, 2012.
{'fraction_non_alphanumeric': 0.09192177572406965, 'fraction_numerical': 0.03368677283604258, 'mean_word_length': 3.6327408256880735, 'pattern_counts': {'":': 0, '<': 23, '<?xml version=': 0, '>': 22, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 19, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'The notions of integral input-to-state stability (iISS) and input-to-state stability (ISS) have been effective in addressing nonlinearities globally without domain restrictions in analysis and design of control systems. In particular, they provide useful tools of module-based methods integrating characteristics of components. This paper applies the framework of module-based analysis to ordinary differential equations which deterministically describe dynamics of prevalence and the duration of epidemics. The objective is to express fundamental properties of models of infectious diseases and vaccination through the language of iISS and ISS. The systematic treatment is expected to facilitate development of effective schemes of controlling the spread of diseases via non-conventional Lyapunov functions.Index Terms-Epidemic models, integral input-to-state stability, Lyapunov functions, positive nonlinear network, small gain theorem.', 'arxivid': '2004.02552', 'author': ['Hiroshi Ito '], 'authoraffiliation': [], 'corpusid': 214802628, 'doi': '10.1007/s00498-020-00272-w', 'github_urls': [], 'n_tokens_mistral': 17454, 'n_tokens_neox': 15106, 'n_words': 8732, 'pdfsha': '23c078d8deed7b64aafed3069e6b13117fd3deba', 'pdfurls': ['https://arxiv.org/pdf/2004.02552v1.pdf'], 'title': ['Interpreting Models of Infectious Diseases in Terms of Integral Input-to-State Stability', 'Interpreting Models of Infectious Diseases in Terms of Integral Input-to-State Stability'], 'venue': []}
arxiv
Concentrations and Assembly Histories of Dark Matter Halos 5 Nov 2001 R H Wechsler Physics Department University of California 95064Santa CruzCA Physics Department University of Michigan 48109Ann ArborMI J S Bullock Department of Astronomy The Ohio State University 43210ColumbusOH J R Primack Physics Department University of California 95064Santa CruzCA A V Kravtsov Department of Astronomy The Ohio State University 43210ColumbusOH Hubble Fellow &amp; A Dekel Racah Institute of Physics The Hebrew University 91904JerusalemIsrael Concentrations and Assembly Histories of Dark Matter Halos 5 Nov 2001 We study the relation between the density profiles of dark matter halos and their mass assembly histories, using a statistical sample of halos in a high-resolution N-body simulation of the ΛCDM cosmology. For each halo at z = 0, we identify its merger-history tree, and determine concentration parameters c vir for all progenitors, thus providing a structural merger tree for each halo. We fit the mass accretion histories by a universal function with one parameter, the formation epoch a c , defined when the log mass accretion rate d log M/d log a falls below a critical value. We find that late forming galaxies tend to be less concentrated, such that c vir "observed" at any epoch a o is strongly correlated with a c via c vir = c 1 a o /a c . Scatter about this relation is mostly due to measurement errors in c vir and a c , implying that the actual spread in c vir for halos of a given mass can be mostly attributed to scatter in a c . Because of the direct connection between halo concentration and velocity rotation curves, and because of probable connections between halo mass assembly history and star formation history, the tight correlation between these properties provides an essential new ingredient for galaxy formation modeling. METHOD We investigate the connection between halo density profiles and their mass assembly histories, using a structural merger tree constructed from a high-resolution N-body simulation of a flat ΛCDM model with Ω m = 0.3, h = 0.7 and σ 8 = 1.0, whose evolution has been simulated with the ART code [2]. The trajectories of 256 3 cold dark matter particles are followed within a cubic, periodic box of comoving size 60h −1 Mpc from redshift z = 40 to the present. We use distinct halo catalogs at 36 output times spaced between z = 7 and 0. NFW density profiles [3], ρ NFW (r) = ρ s /[(r/R s ) (1 + r/R s ) 2 ], are measured for each halo with more than 200 particles, corresponding to halos more massive than 2.2 × 10 11 h −1 M ⊙ . For each halo at z = 0, we identify its full merger history, and determine concentration parameters c vir ≡ R vir /R s for all progenitors, thus providing a structural merger tree for each of ∼ 3000 halos. RESULTS This implies that, for any halo whose mass accretion trajectory follows this functional form, the characteristic formation time is the same regardless of the redshift z o at which the halo is observed. We find that the concentration of a halo, defined as c vir ≡ R vir /R s , is tightly correlated with the characteristic formation epoch as defined above, and that this relation holds at all redshifts when, properly scaled by a o : c vir = c 1 a o /a c ,(2) where c 1 is the typical concentration of halos whose formation time is at the time of measurement, a c = a o . Figure 1 (right) shows that this formula provides a good description of the observed correlation between concentration and formation time for halos at all masses and redshifts. As mentioned, the typical formation time is a function of mass, but there is significant scatter in a c for a fixed mass. The relation defined by Eq. 2 is able to account for the complete mass and redshift dependence of c vir , and for the scatter in c vir measured for fixed mass halos. This correlation is has important consequences for galaxy formation modeling. For further details on this work, see [4]. Figure 1 (Figure 1 : 11left) shows the history of mass growth for the major progenitors of several different halos, spanning a range of masses and concentration parameters. Massive halos tend show substantial mass accumulation up to late times, while the growth curves for less massive halos tend to flatten out earlier.By examining a range of full mass assembly histories for our sample of halos, we find that both average mass accretion histories and mass accretion histories for individual halos are well characterized by asimple function: M (a) = M o e −αz , a = (1+z) −1 . Fits to this equation are shown in Figure 1 (left) for representative individual halos. The single free parameter α can be related to a characteristic epoch for formation, a c , defined as the expansion scale factor a when the logarithmic slope of the accretion rate, d log M/d log a, falls below some specified value, S; the functional form implies a c = α/S. The same formation epoch can be defined equivalently for any "observing" epoch z o of that halo, by replacing a by a/a o , in which case a c = a o α/S. Thus at any such observing redshift, Left: Selected mass accretion trajectories, showing the evolution of the most massive progenitor for individual halos in the simulation (thick). Functional fits to the growth curve of each halo using Eq. 1 are shown as thin smooth lines. Right: Concentration versus scaled formation epoch a c /a o , for halos at z = 0 (triangles), z = 1 (circles), and z = 2 (squares). The 4 panels correspond to different mass ranges. At all masses and redshifts, the concentration parameter c vir is well fit by the functional form c vir = c 1 a c /a o , where c 1 ∼ 4.1 (represented by the solid line in each panel). a o = 1/(1 + z o ) and mass M o = M (z o ), the mass growth is fit by M (a) = M o exp −a c S a o a − 1 . . J S Bullock, T S Kolatt, Y Sigad, R S Somerville, A V Kravtsov, A A Klypin, J R Primack, A Dekel, MNRAS. 321559Bullock, J. S., Kolatt, T. S., Sigad, Y., Somerville, R. S., Kravtsov, A. V., Klypin, A. A., Primack, J. R., & Dekel, A. 2001, MNRAS, 321, 559 . A V Kravtsov, A A Klypin, A M Khokhlov, ApJS. 11173Kravtsov, A. V., Klypin, A. A., & Khokhlov, A. M. 1997, ApJS, 111, 73 . J F Navarro, C S Frenk, S D M White, ApJ. 462563Navarro, J. F., Frenk, C. S., & White, S. D. M. 1996, ApJ, 462, 563 . R H Wechsler, J S Bullock, J R Primack, A V Kravtsov, A Dekel, astro-ph/0108151ApJ. in pressWechsler, R. H., Bullock, J. S.,Primack, J. R., Kravtsov, A. V., & Dekel, A. 2001, ApJ, in press, astro-ph/0108151
{'fraction_non_alphanumeric': 0.04844396749961674, 'fraction_numerical': 0.026061628085236856, 'mean_word_length': 3.931216931216931, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'We study the relation between the density profiles of dark matter halos and their mass assembly histories, using a statistical sample of halos in a high-resolution N-body simulation of the ΛCDM cosmology. For each halo at z = 0, we identify its merger-history tree, and determine concentration parameters c vir for all progenitors, thus providing a structural merger tree for each halo. We fit the mass accretion histories by a universal function with one parameter, the formation epoch a c , defined when the log mass accretion rate d log M/d log a falls below a critical value. We find that late forming galaxies tend to be less concentrated, such that c vir "observed" at any epoch a o is strongly correlated with a c via c vir = c 1 a o /a c . Scatter about this relation is mostly due to measurement errors in c vir and a c , implying that the actual spread in c vir for halos of a given mass can be mostly attributed to scatter in a c . Because of the direct connection between halo concentration and velocity rotation curves, and because of probable connections between halo mass assembly history and star formation history, the tight correlation between these properties provides an essential new ingredient for galaxy formation modeling.', 'arxivid': 'astro-ph/0111069', 'author': ['R H Wechsler \nPhysics Department\nUniversity of California\n95064Santa CruzCA\n\nPhysics Department\nUniversity of Michigan\n48109Ann ArborMI\n', 'J S Bullock \nDepartment of Astronomy\nThe Ohio State University\n43210ColumbusOH\n', 'J R Primack \nPhysics Department\nUniversity of California\n95064Santa CruzCA\n', 'A V Kravtsov \nDepartment of Astronomy\nThe Ohio State University\n43210ColumbusOH\n\nHubble Fellow\n\n', '&amp; A Dekel \nRacah Institute of Physics\nThe Hebrew University\n91904JerusalemIsrael\n'], 'authoraffiliation': ['Physics Department\nUniversity of California\n95064Santa CruzCA', 'Physics Department\nUniversity of Michigan\n48109Ann ArborMI', 'Department of Astronomy\nThe Ohio State University\n43210ColumbusOH', 'Physics Department\nUniversity of California\n95064Santa CruzCA', 'Department of Astronomy\nThe Ohio State University\n43210ColumbusOH', 'Hubble Fellow\n', 'Racah Institute of Physics\nThe Hebrew University\n91904JerusalemIsrael'], 'corpusid': 18616750, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 1947, 'n_tokens_neox': 1685, 'n_words': 1136, 'pdfsha': '57f41acbfc3d485d2545e5dd33842290a7766d5d', 'pdfurls': ['https://arxiv.org/pdf/astro-ph/0111069v1.pdf'], 'title': ['Concentrations and Assembly Histories of Dark Matter Halos', 'Concentrations and Assembly Histories of Dark Matter Halos'], 'venue': []}
arxiv
THE GIT ASPECT OF GENERALIZED KäHLER REDUCTION. I 5 Oct 2018 Yicao Wang THE GIT ASPECT OF GENERALIZED KäHLER REDUCTION. I 5 Oct 2018 We revisit generalized Kähler reduction introduced by Lin and Tolman in[15]from a viewpoint of geometric invariant theory. It is shown that in the strong Hamiltonian case introduced in the present paper, many well-known conclusions of ordinary Kähler reduction can be generalized without much effort to the generalized setting. It is also shown how generalized holomorphic structures arise naturally from the reduction procedure. Introduction Generalized complex geometry, initiated by N. Hitchin [11] and developed in depth by M. Gualtieri [5] [6] [7], is a simultaneous generalization of complex and symplectic geometries. The development of this new geometry thus benefits greatly from the two well-established disciplines. For example, Marsden-Weinstein's famous symplectic reduction is an important construction in symplectic geometry and has been successfully generalized to the generalized complex setting by several authors [1] [15]; in particular, generalized complex reduction introduced in [15] is precisely the analogue of symplectic reduction. Symplectic reduction, when applied to Hamiltonian equivariant Kähler manifolds, is usually referred to as Kähler reduction. Besides the ordinary symplectic content of this procedure, complex structures also play a remarkable role; in particular, in good cases there is the well-known theorem of Kempf-Ness type stating that the symplectic quotient coincides with the complex quotient in the sense of geometric invariant theory (GIT for short) developed by Mumford. The philosophy of Kähler reduction has even been successfully applied formally to infinite-dimensional cases and the Kempf-Ness theorem also has its counterpart-the Kobayashi-Hitchin correspondence. In the generalized complex setting, the analogue of Kähler manifolds are generalized Kähler manifolds, each consisting of two commuting generalized complex structures J 1 and J 2 . The relevant generalized Kähler reduction was investigated in [1] [2] and in [15] independently. Lin-Tolman's work [15] is much more in the original spirit of symplectic reduction. However, in [15] attention was mainly put on one single generalized complex structure J 2 which plays a role as a symplectic structure does in Kähler reduction. The effect of the other generalized complex structure J 1 is thus not very clear. In contrast with Kähler reduction, it is conjectured that J 1 should act like a complex structure in Kähler reduction and there should be an analogue of GIT quotient. Our goal in this paper is basicly to investigate this GIT aspect of generalized Kähler reduction. However, to carry this idea out, we should complexify the underlying action of a compact Lie group G first. There do exist two ordinary complex structures J ± on the generalized Kähler manifold stemming from the bihermitian description, but they are not suitable for this attempt. Motivated by our central Lemma 4.2, we use the generalized complex structure J 1 to complexify the G-action. However, this procedure doesn't apply to general Hamiltonian generalized Kähler manifolds. To circumvent this difficulty we introduce the notion of strong Hamiltonian action of a compact Lie group. For the strong Hamiltonian case, at least in the case that M is compact and G acts locally freely on the zero locus of the moment map, we can really prove that the main results of GIT do hold in this more general setting. The paper is organized as follows: In § 2, we recall the basics of Kähler reduction to motivate our later investigation. In § 3 we collect the most relevant material on generalized geometry. In particular, a very brief review of Courant reduction of [1] is included. Our study really starts from § 4, in which we emphasize that our Lemma 4.2 is actually the generalized Kähler analogue of the basic formula Eq. (2.1) in equivariant Kähler geometry. This motivates our attempt to complexify the group action at the infinitesimal level. We find the Poisson structure β 1 (which is zero in the Kähler case) associated to J 1 is a basic obstruction for this complexification. In § 5, we introduce the notion of strong Hamiltonian action of a compact Lie group. Several nontrivial examples are given. After that, in the compact case, we prove that the infinitesimal action can be integrated to a global one (Thm. 5.7). The goal of § 6 is basically to establish a theorem of Kempf-Ness type in the strong Hamiltonian case, which claims that two kinds of generalized complex quotients are actually the same (Thm. 6.5 and Thm. 6.6). Our investigation follows closely the ideas of Kirwan in [12]. In the same section, we also prove that a generalized holomorphic principal bundle arises naturally from the reduction procedure (Thm. 6.8). This provides an answer to a question posed by the author in [20], which suggests the possibility of constructing generalized holomorphic structures from generalized Kähler reduction. The last section § 7 mainly contains a theorem (Thm. 7.1) stating that a (stable) orbit of the complexified group G C carries a natural structure of Hamiltonian G- Kähler manifold. For simplicity, in the present paper we mainly concentrate on the special case that the group G acts (locally) freely on the zero locus of the moment map. There is certainly a more general story, which we leave for a future work. Reflections on Kähler reduction To motivate our later considerations, in this section we briefly review the ordinary Kähler reduction in equivariant Kähler geometry. We refer the interested readers to F. Kirwan's book [12] for more details. Let G be a compact connected Lie group (with Lie algebra g) acting holomorphically on a Kähler manifold (M, g, J) from the left in a Hamiltonian fashion. This means that G preserves the Kähler form ω = gJ and there is an equivariant map µ : M → g * such that dµ ς = −ι Xς ω where ς ∈ g and X ς is the vector field on M generated by ς. There is a unique complexification G C of G such that its Lie algebra is the complexification of g, i.e. g C = g + √ −1g. 1 The G-action on M can be extended to a G C -action under suitable conditions (e.g. M is compact). To achieve this, at the infinitesimal level one simply uses JX ς as the infinitesimal action generated by √ −1ς for ς ∈ g, which also preserves J. If 0 is a regular value of µ and G acts freely on µ −1 (0), then the quotient µ −1 (0)/G acquires a symplectic structure by Marsden-Weinstein reduction. The complex structure J also descends to this quotient and makes it a Kähler manifold. There is another way to view the reduced complex structure: G C acts freely and holomorphi- cally on the open set M s = G C µ −1 (0), and thus the quotient M s /G C is complex in the natural manner. In this case, the Kempf-Ness theorem says that the two quotient actually coincide. To establish this coincidence, a central observation is that, while X ς is the Hamiltonian vector field associated to µ ς , JX ς is minus the gradient vector field associated to µ ς , i.e. (2.1) JX ς = −g −1 dµ ς . Another fact, which is seldom mentioned in the literature, is that it is almost trivial that M s , as a principal G C -bundle over µ −1 (0)/G, carries a holomorphic structure. However, in the generalized setting, nontrivial generalized holomorphic structures are not easy to construct. So in our later investigation we will pay some attention to finding out whether generalized holomorphic structures arise in a similar manner. 3. Basics of generalized complex geometry 3.1. Courant algebroids and their symmetries. We recall some backgrounds of Courant algebroids and extended actions of Lie groups on a Courant algebroid. Our basic reference is [1]. Generalized geometry arises from the idea of replacing the tangent bundle T of a manifold M by the direct sum T of T and its dual T * , or, more generally, by an exact Courant algebroid. A Courant algebroid E is a real vector bundle E over M, together with an anchor map π to T , 2 a non-degenerate inner product (·, ·) and a so-called Courant bracket [·, ·] c on Γ(E). These structures should satisfy some compatibility axioms we won't mention here. E is called exact, if the short sequence 0 −→ T * π * −→ E π −→ T −→ 0 is exact. In this paper, by "Courant algebroid", we always mean an exact one. Given E, one can always find an isotropic right splitting s : T → E, with a curvature form H ∈ Ω 3 cl (M) defined by H(X, Y, Z) = ([s(X), s(Y )] c , s(Z)), X, Y, Z ∈ Γ(T ). By the bundle isomorphism s + π * : T ⊕ T * → E, the Courant algebroid structure can be transported onto T. Then the inner product (·, ·) is the natural pairing, i.e. (X + ξ, Y + η) = ξ(Y ) + η(X), and the Courant bracket is (3.1) [X + ξ, Y + η] H = [X, Y ] + L X η − ι Y dξ + ι Y ι X H, called the H-twisted Courant bracket. This bracket is not skew-symmetric, actually The pair acts on Y + η ∈ Γ(E) in the following manner: (3.2) [X + ξ, Y + η] H + [Y + η, X + ξ] H = d(ξ(Y ) + η(X)).(ψ, B) · (Y + η) = ψ −1 * (Y ) + ψ * (η − ι Y B). An infinitesimal automorphism is consequently a pair (X, B), where X is a vector field on M and B is a 2-form, satisfying L X H = −dB. (X, B) acts on Y + η ∈ Γ(E) as follows: (X, B) · (Y + η) = L X (Y + η) − ι Y B. Especially, X +ξ ∈ Γ(E) generates an infinitesimal inner automorphism (X, dξ−ι X H) through the Courant bracket (3.1). Let G be a connected Lie group acting on M from the left. Then the infinitesimal action ϕ 0 : g → Γ(T ) is a Lie algebra homomorphism. Since in generalized geometry T is replaced by a Courant algebroid E, we would like to lift the g-action to E. Infinitesimal inner automorphisms of E induced by ϕ(g) through the Courant bracket form a Lie algebra action of g on the total space of E. 3 If this action integrates to a G-action, we shall call it an isotropic trivially extended G-action. There is a fairly general theory of extended G-action in [1]. However, we don't need this generality. In the remainder of the paper, when referring to an extended g-action (G-action), we always mean an isotropic trivially extended one. Additionally, to simplify notation, ϕ will be used to denote either an extended g-action or an extended G-action. Given a splitting of E preserved by the extended action, the extended g-action can be written in the form ϕ(ς) = X ς + ξ ς , where X ς is the vector field generated by ς and ξ (,) : g → Ω 1 (M) is a g-equivariant map such that (3.3) ξ ς (X ζ ) + ξ ζ (X ς ) = 0, dξ ς = ι Xς H. If the underlying G-action on M of an extended G-action is proper and free, the Courant algebroid E descends to the quotient M/G. In fact, let K be the subbundle of E generated by the image of ϕ and K ⊥ ⊂ E the orthogonal of K w.r.t. the inner product. Then K ⊂ K ⊥ and we can obtain a Courant algebroid E red := K ⊥ K /G whose Courant bracket can be derived from the Courant bracket of G-invariant sections of E. If D ⊂ E C is an involutive isotropic subbundle, then D also descends to the quotient under good conditions, e.g. D ∩ K C has constant rank. The reduced version of D is D∩K ⊥ C +K C K C /G. Of(3.4) [JA, JB] c = J[JA, B] c + J[A, JB] c + [A, B] c , for any A, B ∈ Γ(E). Since J and its √ −1-eigenbundle L are equivalent notions, we shall use them interchangeably to denote a generalized complex structure. A generalized complex structure L is an example of complex Lie algebroids. Via the inner product, ∧ · L * can be identified with ∧ ·L , and we have an elliptic differential complex (Γ(∧ ·L ), d L ), inducing the Lie algebroid cohomology associated with the Lie algebroid L. The differential complex can be further twisted by a generalized holomorphic vector bundle F , which is a complex vector bundle equipped with a first-order differential operator∂ : Γ(F ) → Γ(L ⊗ F ) such that for any smooth function f and s ∈ Γ(F )∂ (f s) = d L f ⊗ s + f∂s and∂ 2 = 0. This notion of generalized holomorphic structures has been generalized to the setting of (generalized) principal bundles by the author in [19]. Definition 3.3. Let G be a Lie group. A generalized principal G-bundle over M is a triple (P, E, ϕ) such that (i) p : P → M is an ordinary principal G-bundle, and (ii) E is a Courant algebroid over P and ϕ an extended G-action on E. Note that by the reduction mentioned in § 3.1, there is the Courant algebroid E = K ⊥ K /G on the base manifold M, descending from E. In the following, let Q be a complex Lie group with Lie algebra q and (P, E, ϕ) a generalized principal Q-bundle. Then we have a decomposition q C = q h ⊕ q a , where q h is the holomorphic part and q a its complex conjugate. Denote K a the sunbundle generated by ϕ(q a ). Definition 3.4. Let (P, E, ϕ) be a generalized principal Q-bundle over M and A ⊂ E C a Q-invariant isotropic subbundle such that (i) K a ⊂ A ⊂ K ⊥ C , (ii) A ⊕ A = K ⊥ C , and (iii) A descends to a generalized complex structure L in E in the sense of the reduction theory mentioned in § 3.1. Then A is called an almost generalized holomorphic structure w.r.t. L. If furthermore A is integrable under the Courant bracket, it is called a generalized holomorphic structure. As was proved in [19], given a generalized holomorphic principal Q-bundle, any associated vector bundle of a holomorphic representation of Q is generalized holomorphic in the manner that associated vector bundles of a holomorphic principal Q-bundle are holomorphic. Definition 3.5. A generalized metric on a Courant algebroid E is an orthogonal, self-adjoint operator G such that (Ge, e) > 0 for nonzero e ∈ E. It is necessary that G 2 = id. A generalized metric induces a canonical isotropic splitting: E = G(T * ) ⊕ T * . It is called the metric splitting. Given a generalized metric, we shall always choose the metric splitting to identify E with T. Then G is of the form 0 g −1 g 0 where g is an ordinary Riemannian metric. A generalized metric is an ingredient of a generalized Kähler structure, which is the analogue of Kähler structures in complex geometry. Definition 3.6. A generalized Kähler structure on E is a pair of commuting gener- alized complex structures (J 1 , J 2 ) such that G = −J 1 J 2 is a generalized metric. If necessary, we will use L i to denote the corresponding √ −1-eigenbundle of J i , i = 1, 2. A generalized Kähler structure can also be characterized in terms of more ordinary concepts: There are two complex structures J ± on M compatible with the metric g induced from the generalized metric. Let ω ± = gJ ± . Then in the metric splitting the generalized complex structures and the bihermitian data are related by the Gualtieri map: J 1 = 1 2 −J + − J − ω −1 + − ω −1 − −ω + + ω − J * + + J * − , J 2 = 1 2 −J + + J − ω −1 + + ω −1 − −ω + − ω − J * + − J * − . Let β 1 := − 1 2 (J + − J − )g −1 and β 2 := − 1 2 (J + + J − )g −1 . These are actually Poisson structures associated to J 1 and J 2 respectively. Hamiltonian Generalized Kähler manifolds We first recall the generalized Kähler reduction procedure developed in [15]. Our formulation here is, however, greatly influenced by the works in [1] [2] [4]. In particular, we stick to the metric splitting, which exists naturally on a generalized Kähler manifold M. Let a compact connected Lie group G act on M from the left in the extended manner, preserving the generalized Kähler structure on M and consequently the metric splitting. The notion of (generalized) moment map can be defined in the context of generalized complex manifolds. However, a generalized Kähler manifold consists of two generalized complex structures (J 1 , J 2 ). As a convention, when referring to a moment map, we always mean it is associated to J 2 . µ : M → g * is called a moment map, if (4.1) J 2 (X ς + ξ ς − √ −1dµ ς ) = √ −1(X ς + ξ ς − √ −1dµ ς ) for any ς ∈ g. If this happens, we call M a Hamiltonian G-generalized Kähler manifold. In contrast with ordinary Kähler reduction, J 2 plays the role of a symplectic structure in Kähler reduction and it is expected that J 1 should act like a complex structure in Kähler reduction. Note that Eq. (4.1) is precisely (4.2) J 2 (X ς + ξ ς ) = dµ ς . The following algebraic calculation actually has already appeared in [3] [16]. We include it here only because it provides some motivations for our later considerations. Lemma 4.2. In terms of the bihermitian data, Eq. (4.1) is equivalent to the following two equations: (4.3) J + X + ς = J − X − ς = −g −1 dµ ς , where X ± ς = X ς ± g −1 ξ ς , Proof. The equation J 2 X ς ξ ς = 1 2 −J + + J − ω −1 + + ω −1 − −ω + − ω − J * + − J * − X ς ξ ς = 0 dµ ς written in components is (−J + + J − )X ς − (J + + J − )g −1 ξ ς = 0, and −(ω + (X ς ) + ω − (X ς )) + (J * + − J * − )ξ ς = 2dµ ς . The first is precisely J + X + ς = J − X − ς . Substituting this into the second leads to J + X + ς = −g −1 dµ ς . From the lemma, we have X ς = −β 2 (dµ ς ) and g −1 ξ ς = −β 1 (dµ ς ). That's to say, X ς and g −1 ξ ς are precisely the Hamiltonian vector field of µ ς w.r.t. β 1 and β 2 respectively. The lemma, together with Eq. (4.2), is precisely the analogue of the fact in Hamiltonian equivariant Kähler geometry that X ς is the Hamiltonian vector field of µ ς and JX ς is minus the gradient vector field of µ ς . For later convenience, we denote J + X + ς by Y ς . If G acts freely on µ −1 (0), then µ −1 (0)/G acquires a generalized Kähler structure through the general reduction theory. Let K 0 be the subbundle of E| µ −1 (0) generated by ϕ(g) and dµ ς , ς ∈ g. Then K 0 is isotropic and K ⊥ 0 K 0 /G is the reduced Courant algebroid over µ −1 (0)/G. Since K 0 is J 2 -invariant, there is a natural complex structure J red 2 on K ⊥ 0 K 0 /G. This is the reduced version of J 2 . There is another way to see clearly how the two generalized complex structures descend simultaneously in a compatible way: K ⊥ 0 K 0 is naturally isomorphic to K ⊥ 0 ∩ GK ⊥ 0 , where G = −J 1 J 2 . Since the latter is both G-invariant and J 2 -invariant, it is also J 1 -invariant and therefore one just has to restrict J 1 and J 2 to K ⊥ 0 ∩ GK ⊥ 0 to find the reduced structures J red 1 and J red 2 . Now we turn to the problem of complexifying the extended G-action on M. At the infinitesimal level, the naive choices J ± X ς motivated by the Kähler case won't work because generally J + X ς = J − X ς . But the above lemma provides an alternative choice: We can use Y ς = J + X + ς (= J − X − ς ) instead of J + X ς or J − X ς . This is justified by the fact that (4.4) J 1 (X ς + ξ ς ) = −J 1 J 2 dµ ς = g −1 dµ ς = −J + X + ς = −Y ς . Definition 4.3. The map ϕ C : g C → Γ(E) defined by (4.5) ϕ C (u + √ −1v) = ϕ(u) − J 1 ϕ(v) is called the pre-complexification of ϕ. Remark. The minus sign in (4.5) is to keep accordance with the specified case of Hamiltonian equivariant Kähler manifolds where J + = J − . Proposition 4.4. The pre-complexification ϕ C of ϕ defined above preserves J 1 . Proof. It suffices to verify that J 1 (X ς + ξ ς ) preserves J 1 : For any A ∈ Γ(T), [J 1 (X ς + ξ ς ), J 1 A] H = J 1 [X ς + ξ ς , J 1 A] H + J 1 [J 1 (X ς + ξ ς ), A] H + [X ς + ξ ς , A] H = J 1 [J 1 (X ς + ξ ς ), A] H where Eq.(3.4) and the fact X ς + ξ ς preserves J 1 are used. Remark. The proposition is precisely the analogue of the fact in the Kähler case that the vector field JX ς preserves the complex structure J. Up to now, it seems that we are on the right way to complexifying the G-action. However, we come across some difficulty-The map ϕ C is not necessarily a Lie algebra homomorphism. Let S : g × g → C ∞ (M) be defined by S(ς, ζ) = (J 1 ϕ(ς), ϕ(ζ)) for ς, ζ ∈ g. Obviously S(ς, ζ) = −S(ζ, ς). We have the following characterization of S. Proof. S(ς, ζ) = (J 1 (X ς + ξ ς ), X ζ + ξ ζ ) = (J 1 J 2 dµ ς , J 2 dµ ζ ) = (J 1 dµ ς , dµ ζ ) = {µ ς , µ ζ } β 1 . The appearance of S is a totally new phenomenon compared with the Kähler case where β 1 ≡ 0. S is generally the obstruction for ϕ C to be a Lie algebra homomorphism. This can be seen from the following direct computations: [X ς + ξ ς , J 1 (X ζ + ξ ζ )] H = J 1 [X ς + ξ ς , X ζ + ξ ζ ] H , [J 1 (X ς + ξ ς ), X ζ + ξ ζ )] H = J 1 [X ς + ξ ς , X ζ + ξ ζ ] H + dS(ς, ζ), [J 1 (X ς + ξ ς ), J 1 (X ζ + ξ ζ )] H = −[X ς + ξ ς , X ζ + ξ ζ ] H + J 1 dS(ς, ζ), where Eq. Proof. Since the Poisson structure β 1 is G-invariant, we only need to prove for any smooth function f the Lie bracket of Y ς and β 1 (df ) lies in D. Note that by Eq. (3.4), we have [J 1 (X ς + ξ ς ), J 1 (df )] H = J 1 [X ς + ξ ς , J 1 (df )] H + J 1 [J 1 (X ς + ξ ς ), df ] H + [X ς + ξ ς , df ] H = −[X ς + ξ ς , df ] H + J 1 [J 1 (X ς + ξ ς ), df ] H + [X ς + ξ ς , df ] H = J 1 [J 1 (X ς + ξ ς ), df ] H = −J 1 (dL Yς f ), where we have used the fact that J 1 is preserved by the extended G-action. Therefore, π([J 1 (X ς + ξ ς ), J 1 (df )] H ) = −β 1 (dL Yς f ). However, the left hand side of the above equality is nothing else but −[Y ς , β 1 (df )]. Our conclusion thus holds. Remark. In particular, we have proved that β 1 is preserved by the vector fields Y ς . We feel the following proposition will be of value if one is to consider the interaction between the group action and β 1 . Proposition 4.7. If either X ς or Y ς is tangent to a sympletic leaf L of β 1 at a point x ∈ L, so is the other. Proof. If X ς is tangent to L at x, then X ς = β 1 (df ) at x for some function f defined around x. Therefore, at x X ς + ξ ς = J 1 (df ) + τ for some 1-form τ . So Y ς = −J 1 (X ς + ξ ς ) = df − J 1 (τ ). This shows that Y ς = −β 1 (τ ), i.e. Y ς is also tangent to L at x. The converse can be proved similarly. Strong Hamiltonian actions We have found in the previous section that the Poisson structure β 1 appears as the obstruction for complexfying the extended g-action. To proceed further, we are forced to introduce the following notion of strong Hamiltonian action. The definition has a direct consequence that in the strong Hamiltonian case, the underlying Poisson structure β 1 is preserved by the g C -action. Proof. This is obvious due to dimensional reason and the fact that S is skew-symmetric. Remark. Since Cartan subgroups are a basic ingredient of compact connected Lie groups, strong Hamiltonian S 1 -actions naturally arise on a Hamiltonian G-generalized Kähler manifold. Example 5.4. Let (M, g, I, J, K) be a hyperKähler structure, and ω I , ω J , ω K be the associated Kähler forms. According to the observation in [5], a generalized Kähler structure can be constructed as follows: J 1 = 1 0 −ω K 1 0 1 2 (ω −1 I + ω −1 J ) −ω I − ω J 0 1 0 ω K 1 J 2 = 1 0 ω K 1 0 1 2 (ω −1 I − ω −1 J ) −ω I + ω J 0 1 0 −ω K 1 . Note that we are already in the metric splitting. Suppose M carries a Hamiltonian T k (torus of dimension k)-action with moment map (µ I , µ J , µ K ) : M → t * ⊕ t * ⊕ t * , where t is the Lie algebra of T k . Then for ς ∈ t, we have ι Xς ω I = −dµ I ς , ι Xς ω J = −dµ J ς and ι ς ω K = −dµ K ς . According to computations in [15], µ := µ I − µ J is a moment map associated to J 2 . This T k -action is actually strong Hamiltonian: Note that ω −1 I dµ ς = −KX ς − X ς and ω −1 J dµ ς = −KX ς + X ς . So β 1 (dµ ς ) = 1 2 (ω −1 I + ω −1 J )dµ ς = −KX ς . We thus have {µ ς , µ ζ } β 1 = −dµ ζ (KX ς ) = dµ J ζ (KX ς ) − dµ I ζ (KX ς ) = −ω J (X ζ , KX ς ) + ω I (X ζ , KX ς ) = g(KJX ζ , X ς ) − g(KIX ζ , X ς ) = −g(IX ζ , X ς ) − g(JX ζ , X ς ) = ω I (X ς , X ζ ) + ω J (X ς , X ζ ) = X ς µ I ζ + X ς µ J ζ = 0. The last equality is due to the fact that µ I and µ J are actually T k -invariant. ε = 1 2 z 2 0 (∂ z 1 + 1 2 dz 1 ) ∧ (∂ z 2 − 1 2 dz 2 ). Then J 1 is deformed to another generalized complex structure J ε 1 away from the cylinder |z 0 | = √ 2. A pure spinor 5 of J ε 1 is ϕ ε = e ε · (Ω 1 ∧ Ω 2 ) = (e ε · Ω 1 ) ∧ Ω 2 , where Ω 1 = dz 0 dz 1 dz 2 and Ω 2 = dz 3 dz 4 · · · dz N . Then the pair (J ε 1 , J 2 ) is an S 1invariant generalized Kähler structure, and by the reduction procedure developed in [15] CP N acquires a generalized Kähler structure (J 1 , J 2 ). Actually, the reduction of J 2 is precisely the classical Marsden-Weinstein reduction, giving rise to the Proposition 5.6. If M is a strong Hamiltonian G-generalized Kähler manifold, the pre-complexification ϕ C of ϕ is actually an extended g C -action preserving J 1 . Proof. By definition of Hamiltonian actions, the g-action is isotropic trivially extended. That this continues to hold for ϕ C is a direct result of the fact S(ς, ζ) = (J 1 ϕ(ς), ϕ(ζ)). Since the obstruction S vanishes, ϕ C is thus a Lie algebra homomorphism. Prop. 4.4 shows this extended g C -action preserves J 1 . The next question we shall tackle is whether the extended g C -action can be integrated to a G C -action. We content ourselves with the following result. Then it is routine to check that the pair (ψ t , B t ) acting on E gives the desired global flow. There should be a discrete subgroup Π ofG contained in the center ofG such that G =G/Π, G C =G C /Π. Since when restricted onG theG C -action factors through Π, we actually have an extended G C -action. The uniqueness of this extension is determined by the infinitesimal action. Remark. In ordinary equivariant Kähler geometry, the theorem has a more direct proof (see for example [8]), which uses the well-known fact that the automorphism group of a compact complex manifold is a Lie transformation group. However, this won't work for the present setting if one just recalls the equally well-known fact that the automorphism group of a compact symplectic manifold is not a Lie transformation group. It is also remarkable here that the underlying G C -action in general will preserve neither J + nor J − , and thus is not holomorphic w.r.t. either of the two. If x ∈ µ −1 (0) and a f a Y a = 0 at x for some constants f a , then a f a X + a = a f a X − a = 0 at x due to the fact that Y a = J + X + a = J − X − a . Therefore we have a f a X a = 0 and a f a ς a lies in the Lie algebra of the stabilizer G x (⊂ G) of x. All f a should be zero because G acts locally freely on µ −1 (0). Therefore, vector fields Y a 's are linearly independent pointwise over µ −1 (0). If v ∈ T x µ −1 (0), then at x g(Y a , v) = −dµ ςa (v) = 0 by definition. This completes the proof. Proposition 6.2. The stabilizer G C x in G C of a point x ∈ M s is finite. If x ∈ µ −1 (0), then G C x = G x ; in particular, if G acts freely on µ −1 (0), then G C acts freely on M s . Proof. By the well-known Cartan decomposition G C = P G where P is diffeomorphic to √ −1g via the exponential map, and by the fact that vector fields such as Y ς generate the action of √ −1g, the proof of Prop. 6.1 already implies that the stabilizer of x ∈ µ −1 (0) is finite. Since each point y ∈ M s lies in a G C -orbit which intersects µ −1 (0) at some point x, the stabilizer of y differs from the stabilizer of x only by conjugation and thus is also finite. For x ∈ µ −1 (0), obviously we have G x ⊆ G C x . Let g ∈ G C x \G x . Due to Cartan decomposition, we can write it as g = exp( √ −1ς)k, where 0 = ς ∈ g and k ∈ G. Consider the curve s(t) = exp(− √ −1tς)kx (t ∈ R) in M s and the function h(t) = µ ς (s(t)). Note that dh dt = dµ ς (Y ς ) = −g(Y ς , Y ς ). Since G C acts locally freely on M s , Y ς vanishes nowhere on M s and thus dh dt < 0. Now that kx ∈ µ −1 (0), we have 0 = µ ς (x) − µ ς (kx) = h(1) − h(0) = 1 0 dh dt dt < 0, which is a contradiction! Therefore, we must have G C x = G x . The rest of this proposition is a direct result of this equality. Proposition 6.3. For any point x ∈ µ −1 (0), G C x ∩ µ −1 (0) = Gx. Proof. It suffices to prove G C x ∩ µ −1 (0) ⊆ Gx. If y = gx ∈ µ −1 (0) for some g ∈ G C , without loss of generality we can assume g = exp( √ −1ς). Consider the similar curve s(t) and the function h(t) in the proof of Prop. 6.2 ( k = e). Note that 0 = µ ς (y) − µ ς (x) = h(1) − h(0) = − 1 0 g(Y ς , Y ς ) ≤ 0, and equality holds iff Y ς = 0 along s(t). This immediately implies that X ς = 0 along s(t). Since G acts locally freely on M s , we must have ς = 0 and thus y = x ∈ Gx. Proposition 6.4. If x, y ∈ µ −1 (0) and x / ∈ Gy, then there are disjoint G C -invariant neighbourhoods of x and y in M. Proof. We shall follow closely the ideas of Kirwan in [12]. There is a compact Ginvariant neighbourhood W of x in µ −1 (0) not containing y, because G is compact and x / ∈ Gy. By the proof of Prop. 6.1, exp( √ −1g)W is a neighbourhood of x ∈ M. It suffices to prove that y / ∈ exp( √ −1g)W for exp( √ −1g)W and M \ exp( √ −1g)W will be the neighbourhoods we need. Choose an invariant metric on g and use it to identify g * with ς. We use ς to denote the norm of ς ∈ g. Let U = {exp( √ −1ς)z|ς ∈ g, ς ≤ 1, z ∈ W }, and define ǫ = inf{(X ς , X ς )(w)|w ∈ U, ς ∈ g, ς = 1}. Since G C acts locally freely on M s and U is compact, we have ǫ > 0. Suppose z ∈ W and ς ∈ g such that ς = 1. Consider as in the proof of Prop. 6.2 the function h(t) = µ ς (exp(− √ −1tς)z), t ∈ R. Then h ′ (t) ≤ 0; in particular, h ′ (t) ≤ −ǫ for x ∈ [0, 1]. Since h(0) = 0, for any t ≥ 0 we have that h(t) ≤ 0; moreover, for any t ≥ 1 we have that µ(exp( √ −1tς)z) = µ(exp( √ −1tς)z) · ς ≥ |µ ς (exp( √ −1tς)z)| = |h(t)| ≥ |h(1)| = − 1 0 h ′ (t)dt ≥ ǫ. So we have obtained the conclusion that µ(exp( √ −1ς)z) ≥ ǫ for any z ∈ W and ς ≥ 1. Now we can prove y / ∈ exp( √ −1g)W . If it is not so, there would be sequences ς n ∈ g, v n ∈ W such that lim n→∞ exp( √ −1ς n )v n = y. By continuity, we would have lim n→∞ µ(exp( √ −1ς n )v n ) = lim n→∞ µ(y) = 0. Thus for n sufficiently large, µ(exp( √ −1ς n )v n ) < ǫ. Therefore, for n sufficiently large n, exp( √ −1ς n )v n ∈ U and thus ς n ≤ 1. By compactness, we have subsequences ς n l and v n l , which converge to some ς ( ς ≤ 1) and v ∈ W respectively. Thus y = exp( √ −1ς)v ∈ W ⊂ U. But we already have y / ∈ exp( √ −1g)W ⊇ U. A contradiction! After the above preparation, we can give a theorem of Kempf-Ness type. Theorem 6.5. If G acts locally freely on µ −1 (0), then the natural inclusion i : µ −1 (0) ֒→ M s induces a diffeomorphism µ −1 (0)/G ∼ = M s /G C . Proof. Since M s = G C µ −1 (0), the map j : µ −1 (0)/G ∼ = M s /G C is obviously surjective. Prop. 6.3 implies that j is also injective. From Prop. 6.4, we know M s /G C is a Hausdorff space. Therefore j is a continuous bijection from a compact space to a Hausdorff space, and hence is a homeomorphism. To see j is actually a diffeomorphism, we only need to prove its smoothness locally. We only sketch a proof under the condition that G acts freely on µ −1 (0). The case of locally free action will only involve some notational complexity. By the famous Koszul-Palais slice theorem [10,Appendix B], for x ∈ µ −1 (0), there is a slice S ⊆ µ −1 (0) such that the map Θ : G × S → µ −1 (0), (g, s) → gs is a G-equivariant diffeomorphism from G × S to its image in µ −1 (0). Thus S can be used as a coordinate chart around [x] ∈ µ −1 (0)/G. Θ can be naturally G Cequivariantly extended: Θ C : G C × S → M s , (g, s) → gs. This is obviously a diffeomorphism from G C × S to its image in M s . Thus S can also be used as a coordinate chart around [x] ∈ M s /G C . Therefore j is the identity map in such a chart and consequently smooth. Following the convention of GIT, we denote M s /G C by M//G C . If additionally G acts freely on µ −1 (0), then M s is open in M and can be viewed as a principal G C -bundle over M//G C . Let K be the subbundle of E| Ms generated by ϕ(g) and K = K ⊕ J 1 K. SinceK is isotropic in E| Ms and the extended G C -action preserves J 1 . We are in the situation of [1,Thm. 5.2] and thus J 1 descends to the quotient M//G C . Denote this generalized complex structure by J ♭ 1 . Theorem 6.6. By the diffeomorphism µ −1 (0)/G ∼ = M//G C , the generalized complex structure J red 1 coincides with J ♭ 1 . Proof. By the general theory developed in [1] , the reduced Courant algebroid E red over M//G C isK ⊥ K /G C . Since J 1 preservesK, onK ⊥ K there is a natural mapJ 1 , which gives rise to J ♭ 1 after quotienting by G C . But J ♭ 1 has another convenient description: Recall that G = −J 1 J 2 is the generalized metric. We have a natural isomorphism betweenK ⊥ K andK ⊥ ∩ GK ⊥ , and by this identificationJ 1 is precisely the restriction of J 1 onK ⊥ ∩ GK ⊥ . To prove our theorem, we only need to check on µ −1 (0) it holds that (6.1) K ⊥ 0 ∩ GK ⊥ 0 =K ⊥ ∩ GK ⊥ , where K 0 is the subbundle of E| µ −1 (0) generated by ϕ(g) and dµ ς , ς ∈ g. An element Z + η in K ⊥ 0 ∩ GK ⊥ 0 is characterized by the following equations: ξ ς (Z) + η(X ς ) = 0, dµ ς (Z) = 0, g(X ς , Z) + g(ξ ς , η) = 0, g(η, dµ ς ) = 0 for all ς ∈ g while an element Z + η inK ⊥ ∩ GK ⊥ is characterized by ξ ς (Z) + η(X ς ) = 0, η(Y ς ) = 0, g(X ς , Z) + g(ξ ς , η) = 0. g(Z, Y ς ) = 0 for all ς ∈ g. Noting that g −1 dµ ς = −Y ς , we find the two groups of equations on µ −1 (0) are actually the same. Therefore, Eq. (6.1) does hold. Recall that a generalized complex structure has an important local invariant called type at a point, which is in fact the transverse complex dimension of the structure. Consider J 1 and its reduction J red 1 . We would like to know how the type t(x) of a point x ∈ M s is related to the type t([x]) of its image [x] ∈ M//G C . Concerning this problem, there is a formula in [15]. In the following, we shall prove it in our context. 7 Proposition 6.7. The two numbers t([x]) and t(x) are related by (6.2) t([x]) = t(x) − dimg + 2dim(π(L 1 ) ∩ π(K C )) x . Proof. Note that J red 1 at [x] is actually modelled on the complex linear Dirac structure L 1 ∩K ⊥ C +K C K C ⊂K ⊥ C /K C at x. Therefore, we have (to simplify notation, we omit the subscript x in the following) T ([x]) = 1 2 dim(K ⊥ C /K C ) − dimπ( L 1 ∩K ⊥ C +K C K C ) = dimM − 2dimg − dim π(L 1 ∩K ⊥ C ) π(L 1 ∩K ⊥ C ) ∩ π(K C ) = dimM − 2dimg − dimπ(L 1 ∩K ⊥ C ) + dim(π(L 1 ∩K ⊥ C ) ∩ π(K C )). By Lemma 3.1 in [15], dimπ(L 1 ∩K ⊥ C ) = dimπ(L 1 + K C ) − dimg. Since π(L 1 + K C ) = π(L 1 ) + π(K C ), we have dimπ(L 1 + K C ) = dimπ(L 1 ) + dimg − dim(π(L 1 ) ∩ π(K C )). To prove the final result, we have to show dim(π(L 1 ∩K ⊥ C ) ∩ π(K C )) = dimg + dim(π(L 1 ) ∩ π(K C )). Let K a = L 1 ∩K C . ThenK C = K a ⊕ K a and π(K a ) ⊂ π(L 1 ∩K ⊥ C ). If A ∈ K a and B ∈ K a , for π(A + B) to lie in π(L 1 ∩K ⊥ C ), the sufficient and necessary condition is π(B) ∈ π(L 1 ∩K ⊥ C ). But since B ∈ K a ⊂ L 1 , this happens iff π(B) ∈ π(L 1 ) ∩ π(L 1 ). Note that π(B) = i c i (X ς i − √ −1Y ς i ) for a basic {ς i } of g and some constants c i and i c i (X ς i + √ −1Y ς i ) lies in π(L 1 ). Thus π(B) ∈ π(L 1 ) ∩ π(L 1 ) iff i c i X ς i ∈ π(L 1 ). This completes the proof. If G acts freely on µ −1 (0), M s is actually a generalized principal G C -bundle over M//G C and J 1 induces a generalized holomorphic structure over it. Theorem 6.8. M s viewed as a generalized principal G C -bundle on M//G C is generalized holomorphic. Proof. Let K a := L 1 ∩K C and A := L 1 ∩K ⊥ C . Then K a ⊕ K a =K C , K a ⊂ A ⊂K ⊥ C , and the rank of A is dimM − dimg. We now show A is a generalized holomorphic structure. Obviously, A ⊕ A ⊂K ⊥ C . This inclusion is actually equality due to dimensional reason. Therefore the algebraic When G acts freely on µ −1 (0), µ −1 (0) is then a principal G-bundle over µ −1 (0)/G. Since nontrivial generalized holomorphic structures are not easy to construct, we would like to know when a vector bundle associated to a complex representation of G acquires a generalized holomorphic structure, without complexifying the G-action. Our total argument then implies the following Theorem 6.9. In the strong Hamiltonian case if G acts freely on µ −1 (0), then any associated complex vector bundle of the principal G-bundle µ −1 (0) → µ −1 (0)/G has a natural generalized holomorphic structure. Proof. Let (φ, V ) be a representation of G in a complex vector space V and φ C be the complexification of φ. One simply complexifies the extended G-action on M. Then the (generalized) principal G C -bundle M s → M//G C carries the natural generalized holomorphic structure. Therefore the associated vector bundle V × φ C M s is a generalized holomorphic vector bundle. But as a vector bundle, V × φ µ −1 (0) is the same as V × φ C M s . Geometry on G C -orbits As a concluding section, we consider the geometry of G C -orbits in M s of the former section, which arises from the ambient space. Let i : O ֒→ M s be a G C -orbit in M s . In the case of Kähler reduction, the Kähler structure on M can be pulled back to O such that O is a G-invariant Kähler submanifold. Even more, the moment map µ can also be pulled-back to make the G-action on O Hamiltonian. We refer the interested readers to [9] for details of this material. At first glance, it would be expected that in the strong Hamiltonian generalized Astonishing is that O is actually again a Hamiltonian G-Kähler manifold in a natural way. In the following we describe how this structure arises. O carries a natural complex structures J 0 defined by J 0 X ς = Y ς , J 0 Y ς = −X ς at each x ∈ O. J 0 is just induced from the canonical complex structure on G C . The metric on O cannot be the pull-back of g since this is not Hermitian. We can define a new metric g 0 on O by letting g 0 (X ς , X ζ ) = g 0 (Y ς , Y ζ ) := g(X ς , X ζ ) + g(ξ ς , ξ ζ ), and g 0 (X ς , Y ζ ) = g 0 (Y ζ , X ς ) := g(X ς , Y ς ). By definition J 0 and g 0 are obviously G-invariant. Additionally, the moment map µ can be pulled back to O. By abuse of notation, we still use µ to denote this G-equivariant function from O to g * . Theorem 7.1. (O, J 0 , g 0 , µ) is actually a Hamiltonian G-Kähler manifold. Proof. There is an easy way to see why g 0 is Hermitian w.r.t. J 0 . Let R be the subbundle of E| O generated by elements like X ς + ξ ς and Y ς . T O can be identified with R through the bundle homomorphism χ : T O → R defined by χ(X ς ) = X ς + ξ ς , χ(Y ς ) = Y ς . Then J 0 is nothing else but the restriction of −J 1 on R and g 0 is just the restriction of the generalized metric G on R. The compatibility of J 0 and g 0 is simply that of J 1 and G. To see (O, J 0 , g 0 ) is a Kähler manifold, we first derive an interesting formula: (7.1) dµ ς = −ι Xς ω 0 where ω 0 = g 0 J 0 . This actually means the G-action is Hamiltonian if we have proved ω 0 is closed. Eq. (7.1) can be checked directly. In fact, by Eq. (4.3) and definition, ι X ζ dµ ς = −g(Y ς , X ζ ) = −g 0 (Y ς , X ζ ) = −g 0 (J 0 X ς , X ζ ) = −ω 0 (X ς , X ζ ), and ι Y ζ dµ ς = −g(Y ς , Y ζ ) = −g(X + ς , X + ζ ) = −g 0 (X ς , X ζ ) = −g 0 (J 0 X ς , Y ζ ) = −ω 0 (X ς , Y ζ ) as required. To show ω 0 is closed, we have to prove: (i) dω 0 (X ς , X ζ , X σ ) = 0; (ii) dω 0 (Y ς , X ζ , X σ ) = 0; (iii) dω 0 (Y ς , Y ζ , X σ ) = 0; (iv) dω 0 (Y ς , Y ζ , Y σ ) = 0. We will only prove the last two and the rest two, which can be checked similarly, will be left to the interested readers. Proof of (iii). dω 0 (Y ς , Y ζ , X σ ) = Y ς ω 0 (Y ζ , X σ ) − Y ζ ω 0 (Y ς , X σ ) + X σ ω 0 (Y ς , Y ζ ) − ω 0 ([Y ς , Y ζ ], X σ ) + ω 0 ([Y ς , X σ ], Y ζ ) − ω 0 ([Y ζ , X σ ], Y ς ) = Y ς Y ζ µ σ − Y ζ Y ς µ σ + X σ ω 0 (X ς , X ζ ) − [Y ς , Y ζ ]µ σ + ω 0 ([X ς , X σ ], X ζ ) − ω 0 ([X ζ , X σ ], X ς ) = X σ ω 0 (X ς , X ζ ) − ω 0 ([X σ , X ς ], X ζ ) − ω 0 (X ς , [X σ , X ζ , ]) = 0 where we have used Eq. (7.1) and the fact that [X ς , J 0 X ζ ] = J 0 [X ς , X ζ ]. Proof of (iv). dω 0 (Y ς , Y ζ , Y σ ) = Y ς ω 0 (Y ζ , Y σ ) − Y ζ ω 0 (Y ς , Y σ ) + Y σ ω 0 (Y ς , Y ζ ) − ω 0 ([Y ς , Y ζ ], Y σ ) + ω 0 ([Y ς , Y σ ], Y ζ ) − ω 0 ([Y ζ , Y σ ], Y ς ) = Y ς ω 0 (X ζ , X σ ) − Y ζ ω 0 (X ς , X σ ) + Y σ ω 0 (X ς , X ζ ) + ω 0 ([X ς , X ζ ], Y σ ) − ω 0 ([X ς , X σ ], Y ζ ) + ω 0 ([X ζ , X σ ], Y ς ) = Y ς µ [ζ,σ] − Y ζ µ [ς,σ] + Y σ µ [ς,ζ] − Y σ µ [ς,ζ] + Y ζ µ [ς,σ] − Y ς µ [ζ,σ] = 0 where we have also used the equality [J 0 X ς , J 0 X ζ ] = −[X ς , X ζ ] and the equivariance of µ. In [9], the discussion concerning the geometry on those G C -orbits is aimed at investigating stability conditions in equivariant Kähler geometry. The theorem above shows the same argument could apply to our more general setting. We will turn to the details of this "generalized complex stability" in future works. This study is supported by the Natural Science Foundation of Jiangsu Province, China (BK20150797) and by the China Scholarship Council (201806715027). Different splittings are related by B-field transforms, i.e. e B (X + ξ) = X + ξ + ι X B, where B is a 2-form.A Courant algebroid has more symmetries than the tangent bundle T . In a given splitting, an automorphism of E is representedby a pair (ψ, B), where ψ is a diffeomorphism of M and B is a 2-form on M. These two should satisfy H − ψ * (H) = dB. Definition 3. 1 . 1A map ϕ : g → Γ(E) covering ϕ 0 is called an isotropic trivially extended g-action if (i) ϕ is isotropic,i.e. the image of ϕ is isotropic pointwise w.r.t. the inner product and (ii) ϕ preserves the brackets, i.e. ϕ([ς, ζ]) = [ϕ(ς), ϕ(ζ)] c for ς, ζ ∈ g. Definition 4 . 1 . 41Let M be a generalized Kähler manifold carrying an extended Gaction ϕ preserving the underlying generalized Kähler structure. An equivariant map Proposition 4. 5 . 5S(ς, ζ) = {µ ς , µ ζ } β 1 , i.e. the Poisson bracket of µ ς , µ ζ w.r.t. the Poisson structure β 1 . (3. 2 2) and Eq. (3.4) are used. Note that {X ς , Y ς } ς∈g spans a smooth distribution in the sense of Sussmann 4[17]. This distribution is not necessarily integrable.However, we still have Proposition 4. 6 . 6The smooth distribution D spanned by {X ς , Y ς } ς∈g and the image of β 1 : T * → T is integrable.4 Such a distribution is not necessarily of constant rank. Definition 5. 1 . 1If M is a Hamiltonian G-generalized Kähler manifold, the G-action is called strong Hamiltonian if the map S defined in § 4 vanishes. Example 5. 2 . 2An ordinary Hamiltonian G-Kähler manifold viewed as a Hamiltonian G-generalized Kähler manifold is, of course, strong Hamiltonian. The first nontrivial examples of strong Hamiltonian generalized Kähler manifolds are provided by the following Proposition 5.3. All Hamiltonian S 1 -generalized Kähler manifolds are strong Hamiltonian. Example 5. 5 . 5Let us first construct an SU(N − 3)-invariant generalized Kähler structure on M = CP N where N > 3. This is adapted from a generalized Kähler structure on CP 2 in [1] as follows. Let J 1 , J 2 be the canonical generalized complex structures on C N +1 associated to the canonical complex and symplectic structures. S 1 1acts on C N +1 by scaling, with a moment map µ 0 (z) = |z| 2 − 1. One chooses an S 1 -invariant deformation ε of J 1 while keeping J 2 fixed. For instance, we choose Fubini- Study symplectic form. Note that the standard action of SU(N + 1) on C N +1 commutes with the S 1 -action and preserves J 2 . Thus the SU(N + 1)-action descends to CP N and preserves the Fubini-Study form. This Fubini-Study action is Hamiltonian with a canonical moment map. Let us describe a strong Hamiltonian SU(N − 3)-action on the generalized Kähler manifold (CP N , J 1 , J 2 ). SU(N − 3), as a subgroup of SU(N + 1), acts trivially on the first three components of C N +1 and nontrivially on the remainder components. By our construction, ε is invariant under this SU(N − 3)-action and therefore the pair (J 1 , J 2 ) is also invariant. The moment map µ of this SU(N − 3)-action is just the restriction of the Fubini-Study moment map on the Lie algebra of SU(N − 3). By our construction, only the tranverse complex coordinates induced from J 1 are involved in the moment map. The components of µ are thus Casimir functions w.r.t. β 1 and hence the SU(N − 3)-action is strong Hamiltonian.We will give more nontrivial examples in our future work, and here only turn to the general question of complexifying the extended G-action. Theorem 5. 7 . 7If M is a strong Hamiltonian G-generalized Kähler compact manifold, then the extended g C -action defined in Def. 5.1 integrates to a unique extended G Caction. Proof. LetG be the universal cover of G. The complexification ϕ C actually gives rise to an infinitesimal Lie algebra action of g C on the total space of the Courant algebroid E. Recall the fact that a g C -action on a manifold N integrates to aG Caction if and only if each vector field on N generated by an element in g C is complete [10, Appendix B]. Since we already have the G-action on E, to prove the theorem, it suffices to check the infinitesimal automorphism (Y ς , −ι Yς H), regarded as a vector field on E, generates a global flow. Let ψ t be the flow on M generated by Y ς . It is of course global since M is compact. Define a one-parameter 2-form on M B t := − t 0 ψ * −τ (ι Yς H)dτ, t ∈ R. 6 . 6GIT quotients and generalized holomorphic structures Throughout this section, let M be a compact G-generalized Kähler manifold. To have a good quotient, we assume that we are in the strong Hamiltonian case, and that 0 is a regular value of µ. 6 Consequently 0 is a regular value of µ, the strong Hamiltonian G-action on M can be complexified and we can talk about the extended G C -action and G C -orbits in M. Define M s := {gx|g ∈ G C , x ∈ µ −1 (0)}. From a GIT viewpoint, points in M s can reasonably be called stable and M s be called the stable locus.Proposition 6.1. M s is an open set of M. Proof. If M s contains an open neighbourhood U of µ −1 (0), then M s = ∪ g∈G C gU and is thus an open set of M.Let {ς a } be a basis of g and X a the vector field generated by ς a . Since µ −1 (0) has codimension dimg in M, we only need to prove the vector fields Y a = J + X + a are linearly independent over µ −1 (0) and orthogonal to T µ −1 (0) in T . conditions in Def. 3.4 are all satisfied. Let A, B ∈ Γ(A) and C ∈K C . Since J 1 is integrable, [A, B] H ∈ Γ(L 1 ). Additionally, (C, [A, B] H ) = π(A)(C, B) − ([A, C] H , B) = ([C, A] H , B). But since the extended action of G C preserves J 1 , we must have [C, A] H ∈ Γ(L 1 ). Therefore, (C, [A, B] H ) = 0 and [A, B] H ∈ Γ(A), i.e. A is involutive. Kähler case, O may be a generalized Kähler submanifold in the sense that the two generalized complex structures L 1 and L 2 , as complex Dirac structures, can both be pulled back onto O and induce a generalized Kähler structure on O. However, this is indeed not the fact. As was observed in [18], a generalized Kähler submanifold of M must be invariant under both J + and J − . This cannot happen in general for the orbit O because J + X ς is not tangent to O. L ⊂ E C is involutive under the Courant bracket. We also say J is integrable in this case.particular interest for us is the case of Dirac structures, involutive maximal isotropic subbundles of E C . 3.2. Generalized holomorphic and generalized Kähler structures. Definition 3.2. A generalized complex structure on a Courant algebroid E is a com- plex structure J on E orthogonal w.r.t. the inner product and its √ −1-eigenbundle Ordinary complex and symplectic structures are extreme examples of generalized complex structures. Note that the integrability of J is equivalent to In this paper, V C is used to denote the complexification of a real vector space or vector bundle V . Throughout the paper there are different Courant algebroids, but we always denote the anchor map by π. The context will exclude ambiguities. This still holds even without (i) because [A, B] c + [B, A] c is an exact 1-form, which has no effect at the level of infinitesimal inner automorphism of E. The importance to include (i) in the definition will be clear later. We haven't reviewed the pure spinor description of a generalized complex structure in this paper, for this see[5]. Note that X ς = −β 2 (dµ ς ). In the Kähler case, β 2 is invertible and therefore that (i) 0 is a regular value of µ is equivalent to that (ii) G acts locally freely on µ −1 (0). This equivalence still holds here but needs more technical arguments, which can be found in[16]. Lin-Tolman's formula applies to the more general case of Hamiltonian action. Reduction of Courant algebroids and generalized complex structures. H Bursztyn, G R Cavalcanti, M Gualtieri, Adv. Math. 2112H. Bursztyn, G. R. Cavalcanti, and M. Gualtieri, Reduction of Courant algebroids and gener- alized complex structures, Adv. Math. 211, no. 2, 726-765, 2007. Generalized Kahler and hyperKahler quotients, Poisson geometry in mathematics and physics. H Bursztyn, G R Cavalcanti, M Gualtieri, Contemp. Math. 450Amer. Math. SocH. Bursztyn, G. R. Cavalcanti, and M. Gualtieri, Generalized Kahler and hyperKahler quotients, Poisson geometry in mathematics and physics, Contemp. Math., vol. 450, Amer. Math. Soc., Providence, RI, pp. 61C77, 2008. Topology of generalized complex quotients. T Baird, Y Lin, J. Geom. Phys. 6010T. Baird and Y. Lin, Topology of generalized complex quotients, J. Geom. Phys. Vol. 60, no. 10, 1539-1557, 2010. Reduction of metric structures on Courant algebroid. G R Cavalcanti, J. Symplectic Geom. 43G. R. Cavalcanti, Reduction of metric structures on Courant algebroid, J. Symplectic Geom. 4, no. 3, 317-343, 2006. Generalized complex geometry. M Gualtieri, arXiv:math./0401221Oxford UniversityPhD thesisM. Gualtieri, Generalized complex geometry, PhD thesis, Oxford University, 2003. arXiv: math./0401221. Generalized complex geometry. M Gualtieri, Ann. of Math. 174M. Gualtieri, Generalized complex geometry, Ann. of Math, 174: pp. 75-123, 2011. Generalized Kähler geometry. M Gualtieri, Commun. Math. Phys. 331M. Gualtieri, Generalized Kähler geometry, Commun. Math. Phys. 331, 297C331, 2014. Geometric quantization and multiplicities of group representations. V Guillemin, S Sternberg, Invent. Math. 67V. Guillemin, S. Sternberg, Geometric quantization and multiplicities of group representations. Invent. Math. 67, 515-538, 1982. Convexity properties of the moment mappings. V Guillemin, S Sternberg, Invent. Math. IIV. Guillemin, S. Sternberg, Convexity properties of the moment mappings. II, Invent. Math. 77, 533-546, 1984. Moment Maps, Cobordisms, and Hamiltonian Group Actions. Mathematical Surveys and Monographs. V Guillemin, V Ginzburg, Y Karshon, American Mathematical SocietyV. Guillemin, V. Ginzburg, and Y. Karshon. Moment Maps, Cobordisms, and Hamiltonian Group Actions. Mathematical Surveys and Monographs. American Mathematical Society, 2002. Generalized Calabi-Yau manifolds. N Hitchin, Q. J. Math. 543N. Hitchin, Generalized Calabi-Yau manifolds, Q. J. Math. 54, no. 3, 281-308, 2003. Cohomology of quotientsin symplectic and algebraic geometry. F Kirwan, Mathematical Notes. 31F. Kirwan, Cohomology of quotientsin symplectic and algebraic geometry, Mathematical Notes, 31, Princeton, NJ, 1984. Transformation Groups in Differential Geometry. S Kobayashi, Springer-VerlagS. Kobayashi, Transformation Groups in Differential Geometry, Springer-Verlag, 1972. Convexity properties for generalized moment maps. Y Nitta, math.DG/0710.3924J. Math. Soc. Japan. 614Y. Nitta, Convexity properties for generalized moment maps, J. Math. Soc. Japan, Vol. 61, No. 4, 1171-1204, 2009. math.DG/0710.3924. Symmetries in generalized Kähler geometry. Y Lin, S Tolman, arXiv:math./0509069Commun. Math. Phys. 268Y. Lin, and S. Tolman, Symmetries in generalized Kähler geometry, Commun. Math. Phys. 268, 199-222, 2006. arXiv:math./0509069 Convexity properties for generalized moment maps. Y Nitta, math.DG/0710.3924J. Math. Soc. Japan. 614Y. Nitta, Convexity properties for generalized moment maps, J. Math. Soc. Japan, Vol. 61, No. 4, 1171-1204, 2009. math.DG/0710.3924. Orbits of families of vector fields and integrability of distributions. H Sussmann, Trans. Amer. Math. Soc. 180H. Sussmann, Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc., Vol.180, 171-188, 1973. A note on submanifolds and mappings in generalized complex geometry. I Vaisman, Monatsh. Math. 180I. Vaisman, A note on submanifolds and mappings in generalized complex geometry, Monatsh. Math., Vol.180, Issue 2, 373-390, 2016. generalized holomorphic structures. Y Wang, J. Geom. Phys. 86Y. Wang, generalized holomorphic structures, J. Geom. Phys. 86, 273-283, 2014. Y Wang, arXiv:1708.09724Metric reduction and generalized holomorphic structures. Nanjing 210098, ChinaDepartment of Mathematics, Hohai UniversityY. Wang, Metric reduction and generalized holomorphic structures, arXiv:1708.09724. Department of Mathematics, Hohai University, Nanjing 210098, China
{'fraction_non_alphanumeric': 0.0783309210025139, 'fraction_numerical': 0.023786851527386303, 'mean_word_length': 3.2071148145180675, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 79, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'We revisit generalized Kähler reduction introduced by Lin and Tolman in[15]from a viewpoint of geometric invariant theory. It is shown that in the strong Hamiltonian case introduced in the present paper, many well-known conclusions of ordinary Kähler reduction can be generalized without much effort to the generalized setting. It is also shown how generalized holomorphic structures arise naturally from the reduction procedure.', 'arxivid': '1803.01178', 'author': ['Yicao Wang '], 'authoraffiliation': [], 'corpusid': 119326603, 'doi': '10.1016/j.geomphys.2018.12.014', 'github_urls': [], 'n_tokens_mistral': 18803, 'n_tokens_neox': 16595, 'n_words': 10331, 'pdfsha': '748fcebfa107a1d4ee17a8624b8e019779a23770', 'pdfurls': ['https://arxiv.org/pdf/1803.01178v2.pdf'], 'title': ['THE GIT ASPECT OF GENERALIZED KäHLER REDUCTION. I', 'THE GIT ASPECT OF GENERALIZED KäHLER REDUCTION. I'], 'venue': []}
arxiv
Bounded Distributed Flocking Control of Nonholonomic Mobile Robots Thang Nguyen Department of Computer Science and Engineering Advanced Robotics and Automation (ARA) Lab University of Nevada NV89557RenoUSA Hung M La Department of Computer Science and Engineering Advanced Robotics and Automation (ARA) Lab University of Nevada NV89557RenoUSA Vahid Azimi School of Electrical and Computer Engineering Georgia Institute of Technology 777 Atlantic Drive NW30332-0250AtlantaGAUSA Thanh-Trung Han Faculty of Electrical and Electronics Engineering Ton Duc Thang University, Ho Chi Minh City, Viet Bounded Distributed Flocking Control of Nonholonomic Mobile Robots Chapter 1 There have been numerous studies on the problem of flocking control for multiagent systems whose simplified models are presented in terms of point-mass elements. Meanwhile, full dynamic models pose some challenging problems in addressing the flocking control problem of mobile robots due to their nonholonomic dynamic properties. Taking practical constraints into consideration, we propose a novel approach to distributed flocking control of nonholonomic mobile robots by bounded feedback. The flocking control objectives consist of velocity consensus, collision avoidance, and cohesion maintenance among mobile robots. A flocking control protocol which is based on the information of neighbor mobile robots is constructed. The theoretical analysis is conducted with the help of a Lyapunov-like function and graph theory. Simulation results are shown to demonstrate the efficacy of the proposed distributed flocking control scheme.With point-mass models, the problem of flocking control of multiple agents has been addressed with typical results reported in[7][8][9][10][11]. For a wide range of engineering applications, extensive studies in flocking control of mobile robots have been done in various scenarios[12][13][14].In this chapter, we study the problem of distributed flocking control of mobile robots by bounded feedback, which takes into consideration nonholonomic nature of mobile robots as well as the implementation issue posed by the physical limit of the motor speed. Our flocking control problem employs the full dynamic model of the mobile robot derived in[15]. Similar to[16,17], due to the nonholonomic property of the dynamics of mobile robots, our proposed design framework constructed to achieve velocity consensus is modular. In other words, the consensuses on the linear speed and orientation angles are obtained separately.In this chapter, we are interested in agents with nonholonomic dynamics and boundedness constraints. Specifically, a coordination function is proposed to ensure that the induced attractive and repulsive forces are bounded, and hence can be incorporated in the bounded velocity control. Using the results of Barbalat's lemma and graph theory, the theoretical analysis is conducted, which shows that the maximal value of the coordination function determines the basin of attraction for the flocking convergence.In this chapter, graph theory will be employed as in the case of nearest neighbor communication[7,18]. We will employ the velocity control law reported in[16,17]in a decentralised sense, which helps to avoid collision and maintain a linear speed consensus. In addition, the orientation consensus will be achieved using a modified approach, which is inspired by the one in[19], where the input constraint is taken into account.The organization of the chapter is as follows. Section1.2 summarises some research work in the literature related to the topic in this chapter. In Section 1.3, the multiple-goal control problem for flocking of nonholonomic mobile robots and preliminaries are introduced. Section 1.4 describes main results where a modular design framework is proposed for bounded velocity control and bounded orientation control and the theoretical analyses are introduced. In Section 1.5, a description of an obstacle avoidance scheme is presented. Section 1.6 shows some simulation results. Section 1.7 concludes the chapter by some conclusions.Notations: R and R + are the sets of real numbers and nonnegative real numbers, respectively; for q = [q 1 , . . . , q n ] T , ∇ q = [∂ /∂ q 1 , . . . , ∂ /∂ q n ] T is the del operator [20]; for two vectors a and b, a · b is their scalar product; (a 1 , . . . , a n ) is [a T 1 , . . . , a T n ] T ; | · | is the absolute value of scalars; and · is the Euclidean norm of vectors. INTRODUCTION It is well-known that the collective behavior of self-propelled organisms constitutes flocking [1]. The coherent motion of the flock inspires various research on flocking control of multiagent systems. A typical objective is to achieve a desired collective motion which can be produced by a constructive flocking control procedure. For numerous models, which are described from simplest models such as point-mass models to actual physical models, design protocols have been systematically proposed for multiagent systems [2,3]. Several control strategies were also addressed in noisy environments where the agent's position is affected by noise [4][5][6]. RELATED WORK In many applications, the mission carried out by a single complicated robotic system can be equivalently completed by a coordination of a mobile robotic system with much simpler configurations, whose advantages lie in scalability, flexible deployment, cheaper costs, reliability, etc. Therefore, more sophisticated tasks can be fulfilled using a group of small mobile robots with lower cost and higher efficiency than a complex unit; see [2,14,[21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39] and references therein. Flocking control of mobile robots was widely addressed with different control schemes; see [9,18,21,[40][41][42] and references therein. Recently, a new measuretheoretic approach which systematically provides a framework for obtaining flocking protocols for mobile robots was reported in [13]. The common assumption in many papers is the availability of information of all agents or all-to-all communication. Numerous control protocols for mobile robots have been constructed based on this assumption. This centralized communication control architecture yields inflexibility and large computation costs for the controller of each agent, especially when the number of agents is large. Meanwhile, a distributed control protocol can offer an ease of implementation and less computational burden as each element of the system needs only the information of neighbor agents. In this direction, a range of decentralized control schemes for mobile robots have been proposed [8,10,18,43,44]. For a wide range of engineering applications, cohesion maintenance and collision avoidance (CMCA) properties of a mobile robotic system are of importance. As reported in [21,45], the attractive and repulsive forces cannot be included in the control for CMCA of mobile robots, as it is possible for point-mass agents [7]. In [13], desired attractive and repulsive forces for CMCA of mobile robots was achieved using a new rearrangement strategy. In [7,10,18], the graph theory was employed to generate control protocols that maintain CMCA of multiagent systems with double integrator models. In [19], a distributed flocking control approach was proposed but no constraints on the control inputs are imposed. The work in [16,17] considers the bounded feedback flocking control problem for nonholonomic mobile robots without a flocking desired heading angle. The problem of interest in this chapter is to address bounded control of nonholonomic dynamic mobile vehicles, which also achieves CMCA and obstacle avoidance. We also consider a flocking desired heading angle, which reveals a collective flocking behaviour. PROBLEM FORMULATION Similarly to [16,17], we investigate a collective system of N identical autonomous mobile robots whose respective equations of motion are [15] q i = v i e(θ i ) θ i = w i v i = u i w i = τ i (1.1) where i = 1, ..., N, q i = [x i , y i ] T ∈ R 2 , and θ i ∈ R are respectively the position and the heading angle of the i-th robot in the inertial frame Oxy; v i ∈ R is the linear speed, and e(θ i ) is the unit vector [cos θ i , sin θ i ] T ; w i ∈ R is the angular speed, and u i , τ i ∈ R are control inputs. Following the same vein as in [16], we define 0 < r 0 < R 0 . Then, the flocking control problem for (1.1) is to construct the control inputs u i , τ i as bounded functions of the collective state (q 1 , . . . , q N , θ 1 , . . . , θ N , v 1 , . . . , v N , w 1 , . . . , w N ) in a distributed fashion to satisfy the following multiple goals G1) Velocity consensus: lim t→∞ (q i (t) −q j (t)) = 0, ∀i, j = 1, . . . , N (1.2) G2) Collision avoidance: r i j (t) = q i (t) − q j (t) ≥ r 0 , ∀t ≥ 0, ∀i = j G3) Cohesion maintenance: r i j (t) ≤ R 0 , ∀t ≥ 0, ∀i = j. Similarly to [16,17], we have the following definition. Definition 1. A controlζ = g(ζ , y), u = c(ζ , y), (ζ , y) ∈ R d × R m of a systemẋ = f (x, u), y = h(x, u) is said to be bounded if there is a finite constant M > 0 such that c(ζ , y) ≤ M, ∀(ζ , y) ∈ R d × R m . To achieve the goals G2) and G3), we consider the coordination function U : R + → R + which satisfies the following properties: P1) there is a constant U M > 0 such that 0 ≤ U(r) ≤ U M , ∀r ∈ R (1.3) P2) U(r) is continuously differentiable on [r 0 , R 0 ]; P3) lim r→r + 0 U(r) = U M ; and P4) lim r→R − 0 U(r) = U M . For a link between agents i and j of the flock, we aim to maintain r i j (t) ∈ [r 0 , R 0 ]. Without loss of generality, we assume that U(0) = 0 and hence U(r) is well defined for r ii = 0 [16]. We are interested in the function U with the dead zone [a, A] since even distribution of agents may not be achievable by a common coordination function U. Accordingly, we use the zone [a, A] for free alignment. A function U satisfying the above requirements is shown in Figure 1 For bounded control, we shall use the linear saturation functions σ 1 , and σ 2 , which are continuous and nondecreasing functions and satisfy, for given positive constants L i ≤ M i , i = 1, 2 [16] i) σ i (−s) = −σ i (s) for all s; ii) σ i (s) = s for s ≤ L i ; and iii) |σ i (s)| ≤ M i , ∀s ∈ R. Similarly to other works on distributed for multiagent systems [7,10,17,18], the graph theory will be utilised to address our problem. A digraph associated with (1.1) is called G (t) = (V , E (t)) where V = 1, . . . , N and E ⊆ V × V . The set V is denoted as the node set of G (t) and the set E (t) is defined as the edge set of G (t). In addition, N i (t) denotes the neighbor set of the node i for i = 1, . . . , N. As in [10], the description of the edge E (t) is presented as follows. Given any R > 0, ε 2 ∈ (0, R), and ε 1 ∈ (0, R−ε 2 ), for any t ≥ 0, E (t) = {(i, j)|i, j ∈ V } is defined such that 1. E (0) = {(i, j)|ε 1 < q i (0) − q j (0) < (R − ε 2 )}; 2. if q i (0) − q j (0) ≥ R, then (i, j) / ∈ E (t); 3. for i = 1, . . . , N, j = 1, . . . , N, if (i, j) / ∈ E (t − ) and q i (t) − q j (t) < R − ε 2 , then (i, j) ∈ E (t); 4. for i = 1, . . . , N, j = 1, . . . , N, if (i, j) ∈ E (t − ) and q i (t) − q j (t) < R, then (i, j) ∈ E (t). As in [17], the following results will be employed for the main results. Lemma 1. Let σ : R → R be a function satisfying σ (−s) = −σ (s), ∀s ∈ R. Then, for all a i , b i , it holds true that 1 2 N ∑ i=1 ∑ j∈N i (t) (a i − a j )σ (b i − b j ) = N ∑ i=1 ∑ j∈N i (t) a i σ (b i − b j ). (1.4) Proof: Since σ (−s) = −σ (s) and G (t) is an undirected graph, we have N ∑ i=1 ∑ j∈N i (t) a j σ (b i − b j ) = − N ∑ i=1 ∑ j∈N i (t) a j σ (b j − b i ) = − N ∑ i=1 ∑ j∈N i (t) a i σ (b i − b j ). (1.5) Hence, N ∑ i=1 ∑ j∈N i (t) (a i − a j )σ (b i − b j ) = N ∑ i=1 ∑ j∈N i (t) a i σ (b i − b j ) − N ∑ i=1 ∑ j∈N i (t) a j σ (b i − b j ) = 2 N ∑ i=1 ∑ j∈N i (t) a i σ (b i − b j ) (1.6) which implies (1.4). Remark 1. Lemma 1 plays an important role in the theoretical analysis of the main results. Here, the lemma is similar to the one in [46]. However, [46] considers all-to-all communication in the multiagent system. Our problem in this chapter is focused on the distributed fashion, which requires the employment of the neighbour set N i (t) of robot i. Lemma 2. The linear saturation functions σ i , i = 1, 2, 3 satisfy (σ i (θ 1 ) − σ i (θ 2 ))σ i (θ 1 − θ 2 ) ≥ 0, ∀θ 1 , θ 2 . (1.7) Proof: Without loss of generality, suppose that θ 1 ≥ θ 2 . Since σ i are nondecreasing functions, this implies that σ i (θ 1 ) − σ i (θ 2 ) ≥ 0. (1.8) Furthermore, as σ i (0) = 0, θ 1 ≥ θ 2 and the nondecreasing property of σ i imply that σ i (θ 1 − θ 2 ) ≥ 0. (1.9) Multiplying (1.8) and (1.9) side-by-side, we obtain (1.7). MAIN RESULTS Our constructive strategy is to design u i to achieve consensus on v i , and τ i to achieve consensus on θ i . The design for u i is derived from [17], while the construction for τ i is built based on the approach in [19]. Note that U(r i j ) = U( q i − q j ), which is the symmetric function of q i and q j . As a result, we write U(q i , q j ) with the understanding that U(q i , q j ) = U(q j , q i ). The control protocols u i and τ i are constructed based on Lyapunov theory. Specifically, a positive definite function V is presented such that the time derivative of V is a negative definite function. Regarding the distribution control problem, the graph theory will be employed to show the connectivity preservation for our multiagent system. Then, we apply the LaSalle's invariance principle [47] to conclude the desired consensuses. Similarly to [17], the initial state of the collective system of agents (1.1) is chosen such that the graph G (0) is connected. The parameters of the graph G (0) are chosen as follows R = R 0 , (1.10) r 0 ≤ ε 1 < a, (1.11) 0 < ε 2 ≤ R 0 − a. (1.12) Speed consensus and connectivity preservation The derivation of this subsection is essentially similar to the control design for the linear speed in [17]; hence, it is presented here for completeness. Consider the energy function for system (1.1) V 1 = 1 2 N ∑ i=1 ∑ j∈N i (t) U(q i , q j ) + 1 2 N ∑ i=1 v 2 i . (1.13) We assume that U(r) is designed such that U(R 0 ) = U M > V 1max , (1.14) where V 1max 1 2 N ∑ i=1 v 2 (0) + N(N − 1) 2 U(R 0 − ε 2 ). (1.15) Let m 0 be the number of the links of the initial graph. The simplest connected graph of N agents is a tree whose number of links is n − 1. Hence, m 0 ≥ n − 1. Let V 1 (0) ≤ V 1max − (N − 1)(N − 2) 2 U(R 0 − ε 2 ). (1.16) Note that U(q i , q j ) is a symmetric function of q i and q j . We compute the derivative of V 1 with respect to (1.1) V 1 = N ∑ i=1 ∑ j∈N i (t) ∇ q i U(q i , q j ) ·q i + N ∑ i=1 v i u i = N ∑ i=1 v i ∑ j∈N i (t) ∇ q i U(q i , q j ) · e(θ i ) + u i . (1.17) From (1.17), a control law for the speed consensus protocol is chosen as u i = − ∑ j∈N i (t) ∇ q i U(q i , q j ) · e(θ i ) − ∑ j∈N i (t) σ 1 (v i − v j ) (1.18) where σ 1 is the linear saturation function introduced in Section 1.3. Substituting (1.18) into (1.17), we obtaiṅ V 1 = − N ∑ i=1 ∑ j∈N i (t) v i σ 1 (v i − v j ). (1.19) We have the following speed consensus theorem [17]. Theorem 1. Suppose that the collective system (1.1) subject to the protocol (1.18) is initiated such that V 1 (0) < V 1max . Then, the following properties hold: i) G (t) is connected for all t ≥ 0 and there exists t k such that for t ≥ t k , G (t) = G (t k ) ii) Collision avoidance is guaranteed, i.e. q i − q j > r 0 for all i, j ∈ N and i = j. iii) lim t→∞ (v i (t) − v j (t)) = 0 Proof: Assume that G (t) switches at time t k (k = 1, 2, . . . ). Hence, G(t) = G(0) for all t ∈ [0,t 1 ) . In other words, G (t) = G(0), t ∈ [0,t 1 ) G (t 1 ) = G(0). (1.20) We prove that G(0) ⊂ G(t 1 ). Using control law (1.18), we havė V 1 = − N ∑ i=1 v i ∑ j∈N i (t) σ 1 (v i − v j ). (1.21) According to Lemma 1, V 1 = − 1 2 N ∑ i=1 ∑ j∈N i (t) (v i − v j )σ 1 (v i − v j ). (1.22) Because σ 1 (s) is an odd function, (v i − v j )σ 1 (v i − v j ) ≥ 0. Thus,V 1 ≤ 0, which deduces V 1 (t) ≤ V 1 (0) < V 1max < U M for [0,t 1 ). (1.23) From the definition of U(r), U(R 0 ) > V 1max ≥ V 1 (0). Hence for any (i, j) ∈ G (t) for t ∈ [0,t 1 ) U(q i , q j ) ≤ V 1 (t) < U M = U(r 0 ) = U(R 0 ). (1.24) By the continuity of U(r), (1.24) shows that r 0 < q i − q j < R 0 . This implies that no existing links are deleted at time t 1 and collision avoidance is achieved. As a result, new links must be added to the current graph at the switching time t 1 . We assume that there are m 1 new links being added to the network at time t 1 . Since the number of the current links before switching is m 0 ≥ N − 1 and the complete graph possesses N(N−1) 2 edges, m 1 ≤ N(N−1) 2 − (N − 1) = (N−1)(N−2) 2 . Hence, we have V 1 (t 1 ) = V 1 (t − 1 ) + m 1 U(R 0 − ε 2 ). (1.25) Due to (1.16), V 1 (t − 1 ) ≤ V 1 (0) < V 1max − (N − 1)(N − 2) 2 U(R 0 − ε 2 ).(1.26) Thus, V 1 (t 1 ) < V 1max .(1.27) By induction, for t ∈ [t k−1 ,t k ), V 1 = − 1 2 N ∑ i=1 ∑ j∈N i (t) (v i − v j )σ 1 (v i − v j ) ≤ 0, (1.28) and therefore V 1 (t) ≤ V 1 (t k−1 ) ≤ V 1max . This shows that no edges are lost at time t k and V 1 (t k ) ≤ V 1max . As a result, the size of the set of the links of G (t) forms an increasing sequence, bounded above by N(N−1) 2 , which is the number of the links of a complete graph. Thus, there exists a finite integer k > 0 such that G (t) = G (t k ), t ∈ [t k , ∞). (1.29) Hence, for t ≥ t k , we havė V 1 = − 1 2 N ∑ i=1 ∑ j∈N i (t k ) (v i − v j )σ 1 (v i − v j ) ≤ 0. (1.30) Next, we will show that the linear velocities of all agents converge to the same value. Using the fact that U(q i , q j ) ≤ V 1 (t) ≤ V 1max < U M and the properties of U, we deduce q i − d j > r 0 . This shows that no collision takes place among agents. Since 0 ≤ V 1 (t) ≤ V 1max andV 1 ≤ 0, by Barbalat's lemma, lim t→∞V1 (t) = 0. Because the graph G (t) is connected for all t and sσ 1 (s) ≥ 0 for all s, from (1.30), lim t→∞ (v i − v j ) = 0, for all i, j = 1, 2, . . . , N. (1.31) Remark 2. The proof of the theorem follows similar approaches as in [10,48], where graph theory was employed as a means for proving the connectivity of mobile networks. The potential in this chapter is similar to the one in [10] in the sense that it is bounded. In contrast, the potential function used in [48] goes to infinity at singularities. Note that the mobile robots in this work are nonholonomic while [10,48] addressed double integrator systems. Theorem 1 shows that the design (1.18) achieves speed consensus and the goals G2) and G3). In the next subsection, we will design τ i for orientation consensus completing the goal G1). Orientation Consensus Motivated by the orientation consensus design method presented in [19], we shall develop a bounded control approach which employs a saturation function in Section 1.3. Define the orientation trajectory error for agent i as e i = θ i − θ r , (1.32) where θ r is the desired orientation of the flock. Thus, the angle difference between two agents i and j is θ i − θ j = θ i − θ r − (θ j − θ r ) = e i − e j .(1.33) Similarly to [19], the following lemma is employed for our convergence analysis. Lemma 3. Suppose that the flock possesses a graph G (t), then the trajectory error signals of the group have the following property: 1 2 N ∑ i=1 ∑ j∈N i (t) (e i − e j )(ė i −ė j ) = N ∑ i=1 ∑ j∈N i (t) e i (ė i −ė j ). (1.34) Proof: The proof is similar to the one employed in Lemma 1. We have the following orientation consensus theorem. Theorem 2. Assume that the desired orientation θ r and its first and second derivation are bounded, and the collective system (1.1) is subject to the following protocol τ i =θ r − σ 2 (θ i −θ r ) − k θ n i + 1 [(n i + 1)θ i − ∑ j∈N i (t) θ j − θ r ],(1. 35) where n i is the number of the neighbors of robot i and k θ is a positive parameter. Then, all the mobile robots eventually reach consensus on the heading angles θ i in the sense that lim t→∞ (θ i (t) − θ j (t)) = 0, ∀i, j. (1.36) Proof: Consider the following Lyapunov function candidate V 2 (t) = 1 2 N ∑ i=1 k θ n i + 1 e 2 i + 1 2 N ∑ i=1ė 2 i + 1 4 N ∑ i=1 ∑ j∈N i (t) k θ n i + 1 (θ i − θ j ) 2 . (1.37) According to Theorem 1, there exists t k such that for t ≥ t k , G (t) = G (t k ). The derivative of V 2 (t) with respect to t for t ≥ t k is given aṡ V 2 (t) = N ∑ i=1 k θ n i + 1 e iėi + N ∑ i=1ė i (τ i −θ r ) + N ∑ i=1 ∑ j∈N i (t) k θ n i + 1 (θ i − θ j )(θ i −θ j ) = N ∑ i=1 k θ n i + 1 e iėi − N ∑ i=1 k θ n i + 1ė i e i − N ∑ i=1ė i σ 2 (ė i ) − N ∑ i=1 ∑ j∈N i (t) k θ n i + 1ė i (θ i − θ j ) + 1 2 N ∑ i=1 ∑ j∈N i (t) k θ n i + 1 (θ i − θ j )(θ i −θ j ) = − N ∑ i=1ė i σ 2 (ė i ) − N ∑ i=1 ∑ j∈N i (t) k θ n i + 1ė i (e i − e j ) + 1 2 N ∑ i=1 ∑ j∈N i (t) k θ n i + 1 (e i − e j )(ė i −ė j ). (1.38) Using Lemma 3, we obtaiṅ V 2 (t) = − N ∑ i=1ė i σ 2 (ė i ) ≤ 0 (1.39) since σ 2 (.) is a nondecreasing function defined in Section 1.3. Since θ i , θ r ∈ [−pi, pi] andθ r is bounded, the control law (1.35) impliesë i is bounded. By the Barbalat's lemma, from (1.39),ė i → 0. Also, sinceθ r is bounded,ë i → 0. Therefore, the control law (1.35) implies that θ i → θ j , which proves the theorem. Remark 4. The boundedness of the control law (1.35) is guaranteed by the properties of linear saturation function σ 2 and the fact that θ i , θ j , θ r ∈ [−π, π]. This demonstrates that the proposed control scheme meets the requirement on the physical limits of the control inputs. Remark 5. It should be noted that our control law for orientation consensus is similar to the one in [19] but here the boundedness of the control input is taken into account. The scheme in this chapter also shares the same objective as the one in [17] but offers a more simple form and implementation. Combining Theorems 1 and 2, we have the following bounded flocking theorem. Theorem 3. Suppose that the collective system (1.1) is subject to the bounded protocols (1.18) and (1.35). Suppose further that the initial configuration of the collective system (1.1) is such that N (0) is connected. Then, all the multiple flocking goals of velocity consensus, cohesion maintenance, and collision avoidance are achieved. Proof: The proof is straightforward from the results of Theorems 3.1 and 3.2. Avoidance of Obstacles The problem of obstacle avoidance has been extensively studied in the literature [17,19,22]. In this section, we employ the idea from [19] to derive our control algorithm in which the agents are able to pass obstacles. It is shown that a convex obstacle can be extrapolated by a round shape or a rectangle [17,19]. In [19], a convex obstacle is presented by a circle, which is used in our work. Let q r = [x r , y r ] T be the coordinate of the robot, q jobs = [x obs , y obs ] T be the projection point of the robot onto obstacle j = 1, 2, . . . , n obs where n obs is the number of obstacles, andŌ k = [x k , y k ] T be the centre of the obstacle. From [19], q jobs = r q r −Ō k q r + (1 − r q r −Ō k )Ō r (1.40) where r is the radius of the obstacle. The projection point has the following velocity v jobs = vr sin α q r − q jobs (1.41) where v is the velocity of the agent, α is the angle between the heading of the robot and the straight line which connects the robot to the centre of the obstacle. The projection point moves in the direction [19] θ jobs = − π 2 + α + θ , α > 0 (1.42) θ jobs = π 2 + α + θ , otherwise. (1.43) The fact that the projection point possesses a position, velocity, and orientation enables it to be an agent. The above descriptions of the robot and obstacle are demonstrated in Figure 1.2. We have the following orientation consensus theorem. Theorem 4. The following control protocol guarantees that the robot avoids obstacles with arbitrary boundary shapes: 45) where N obs is the set of obstacles, n obs is the number of obstacles and Proof: The proof can be derived using the same approach as in [19]. u i = − ∑ j∈N obs ∇ q i U(q i , q j ) · e(θ i ) − ∑ j∈N obs σ 1 (v i − v j ) (1.44) τ i =θ r − σ 2 (θ i −θ r ) − k θ n obs + 1 [(n obs + 1)θ i − ∑ j∈N i (t) θ j − θ r ],(1. Remark 6. It should be noted that the speed control law in (1.44) enjoys the boundedness due to the saturation function σ 1 (.), which is different from the one in [19]. Since θ i , θ r ∈ (−π, π] and the saturation function σ 2 (.) is bounded, (1.45) reveals that the heading control law is also bounded. SIMULATION We conducted simulation for a multi-agent system of 15 mobile robots of the model (1.1). A bump function is used to generate the smooth coordination function U. As the control (1.18) invokes the gradient forces ∇ q i U, we designed the coordination function in the form U(r) = r 0 ϕ(s)ds (1.47) where ϕ is a compact support function given by The initial values of θ i and v i are randomly chosen. The initial value of w i is 0. The desired orientation of the flock is θ r = π/2. We obtained the simulation results shown in Figures 1.9-1.7. It is shown in Figures 1.3 and 1.5, the heading angles converge to θ r and angular speeds of all agents converge to 0 after t = 40s. The linear speeds are depicted in Figure 1.4, where the convergence of all agents takes place after t = 45s. In Figure 1 Next, we consider a simulation for a multiagent system with an obstacle. The obstacle is a circle whose coordinate is [12, −1] with a radius of 1. The configuration of the multiagent system is the same as in the obstacle-free case above. The desired group heading is chosen as θ r = π/4. An agent is chosen as a leader of the group. When the collision avoidance mechanism of the leader is inactive, a speed control law is designed to drive the group at a constant speed, that is ϕ(s) =        p 1 exp −(s−s 0 ) 2 ((a−r 0 )/2) 2 −(s−s 0 ) 2 if s ∈ (r 0 , a) p 2 exp −(s−s 1 ) 2 ((R 0 −A)/2) 2 −(s−s 1 ) 2 if s ∈ (A, R 0 )u l = −σ 1 (v l − v r ) (1.49) where u l is the speed control of the leader, v l is its linear speed, and v r = 0.2. During the first 50 seconds of their evolution, the mobile robots encounter the obstacle. The control laws (1.44) and (1.45) enable them to avoid potential collisions with the obstacle and with other neibouring agents. The orientations of the robots in Figure 1.10 converge to π/4 after 60 seconds. Similarly, in Figure 1.11, the linear speeds of the agents converge after 50 seconds. In Figure 1.12, the angular speeds converge faster to 0 after 50 seconds. Figs. 1.13 and 1.14 demonstrate that the control inputs both are bounded. Finally, Figure 1.15 shows that no collision takes place during the evolution of the robots. The evolution of the multiagent system is shown in Figure 1.16 in which the robots cooperate to form a flocking and avoid the obstacle. CONCLUSIONS This chapter has presented a bounded decentralized control protocol for the flocking problem of mobile robots by a systematic fashion, where the control laws only require the information of neighbor agents. The proposed scheme is modular in designing the speed control and steering control separately. Theoretical and numerical results have shown that using our proposed method, a collective system of mobile robots achieves all the multiple objectives of the flocking control: velocity consensus, cohesion maintenance, and collision avoidance. Future work would consider the shape and size of each mobile robot and obstacle avoidance for which a similar context was studied in [49]. Noisy and uncertain environments can affect the performance of the proposed scheme. Robustness analysis and improved methods can be proposed to address this issue. Acknowledgement This work was partially supported by the National Science Foundation under grant NSF-NRI #1426828, the National Aeronautics and Space Administration (NASA) under Grant No. NNX15AI02H issued through the Nevada NASA Research Infrastructure Development Seed Grant, and the University of Nevada, Reno. .1. Figure 1 . 1 11Coordination function (extracted from[16]). Remark 3 . 3The first sum in(1.18) consists of the gradients of U(q i , q j ) and the unit vector e(θ i ) which are bounded by definition. The second sum in (1.18) is comprised of σ 1 (.), which is a linear saturation function defined in Section 1.3. Hence, as a whole, the control law (1.18) for each agent is bounded. This satisfies our objective on the boundedness of the control input u i . Figure 1 . 2 12Illustration of a robot and a convex obstacle. 0 = r 0 +a 2 , s 1 = R 0 +A 2 , and p 1 , p 2 , a, A, r 0 and R 0 are design parameters. The parameters of the coordinate function are r 0 = 1, a = 3, A = 6, R 0 = 8, and U M = 15. The parameter for the control law (1.35) is k θ = 1.5. The initial positions of 15 mobile robots are randomly distributed on three circles. Their coordinates arex(i) = Γ sin(z π (i − 1)/Γ + π) y(i) = Γ cos(z π (i − 1)/Γ + π) .5, the angular speeds converge after 40 seconds. Hence, Figure 1.3 and Figure 1.4 demonstrate that consensuses on orientation and linear speed of the mobile robots have been obtained. The speed control and steering control signals are shown to be bounded in Figure 1.6 and Figure 1.7.The minimum distance among agents is described inFigure 1.8, which shows that collision avoidance is guaranteed. The flocking behavior is shown inFigure 1.9, where no collision occurred. Figure 1.3 Orientation consensus Figure 1 . 8 18The minimum distance among agents Figure 1 .Figure 1 .Figure 1 .Figure 1 . 111111 Linear speed consensus for the obstacle avoidance case 13 Speed control for the obstacle avoidance case 15 The minimum distance among agents for the obstacle avoidance case 16 Distributed flocking of 15 mobile robots for the obstacle avoidance case Flocks, herds, and schools: A qualitative theory of flocking. J Toner, Y Tu, Phys Rev E. 58Toner J, Tu Y. Flocks, herds, and schools: A qualitative theory of flocking. Phys Rev E. 1998;58:4828-4858. Distributed Sensor Fusion for Scalar Field Mapping Using Mobile Sensor Networks. H M La, W Sheng, IEEE Trans Cybernetics. 432La HM, Sheng W. Distributed Sensor Fusion for Scalar Field Mapping Using Mobile Sensor Networks. IEEE Trans Cybernetics. 2013 April;43(2):766- 778. Adaptive consensus algorithms for real-time operation of multi-agent systems affected by switching network events. F Muñoz, E S Espinoza Quesada, H M La, 10.1002/rnc.3687International Journal of Robust and Nonlinear Control. 279Rnc.Muñoz F, Espinoza Quesada ES, La HM, et al. Adaptive consensus algo- rithms for real-time operation of multi-agent systems affected by switch- ing network events. International Journal of Robust and Nonlinear Con- trol. 2017;27(9):1566-1588. Rnc.3687. Available from: http://dx.doi.org/ 10.1002/rnc.3687. Flocking control of multiple agents in noisy environments. H M La, W Sheng, IEEE International Conference on Robotics and Automation. La HM, Sheng W. Flocking control of multiple agents in noisy environments. In: 2010 IEEE International Conference on Robotics and Automation; 2010. p. 4964-4969. Multi-Agent Motion Control in Cluttered and Noisy Environments. H M La, W Sheng, Journal of Communications. 81La HM, Sheng W. Multi-Agent Motion Control in Cluttered and Noisy En- vironments. Journal of Communications. 2013;8(1):32-46. Distributed formation control for autonomous robots following desired shapes in noisy environment. A D Dang, H M La, J Horn, 2016 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI). Dang AD, La HM, Horn J. Distributed formation control for autonomous robots following desired shapes in noisy environment. In: 2016 IEEE In- ternational Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI); 2016. p. 285-290. Flocking for multi-agent dynamic systems: algorithms and theory. R Olfati-Saber, IEEE Trans Autom Control. 513Olfati-Saber R. Flocking for multi-agent dynamic systems: algorithms and theory. IEEE Trans Autom Control. 2006 Mar;51(3):401-420. Flocking in fixed and switching networks. H G Tanner, A Jadbabaie, G J Pappas, IEEE Transactions on Automatic Control. 525Tanner HG, Jadbabaie A, Pappas GJ. Flocking in fixed and switching net- works. IEEE Transactions on Automatic Control. 2007;52(5):863-868. Emergent behavior in flocks. F Cucker, S Smale, IEEE Trans Autom Control. 525Cucker F, Smale S. Emergent behavior in flocks. IEEE Trans Autom Control. 2007 May;52(5). Flocking with connectivity preservation of multiple double integrator systems subject to external disturbances by a distributed control law. Y Dong, J Huang, Automatica. 55Dong Y, Huang J. Flocking with connectivity preservation of multiple double integrator systems subject to external disturbances by a distributed control law. Automatica. 2015;55:197 -203. Fully distributed flocking with a moving leader for Lagrange networks with parametric uncertainties. S Ghapani, J Mei, W Ren, Automatica. 67Ghapani S, Mei J, Ren W, et al. Fully distributed flocking with a moving leader for Lagrange networks with parametric uncertainties. Automatica. 2016;67:67-76. Distributed geodesic control laws for flocking of nonholonomic agents. N Moshtagh, A Jadbabaie, IEEE Transactions on Automatic Control. 524Moshtagh N, Jadbabaie A. Distributed geodesic control laws for flock- ing of nonholonomic agents. IEEE Transactions on Automatic Control. 2007;52(4):681-686. Styled-velocity flocking of autonomous vehicles: a systematic design. T T Han, S S Ge, IEEE Trans Autom Control. 608Han TT, Ge SS. Styled-velocity flocking of autonomous vehicles: a system- atic design. IEEE Trans Autom Control. 2015 Aug;60(8):2015-2030. Cooperative and Active Sensing in Mobile Sensor Networks for Scalar Field Mapping. H M La, W Sheng, J Chen, IEEE Trans Systems, Man, and Cybernetics: Systems. 451La HM, Sheng W, Chen J. Cooperative and Active Sensing in Mobile Sensor Networks for Scalar Field Mapping. IEEE Trans Systems, Man, and Cyber- netics: Systems. 2015 Jan;45(1):1-12. Stabilized feedback control of unicycle mobile robots. K Amar, S Mohamed, Int J Adv Robot Syst. 10187Amar K, Mohamed S. Stabilized feedback control of unicycle mobile robots. Int J Adv Robot Syst. 2013;10(187):1-8. Swarm Intelligence -From Concepts to Applications. Swarm Intelligence -From Concepts to Applications Flocking of mobile robots by bounded feedback. T T Han, H M La, B H Dinh, 2016 IEEE International Conference on Automation Science and Engineering. Han TT, La HM, Dinh BH. Flocking of mobile robots by bounded feed- back. In: 2016 IEEE International Conference on Automation Science and Engineering (CASE); 2016. p. 689-694. Distributed flocking control of mobile robots by bounded feedback. T Nguyen, T T Han, H M La, 54th Annual Allerton Conference on Communication, Control, and Computing. AllertonNguyen T, Han TT, La HM. Distributed flocking control of mobile robots by bounded feedback. In: 2016 54th Annual Allerton Conference on Communi- cation, Control, and Computing (Allerton); 2016. p. 563-568. Coordination of groups of mobile autonomous agents using nearest neighbor rules. A Jadbabaie, J Lin, A S Morse, IEEE Trans Autom Control. 486Jadbabaie A, Lin J, Morse AS. Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans Autom Control. 2003 Jun;48(6):988-1001. Avoidance of Multiple Obstacles for a Mobile Robot With Nonholonomic Constraints. In: ASME proceedings of Dynamic Systems and Control. Y Liang, H H Lee, Liang Y, Lee HH. Avoidance of Multiple Obstacles for a Mobile Robot With Nonholonomic Constraints. In: ASME proceedings of Dynamic Systems and Control; 2005. p. 1657-1663. K F Riley, M P Hobson, S J Bence, Mathematical Methods for Physics and Engineering. CambridgeCambridge University Press3rd edRiley KF, Hobson MP, Bence SJ. Mathematical Methods for Physics and Engineering. 3rd ed. Cambridge: Cambridge University Press; 2006. Flocking in teams of nonholonomic agents. H G Tanner, A Jadbabaie, G J Pappas, Cooperative Control. Kumar V, Leonard N, Morse SSpringer309Tanner HG, Jadbabaie A, Pappas GJ. Flocking in teams of nonholonomic agents. In: Kumar V, Leonard N, Morse S, editors. Cooperative Control. vol. 309 of Lecture Notes in Control and Information Sciences. Springer; 2004. p. 229-239. Decentralized formation control and obstacle avoidance for multiple robots with nonholonomic constraints. Y Liang, H H Lee, American Control ConferenceLiang Y, Lee HH. Decentralized formation control and obstacle avoidance for multiple robots with nonholonomic constraints. In: 2006 American Control Conference; 2006. p. 5596-5601. Formation Control of Multiple Rectangular Agents with Limited Communication Ranges. T Nguyen, H M La, 10th International Symposium on Advances in Visual Computing. Springer International PublishingNguyen T, La HM. Formation Control of Multiple Rectangular Agents with Limited Communication Ranges. In: 10th International Symposium on Ad- vances in Visual Computing. Springer International Publishing; 2014. p. 915- 924. On the Formation Control of a Multi Vehicle System. T Nguyen, H M La, M Jafari, ISSAT International Conference on Modeling of Complex Systems and Environments. Nguyen T, La HM, Jafari M. On the Formation Control of a Multi Vehi- cle System. In: ISSAT International Conference on Modeling of Complex Systems and Environments; 2015. . Decentralized cooperative control-a multivehicle platform for research in networked embedded systems. D Cruz, J Mcclintock, B Perteet, IEEE control systems. 273Cruz D, McClintock J, Perteet B, et al. Decentralized cooperative control-a multivehicle platform for research in networked embedded systems. IEEE control systems. 2007;27(3):58-78. Distributed Formation Control of Nonholonomic Mobile Robots by Bounded Feedback in the Presence of Obstacles. T Nguyen, H M La, arXiv:170404566arXiv preprintNguyen T, La HM. Distributed Formation Control of Nonholonomic Mobile Robots by Bounded Feedback in the Presence of Obstacles. arXiv preprint arXiv:170404566. 2017;. A Distributed Control Framework for a Team of Unmanned Aerial Vehicles for Dynamic Wildfire Tracking. H X Pham, H M La, D Feil-Seifer, arXiv:170402630arXiv preprintPham HX, La HM, Feil-Seifer D, et al. A Distributed Control Framework for a Team of Unmanned Aerial Vehicles for Dynamic Wildfire Tracking. arXiv preprint arXiv:170402630. 2017;. Flocking control of a mobile sensor network to track and observe a moving target. H M La, W Sheng, IEEE International Conference on Robotics and Automation. La HM, Sheng W. Flocking control of a mobile sensor network to track and observe a moving target. In: 2009 IEEE International Conference on Robotics and Automation; 2009. p. 3129-3134. Moving targets tracking and observing in a distributed mobile sensor network. H M La, W Sheng, American Control ConferenceLa HM, Sheng W. Moving targets tracking and observing in a distributed mobile sensor network. In: 2009 American Control Conference; 2009. p. 3319-3324. Adaptive flocking control for dynamic target tracking in mobile sensor networks. H M La, W Sheng, IEEE. La HM, Sheng W. Adaptive flocking control for dynamic target tracking in mobile sensor networks. In: 2009 IEEE/RSJ International Conference on Intelligent Robots and Systems; 2009. p. 4843-4848. Compressive and collaborative mobile sensing for scalar field mapping in robotic networks. M T Nguyen, H M La, K A Teague, 53rd Annual Allerton Conference on Communication, Control, and Computing. AllertonNguyen MT, La HM, Teague KA. Compressive and collaborative mobile sensing for scalar field mapping in robotic networks. In: 2015 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton); Cooperative and active sensing in mobile sensor networks for scalar field mapping. H M La, W Sheng, J Chen, 2013 IEEE International Conference on Automation Science and Engineering. La HM, Sheng W, Chen J. Cooperative and active sensing in mobile sensor networks for scalar field mapping. In: 2013 IEEE International Conference on Automation Science and Engineering (CASE); 2013. p. 831-836. Cooperative sensing in mobile sensor networks based on distributed consensus. H M La, W Sheng, SPIE Optical Engineering+ Applications. International Society for Optics and PhotonicsLa HM, Sheng W. Cooperative sensing in mobile sensor networks based on distributed consensus. In: SPIE Optical Engineering+ Applications. Interna- tional Society for Optics and Photonics; 2011. p. 81370Y-81370Y. Optimal flocking control for a mobile sensor network based a moving target tracking. H M La, T H Nguyen, C H Nguyen, 2009 IEEE International Conference on Systems, Man and Cybernetics. La HM, Nguyen TH, Nguyen CH, et al. Optimal flocking control for a mobile sensor network based a moving target tracking. In: 2009 IEEE International Conference on Systems, Man and Cybernetics; 2009. p. 4801-4806. Hybrid system of reinforcement learning and flocking control in multi-robot domain. H M La, R S Lim, W Sheng, Proc. Conf. Theoretical Appl. Conf. Theoretical ApplLa HM, Lim RS, Sheng W. Hybrid system of reinforcement learning and flocking control in multi-robot domain. In: Proc. Conf. Theoretical Appl. Comput. Sci.; 2010. p. 7-13. Cooperative flocking and learning in multirobot systems for predator avoidance. H M La, R S Lim, W Sheng, IEEE International Conference on Cyber Technology in Automation, Control and Intelligent Systems. La HM, Lim RS, Sheng W, et al. Cooperative flocking and learning in multi- robot systems for predator avoidance. In: 2013 IEEE International Confer- ence on Cyber Technology in Automation, Control and Intelligent Systems; 2013. p. 337-342. Adaptive Flocking Control of Multiple Unmanned Ground Vehicles by Using a UAV. M Jafari, S Sengupta, H M La, G Bebis, R Boyle, B Parvin, 10.1007/978-3-319-27863-6_58Springer International PublishingChameditorsJafari M, Sengupta S, La HM. In: Bebis G, Boyle R, Parvin B, et al., edi- tors. Adaptive Flocking Control of Multiple Unmanned Ground Vehicles by Using a UAV. Cham: Springer International Publishing; 2015. p. 628-637. Available from: http://dx.doi.org/10.1007/978-3-319-27863-6 58. Multi-Robot Swarm for Cooperative Scalar Field Mapping. Handbook of Research on Design, Control, and Modeling of Swarm Robotics. H M La, La HM. Multi-Robot Swarm for Cooperative Scalar Field Mapping. Hand- book of Research on Design, Control, and Modeling of Swarm Robotics. 2015;p. 383-395. Flocking control algorithms for multiple agents in cluttered and noisy environments. H M La, W Sheng, SpringerBerlin HeidelbergIn: Bio-Inspired Self-Organizing Robotic SystemsLa HM, Sheng W. Flocking control algorithms for multiple agents in cluttered and noisy environments. In: Bio-Inspired Self-Organizing Robotic Systems. Springer Berlin Heidelberg; 2011. p. 53-79. Asymptotic flocking dynamics for the kinetic Cucker-Smale model. J A Carrillo, M Fornasier, J Rosado, SIAM Journal on Mathematical Analysis. 421Carrillo JA, Fornasier M, Rosado J, et al. Asymptotic flocking dynamics for the kinetic Cucker-Smale model. SIAM Journal on Mathematical Analysis. 2010;42(1):218-236. Multirobot cooperative learning for predator avoidance. H M La, R Lim, W Sheng, IEEE Trans Control Syst Technol. 231La HM, Lim R, Sheng W. Multirobot cooperative learning for predator avoid- ance. IEEE Trans Control Syst Technol. 2015 Jan;23(1):52-63. Stabilization of planar collective motion: all-to-all communication. R Sepulchre, D A Paley, N E Leonard, IEEE Trans Autom Control. 525Sepulchre R, Paley DA, Leonard NE. Stabilization of planar collec- tive motion: all-to-all communication. IEEE Trans Autom Control. 2007 May;52(5):811-824. Hybrid control for connectivity preserving flocking. M M Zavlanos, H G Tanner, A Jadbabaie, IEEE Transactions on Automatic Control. 5412Zavlanos MM, Tanner HG, Jadbabaie A, et al. Hybrid control for con- nectivity preserving flocking. IEEE Transactions on Automatic Control. 2009;54(12):2869-2875. Flocking of multiple mobile robots based on backstepping. W Dong, IEEE Transactions on Systems, Man, and Cybernetics. 412Part B (Cybernetics).Dong W. Flocking of multiple mobile robots based on backstepping. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics). 2011;41(2):414-424. Vision-based, distributed control laws for motion coordination of nonholonomic robots. N Moshtagh, N Michael, A Jadbabaie, IEEE Trans Robotics. 254Moshtagh N, Michael N, Jadbabaie A, et al. Vision-based, distributed control laws for motion coordination of nonholonomic robots. IEEE Trans Robotics. 2009 Aug;25(4):851-860. Flocking of mobile robots by bounded feedback. T T Han, H M La, B H Dinh, 2016 IEEE International Conference on Automation Science and Engineering. Han TT, La HM, Dinh BH. Flocking of mobile robots by bounded feed- back. In: 2016 IEEE International Conference on Automation Science and Engineering (CASE); 2016. p. 689-694. Nonlinear Systems. H K Khalil, Prentice HallUpper Saddle River, NJ3rd edKhalil HK. Nonlinear Systems. 3rd ed. Upper Saddle River, NJ: Prentice Hall; 2002. Flocking while preserving network connectivity. M M Zavlanos, A Jadbabaie, G J Pappas, 46th IEEE Conference on Decision and Control. Zavlanos MM, Jadbabaie A, Pappas GJ. Flocking while preserving network connectivity. In: 2007 46th IEEE Conference on Decision and Control; 2007. p. 2919-2924. Formation Control and Obstacle Avoidance of Multiple Rectangular Agents with Limited Communication Ranges. T Nguyen, H La, T D Le, IEEE Transactions on Control of Network Systems. Nguyen T, La H, Le TD, et al. Formation Control and Obstacle Avoidance of Multiple Rectangular Agents with Limited Communication Ranges. IEEE Transactions on Control of Network Systems. 2016;.
{'fraction_non_alphanumeric': 0.07236367610097257, 'fraction_numerical': 0.03923068517101956, 'mean_word_length': 3.768236765318883, 'pattern_counts': {'":': 0, '<': 17, '<?xml version=': 0, '>': 9, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 52, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': "There have been numerous studies on the problem of flocking control for multiagent systems whose simplified models are presented in terms of point-mass elements. Meanwhile, full dynamic models pose some challenging problems in addressing the flocking control problem of mobile robots due to their nonholonomic dynamic properties. Taking practical constraints into consideration, we propose a novel approach to distributed flocking control of nonholonomic mobile robots by bounded feedback. The flocking control objectives consist of velocity consensus, collision avoidance, and cohesion maintenance among mobile robots. A flocking control protocol which is based on the information of neighbor mobile robots is constructed. The theoretical analysis is conducted with the help of a Lyapunov-like function and graph theory. Simulation results are shown to demonstrate the efficacy of the proposed distributed flocking control scheme.With point-mass models, the problem of flocking control of multiple agents has been addressed with typical results reported in[7][8][9][10][11]. For a wide range of engineering applications, extensive studies in flocking control of mobile robots have been done in various scenarios[12][13][14].In this chapter, we study the problem of distributed flocking control of mobile robots by bounded feedback, which takes into consideration nonholonomic nature of mobile robots as well as the implementation issue posed by the physical limit of the motor speed. Our flocking control problem employs the full dynamic model of the mobile robot derived in[15]. Similar to[16,17], due to the nonholonomic property of the dynamics of mobile robots, our proposed design framework constructed to achieve velocity consensus is modular. In other words, the consensuses on the linear speed and orientation angles are obtained separately.In this chapter, we are interested in agents with nonholonomic dynamics and boundedness constraints. Specifically, a coordination function is proposed to ensure that the induced attractive and repulsive forces are bounded, and hence can be incorporated in the bounded velocity control. Using the results of Barbalat's lemma and graph theory, the theoretical analysis is conducted, which shows that the maximal value of the coordination function determines the basin of attraction for the flocking convergence.In this chapter, graph theory will be employed as in the case of nearest neighbor communication[7,18]. We will employ the velocity control law reported in[16,17]in a decentralised sense, which helps to avoid collision and maintain a linear speed consensus. In addition, the orientation consensus will be achieved using a modified approach, which is inspired by the one in[19], where the input constraint is taken into account.The organization of the chapter is as follows. Section1.2 summarises some research work in the literature related to the topic in this chapter. In Section 1.3, the multiple-goal control problem for flocking of nonholonomic mobile robots and preliminaries are introduced. Section 1.4 describes main results where a modular design framework is proposed for bounded velocity control and bounded orientation control and the theoretical analyses are introduced. In Section 1.5, a description of an obstacle avoidance scheme is presented. Section 1.6 shows some simulation results. Section 1.7 concludes the chapter by some conclusions.Notations: R and R + are the sets of real numbers and nonnegative real numbers, respectively; for q = [q 1 , . . . , q n ] T , ∇ q = [∂ /∂ q 1 , . . . , ∂ /∂ q n ] T is the del operator [20]; for two vectors a and b, a · b is their scalar product; (a 1 , . . . , a n ) is [a T 1 , . . . , a T n ] T ; | · | is the absolute value of scalars; and · is the Euclidean norm of vectors.", 'arxivid': '1704.04722', 'author': ['Thang Nguyen \nDepartment of Computer Science and Engineering\nAdvanced Robotics and Automation (ARA) Lab\nUniversity of Nevada\nNV89557RenoUSA\n', 'Hung M La \nDepartment of Computer Science and Engineering\nAdvanced Robotics and Automation (ARA) Lab\nUniversity of Nevada\nNV89557RenoUSA\n', 'Vahid Azimi \nSchool of Electrical and Computer Engineering\nGeorgia Institute of Technology\n777 Atlantic Drive NW30332-0250AtlantaGAUSA\n', 'Thanh-Trung Han \nFaculty of Electrical and Electronics Engineering\nTon Duc Thang University, Ho Chi Minh City, Viet\n'], 'authoraffiliation': ['Department of Computer Science and Engineering\nAdvanced Robotics and Automation (ARA) Lab\nUniversity of Nevada\nNV89557RenoUSA', 'Department of Computer Science and Engineering\nAdvanced Robotics and Automation (ARA) Lab\nUniversity of Nevada\nNV89557RenoUSA', 'School of Electrical and Computer Engineering\nGeorgia Institute of Technology\n777 Atlantic Drive NW30332-0250AtlantaGAUSA', 'Faculty of Electrical and Electronics Engineering\nTon Duc Thang University, Ho Chi Minh City, Viet'], 'corpusid': 38441329, 'doi': '10.1049/pbce119f_ch11', 'github_urls': [], 'n_tokens_mistral': 15333, 'n_tokens_neox': 13064, 'n_words': 8071, 'pdfsha': 'e669d43a947bf412dbdcce4f44da42c13fd795d4', 'pdfurls': ['https://arxiv.org/pdf/1704.04722v2.pdf'], 'title': ['Bounded Distributed Flocking Control of Nonholonomic Mobile Robots', 'Bounded Distributed Flocking Control of Nonholonomic Mobile Robots'], 'venue': []}
arxiv
m-IRREDUCIBLE NUMERICAL SEMIGROUPS 17 Jun 2010 V Blanco [email protected] J C Rosales m-IRREDUCIBLE NUMERICAL SEMIGROUPS 17 Jun 2010 In this paper we introduce the notion of m-irreducibility that extends the standard concept of irreducibility of a numerical semigroup when the multiplicity is fixed. We analyze the structure of the set of m-irreducible numerical semigroups, we give some properties of these numerical semigroups and we present algorithms to compute the decomposition of a numerical semigroups with multiplicity m into m-irreducible numerical semigroups. Introduction A numerical semigroup is a subset S of N (here N denotes the set of nonnegative integers) closed under addition, containing zero and such that N\S is finite. A numerical semigroup is irreducible if it cannot be expressed as an intersection of two numerical semigroups containing it properly. This notion was introduced in [8] where it is also shown that the family of irreducible numerical semigroups is the union of two families of numerical semigroups with special importance in this theory: symmetric and pseudo-symmetric numerical semigroups. Every numerical semigroup can be expressed as a finite intersection of irreducible numerical semigroups. In [7], the authors give an algorithm that allows to compute, for a numerical semigroup S, a finite set of irreducible numerical semigroups, S 1 , . . . , S n , such that S = S 1 ∩ · · · ∩ S n . Furthermore, the above decomposition, given by that algorithm, is minimal in the sense that n is the smallest possible positive integer with that property. In [5], upper and lower bounds are given for that n in terms of S. If S is a numerical semigroup, the least positive integer belonging to S is called the multiplicity of S and we denote it by m(S). Let m be a positive integer and S(m) the set of numerical semigroups with multiplicity m. It is clear that (S(m), ∩) is a semilattice, that is, it is a semigroup where all its elements are idempotent. The search of the generators of this semilattice leads us to the following definition. An element S in S(m) is m-irreducible if it cannot be expressed as the intersection of two elements in S(m) containing it properly. In Section 2 we prove that every element in S(m) can be expressed as a finite intersection of m-irreducible numerical semigroups. Furthermore, we prove that {S : S is a m-irreducible numerical semigroup} coincides with {{x ∈ N : x ≥ m} ∪ {0}} ∪ {{x ∈ N : x ≥ m and x = i} ∪ {0} : i = m + 1, . . . , 2m − 1} ∪ {S : S is an irreducible numerical semigroup and m(S) = m}. If S is a numerical semigroup, the largest integer not belonging to S is called the Frobenius number of S and we denote it by F(S). We say that a positive integer, x, is a gap of S if x ∈ S. We denote by G(S) the set of all the gaps of S. The cardinal of G(S) is called the genus of S, and we denote it by g(S). We can find in the literature several characterization for irreducible numerical semigroups. In [9], the authors state that the irreducible numerical semigroups are those with minimum genus among all the numerical semigroups with the same Frobenius number. In Section 3 we prove that this characterization is also valid for m-irreducible numerical semigroups. In fact, we prove that an element in S(m) is m-irreducible if and only if it has minimum genus in the set of all the elements in S(m) with the same Frobenius number. Given a numerical semigroup S with multiplicity m, we denote by O m (S) = {S ′ ∈ S(m) : S ⊆ S ′ } the set of oversemigroups with multiplicity m of S. The aim of Section 4 is to provide an algorithm to compute O m (S) when S is given. These results are used in Section 5 to give an algorithm to compute a decomposition, as an intersection of m-irreducible numerical semigroups, of a numerical semigroup with multiplicity m. Moreover, the decomposition given by the algorithm is minimal in the sense that it uses the smallest number of m-irreducible numerical semigroups for taking part of the decomposition. For concluding this introduction, observe that although this work is performed by a "semigroupist" point of view, it can be used in Commutative Ring Theory. In fact, let M be a submonoid of (N, +), K a field, and K[[t]] the ring of formal power series over K. It is well-known (see for instance, [2]) that K[[M ]] = { s∈M a s t s : a s ∈ K} is a subring of K[[t]] , called the ring of the semigroup associated to M . All the above invariants, as the multiplicity, the genus (degree of singularity) and the Frobenius number (conductor minus one) have their corresponding interpretation in this context (see [2]). Moreover, in [4] it is shown that a numerical semigroup is symmetric if and only if K[[S]] is a Gorenstein ring and in [3] m-irreducible numerical semigroups We begin this section showing that every numerical semigroup with multiplicity m can be expressed as a finite intersection of m-irreducible numerical semigroups. Note that {x ∈ N : x ≥ m} ∪ {0} is the maximum (with respect to the inclusion ordering) of S(m) and then, this semigroup is m-irreducible. Proposition 1. Let S ∈ S(m). Then, there exist S 1 , . . . , S k m-irreducible numerical semigroups such that S = S 1 ∩ · · · ∩ S k . Proof. We prove the result by induction over g(S). If g(S) = m − 1, then S = {x ∈ N : x ≥ m} ∪ {0} and then m-irreducible. Assume that g(S) ≥ m and that S is no m-irreducible. Then there exist S 1 and S 2 in S(m) such that S S 1 , S S 2 , and S = S 1 ∩ S 2 . By the induction hypothesis, there exist S 11 , . . . , S 1p , S 21 , . . . , S 2q m-irreducible numerical semigroups such that S 1 = S 11 ∩ · · · ∩ S 1p and S 2 = S 21 ∩ · · · ∩ S 2q . Then, S = S 11 ∩ · · · ∩ S 1p ∩ S 21 ∩ · · · ∩ S 2q is a decomposition of S into m-irreducible numerical semigroups. Our next goal for this section is to prove Theorem 3 which states that the m-irreducible numerical semigroups are those maximal elements in S(m) with the additional condition that they have certain Frobenius number. In order to prove that result, we first introduce some notation and previous results. Given two positive integer m and F , we denote by S(m, F ) the set of numerical semigroups with multiplicity m and Frobenius number F . Note that S(m, F ) = ∅ if and only if F ≥ m − 1 and F is not a multiple of m. Denote by S * (m, F ) the set of maximal elements in S(m, F ) (with respect to the inclusion ordering). The following result has an immediate proof and it is left to the reader Lemma 2. (1) If S = N is a numerical semigroup, then S ∪ {F(S)} is also a numerical semigroup. (2) Let S 1 , . . . , S n be numerical semigroups and S = S 1 ∩· · ·∩S n . Then, F(S) = max{F(S 1 ), . . . , F(S n )}. We are now ready to prove the announced result. (Sufficiency) Let S ∈ S * (m, F(S)). If S is not m-irreducible then, there exist S 1 and S 2 in S(m) such that S S 1 , S S 2 and S = S 1 ∩ S 2 . By Lemma 2, F(S) ∈ {F(S 1 ), F(S 2 )}. Assume without loss of generality that F(S) = F(S 1 ). Then, S 1 ∈ S(m, F(S)) and S S 1 contradicting the maximality of S. In what follows we intend to give a proof for Proposition 6 where we describe explicitly the elements in S * (m, F ). Before that, we give some previous results. The following lemma is well-known and appears in [9]. Lemma 4. Let S be a numerical semigroup and assume that h = max{x ∈ Z\S : F(S) − x ∈ S, x = F 2 } exists, then S ∪ {h} is a numerical semigroup with Frobenius number F(S). In [8] it is shown that a numerical semigroup S is irreducible if and only if it is maximal in the set of numerical semigroups with Frobenius number F(S). As a direct consequence of the above lemma we get the following result. Lemma 5. A numerical semigroup is irreducible if and only if {x ∈ Z\S : F(S) − x ∈ S and x = F(S) 2 } = ∅. We are ready to describe the structure of S * (m, F ). Here, we denote by a ≡ b (mod c) if a − b is multiple of c. Proposition 6. Let m and F be positive integers such that F ≥ m − 1 and F ≡ 0 (mod m). ( 1) If F = m − 1, then S * (m, F ) = {x ∈ N : x ≥ m} ∪ {0} . (2) If m < F < 2m, then S * (m, F ) = {x ∈ N : x ≥ m, x = F } ∪ {0} . (3) If F > 2m, then S * (m, F ) = S ∈ S(m, F ) : S is irreducible . Proof. (1) and (2) are trivial. Let us then prove (3). It is clear, by the comment before Lemma 5, that ( {S ∈ S(F, m) : S is irreducible ⊆ S * (m, F ). We prove the other inclusion. Let S ∈ S * (m, F ). If S is not irreducible, we know, by Lemma 5, that h = max{x ∈ S : F(S) − x ∈ S, x = F 2 } exists,1) S = {x ∈ N : x ≥ m} ∪ {0}. (2) S = {x ∈ N : x ≥ m, x = f } ∪ {0} with f ∈ {m + 1, . . . , 2m − 1}. (3) S is an irreducible numerical semigroup. The gaps of a m-irreducible numerical semigroup Let q be a rational number. We denote by ⌈q⌉ = min{z ∈ Z : q ≤ z} the ceiling part of q. It is wellknown (see for instance [9]) that if S is a numerical semigroup then g(S) ≥ F(S) + 1 2 . The following result is easily deduced from Lemma 5 and appears in [9]. Lemma 8. A numerical semigroup S is irreducible if and only if g(S) = F(S) + 1 2 . As a consequence of Theorem 3, Proposition 6 and Lemma 8 we get the following result. Proposition 9. If S is a m-irreducible numerical semigroup, then g(S) =        m − 1 if F(S) = m − 1, m if m < F(S) < 2m, F(S) + 1 2 if F(S) > 2m . By Lemma 8, the irreducible numerical semigroups are those with the smallest possible number of gaps once the Frobenius number is fixed. In the following result we prove that this property is extendable for m-irreducible numerical semigroups. Let m and F be two positive integers such that F ≥ m − 1 and F ≡ 0 (mod m). Proof. (Sufficiency) It is clear that if g(S) = g(m, F ) then S is maximal in S(m, F ). Hence, S ∈ S * (m, F ) and by Theorem 3 we have that S is m-irreducible. (Necessity) By Theorem 3 and Proposition 6 we distinguish the following three cases: (1) If S = {x ∈ N : x ≥ m} ∪ {0}, it is clear that g(S) = m − 1 = g(m, F ). (2) If S = {x ∈ N : x ≥ m, x = F } ∪ {0} with m < F < 2m, then, clearly g(S) = m = g(m, F ). (3) If S is irreducible and F > 2m, then by Lemma 8, g(S) = g(m, F ). From the above theorem we have that the m-irreducible numerical semigroups are exactly those with minimum genus among all the numerical semigroups with its same multiplicity and Frobenius number. Let S be a numerical semigroup. We say that an element x ∈ N\S is an special gap of S if S ∪ {x} is a numerical semigroup. We denote by SG(S) the set of all the special gaps of S. Let A be a set, we denote by |A| the cardinal of A. The following result appears in [9]. Lemma 12. Let S be a numerical semigroup. (1) S is irreducible if and only if |SG(S)| ≤ 1. (2) If T is a numerical semigroup such that S T and x = max(T \S), then x ∈ SG(S). From the above lemma we have that the irreducible numerical semigroups are characterized by the cardinal of the set of its special gaps. Next, we show that the m-irreducible numerical semigroups are characterized by the cardinal of its special gaps that are greater than m. In the rest of the section we denote by T a numerical semigroup and n ∈ T \{0}. The Apery set [1] of T with respect to n is Ap(T, n) = {t ∈ T : t − n ∈ T }. Our goal is to show how to determine SG(T ) when Ap(T, n) is known,. This process will be useful for the rest of this paper. It is well-known and easy to prove that Ap(T, n) = {w(0), . . . , w(n − 1)}, where w(i) is the least element in T congruent with i modulo n, If we denote by a mod b the remainder of division of a by b, then a positive integer, x, is in T if and only if w(x mod n) ≤ x (see for instance [9]). By using the notation introduced in [6], an integer x is a pseudo-Frobenius number of T if x ∈ T and x + t ∈ T for all t ∈ T \{0}. We denote by PF(T ) the set of all pseudo-Frobenius number of T . The cardinality of the above set is an important invariant of T , called the type of T (see [2]). The relationship between P F (T ) and SG(T ) is shown in the following lemma, whose proof is immediate. Over N, we can define the following order relation: a ≤ T b if b − a ∈ T The following result appears in [6] and characterizes the set of pseudo-Frobenius numbers of a numerical semigroup in terms of one of its Apéry sets. Lemma 15. With the above notation PF(T ) = {w − n : w ∈ Maximals ≤T Ap(T, n)}. Note that w(i) ∈ Maximals ≤T Ap(T, n) if and only if w(k) − w(i) ∈ Ap(T, n) for all k = i. We finish this section by giving an algorithm that allows to determine the special gaps of a numerical semigroup when the Apéry set with respect a non-zero element of the semigroup of is given. Algorithm 1 shows a pseudocode for this computation. Given two numerical semigroups S and S ′ in S(m) with S S ′ , we define recursively the following sequence of elements in S(m): • S 0 = S. • S n+1 = S n if S n = S ′ , S n ∪ {max(S ′ \S n )} otherwise. It is clear that if |S ′ \S| = k, then S = S 0 S 1 · · · S k = S ′ . With this idea it is not difficult to design an algorithm that allows to compute all the elements in O m (S) for a given numerical semigroup with multiplicity m, S. The main idea for this procedure is that if S ′ ∈ O m (S) (we start with S ′ = S) and {x 1 , . . . , , x r } = {x ∈ SG(S ′ ) : x > m} then S ′ ∪{x 1 }, . . . , S ′ ∪{x r } are also in O m (S). Algorithm 2 shows a pseudocode for this procedure. Algorithm 2: Computing the oversemigroups with multiplicity m of a numerical semigroup. Input : A numerical semigroup S with multiplicity m Initialization: A = {S} and B = {S}. while B = ∅ do • For each S ′ ∈ B compute D(S ′ ) = {x ∈ SG(S ′ ) : x > m}. • Set A := A ∪ { S ′ ∈B {S ′ ∪ {x} : x ∈ D(S ′ )}}. • Set B := S ′ ∈B {S ′ ∪ {x} : x ∈ D(S ′ )}. Output : A = O m (S). Observe that the difficulty of the above algorithm lies on the computation of {x ∈ SG(S ′ ) : x > m}. Recall that if we know the Apéry set Ap(S, m), by using Algorithm 1 we can easily compute the set {x ∈ SG(S ′ ) : x > m}. The following result states that if we know Ap(S, m), then we can also compute Ap(S ′ , m) for all those oversemigroups S ′ that appears when we run Algorithm 2. Observe that a numerical semigroup with multiplicity m, S, is completely determined by Ap(S, m). Hence, S is also completely determined by a (m − 1)-tuple (w(1), . . . , w(m − 1)) in N m−1 where w(i) is the least element in S congruent with i modulo m. We will call in the rest of the paper to (w(1), . . . , w(m − 1)) the coordinates of S. We denote by [m − 1] = {1, . . . , m − 1}, and for each i ∈ [m − 1] by e i , the (m − 1)-tuple having a 1 as its ith entry and zeros otherwise. With this notation, the following result is a reformulation of Lemma 18. Output : A = Coordinates of O m (S). We finish the section by illustrating the above algorithm with an example. Example 20. Let S = {0, 5, 7, 9, 10, 12, 14, →}. It is clear that S is a numerical semigroup with multiplicity 5 and that (16,7,18,9) are its coordinates. • A = {(16, 7, 18, 9), (11,7,18,9), (16,7,13,9), (11,7,13,9), (16,7,8,9)} and B = {(11, 7,13,9), (16,7,8,9)}. D(11, 7, 13, 9) = {1, 3} and D(16, 7, 8, 9) (16,7,18,9), (11,7,18,9), (16,7,13,9), (11,7,13,9), (16,7,8,9), (6,7,13,9), (11,7,8,9)} and B = { (6,7,13,9), (11,7,8,9)}. D (6,7,13,9) = {3} and D (11,7,8,9) (16,7,18,9), (11,7,18,9), (16,7,13,9), (11,7,13,9), (16,7,8,9), (6,7,13,9), (11,7,8,9), (6,7,8,9)} and B = { (6,7,8,9)}. D (6,7,8,9) (16,7,18,9), (11,7,18,9), (16,7,13,9), (11,7,13,9), (16,7,8,9), (6,7,13,9), (11,7,8,9), (6,7,8,9)} and B = ∅. And then, the coordinates of all the oversemigroups with multiplicity 5 of S are {(16, 7, 18, 9), (11,7,18,9), (16,7,13,9), (11,7,13,9), (16,7,8,9), (6,7,13,9), (11,7,8,9), (6,7,8,9) As a consequence of the results in Section 3 and Theorem 13 we have justified the following algorithm that computes the minimal elements in J m (S). Note that the decomposition described in Lemma 21 is not necessarily minimal, in the sense that the smallest number of m-irreducible numerical semigroups are taking part of the decomposition. An example of this fact is shown in Example 27. = {1}. • A = {= {1}. • A = {= ∅. • A = { To compute the minimal decomposition are necessary the two following results. Lemma 23. Let S ∈ S(m). If S = S 1 ∩ · · · S n with S 1 , . . . , S m ∈ J m (S), then there exist S ′ 1 , . . . , S ′ n ∈ Minimals ⊆ (J m (S)) such that S = S ′ 1 ∩ · · · ∩ S ′ n . Proof. Take, for each i ∈ {1, . . . , n} S ′ i the element in Minimals ⊆ (J m (S)) such that S ′ i ⊆ S i . Lemma 24. Let S ∈ S(m) and S 1 , . . . , S n ∈ O m (S). Then, S = S 1 ∩ · · · ∩ S n if and only if for all h ∈ {x ∈ SG(S) : x > m} there exists i ∈ {1, . . . , n} such that h ∈ S i . Proof. It is a direct consequence of Lemma 12. Remark 25. Note that as a direct consequence of the above lemma we have that if S ∈ S(m), then S can be expressed as an intersection less or equal than |{x ∈ SG(S) : x > m}| m-irreducible numerical semigroups. In fact, by Lemma 14 and Corollary 1.23 in [9] we get that we can decompose S into less or equal that m − 1 m-irreducible numerical semigroups. Assume that Minimals ⊆ (J m (S)) = {S 1 , . . . , S n }. For each i ∈ {1, . . . , n}, let P (S i ) = {h ∈ SG(S) : h > m and h ∈ S i }. By Lemma 24, we know that S = S i1 ∩ · · · ∩ S ir if and only if P (S i1 ) ∪ · · · ∪ P (S ir ) = {x ∈ SG(S) : x > m}. This comment and Lemma 23 justify the following algorithm that allows to compute from a given numerical semigroup with multiplicity m, a minimal decomposition as intersection of m-irreducible numerical semigroups. (11,22,13,9) and (11, 17, 28, 14) are enough to decompose S into 5-irreducible numerical semigroups. In the above example, both 5-irreducible numerical semigroups in the decomposition are also irreducible. In the next example, we show that it is not true in general. Theorem 3 . 3Let S ∈ S(m). Then, S is m-irreducible if and only if S ∈ S * (m, F(S)). Proof. (Necessity) If S = {x ∈ N : x ≥ m} ∪ {0}, clearly S ∈ S * (m, F(S)). Assume that S = {x ∈ N : x ≥ m} ∪ {0}, then F(S) > m. By Lemma 2 we have that S ∪ {F(S)} ∈ S(m). If S ∈ S * (m, F(S)) there exists T ∈ S(m, F(S)) such that S T . It is clear that S = T ∩ (S ∪ {F(S)}) contradicting that S is m-irreducible. and by Lemma 4, that S ∪ {h} is numerical semigroup with Frobenius number F . Furthermore, h > F 2 > m and then, S ∪ {h} ∈ S(m, F ) contradicting the maximality of S.As a consequence of Theorem 3 and Proposition 6 we get this corollary. Corollary 7 . 7A numerical semigroup, S, with multiplicity m is m-irreducible if and only if one of the following conditions holds: genus among all the numerical semigroups with multiplicity m and Frobenius number F . Theorem 10. Let S be a numerical semigroup with multiplicity m and Frobenius number F . Then, S is m-irreducible if and only if g(S) = g(m, F ). Corollary 11 . 11Let S be a numerical semigroup with multiplicity m. Then, S is m-irreducible if and only if g(S) ∈ m − 1, m, F(S) + 1 2 . Proof. (Necessity) It is a direct consequence of Proposition 9. (Sufficiency) If g(S) = m − 1, then S = {x ∈ N : x ≥ m} ∪ {0} and by Corollary 7 we have that S is m-irreducible. If g(S) = m then we deduce that m < F(S) < 2m, and S = {x ∈ N : x ≥ m, x = F(S)} ∪ {0} and again by applying Corollary 7, S is m-irreducible. by Corollary 7 and Lemma 8, we conclude that S is m-irreducible. Theorem 13 . 13Let S be a numerical semigroup with multiplicity m. Then, S is m-irreducible if and only if |{x ∈ SG(S) : x > m}| ≤ 1.Proof. (Necessity) If S is a m-irreducible numerical semigroup, then, by Theorem 3 and Proposition 6, one of the following condition holds:(1) If S = {x ∈ N : x ≥ m} ∪ {0}, it is clear that |{x ∈ SG(S) : x > m}| = 0 ≤ 1. (2) If S = {x ∈ N : x ≥ m, x = F(S)} ∪ {0} with m < F(S) < 2m,then, clearly |{x ∈ SG(S) : x > m}| = |{F(S)}| = 1. (3) If S is irreducible and F(S) > 2m, then by Lemma 12, |{x ∈ SG(S) : x > m}| ≤ 1. (Sufficiency) If S is not an m-irreducible numerical semigroup then there exist S 1 , S 2 ∈ S(m) such that S S 1 , S S 2 and S = S 1 ∩ S 2 . For i ∈ {1, 2}, let x i = max(S i \S). Because S = S 1 ∩ S 2 we have that x 1 = x 2 and by Lemma 12, we conclude that |{x ∈ SG(S) : x > m}| ≥ 2 contradicting the hypothesis. Lemma 14 . 14With the above conditions SG(T ) = {x ∈ PF(T ) : 2x ∈ T }. Algorithm 1 : 1Computing the special gaps when an Apéry set is given. Input : A numerical semigroup T and its Apéry set with respect to n ∈ T \{0}, Ap(T, n) = {w(0), . . . , w(n − 1)}. Compute M = {w(i) − n : w(k) − w(i) ∈ Ap(T, n), for all k = i}. Output: SG(T ) = {m ∈ M : 2m ≥ w(2m mod n)}. Next, we illustrate the usage of the above algorithm in a simple example. Example 16. Let us compute the special gaps of the numerical semigroup S = {0, 5, 7, 9, 10, 12, 14, →} by using Algorithm 1. It is clear that Ap(S, 5) = {0, 7, 9, 16, 18}. Then, M = {16 − 5 = 11, 18 − 5 = 13} and SG(S) = {11, 13}. 4. The oversemigroups of multiplicity m of a numerical semigroup Let S be a numerical semigroups with multiplicity m. We denote by O m (S) = {S ′ ∈ S(m) : S ⊆ S ′ } the set of oversemigroups with multiplicity m of S.The elements in O m (S) can be obtained recursively. The main idea for this construction is the following result that can be directly obtained from Lemma 12.Lemma 17. Let S and S ′ be elements in S(m) such that S S ′ and let x = max(S ′ \S). Then S ∪ {x} ∈ S(m). Lemma 18 . 18Let T ∈ S(m) and Ap(T, m) = {w(0), . . . , w(m − 1)}. If x ∈ SG(T ), then Ap(T ∪ {x}, m) = (Ap(T, m)\{w(x mod m)}) ∪ {w(x mod m) − m} Proof. Note that by Lemma 15 we know that x = w(x mod m) − m. Lemma 19 . 19Let T ∈ S(m) and x ∈ SG(T ) such that x > m. If (x 1 , . . . , x m−1 ) are the coordinates of T , then (x 1 , . . . , x m−1 ) − m · e x mod m are the coordinates of T ∪ {x}. The reader can easily check that the following algorithm computes, from the coordinates of a numerical semigroup with multiplicity m, S, the set of all the coordinates of the elements in O m (S). Algorithm 3: The coordinates of all the oversemigroups with multiplicity m of a numerical semigroup. Input : The coordinates x = (x 1 , . . . , x m−1 ) of a numerical semigroup S with multiplicity m Initialization: A = {x} and B = {x}. while B = ∅ do • For each y = (y 1 , . . . , y m−1 ) ∈ B compute D(y) = {i ∈ [m − 1] : y i > 2m and y k − y i ∈ {y 1 , . . . , y m−1 } for all k ∈ [m − 1]}. • Set A := A ∪ { y∈B {y − m · e i : i ∈ D(y)}}. • Set B := y∈B {y − m · e i : i ∈ D(y)}. • We start by initializing A = {(16, 7, 18, 9)} and B = {(16, 7, 18, 9)}. D(16, 7, 18, 9) = {1, 3}. • A = {(16, 7, 18, 9), (11, 7, 18, 9), (16, 7, 13, 9)} and B = {(11, 7, 18, 9), (16, 7, 13, 9)}. D(11, 7, 18, 9) = {3} and D(16, 7, 13, 9) = {1, 3}. } 5 . 5Decomposition into m-irreducible numerical semigroups Let S be a numerical semigroup with multiplicity m. We denote by J m (S) = {S ′ ∈ O m (S) : S ′ is m-irreducible}. From Proposition 1 we deduce that S = S ′ ∈Jm(S) S ′ . The following result has an immediate proof. Lemma 21. Let S ∈ S(m) and {S 1 , . . . , S n } the set of all the minimal elements (with respect to the inclusion ordering) of J m (S). Then S = S 1 ∩ · · · ∩ S n . Algorithm 4 : 4Computation of Minimals ⊆ (J m (S)). Input : A numerical semigroup S with multiplicity m Initialization: A = {S} and B = ∅. while A = ∅ do For each S ′ ∈ A compute D(S ′ ) = {x ∈ SG(S ′ ) : x > m}. if |D(S ′ )| ≤ 1 then Set B := B ∪ {S ′ } and A := A\{S ′ }. else Set A := (A\{S ′ }) ∪ {S ′ ∪ {x} : x ∈ D(S ′ ) and S ′ ∪ {x} does not contain any element of B}. Output : B = Minimals ⊆ (J m (S)) We illustrate the above algorithm with the following example. Note that if S and T are elements in S(m) with coordinates x and y, respectively, then S ⊆ T if and only if y ≤ x. Example 22. Let S = {0, 5, 7, 9, 10, 12, 14, →} the numerical semigroup given in Example 20. We compute the set Minimals ⊆ (J m (S)). Recall that (16, 7, 18, 9) are the coordinates of S. Then, by running Algorithm 4 we obtain: • A = {(16, 7, 18, 9)} and B = ∅. D(16, 7, 18, 9) = {11, 13}. • A = {(11, 7, 18, 9), (16, 7, 13, 9)} and B = ∅. D(11, 7, 18, 9) = {13}. • A = {(16, 7, 13, 9)} and B = {(11, 7, 18, 9)}. D(16, 7, 13, 9) = {11, 8}. • A = {(16, 7, 8, 9)} and B = {(11, 7, 18, 9)}. D(16, 7, 8, 9) = {11}. • A = ∅ and B = {(11, 7, 18, 9), (16, 7, 8, 9)}. Algorithm 5 : 5Computation of a minimal decomposition of a numerical semigroup with multiplicity m in m-irreducible numerical semigroups.Input : A numerical semigroup S with multiplicity m (1) Compute Minimals ⊆ (J m (S)) by Algorithm 4.(2) For each S ′ ∈ Minimals ⊆ (J m (S)), compute P (S ′ ) = {h ∈ SG(S) : h > m and h ∈ S ′ }. (3) Choose A ⊆ Minimals ⊆ (J m (S))with minimal cardinality and such thatS ′ ∈A P (S ′ ) = {x ∈ SG(S) : x > m}. Output: A = {S 1 , . . . , S n } such that S = S 1 ∩ · · · ∩ S n is a minimal decomposition of S in m-irreducible numerical semigroups.The following two examples shows the usage of the above methodology.Example 26. Let S = {0, 5, 10, 11, 14, 15, 16, 19, 20, 21, 22, 24, →}. The coordinates for S are x = (11, 22, 28, 14). By using Algorithm 4, we obtain that the coordinates of the elements in Minimals ⊆ (J m (S)) are (11, 22, 8, 14), (11, 22, 13, 9), (11, 17, 28, 14) and then, we have a decomposition into the three above 5-irreducible numerical semigroups of S. However, if we apply Algorithm 5: • P (11, 22, 8, 14) = {17}, • P (11, 22, 13, 9) = {17}, • P (11, 17, 28, 14) = {23}, while {x ∈ SG(S) : x > 5} = {17, 23}, so the rings K[[S]] when S is pseudo-symmetric are called Kunz rings. Then, as a consequence of the results in this paper we have that, given a numerical semigroup S we can decompose the ring K[[S]] as intersection of rings with the same multiplicity where some of them are Gorenstein, some other are Kunz and others are rings associated to numerical semigroups with special simplicity: {x ∈ N : x ≥ m} ∪ {0} and {x ∈ N : x ≥ m and x = i} ∪ {0} for i ∈ {m + 1, . . . , 2m − 1}. Example 27. Let S = {0, 9, 17, 18, 24, →}. The coordinates for S are x = (28, 29, 30, 31, 32, 24, 25, 17). By using Algorithm 4, we obtain that the coordinates of the elements in Minimals ⊆ (J m (S)) are: (10, 20, 21, 31, 14, 15, 16, 17), (10, 20, 12, 22, 32, 15, 16, 17), (19, 11, 21, 13, 32, 15, 16, 17), (19, 11, 30, 13, 14, 15, 16, 17), (19, 20, 12, 13, 32, 15, 16, 17), (28, 11, 12, 13, 14, 15, 16, 17), (10, 20, 30, 13, 14, 15, 16, 17), (10, 11, 12, 13, 14, 24, 16, 17), (19, 29, 12, 13, 14, 15, 16, 17), (10, 11, 12, 13, 14, 15, 25, 17), (10, 11, 21, 22, 32, 15, 16, 17) and (19, 20, 12, 31, 14, 15, 16, 17), and then, we have a decomposition into the above 9-irreducible numerical semigroups of S. However, if we apply Algorithm 5 we obtain these 12 minimal 9-irreducible oversemigroups of S:•P (10, 20, 21, 31, 14, 15, 16, 17) = {22}, • P (10, 20, 12, 22, 32, 15, 16, 17) = {23}, • P (19, 11, 21, 13, 32, 15, 16, 17) = {23}, • P (19, 11, 30, 13, 14, 15, 16, 17) = {21}, 17) are enough to decompose S into 9-irreducible numerical semigroups. Note that the standard decomposition into irreducible numerical semigroups is given by the coordinates. • P ; = {23} , • P ; = {19} , • P ; = {21} , • P ; = {15} , • P ; = {20} , • P ; = {16} , • P ; = {23} , • , ) = {22}. Because {x ∈ SG(S) : x > 9} = {15. 111719, 11, 30, 13, 14, 15, 16, 17), (10, 20, 21, 31, 14, 15, 16, 17) and (19, 11, 21, 13, 32, 15, 16,. and (19, 20, 12, 31, 14, 15, 16, 17), where (9, 18, 19, 12, 13, 22, 31) has multiplicity 8, and then, it is not a decomposition into 9-irreducibles• P (19, 20, 12, 13, 32, 15, 16, 17) = {23}, • P (28, 11, 12, 13, 14, 15, 16, 17) = {19}, • P (10, 20, 30, 13, 14, 15, 16, 17) = {21}, • P (10, 11, 12, 13, 14, 24, 16, 17) = {15}, • P (19, 29, 12, 13, 14, 15, 16, 17) = {20}, • P (10, 11, 12, 13, 14, 15, 25, 17) = {16}, • P (10, 11, 21, 22, 32, 15, 16, 17) = {23}, • P (19, 20, 12, 31, 14, 15, 16, 17) = {22}. Because {x ∈ SG(S) : x > 9} = {15, 16, 19, 20, 21, 22, 23}, then (10, 11, 12, 13, 14, 24, 16, 17), (10, 11, 12, 13, 14, 15, 25, 17), (28, 11, 12, 13, 14, 15, 16, 17), (19, 29, 12, 13, 14, 15, 16, 17), (19, 11, 30, 13, 14, 15, 16, 17), (10, 20, 21, 31, 14, 15, 16, 17) and (19, 11, 21, 13, 32, 15, 16, 17) are enough to de- compose S into 9-irreducible numerical semigroups. Note that the standard decomposition into irreducible numerical semigroups is given by the coordinates (9, 18, 19, 12, 13, 22, 31), (10, 11, 12, 13, 14, 15, 25, 17), (28, 11, 12, 13, 14, 15, 16, 17), (19, 11, 30, 13, 14, 15, 16, 17), (19, 29, 12, 13, 14, 15, 16, 17) and (19, 20, 12, 31, 14, 15, 16, 17), where (9, 18, 19, 12, 13, 22, 31) has multiplicity 8, and then, it is not a decomposition into 9-irreducibles. Note that analogously to the extension done in this paper for irreducible numerical semigroups when the multiplicity is fixed. we could also extend the notions of symmetry and pseudosymmetry as follows: • S ∈ S(m) is m-symmetric if S is m-irreducible and F(S) is odd. Remark 28Remark 28. Note that analogously to the extension done in this paper for irreducible numerical semigroups when the multiplicity is fixed, we could also extend the notions of symmetry and pseudosymmetry as follows: • S ∈ S(m) is m-symmetric if S is m-irreducible and F(S) is odd. S(m) is m-pseudosymmetric if S is m-irreducible and F(S) is even. • S ∈, • S ∈ S(m) is m-pseudosymmetric if S is m-irreducible and F(S) is even. Moreover, by using Proposition 6, we can describe, for a given multiplicity m, all the m-symmetric and m-pseudosymmetric numerical semigroups in terms of the Frobenius number. Denote by Symm(m) and PSymm(m) the set of m-symmetric and m-pseudosymmetric numerical semigroups, respectively. Let S ∈ S(m) and F(S) = F. Moreover, by using Proposition 6, we can describe, for a given multiplicity m, all the m-symmetric and m-pseudosymmetric numerical semigroups in terms of the Frobenius number. Denote by Symm(m) and PSymm(m) the set of m-symmetric and m-pseudosymmetric numerical semigroups, respectively. Let S ∈ S(m) and F(S) = F . If F = m−1, then S ∈ Symm(m) (resp. S ∈ PSymm(m)) if and only if S = {x ∈ N : x ≥ m}∪{0} and m is even (resp. m is odd). If F = m−1, then S ∈ Symm(m) (resp. S ∈ PSymm(m)) if and only if S = {x ∈ N : x ≥ m}∪{0} and m is even (resp. m is odd). If M &lt; F &lt; 2m, then S ∈ Symm(m) (resp. S ∈ PSymm(m)) if and only if S = {x ∈ N : x ≥ m, x = F } ∪ {0} and F is odd. resp. F is evenIf m < F < 2m, then S ∈ Symm(m) (resp. S ∈ PSymm(m)) if and only if S = {x ∈ N : x ≥ m, x = F } ∪ {0} and F is odd (resp. F is even). If F &gt; 2m, then S ∈ Symm(m) (resp. S ∈ PSymm(m)) if and only if S is symmetric. resp. S is pseudosymmetricIf F > 2m, then S ∈ Symm(m) (resp. S ∈ PSymm(m)) if and only if S is symmetric (resp. S is pseudosymmetric). Sur les branches superlinéaires des courbes algébriques. R Apéry, C. R. Acad. Sci. Paris. 222Apéry, R. (1946). Sur les branches superlinéaires des courbes algébriques. C. R. Acad. Sci. Paris 222, 1198-2000. Maximality properties in numerical semigroups and applications to one-dimensional analitically irreducible local domains. V Barucci, D E Dobbs, M Fontana, Memoirs of the American Mathematical Society. 125598Barucci, V., Dobbs, D.E., and Fontana, M. (1997). Maximality properties in numerical semigroups and applications to one-dimensional analitically irreducible local domains. Memoirs of the American Mathematical Society. Vol.125, n.598. One-dimensional almost Gorenstein rings. V Barucci, R Froberg, Journal of Algebra. 188Barucci, V. and Froberg, R. (1997). One-dimensional almost Gorenstein rings. Journal of Algebra 188, p.418-442. The value-semigroup of a one-dimensional Gorenstein ring. E Kunz, Proc. Amer. Math. Soc. 25Kunz, E. (1973). The value-semigroup of a one-dimensional Gorenstein ring, Proc. Amer. Math. Soc. 25 (1973), 748-751. Decomposition of a numerical semigroup as an intersection of irreducible numerical semigroups. J C Rosales, M B Branco, B. Belg. Math. Soc-Sim. 9Rosales, J.C, and Branco, M.B. (2002). Decomposition of a numerical semigroup as an intersection of irreducible nu- merical semigroups. B. Belg. Math. Soc-Sim. 9 (2002), 373-381. Numerical semigroups that can be expressed as an intersection of symmetric numerical semigroups. J C Rosales, M B Branco, J. Pure Appl. Algebra. 1712-3Rosales, J.C, and Branco, M.B. (2002). Numerical semigroups that can be expressed as an intersection of symmetric numerical semigroups. J. Pure Appl. Algebra 171 (2-3) (2002), 303-314. The oversemigroups of a numerical semigroup. J C Rosales, P A García-Sánchez, J I García-García, J A Jimenez-Madrid, Semigroup Forum. 67Rosales, J.C., García-Sánchez, P.A., García-García, J.I. and Jimenez-Madrid, J.A. (2003). The oversemigroups of a numerical semigroup. Semigroup Forum 67 (2003), 145-158. Irreducible numerical semigroups. J C Rosales, M B Branco, Pacific J. Math. 209Rosales, J.C, and Branco, M.B. (2003). Irreducible numerical semigroups, Pacific J. Math. 209 (2003), 131-143. Numerical semigroups. J C Rosales, P A García-Sanchez, 978-1- 4419-0159-0SpringerNew York, NYRosales, J.C. and García-Sanchez, P.A. (2009). Numerical semigroups, Springer, New York, NY, 2009. ISBN: 978-1- 4419-0159-0. Universidad de Granada E-mail address: † [email protected]; ‡ jrosales@ugr. Departamento deÁlgebraDepartamento deÁlgebra. Universidad de Granada E-mail address: † [email protected]; ‡ [email protected]
{'fraction_non_alphanumeric': 0.12301360809877136, 'fraction_numerical': 0.057646670071194686, 'mean_word_length': 3.2445492796123934, 'pattern_counts': {'":': 0, '<': 12, '<?xml version=': 0, '>': 29, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 4, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 88, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'In this paper we introduce the notion of m-irreducibility that extends the standard concept of irreducibility of a numerical semigroup when the multiplicity is fixed. We analyze the structure of the set of m-irreducible numerical semigroups, we give some properties of these numerical semigroups and we present algorithms to compute the decomposition of a numerical semigroups with multiplicity m into m-irreducible numerical semigroups.', 'arxivid': '1006.3493', 'author': ['V Blanco [email protected] ', 'J C Rosales '], 'authoraffiliation': [], 'corpusid': 119164659, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 13654, 'n_tokens_neox': 11490, 'n_words': 6417, 'pdfsha': '5871b0b0fb363ec446541ca57204c88de93dc297', 'pdfurls': ['https://arxiv.org/pdf/1006.3493v1.pdf'], 'title': ['m-IRREDUCIBLE NUMERICAL SEMIGROUPS', 'm-IRREDUCIBLE NUMERICAL SEMIGROUPS'], 'venue': []}
arxiv
Knot Invariants from Four-Dimensional Gauge Theory 23 Jun 2011 Davide Gaiotto School of Natural Sciences Institute for Advanced Study 1 Einstein Drive08540PrincetonNJUSA Edward Witten School of Natural Sciences Institute for Advanced Study 1 Einstein Drive08540PrincetonNJUSA Department of Physics Stanford University 94305Palo AltoCA Knot Invariants from Four-Dimensional Gauge Theory 23 Jun 2011 It has been argued based on electric-magnetic duality and other ingredients that the Jones polynomial of a knot in three dimensions can be computed by counting the solutions of certain gauge theory equations in four dimensions. Here, we attempt to verify this directly by analyzing the equations and counting their solutions, without reference to any quantum dualities. After suitably perturbing the equations to make their behavior more generic, we are able to get a fairly clear understanding of how the Jones polynomial emerges. The main ingredient in the argument is a link between the four-dimensional gauge theory equations in question and conformal blocks for degenerate representations of the Virasoro algebra in two dimensions. Along the way we get a better understanding of how our subject is related to a variety of new and old topics in mathematical physics, ranging from the Bethe ansatz for the Gaudin spin chain to the M -theory description of BPS monopoles and the relation between Chern-Simons gauge theory and Virasoro conformal blocks. 7 An Effective Superpotential For Monopoles 84 7.1 Overview Of Results 84 7.2 Coordinates for Monopoles 88 7.3 Realization Via M -Theory And Branes 90 7.3.1 M -Theory Preliminaries 91 7.3.2 Reduction To Gauge Theory 92 7.3.3 Reducing On A Half Space 93 7.3.4 The Instanton 93 8 Opers And Branes 95 8.1 Back to t = 1 95 8.2 General t 96 8.3 S-Duality 98 -ii -8.4 Monodromy Defects 99 8.5 A Selfdual Brane 103 8.6 Application To The Gaudin Model 104Hence the number of four-dimensional solutions which flow from J I [ z i ; t 0 ] at some given time t 0 after the braiding occurs to the critical point J in the future are the coefficients N J I in the expansion(4.20)But since J I [ z i ; t 0 ] parametrizes flows on the interval (−∞, t 0 ] that start at I, a flow from J I [ z i ; t 0 ] to J on the interval [t 0 , ∞) is equivalent to a flow from I to J defined on the whole real line. So the N J I are the same as the desired invariants N J I : Contents Introduction The Jones polynomial [1] is an invariant of knots that has multiple relations to many aspects of mathematical physics, including integrable lattice statistical mechanics, twodimensional conformal field theory and associated representations of braid groups, and three-dimensional Chern-Simons gauge theory. Khovanov homology [2] is a more recent topological theory in four dimensions; in this theory, a knot is viewed as an object in threedimensional space and the invariant associated to a knot is a vector space (of physical states) rather than a number. The relation between the two theories is that the four-dimensional theory associated to Khovanov homology, when compactified on a circle, reduces to the three-dimensional theory that yields the Jones polynomial. Khovanov homology has been interpreted physically [3] in terms of topological strings, building on earlier work on BPS states of open strings [4]. See [5]- [13] for a sampling of additional developments. An alternative but closely related physical interpretation of Khovanov homology has been given in [14], where more detailed references can be found concerning the Jones polynomial, Khovanov homology, and their relations to mathematical physics. According to this more recent proposal, the Jones polynomial can be computed by counting the solutions of certain elliptic partial differential equations in 4 dimensions, and Khovanov homology can then be constructed by counting the solutions of related equations in 4+1 dimensions. The reasoning that led to this proposal relied on electric-magnetic duality of N = 4 super Yang-Mills theory in four dimensions to transform one description that is rather "quantum" in nature (being closely related to Chern-Simons gauge theory on a bounding three-manifold) to another that is "semiclassical" in the sense that the partition function can be computed just by suitably counting the classical solutions of certain differential equations. Instead of relying on electric-magnetic duality to predict this perhaps mysterious result, can we understand it by a direct study of the equations? This is the goal of the present paper. We will gain a reasonable degree of understanding of the Jones polynomial and a good foundation for understanding Khovanov homology. R + Figure 1. A knot has been placed at the boundary of the four-manifold M 4 = W × R + . A Brief Review The four-dimensional equations in question can be described as follows. The gauge group is a compact Lie group 1 G. The fields in the equations are a gauge field A which is a connection on a G-bundle E → M 4 , with M 4 an oriented Riemannian four-manifold, and another field φ that is a one-form valued in the adjoint representation of G. The equations, which were first studied in relation to the geometric Langlands correspondence [15], read (F − φ ∧ φ + t d A φ) + = 0 (F − φ ∧ φ − t −1 d A φ) − = 0 d A ⋆ φ = 0,(1.1) where the selfdual and anti-selfdual projections of a two-form b are denoted b ± ; d A = d + [A, · ] is the gauge-covariant exterior derivative; F = dA + A ∧ A is the Yang-Mills field strength; ⋆ is the Hodge star operator; and t is a real parameter. (Actually, t takes values in RP 1 = R ∪ ∞; for t → 0 or t → ∞, one multiples the second equation by t or the first by t −1 .) To study knot invariants, one specializes to M 4 = W × R + , where W is a three-manifold and R + is the half-line y ≥ 0 ( fig. 1). The boundary condition at y = 0 is slightly subtle but can be easily described in the absence of knots. Suppose first that t = 1. Since the boundary condition is local, we can specialize to W = R 3 in describing it. (In any case, that is the main example for the present paper.) Consider a classical solution that is invariant under translations along R 3 and such that A and the part of φ normal to the boundary vanish. The equations then reduce to Nahm's equations for φ, the part of φ tangent to the boundary: d φ dy + φ × φ = 0. (1.2) These equations have a singular solution, first introduced by Nahm in his work on monopoles. Pick an embedding ρ : su(2) → g (where su (2) and g are the Lie algebras of SU (2) and G, respectively), given by a triple of elements t ∈ g obeying [t 1 , t 2 ] = t 3 , and cyclic permutations. Then the solution is φ = t y . (1.3) Though any ρ gives a solution, the case we want is that ρ is a principal embedding. For G = SU (N ), this means that ρ is an irreducible embedding of SU (2) in G; for any G, it means that the raising operator t + = t 1 + it 2 is a "regular" element of the complexified Lie algebra g C (this means that the subalgebra of g C that commutes with t + has the minimum possible dimension). Then one can define a boundary condition by allowing precisely those solutions of (1.1) that can be approximated for y → 0 by the model solution (1.3) with the regular Nahm pole. This has an analog for any t; the starting point, as explained in an appendix, is to set the tangential part A of the gauge field to be a specific multiple of φ, so that the equations reduce again to Nahm's equations. When a link L ⊂ W is included, this boundary condition is modified along L. A link is simply the union L = ∪ i K i of disjoint embedded circles K i . The K i are labeled by representations R ∨ i of the Langlands or GNO dual group G ∨ to G, and in this description the knots enter the formalism only via the way they enter the boundary conditions. Roughly speaking, the modification is made by requiring the presence of singular BPS monopoles supported along the K i with magnetic charges given by the R ∨ i . The G-bundle E → M 4 has an instanton number P defined in the usual way as a multiple of M 4 Tr F ∧ F . (The definition of P as a topological invariant involves some subtleties that are described in [14]; roughly speaking, the boundary conditions at the finite and infinite ends of R + give suitable trivializations of E, enabling one to define the instanton number. 2 ) For each value n of the instanton number, one defines an integer a n by "counting" (with signs that are determined by the sign of the fermion determinant of N = 4 super Yang-Mills theory) the number of solutions of the supersymmetric equations (1.1) with instanton number n. Then the partition function of a certain version of twisted N = 4 super Yang-Mills theory on M 4 is Z(q) = n a n q n , (1.4) where the definition of q in terms of parameters of N = 4 super Yang-Mills theory was explained in [14]. To get the Jones polynomial and its analogs for other groups and representations, one specializes to W = R 3 and takes the boundary condition at y = ∞ to be simply A, φ → 0. Then for example for G ∨ = SU (2) and R ∨ the two-dimensional representation of SU (2), Z(q) is supposed to become the Jones polynomial. Since W = R 3 is the case relevant to the Jones polynomial, it will be the main example in the present paper. However, many of our considerations apply also for W = R × C where C is a Riemann surface, so we will consider this case as well. A slight generalization of the above-described procedure is to modify the boundary condition at infinity so that A and φ y vanish but φ approaches, up to a gauge transformation, a specified triple a of elements of t, the Lie algebra of a maximal torus T of G. Physically, this means that one takes the vacuum at infinity to be specified by a given point on the Coulomb branch. (In the presence of the Nahm pole boundary condition, turning on φ y or the other two scalars of N = 4 super Yang-Mills theory -called σ, σ in [14] -would break supersymmetry; so a are the only useful Coulomb branch parameters.) Continuously turning on Coulomb branch parameters should not affect the counting of solutions of an elliptic equation, so this procedure should give a slightly more general way to compute the Jones polynomial. To describe the basic solution of the equations (1.1) with a specified choice of a at infinity, one looks for a solution that still has A = φ y = 0 and is still invariant under translations along R 3 , but now obeys lim y→∞ φ = g ag −1 , for some g ∈ G. The equations still reduce to Nahm's equations (1.3). A general theorem [16] says that for any simple Lie group G, and any specified choice of a, there is a unique solution of Nahm's equations with a regular Nahm pole at y = 0 and the required behavior for y → ∞. This solution describes the ground state at the given point on the Coulomb branch in the absence of any 't Hooft operators on the boundary. Lift To Khovanov Homology Though our main focus will be to recover the Jones polynomial from this framework, we will also briefly sketch how Khovanov homology is supposed to arise. A primary purpose of this is to explain the extent to which the particular values t = ±1 are or are not special, since this will be important later. To get Khovanov homology instead of the Jones polynomial, we are supposed to "categorify" the above-described situation, which is just a fancy way to say that we must obtain everything that has been described so far from a theory in one dimension higher. For this, let x 1 , x 2 , x 3 be local coordinates on W and decompose φ as φ = φ · d x + φ y dy, where φ y is the component of φ in the y direction. Categorification is accomplished by introducing a new time coordinate x 0 and replacing φ y by the covariant derivative D/Dx 0 . This replacement makes sense in that, since φ y only appears in (1.1) inside commutators and covariant derivatives, the replacement does give a differential equation (rather than a differential operator), now on the five-manifold R × W × R + . Moreover this differential equation, whose details are described in section 5 of [14], is elliptic so problems of counting its solutions make sense. 3 This five-dimensional lift of the four-dimensional equations (1.1) also has a surprising four-dimensional symmetry, provided we set t = ±1. The original four-dimensional symmetry relating the different directions in M 4 = W × R + has been spoiled by the replacement φ y → D/Dx 0 . But at t = ±1, the five-dimensional equations acquire a new four-dimensional symmetry: one can replace R × W by a general oriented Riemannian four-manifold M , without additional structure, and formulate these equations on M × R + . For studying the Jones polynomial, the values t = ±1 are not particularly distinguished; the counting of solutions of the elliptic equations (1.1) is independent of t. Moreover, categorification -the substitution φ y → D/Dx 0 -is not limited to t = ±1. What is special about t = ±1 is the four-dimensional symmetry of the categorified theory, which is likely to have important implications for Khovanov homology and its analogs on other manifolds. From a physical point of view, the five-dimensional lift of the equations (1.1) are BPS conditions of a certain twisted version of five-dimensional super Yang-Mills theory, formulated on R × W × R + ; they describe configurations that are invariant under one of the supercharges, which we will call Q. This operator obeys Q 2 = 0, and the space of supersymmetric ground states is the same as the cohomology of Q. This is the candidate for Khovanov homology. Mathematically, the five-dimensional equations can be interpreted as Morse theory flow equations, and the space of supersymmetric ground states is the analog of Floer homology for this situation. Physically, to construct the space of supersymmetric ground states, one starts with time-independent solutions of the five-dimensional equations -these are simply the solutions of the original uncategorified equations (1.1) in four dimensions. Expanding around any one of these solutions, one can construct an approximate supersymmetric state, and these furnish a basis for the space of supersymmetric states in the classical approximation. Then one computes quantum corrections by taking account of tunneling between classical vacua; the tunneling events are solutions of the full five-dimensional equations. From this point of view, the link between Khovanov homology and the Jones polynomial comes from the fact that the classical solutions that give a basis for the classical approximation to Khovanov homology are the same ones that must be counted to compute the Jones polynomial. Methods Used In This Paper A priori, to count the solutions of the nonlinear partial differential equations (1.1) is a daunting problem. Our attempts to simplify this problem are based on three ideas. The first is a standard idea in topological field theory. We consider knots in W = R × C, where C (which may be simply R 2 ) is a two-manifold, and we parametrize R by x 1 . We stretch our knots in the x 1 direction, so that except at a few exceptional values of x 1 where the number of strands changes, the boundary conditions are nearly independent of x 1 ( fig. 2). We hope that, away from the exceptional values of x 1 , the solutions can be approximated by solutions that are independent of x 1 . Once one drops x 1 , the equations reduce to equations in three dimensions. The reduced equations preserve more supersymmetry and one may hope to understand their solutions. After finding the three-dimensional solutions, to recover a four-dimensional picture, we have to take into account an adiabatic variation of the parameters in the three-dimensional equations. This is because our knot, even after stretching, is not quite independent of x 1 . We also have to consider the jumping that occurs when the number of strands changes. Actually, there is an important special case in which one only has to consider the adiabatic variation of parameters. This is the case ( fig. 3) that R × C is replaced by S 1 × C (or I × C where I is a closed interval, though this introduces questions about boundary conditions) R + x 1 Figure 2. Stretching a knot in one direction -here taken to be the x 1 direction -to reduce to a situation that almost everywhere is nearly independent of one coordinate. After much stretching, the knot is everywhere nearly independent of x 1 , except near the finite set of critical values of x 1 at which a pair of strands appears or disappears. (In the figure, these occur only at the top and bottom.) x 1 Figure 3. A braid in I × C; by gluing together the top and bottom, one can make a closed braid in S 1 × C. After much stretching, a braid can be described by adiabatic evolution in x 1 , with no exceptional values where this description breaks down. and the link is replaced by a braid. In this situation, one would study not the Jones polynomial but its associated braid group representations, which are also of great interest. One important thing to mention about this program is that it is not guaranteed to work. As one stretches a knot in the x 1 direction, the solution might simultaneously "spread" in the y direction ( fig. 4) so that even after stretching, the solution might not approach an x 1 -independent limit. In fact, we will find that this happens under some conditions. One of our main tasks will be to understand conditions under which the sort of behavior suggested in fig. 4 does not occur. In carrying out the program that we have just described, we start in section 2 at t = ±1 Figure 4. As a knot is stretched along the boundary, a solution of the supersymmetric equations might become delocalized in the y direction, normal to the boundary. This is schematically indicated here; the shaded region indicates the spatial extent of a solution -that is, of the region over which the chosen solution deviates significantly from the one that describes the vacuum in the absence of knots -and its thickness is proportional to the amount that the knot has been stretched. because these are special values for Khovanov homology (as we recalled in section 1.1.1 above) and also because some simplifications in the three-dimensional equations at t = ±1 were already found in section 3.6 of [14]. Because we encounter some puzzling phenomena (which we will ultimately understand along the lines of fig. 4), we look for some additional simplifications. In doing so, we primarily exploit two ideas. The first idea is to modify the boundary conditions to incorporate gauge symmetry breaking. The basic idea was already explained at the end of section 1.1: instead of asking for φ to vanish at infinity, we ask for φ → g ag −1 , where g ∈ G and the three components of a = (a 1 , a 2 , a 3 ) take values in a Cartan subalgebra of the Lie algebra g of G. (In the more general case W = R × C, we would similarly modify the boundary condition to require that the component of φ in the R direction is in a specified conjugacy class at infinity.) Continuously changing the boundary conditions in this way should not change the counting of solutions that leads to the Jones polynomial. On the other hand, in such counting problems one often finds that perturbing to a more generic situation can make things easier. Moreover, in the present case, taking a to be generic reduces the nonabelian gauge theory that we are studying to an abelian theory at low energies. If we scale up our knots so that all relevant directions of the K i are large compared to 1/| a|, then we can reasonably hope to find some sort of effective abelian description of the relevant phenomena. The second idea that we exploit is perhaps even more obvious. Since the equations that arise at t = 1 with the Nahm pole boundary conditions described above are rather special, we perturb the value of t and/or the Nahm pole boundary conditions to something more generic. This proves to be very fruitful, especially when combined with gauge symmetry z 1 z 4 z 2 z 3 Figure 5. In a time-independent situation, we look for solutions on a three-manifold M 3 = C ×R + , where C (taken here to be a two-sphere) is a Riemann surface. Knots are placed at points on the boundary of M 3 , labeled here as z 1 , . . . , z 4 . breaking. Outline And Results In section 2, we analyze the three-dimensional reduction of equations (1.1) at t = 1. We get an interesting description in terms of Higgs bundles with some additional structure, but it becomes clear that the program suggested in fig. 2 will encounter some difficulties at t = 1. In section 3, we perturb the equations to t = 1 and find that this offers a much more promising framework for understanding the Jones polynomial. The equations for generic t have surprising and useful relations to a variety of topics in mathematical physics, including the Bethe equations for an integrable spin system known as the Gaudin model, and certain special "degenerate" conformal blocks of the Virasoro algebra; the rest of the paper is based on these relations. In section 4, we discuss the general framework for constructing braid group representations from adiabatic evolution of the parameters governing time-independent solutions. The general framework is a little abstract, but in section 5, we show that in our particular problem, it can be made very concrete using the free field representation of certain Virasoro conformal blocks. In section 6, we implement that idea in detail. This finally enables us to understand how the Jones polynomial and the braid group representations associated to it can be recovered by counting solutions of the four-dimensional BPS equations (1.1). Section 7 is devoted to describing an effective superpotential for BPS monopoles that can be used to understand some of the subtle results of sections 2 and 3. In section 8, we place some structures encountered in this paper in a wider context of mathematical physics. Three appendices fill in details of the derivations. 2 Analysis At t=1 Some Preliminaries As explained in the introduction, after stretching a knot along the first factor of M 4 = R × C × R + , we want to find the solutions that are independent of the first coordinate (which we call x 1 ), that is the solutions that obey reduced equations on the three-manifold 4 M 3 = C × R + . Here C is a Riemann surface (which may be simply R 2 ), and R + is the half-line y ≥ 0. If knots are present, we take their support to be of the time-independent form R × z i × {0}, where the z i are points in C, and {0} is the endpoint y = 0 of R + . The picture is sketched in fig. 5. Solutions that can be derived from three dimensions preserve more than the generic amount of supersymmetry -they preserve four supercharges, to be precise -and accordingly, as described in section 3.6 of [14], their structure simplifies. In all cases that we will encounter in the present paper, the reduced equations can be usefully described in terms of three differential operators D i , i = 1, 2, 3. The equations say that the D i commute [D i , D j ] = 0,(2.1) and obey a moment map condition 3 i=1 [D i , D † i ] = 0, (2.2) where D † i is the adjoint of D i in a natural sense. The commutativity constraint (2.1) is invariant under complex-valued gauge transformations D i → gD i g −1 , where g is a G C -valued gauge transformation (G C is the complexification of G), while the moment map condition (2.2) is only invariant under G-valued gauge transformations. What will make our problem tractable is that solutions of the combined system of equations modulo G-valued gauge transformations are equivalent to solutions of just the commutativity constraint (2.1) modulo G C -valued gauge transformations. But solutions of the commutativity constraint modulo complex gauge transformations can be described in terms of holomorphic quantities, so it is possible to understand them. Different instances of this structure vary by the construction of the D i , which depends on the choice of t, and also on the boundary conditions that we assume at y = 0 and at y = ∞. The most basic case considered in [14] is that t = 1 (or −1) and the boundary condition is given by a Nahm pole in the part of φ tangential to the boundary. That boundary condition sets to zero φ y , the normal part of φ, at y = 0. If we also require φ y to vanish for y → ∞, then a simple vanishing argument shows that in a solution that is independent of x 1 , φ y is identically zero; similarly, A 1 , the component of A in the x 1 direction, vanishes in a three-dimensional solution. Once φ y and A 1 are set to zero, the equations can be put in the above-described form, as shown in detail in [14], section 3.6, 5 4 To minimize confusion, we note the following. In this paper, we use two different decompositions of M4 = R × C × R+ as the product of a three-manifold and a one-manifold: we write M4 = W × R+ with W = R × C, but also M4 = R × M3 with M3 = C × R+. 5 Our notation differs from the notation used there by a relabeling of the coordinates x i → x i+1 (whose purpose is to make "room" for a new time coordinate x 0 upon categorification). Also, for later convenience we permute the Di in an obvious way. with D 1 = D Dx 2 + i D Dx 3 D 2 = [φ 2 − iφ 3 , · ] D 3 = D Dy − i[φ 1 , · ] (2.3) and the moment map condition 0 = 3 i=1 [D i , D i † ] = F 23 − [φ 2 , φ 3 ] − D y φ 1 . (2.4) In writing these formulas, we have simply taken C = R 2 with coordinates x 2 , x 3 . However, it is helpful to introduce a complex coordinate z = x 2 + ix 3 , and to write φ 2 dx 2 + φ 3 dx 3 = ϕdz + ϕdz; also we introduce a complex connection A y = A y − iφ 1 for parallel transport in the y direction and write D y = d y + [A y , · ]. Then we can write D 1 = 2 D Dz D 2 = 2[ϕ, · ] D 3 = D Dy . (2.5) With this way of writing the D i , they make sense on an arbitrary Riemann surface C, with D/Dz understood as the ∂ operator and ϕ as a (1, 0)-form on C. As one would expect in a geometry that preserves four supercharges, the commutativity constraint [D i , D j ] = 0 can be derived from a superpotential, namely [15]; actually, these equations are a hybrid of the equations of Nahm, Hitchin, and Bogomolny. They reduce to Nahm's equations if we drop the dependence on z, to Hitchin's equations if we drop the dependence on y, and to the Bogomolny equations if we set ϕ = 0. This is not just an analogy: we can borrow standard strategies from the theory of moduli spaces of Nahm, Hitchin, or Bogomolny equations. W = 1 4πi C×R + Tr ϕF yz ,(2. The equation [D 1 , D 2 ] = 0, taken for fixed y, defines a Higgs bundle (E, ϕ) in the sense of Hitchin. The fact that D 1 and D 2 commute with D 3 simply means that the Higgs bundle is independent of y, up to a complex-valued gauge transformation. When we specialize to C = R 2 ∼ = C, we get a Higgs bundle on C that can be understood as a Higgs bundle on CP 1 = C ∪ ∞, possibly with a singularity at infinity. This case will be considered in section 2.5. There is another natural way for singularities of the Higgs bundle to arise. As in section 6 of [14], one may include surface defects supported on codimension two submanifolds in M 4 . Taking these to be of the form R × q i × R + , where the q i are points in C, the construction summarized above still applies and the Higgs bundles acquire singularities at the points q i . We consider this situation in section 8.4. Additional structure arises from boundary conditions at y = 0 and y = ∞. We will discuss the consequences of the Nahm pole at y = 0 in section 2.2. The analogy to Nahm's equations will be useful: we can extract some holomorphic data from the commuting pair D 2 ,D 3 for any given point in C. As D 2 ,D 3 commute with D 1 , this data varies holomorphically on C, or possibly meromorphically in the presence of singularities. Finally, an analogy with the Bogomolny equations will help us understand the physical content of our solutions, especially when we turn on gauge symmetry breaking for y → ∞. Indeed, the holomorphic data in the commuting pair D 1 ,D 3 is analogous to the data which specifies the position of BPS monopoles in a solution of the Bogomolny equations. The Boundary Condition Our next task is to analyze the boundary conditions, first in the absence of singular monopoles. The model solution (1.3) at t = 1 has a singularity in D 2 and D 3 , but not in D 1 , simply because D 1 does not contain the scalar fields. For g = su (2), with a standard choice of the Lie algebra elements t, this solution is explicitly ϕ = 1 y 0 1 0 0 , A z = 0, A y = 1 2y 1 0 0 −1 . (2.7) We consider the matrices here to act on the fiber of a trivial rank two complex vector bundle E → R 2 × R + . For G = SO(3), on a general Riemann surface, there could be a global obstruction to defining E as a rank two bundle and one would then consider instead the corresponding adjoint bundle ad(E). Because our considerations will be local along the Riemann surface C, a reformulation in terms of the adjoint representation does not change much, and we will omit this. The solution (2.7) can be written ϕ = gϕ 1 g −1 , D y = g d dy g −1 (2.8) with ϕ 1 = 0 1 0 0 (2.9) and g a G C -valued gauge transformation that is singular at y = 0 g = y −1/2 0 0 y 1/2 . (2.10) In other words, the solution is obtained by the complex-valued gauge transformation g from a trivial solution ϕ = ϕ 1 , A z = A y = 0. We want to consider solutions which look like the model solution near y = 0, up to a gauge transformation. We would like to express this constraint in terms of the Higgs bundle data (E, ϕ) away from the boundary. For that purpose, it is useful to consider the behavior of a local holomorphic section s of the gauge bundle E that is invariant under parallel transport in the y direction. We will first do the calculation very explicitly for the model solution, and then identify which features are valid more generally. Let s be a section of the gauge bundle E that obeys Ds Dy = 0. (2.11) Since D/Dy = g(d/dy)g −1 , the solutions of this equation are of the form s = gs 0 , where s 0 is independent of y. Thus a general solution takes the form s = ay −1/2 by 1/2 , (2.12) with constants a, b. In particular, a generic vector in E, when parallel transported in the y direction to y = 0, will blow up as y −1/2 . There is a one-dimensional subspace consisting of solutions of Ds/Dy = 0 that actually vanish as y 1/2 for y → 0. This subspace is simply characterized by the condition a = 0. We write E y for the restriction of E → C × R + to C × {y} for any fixed y > 0. Parallel transport using D y gives a natural identification of the E y for all y, and we write simply E for E y , regarded as a bundle over C. Similarly, the restriction of ϕ to C × y is independent of y, up to parallel transport by D y . So by restriction to C × y, we get a Higgs bundle (E, ϕ) → C. The "small" sections of E -the solutions s of Ds/Dy = 0 that vanish for y → 0generate a rank one sub-bundle L ⊂ E. In the model solution, it is simply the sub-bundle of sections of E of the form 0 b . L is a holomorphic sub-bundle of E → C; a section of L is holomorphic if it is annihilated by D 1 = 2 D/Dz. Concretely, in the model solution, a section 0 b of L is holomorphic if b is a holomorphic function of z. The fundamental reason that L is holomorphic is that, as [D 2 , D 3 ] = 0, we can ask for a small solution of D 3 s = 0 to also be annihilated by D 2 = 2D/Dz. However, we cannot also ask for a small solution to be annihilated by ϕ = D 1 /2. This is clear from the above formulas; in the model solution, ϕ annihilates a vector if the bottom component vanishes, not if the top component vanishes. To measure the failure of L to be ϕ-invariant, we can proceed as follows. If L is a trivial line bundle, which will be the case in our applications, then we can pick a section s of L that is everywhere nonzero. For our model solution, we just pick s = 0 1 . Then we define κ = s ∧ ϕs. (2.13) For the model solution, we see that κ = 1, and in particular κ is everywhere nonzero. Nonvanishing of κ means that ϕs is not a multiple of s, so L is not invariant under multiplication by ϕ. (If R 2 is replaced by a general Riemann surface C, it might be impossible to pick an s that is globally nonzero, but one can still pick a local section s of L and measure the failure of L to be ϕ-invariant by computing κ = s ∧ ϕs. Whether κ vanishes does not depend on the choice of s as long as it is nonzero.) Three basic properties of s which held for the model solution remain true for any solution with a regular Nahm pole, since they are unaffected by the subleading behavior of D y and ϕ as y → 0: • A generic vector in E, when parallel transported in the y direction to y = 0, will blow up as y −1/2 . • The sections s that under parallel transport to y = 0 actually vanish span a rank 1 holomorphic sub-bundle L ⊂ E. • Finally, we cannot also ask for a small section to be annihilated by ϕ = D 1 /2. On the contrary, in a solution that can be approximated by the model solution near y = 0, ϕ does not annihilate a small section at any point in C. Consequently, a solution of the full equations with a regular Nahm pole for y → 0 gives not just a Higgs bundle (E, ϕ). Rather, there is some additional structure: E is endowed with a holomorphic line sub-bundle L which is nowhere stabilized by ϕ. On R 2 , L is inevitably trivial, so we can simply say that an everywhere non-zero holomorphic section s of E exists with s ∧ ϕs = 1. Notice that this condition is far from sufficient to determine ϕ. If s = 0 1 , then the condition s ∧ ϕs = 1 fixes the upper right matrix element of ϕ and puts no condition on the others. The reason that only one matrix element of ϕ is fixed in terms of s is that when we make a gauge transformation that behaves like (2.10) for y → 0, ϕ acquires a singularity that only depends on its upper right matrix element. In an appropriate situation (on a Riemann surface C of higher genus, or on R 2 in the presence of singular monopoles, as introduced shortly), there can be a nontrivial moduli space of triples (E, ϕ, L) with E and L fixed and only ϕ varying. This is explained in section 8. For any triple (E, ϕ, L), we expect that the full system of equations can be solved by a complex gauge transformation. Suppose we are given a y-independent Higgs bundle (E, ϕ), and a holomorphic sub-bundle L ⊂ E which is nowhere stabilized by ϕ. In a basis of E given by ϕs and s, we make the gauge transformation (2.10). This will reproduce the Nahm pole singularity at y = 0, but generically it will not give a solution of the moment map condition (2.4). By further making a smooth complex gauge transformation, one can hope to get a solution of the moment map condition. Adding Singular Monopoles In order to add singular monopoles at the boundary, we need to replace the Nahm model solution with a more general singular solution, given in [14], section 3.6. Let us consider the case of a single singular monopole, located at y = z = 0. The model solution has the same singularity as before for y → 0 at z = 0, but has a more complicated form near y = z = 0. It can be obtained by a complex-valued gauge transformation, described explicitly in [14], from a solution of the commutativity constraint with ϕ = 0 z k 0 0 , A z = A y = 0. (2.14) To express this in our present language, L is still spanned by sections that in the gauge (2.14) are multiples of s = 0 1 ; this is because g has similar behavior as before for y → 0. We now have s ∧ ϕs = z k . (2.15) The zero of s ∧ ϕs is interpreted as the position of the singular monopole, and its degree is the charge. This interpretation suggests immediately what a solution with several singular monopoles should mean. We consider a solution described by a Higgs bundle (E, ϕ) with a sub-bundle L ⊂ E that is generically not ϕ-invariant. If s ∧ ϕs = s a=1 (z − z a ) ka ,(2.16) then we say that the solution has singular monopoles of charges k a at the locations z a . One hopes to be able to prove that given such data, there is a unique solution whose singularity near each z = z a agrees with that of the singular model solution. Though we mainly consider su(2) in the present paper, we can readily generalize these statements to a more general Lie algebra. For simplicity, take G = SU (N ) and view E as a complex vector bundle of rank N . Consider a Nahm pole based on the principal embedding of su(2) → su(N ). The eigenspaces of t 3 (in the fundamental n-dimensional representation of su(N )) are one-dimensional, and we have a line bundle L defined by sections which decrease as fast as possible as y → 0. The only constraint on L is that, away from the positions of singular monopoles, L ⊕ ϕL ⊕ · · · ⊕ ϕ n−1 L = E (2.17) When we specialize to C = R 2 ∼ = C, we can define the line sub-bundle L by a specific section s of E, defined up to rescaling, such that the sections s, ϕs, · · · , ϕ N −1 s are linearly independent. This constraint is relaxed at the location of singular monopoles, in a way which depends on their charges. An equivalent description in terms of zeroes of matrix elements of ϕ is given in [14], eqn. (3.59). This can be extended to the case of a general Nahm pole at y = 0, not necessarily associated to a principal embedding of su (2). (This extension will not be studied in the present paper.) In general, for y → 0, there are local sections growing as either integer or half-integer powers of y. We can define a flag of holomorphic sub-bundles E n of E by looking at local sections which grow at most as a given power y (n−N )/2 as y → 0. Clearly, E n+1 ⊂ E n ; also E 0 = E, and E n = 0 for large enough n. Upon rescaling by y −(n−N )/2 , a generic vector in E n has a finite limit as y → 0, and this limit is an eigenvector of t 3 with eigenvalue (N − n)/2. The kernel of this map is E n+1 ; hence the y → 0 limit identifies the quotient spaces E n /E n+1 with the eigenspaces of t 3 . Multiplication by ϕ gives maps E n → E n−2 . As ϕ ∼ t + y , the holomorphic maps φ n : E n /E n+1 → E n−2 /E n−1 can be identified with the action of t + on the eigenspaces of t 3 . Solutions Without Symmetry Breaking Now we want to use the ideas that have just been described to determine some moduli spaces of solutions of the supersymmetric equations (2.1), (2.2). We will do this for G = SO(3), so that the dual group whose representations label the singular monopoles is G ∨ = SU (2). First we work at the origin of the Coulomb branch; this means that we consider solutions such that the scalar fields φ vanish for y → ∞. In particular, ϕ must vanish at infinity. Since the equation D y ϕ = 0 means that the conjugacy class of ϕ is y-independent, ϕ can only vanish at infinity if it is everywhere nilpotent. This means that by a complex gauge transformation, we can make ϕ upper triangular: ϕ = 0 p(z) 0 0 , (2.18) with some polynomial p(z). We cannot, however, put s in a standard form at the same time. So we simply take s = P (z) Q(z) ,(2.19) with polynomials P, Q. Without changing ϕ, we can make an upper triangular gauge transformation, shifting P by a polynomial multiple of Q: P (z) → P (z) + U (z)Q(z). (2.20) This is the only freedom, apart from a rescaling of s by a complex constant. Now let us ask how we can pick p, Q, and P to describe a configuration with singular monopoles of charges k a at the points z a in the boundary. Since s∧ϕs = pQ 2 , the condition that we need is p(z)Q(z) 2 = a (z − z a ) ka := K(z). (2.21) If the singular monopoles all have minimal charge, k a = 1, we can only obey this with p = K and Q = ±1. Then we can set P (z) = 0 by a transformation (2.20), and changing the sign of Q multiplies s by an inessential constant. So at the origin of the Coulomb branch, there is a unique solution with singular monopoles of specified locations and minimal magnetic charge. This fact is actually an obstruction to the program that was described in section 1.2 (see fig. 2). The case of singular monopoles of minimal charge is supposed to be dual to the Jones polynomial, which is the invariant computed in Chern-Simons gauge theory for G ∨ = SU (2) with all knots labeled by the two-dimensional representation of SU (2). If there is only one solution, this means that the physical Hilbert space associated to this problem is one-dimensional. The Jones representations of the braid group would then be of rank 1. This is certainly not the case. We must be running into some version of the problem that was indicated in fig. 4. We will get a clearer picture of what is happening after including symmetry breaking in section 2.4, and after deforming to t = 1 in section 3. This will ultimately enable us to circumvent the obstacle just described. What happens if some singular monopoles have non-minimal charge? In this case, we encounter moduli spaces of solutions. In general we solve (2.21) by In implementing the program described in section 1.2, if some knots are labeled by integers k a > 1 -in other words, by representations of G ∨ = SU (2) of dimension k a + 1 > 2 -one would have to handle the evolution of the four-dimensional solution as a path in the moduli space of three-dimensional solutions. (This would be done by quantizing the moduli space to get an appropriate space of physical states, in which the evolution would take place.) Instead of following that route, we will perturb the equations that we have just analyzed to more generic ones, with symmetry breaking at infinity or with t = 1. This will eventually reduce all of our moduli spaces to finite collections of points. p(z) = a (z − z a ) ma , Q = a (z − z a ) ra ,(2. "Real" Symmetry Breaking Our first approach to getting a clearer understanding of the solutions -and a more useful reduction to three dimensions -will be to move on the Coulomb branch. The basic idea was already explained at the end of section 1.1. We fix some constant, nonzero expectation values a for the tangential scalar fields φ, and consider only solutions of the supersymmetric equations (1.1) such that lim y→∞ φ = g ag −1 , for some g ∈ G. From our present point of view, we want to analyze the reduced three-dimensional equations in the context of symmetry breaking. The reduction splits off the x 1 direction, so we write φ · dx = φ 1 dx 1 + ϕ dz + ϕ dz. The effects of an expectation value for φ 1 or for ϕ will be quite different. We first consider the case of turning on φ 1 only. As we will see, this has the effect of making the solutions somewhat more physically transparent, without adding or removing solutions. For G = SO(3), we can pick a gauge where φ 1 is constant at large y. This means that for large y, A y = A y − iφ 1 approaches a constant matrix at infinity, say such that A y = a 1 0 0 −a 1 . (2.24) with a 1 > 0. This expectation value breaks SO(3) to U (1). We pick a generator of the unbroken U (1), normalized so that the off-diagonal components of an adjoint-valued field, which are the fields of minimum electric charge for G = SO (3), have charges ±1: Q = −1/2 0 0 1/2 (2.25) The choice of sign will be convenient. The equation D y ϕ = 0 implies that the lower-triangular component of ϕ must be zero, since otherwise it would grow exponentially fast at infinity. The diagonal component must also be zero, since otherwise ϕ would have a nonzero limit at infinity. Hence symmetry breaking provides a natural frame in which ϕ is strictly upper-triangular for y → ∞, just as the boundary condition did for y → 0. But ϕ decays exponentially fast at large y. In the low energy effective U (1) theory, a field of electric charge 1 is a section of a line bundle that we will call M. The first Chern class of M, integrated over the plane given by y = y 0 , for some large constant y 0 , is an integer that we will call the magnetic charge m. The upper triangular matrix element of ϕ ϕ ∼ 0 p 0 0 (2.26) is a section of M −1 , and hence the magnetic charge, which is minus the first Chern class of M −1 , is minus the number of zeroes of p. This is the same as minus the number of zeroes of p (defined in (2.18)), which is equivalent to p by a complex gauge transformation. So in the notation of (2.23), the magnetic charge is m = − a m a = a (−k a + 2r a ) . (2.27) This means that for a given configuration of singular monopoles or 't Hooft operators at y = 0, the smallest possible value of m is − a k a . Every time that we add a zero to Q, m increases by 2. We propose that this fact can be interpreted in terms of smooth BPS monopoles. If we simply set ϕ = 0, the extended Bogomolny equations that we have been studying reduce to the usual Bogomolny equations. Far from the boundary, the Bogomolny equations are a very good approximation, since ϕ is so small. The Bogomolny equations on R 3 admit smooth monopole solutions of charge 2. (We measure magnetic charge in units such that the minimum magnetic charge allowed by Dirac quantization is ±1. This is also the magnetic charge of a minimum charge singular monopole or 't Hooft operator, but smooth monopoles have even charge.) Our proposal is that if Q has a zero of order r a at z = z a , then there are r a smooth monopoles located at z = z a . This statement has a precise meaning only if the monopoles are located at very large y, where the extended Bogomolny equations reduce to the ordinary ones, and have smooth monopoles as solutions. However, we have found that if Q has a zero of order r a at z = z a , then there are precisely r a complex moduli associated to the point z = z a in the complex z-plane. We propose that these moduli are the positions in the y direction of r a smooth monopoles that are located at z = z a , along with conjugate angles. Again, the precise meaning of this statement holds when the y positions in question are large. We will make this interpretation quantitative in section 7, but for now we consider qualitative arguments. A first motivation for this proposal comes from m; we interpret the formula (2.27) as the sum of the magnetic charges of the singular monopoles at the boundary (− a k a ) plus a contribution of 2 for every smooth monopole (contributing 2 a r a in toto). The fact that the charge of the singular monopoles is negative is worthy of note. Prior to symmetry breaking, the charge of the singular monopoles does not have a meaningful sign -it is dual to a representation of G ∨ = SU (2) -but after symmetry breaking, this sign makes sense and with our normalization it is negative. To learn more, we recall some facts about the Bogomolny equations. For the Bogomolny equations on R 3 , the holomorphic data are the commuting operators D 1 and D 3 , defined exactly as in eqn. (2.5). Localized monopole solutions of the Bogomolny equations manifest themselves in the holomorphic data through "bound states" in the parallel transport by D 3 , i.e. through normalizable solutions to Ds Dy = 0. (2.28) Such solutions only appear at specific positions in the z-plane, which are interpreted as the z values of the monopole locations, because generically, if we pick s to decay exponentially for y → −∞, it will grow exponentially for y → +∞. In our setup, we are limited to y ≥ 0, and an analogous normalizable solution exists precisely if the "small" section s that vanishes for y → 0 also decays exponentially at large y. If ϕ also decays exponentially as well, then s ∧ ϕs vanishes for y → ∞ and hence for all y. So sections s that vanish at both ends can arise only only at zeroes of K(z). Indeed, if s = P Q (2.29) near infinity, then given the form of (2.24), vanishing of s for y → ∞ is equivalent to Q = 0. Zeroes of Q are the same as zeroes of Q. (Indeed, s ∧ ϕs is independent of y; its zeroes are the zeroes of Q if y is small or of Q if y is large.) So the relation to the ordinary Bogomolny equations does indeed suggest that there are r a smooth monopoles at each point z = z a at which Q vanishes. The position of the smooth monopoles in the y direction should be encoded in the values of P and its first r a − 1 derivatives. We will make this picture more precise in section 7; for now we simply observe that the normalizable small section s(z a ) behaves as P (z a )e −m 1 y , which suggests that increasing |P (z a )| moves the smooth monopoles towards large y. The limit P (z a ) → 0 is somewhat singular, as s is supposed to be everywhere nonzero. The physical picture suggests that the limit will correspond to a monopole bubbling situation, where the smooth monopole is pushed to the boundary, and screens the singular monopole's charge. Monopole bubbling is a phenomenon (originally described in [18] and rediscovered in section 10.2 of [15]) in which an 't Hooft operator absorbs a smooth BPS monopole, lowering the magnitude of its magnetic charge. "Complex" Symmetry Breaking Now we consider the case of symmetry breaking in ϕ. Suppose, for G = SO (3), that the eigenvalues of ϕ for y → ∞ are ±a. Then, as ϕ commutes with D y , its eigenvalues are ±a everywhere. By a complex gauge transformation, we can reduce to the case that ϕ is a constant diagonal matrix: ϕ = a 0 0 −a . (2.30) Writing as usual s = P Q for the small section, we find s ∧ ϕs = 2aP Q. After putting ϕ in the form (2.30), we can still make a gauge transformation by diag(λ, λ −1 ), mapping P → λP, Q → λ −1 Q. (2.31) In the presence of singular monopoles of charges k a located at z = z a (and y = 0), the condition we want to satisfy is s ∧ ϕs = d a=1 (z − z a ) ka := K(z). This becomes 2aP Q = K(z). (2.32) Solutions of these equations are associated to factorizations of K(z) and (modulo a transformation (2.31)) have no moduli. In the case of d boundary 't Hooft operators that all have k a = 1, corresponding to minimum magnetic charge, the number of solutions is precisely 2 d . The solutions correspond simply to the possible ways to distribute the factors of K(z) between P and Q. The number 2 d has a natural interpretation. On the Coulomb branch, we might expect a minimum charge 't Hooft operator to have two possible states, with its magnetic charge being aligned or anti-aligned with the symmetry breaking. The two states correspond to a zero in P or a zero in Q. In the dual description by Chern-Simons theory with Wilson operators, a minimum charge 't Hooft operator corresponds to a Wilson operator in the two-dimensional representation of SU (2); such an operator again represents two quantum states, with positive or negative electric charge along the axis of symmetry breaking. To confirm that the 2 d solutions correspond to two possible choices of the magnetic charge for each 't Hooft operator, let us compute the magnetic charges of these solutions. Suppose that P is of degree d 1 and Q of degree d 2 , where d 1 + d 2 = d. The ratio P/Q does not depend on the normalization of the small section s. This ratio has electric charge −1 in the low energy abelian gauge theory; it is a section of the line bundle M −1 . On the other hand, concretely, P/Q has d 1 zeroes and d 2 poles on the z-plane. (P/Q has neither a pole nor a zero at z = ∞, when understood as a section of M −1 . In fact, by a complex gauge transformation, the solution can be put near z = ∞ in the form of the original Nahm pole solution (1.3), and in particular is independent of z. This trivializes M near z = ∞ and makes P/Q independent of z.) So the line bundle M has degree d 2 − d 1 and hence m = d 2 − d 1 . (2.33) From this, we see that the 't Hooft operator at z = z a contributes either −1 or +1 to m, depending on whether we place the factor of z − z a in P or in Q. To get some insight into why there are more solutions for a = 0 than there are for a = 0, consider a slightly more general ansatz for ϕ: ϕ = a p(z) 0 −a . (2.34) For a = 0, the polynomial p(z) can be removed by an upper triangular gauge transformation, but now there is a smooth limit for a = 0. The condition s ∧ ϕs = K becomes 2aP Q + pQ 2 = d a=1 (z − z a ). (2.35) For a = 0, p is irrelevant, as it can be eliminated by P → P − pQ/2a. For a = 0, the equation (2.35) has 2 d solutions (modulo a complex gauge transformation that preserves the form of ϕ), corresponding to factorizations of K(z) as (2aP + pQ)Q. But all of these solutions have P ∼ 1/a except for the one solution with Q = 1 (and P = 0) that we found already in section 2.3. We interpret this as follows. In the presence of a minimum charge 't Hooft operator at a boundary point z = z a , there is always a magnetic charge −1 localized near the boundary. The magnetic charge that is localized near the boundary is the same for all solutions because the form of the solution near the boundary is always given by a standard model solution (the one described in section 3.6 of [14]), independent of everything else. In the case of a solution of (2.32) with Q(z a ) = 0 and (therefore) P (z a ) = 0, there is in addition a smooth BPS monopole, with magnetic charge 2, located at z = z a and at a value of y that depends on P (z a ). So the total magnetic charge associated to z = z a is −1 + 2 = 1. As a → 0, P (z a ) → ∞ and the smooth monopole disappears to y = ∞. In this way, all of the 2 d solutions become equivalent for a → 0, even though they are different for a = 0. As always, such a description in terms of smooth BPS monopoles is only precise in the limit that the monopoles are located at large values of y, so that the Bogomolny equations are a good approximation near their positions. As we explain most fully in section 7, this is the case exactly when a is very small (compared to the inverse distances 1/(z a − z b ) between the 't Hooft operators) so in particular the description by BPS monopoles becomes precise for a → 0. In the opposite case that a is large compared to the inverse distances, the symmetry breaking is strong and the different singular monopoles on the boundary are so far separated that they do not significantly influence each other. In this case, the essential statement is simply that a single 't Hooft operator of minimal charge, in the presence of symmetry breaking with a = 0, has two possible states, in which it looks like an 't Hooft operator of charge 1 or −1 in the effective low energy theory. Given this, a system of d widely separated 't Hooft operators of minimal charge naturally has 2 d possible states. Implications The counting of solutions for a = 0 circumvents a difficulty that we encountered in section 2.3. With 2 d classical solutions, the physical Hilbert space in the presence of d 't Hooft operators of minimal charge will have dimension 2 d . These 2 d states are potentially dual to the states of d Wilson operators labeled by the two-dimensional representation of SU (2). And the representations of the braid group on d strands that are associated to the Jones polynomial can certainly be realized in a vector space of dimension 2 d . Given this, it is reasonable to expect that the stretching strategy sketched in fig. 2 of section 1.2 can work if a is generic, but that for a = 0, an attempt to simplify a classical solution by stretching in one direction leads to behavior along the lines suggested in fig. 4. More generally, in the presence of an 't Hooft operator on the boundary of charge k a , we would hope to find k a + 1 states of magnetic charge −k a , −k a + 2, −k a + 4, . . . , k a . Indeed, such an 't Hooft operator in SO(3) gauge theory is dual to a Wilson operator in SU (2) gauge theory associated to the representation of spin k a /2; this representation has dimension k a + 1, and its weights are as indicated. We get the right number of solutions with the right magnetic charges if in solving eqn. (2.32), we allow arbitrary factorizations of K(z) with the zeroes split between P and Q in an arbitrary fashion. Unfortunately, we do not have a simple interpretation of the factorizations for which both P and Q vanish at z = z a . After all, s = P Q is supposed to be everywhere nonzero. It seems possible that the factorizations in which P and Q have a common zero should be interpreted in terms of monopole bubbling, that is, as solutions of the equations in the presence of an 't Hooft operator of reduced charge k a − 2, k a − 4, etc. To develop this idea in detail is beyond the scope of the present paper. Going back to the simpler case that all k a are equal to 1, since the space of physical states has a promising dimension 2 d , the next step could be to try to compute the action of the braid group and to extract the Jones polynomial. Before attempting that, we will describe a further deformation of the problem, which turns out to be illuminating. Among other things, with this deformation, we will get a nice behavior regardless of the charges of the 't Hooft operators. Analysis At General t Some Basics Still searching for a useful description for 't Hooft operators of arbitrary magnetic charges, we consider the possibility of deforming the equation that we are trying to solve to one that might behave more conveniently upon stretching along one direction. A strong hint that there is a useful deformation comes by considering the relation of the extended Bogomolny equations (2.3), (2.4) to Hitchin's equations [20]. Hitchin's equations are equations on an oriented two-manifold C for a connection A and an adjoint-valued one-form φ: F − φ ∧ φ = 0 d A ⋆ φ = 0 d A φ = 0. (3.1) The moduli space of solutions of these equations, up to gauge transformation, is a hyper-Kahler manifold M H (G, C). As a hyper-Kahler manifold, M H (G, C) has a family of complex structures parametrized by a copy of CP 1 . The complex structures of M H (G, C) have a simple interpretation. They correspond to ways of splitting the three real equations in (3.1) into two parts: two real equations, which can be combined to a single complex equation; and a third real equation that is "orthogonal" to the first two. The complex equation then describes the holomorphic data parametrizing M H (G, C) in one of its complex structures, and the remaining real equation is a moment map condition. An exceptional split corresponds to the case that the complex equation is made by combining together the last two equations in (3.1) to get the Higgs bundle equation ∂ A ϕ = 0. (3.2) A more generic split involves a complex parameter ζ. For any ζ, set For generic ζ, the equation (3.3) simply says that the complex connection A ζ = A ζ z dz + A ζ z dz is flat. Once the equation (3.3) is expressed in terms of A ζ , it has no explicit dependence on ζ. Thus, M H (G, C) when regarded simply as a complex manifold in complex structure I ζ , without worrying about its Kahler metric, is independent of ζ for generic ζ. The exceptional values of ζ are 0 and ∞. For example, for ζ → 0, we must multiply D ζ z by ζ, whence it reduces to −[φ z , · ]. Meanwhile, for ζ = 0, D z = D z . So the ζ → 0 limit of eqn. (3.4) is the Higgs bundle equation (3.2). By similar reasoning, the ζ → ∞ limit of (3.4) is the complex conjugate of (3.2). The Higgs bundle equation has played a prominent role in our analysis because in section (2.1), we derived the holomorphic data at y = 0 from the equation [D 1 , D 2 ] = 0, which was none other than the Higgs bundle equation. (The moment map condition in that analysis condition was not the usual moment map equation for Higgs bundles -namely the first equation in (3.1) -but rather it was the three-dimensional equation (2.4).) At this point, we are led to wonder whether we could modify the construction that led to (2.3) so that the holomorphic data at y = 0 will be given not by a Higgs bundle but by the solution of a different complex linear combination of Hitchin's equations. It is in fact possible to do so. Indeed the path to doing so is not quite uniquely determined. D ζ z = D Dz − ζ −1 [φ z , · ] D ζ z = D Dz + ζ[φ z , · ]. There is one particularly nice parameter by which we can vary the underlying fourdimensional equations. This is simply the parameter t in eqn. (1.1). In addition to changing the equations, we should consider the possibility of changing the boundary conditions. Of the six fields ( A, φ) (that is, the components of A and φ that are tangent to the boundary), only φ has a singularity at y = 0 in the basic Nahm pole solution (1.3). However, once one drops the dependence on the spatial coordinates x, N = 4 super Yang-Mills theory has an SO(6) symmetry that rotates the six components of A and φ. Therefore, one can obey the classical Yang-Mills equations with a solution obtained by applying an SO(6) rotation to the Nahm pole (such a rotated Nahm pole was studied in [19], section 4). Of course, the rotation in general will change the unbroken supersymmetry. It turns out, however (see Appendices A and B for more detail), that as long as the rotation matrix is contained in a certain SU (3) subgroup of SO(6), the unbroken supersymmetry is unchanged, and if it is contained in a certain U (3) subgroup, then the unbroken supersymmetry changes in a way that corresponds to a change in the parameter t in the four-dimensional equation (1.1). So there is considerable freedom in rotating the boundary condition, with or without a change in t. After making a choice along the lines just described, the deformed equations and boundary conditions have a three-dimensional reduction that takes the familiar form of (2.1) and (2.2), but with a different definition of the D i . Deferring most of the details to the appendices, we will summarize some formulas that arise in an illuminating special case. If one wishes to preserve the SO(3) symmetry of rotations of the boundary, then the rotation of the Nahm pole can only depend on a single parameter: the polar part of A must be a constant ζ times the polar part of φ. It turns out that such a boundary condition is compatible with the four-dimensional equations (1.1), with a modified value of t that depends on ζ. In the reduction to three dimensions, we must now impose A 1 − ζφ 1 = 0 = φ y . The three commuting differential operators D i then take the form D 1 = 2 D Dz + 2ζ[ϕ, · ] D 2 = −2ζ D Dz + 2[ϕ, · ] D 3 = D Dy . (3.5) Here, as described in the appendix, A y is an appropriate linear combination of A y and φ 1 . The condition that the D i commute must be supplemented by a moment map condition, which is also described in the appendix. As one would anticipate for a geometry that preserves four supercharges, the commutativity constraint [D i , D j ] = 0 can be derived from a superpotential, which in fact is a multiple of the Chern-Simons function: W = 1 4πi W Tr A ∧ dA + 2 3 A ∧ A ∧ A . (3.6) To be more exact, in varying W to derive the equations [D i , D j ] = 0, one imposes a constraint on the variation of A at y = 0, so as to avoid delta function terms in the variation of W. The holomorphic data away from y = 0 are now easy to describe. We recover the picture studied in section 2 if ζ = 0, but as soon as ζ is nonzero (and not infinite), the three commuting operators D i simply describe a complex flat connection on C × R + . The covariant derivatives for this flat connection are D z = D 1 /2, D z = −D 2 /2ζ, D y = D 3 . So commuting operators D i simply describe a complex flat connection. We will call the flat connection A, irrespective of how it was defined in terms of A and φ. A is constrained by a moment map condition, which does not quite coincide with the most commonly studied moment map condition for a complex flat connection [21], though it is qualitatively similar; we expect it to have a unique solution for any ζ. The holomorphic data away from y = 0 are hence simply a complex flat connection on C × R + , or equivalently, since this space is contractible to C, a complex flat connection on C. We will mainly be interested in the case that C is R 2 or CP 1 . In either case, C is simply-connnected, so a complex flat connection on C is trivial. One may wonder therefore how anything of interest can happen. The answer is that most of the structure of interest will come from the boundary condition at y = 0. In the case of symmetry breaking, there is also some interesting structure in the behavior at y = ∞. Nahm Poles and Opers The first point is to understand how a complex flat connection can have a Nahm pole. The answer is that the pole appears in A y and A z , but not in A z . The model example of a flat connection with a Nahm pole is A z = t + y A z = 0 A y = t 3 y . (3.7) This describes a flat connection as long as [t 3 , t + ] = t + . We are interested in the case that t + = t 1 + it 2 , where the t i , i = 1, 2, 3 generate a principal su(2) subalgebra of g. Being flat, this connection can be described by a formula d + A = gdg −1 . For example, for su(2), we can take explicitly g = y −1/2 −zy −1/2 0 y 1/2 ,(3.8) which leads to A z = 0 1 0 0 1 y A z = 0 A y = 1 0 0 −1 1 2y . (3.9) Alternatively, the model solution can be generated from the non-singular flat connection A z = 0 1 0 0 , A z = A y = 0 (3.10) by the same singular gauge transformation as in (2.10): g = y −1/2 0 0 y 1/2 . (3.11) We now proceed rather as we did in the Higgs bundle case to explain the condition that must be placed on a complex flat bundle E so that it can be placed in the form (3.9) near y = 0, modulo less singular terms. We write simply E for the restriction of E to C = C × y for some fixed y > 0. We consider solutions s of the equation D y s = 0 that vanish as y 1/2 for s → 0. Sections s obeying these conditions span a rank one sub-bundle L ⊂ E. In the case of the model solution, any such s is a multiple of s = y 1/2 0 1 ,(3.12) so as in the Higgs bundle case, L is simply spanned by sections whose upper component vanishes. Also as before, if we regard E as a flat bundle over C, then L is a holomorphic subbundle; indeed, the object s that we have just defined is a holomorphic section of L, since it is certainly annihilated by D z . However, it is not true that s is annihilated by D z . On the contrary, a look at the previous formulas shows at once that s ∧ D z s = 1. (3.13) This brings us to the mathematical notion of an "oper." (For an explanation of this notion as well as a review of many related ideas that will enter our story later, see [22] or [23].) For G = SU (2), an oper is a flat rank two complex bundle E bundle over a Riemann surface C, with structure group SL(2, C), together with a holomorphic line sub-bundle L ⊂ E with the following property: L is nowhere invariant under parallel transport by D z . The last statement means the following. If s is a local nonzero holomorphic section of L, then D z s, which will be E-valued since D z is a connection on E, is nowhere L-valued. An equivalent statement, since L is spanned by multiples of s, is that D z s is nowhere a multiple of s. Alternatively, s ∧ D z s is everywhere nonzero. The last statement does not depend on the choice of the nonzero section s, since if we replace s by f s (where f is a nonzero local holomorphic function on C), we have s ∧ D z s → f 2 s ∧ D z s. We have extracted an oper structure from the Nahm pole boundary conditions; conversely let us see that given an oper on C, that is a pair (E, L) obeying the conditions just described, we can construct a solution of the Nahm pole boundary conditions. Go to a gauge in which L is spanned by vectors whose upper component vanishes. Holomorphy of L means that A z is lower triangular in this gauge. The oper condition s ∧ D z s = 0 for any nonzero local section of s implies that in this gauge, the upper right matrix element of A z is nonzero. By a further diagonal gauge transformation, we can set this matrix element to 1. Then we pull back the flat bundle E with connection A from C to C × R + (to get a flat connection on C × R + with no dependence on y) and make the singular gauge transformation (3.11). Having an upper right matrix element of 1 means that after the singular gauge transformation, A z has the singular behavior of the model solution (3.9); being lower triangular, A z acquires no singularity. Finally, the gauge transformation gives A y precisely the form of the model solution. So we have shown, at least for the case that G has rank 1, that two-dimensional opers correspond precisely to solutions of the Nahm pole boundary conditions in three dimensions modulo less singular terms. Conjecturally, by a further smooth complex-valued gauge transformation, one can satisfy the moment map condition. Some Further Remarks We add the following technical remarks. Since we want to be able to consider 't Hooft operators of minimum charge, we will take the gauge group in the rank 1 case to be G = SO(3), rather than SU (2). Accordingly, we should restate the above derivation in terms of the adjoint bundle ad(E) rather than E. Because our considerations have been local on C, rewriting the construction in terms of the adjoint bundle does not change very much and we will omit it. (The main difference is that what can be naturally defined globally is in general not L but L 2 , which is a sub-bundle of ad(E).) Also, everything we have said for G of rank 1 has an analog for any semi-simple G, somewhat as we indicated in the Higgs bundle case at the end of section 2.2. For example, for G = SU (n), an oper is a flat complex bundle E → C of rank n with SL(n, C)-valued holonomies together with a line sub-bundle L ⊂ E that is holomorphic and has the property that if s is a local nonzero holomorphic section of L, then s, D z s, . . . , D n−1 z s furnish a local trivialization of E. The equations [D i , D j ] = 0 together with the Nahm pole boundary condition determine such an oper structure, by arguments similar to those we have already given. Opers With Singularities Now we would like to modify the Nahm pole boundary condition to incorporate additional singularities -which we will associate with 't Hooft operators -on the boundary at y = 0. The type of singularity that we want can be guessed by analogy with the discussion of Higgs bundles. We will still have a flat G C bundle E → C×R + , and near a generic boundary point, the flat connection will look like the model solution (3.7), up to a unitary (that is, G-valued rather than G C -valued) gauge transformation. We can still define a holomorphic line sub-bundle L ⊂ E by considering sections s obeying D y s = 0 and vanishing for y → 0; and we still require that if s is a local holomorphic section of s, then s ∧ D z s is generically nonzero. The only difference is that now we assume the existence of exceptional points z a , a = 1, . . . , d, at which s ∧ D z s vanishes. In fact, we specify positive integers k a and require that s ∧ D z s vanishes in order k a for z → z a : s ∧ D z s ∼ (z − z a ) ka . (3.14) This is analogous to requiring s ∧ ϕs ∼ (z − z a ) ka in the Higgs bundle case. To get a precise problem of classical or quantum gauge theory with this sort of boundary behavior, what remains is to specify precisely what sort of singularity a solution of the moment map condition i [D i , D † i ] = 0 (or the four-or five-dimensional equations that can be dimensionally reduced to it) is supposed to have at z = z a . As in most such problems, to do this one finds a model solution for the case of only one singularity, at, say, z = 0 (and y = 0) and with an arbitrary k. Then one asks that the singular behavior of the solution near each of the points z = z a , y = 0 should coincide with that of the model solution, for k = k a . For the case of Higgs bundles, the appropriate model solutions were found (for G of rank 1) in section 3.6 of [14], but for the generalization considered here, at present we are only able to find the model solutions numerically. They are described in Appendix C. The objects that we have described so far correspond to solutions of the flatness and moment map conditions on C × R + with boundary conditions associated to Nahm poles or opers, except at finitely many boundary points where the oper condition is corrected. In particular, as soon as one gets away from y = 0, one simply has a flat bundle (with a moment map condition). The monodromy of the flat bundle around the points z = z a is therefore trivial, so the exceptional behavior at the points z = z a only affects the oper property of the pair (E, L), not the flatness of E. Singularities of this kind are called oper singularities with trivial monodromy. Oper Singularities And Bethe Equations Let us now make concrete (referring to [22] for much more detail) what sort of an object is an oper with monodromy-free singularities. We will make this analysis for the case that C is simply-connected, so that there are no moduli in the choice of the flat bundle E → C. So C will be either R 2 or CP 1 ; that is, it will be the complex z-plane with or without an added point at infinity. There are two reasons for assuming C to be simply-connected: this is the most relevant case for understanding the Jones polynomial; and also, eliminating the choice of E from the discussion will make it easier to focus on the opers and their singularities. Since C is simply-connected, a flat bundle over C is trivial. So we can go to a gauge with A z = A z = 0. The line sub-bundle L of E is inevitably trivial for C = R 2 , or trivial after omitting the point z = ∞ for C = CP 1 . So it is globally generated by a section s, but we cannot put s in a simple form while also setting A z = A z = 0. Instead, we take s = P (z) Q(z) ,(3.15) with polynomials P and Q. P and Q are uniquely determined up to a linear transformation P Q → M P Q , M ∈ GL(2, C). (3.16) Now s ∧ D z s reduces to P ∂ z Q − Q∂ z P . So if the polynomial K(z) = d a=1 (z = z a ) ka encodes the positions and charges of the oper singularities, then the equation we would like to solve is P ∂ z Q − Q∂ z P = K(z), (3.17) modulo the action of SL(2, C). (Choosing K to have leading coefficient 1 has reduced GL(2, C) to SL(2, C).) It is convenient to fix the action of two of the three generators of SL(2, C) by requiring that the degree of the polynomial Q is less 6 than the degree of P (if two polynomials have the same degree, a linear combination of them has smaller degree and we call this Q), and that Q has leading coefficient 1, Q(z) = q i=1 (z − w i ),(3.18) for some w i . These conditions leave only the freedom to add to P a multiple of Q. We can recast (3.17) as ∂ z P Q = − K(z) Q 2 (3.19) The left hand side of this equation has zero residues at the zeroes w i of Q(z). The right hand side must also have zero residues. This gives the constraints a k a w i − z a = j =i 2 w i − w j , i = 1, . . . , q. (3.20) Vice-versa, given a solution of these equations, the residues of K/Q 2 are zero; hence K/Q 2 dz is a rational function P/Q, and P is fixed up to a constant multiple of Q, which is the expected indeterminacy. If C = R 2 , opers with the desired monodromy-free singularities simply correspond to the solutions of the equations (3.20). For C = CP 1 , we must further ensure that the oper does not have an additional singularity at infinity. The condition for this turns out to be that the degree q of the polynomial Q is just one-half of the degree of K: q = k 2 , k = a k a . (3.21) To determine whether the oper has a singularity at infinity, let p be the degree of P and define P Q = z −p P Q . We view P , Q as polynomials in v = 1/z. The condition that the degree of P exceeds the degree q of Q implies that there is no cancellation of the leading power of z on the left hand side of (3.19) and hence that p + q = k + 1. Since q < p, it follows that q ≤ k 2 (3.22) in general. The condition that the oper has no singularity at z = ∞ or v = 0 is that P ∂ v Q − Q∂ v P v=0 = 0, and it is not hard to see that this coincides with (3.21). To get farther, we need the theory of integrable systems. Rather "miraculously," the equations (3.20) are the Bethe equations of an integrable model, which is the Gaudin model or a certain large impurity limit of the XXX spin chain. (The connection between opers with monodromy-free singularities and the Gaudin model is reviewed in [22], following earlier developments such as [24,25]. For more on this, see section 8.6.) For a = 1, . . . , d, let R a be a copy of the representation of SU (2) of spin j a = k a /2, and let H = ⊗ d a=1 R a . The Hamiltonians of the Gaudin model are the commuting operators on H given by H a = b =a T a · T b z a − z b . (3.23) Here T a are the generators of su(2) acting on R a , and T a · T b is the inner product of T a and T b (defined with the quadratic form such that T a · T a = j a (j a + 1)). Actually, what we have written in (3.23) are the Hamiltonians for the Gaudin model for G ∨ = SU (2). (We call this group G ∨ as it is naturally dual to the gauge group G that appears in the rest of our analysis.) There is a Gaudin model for any G ∨ , and it bears the same relation to opers that we are about to describe for SU (2), but if G ∨ has rank bigger than 1, then the H a are only part of a complete set of commuting Hamiltonians. Since the Gaudin Hamiltonians commute, they can be simultaneously diagonalized. Moreover, since they commute with the action of G ∨ , their joint eigenvectors can be organized in irreducible representations of G ∨ . Because of the G ∨ action, to understand all of the joint eigenvectors of the Gaudin Hamiltonians, it suffices to understand those joint eigenvectors that are also highest weight vectors for the action of G ∨ . In the theory of the Bethe ansatz for the Gaudin model of G ∨ = SU (2), it is shown that solutions of the Bethe equations (3.20) correspond to the joint eigenvectors that are also highest weight vectors for the action of G ∨ . In this correspondence, the weight w is related to the degree q of Q by w = k 2 − q. (3.24) In particular, if we want G ∨ -invariant joint eigenvectors of the Gaudin Hamiltonians, we need w = 0 and q = k/2; but as we observed in (3.21), this is the condition that the corresponding oper extends over CP 1 with no singularity at infinity. So the number of opers on CP 1 with monodromy-free singularities is the same as the number of SU (2)-invariant joint eigenvectors of the Gaudin Hamiltonians. But the joint eigenvectors of the commuting Gaudin Hamiltonians are a basis for H, and similarly the G ∨ -invariant joint eigenvectors are a basis for H G ∨ , the G ∨ -invariant part of H. So the number of opers that obey the conditions that we have imposed is precisely the dimension of H G ∨ . This result is our first concrete success in comparing the counting of BPS solutions in G gauge theory to Chern-Simons theory with gauge group G ∨ . Consider Chern-Simons theory on CP 1 with charges in the representations R a , a = 1, . . . , d. We place these charges at points z a ∈ CP 1 . In the classical limit, the space of physical states is just the G ∨ -invariant part of H = ⊗ a R a ; the restriction to G ∨ -invariant states is the Gauss law constraint. This also gives the right answer for the dimension of the physical Hilbert space of Chern-Simons theory if the Chern-Simons coupling parameter k ∨ is generic. On the other hand, in the dual description in which the Hilbert space is constructed starting with time-independent solutions in G gauge theory, the states should correspond, 7 from the arguments we have given, to opers on CP 1 with singularities of charge k a at the z a . Since the number of these opers is the same as the dimension of H G ∨ , we have at least succeeded in reconciling the dimensions of the spaces of physical states in the two descriptions. This gives an indication that with the help of the deformation that we have exploited in the present section to ζ = 0, the program of section 1.2 based on stretching a knot in one direction can actually work. The counting of states is less transparent if we take C = R 2 rather than CP 1 . Qualitatively, it is clear that the number of physical states in Chern-Simons theory is larger on R 2 than on CP 1 , because, as the flux can escape to infinity, a physical state need not be completely gauge-invariant. However, to understand the condition that should be imposed at infinity is rather delicate, and it is hard to understand in G ∨ Chern-Simons theory the result that seems to come from the opers: physical states on R 2 correspond to highest weight vectors in H. After incorporating symmetry breaking in section 3.5, the comparison between the two descriptions will be simpler. Relation To Conformal Field Theory In arriving at the Gaudin model, we have accomplished much more than simply getting a number of classical solutions that is reminiscent of known constructions of the Jones polynomial. The Jones representations of the braid group can be described [26] as the monodromy of the Knizhnik-Zamolodchikov equations [27]. These equations are as follows. Express the usual parameter q that enters the Jones polynomial as q = exp(2πi/(k ∨ + 2)). Let B be the space of distinct d-plets z 1 , . . . , z d ∈ C. And let H * be the trivial bundle over B with fiber H = ⊗ d a=1 R a . The Knizhnik-Zamolodchikov equations are the following system of equations for a section Θ of H * : ∂ ∂z a + H a k ∨ + 2 Θ = 0. (3.25) The H a are the Gaudin Hamiltonians (3.23). The Knizhnik-Zamolodchikov equations describe parallel transport of the section Θ of H * with respect to a certain flat connection, which is implicitly defined in (3.25). To verify flatness of the connection, one uses the fact that the H a commute and also the relation ∂H b /∂z a = ∂H a /∂z b . The solutions Θ of the Knizhnik-Zamolodchikov equation are conformal blocks of two-dimensional current algebra with symmetry group G ∨ ; they are important in two-dimensional conformal field theory. Since opers with monodromy-free singularities correspond to a basis for H, we will, in our approach to the Jones polynomial, eventually be using gauge theory to construct a flat connection on the bundle H * ; moreover, as we hope to recover the Jones representations of the braid group, this flat connection should be gauge-equivalent to the one defined by the Knizhnik-Zamolodchikov equations. Actually, this tempting-sounding route is not the one we will follow. The very same Jones representations of the braid group have another realization in conformal field theory in terms of Virasoro conformal blocks for correlators of a product of degenerate fields and this will prove more useful. Symmetry Breaking Again Just as in section 2.4, we can gain some further clarity by moving away from the origin of the Coulomb branch. As always, we do so by turning on constant and commuting expectation values for φ near y = ∞. In the present context, this means that the connection form A does not vanish at infinity, but is a one-form with constant coefficients; moreover, these coefficients commute with each other. "Real" Symmetry Breaking First we consider the case that only A y has an expectation value at infinity. This expectation value arises from the value of φ 1 at infinity, and so just as in (2.24) we have A y = a 1 0 0 −a 1 , y → ∞,(3.26) where we can take a 1 > 0. The condition that A z and A z should have no exponential growth at infinity tells us that they must be upper triangular in a gauge in which A y looks like (3.26) for y → ∞. Thus, near y = ∞, the real symmetry breaking gives us a natural way to put the whole flat connection in a triangular form. An oper that is endowed with a covariantly constant reduction of its structure group to the group of upper triangular matrices -that is, to a Borel subgroup -is called a Miura oper. This notion is described in detail in [22]. Any oper bundle without monodromy can be given a Miura oper structure; in fact, there is a one-parameter family of ways to do so. Concretely, if the rank two flat bundle E → C has trivial monodromy, then the associated bundle of CP 1 's (whose fibers are obtained by projectivizing the fibers of E) also carries a flat connection without monodromy. Let us call this bundle B. Picking an arbitrary section of B over some given point p ∈ C and parallel transporting it, we get a covariantly constant section of B which turns the underlying oper into a Miura oper. This procedure introduces one complex modulus -the choice of a point in the fiber of B over the starting point p. This means that a Miura oper without monodromy depends on a complex modulus. (When -as in section 3.5.2 -we introduce symmetry breaking in a complex direction, this modulus will disappear, because there will be no freedom to make a gauge rotation of A y at infinity relative to A z .) The Bethe roots have a particularly nice interpretation in the case of a Miura oper. To explain this most simply, let us go back to the case that C = R 2 and use a gauge with A z = A z = 0. The behavior for y → ∞ singles out a sub-bundle L of the rank 2 bundle E that is invariant under parallel transport; we may call it a flat sub-bundle. In a gauge with the asymptotic behavior (3.26), L is generated by a covariantly constant section s that vanishes for y → ∞. After a complex gauge transformation to set A z = A z = 0, s is simply constant; we may as well take s = 1 0 . (3.27) On the other hand, the behavior for y → 0 determines a holomorphic (not flat) sub-bundle L ⊂ E, generated as before (in a gauge A z = A z = 0) by s = P Q . (3.28) The choice of the Miura structure s gives a way to pick a natural linear combination of the components of s, namely s ∧ s = Q. (3.29) The Q determined this way is not necessarily the one that we used in section 3.4, where we took Q to be a linear combination of components of s with minimum degree; now Q is simply determined by the Miura structure. With our new choice, the zeroes of Q have a simple interpretation: they are the points at which s is a multiple of s. In other words, the sub-bundle determined by the behavior for y → ∞ is generically different from the sub-bundle determined by the behavior for y → 0. The zeroes of Q -which are called Bethe roots -are precisely the points at which these coincide. Another way to say the same thing is that the Bethe roots are the values of z at which there is a solution of D y s = 0 that vanishes for both y → 0 and y → ∞. As we discussed in section 2.4, in the context of the Bogomolny equations one would say that there are smooth BPS monopoles at those values of z (and some values of y). For a concrete example, suppose that there are no 't Hooft operators at all. The flat bundle E → R 2 is completely trivial and it has up to isomorphism a unique oper structure with s = z 1 . (3.30) In the absence of symmetry breaking, our convention that Q is the linear combination of components of s with smaller degree leads to Q = 1, and hence (up to the freedom of adding to P a multiple of Q and rescaling it) P = z. In the presence of real symmetry breaking, the Miura structure gives a distinguished choice (3.29) of Q which has no reason to be a constant. If Q is not constant, then by adding to P a multiple of Q and rescaling it, we can set P = 1, so s = P Q = 1 z − w , (3.31) for some w. The polynomial K = P Q ′ − QP ′ is 1, consistent with the absence of any 't Hooft operators. We have found, in the presence of real symmetry breaking, holomorphic data corresponding to a one-parameter family of solutions depending on the choice of a point w ∈ R 2 . This is a solution with no 't Hooft operator and a single Bethe root. We will call it a bare Miura oper. In the right context, when lifted back to four dimensions, we will interpret this solution later as a "string" that is localized at z = w and at a value of y that depends on ζ. In general, the degree of K is at most one less than the degree of Q (this bound is achieved precisely if P = 1), so the number of Bethe roots is at most one more than the degreee k = a k a of K. "Complex" Symmetry Breaking If we give expectation values at infinity to all components of φ, while requiring A to vanish at infinity, then the complex connection A is constant and diagonal for y → ∞, A z ∼ 1 ζ a 0 0 −a A z ∼ ζ a 0 0 −a A y ∼ a 1 0 0 −a 1 ,(3.32) where a is a complex number. The factors of ζ arise in the change of variables from ϕ to A. At infinity in the z direction, for any y, the solution reduces to the unique solution [18] of Nahm's equations which has a Nahm pole at y = 0 and behaves as (3.32) at y = ∞. In particular, the connection form is constant for z → ∞ with fixed y. For this form of the connection, we can write the small holomorphic section s as s = e −ζa ′ z 0 0 e ζa ′ z s 0 (3.33) where a ′ (which equals a for y → ∞) is the constant value of A z /ζ at large z with fixed y, and s 0 has a finite limit at large z and fixed y. In order to analyze the holomorphic data in such solutions, it is unnatural to gauge the connection away. The information we want would be hidden in the behavior of the necessary gauge transformation at infinity. Rather, we will pick a complex gauge transformation which brings the connection exactly (not just asymptotically) to the form A z = 1 ζ a 0 0 −a A z = ζ a 0 0 −a A y = a 1 0 0 −a 1 (3.34) As we will now see, this can be done by a relatively simple type of gauge transformation. Consider any gauge field on R 2 × R + , such as one that has the asymptotic form discussed above, such that the connection form approaches a nonzero constant for z → ∞ with fixed y. If we were to compactify R 2 to CP 1 , we would say that such a connection has an irregular singularity at z = ∞ with a double pole. Connections with irregular singularities have Stokes phenomena. If one only considers gauge transformations with no essential singularities (that is, with polynomial growth only at infinity), the Stokes data is gaugeinvariant. Two flat connections with an irregular singularity are gauge-equivalent by a gauge transformation with only polynomial growth at infinity if and only if they have the same monodromy and the same Stokes data. As our connection has only has a double pole at z = ∞, it has only two Stokes sectors. Stokes theory tells us that the monodromy around z = ∞ can be decomposed into the product of a diagonal formal monodromy matrix, and a sequence of Stokes matrices, which are alternatingly upper and lower triangular with ones on the diagonal. With only two Stokes sectors, the expression for the monodromy is M = 1 b 0 1 1 0 b 1 µ 0 0 µ −1 ,(3.35) with constants b, b, and µ. As we are on R 2 , which is simply-connected, the monodromy at infinity must be M = 1. This together with the form (3.35) of the monodromy implies that the three factors in (3.35) -the Stokes matrices and the formal monodromy -must all equal 1. Hence we can bring our connection to the constant diagonal form (3.34) by a gauge transformation which grows only polynomially at infinity. In particular, s will now take the form s = e −ζa z 0 0 e ζa z P (z) Q(z) (3.36) with polynomials P and Q. The equation s ∧ D z s = K(z) gives (P ∂ z Q − Q∂ z P ) − 2aP Q ζ = K(z) (3.37) which can be converted to the requirement that Ke 2az /Q 2 has no residues at the zeroes w i of Q(z). This becomes 2a ζ + a k a w i − z a = j 2 w i − w j (3.38) We are interested in solutions of (3.37) modulo a rescaling P → λP , Q → λ −1 Q (which corresponds to an automorphism of the diagonal flat connection), so we actually only care about the zeroes of Q(z). These are, again, Bethe equations, this time for the Gaudin model with an irregular singularity at z = ∞ [28][29][30]. It is shown in [30] that solutions of these Bethe equations are in one-to-one correspondence with eigenvectors of the spin chain, making in all a (k a + 1) solutions. It may also be possible to extract this result from the theory of the XXX spin chain, which is more familiar than the Gaudin model. The Bethe equations of the Gaudin model arise from a specific "large impurity limit" of those of the XXX chain. The deformation parameter a maps to the twist of the XXX chain, which is known to break the global SU (2) symmetry to U (1), and simplify the counting of solutions to Bethe equations: instead of a solution for each eigenvector of the spin chain Hamiltonian that is an SU (2) highest weight, one gets a solution for each eigenvector. As a simple example, consider the case of a single 't Hooft operator of charge 2, so K(z) = z 2 . Just as in section 2.5.1, there is a solution of (3.37) with P = 1 and Q a quadratic polynomial in z, and a solution with Q = 1 and P a quadratic polynomial. What happens if P and Q are both linear in z? Again as in section 2.5.1, we can solve (3.37) with P = Q = z −ζ/2a, but this solution does not correspond to an oper, since P and Q have a common zero. The novelty is that there is also an acceptable solution with P = −(ζ/2a)(z − ζ/a), Q = z + ζ/a, corresponding to a solution of the Bethe equations (3.38) with a single Bethe root. This gives a total of 2 + 1 = 3 opers obeying the necessary conditions. The last solution disappears if we take a = 0 and has a common zero for P and Q if we take ζ → 0. Opers And Stress Tensors In our analysis of opers with trivial monodromy, we have used a gauge in which the connection is trivial, A z = A z = 0. Correspondingly, we had to make a general ansatz for the small section s that generates the holomorphic sub-bundle L ⊂ E: s = P Q . (3.39) Here we will make a gauge transformation to put s in the standard form with upper component vanishing, and see what we can say about A z . This will have two benefits. We will begin to understand the relation of opers to conformal field theory. And we will get a description that is more general, not limited to the case (which however is particularly important in the present paper) of oper bundles of trivial monodromy. To put s in a simple form by a smooth gauge transformation would make A z and A z both nonzero. It turns out to be more helpful to keep A z = 0. To also keep A z regular would require that our gauge transformation should be holomorphic, which is too restrictive a condition. Instead we will consider meromorphic gauge transformations, which will keep A z = 0, put s in a standard form, and generate poles in A z . The most obvious meromorphic gauge transformation that puts s in a standard form is h = Q −P 0 Q −1 . (3.40) We have hs = 0 1 h∂ z h −1 = ∂ z + − Q ′ Q −K 0 Q ′ Q , (3.41) where as before K = P Q ′ − QP ′ . It turns out to be more convenient to go to a gauge in which the upper right matrix element of A z is −1, at the cost of mapping s to a multiple of itself (this leaves unchanged the line bundle generated by s). We make a further gauge transformation by h = 1/ √ K 0 0 √ K . (3.42) The possible double-valuedness of √ K is of no concern, for the following reason. If G ∨ = SO(3), so that G = SU (2), then K is a perfect square as all its zeroes are of even degree. If instead G ∨ = SU (2), then G = SO(3), and we should really be writing all formulas in the adjoint representation, rather than the two-dimensional representation; accordingly, the sign of a gauge transformation is irrelevant. After a gauge transformation by h, A z takes the form A z = −v −1 0 v (3.43) where we have set v = − K ′ 2K + Q ′ Q = − a k a /2 z − z a + i 1 z − w i . (3.44) In the last step, we used K = a (z − z a ) ka , Q = i (z − w i ). Finally, a lower triangular gauge transformation 1 0 v 1 (3.45) leads to our final result for A z : A z = 0 −1 t 0 , (3.46) with t = −v ′ − v 2 . (3.47) In general, t has poles at both the z a and the w i . Near z = z a , t ∼ − j a (j a + 1) (z − z a ) 2 + c a z − z a + . . . , j a = k a /2. (3.48) Near z = w i , t ∼ 1 z − w i   a k a w i − z a − j =i 2 w i − w j   . (3.49) Thus, t has no singularity at z = w i if and only if the Bethe equations (3.20) are satisfied. To get some more insight, set D z = ∂ z + [A z , · ] and look for a flat section, that is a holomorphic solution of D z f f = 0. We find that f = f ′ and ∂ 2 ∂z 2 + t(z) f = 0. (3.50) We have carried out this derivation using a particular local coordinate z, but the notion that we started with -a flat bundle with an oper structure -did not depend on the local coordinate. So eqn. (3.50) must be covariant under a change of the local coordinate, with a suitable transformation for t. A short calculation (or a more careful study of the above derivation) shows that under a change of local coordinate z → z, and a suitable transformation of t, the object t transforms like a stress tensor in two-dimensional conformal field theory. In other words, it transforms not as a quadratic differential, as one might naively think from its pairing with the second derivative ∂ 2 /∂z 2 in (3.50), but with an "anomalous" term involving the Schwarzian derivative 1 2 {z, z}. The double pole in t at z = z a is as if there is a primary field inserted at z a . If we had started from A z = ζ −1 diag(a, −a) and used the same sequence of gauge transformations, with K = P Q ′ − QP ′ − 2aζ −1 P Q, we would have arrived to the same formulas, but with an extra constant term in v: v = − a ζ − a k a /2 z − z a + i 1 z − w i . (3.51) Now t(z) has a pole of order four at z = ∞. This would correspond in conformal field theory to the insertion at infinity of a somewhat unusual operator [36]. Finally, the Bethe equations have the following interesting property. They describe stationary points at fixed z a (and a) of a Yang-Yang function: W(w i , z a ) = − i<j log((w i − w j ) 2 ) + i,a k a log(w i − z a ) − 1 4 a<b k a k b log((z a − z b ) 2 ) + 2a ζ i w i − a ζ a k a z a (3.52) In other words, the Bethe equations can be written as ∂W ∂w i = 0. (3.53) The Yang-Yang function has another interesting property: the coefficients of the single poles in t at z = z a -sometimes called accessory parameters -are given by c a = ∂W ∂z a . (3.54) Some terms in W which are independent of the w i have been included to insure that this relation is satisfied. A similar relation holds for a∂ a W. Opers and Virasoro Conformal Blocks In this section, we will show how these formulae arise naturally in the semiclassical limit of Virasoro conformal blocks. The semiclassical limit is defined as a limit in which the central charge c of the Virasoro algebra goes to infinity, while the conformal dimensions of operators also scale in the same way as c. There is a useful way to parametrize the central charge: c = 1 + 6Q 2 , with Q = b + b −1 . In this parametrization, we take b → 0 to get a semiclassical limit. The conformal dimensions of operators are conveniently parametrized as ∆ = α(Q − α). The parameter α is often referred to as "momentum." We keep bα = η fixed as b → 0. Then the insertion of an energy-momentum tensor T in a correlation function scales as b −2 , and we can define the finite limit t = b 2 T . We propose to identify this t, inserted in certain conformal blocks, with the t of section 3.6. The quantum stress-tensor T (z) has an anomaly under conformal transformations; it shifts by a multiple c 12 {z, z} of the Schwartzian derivative. Hence t = b 2 T has a conformal anomaly that is independent of b for b → 0. The behavior of t near z = z a in the previous section corresponds to the behavior near a Virasoro primary field with α a = − ka 2b . These operators are very special: correlation functions and conformal blocks which involve only operators of this type can be described very easily by a free-field realization. We can describe such a realization in close parallel to the discussion in the previous section. Let χ be a two-dimensional free field with two-point function χ(z)χ(z ′ ) = − 1 2 ln(z − z ′ ). A standard way to construct an energy-momentum tensor of central charge c = 1+6Q 2 is to take T = − : ∂χ∂χ : +Q∂ 2 χ. (3.55) If we define v = −b∂χ, this definition reduces to (3.47) in the limit b → 0. This is the first hint that a free-field realization can be useful for us. Operators of dimension ∆ = α(Q − α) can be readily described as normal-ordered exponentials of the free boson, V α (z) =: e 2αχ(z) : . (3.56) A second hint comes from (3.44): the quantity v is the semiclassical limit of the expectation value of −b∂χ in the presence of chiral vertex operators of momenta −k a /2b at z = z a and of momenta 1/b at z = w i . The operators V 1/b (w i ) have dimension 1. They are usually called screening operators in free-field realizations [32,33], and are naturally integrated over curves. The singular part of the stress tensor near the location of a screening operator T (z)V 1/b (w i ) ∼ 1 (z − w i ) 2 V 1/b (w i ) + 1 z − w i ∂ w i V 1/b (w i ) + · · · = ∂ w i 1 z − w i V 1/b (w i ) · · · (3.57) is a total derivative, and drops off upon integrating over the position of the screening operator. We can easily compute the following free field correlation function: i V 1/b (w i ) a V −ka/2b (z a ) free = i<j (w i − w j ) − 2 b 2 i,a (w i − z a ) ka b 2 a<b (z a − z b ) − 1 2b 2 kak b = exp 1 b 2 W(w, z) . (3.58) The exponent on the right is the Yang-Yang function (3.52)! (For the moment, the terms proportional to a are absent as we have not included symmetry breaking.) Consider the integral a V −ka/2b (z a ) Γ = Γ i V 1/b (w i ) a V −ka/2b (z a ) free i dw i , (3.59) where Γ is any integration cycle for which the integral converges. Because of (3.58), this is equivalent to a V −ka/2b (z a ) Γ = Γ exp(W(z a , w i )/b 2 ) i dw i . (3.60) A Virasoro conformal block for the expectation value of a product of primary fields is a candidate correlation function that is compatible with the Virasoro Ward identity: T (z) a V −ka/2b (z a ) Γ = a ∆ a (z − z a ) 2 + 1 z − z a ∂ ∂z a a V −ka/2b (z a ) Γ (3.61) The functions defined in (3.59) or (3.60) have this property for an arbitrary choice of the number of w's and the integration cycle Γ; this is proved using the definition (3.59) and the fact that the screening charges are primary fields of dimension 1. (The function T (z) a V −ka/2b (z a ) Γ is defined by the integral (3.59) with an insertion of T (z) in the free field correlation function on the right hand side.) What we have just described is the free-field realization of the conformal blocks [32,33]. The space of possible integration cycles Γ for the integral (3.60) has a natural flat connection (the Gauss-Manin connection) as the points z a , a = 1, . . . , d vary. Hence the functions a V −ka/2b (z a ) Γ -for any fixed number of w's -furnish a representation of the braid group on d strands. It is known [34,35] that these are precisely the representations of the braid group that are associated to the Jones polynomial and its generalizations. This relation of the Jones polynomial to conformal field theory will be more useful for the present paper than the relation via the Knizhnik-Zamolodchikov equation, which was noted in section 3.4.1. The functions a V −ka/2b (z a ) Γ are not all possible conformal blocks for a product of primary fields with the dimensions of the V −ka/2b ; rather, they are all such conformal blocks if the operators V −ka/2b are degenerate primary fields in the sense introduced in [31]. Alternatively, these functions are all possible conformal blocks if the oper derived from the small b limit of t = b 2 T is supposed to have trivial monodromy at the points z = z a . These concepts and the relation between them are described in section 3.7.2. The Irregular Case Now let us incorporate the complex symmetry breaking parameter a in the above discussion. We can certainly in the integral (3.60) over the w's modify the exponent to include the terms proportional to a in the Yang-Yang function (3.52). But what does this mean in conformal field theory? We need to replace the free field correlation function (3.58) by i V 1/b (w i ) a V −ka/2b (z a ) free = i =j (w i − w j ) − 2 b 2 i,a (w i − z a ) ka b 2 a =b (z a − z b ) − 1 b 2 kak b e a ζb 2 (2 i w i − a za) . (3.62) What is the conformal field theory interpretation of this formula? Almost by construction, the right hand side is the free-field correlation function of the given product of fields with peculiar boundary conditions for χ at infinity, χ ∼ az/bζ. Alternatively, we have inserted at infinity an "irregular vertex operator", i.e. the L → ∞ limit of exp − 2a ζb L 2 ∂χ(L) . (3.63) In the presence of such an irregular vertex operator, the stress-tensor has the expected degree four pole. Nothing changes in the above formulae, except that the choice of possible integration contours is enlarged. The result of the integral is a conformal block with an irregular puncture at infinity, as defined in [36], in addition to the standard punctures of momenta −k a /2b. In the context of free fermions, operators associated to irregular singularities were originally defined in [37,38]. Degenerate Primary Fields And Trivial Monodromy Now we will review some standard facts about representations of the Virasoro algebra. This will enable us to explain what is special about the particular conformal blocks that come from the free field representation. For a generic value of the conformal dimension ∆, the Verma module defined as the span of all possible Virasoro descendants of a highest weight vector of dimension ∆ is irreducible. For a set of special values α = α r,s = − (r − 1)b 2 − s − 1 2b , r, s = 1, 2, 3, . . . (3.64) or α = Q − α r,s , this is not true: a certain descendant at level rs is again a highest weight vector, and has zero norm. In a unitary conformal field theory, the descendant in question will vanish, and even in a non-unitary theory, it might vanish. The primary field whose descendant vanishes is called a degenerate primary field. We write Φ r,s for such a field. The vanishing descendant of a degenerate primary field will certainly decouple in correlation functions. We call conformal blocks obeying such a relation degenerate conformal blocks. They satisfy a condition known as a "degenerate fusion rule." (It can be proved using the differential equation that we mention shortly.) In the OPE of an operator Φ r,s and an operator of momentum α, only operators of momentum α − r ′ b 2 − s ′ 2b appear, with r ′ = r − 1, r − 3, · · · , 1 − r s ′ = s − 1, s − 2 · · · , 1 − s. (3.65) A conformal block with an insertion of momentum α r,s satisfies the null-vector decoupling condition if and only if its correlation functions satisfy a certain differential equation of order rs. An important special case is r = 2, s = 1; the equation is ∂ 2 Φ 2,1 (z) + b 2 : T (z)Φ 2,1 (z) := 0. (3.66) This reduces in the semiclassical limit to the differential equation (3.50) associated to an oper. (For b → 0, the normal ordering in (3.66) is irrelevant; the only part of b 2 T that survives for b → 0 is the response to the "heavy" fields with momenta of order 1/b.) Moreover, in the semiclassical limit, the expectation value of t = b 2 T will always be such that the monodromy of the differential equation around points with additional Φ 1,s insertions is trivial. Indeed, the degenerate fusion rule implies that the OPE of Φ 2,1 and Φ 1,s contains only one primary field Φ 2,s ; from this, it follows that the monodromy of Φ 2,1 around a Φ 1,s puncture is trivial. This is a quantum version of the trivial monodromy condition on the differential equation (3.50) associated to the oper. On the other hand, the singularity of t near a Φ 1,s insertion is precisely that which we have exhibited in (3.48) (with j = s/2). The upshot of this is that the semiclassical limit of a conformal block for a correlation function d a=1 Φ 1,ka (z a ) determines an oper with precisely the sort of monodromy-free singularities that we extracted from three-dimensional gauge theory in section 3.3. The conformal blocks constructed from the free field formula (3.59) describe correlation functions of degenerate primary fields, simply because the free field vertex operators of momenta − ka 2b are degenerate. Related to this, it is possible to show that the conformal blocks which are given by the free-field realization do satisfy the degenerate fusion constraints. The corresponding differential equations are equivalent to the Picard-Fuchs equations satisfied by the free-field integrals, or to the natural flat connection on the space of integration cycles. They are a close analogue to the Knizhnik-Zamolodchikov equations. The free field realization gives all the conformal blocks for the correlation function d a=1 Φ 1,ka (z a ) that are allowed by the fusion rules, so there are no more to be had. The interpretation of opers with monodromy-free singularities in terms of correlation functions of degenerate conformal fields gives an intuitive explanation to the Bethe equations. Naturally, t should have no poles at the points w i , because no conformal fields are inserted there. z 1 z 4 z 2 z 3 Four-Dimensional Solutions and Parallel Transport Introduction We now turn to the problem of analyzing time-dependent solutions of the original BPS equations (1.1). Even though this is a problem of classical partial differential equations, a quantum mechanical view is helpful. We start with a time-independent situation -twisted N = 4 super Yang-Mills on a four-manifold M 4 = R × M 3 , where M 3 is a three-manifold and we think of R as the time direction. In our application, M 3 = C × R + , with C a Riemann surface. If knots are present, we assume initially that they are time-independent. In this situation, which is depicted again for convenience in fig. 6, we want to find the BPS states -quantum ground states. The first approximation, already analyzed in sections 2 and 3, is to find time-independent classical solutions. In going from classical solutions to BPS states, we will ignore the noncompactness of M 3 . This means that we will ignore the existence of a continuum of non-BPS excitations. If there is only a finite set of classical solutions of the BPS equations (and they are nondegenerate -there are no zero modes in expanding around such a solution), then the classical approximation to the space of BPS states is very simple. Let I be the set of classical solutions. Then for each I ∈ I, there is in perturbation theory a quantum ground state ψ I that is localized near I. In perturbation theory, the ψ I form a basis for the space H of BPS states. Nonperturbatively, in problems of this general type, instanton effects might lift some of these approximate ground states away from zero energy. However, we are actually here dealing with a problem in which this does not occur. This is because in the timeindependent case, even with knots present, our problem has four supercharges, and an instanton (that is, a classical solution with non-trivial time dependence) violates at least two of them. This leads to the existence of two fermion zero modes in an instanton background, which is one too many to contribute to a matrix element of the supercharge Q between approximate ground states ψ I and ψ J . Relation To Morse Theory A more explicit understanding of why instantons do not lift the classical vacuum degeneracies comes from the relation of this problem to Morse theory. 8 Supersymmetric quantum mechanics related to Morse theory [42] is, in general, a theory of maps from R to U , where U is a Riemannian manifold with metric tensor g endowed with a real-valued function h that we call the superpotential. For generic U and h, the model has two supercharges, one of which is conjugate to the exterior derivative: Q = e h de −h . (4.1) The classical vacua correspond to critical points of h. If h is a Morse function -that is, its critical points are all isolated and nondegenerate -then in perturbation theory, each critical point I corresponds to an approximate quantum ground state ψ I . The fermion number q I of ψ I is equal to the Morse index of the critical point I (the number of negative eigenvalues of the matrix of second derivatives of the function h at I). Since Q increases the fermion number by one unit, quantum corrections inducing non-zero matrix elements ψ J |Q|ψ I are possible only if q J = q I + 1. (4.2) Such nonzero matrix elements can be computed by counting, in a suitable sense, the instanton solutions that interpolate between the critical point I in the far past and the critical point J in the far future. The relevant "instanton" equations, in other words the conditions for a map R → U to be Q-invariant, are the gradient flow equations of Morse theory: dx i dt = −g ij ∂h ∂x j . (4.3) The problem we are studying of twisted N = 4 super Yang-Mills theory on R × M 3 (we primarily take M 3 = C × R + but the following remarks are more general) is an infinite-dimensional problem of this sort, 9 with U being the space of complex-valued connections on M 3 . We view the Chern-Simons function W = 1 4πi M 3 Tr A ∧ dA + 2 3 A ∧ A ∧ A (4.4) as a holomorphic function on the complex manifold U . Holomorphy means that the onedimensional sigma model with target U and superpotential W has four supercharges (this actually depends on the fact that the metric of U is Kahler and is also true in the gaugeinvariant case mentioned in footnote 9). We actually want to study this model in the context of a twisting of N = 4 super Yang-Mills theory in which a particular supercharge Q is distinguished. This supercharge depends on a twisting parameter t [15] and is an infinite-dimensional version of Q = e h de −h where h is the ordinary Morse function h = Re (e iα W),(4.5) and t = 1 − sin α cos α (4.6) For a Morse function of this type, the gradient flow equation becomes dw j dt = −g ji e iα 2 ∂W ∂w i , (4.7) where the w i are local holomorphic coordinates on U . A down-to-earth manifestation of the relation of our problem to Morse theory is that the underlying four-dimensional supersymmetric equations (1.1) are the gradient flow equations (4.3) for the Morse function h on the infinite-dimensional manifold U . (This is one of the main ideas in [40,41], where the gauge invariance mentioned in footnote 9 has been taken into account.) Now we can give a more explicit explanation of why nonperturbative effects in our problem will not spoil the supersymmetry of any of the approximate quantum ground states ψ I . In general, for a Morse function that is the real part of a holomorphic function, isolated critical points all have the same (middle-dimensional) Morse index and thus the same value of the fermion number q. Hence the condition (4.2) is never satisfied. Another route to the same result is as follows. In general, gradient flow for a Morse function such as h that is the real part of a holomorphic function has a conserved quantity, namely the imaginary part of the relevant holomorphic function, in our case j = Im (e iα W). (4.8) 9 To be more precise, our problem is a gauge-invariant version of a such a problem -corresponding to a supersymmetric sigma-model with target U coupled to gauge fields that gauge a symmetry of U. The gauge group in our case is the group of maps from M3 to the finite-dimensional group G, while U is the space of complex-valued connections on M3. However, in our problem the gauge group acts freely on U; this is ensured by the Nahm pole boundary condition. As a result, the gauge-invariance will not play a major role. In effect, for our purposes, we can replace U by its quotient by the group of complex gauge transformations and reduce to the case that there are no gauge fields. For generic α, all critical points have distinct values of j and hence distinct critical points cannot be connected by a solution of the gradient flow equation. Hence there are no instantons that might spoil the supersymmetry of the states ψ I . Time-Dependence As explained in section 1.2, we do not literally want to consider a time-independent situation; rather, we want to allow for a slow time-dependence of the positions of the knots. Let B be the space of distinct points z 1 , . . . , z d ∈ C. The space H of BPS states is the fiber of a bundle H over B. This bundle carries a natural flat connection. This is a general property of topological field theory, but the Morse theory interpretation leads to a particularly nice description. In supersymmetric quantum mechanics related to Morse theory, since the supercharge Q = e h de −h is conjugate to the exterior derivative d, the ground states ψ I associated to critical points must have an interpretation in terms of the cohomology or dually the homology of U . There is a standard way to understand this in Morse theory. To a critical point I, one associates the downward-flowing cycle J I consisting of all points in U that can be reached by gradient flow starting at I. In other words, one considers solutions of the gradient flow equation on a half-line (−∞, 0], with initial conditions that the flow starts at I at t = −∞. J I parametrizes the values at t = 0 of such flows. The case that h is the real part of a holomorphic function has special features, and is particularly simple, so let us focus on that case. We make the further simplifying assumption that the critical points of h are isolated and irreducible. U is inevitably not compact (or it would not admit a non-constant holomorphic function). In this very special situation, the J I are called Lefschetz thimbles. The thimbles J I are closed if the angle α used in defining h is sufficiently generic (to prevent the existence of gradient flows between distinct critical points), but they are not compact. So they do not represent classes in the ordinary homology of U . However, as J I is defined by downward gradient flow with respect to h, one has h → −∞ at infinity along J I . As a result, the J I are elements of a certain relative homology group -the homology H(U , U < ) of U relative to the region U < where h goes to −∞. (In the notation, we do not indicate the dimension of a homology cycle, because this relative homology is nonzero only in the middle dimension. That is related to the fact that the critical points all have a middle-dimensional Morse index.) The space H of supersymmetric ground states can be identified with the relative homology H(U , U < ). In this correspondence, the quantum ground state ψ I associated to a critical point I maps to the element J I of H(U , U < ). For an explanation of this from a physical point of view (in the context of supersymmetric quantum mechanics related to Morse theory), see [43] or [41]. The interpretation in terms of relative homology means that H has an integral structure and hence a natural flat connection. To give it a fancy name, this flat connection is the Gauss-Manin connection on the relative homology. This connection is trivial for generic values of α and the z i : the ψ I are flat sections, and the connection on H is fully described by the smooth evolution of the classical critical points and corresponding thimbles. (In transporting the thimbles, one must keep track of their orientations; the sign of the relative homology class associated to J I depends on the orientation of J I .) Crucially, there are codimension one walls in the space S 1 × B of parameters α and z i where the thimbles fail to be closed (and so do not define elements of the relative homology), and the map from the critical points to quantum states jumps discontinuously. This can occur if there are gradient flow lines from I to J; in this case, J I is not closed as it contains points arbitrarily close to J, but not J itself. We write ℓ J I for the locus in S 1 × B on which the following necessary conditions are obeyed for flows from I to J: the value of the conserved quantity j is equal at I and J, while h(I) > h(J). The first condition is a single real condition, while the second is just an inequality. So ℓ J I is of real codimension 1, and we call it a Stokes wall. In crossing a Stokes wall ℓ J I , only the thimble J I becomes ill-defined. It jumps by a multiple of J J : J I → J I + m IJ J J , (4.9) where m IJ is the "number" of gradient flow lines from I to J counted in an appropriate sense. A given line contributes 1 or −1 to the sign depending on the direction in which the difference between the values of j = Im(e iα W(w, z)) at I and J passes through zero. For an elementary explanation of such matters, see section 2 of [40]. The correspondence between states ψ I and thimbles J I means that the ψ I have the same jumping in crossing Stokes walls. In the ψ I basis, the connection is trivial except across the Stokes walls, where the transport matrix is a triangular "Stokes factor" S[ℓ J I ] = 1 + m IJ e J I (4.10) Here e J I is the matrix whose only non-zero element is 1 at position J, I. In particular, the parallel transport along a path P in S 1 ×B is a path-ordered product of factors of the following kind: (a) between two Stokes walls, one has only the "formal monodromy" which expresses the permutations of the classical critical points, with minus signs that keep track of the orientations of the thimbles; (b) every time one crosses a Stokes wall, the monodromy acquires a corresponding Stokes factor. One can visualize the matrix elements n J I [P] of the transport matrix for the path P as counting paths from a critical point I to a critical point J, where away from Stokes walls, one has to follow a critical point continuously, but in crossing a wall, one is allowed to "jump" along a gradient flow trajectory from one critical point to the next. A matrix element of the transport matrix is given by a sum of contributions of hybrid paths of this type, with each path contributing 1 or −1 depending on how the orientation of a thimble evolves along the given path. Incidentally, the fact that the monodromy for parallel transport along a path P depends only on the homotopy class of P implies wall-crossing formulas for the numbers m IJ that control the jumping. More generally, the whole picture can be interpreted in terms of BPS states in an LG model based on W, but we will not pursue that interpretation in this paper. Let us collect a few properties of the matrix elements n J I [P]: • n J I [P] only depends on the homotopy class of P; • The composition of paths maps to matrix multiplication: n K I [P 1 •P 2 ] = J n J I [P 1 ]n K J [P 2 ]; • For an infinitesimal path δP from z to z +δz it is almost always true that n J I [δP] = δ J I -this fails only in crossing a Stokes wall; • In crossing a Stokes wall, n J I [δP] − δ J I = m IJ , where m IJ is computed by a count of flow lines. The Dual Basis If Γ is any cycle in the relative homology H(U , U < ), then as the thimbles are a basis for this relative homology, Γ is equivalent in relative homology to a linear combination of thimbles, Γ = I c I J I . (4.11) How can the coefficients c I be determined? If we had, in some sense, a dual basis K I to the J I 's with pairings K J , J I = δ J I , (4.12) then we would identify the coefficients in (4.11) as c I = K I , Γ . (4.13) The Poincaré dual of the relative homology H(U , U < ) is the opposite relative homology H(U , U > ) of U relative to the region U > with h → +∞. A natural basis of H(U , U > ) is given by the upward-flowing thimble K I associated to critical points. For each critical point I, K I is defined as the boundary values at t = 0 of solutions of the gradient flow equation on the half-line [0, ∞) that approach the point I for t → +∞. Since h decreases along gradient flow lines, the smallest value it assumes along such a flow is its value at t = ∞, which is its value at the critical point I. So h is bounded below along K I , but possibly not bounded above, and K I takes values in H(U , U > ). As for the pairing (4.12), from the definitions of the K's and J 's, K J , J I counts flows on the full real line (−∞, ∞) that start at J at t = −∞ and end at I at t = +∞. For J = I there are no such flows (for generic α where the thimbles are well-defined). For J = I, since h strictly decreases along a nonconstant flow, the only flow is the constant one that sits at J for all times. For a certain natural relative orientation of the J 's and K's, the contribution of the constant flow to the pairing is +1. Non-Single Valued Superpotentials and q-Grading In the framework that we have presented so far, the matrices which represent the action of the braid group on H have integer-valued entries, since the relative homology has an integral structure. Now we want to consider a situation where W is not single-valued. To be more precise, we will consider a holomorphic function like the Chern-Simons functional W, whose real part is single-valued, but whose imaginary part is well defined only modulo 2πZ. We can reduce to the framework which we have employed so far by replacing U by the smallest cover U on which W is single-valued. Passing to U comes at the cost that now the number of critical points will be infinite, since each critical point in U has infinitely many preimages in U . A critical point I in U is the same as a critical point I in U together with a choice of a branch of W at I. Locally, we can pick an arbitrary preimage I 0 of a critical point I, and denote as I n the critical point for which the value of W is shifted by −2πin compared to the value at I 0 . The Stokes factors and transport matrices are now matrices of infinite size, but their matrix elements can be computed by the techniques we have described, and are integervalued. As the calculations only depend on the gradient of W, these matrices commute with the deck transformation I n → I n+1 . It is convenient to introduce a variable q taking values in C * . Then if we simply write q n ψ I as a symbolic shorthand for ψ In , we can replace infinitedimensional matrices whose entries are integers with finite-dimensional matrices whose entries are Laurent polynomials in q with integer coefficients. We denote such a matrix as S[ℓ J I ; q]. In order to keep track of the lift of W as we move in B, the formal monodromy matrices which encode the permutation of critical points and the Stokes matrices associated to gradient flows are valued in powers of q, to keep track of the change of W along a path. The matrix elements of the transport matrix are now obtained by summing over hybrid paths (continuous evolution away from Stokes walls and gradient flow across Stokes walls) with weight ±q n , where −2πin is the change of superpotential along the path and as usual the sign involves the orientation of the thimbles. An alternative description of all this is as follows. Since U admits the non-single-valued superpotential W, its first Betti number is positive and one can introduce a "theta-angle" θ in the supersymmetric quantum mechanics, weighting by e inθ a path in which W jumps by −2πin. This has the effect of replacing the relative homology of U with a twisted version of the relative homology, valued in a flat line bundle of monodromy q = e iθ . Then we consider the Gauss-Manin connection for homology twisted by this flat bundle. The holonomy matrices for this connection have entries that are Laurent polynomials in q with integer coefficients and they can be computed as just described. Classical Description of Counting of Four-Dimensional Solutions Now, let us consider the problem that we are really interested in -time-dependent solutions of the supersymmetric equations (1.1) on R×C ×R + . The solutions will be time-dependent because the boundary conditions are time-dependent -we allow the positions z i of singular monopoles on the boundary of M 3 = C × R + to vary with time. Although we typically assume an adiabatic evolution of the monopole positions, the counting of four-dimensional solutions is topological, and the adiabatic assumption is not necessary. In the simplest setup, with singular monopole strands at the boundary braided in time, the superpotential depends holomorphically on some parameters (the positions of the strands in C), and the parameters evolve in time. Schematically, the equations take the form of a "forced gradient flow" dw i dt = −e iα g ij ∂W(w, z(t)) ∂w j (4.14) We suppose that the singular monopoles begin at positions z i = (z 1 , . . . , z k ) near time t = −∞ and end at positions z f = (z ′ 1 , . . . , z ′ k ) near t = +∞. In fact, we can assume that the positions z i (t) of the singular monopoles are constant except in a bounded interval −T < t < T , for some T , during which they follow a path P in their parameter space B. In such a situation, we can look for solutions of the forced gradient flow equation that begin at a specified critical point I of W(w, z i ), and end at a specified critical point J of W(w, z f ). The "number" of such solutions, with each solution weighted by the sign of the fermion determinant, is a topological invariant -unchanged under deformations of the path P or the metric on U . We will call this invariant N J I . (For a reason that will be clear momentarily, we really only want to define N J I if z i and z f are not on Stokes walls.) In section 4.1.2, we already associated an integer invariant n J I [P] to this situation. n J I [P] was a matrix element of the Gauss-Manin connection for transport along the path P from z i to z f . One can think of n J I [P] as the expansion coefficients when a thimble J I in the relative homology H(U , U < ) z i is transported along the path P using the Gauss-Manin connection, and then expressed in terms of the thimbles J ′ J that furnish a basis of H(U , U < ) z f : J I = J n J I J ′ J . (4.15) We claim that in fact N J I = n J I . dw i dt = −e iα g ij ∂W(w, z i ) ∂w j (4.19) which start from Γ and asymptote to the critical point J in the future. The relative homology is defined over Z, and an integral relative homology class such as J I has no continuous deformations. So clearly, as long as the continuous deformation from J I [ z i ] to J I [ z; t 0 ] induced by the flow equations (4.14) lives at any given time in H m (U , U < ) z(t) , it coincides with the natural transport along P by the Gauss-Manin connection. In this case, (4.21) is equivalent to the desired result N J I = n J I . To show that J I [ z; t 0 ] lies in H m (U , U < ) z(t 0 ) for any t 0 , we are supposed to prove that Re W(w, z(t 0 )) goes to −∞ at infinity along J I [t 0 ]. Indeed, if a sequence of forced gradient flows on the semi-infinite interval (−∞, t 0 ] goes to infinity, it does so by diverging for t → t 0 , in which case Re(W(w, z(t 0 )) (whose gradient drives the flow for t → t 0 ) must go to −∞. An alternative approach to (4.16) is the following. Suppose that z f = z i and P is the trivial path between them. Then N J I = n J I = δ J I . As we vary z f , both N J I and n J I may jump in and only in crossing Stokes walls; they jump in exactly the same way, so they remain equal. We have already describing the jumping of n J I . The jumping of N J I occurs because in crossing a Stokes wall, a time-dependent solution may disappear to infinity, as follows. Suppose that, for z f on some Stokes wall, there is a jump in n J I , resulting from an ordinary gradient flow from some critical point J ′ of W(w, z f ) to J. Such a flow produces a jump n J I → n J I ± n J ′ I ,(4.22) where the sign depends on the direction in which one crosses the Stokes wall. To see a corresponding jump in N J I , one looks for forced gradient trajectories from I to J that consist of a forced gradient trajectory from I to J ′ followed, at some time very far in the future, by the same ordinary gradient trajectory from J ′ to J that causes the jump of n J I . A two-step forced trajectory of this kind exists if z f is near the Stokes wall and on the proper side of it; the time at which the second step of the flow occurs diverges as z f crosses the Stokes wall. This leads to the disappearance of the two-step solution and a jump of N J I that just matches the jump of n J I . From Braiding Of Thimbles To Free Field Integrals According to the reasoning in section 4, to understand the braid group representations associated to the Jones polynomial, we are supposed to compute a natural monodromy action on the middle-dimensional relative homology of an infinite-dimensional space U of connections on a three-manifold M 3 = C × R + . This may sound hopelessly abstract. We will now show how it can be turned into something concrete and calculable. From Thimbles to Integrals A convenient way to describe the evolution of states in H, including the q-grading, is to view the homology cycles as integration cycles. Instead of looking at the evolution of the homology cycles, it is equivalent to look at the evolution of the integrals. Of course, we need an integrand which can be integrated on the thimbles J I , which are not compact. A function that fills the bill is e W/ε , where ε is chosen in a suitable half-plane. Since h = Re e iα W (5.1) goes to minus infinity along a thimble, the condition we want is Re e iα ε > 0. This will ensure the convergence of the integrals I Γ = Γ e W(w)/ε dΩ (5.2) where dΩ is a holomorphic volume form on U (which will be kept fixed in what follows) and the integration cycle Γ is a thimble, or more generally any cycle in the relative homology H(U , U < ). The thimbles are particularly nice integration cycles, because the ε → 0 limit of the integral over a thimble is very simple. On a thimble J I defined by gradient flow from a critical point I, the function h has a unique maximum, namely the critical point. So for ε → 0, the integral over a thimble is I I := J I e W(w)/ε dΩ ∼ exp(W I /ε) ε −dim J I /2 c 0 + . . . ,(5.3) where W I is the value of W at the critical point I. This formula is valid throughout the half-plane Re e iα ε > 0, but actually as long as α is not on a Stokes wall, this asymptotics holds in a slightly larger sector in the complex plane. 10 This property uniquely characterized the basis of integrals I I among all the possible I Γ : if we were to take a linear combination of several I I , the asymptotics would fail at some ray in the extended half-plane where two critical points exchange dominance. This characterization is familiar in Stokes theory, and motivated the terminology "Stokes walls." As we vary the parameters of W, the homology H(U , U < ) will vary continuously. If we vary the integration cycle Γ continuously, the integral will vary holomorphically in the parameters of W. The monodromy of the cycles Γ is the same as the monodromy of the integrals I Γ . Two Chern-Simons Theories In our present context, the thimble integral (5.2) is a Chern-Simons path integral on the three manifold M 3 = C × R + , albeit on an unusual integration cycle. Such an integral can be concretely expressed in terms of N = 4 gauge theory on M 3 × R + . (In this section only, we write R or R + for the x 1 direction to distinguish it from the y direction R + .) This statement was one of the main conclusions of [40,41]. In our case, since M 3 = C × R + , the four manifold is M 4 = C × R + × R + . This is the second Chern-Simons theory to appear in this paper, at least implicitly. Our whole analysis concerns the calculation of the Jones polynomial, in a gauge theory setup which is S-dual to a setup which computes the Jones polynomial by Chern-Simons theory on W = C × R. In that "original" Chern-Simons theory, the knot is a Wilson loop, the gauge group is G ∨ , and the coupling parameter is k ∨ . The Jones polynomial is a Laurent polynomial in q = exp(2πi/(k ∨ + 2)), (5.4) where 2 is the dual Coxeter number of SU (2). As explained in [14], Chern-Simons theory on a three-manifold W can be computed via topologically twisted N = 4 super Yang-Mills theory on W × R + if one relates k ∨ to the twisting parameter Ψ ∨ of the N = 4 theory by Ψ ∨ = ±(k ∨ + 2). (5.5) In this description one uses a D3-NS5 boundary condition at the origin in R + . The sign ± depends on the relative choice of orientation between M 4 and W . One can also apply S-duality, converting the gauge group from G ∨ = SU (2) to G = SO(3) and converting the D3-NS5 boundary condition to a D3-D5 boundary condition; this boundary condition involves a Nahm pole, as explained in [19]. In this new description, which has been the starting point of the present paper, the twisting parameter is Ψ = −1/Ψ ∨ . Because this dual description is difficult, we have tried to simplify it, as first explained in section 1.2, by "stretching" W in one direction. Thus we approximated W by R×C, and looked for solutions of the BPS equations on W ×R + that are "pulled back" from M 3 = C ×R + . If this dual description is formulated as supersymmetric quantum mechanics, with R as the time direction and the field variables being gauge fields on M 3 = C × R + , then the superpotential is Ψ (not Ψ ∨ ) times the Chern-Simons function. We then relate the braiding of solutions on M 3 to the braiding of thimbles for Chern-Simons theory on M 3 , which we express by N = 4 theory on M 3 × R + . In this description, we impose D3-NS5 boundary conditions at the origin of R + (and D3-D5 at the origin of R + ). The level k of this Chern-Simons description is related to the twisting parameter of the N = 4 theory by Ψ = ∓(k + 2). (5.6) The two Chern-Simons descriptions are related by S-duality, together with the exchange of the roles of R + and R + . The opposite sign in (5.5) and (5.6) is due to the fact that exchanging R + and R + reverses the orientation of M 4 . Combining this with Ψ = −1/Ψ ∨ and Ψ ∨ = k ∨ + 2, we find that the relation between the level k ∨ in the Chern-Simons description that is related to the Jones polynomial in the traditional way and the level k in the Chern-Simons description that relates the Jones polynomial to Nahm poles and opers is k + 2 = 1/(k ∨ + 2). (5.7) This four-dimensional setup, with the N = 4 theory on a manifold M 4 = C × R + × R + with a "corner," is rather interesting, and we believe it deserves to be explored further. We will not do so in this paper. From Chern-Simons to Conformal Blocks For this paper, more useful than the relation of Chern-Simons theory on M 3 = C × R + to four dimensions is its relation to conformal blocks on C. The most familiar version of this statement [44] is that a path integral with A z fixed and A z varying gives a WZW conformal block (that is, a conformal block in two-dimensional current algebra, with the symmetry group G being the same as the gauge group of the Chern-Simons theory). Local variations of the fixed value of A z insert a holomorphic current J(z) in the conformal block: δ log Z = Tr J z δA z (5.8) This statement has an analog [45] that leads to Virasoro conformal blocks in the case of SU (2) or SO(3) gauge theory, or to more general W -algebra conformal blocks in the case of gauge groups of higher rank [46,47]. This analog involves a different boundary condition in which, at the price of breaking some gauge symmetry at the boundary, one fixes some parts of A z , and some parts of A z . For gauge group SU (2), the boundary condition which leads to Virasoro conformal blocks is simply stated: A z = * 1 * * A z = × 0 * × (5.9) Here we denote as * the elements which are free to vary, and as × elements which are fixed. The connection just described is an oper! (Since A z is lower-triangular, a bundle with this connection has a holomorphic sub-bundle L whose sections are of the form 0 * , and because the upper right matrix element of A z nowhere vanishes, this sub-bundle is nowhere preserved by D z = ∂ z + [A z , · ].) So the complex boundary condition (5.9) is the one that is induced by the Nahm pole boundary condition studied in section 3. To explain the relation to Virasoro conformal blocks, we note that the Nahm pole boundary condition depends on a choice of complex structure on C. Once a complex structure is picked with a local complex coordinate z, nearby complex structures can be described by a Beltrami differential µ z z . The relation is that in the new complex structure, the holomorphic fields on the space of complex connections are not A z but A z − µ z z A z . Making this deformation is equivalent to replacing the "0" in the boundary condition for A z in (5.9) with µ z z (so that now A z − µ z z A z is lower triangular). Hence a local variation of the boundary condition associated to a change in complex structure inserts a holomorphic stress tensor 10) and this leads to the relation between Chern-Simons theory with the oper boundary condition and Virasoro conformal blocks. δ log Z = T zz δµ z z ,(5. In the semiclassical limit, the operator T zz reduces to the classical stress tensor t(z)[A] of the oper. More precisely, the identification of parameters from [45] is that if we define We have here assumed that G is SU (2) or SO (3). For general gauge group G, both Nahm poles and W -algebras are labeled by an su(2) embedding in the Lie algebra g of G. Inspection confirms that the boundary conditions used to define a general W -algebra conformal block are induced by the corresponding Nahm pole. − b −2 = k + 2,(5. Analog For Liouville Though we will not need this fact in the present paper, we should remark that the relation between Virasoro conformal blocks and Chern-Simons theory has a simple extension to a relation beween Liouville theory and Chern-Simons theory. To do Liouville theory on a Riemann surface C, one considers Chern-Simons theory on C × I where I is a unit interval. At one end of I, one imposes the Nahm pole boundary condition and at the other end, one imposes a variant of the Nahm pole boundary condition with z and z exchanged. Liouville partition functions and correlation functions are built by combining holomorphic and antiholomorphic Virasoro conformal blocks, which arise naturally from Chern-Simons on C × I with boundary conditions just stated. (For the case of a compact symmetry group, it is already known that Chern-Simons on C × I reproduces the WZW model on C.) By slightly extending arguments that we present presently, light degenerate fields and generic primary fields of Liouville theory, inserted at a point p ∈ C, correspond to Wilson operators or monodromy defects on p × I. In the classical limit, the correspondence between Chern-Simons and Liouville theory means the following. A classical solution of Chern-Simons theory on C × I with boundary conditions as above is a flat bundle on C whose holomorphic and antiholomorphic structures both obey the oper condition. Indeed, a classical solution of Liouville theory corresponds to a metric on C of constant negative curvature. If ω and e are the vierbein and spin connection of this metric, then we can define a corresponding SL(2, R) flat connection A = ωt 3 + e z t + + e z t − . With a standard representation of the t i , A z is upper triangular with an upper right matrix element that is everywhere nonzero, and A z is lower triangular with a lower left matrix element that is everywhere nonzero. So both the holomorphic and antiholomorphic structures defined by this flat connection satisfy oper conditions, as expected in the Chern-Simons description. (The antiholomorphic oper structure is defined with the roles of "upper triangular" and "lower triangular" matrices reversed.) So this gives the mapping between the two theories at the classical level. Wilson Line Operators To further understand the mapping from three-dimensional Chern-Simons theory with oper boundary conditions to Virasoro conformal blocks in two dimensions, we will explore the interpretation of Wilson line operators. First let us recall what happens if one uses standard boundary conditions that relate Chern-Simons theory to current algebra. In this case, a Wilson line operator ending on the boundary of a three-manifold M 3 represents insertion of a conformal primary field at that boundary point in the WZW conformal block. If the Wilson line operator transforms in a finite-dimensional representation R of G, then the corresponding conformal primary field transforms in the same representation. This is consistent with the fact that a Wilson line operator ending on the boundary is not gaugeinvariant, but transforms in the representation R, just like the corresponding primary field of the WZW model. What is the analogous interpretation of a Wilson line operator that ends on a boundary at which one imposes Nahm pole boundary conditions? The Nahm pole breaks the gauge symmetry at the boundary, so we have to pick a component of the Wilson line operator. As we discussed in section 3.2, a generic vector diverges when parallel transported to the boundary. Given a Wilson line operator ending at y = 0, the most easily defined gauge invariant information is the coefficient of the most negative power of y. This is extracted simply by contracting with an appropriate power of the small section s. Actually, we will find useful a rescaled version of s, namely s = K(z) −1/2 s(z), which satisfies D 2 z + t(z) s = 0 D z s = 0 D y s = 0 (5.13) and has definite conformal dimension −1/2. In the gauge (3.46), s = 0 1 . In the two-dimensional representation of SU (2), we would consider an operator P exp − γ A s(z),(5.14) where γ is a path ending on the boundary at y = 0. In the classical limit, under conformal transformations of the boundary, this has the same conformal dimension as s, i.e. −1/2. For a spin k/2 representation, one must contract with k powers of s and the classical limit of the dimension is −k/2. We want to argue now that a Chern-Simons path integral with a spin k/2 Wilson operator ending on the boundary gives a Virasoro conformal block with the insertion of a "light" degenerate field of Liouville momentum −bk/2. Such a field has the correct classical dimension −k/2 in the b → 0 limit, and furthermore the exact formula for its quantum dimension − k 2 b b + 1 b + k 2 b = − k 2 − k(k + 2) 4 b 2 = − k 2 + k(k + 2) 4(k + 2) (5.15) is the sum of the classical dimension of s k and the dimension of a spin k/2 operator in a WZW model (of level k + 2 = −1/b 2 as in (5.11)). Furthermore, the operator (5.14) satisfies classically the correct differential equation: the k = 1 operator is annihilated by ∂ 2 z + t(z), etc. Part of what makes possible the correspondence between spin k/2 Wilson lines and degenerate conformal fields possible is that the degenerate fields satisfy fusion rules which coincide with the fusion rules of spin k/2 operators in the WZW model. This last fact is part of the input in the statement that the braid group representations associated to the Jones polynomial can be computed by the braiding of either primary fields of the WZW model or degenerate conformal fields of the Virasoro algebra. Singular Monopoles and "Heavy" Degenerate Fields Next, we would like to identify in the Chern-Simons description of Virasoro conformal blocks the "heavy" degenerate fields of Liouville momentum −k/2b. These are the degenerate fields whose conformal dimension diverges for b → 0. We claim that they correspond to the insertion of singular monopoles at the boundary. The main insight of section 3.3 was that at a Nahm boundary with singular monopoles (and generic ζ), the connection is an oper with singularities of trivial monodromy. We observed that the classical stress tensor of such an oper has poles that agree with the semiclassical limit of the quantum stress tensor in the presence of a heavy degenerate field. Moreover, the trivial monodromy condition holds quantum-mechanically as well: a light degenerate field has no monodromy around a heavy degenerate field, as they fuse in a unique channel. In the Chern-Simons setup, the classical trivial monodromy condition follows naturally from the fact that the singular monopole does not extend in the bulk. This was part of our derivation in section 3.3. Quantum mechanically, we need to consider the behavior when a light degenerate field -represented in three dimensions by an expression such as P exp − γ A s -approaches the singular monopole. The Wilson loop itself is topological, and the small section s has trivial monodromy around the singular monopole. Putting The Pieces Together We can now finally establish a link between the solutions of the four-dimensional BPS equations (1.1) that we started with and the braid group representations associated to the Jones polynomial. The time-independent solutions of the BPS equations correspond to opers with trivial monodromy. We have identified the braiding of the corresponding quantum states with the braiding of complex integration cycles for Chern-Simons theory, and then with the braiding of degenerate Virasoro conformal blocks. These are known [34,35] to be the braid group representations associated to Jones polynomials. So we have arrived at our goal, though in a form that may sound a little abstract. We can put this result in a perfectly concrete form using the free field representation of the conformal blocks. Opers with trivial monodromy are also associated to critical points of the Yang-Yang function for the Bethe equations; this is the logarithm of the integrand in the free field realization of conformal blocks. We can derive a degenerate Virasoro conformal block either from an infinite-dimensional thimble associated to an oper with trivial monodromy, or from a finite-dimensional thimble associated to a critical point of the Yang-Yang function. Either way, we get a conformal block with definite and uniform semiclassical limit in a sector of angular width greater than π in the b 2 plane. As those are unique, the two bases of conformal blocks must coincide. Hence the braiding representations associated to the four-dimensional gauge theory coincide with the braiding representations of integration cycles in the space of Bethe parameters w i . This is not as surprising as it may seem if we turn on a symmetry breaking parameter: then we have interpreted the w i as positions of bulk BPS monopoles, and our claim possibly amounts to the statement that the four-dimensional nonabelian gauge theory on the Coulomb branch reduces to a theory of massive monopoles and abelian gauge fields. We will develop this point of view further in section 7. Braiding Representations of Integration Cycles Overview A highlight of what we have learned so far is the existence of a natural map from the braid group representations derived from the four-dimensional gauge theory equations (1.1) to the braid group representations associated to correlation functions of Virasoro degenerate fields. Those braid group representations can be effectively studied using the free field representation, which we reviewed in section 3.7. Making this explicit will be our goal here. We consider a degenerate correlation function d a=1 V −ka/2b (z a ) . We assume that the z a are distinct points in C. To represent conformal blocks, we introduce q variables w i ∈ C, which we assume to be distinct from each other and from the z a . The allowed values of q have been analyzed in section 3. Moreover, we consider the w i to be indistinguishable, in the sense that configurations that differ by permuting them are equivalent. We write M for the space of such distinct and indistinguishable variables w i ∈ C\{z 1 , . . . , z d }. We also write M for the smallest cover of M on which the Yang-Yang function W of eqn. (3.52) is single-valued. In the free field representation, degenerate conformal blocks are written in the form Γ exp W/b 2 dw 1 . . . dw q . (6.1) Γ is a middle-dimensional cycle in M, chosen so that the integral converges. 11 Morse theory offers a systematic way to produce all such integration cycles: a basis of integration cycles is given by the thimbles associated to critical points of W. Cycles of this kind are never compact; they have noncompact ends on which the Morse function h = Re W goes to −∞. In our problem, this happens when one of the w i either approaches one of the z a or, in the presence of symmetry breaking, goes to infinity in the correct direction. In simple situations, instead of using Morse theory, one can describe integration cycles by hand. In constructing a cycle Γ by hand, one has to make sure that the Morse function really goes to −∞ at infinity along Γ. For example, this will fail if too many w i approach simultaneously the same z a . With symmetry breaking, some of the important integration cycles have ends at w = ∞ and the use of noncompact integration cycles is unavoidable. However, in the absence of symmetry breaking, the noncompact integration cycles produced by Morse theory have their ends at w i → z a , for various i and a, and are equivalent in the appropriate twisted relative homology to compact cycles in which the w i wrap around the z a in a suitable fashion. (For an example, see fig. 7 below.) In the extensive literature on integration cycles in free-field realizations of conformal blocks [32,33] and their application to the Jones polynomial [34,35], compact integration cycles are often used. Symmetry breaking, or in other words the introduction of an irregular singularity at infinity, has not been considered in this context, as far as we know. The use of Morse theory has advantages and disadvantages. The main disadvantage is that the thimbles do not correspond to a standard BPZ basis of conformal blocks defined by fusing the degenerate fields in specific channels. The main advantage is that in the basis of thimbles, the braid group is manifestly represented by matrices whose entries are Laurent polynomials in q with integer coefficients. This property, which was explained in section 4. 1.4, is important vis-a-vis the Jones polynomial and Khovanov homology. In what follows, we will first analyze a few important examples with a small number of degenerate insertions z a and Bethe roots w i , with or without symmetry breaking. Then in section 6.5, we analyze the general case in the presence of symmetry breaking. From that analysis, we get the experience we need to deduce a general description of the Jones polynomial -not just the associated braid group representations. This is presented in section 6.7. SU (2) Versus SO(3) We pause for a technical remark concerning the assertion that the entries of the braiding matrices are Laurent polynomials in q. 11 As explained in [34], the cycle Γ should actually be odd under the exchange of any pair of w's, to compensate for the sign change of the differential form dw1 ∧ · · · ∧ dwq under permutations. This means that the appropriate relative homology is actually the part that is antisymmetric under permutations of the w's. To be concrete, suppose that there are two w's and we find a solution of the Bethe equations at which the w's equal α and β up to permutation. Then we can define a cycle C ′ associated to the critical point w1 = α, w2 = β, and a cycle C ′′ associated to the critical point w1 = β, w2 = α. The difference C ′ − C ′′ is an element of the antisymmetric part of the homology. In practice, we can omit to explicitly form such differences and also ignore minus signs arising from permutations of the factors in dw1 ∧ · · · ∧ dwn. If the gauge group is G = SU (2), meaning that the dual group is G ∨ = SO(3) and the charges k a of the singular monopoles are all even, then the Yang-Yang function W as defined in (3.52) is well-defined mod 2πi. Hence a change of branch of W multiplies exp(W/b 2 ) by an integer power of q = exp(−2πi/b 2 ), and the braiding matrices are Laurent polynomials in q with integer coefficients. If instead G = SO(3), G ∨ = SU (2), then some of the k a may be odd. (Indeed, we will do our detailed computations for the case that all k a are 1.) Then W is well-defined mod 2πi if the w i are varied for fixed positions z a of the knots, but is only well-defined mod 2πi/4 when the z a are varied. Consequently, for G = SO(3), the braid matrices will actually be Laurent polynomials in q 1/4 . Of course, we could eliminate this by writing the formulas in terms of q = q 1/4 , but we prefer not to do so since q as we define it is the natural instanton counting parameter in four dimensions. The underlying reason for the difference between G = SU (2) and G = SO (3) is that the Chern-Simons function, normalized as we have done in (3.6), is gauge-invariant mod 2πi in SU (2) gauge theory, but gauge-invariant mod 2πi/4 in SO(3) gauge theory. The last statement holds on any sufficiently rich three-manifold M 3 , or on any M 3 if singular monopoles of odd charge are present. In the latter case, one considers only gauge transformations that are trivial at the position of the singular monopole. A Single Critical Point We begin with the two examples in which the Yang-Yang function has only a single critical point. Braiding of Two Primaries With Minimal Charge The first example arises in the absence of symmetry breaking, with two z a of charge k = 1 and one w. An obvious integration cycle C 12 is a segment joining the two z a . Now, let us compare it with the thimble. The Bethe equation 1 w − z 1 + 1 w − z 2 = 0 (6.2) has a unique solution w = 1 2 (z 1 + z 2 ). The thimble flows down from w to the z i along a straight line, and coincides with the obvious cycle C 12 . The critical value of the Yang-Yang function W = − 1 2 log(z 1 − z 2 ) + log(w − z 1 ) + log(w − z 2 ) = 3 2 log(z 1 − z 2 ) + const. (6.3) is such that exp(W/b 2 ) agrees with the OPE coefficient of the two degenerate fields in the identity channel V −1/2b (z 1 )V −1/2b (z 2 ) ∼ (z 1 −z 2 ) 3 2b 2 . (Recall that for small b, the dimension of V −1/2b is −3/4b 2 + O(1) .) This is expected, because with two z a and one w the oper has no singularity at infinity, in view of the discussion of eqn. (3.22), and hence describes the fusing of two fields to the identity. In order to fuse the two degenerate fields in a channel of momentum −b we would consider something even simpler: a case without any w's. In this channel, the conformal block is simply the free field correlation function V −1/2b (z 1 )V −1/2b (z 2 ) with no integral at all. With no w's, the discussion of (3.22) shows that the oper does have a singularity at infinity -corresponding to fusion of the two k = 1 degenerate primary fields to a k = 2 degenerate primary field. To be precise, in defining an integration cycle such as C 12 , we should specify a choice of branch of W. (Differently put, the cycle is supposed to be defined in the covering space M.) If we braid z 1 around z 2 , C 12 evolves continuously, but we may end up with a different branch of W. As the charges at z 1 and z 2 are identical, a basic braiding move is to exchange the position of z 1 and z 2 , and then relabel them. In the absence of w, we would only have the factor (z 1 − z 2 ) − 1 2b 2 coming from the part of W which only depends on z a , so a braiding which exchanges z 1 and z 2 counterclockwise would give a factor of e − iπ 2b 2 = q 1 4 . In the presence of one w, we can get the result by following the value of W at the midpoint of C 12 : C 12 → −q − 3 4 C 12 . (6.4) The minus sign follows from the change in orientation of C 12 . It is interesting to compare the thimble with closed contours which are commonly used in order to describe BPZ conformal blocks, as depicted in fig. 7. As illustrated in the picture, these contours are equivalent in homology to the thimble times a Laurent polynomial in q. From the point of view of the Stokes matrices, they are not as elementary as the thimble. Figure 9. The cycles C 1 , C 2 and C 12 z 2 1 z C 2 C 12 C 1 One Primary Field With Symmetry Breaking The other basic example with one critical point occurs in the presence of symmetry breaking with a single degenerate field of k = 1, and a single Bethe root. The Bethe equation reads 1 w − z = c c = − 2a ζ . (6.5) So w = z + 1 c . For convenience, we will take the constant c to be real and positive. The thimble again coincides ( fig. 8) with the most natural integration cycle C, along a ray starting at z and parallel to the positive real axis, passing through w. This example illustrates that an integration cycle in the presence of symmetry breaking may end at infinity. The cycle that we have just described cannot be replaced with an equivalent compact cycle. Two Critical Points Next we can consider examples with only one Bethe root w, but two critical points. This occurs with two primaries of k = 1 in the presence of symmetry breaking, or with three such primaries in the absence of symmetry breaking. The First Example We consider first the case of two primaries with symmetry breaking. We denote the positions of the primaries as z 1 and z 2 and we continue to assume that the symmetry breaking parameter c is positive. As long as z 1 − z 2 is not real, there is a symmetric choice of basic integration cycles C a , a = 1, 2: rays which start at z a and are parallel to the positive real axis. The difference C 1 − C 2 = C 12 is a segment from z 1 to z 2 . See fig. 9. Any two of these three cycles can be thimbles, depending on the relative values of c and z 1 − z 2 . Since there are always only two thimbles, it is never the case that all three of C 1 , C 2 , and C 12 are thimbles. In this simple example, we already have the basic ingredients of the general braid group representation. The elementary move is to exchange the two z a , either clockwise or Figure 10. The cycles C 1 , C 2 and C 12 after a braiding operation. counterclockwise. We start with a configuration in which the z a have distinct imaginary parts, so that the C a are well-defined. We may as well take the real parts of the z a to be zero. We want to define the branches of W along C 1 , C 2 , and C 12 so that it is true that C 12 = C 1 − C 2 . Picking any branch of W on C 12 , we define W on C 1 and on C 2 so that at the unique point where C 1 intersects C 12 or where C 2 intersects C 12 , the definitions agree. This will ensure that C 12 = C 1 − C 2 . Although C 1 and C 2 do not intersect, we can deform them slightly so that they meet at a reference point far to the right, and then the two values of W will agree at this reference point. (This is ensured by the fact that one can define W to be single-valued in the semi-infinite rectangle bounded by C 1 , C 2 , and C 12 .) z C 12 C 1 1 C 2 z 2 Suppose that Im (z 1 − z 2 ) > 0. Then we can cross to Im (z 1 − z 2 ) < 0 in two ways, with z 1 passing either to the left or to the right of z 2 . The two operations are inverses, so it will suffice to consider one in detail. If z 1 passes to the right of z 2 , then C 1 evolves continuously as a ray parallel to the positive real axis. On the other hand, C 2 does not. We can use instead C 12 , defined continuously as a segment from z 1 to z 2 . In the basis of C 1 and C 12 , this braiding move is diagonal: we only have to keep track of the branches of W and orientation of cycles. We want to express the final result in the basis of C 1 and C 2 . In the half-braiding, C 1 is multiplied by a factor of q − 1 4 : z 1 is transported clockwise around z 2 and the reference point does not move significantly. After the half-braiding, we rename C 1 as C 2 : C 1 → q − 1 4 C 2 . (6.6) On the other hand, C 2 becomes the cycle in fig. 10. It can be deformed to the sum of three pieces that zig-zag to and from Re z = ∞, as in fig. 11. Each of the three pieces is equivalent to a power of q times one of the original cycles C 1 and C 2 . Just as in (6.6), one piece (after again exchanging the labels of C 1 and C 2 ) is q − 1 4 C 2 . The other two pieces are images of −q − 1 4 C 2 and q − 1 4 C 1 under a deck transformation. Hence the braiding Figure 11. The cycle C 2 , after the braiding operation, has been copied from fig. 10: it starts at z 2 , curves below z 1 to the left and then goes to Re z = ∞. It is equivalent in homology to a zig-zag cycle, labeled C 2 in the figure, which, starting at z 2 , heads directly to Re z = ∞ before doubling back around z 1 and returning to Re z = ∞. Thus C 2 is the sum of three pieces, each of which heads to or from Re z = ∞; each piece is equivalent to a power of q times an elementary cycle C 1 or C 2 (a ray starting at z 1 or z 2 and parallel to the positive z axis). z 2 z 1 C 2 C 2 transformation is C 2 → q − 1 4 C 2 − q 3 4 C 2 + q 3 4 C 1 . (6.7) Notice that with these transformation rules, C 12 → −q 3 4 C 12 . (6.8) This is the same result that we found in the eqn. (6.4) (the sign of the exponent is reversed because in deriving (6.8), we braided z 1 clockwise around z 2 ); symmetry breaking does not affect the fact that C 12 represents the conformal block in which the two degenerate fields fuse to the identity. The braiding matrix has eigenvalues −q 3 4 and q − 1 4 , which correspond to the two possible fusion channels. The linear combination C 1 + q −1 C 2 transforms as C 1 + q −1 C 2 → q − 1 4 (C 1 + q −1 C 2 ). (6.9) and hence it represents the fusion in the channel of momentum −b. Now, other choices of normalization of C 1 and C 2 may occur more naturally in various situations. If we change the relative normalization between C 1 and C 2 , say by setting C 1 = q −s/2 C 1 and C 2 = q s/2 C 2 , we can get braiding formulae B 12 : C 1 → q −s− 1 4 C 2 C 2 → q − 1 4 − q 3 4 C 2 + q s+ 3 4 C 1 . (6.10) We will find the choice s = − 1 2 to be useful momentarily, so whenever we write C a we assume that choice of s. The behavior of the C a as a function of c(z 1 − z 2 ) is depicted in fig. 12. The C a fail to be well-defined when c(z 1 − z 2 ) is real. In this very simple example, the integral over the Figure 12. The cycles (C 1 , C 2 ) (defined as rays in the direction of Re cz that start at z = z 1 or z 2 ) provide a local basis for homology, but the definition of the cycles jumps by the braiding matrix B 12 across the line on which Z = c(z 1 − z 2 ) is real. z 1 z 2 = B 12 B 12 C 1 , C 2 C a can be expressed explicitly in terms of familiar functions: I a = Ca (w − z 1 ) 1 b 2 (w − z 2 ) 1 b 2 (z 1 − z 2 ) − 1 2b 2 e cz 1 2 + cz 2 2 −cw dw = c −1− 3 2b 2 Ca (W − Z) 1 b 2 W 1 b 2 Z − 1 2b 2 e Z 2 −W dW (6.11) Here we defined W = c(w−z 2 ) and Z = c(z 1 −z 2 ). This integral can be explicitly written in terms of Bessel functions. The integrals over C 1 and C 2 give a basis of Bessel functions with specific asymptotic behavior at large Z, and the braiding matrix B captures the Stokes phenomena of Bessel functions: I 1 = 1 √ πc 1+ 3 2b 2 Z 1 2b 2 + 1 2 Γ 1 + 1 b 2 K 1 2 + 1 b 2 Z 2 I 2 = q 1/4 √ πc 1+ 3 2b 2 Z 1 2b 2 + 1 2 Γ 1 + 1 b 2 K 1 2 + 1 b 2 − Z 2 (6.12) In the large Z limit, the integral along C a is controlled by the region where w is close to z a . It is useful to pick a branch of the logarithms such that for w near z 1 (or z 2 ), the sum of the logarithms in the superpotential approaches 1 4 log(z 1 − z 2 ) 2 . This is the same as the change in normalization between C a and C a for s = − 1 2 . This basis of C a will be useful whenever we are at strong symmetry breaking. In this basis B 12 := C 1 → q 1 4 C 2 C 2 → q − 1 4 − q 3 4 C 2 + q 1 4 C 1 . (6.13) So far, we have analyzed this problem using cycles C 1 , C 2 , and C 12 that are visible by inspection. Let us compare these cycles with the thimbles. The Bethe equation with two z's and one w is 1 w − z 1 + 1 w − z 2 = c (6.14) and it has two solutions. There are two regimes of interest. If |Z| >> 1, then the critical points are approximately w = z a + 1/c, a = 1, 2. For each of the two critical points, assuming that c > 0 and Re Z = 0, both w − z 1 and w − z 2 have positive real part, and it is natural in defining W to pick branches of log(w − z 1 ) and log(w − z 2 ) such that the imaginary parts are bounded by ±π/2. The thimbles defined this way coincide with the C a we defined above (and not with the C a ). The advantage of this choice is that it extends naturally to the general case of many fields with symmetry breaking, which we will treat in section 6.5. If |Z| << 1, then we have approximate critical points w = 1 2 (z 1 + z 2 ) and w = 2/c. The first critical point sits between the two z a , and the associated thimble is C 12 . The second critical point is associated to C 1 if Re Z > 0, C 2 otherwise. (C 12 , C 2 ) (C 1 , C 2 ) (C 1 , C 12 ) (C 1 , C 2 ) Z = 0 Z = 2i Z = −2i We depict the Stokes walls for the system of thimbles in fig. 13. There are regions in parameter space where the basis of thimbles consists of any two of C 1 , C 2 and C 12 , up to a choice of branches of the superpotential. Pairs of regions meet along Stokes walls, and triples of regions meet at the points Z = ±2i where the two critical points coincide. This is a general feature; in a generic problem of this type, there will always be loci of complex codimension 1 where two critical points coincide. In a plane transverse to such a locus, W can be modeled by a simple cubic function W = w 3 + δw of one variable w, with a parameter δ; this function has two critical points that coincide for δ = 0, where three Stokes walls meet. The local Stokes behavior is universal; it corresponds to the behavior of the Airy function. As we explained in our general discussion of section 5.1, the asymptotic behavior of integrals for b → 0 is clearest in the basis of thimbles. For the present problem, fig. 13 captures the relevant information. There are four regions: the upper and lower regions correspond to the system of thimbles we saw in the |Z| >> 1 limit, the two intermediate regions to the system of thimbles which we saw in the |Z| << 1 limit. The formulas (6.10) or (6.13) describe the braiding matrix B 12 that compares the region at the top of fig. 13, where the thimbles are C 1 and C 2 , to the region at the bottom, where again the thimbles are C 1 and C 2 . We did this computation by inspection, not by analyzing the Stokes lines. In deriving (6.13), we assumed that z 1 moves half-way around z 2 in a clockwise direction; this is equivalent to moving from the top to the bottom of fig. 13 with Re Z > 0. From fig. 13, we see that in this process, we will cross two Stokes lines. This means that in the basis of thimbles, the braiding matrix B 12 "decomposes" into a sequence of two elementary moves, each associated to one Stokes line. Each elementary move involves a gradient flow in which the Bethe root w flows from a critical point just to the right of z 2 to a critical point just to the right of z 1 . There are two such flows, differing by whether w passes above or below z 1 . The two flows, which occur at slightly different values of Im Z, are sketched in fig. 14. In the formula B 12 C 2 = (q −1/4 − q 3/4 ) C 2 + q 1/4 C 1 of eqn. (6.13), the term q 1/4 C 1 on the right hand side is the contribution of the formal monodromy alone, while the other two terms are contributions from the two gradient flow solutions. In the small Z region, braiding around Z = 0 is represented in the basis of thimbles by a triangular matrix: C 12 is an eigenvector. There is a general reason for this fact; the integral over C 12 is smaller than the integral over any other cycle either in the Z → 0 limit, or in the b → 0 limit. In either limit, the integral over C 12 is controlled by the usual saddle point approximation, and the critical point associated to C 12 is the one at which the Morse function is the smallest. So C 12 must be an eigenvector of the monodromy. With any number of z's and w's, the set of thimbles which has one w in between a given pair of very close z a span the "small" eigenspace of conformal blocks where the two degenerate fields of momentum − 1 2b fuse to the identity. The Second Example The second example with two critical points arises if there are three singular monopoles of minimum charge and a single Bethe root w. Placing the singular monopoles at z 1 , z 2 , z 3 , there are three obvious possible integration cycles: a straight path connecting z a to z b for any a, b. We can pick the branch of the superpotential in such a way that these three cycles add to zero. This definition makes sense if the three points are not aligned. The parameter space of z a is then split into two halves: either z 1 , z 2 and z 3 form a triangle with positive orientation, or they form a triangle with negative orientation. We will denote the three natural cycles in either case as C ± ab . So C + 12 + C + 23 + C + 31 = 0 C − 12 + C − 23 + C − 31 = 0 (6.15) but the two bases are related in an interesting way across the loci where the z a are collinear. There are three such loci, where one of the three z a passes between the other two. For example, if z 2 passes between z 1 and z 3 , C + 12 and C + 23 will be related to C − 12 and C − 23 simply by a change of branch of W, while the transformation of C + 13 then follows from (6.15). The Bethe equation 1 w − z 1 + 1 w − z 2 + 1 w − z 3 = 0 (6.16) is equivalent to a quadratic equation for w, so it has two solutions, corresponding to two independent thimbles. The thimbles are equivalent to two of the paths joining a pair of z's, but which pairs appear depends on the choice of the z's. If the z's are collinear, then the Bethe roots are located in the segments between adjacent z's, and the thimbles coincide with those segments. For example, if z 2 is between z 1 and z 3 , the two thimbles are C ± 12 and C ± 23 , up to powers of q. z 1 z 2 = z 1 z 3 = z 3 = z 2 As usual, the thimble joining z a and z b corresponds to the conformal block where the corresponding two degenerate fields fuse to the identity. The relations (6.15) correspond to the elementary "skein relation" between the three different ways to fuse two of the three primary fields to the identity. In other words, whenever z a and z b are close together, there is a thimble which joins them and is an eigenvector of the braiding of z a and z b , and that braiding matrix is triangular. We depict the Stokes walls in fig. 15. The integrals I ab can be readily evaluated in terms of hypergeometric functions. A Final Example There is one more example that is both instructive and relevant to understanding the general picture. This is the case of two z a of charge 1, accompanied by two w i . This can only happen in the presence of complex symmetry breaking. It is easy to see that the Bethe equations (3.37) have only a unique solution. Q and K are both of degree 2, so P is of degree 0 and can be set to 1. This leads to linear equations that uniquely determine the coefficients in Q. Since the solution of the Bethe equations is unique, there is only one conformal block and the monodromy in braiding the z a can only be multiplication by a function of q. If z 1 , and z 2 are well-separated, i.e. c(z 1 − z 2 ) has large absolute value, then the solution of the Bethe equations is easily described: w 1 ∼ z 1 + 1/c and w 2 ∼ z 2 + 1/c. There is an obvious integration cycle, with w 1 and w 2 integrated respectively over the rays from z 1 and z 2 to infinity in the c direction; these rays were labeled C 1 and C 2 in fig. 9. We can denote this cycle as C = C 1 × C 2 . The unique thimble for the problem is a small deformation of C if the imaginary part of Z = c(z 1 − z 2 ) is large. As we deform Z, the unique thimble will deform continuously. Here Morse theory is rather useful: it is rather tricky to verify by hand that C goes back to itself under braiding of z 1 and z 2 , as it requires a contour deformation which does not keep C in a simple product form. But the evolution of the thimble provides us implicitly with such a deformation. It is likewise far from obvious at first sight that the conformal block produced by the contour integral over w 1 and w 2 will be a simple function with abelian monodromy in the Z-plane. Nevertheless, this must be the case. The effect of braiding is easily computed at large c, by looking at the saddle point estimate for the integral. At the saddle, w 1 ∼ z 1 + 1/c and w 2 ∼ z 2 + 1/c remains true along the whole braiding, as long as it is executed at large |Z|. The Hessian of W is close to a large, Z-independent, multiple (∼ |c| 2 ) of the identity matrix, so the braiding phase at large Z is controlled by the value of exp(W/b 2 ) at the critical point. The relevant factors are (z 1 − z 2 ) −b −2 /2 (w 1 − z 2 ) b −2 (w 2 − z 1 ) b −2 (w 1 − w 2 ) −2b −2 ∼ (z 1 − z 2 ) −b −2 /2 . (6.17) Hence we recover the same braiding phase as in a setup with two z a of charge 1 and no w i . This last statement will be part of the input for constructing an effective abelian description in section 6.7. In an abelian theory, one would expect that the braiding of two objects depends only on the product of their charges. So braiding of two objects both of charge −1 (z's unaccompanied by w's) or two objects both of charge 1 (z's accompanied by w's) gives the same result. Braiding of an object of charge 1 with an object of charge −1 is less simple, since non-trivial gradient flows enter the picture, as we saw in section 6.3.1. General Picture With Symmetry Breaking In this section, we develop a general picture with symmetry breaking. We take the symmetry breaking parameter c to be real and positive; we consider any number of singular monopoles at positions z a , a = 1, . . . , d, and any number of Bethe roots w i , i = 1, . . . , q. At first we will set all the charges k a to 1. We can produce a basis of integration cycles by hand. We assume that the imaginary parts of the z a are distinct, and ordered so that Im(z a − z a ′ ) > 0 if a ′ > a. We define the rays C a which start at z a and are parallel to the positive real axis. Our integration cycles will be products C a 1 a 2 ···aq := C a 1 ×C a 2 ×· · · ×C aq for every subset of q distinct z's. This gives d q integration cycles, which in the region of large c can be interpreted as thimbles. Indeed the Bethe equations a 1 w i − z a = c + j =i 2 w i − w j (6.18) have approximate solutions for large c with each w i equal approximately to z a i + 1/c, with a i = a j for i = j. The corresponding thimbles are precisely the C a 1 a 2 ...aq . Summing over all q, we get 2 d critical points or conformal blocks in all, as expected. z Figure 16. An integration cycle for one singular monopole of non-minimal charge. Figure 17. An integration cycle for several singular monopoles of non-minimal charge. To z a we attach q a of the w's, for suitable q a . In the picture, the q a are 1, 3, and 2. The braid group representation is rather simple in this basis. We let the z a move around in the complex plane. The Stokes walls are approximately at the locus where the real parts of two z a coincide. Morse flows will occur if and only if a z a unaccompanied by a w i passes to the right of a z a ′ accompanied by a w i . The resulting behavior involves only the two z's that are crossing and is precisely what we analyzed in detail in section 6.3.1. The braiding matrix when z a crosses z a ′ acts non-trivially only on C a and C a ′ (and any cycle C a 1 a 2 ...aq that contains one or both of these), and takes the same form as (6.13). As in fig. 14, the braiding matrix for this process can be interpreted as resulting from a pair of Morse theory flows. It is tedious, but elementary, to check that the braid group relations are satisfied, as they should be. Degenerate Fields Of Any Charge Now, still with complex symmetry breaking, we will relax the constraint that all k a equal 1. First, we can consider a single z of charge k, with q Bethe roots. The Bethe equations only have solutions if k ≥ q. (This is clear from eqn. (3.37) for the opers.) Actually, according to the theory of Bethe equations, the solution is unique, for given q, and corresponds to a unique thimble or integration cycle. Indeed, it is easy to describe this unique possible integration cycle C (q) : one integrates each w i from z to +∞, along distinct, non-intersecting paths, as in fig. 16. We can then immediately describe a basis of integration cycles in the most general case, with any number of z a of charge k a , and the number of w i being q ≤ a k a . We get a unique cycle for each way to decompose q = a q a with q a ≤ k a : for each a, we integrate q a of the w i from z a to +∞, along distinct, non-intersecting paths, as in fig. 17. This gives an integration cycle a C (qa) a . At strong symmetry breaking (or equivalently, if the z a are well separated along the imaginary axis), these integration cycles correspond to the thimbles associated to the unique solutions of the Bethe equations with q a of the w i near z a . There is no obstruction, in principle, to derive the braid group representation in this basis. For a homological approach -essentially corresponding to the free field realizationto the construction of braid group representations for any k a , though without symmetry breaking, see [48]. Turning Off Symmetry Breaking Since symmetry breaking is so useful in simplifying our analysis, the question arises of verifying that symmetry breaking does not affect the values of knot or link invariants. The description of integration cycles in the previous section is applicable as long as the symmetry breaking parameter c is non-zero. But for c → 0, we lose some integration cycles, as the w i cannot go to infinity any more. In terms of critical points of W, the behavior for small c is easy to describe. For small c, for each solution of the Bethe equations, the Bethe roots split naturally into two subsets. Some number q 0 of Bethe roots, which we call w (0) i , i = 1, . . . , q 0 , remain of order 1 in the limit c → 0, while the remaining q ∞ Bethe roots, which we call w (∞) j , j = 1, . . . , q ∞ , are of order c −1 . In this situation, the Bethe equations for the w The counting can be carried out as follows, in terms of Wilson operators of the dual SU (2) gauge theory. The singular monopoles of charge k a correspond to representations R a of SU (2) of spin k a /2 and dimension k a + 1. The tensor product R = ⊗ a R a has dimension a (k a + 1). This representation can be decomposed as a direct sum of SU (2) modules of spin k eff /2 (where k eff is bounded by 0 ≤ k eff ≤ a k a , just as in (6.19)). The states that transform with this spin are as numerous as the solutions of the Bethe equations with a k a − 2q 0 = k eff /2 (and all possible values of q). Now consider braiding of the z a . For sufficiently small c, when one crosses a Stokes wall, there are Morse flows in which q ∞ becomes smaller, but no such flows in which q ∞ becomes larger. The reason for this is that for c → 0, the values of the Morse function h = Re W at a critical point are greater the greater is q ∞ . (This is because some contributions to h are of order ln(1/|c|) for c → 0. With the help of (6.19), one can show that the coefficient of ln(1/|c|) is an increasing function of q ∞ .) As a result, decomposing the space H of physical states according to the value of q ∞ , the monodromy matrix is block triangular: B ∼        * * * * * * * * * * 0 0 * * * 0 0 * * * 0 0 0 0 *        . (6.20) The diagonal blocks (which are of rank 2, 2, and 1 in the example given) are the monodromy representations that one would have in the absence of symmetry breaking for given q 0 = q − q ∞ . To the extent that one can compute knot or link invariants by taking traces of braid group representations, the off-diagonal blocks in (6.20) are not important as they do not contribute to traces. Actually, to compute the Jones polynomial and related invariants of knots and links, one needs in addition to the braid group representations an additional "fusion" operation in which a pair of z a of the same charge is created or annihilated. The additional information that we need to ensure that knot invariants are unaffected by symmetry breaking and do not change upon setting c = 0 is that fusion never involves creating or annihilating any w's at infinity. This is natural because of the local nature of the fusion operation. A Clarification A careful reader might notice a small sleight of hand in this derivation. The inequality k eff ≥ 0 was deduced from (3.22), but the original derivation of (3.22) was based on picking Q to have a smaller degree than P . The Bethe equations at c = 0 certainly have solutions in which this is not the case. Why are we entitled to restrict to this case? Suppose that at c = 0, we find a pair P 0 , Q 0 obeying the oper condition P dQ dz − dP dz Q = K(z). (6.21) Now suppose that we turn on very weak symmetry breaking. The oper equation becomes P dQ dz − dP dz Q − cP Q = K(z). (6.22) We hope that as c is turned on, there is a pair (P (z; c), Q(z; c)) obeying (6.22) and such that Q has an expansion Q(z; c) = Q 0 + cQ 1 + c 2 Q 2 + . . . . (6.23) This will ensure that the Q(z; c) has roots that approach the roots of Q 0 as c → 0, plus possible additional roots that go to infinity for c → 0. One might expect that P would have an expansion of the same form, but this is not the case. The degrees p and q of polynomials P, Q obeying (6.21) satisfy p + q = k + 1, but as soon as c = 0, the relation becomes p + q = k. So the degree of P must drop as soon as c = 0. The way that this happens is that the expansion for P is actually P (z; c) = c −1 P −1 + P 0 + cP 1 + . . . (6.24) where we write P 0 for the coefficient of c 0 , as this polynomial does not coincide with P 0 . Plugging (6.23) and (6.24) in (6.22), we learn from the term of order c −1 in the equation that P −1 is a multiple of Q 0 , and this multiple must be nonzero or else the term of order c 0 in the equation would force p + q ≥ k + 1. So p ≥ q 0 , and this together with p + q = k and q ≥ q 0 implies that q 0 ≤ k/2, as desired. The moral of the story is that solutions of the Bethe equations for c = 0 with q > k/2 do exist, but they are unstable to symmetry breaking. Various forms of this statement are known in the literature on integrable systems. Three-Dimensional Interpretation To apply our results to knots and not just to braids, it will help to understand the threedimensional interpretation of what we have computed so far. We consider knots in R 3 , so the four-manifold on which we are trying to count solutions of eqns. (1.1) is M 4 = R 3 ×R + . We describe R 3 with Euclidean coordinates x 1 , x 2 , x 3 . The adiabatic evolution considered so far has been in the x 1 direction, while we have combined the other coordinates to a complex variable z = x 2 + ix 3 . As usual, we take the gauge group to be G = SO(3). We will focus on the case of strong symmetry breaking. The symmetry breaking involves the choice of an expectation value φ = diag( a, − a), where a is a vector in R 3 . As long as the complex symmetry breaking is nonzero, this vector does not point in the x 1 direction, that is, the direction that we chose for the adiabatic evolution. Topologically, if the directions are not the same, we may as well think of them as orthogonal: we consider adiabatic evolution in the x 1 direction, and symmetry breaking with c = −2 a/ζ pointing in the positive x 2 direction. This will be strong complex symmetry breaking with real, positive c, in the terminology that we have used so far. We will concentrate on the case that the strands have minimum magnetic charge only, and thus are dual to the two-dimensional representation of SU (2). At a generic time, each strand has two possible states: it is or it is not accompanied by a Bethe root w i . In the low energy effective abelian gauge theory, the strand has magnetic charge 1 if accompanied by a Bethe root, and otherwise −1. The magnetic charge of a given strand changes when the Bethe root accompanying that strand moves to another strand. In the context of adiabatic evolution, this results from a Morse theory flow in which a Bethe root moves from one strand to another. This happens at a value of x 1 at which one crosses a Stokes wall. The lesson of section 6.3.1 is that (in the limit of strong symmetry breaking) one crosses a Stokes wall at a time (that is Figure 18. A three-dimensional picture of the process that leads to a Morse theory flow. Just two strands are pictured here. The x 1 direction is plotted vertically and the x 2 direction runs into the paper. We are looking at the picture from along the negative x 2 axis. The coordinates x 2 and x 3 combine to a complex variable z = x 2 + ix 3 . For a particular choice of the direction of complex symmetry breaking, a non-trivial Morse theory flow can occur only at values of x 1 at which the two strands have the same value of x 3 and thus project to the same point in the x 1 − x 3 plane. In the language of knot theory, we make a two-dimensional picture by projecting a knot or link to the x 1 − x 3 plane. In this projection, there are crossing points, and these are the points at which a non-trivial Morse flow may occcur. a value of x 1 ) at which two strands have the same value of x 3 = Im z. So the crossing of a Stokes wall occurs when two strands have common values of x 1 and x 3 , and thus differ only in x 2 . Differently put, this happens when the two strands are separated in the direction of symmetry breaking. x 1 x 2 x 3 A three-dimensional picture clarifies things ( fig. 18). When two strands align along the x 2 direction, a Morse flow can occur. The Morse flow occurs on a time scale fast compared to the adiabatic evolution, so in the adiabatic picture it is essentially instantaneous. The flow involves a Bethe root moving towards the positive x 2 direction. The flow can only occur between strands of opposite charge, and will allow positive charge to move towards positive x 2 only. In general, given any knot, we can usefully project it to the x 1 −x 3 plane, and look at it from the negative x 2 direction. We suppose that the embedding of the knot in R 3 is generic enough so that its tangent vector always has a non-zero projection to the x 1 − x 3 plane, and moreover so that the projection of the knot to the plane has only simple crossings; finally we will assume that the function x 1 has only simple maxima and minima along the knot. A simple example of a knot projection is given in fig. 19. In such a knot projection, the low energy abelian description is valid away from crossings, so away from crossings and local maxima and minima, which we discuss in section 6.7.1, each strand can be labeled $\mathbb{R}_+$ Figure 19. A simple example of a knot projection, with only simple crossings and simple maxima and minima of the height function. This figure also illustrates the fact that the projection to a plane of an oriented knot allows one to define an integer invariant p = (1/2π) ds dθ/ds that equals the total change in moving around the knot of the angle θ defined by the tangent vector to the knot. For the example shown, p = 2. p is the only invariant of a knot projection that can be written as a local integral along the knot. It depends on the choice of projection and is not an invariant of the knot per se. Our previous calculations assign a weight to each possible crossing. These weights are just q ±1/4 if the charges are unchanged at the crossing. When the charge jumps, the weight is q ±1/4 − q ∓3/4 . The weights are summarized in fig. 20. We have arrived at a known vertex model representation of the braid group representations associated to the Jones polynomials. See for example the R-matrix on page 125 of [49] (where A is our q 1/4 ) or see [50], especially pp. 1777-8, or fig. 10 of [51]. Creation And Annihilation Of Strands In order to reproduce the knot invariants, we need to understand the loci where the adiabatic approximation is invalid, because two strands are created or annihilated. Although the adiabatic approximation is invalid near such points, the low energy abelian description remains valid. As an immediate consequence, conservation of charge in the abelian theory makes it clear that only pairs of strands with opposite charge can be created or annihilated. The map from line operators in a microscopic theory to line operators in an effective low energy description is akin to the corresponding map for local operators, but it has a little twist: the coefficients are not c-numbers, but rather quantum mechanical vector spaces that have to be transported along the line. In the present case, the vector spaces are one-dimensional (in the U (1) theory, an 't Hooft operator has no structure except its charge), but we can still get an overall factor from parallel transport. This factor has to be written locally along the loop, and must also be consistent with topological invariance. For a knot without any additional structure there is no topological invariant that can be written as a local integral along the knot, but once one is given a projection of the knot to a plane -in our case the x 1 − x 3 plane -there is precisely one such invariant, the total winding number of the tangent vector to the knot ( fig. 19). This can be written as (1/2π) ds dθ/ds, where the knot is parametrized by a variable s, and θ is the angular direction of the tangent vector to the knot in the x 1 − x 3 plane. To define the sign of this invariant, one needs an orientation of the knot, which in our case comes from the direction of flow of magnetic charge. When the microscopic SU (2) theory is approximated at long distances as an effective U (1) theory, the effective action for an 't Hooft operator may acquire a term −iη ds dθ/ds, with a universal coefficient η. If the tangent direction to a knot changes by an angle ∆θ, this will contribute a factor exp(iη∆θ) (6.25) In creation or annihilation of a pair of strands, the change in the tangent angle is ∆θ = π or −π, depending on whether the positive charge bends to the left or to the right. This effect will associate a universal factor exp(±iπη) to each creation or annihilation event, depending on the direction of flow of charge. Up to sign, there is a unique choice of η that leads to a knot invariant, namely exp(iπη) = ∓iq 1/4 . The choice of sign does not matter, since every knot has an even number of creation and annihilation events; we will take exp(iπη) = −iq 1/4 . The weights for creation and annihilation events with this value of η are shown in fig. 21. The value of η could possibly be computed directly by studying the four-dimensional gauge theory BPS equations near the abelian limit. The factor of q 1/4 should express the difference between instanton number computed microscopically in the SO(3) theory and instanton number computed in the low energy U (1) theory. The factor of ∓i should involve a comparison between fermion determinants for SO(3) and for U (1). Globally, the contribution of a given classical solution to the Jones polynomial is proportional to the sign of the fermion determinant, a subtle invariant that may receive contributions from charged modes in the microscopic SO(3) theory; to write this sign as a product of local factors, one apparently must use factors of i ±1 , with overall signs that depend on how one trivializes the determinant line bundle. The value of η can actually be deduced by combining the information which is available in the abelian description and information available from the conformal block description. When a pair of strands is created or annihilated, we expect them to be fused to the identity. This means that the two nearby strands, located at say z 1 and z 2 , are accompanied by a Bethe root w, and that the integration cycle for this Bethe root is the thimble C 12 connecting z 1 and z 2 . On the other hand, in section 6.3.1, we also defined integration cycles C 1 and C 2 with the property that the strands at z 1 and at z 2 have definite magnetic charges. (For example, in C 1 , the Bethe root accompanies z 1 , so the charges of the two strands are respectively 1 and −1. We order z 1 and z 2 in order of decreasing x 3 = Im z.) The relation among these cycles turned out to be C 12 = C 1 − C 2 = q −1/4 C 1 − q 1/4 C 2 . (6.26) The ratio of the ampitude to create a pair of charges (1, −1) to the amplitude to create a pair of charges (−1, 1) is the ratio of the coefficients on the right hand side of (6.26), or −q −1/2 . On the other hand, in the abelian description, this ratio is exp(2πiη). So exp(iπη) = ∓iq −1/4 , as shown in fig. 21. We have arrived to what is essentially a standard vertex model algorithm for calculating the Jones polynomial: given a knot or a link, pick a projection to the x 1 −x 3 plane such that there are only simple crossings and the function x 1 only has simple maxima and minima. Divide the link into segments separated by the maxima, minima and crossings. Label the segments by ± and sum over all labelings, weighting each labeling with the product Figure 22. An unknot projected to the plane in the most obvious way. There are no crossings, but there is a creation event and an annihilation event. In the vertex model, the invariant for the unknot is computed by summing over all ways to label the two sides of the knot (that is, the segments between crossing, creation, and annihilation events) by charges + or −. Each labeling is weighted by the product of the appropriate local factors. In the present example, only the two choices shown make nonzero contributions, leading at once to the result −q 1/2 − q −1/2 . of the local weights at crossings, maxima, and minima. Some simple examples are given momentarily. − + − + + −q −1/2 = −q 1/2 Some Examples And Some Topological Details For the simplest example of the use of the vertex model, we compute the expectation value of an unknot, projected to the plane in an obvious way ( fig. 22). Summing over the two possible labelings of the diagram, we arrive at the result −q 1/2 − q −1/2 . For a slightly less trivial example, we consider the two projections of a single strand to the x 1 − x 3 plane shown in fig. 23. In either (a) or (b), the knot projections shown on the left or right can be deformed into one another, so one might expect them to be equivalent. But in the present context, this is actually not the case. The twist of (a) relative to (b) introduces a factor of −q 3/4 , which can be evaluated by making use of the weights of the vertex model. It is instructive to actually do this; the same factor −q 3/4 arises whether the magnetic charge of the strand is +1 or −1, but the details of the computation are quite different in the two cases. Since the factor −q 3/4 does not depend on the magnetic charge carried by a given strand, it is universal: adding a twist of the type shown in the figure to any strand in an arbitrary knot or link multiplies the associated invariant by −q 3/4 . A similar twist of the opposite handedness multiplies the invariant by −q −3/4 , for similar reasons. This factor means that the invariant associated to a knot (or link) by the quantum field theory depends on a choice of "framing." A framed knot is a knot that is slightly thickened into a ribbon. One keeps track of how the ribbon is twisted and (in the present context) adding a twist multiplies the invariant by a factor of −q 3/4 . A knot that is presented with a projection to a plane comes with a natural framing, given by a slight thickening in the vertical direction, normal to the plane. A little thought (or experimentation with a strip of paper) shows that although the pictures on the left and right of fig. 23(a) or (b) are topologically equivalent if one ignores the framings, they do differ by one unit of framing. In the context of three-dimensional knot invariants that are associated to two-dimensional conformal field theory, conformal primary fields in two dimensions are associated to line operators in three dimensions. If a conformal primary has dimension h, then in a unit change in framing, the corresponding line operator is multiplied by exp(2πih). The factor −q 3/4 is indeed exp(2πih), where the conformal dimension of the degenerate Virasoro primary field V −k/2b is h V (k) = − k 2 − k(k + 2) 4b 2 ,(6.27) and in addition q = exp(−2πi/b 2 ), and the vertex weights that we have described are for the minimum charge case k = 1. At this stage, an important detail arises. The vertex weights that we have described are appropriate for a certain natural normalization of the Jones polynomial, which has been used in the literature, for instance in [49]. However, a slightly different normalization arises in SU (2) Chern-Simons theory. In Chern-Simons theory, the expectation value of an unknot labeled by the two-dimensional representation of SU (2) is q 1/2 + q −1/2 , which differs in sign from what we deduced in fig. 22 using the vertex weights. (The sign is easily checked in Chern-Simons theory. Since q = exp(2πi/(k ∨ + 2)) in SU (2) Chern-Simons theory, where k ∨ is the level, the weak coupling limit k ∨ → ∞ corresponds to q = 1; for q = 1, the expectation value of a Wilson loop in any representation is simply the dimension of the representation, or +2 for the two-dimensional representation.) Similarly, the dimension of a primary field related to a representation of SU (2) of spin j = k/2 is h CS (k) = k(k + 2) 4(k ∨ + 2) ,(6.28) so that the factor acquired in a unit change of framing is exp(2πih CS ) = q 3/4 , without the minus sign found in fig. 23. Clearly the discrepancy in sign reflects the fact that h V (k) − h CS (k) = −k/2, independent of b and k ∨ . The comparison with Chern-Simons theory is not necessarily a problem for the present paper, in which we have simply started with the four-dimensional gauge theory equations (1.1). However, one would like to know the best interpretation of these signs in the context of the duality presented in [14] between Wilson operators of Chern-Simons theory and singular monopoles at the boundary. We believe that the interpretation may be that the dual of a Wilson operator of spin k/2 in Chern-Simons theory is actually a boundary 't Hooft operator that carries angular momentum k/2 and is fermionic when k is odd. (For 't Hooft operators defined on the boundary of a four-manifold, the relevant rotation group is SO(2) or rather its double cover Spin(2); this group is abelian and has one-dimensional representations, labeled by the angular momentum k/2.) The fermi statistics for odd k would give a minus sign for every crossing (relative to what is presented in fig. 20) and a minus sign for every closed loop; including these signs brings the results obtained by braiding of Virasoro degenerate fields in agreement with the results obtained by braiding in Chern-Simons theory. Finally, it is instructive to compare the computation in fig. 23 to an equivalent computation if the gauge group were simply G = U (1) instead of SO(3). There would be two differences. First, the factors in fig. 21 associated to creation and annihilation of a pair of strands would simply be 1. (Those factors come entirely from integrating out massive degrees of freedom of the SO(3) theory, in reducing to an effective abelian description at low energies.) Second, charge exchange processes are absent for G = U (1) (as there are no smooth monopoles), so the second contribution in fig. 23(b) would be absent. A look back to fig. 20 shows that for G = U (1), evaluation of either fig. 23(a) or (b) gives a simple factor of q 1/4 , instead of −q 3/4 . Two comments are in order: • The minus sign of fig. 23 is absent for G = U (1) (and similarly the minus sign in fig. 22 is absent). • In Chern-Simons theory, the power of q is the quadratic Casimir invariant of the relevant representation of G ∨ . The quadratic Casimir of a representation of highest weight j is j 2 for G ∨ = U (1) and j(j + 1) for G ∨ = SU (2). For j = 1/2, this gives q 1/4 or q 3/4 in the abelian and nonabelian cases, respectively. x 2 x 3 x 1 Figure 24. To make the charge exchange process more visible, we have exchanged the coordinate axes relative to fig. 18. The x 2 axis now runs vertically while x 1 runs horizontally. Charge exchange occurs at values of x 1 at which two strands differ only in the value of x 2 , so with the coordinate axes aligned as in this picture, the charge exchange invoves a flow of charge in the vertical direction, represented by the dotted line. We take this to represent the propagation of a BPS soliton. The soliton propagates along the axis of symmetry breaking, so it is described by a solution with real symmetry breaking only -in fact, by the bare Miura oper of equtation (3.31) with no 't Hooft operator and a single Bethe root. Gradient Flow And Strings We can give a more concrete physical interpretation to the gradient flows which occur when strands of appropriate charge cross. Propagation of magnetic charge from one strand to another can be described by motion of a magnetic monopole between the two strands. If we rotate the coordinates and think of the x 2 direction as "time," we should be able to see the relevant object as a time-independent solution in the presence of only real symmetry breaking. In fact, we have already described precisely the necessary solution: it is associated to the bare Miura oper with a single Bethe root and no singular monopole that was described in eqn. (3.31). So we visualize the charge exchange process as propagation of an object in the x 2 direction, as in fig. 24. This picture is oversimplified, as it ignores the existence of a fourth dimension, normal to the boundary R 3 that contains the knots. An alternative picture showing the role of the y direction is given in fig. 25. When propagating in the x 2 direction, the soliton settles at a value of y that is given by the solution for the bare Miura oper. (How to compute this value is explained most fully in section 7.) Of course, this description is only good if the soliton propagates far enough in x 2 that it has "time" to reach the equilibrium value of y. So it is only good if the strands are sufficiently far separated, or the symmetry breaking is strong enough. We have gained an intuitive picture of the solutions of the four-dimensional BPS B A x 2 Figure 25. An alternative view of the charge exchange process that was pictured in fig. 24, to show the role of the y direction (here depicted as the direction normal to the plane that contains the two strands). A soliton propagating between two boundary points A and B that are separated by a long distance in the x 2 direction will bend away from the boundary, to reach a value of y corresponding to the solution for the bare Miura oper. equations at strong symmetry breaking: they describe smooth monopole configurations stretched between the singular monopole strands. The contribution of each configuration to the knot invariant is then the product of two types of factor. One type arises from the map from the microscopic nonabelian theory to the abelian theory, while the other is computed in the abelian theory. Factors of the first type appear where pairs of strands are created or annihilated and where a smooth monopole is emitted or absorbed by a boundary singular monopole. In the abelian description, the smooth monopoles also look like singular monopoles, of charge 2, but not attached to the boundary. The whole configuration looks like a web of monopole strands, to which the abelian theory associates an overall power of q. Analog in the Dual Chern-Simons Theory It is entertaining to carry this picture all the way back to three-dimensional Chern-Simons theory, before all the dualities which brought us to the four-dimensional description studied in the present paper. A key step in relating the two pictures is S-duality, which luckily is very transparent at strong symmetry breaking, as it reduces to electric-magnetic duality in the abelian gauge theory. We get immediately a sum over configurations of massive W -bosons stretched between Wilson lines, and only carrying electric charge towards the positive x 2 direction. This suggests to look for a gauge condition in nonabelian Chern-Simons theory which would have this effect. It is easy to indentify it: one can pick a "partial" axial gauge fixing which reduces the A 2 component of the gauge field to the Cartan subalgebra. We write the nonabelian gauge field as A = Bt 3 + W + t + + W − t − , and impose the A 2 = 0 gauge condition for W ± ; it will not be necessary here to make a gauge choice for the diagonal gauge field B. The action for charged W -bosons then reduces to k ∨ 4π d 3 x W + D 2 W − , D 2 = ∂ ∂x 2 + [B 2 , · ]. (6.29) The equation for the propagator is D 2 G(x, y) = 2π k ∨ δ 3 (x − y) (6.30) and has a solution 2π k ∨ exp y 2 x 2 B 2 δ(x 1 − y 1 )δ(x 3 − y 3 )θ(x 2 − y 2 ),(6.31) which only describes propagation of charge towards the positive x 2 direction. It is pretty clear that such a partial gauge fixing will lead to a version of the vertex model, though it may be tricky to compute the precise vertex weights. The necessary computations are likely to be somewhat similar to those involved in studying Chern-Simons theory in an ordinary axial gauge -for example, see [52,53] -or in a certain almost axial gauge [54]. Somewhat analogous is the use of a complex version of axial gauge to derive the Knizhnik-Zamolodchikov equations [55]. Up to Six Dimensions With an eye to future categorification, it is useful to lift this abelian picture all the way to six dimensions. After all, in the presence of strong symmetry breaking, the six-dimensional (0, 2) theory is not that mysterious. It is a theory of self-dual two-forms coupled to heavy dynamical BPS strings. The six-dimensional setup which leads to knot invariants involves the six-dimensional theory on the product R × M 3 × D, where D is a copy of R 2 with the geometry of a semi-infinite cigar (see eqn. (7.28)). The knot itself is represented by a knotted two-dimensional defect placed at the tip of the cigar. The q-grading comes from the conserved angular momentum derived from the rotational symmetry of D. In the absence of symmetry breaking, the fact that the metric on D is cigar-like rather than being the Euclidean metric is important in order to get a well-defined space of states for the defect. A defect placed in flat six-dimensional space would be strongly coupled to the bulk SCFT. On the other hand, in the presence of symmetry breaking the IR physics is free, and we can hope to recover the space of states from bound states of the dynamical heavy strings and the defect. The cigar geometry would then not play a significant role, and D can be replaced by a flat R 2 . Now, we will specialize to M 3 = R 3 , and for brevity we will take the knot to be timeindependent. The BPS condition for a time-independent dynamical string is very simple: it must be straight, and aligned with the symmetry breaking (which we will still take to be the x 2 direction). This is literally the six-dimensional lift of the condition satisfied by the smooth monopoles, and leads to the same vertex-model picture when projected on x 2 . The defect strands have two possible ground states in the abelian low energy description, of opposite two-form charge. A junction with a dynamical string allow the two-form charge to jump. The interesting question is to reproduce the weights of the vertex model from sixdimensional considerations. This includes both contributions from the abelian theory of self-dual two-forms sourced by the strings, and contributions from the worldvolume theory of the strings. We will not compute the former here, but we can give some insight on the latter. As summarized in fig. 20, the vertex model weight for a charge exchange process contributing to the Jones polynomial is q ±1/4 − q ∓3/4 . (6.32) In the six-dimensional picture, the Jones polynomial is supposed to come from a sum over BPS states, weighted by q P (−1) F , where P is the conserved charge that corresponds to rotation of D, while F , which one might loosely call fermion number, is a certain Rsymmetry generator. We interpret the relative factor −q ±1 between the two terms in (6.32) to mean that a BPS string connecting two strands in a knot has two physical states, differing by 1 in both angular momentum (to account for the factor of q ±1 ) and fermion number (to account for the minus sign). To understand this, we can focus on the two crossing strands, and the string stretched between them. Crucially, a single strand is half-BPS, two non-parallel strands are quarter-BPS, but the configuration of two strands exchanging a BPS string breaks symmetry further down to eighth-BPS. So the string breaks two supercharges, and hence it must have two ground states, exchanged by the action of the broken supercharges. This immediately gives the desired difference in quantum numbers. Hence the space of approximate ground states for the system can be described as follows. Let S be the set of configurations of the vertex model (labelings of strands by + or −, with smooth strings attached at crossings where a label jumps). For every smooth string, introduce a two-dimensional Hilbert space with quantum numbers derived from the last paragraph. To each s ∈ S, introduce a Hilbert space H s that is defined as the tensor product of the two-dimensional factors associated to the smooth strings. Then an approximation H 0 to the space of BPS states is H 0 = ⊕ s∈S H s . The grading of H 0 (by angular momentum and fermion number) is affected by the abelian field configurations sourced by the system of strings; for example, the self-dual abelian tensor fields can carry angular momentum. In order to compute Khovanov homology, one needs to evaluate the differential acting in this space of approximate ground states, by searching for instanton configurations which interpolate between different states in the past and future. The BPS condition for a timedependent BPS string is still rather transparent. The worldsheet should be holomorphic in complex coordinates x 0 + ix 2 and x 1 + ix 3 . Notice that a time-independent string stretched along the x 2 direction (and thus parametrized by x 0 and x 2 , with x 1 and x 3 fixed) indeed has a holomorphic worldvolume. Thus we expect to be able to build the differential for Khovanov homology from the data of holomophic curves in R × M 3 which end on the knot. This avenue seems promising for future development. An Effective Superpotential For Monopoles Overview Of Results Starting in section 2.4, we interpreted solutions of three-dimensional supersymmetric equations in terms of configurations containing smooth BPS monopoles. However, the considerations were purely qualitative. In this section, we will make the reasoning quantitative. We will construct an effective superpotential for smooth BPS monopoles on R 2 ×R + interacting with a Nahm pole and singular monopoles on the boundary. This effective superpotential will account in a direct way for all qualitative results from sections 2 and 3 about what solutions to our equations do or do not exist for various values of the parameters. Also, by integrating out some massive fields, we will be able to recover the Yang-Yang function (3.52) that has been one of our main tools. Let us first consider our underlying supersymmetic equations (F − φ ∧ φ + t d A φ) + = 0 (F − φ ∧ φ − t −1 d A φ) − = 0 d A ⋆ φ = 0, (7.1) on R × R 3 (where the first factor is the "time" direction) and ask how the solutions of Bogomolny equations for smooth monopoles can be embedded as solutions of these equations. This is possible precisely if t = 1 (or −1), the only nonzero component of φ is the time component, which here we will call φ t (rather than φ 1 , as before), and we also set the time component of A to zero. Then the equations (7.1) reduce to the Bogomolny equations F = ⋆d A φ t . (7.2) For the Bogomolny equations to have smooth monopole solutions, the field φ t must have an expectation value at infinity, which means that the real symmetry breaking parameter a 1 of section 2.4 must be nonzero. On the other hand, the complex symmetry breaking parameter a of section 2.5 must vanish (or φ t would not be the only nonzero component of φ). The basic solution of the Bogomolny equations on R 3 is the one-monopole solution for G = SO(3) or SU (2). It has has magnetic charge m = 2 (in units in which the basic singular monopole has charge m = 1). The moduli space of the one-monopole solution is P = R 3 × S 1 , where R 3 measures the center of mass position of the monopole on the spatial manifold R 3 , and S 1 is parametrized by a collective coordinate ϑ for the U (1) gauge symmetry that is unbroken at infinity. Of course, P is a hyper-Kahler manifold; this follows from the unbroken supersymmetry of the Bogomolny equations on R 3 . However, since we will be considering perturbations that break some of the supersymmetry, for our purposes it is more useful to merely look at P as a complex manifold in one of its complex structures. We decompose R 3 as R 2 × R (which we will eventually replace with R 2 × R + ). The motion of the smooth monopole along R 2 is parametrized by a chiral superfield W . And the position y of the smooth monopole in the R direction combines with the collective coordinate ϑ to a second chiral superfield Y = a 1 y + iϑ. (7.3) Now we want to construct an effective superpotential W(W, Y ) that describes this situation. If a = ζ = 0, there is a smooth monopole solution for every value of W and Y , which means that every value of W and Y is a critical point of W, so W must vanish (modulo an irrelevant constant). On the other hand, if either a or ζ is nonzero, then there is no supersymmetric monopole solution, meaning that W has no critical point. But turning on a and ζ preserves the symmetries of adding a constant to W or Y . So W must be invariant modulo an additive constant under constant shifts of W or Y ; in other words, W must be a linear function of W and Y . In fact, the form of W is W = aW + ζY. (7.4) This follows from the following considerations. Invariance under rotations of R 2 implies that a and W can only appear as the product aW . On the other hand, although at ζ = 0, the superpotential W is given microscopically by the single-valued function (2.6), as soon as ζ becomes nonzero it is given by the Chern-Simons function (3.6), which is only singlevalued mod 2πiZ. Since Y is similarly single-valued mod 2πiZ (because of the angular nature of ϑ), the effective superpotential (7.4) has a multivaluedness that just matches that of the microscopic description. Now let us turn off a and ζ but replace R 2 × R by R 2 × R + , where R + is the half-line y ≥ 0 and we assume the usual Nahm pole at y = 0. We assume symmetry breaking for y → ∞ with φ t → diag(a 1 , −a 1 ) while of course A and (at a = 0) φ vanish for y → ∞. In any solution, the fields approach these asymptotic values exponentially fast for y → ∞. This reflects the fact that near y = ∞, the gauge symmetry is reduced from SU (2) to U (1) by the expectation value of φ 1 , and all charged fields have masses proportional to |a 1 |. In particular, in the absence of singular or smooth monopoles, the solution for the full system (7.1) is given by a solution of Nahm's equations (the relevant solution is described in [16]) in which the fields approach their vacuum values exponentially fast for y → ∞. Next, still with a = ζ = 0, let us add a smooth monopole with positions W , Y . There is not an exact solution for the smooth monopole, because the Nahm pole forces charged components of φ to have nonzero values that "repel" the monopole to y = ∞. However, these charged fields are exponentially small for large y, so a smooth monopole located at large y is exponentially close to being a solution. This being so, a smooth monopole that is located at large y must be governed by an effective superpotential. This superpotential must be a single-valued function of Y that vanishes exponentially for y → ∞. These conditions are satisfied by a linear combination of exponentials exp(−nY ), n = 1, 2, 3 . . . . However, it will soon become clear that the expected qualitative picture emerges if and only if at a = ζ = 0, W is linear in exp(−Y ): W = Λ exp(−Y ),(7.5) for some constant Λ. In principle, it should be possible to compute this result by evaluating the microscopic superpotential (2.6) for an approximate solution consisting of a smooth monopole at large y in the presence of a Nahm pole. The exponentially small term should come from W boson exchange between the boundary and the monopole. Instead of attempting such a computation, we will take a shortcut in this paper: we will consider a representation of the smooth monopole and the Nahm pole by a configuration of branes, and in that context the exponential superpotential (7.5) will emerge from a simple brane instanton. Postponing that analysis to section 7.3, let us discuss the implications of (7.5). First we assume that a = ζ = 0, so that (7.5) is the full superpotential. We see at once that W has no critical point so there is no supersymmetric solution in the presence of the smooth monopole. This is in full accord with the analysis in section 2.4. With real symmetry breaking only and ζ = 0, a supersymmetric solution is expected to be unique (this remains true even in the presence of singular monopoles at y = 0, which we have not yet included in W) and does not require smooth monopoles. Now let us see what happens when we turn on a and ζ. We construct the full superpotential by simply adding the various terms that we have found so far: W = aW + ζY + Λ exp(−Y ). (7.6) The justification for including a and ζ in this way is the same as before (Λ may now depend on ζ). Let us look for critical points. We see at once that there is no critical point unless a = 0 and ζ = 0. If these conditions are satisfied, there is a one-parameter family of critical points, parametrized by W , with exp(−Y ) = ζ Λ . (7.7) (The uniqueness of the solution for exp(−Y ) holds precisely because in (7.5) we took W to be linear in exp(−Y ).) We found a similar result in section 3.5.1 where we described in eqn. (3.31) a family of solutions that exist precisely if a = 0 and ζ = 0. The solution in question corresponded to a Miura oper with only real symmetry breaking, a single Bethe root at an arbitrary point w ∈ R 2 , and no singular monopoles. We interpret the Bethe root w as the value of the superfield W -in other words, the position of the smooth monopole in R 2 -while the position of the smooth monopole in the R + direction is determined in (7.7). The reason that this description makes sense is that for small ζ, the smooth monopole is located at large y, where the effective superpotential W is valid. The next step is to include singular monopoles. As in section 2.2.1, in the presence of singular monopoles of charge k a located at positions z a , a = 1, . . . , d in the complex plane, it is convenient to introduce the polynomial K(z) = d a=1 (z − z a ) ka . In section 7.2, we will argue that the only effect of the singular monopoles is to multiply the exponential term by K(W ), so that the superpotential becomes W = aW + ζY + ΛK(W ) exp(−Y ). (7.8) Now let us examine the implications of this formula. Suppose first that ζ = 0. Then the conditions for a critical point are K(W ) = 0 = a + ΛK ′ (W ) exp(−Y ). (7.9) The first condition says that W must equal one of the z a . For a = 0 and all k a = 1, the second condition then has no solutions. If k a > 1 for some a, still with a = 0, the second condition is satisfied for arbitrary Y . All this is in keeping with what we found in sections 2.3 and 2.4. Now suppose that a = 0. If k a = 1, the second condition in (7.9) determines Y uniquely, and if a is small, then Y is large so that the analysis is valid. For k a > 1, the second condition in (7.9) cannot be satisfied. All these statements match what was found in section 2.5 from a quite different point of view. 12 Now let us consider the case that ζ = 0. The most illuminating way to proceed is to integrate out the massive field Y to generate an effective superpotential for W . For fixed W , the condition ∂W/∂Y = 0 has the unique solution exp(−Y ) = ζ/ΛK(W ). Setting Y to this value and evaluating W, we find (modulo an irrelevant constant) W = aW + ζ log K(W ) = aW + ζ a k a log(W − z a ). (7.10) But this is the Yang-Yang function (3.52) for the case of a single Bethe root W = w, modulo terms that depend only on the z a and not on W ; the present derivation is not sensitive to those terms. At this point, the reader hopefully would like to see a similar derivation leading to the general Yang-Yang function with any number of Bethe roots. For a general case with q smooth monopoles, we describe the positions of the i th smooth monopole by chiral superfields W i , Y i , i = 1, . . . , q. The definition of these fields is somewhat subtle and is discussed in section 7.2. The expectation values of the W i will turn out to be the Bethe roots w i of section 3.4. As in that discussion, it is convenient to introduce the polynomial Q(z) = q i=1 (z − W i ). It turns out that the generalization of the superpotential (7.8) to an arbitrary number of smooth monopoles is W = a i W i + ζ i Y i + Λ i K(W i ) Q ′ (W i ) exp(−Y i ). (7.11) 12 In section 2.5, we found, for general ka and a = 0, a solution with no smooth monopoles at W = za and a solution with ka of them. (These correspond to the two ways of solving P Q = (z − za) ka such that P and Q have no common zero at z = za.) Since (7.8) is the superpotential for just one smooth monopole, it describes a solution with ka smooth monopoles only if ka = 1. The general analysis for arbitrary ka can be made and matched to section 2.5 using the superpotential (7.11) for an arbitrary number of smooth monopoles. To recover the qualitative results of section 2, we first set ζ = 0, a = 0. To find a critical point, the W i must each equal zeroes z a of K. Assuming that the charges k a are all 1, no more than one of the W i may equal the same z a (otherwise a zero of Q ′ (W i ) cancels a zero of K(W i ) and the condition ∂W/∂Y i = 0 is not obeyed). Summing over all values of q, there are a total of 2 d solutions -each z a may or may not be equal to one of the W i . If we take a → 0, then all but one of these solutions (the one with no smooth monopoles at all) disappear, with Y i ∼ log(1/a). For ζ = 0, just as in the derivation of (7.10), it is convenient to integrate out the massive fields Y i . Modulo terms that do not depend on the W i , the superpotential that we arrive at is precisely the Yang-Yang function: W = i aW i +ζ i log(K(W i )/Q ′ (W i )) = i aW i +ζ i,a log(W i −z a )−2ζ i<j log(W i −W j ). (7.12) The attentive reader may notice one gap in what we have said. In the case ζ = 0, we have not analyzed the problem with k a > 1 for some a. To do this, it is important to consider the case that W i = W j = z a for some i, j, but the coordinates that we have used to describe the monopole moduli space are actually not adequate when W i = W j . We explain a better description momentarily. Coordinates for Monopoles The moduli space of several smooth BPS monopoles on R 3 is the subject of a rich mathematical theory [56]. From this theory we only need a small part: when the monopoles are widely separated in space (or equivalently when the symmetry breaking is strong), an effective abelian description of the moduli space is possible. In this description, the monopoles are regarded as "point" Dirac monopoles that interact with each other via the abelian gauge multiplet. Each monopole has a position in R 3 and an angular coordinate ϑ that is a collective coordinate for charge rotations. As in section 7.1, once we pick a particular complex structure on the moduli space, the position and angular coordinate of each monopole combine to a pair of chiral superfields W , Y. However, there is a subtlety in the definition of Y, which is the reason that we have changed our notation from section 7.1. The angular part of the coordinate Y parametrizes the freedom to do a U (1) gauge transformation on the smooth monopole solution before "gluing" it to the abelian solution. Hence e Y is an holomorphic section of the U (1) gauge bundle over the W -plane. This is the reason that in the presence of boundary singular monopoles, the exponential superpotential exp(−Y) for a single smooth monopole needs a prefactor K(W ): the superpotential should be a function, but the singular monopoles make e −Y into the section of a bundle ⊗ a O(z a ) −ka , where the exponents are the charges of the singular monopoles in the abelian effective field theory. So we compensate for this by multiplying by K(W ). A more intuitive explanation is that the superpotential encodes the interaction of the BPS monopole with the off-diagonal part of the complex Higgs field ϕ, which has a zero of order k a at z a . So the superpotential acquires a factor (z − z a ) ka . On the other hand, in the abelian theory, the BPS monopole behaves like a singular monopole. At any given value of y, one can restrict the U (1) gauge bundle of the low energy description to the W plane. As one increases y so that one passes the location of a BPS monopole, the U (1) gauge bundle on the W plane jumps. In other words, a second BPS monopole to the right of the first will feel the presence of a singular monopole of charge +2 at the location W 1 of the first monopole. Hence we expect a superpotential W = K(W 1 ) exp(−Y 1 ) + K(W 2 ) (W 2 − W 1 ) 2 exp(−Y 2 ) (7.13) and similarly for several monopoles with increasing values of the real parts of Y i : W = i K(W i ) j<i (W i − W j ) 2 exp(−Y i ) (7.14) This superpotential is equivalent to the relevant part of (7.11) as long as the W i are distinct. The two are related by the change of variables exp(−Y i ) = j<i (W i − W j ) j>i (W i − W j ) exp(−Y i ) (7.15) This re-definition does not affect the part of the superpotential linear in a and ζ, since i Y i = i Y i . However, it is the Y i , not the Y i whose real parts are the actual positions of the monopoles in the y direction; moreover, the difference between the Y i and the Y i is divergent when two or more W i coincide. The superpotential expressed in terms of the Y i reproduces correctly the counting of solutions at ζ = 0 for arbitrary values of the k a . For a = 0, if K(z) = z ka the superpotential is W = q i=1 W ka i j<i (W i − W j ) 2 exp(−Y i ) (7.16) This function is extremized for arbitrary Y i if the W i are all zero and the number q of smooth monopoles is no greater than k/2, because the prefactors have a zero of order at least 2 when the W i are all zero. This reproduces what we found in section 2.3. If we do the same computation in terms of the Y i , we would seem to get solutions even when the number of monopoles at the origin is greater then k a /2, but the Y i are not good variables when the W i coincide. We will leave the case a = 0, ζ = 0 to the reader. For ζ = 0, the difference between the Y i and the Y i is not important. It is interesting to match the coordinates in the low energy description to the exact nonabelian description of the monopole moduli space. The exact monopole moduli space is parametrized [57] by a scattering matrix for the operator D y = D y +i[φ t , · ]. The scattering matrix takes the form S = Q(z) P (z) P (z) R(z) QR − P P = 1 (7.17) where Q is a monic polynomial of order q, P and P are polynomials of degree up to q − 1, and R is a polynomial of degree up to q − 2. P and Q are necessarily coprime (this follows from the condition QR − P P = 1), and both P and R are uniquely determined by Q and P . One can define coordinates W i , Y i on the monopole moduli space by Q(W i ) = 0 and P (W i ) = exp Y i . However, this definition does not work well if the W i are not distinct. To find a parametrization that works better as long as the symmetry breaking is strong, we can proceed as follows. For a single monopole, the scattering matrix takes the form. S 1 = (z − W 1 ) e Y 1 −e −Y 1 0 (7.18) This expression can be matched naturally to the fact that at low energies, the smooth monopole can be approximated as a singular Dirac monopole. It tells us that there are two solutions ψ ± of the equation D y ψ = 0 that behave as ψ + ∼ e a 1 y/2 1 0 y << 0 ψ + ∼ (z − W 1 )e a 1 y/2 1 0 + e Y 1 −a 1 y/2 0 1 y >> 0 (7.19) ψ − ∼ e −a 1 y/2 0 1 y >> 0 ψ − ∼ (z − W 1 )e −a 1 y/2 0 1 + e −Y 1 +a 1 y/2 1 0 y << 0 (7.20) Here ψ + (ψ − ) is the unique solution which is small for y << 0 (y >> 0). A singular monopole solution in the abelian theory would have given the same exponential growth for y >> 0 (y << 0), and the subexponential correction is due to the exponentially decreasing corrections for the smooth monopole solution. For a configuration of many well-separated smooth monopoles, the scattering matrix is a product S = S q S q−1 · · · S 1 . (7.21) where S a is the scattering matrix due to the a th monopole and the monopoles are taken to be ordered in the y direction. It is natural to parametrize the moduli space by S a = (z − W a ) e Ya −e −Ya 0 . (7.22) This enables us to write P and Q in terms of the W i and Y i , and finally, using P (W i ) = exp(Y i ), to express the Y i in terms of the W i and the Y i . Realization Via M -Theory And Branes Here we will explain an M -theory approach to understanding the exponential superpotential (7.5). We begin with some preliminaries. M -Theory Preliminaries The six-dimensional (0, 2) model of type A 1 can be realized on a pair of parallel M5branes. Thus, we begin on R 11 with coordinates x 0 , . . . , x 10 , and we consider two M5-branes parametrized by x 0 , . . . , x 5 and located at x 6 = · · · = x 10 = 0. This system preserves 16 global supersymmetries. Their generators can be understood as eleven-dimensional spinors ε that obey Γ 0 Γ 1 · · · Γ 5 ε = Γ 6 Γ 7 · · · Γ 10 ε = ε. (7.23) Here the gamma matrices obey {Γ µ , Γ ν } = 2g µν . To simplify the picture, we introduce symmetry breaking, separating the two M5branes in, say, the x 6 direction. So now we place one at x 6 = 0 and the other at x 6 = L, for some L. At low energies, this system is described by a pair of abelian tensor multiplets, coupled to BPS strings. The strings arise from M2-branes stretched between the two M5-branes. The string tension is T = T M2 L, where T M2 is the M2-brane tension. We will consider a string whose world-volume is parametrized by x 0 and x 1 , and that is located at specified values of x 2 , . . . , x 5 . The string is of course represented by an M2brane that stretches from x 6 = 0 to x 6 = L, though this direction will just factor out of the following analysis. The string described in the last paragraph preserves those supersymmetries whose generator obeys Γ 0 Γ 1 Γ 6 ε = ε (7.24) as well as (7.23). Altogether, there are eight unbroken supersymmetries, corresponding to N = 4 supersymmetry in the two-dimensional sense. There is an SO(4) symmetry group rotating the x 2 , . . . , x 5 coordinates, and a second SO(4) symmetry, which we will call SO(4) R , that rotates x 7 , . . . , x 10 . However, we will soon modify the construction in a way that will break half of the supersymmetry and also reduce SO(4)×SO(4) R to a maximal torus. So it will help to focus on the relevant symmetries to begin with. We consider the N = 2 subalgebra consisting of supersymmetries that (in addition to the conditions already given) are invariant under a combined rotation of (say) the x 4 − x 5 plane together with an R-symmetry rotation of the x 9 − x 10 plane: (Γ 4 Γ 5 + Γ 9 Γ 10 ) ε = 0. (7.25) Given (7.24), this is equivalent to (Γ 2 Γ 3 + Γ 7 Γ 8 ) ε = 0. (7.26) We note that the equations (7.25) and (7.26) are exchanged if we exchange W = x 2 + ix 3 with Z = x 4 + ix 5 , and similarly exchange x 7 , x 8 with x 9 , x 10 . So in particular, our conditions on ε are symmetrical between W and Z. The N = 2 supersymmetry algebra singled out by the above conditions has a U (1) × U (1) group of R-symmetries generated by J = Γ 7 Γ 8 and J ′ = Γ 9 Γ 10 . We want to compare J and J ′ to R-symmetry generators that we will call J + and J − that only act, respectively, on supersymmetry generators that have positive or negative two-dimensional chirality, in other words, that obey χε = ±ε, where χ = Γ 0 Γ 1 is the two-dimensional chirality. The conditions given above can be combined to give χε = JJ ′ ε (7.27) and this implies that (with a suitable choice of sign for J + and J − , and normalizing them so that they square to 1 on supersymmetry generators of the appropriate chirality) J and J ′ can be expressed as J = J + + J − , J ′ = J + − J − . In particular, an exchange J ↔ J ′ amounts to J ± → ±J ± , an operation known as the mirror symmetry automorphism of the N = 2 algebra. The automorphism of the abovedescribed N = 2 algebra that exchanges W and Z also exchanges J and J ′ , so it is a mirror symmetry. Hence, if we view W as a chiral superfield in the two-dimensional worldsheet theory of the string, we must view Z as a twisted chiral superfield, in the sense of [58]. Reduction To Gauge Theory So far we have half-BPS strings, but no gauge theory description of them. To get a gauge theory description, we compactify one direction, say the x 5 direction, on a circle of radius R. M -theory on a circle reduces at long distances to Type IIA superstring theory. The M5-branes become D4-branes and the theory on the D4-branes is at long distances a U (2) gauge theory, broken to U (1) × U (1) by the separation between the D4-branes. What is relevant to us is the SU (2) subgroup, broken at low energies to U (1). The string that was originally described via a stretched M2-brane is now represented by a D2-brane stretched between the two D4-branes. This situation has been studied in [59]. The D2-brane stretched between two D4branes carries magnetic charge and corresponds to a smooth BPS monopole in the low energy SU (2) gauge theory. Before compactifying the x 5 direction, the low energy theory along the string was described by the chiral superfield W = x 2 + ix 3 and the twisted chiral superfield Z = x 4 + ix 5 . After the compactification, we can replace Z by the single-valued field exp(Z/R), which is still a twisted chiral superfield. However, for matching to the theory of BPS monopoles, another variable is more useful. The moduli of the BPS monopole corresponding to the D2-brane are the positions of the underlying M2-brane in x 2 , x 3 , and x 4 , and the dual of its position in x 5 . The duality in question is a T -duality in the two-dimensional effective field theory governing the string. This T -duality is a mirror symmetry in the two-dimensional sense. It replaces x 5 /R with a new angular coordinate ϑ. So while Z/R = x 4 /R + ix 5 /R is a twisted chiral superfield, Y = x 4 /R + iϑ is an ordinary chiral superfield, just like W . Of course, the single-valued chiral superfield is not Y but e Y . We conclude with two comments: • The fact that the angular coordinate ϑ of the BPS monopole is T -dual to the angular position x 5 is part of the relation between M5-branes on a circle and D4-brane gauge theory. The symmetry that rotates x 5 becomes instanton number in the 4 + 1dimensional gauge theory, while the symmetry that rotates ϑ is electric charge. • The single-valued field Ω = exp(Z/R) can be understood as a map to C * ; it can be neither zero nor infinity. In section 7.3.3, we modify the problem to make it possible to have Ω = 0. Reducing On A Half Space To get a non-trivial superpotential, we will have to break some of the translation symmetries of the problem. In fact, we are interested in gauge theory on a half-space, so we want to restrict y = x 4 to be non-negative. The gauge theory problem studied in the present paper arises if x 4 and x 5 parametrize not R×S 1 , as is the case in our presentation so far, but rather a copy of R 2 with a cigar-like metric ds 2 = dy 2 + f (y) dx 5 2 . (7.28) Here f (y) ∼ y 2 /R 2 for y → 0 and f (y) → 1 for y → ∞. In fact, this was the starting point in the derivation in [14]. As a complex manifold, R 2 is the same as C, so we can now parametrize the x 4 and x 5 directions by a C-valued chiral superfield Ω, which asymptotically at large y (but only there) can be written Ω = exp(Z/R), Z = x 4 + ix 5 . (7.29) In contrast to the concluding remark of section 7.3.2, Ω is now C-valued rather than C *valued, and in particular there is no problem in having Ω = 0. We will make use of this shortly in generating a superpotential. The Instanton We now want to describe an M2-brane instanton that will generate the superpotential that we are looking for. The instanton is supposed to correct the physics of a string that is parametrized by a worldsheet coordinate X = x 0 + ix 1 . The string is located at definite values of W and Ω, say W = W 0 , Ω = Ω 0 . We can understand qualitatively what sort of instanton can generate a superpotential. We consider a two-dimensional model with N = 2 supersymmetry whose chiral ring is generated at the classical level by the chiral superfield W and whose twisted chiral ring is generated by the twisted chiral superfield Ω. The chiral ring is the ring of observables of a twisted B-model, and the superpotential that we want to generate will give a deformation of this chiral ring. A hypothetical superpotential must be generated by configurations that preserve the B-model supersymmetry. These are configuations in which the chiral superfield W is constant, while the twisted chiral superfield Ω is holomorphic as a function of the worldsheet coordinate. (It is a familiar fact that B-model supersymmetry requires a chiral superfield to be constant. The fact that B-model supersymmetry allows a twisted chiral superfield to be holomorphic is mirror to the perhaps more familiar fact that A-model supersymmetry allows a chiral superfield to be holomorphic.) The instanton that generates the superpotential is accordingly given by W = W 0 while Ω is a nontrivial but simple holomorphic function of X: (7.30) (Note that this formula only makes sense because Ω is allowed to vanish.) Here X 0 is a constant, which we interpret as the instanton position; as always in instanton physics, to calculate physical amplitudes, it is necessary to integrate over the instanton moduli, which here mean X 0 as well as some fermionic moduli associated to the supercharges under which the instanton solution is not invariant. The worldvolume of an M2-brane instanton is supposed to be a three-manifold. The three-manifold we want is just the product of the two-manifold S that was defined in (7.30) with the one-manifold 0 ≤ x 6 ≤ L (all at x 7 = · · · = x 10 = 0). Ω Ω 0 = X − X 0 , Since it is invariant under B-model supersymmetry, and has no moduli except what follows from translation invariance and supersymmetry (the parameter Ω 0 corresponds roughly to a constant value that Ω would have in the absence of the instanton), this sort of instanton will generate a superpotential. To understand just what superpotential will be generated, we use the asymptotic formula (7.29) and look at the disturbance in the string that is generated by the instanton, at great distances. For large values of y = x 4 , we can write exp((y + ix 5 )/R) = Ω 0 (X − X 0 ). (7.31) We see that as X circles once around X 0 in the clockwise direction (at large values of |X − X 0 | so that the formula (7.31) is valid), x 5 increases by 2πR. To produce this effect, the operator inserted at X = X 0 must be a twist field. As the T -dual of x 5 is the angular variable ϑ, a twist field is exp(−iϑ), and this must be the ϑ-dependence of a superpotential that captures the effects of the instanton. The holomorphic expression must therefore be W = Λ exp(−Y ), where Y = y + iϑ, for some constant Λ. This is precisely the result claimed in (7.5). For further confirmation, and also to check the sign in the exponent of W, let us consider the behavior of the field y at large distances, far from X = X 0 . At long distances, the fluctuations in y are described by a free-field path integral Dy exp − 1 4πR 2 dx 0 dx 1 |∇y| 2 . (7.32) When the operator exp(−y/R) is inserted in such a path integral at a point X = X 0 , the result is that at large distances, y/R grows as | log(X − X 0 )|. But this is exactly what we see in (7.31). A final comment is that if the worldvolume dimension of the string were bigger than 2, we would have considered the instanton as a fluctuation around a vacuum defined by a limiting value of y (and all the other worldvolume fields) for X → ∞, and we would have asked for the instanton to approach this limiting value at infinity. In two dimensions, because of the usual infrared divergences -which appear, for instance, in the logarithmic growth mentioned in the last paragraph -such a formulation is not valid. Opers And Branes The purpose of the present section is to place some of the ingredients that have appeared in the present paper in a wider context. We continue to study N = 4 super Yang-Mills theory, with a twist that preserves half the supersymmetry, on the four-manifold M 4 = R × C × R + , and with the usual Nahm pole boundary condition at the finite end of R + . The novelty, compared to what has been said so far, is that we will view the problem from the point of view of compactification on C from four to two dimensions. In general [60,61], assuming for simplicity that C has genus at least 2 (we relax this condition in section 8.4), compactification on C gives at low energies a two-dimensional sigma-model in which the target is M H , the moduli space of solutions of Hitchin's equations [20]. The Nahm pole boundary condition must reduce at low energies to a brane in this sigma-model, and this brane must be half-BPS because the Nahm pole boundary condition is half-BPS in four dimensions. Back to t = 1 We begin by analyzing the case t = 1. For simplicity, we take G to be SU (2) or SO(3). In section 2.2, we found that at t = 1, the Nahm pole boundary condition (in the absence of singular monopoles) describes a Higgs bundle (E, ϕ) → C that is endowed with a line sub-bundle L ⊂ E that is nowhere ϕ-invariant. Viewing E as a rank 2 complex bundle of trivial determinant, the inclusion L ⊂ E is part of an exact sequence: 0 → L → E → L −1 → 0. (8.1) Here we use the fact that, as E has trivial determinant, the quotient E/L must be isomorphic to L −1 . We view ϕ as a holomorphic map E → E ⊗ K, where K is the canonical bundle of C. We can restrict ϕ to L, to get a holomorphic map L → E ⊗ K, and then using the projection E → L −1 , we get a map ϕ : L → L −1 ⊗ K. The condition that L is nowhere ϕ-invariant means precisely that the map ϕ : L → L −1 ⊗ K is everywhere nonzero. In other words, this map is an isomorphism. Tensoring with L, we learn that L 2 is isomorphic to K, so that L is a square root K 1/2 of K. If G = SU (2), a solution of the Nahm pole boundary condition involves a choice of K 1/2 , while if G = SO(3), since we really should be working with the adjoint bundle ad(E) rather than E, the choice of K 1/2 does not matter. In what follows, we assume that either G = SO(3) or we have picked a particular square root of K. It is possible to make a non-trivial extension 0 → K 1/2 → E → K −1/2 → 0, and we will exploit this fact in section 8.2. However, for Higgs bundles, we want E to be a direct sum K 1/2 ⊕ K −1/2 , since in the case of a non-trivial extension, the Higgs fields that we are about to write would not exist. If we write E in column form E = K −1/2 K 1/2 . (8.2) then up to an automorphism of E, a possible Higgs field ϕ takes the form ϕ = 0 1 q 0 , (8.3) where q is a quadratic differential. To be more exact, we assume the upper right matrix element of ϕ to be nonzero as otherwise L would be ϕ-invariant (and the Higgs bundle (E, ϕ) would be unstable, as explained in [20]). Given this, by a bundle automorphism diag(λ, λ −1 ), we can take the upper right matrix element to be 1, and by a lower triangular bundle automorphism, we can make the diagonal matrix elements of ϕ vanish. Finally, for E as in (8.2), the lower left matrix element of ϕ is a quadratic differential (an element of H 0 (C, K 2 )), which we call q. Let T ⊂ M H be the submanifold parametrizing the Higgs bundles (E, ϕ) described in the last paragraph. At t = 1, the brane in M H defined by the Nahm pole is supported on T . What sort of subvariety is T ? As in [20], let I be the complex structure on M H in which it parametrizes Higgs bundles, J the complex structure in which M H parametrizes flat bundles with connection A = A + iφ, and K = IJ. The Hitchin fibration is the map from M H to the space of quadratic differentials that maps (E, ϕ) to Tr ϕ 2 . This map is holomorphic in complex structure I. For the Higgs field in (8.2), we have Tr ϕ 2 = 2q, so there is a unique such ϕ for every desired value of Tr ϕ 2 . Accordingly, T is a holomorphic section of the Hitchin fibration; in fact it is the holomorphic section constructed in [20]. Actually, T is complex Lagrangian from the point of view of complex structure I. That assertion means that the complex symplectic form Ω I = 1 4π C dz dz Tr δA z δφ z (8.4) vanishes when restricted to T . This is the case since, as the holomorphic type of E is fixed for all Higgs bundles that represent points in T , δA z is zero (up to a gauge transformation) when restricted to T . Since T is complex Lagrangian in complex structure I, we can identify as follows the supersymmetry of the half-BPS brane produced by the Nahm pole. This is a brane of type (B, A, A), that is, it is a B-brane in complex structure I, but an A-brane from the point of view of J or K. General t At general t, we are dealing with a flat bundle rather than a Higgs bundle. The Nahm pole still gives a line sub-bundle L ⊂ E, so we still have an exact sequence 0 → L → E → L −1 → 0,(8.5) as in (8.1). The covariant derivative D/Dz now gives a holomorphic map E → E ⊗ K. We can still restrict this map to L and project the image to L −1 ⊗ K, to get a linear map D/Dz : L → L −1 ⊗ K. The condition that L is nowhere invariant under D/Dz implies, just as in section 8.1, that this map is an isomorphism from L to L −1 ⊗ K, and again we conclude that L = K 1/2 . The difference from section 8.1 is that now the bundle E is not a direct sum K 1/2 ⊕ K −1/2 but a non-trivial extension. Indeed, as we assume that the genus of C exceeds 1, a bundle that holomorphically is a direct sum K 1/2 ⊕ K −1/2 would not admit a flat connection. Non-trivial extensions of K −1/2 by K 1/2 are all isomorphic; this is so because such an extension is determined by an element of H 1 (C, K) ∼ = C, and the choice of a nonzero element does not matter, up to a bundle automorphism. The simplest example of a flat bundle that from a holomorphic point of view is the extension described in the last paragraph can be found by placing on C a Kahler metric of scalar curvature R = −1. Let ω be the spin connection of such a metric and e the vierbein. The flat connection is A = ωt 3 + e z t − + e z t + . (8.6) In differential geometry, since ω is the spin connection, the flat bundle E is the spin bundle of C, or more precisely the direct sum K 1/2 ⊕ K −1/2 of the two spin bundles of opposite chirality. But in this basis, the complex structure of E is defined by the (0, 1) part of A, which is A z = ω z t 3 + e z t − ; this is lower triangular, but not diagonal, so E is an extension rather than a direct sum. Having found a single flat connection A on the bundle E, it is straightforward to find them all. We do not want to change A z (since the holomorphic structure of E is supposed to be unchanged), but we can change A z by A z → A z + λ z , where (to preserve flatness) λ z is annihilated by D z . For E as described in the last paragraph, the relevant choice is λ z = qt − , where q is a quadratic differential. Mathematically, a flat bundle E → C that from a holomorphic point of view fits in a non-split exact sequence (8.5) is called an oper; see [22] for a detailed explanation. We have learned that, at general t, the brane defined by the Nahm pole boundary condition is supported on the variety of opers. Actually, we should be more precise, because the Nahm pole boundary condition depends in general on a parameter ζ that was introduced in section 3.1, and the complex connection A that obeys the oper condition is in general not A + iφ but the more general connection A ζ defined in eqn. (3.3). We write V ζ for the subvariety of M H defined by requiring that A ζ obeys the oper condition. When restricted to V ζ , A ζ z is fixed, up to a gauge transformation, so the complex symplectic form Ω I ζ = 1 4π C dz dz Tr δA ζ z δA ζ z (8.7) vanishes. Accordingly, the brane V ζ is a complex Lagrangian brane, just as in section 8.1, but now in a rotated complex structure. In the context of the present paper, the rotated complex structure is I ζ , defined in section 3.1. V ζ might be called a brane of type (B, A, A) ζ , being related to complex structure I ζ as a (B, A, A) brane is to complex structure I. As long as ζ = 0, ∞, the complex structures I ζ are all equivalent. If we simply set ζ = i, we get the usual variety of opers for complex structure J; alternatively, in the limit ζ → 0, V ζ reduces to the holomorphic section V of the Hitchin fibration, described in section 8.1. S-Duality A particularly simple boundary condition in N = 4 super Yang-Mills theory is the Neumann boundary condition for gauge fields, extended to the whole supermultiplet in a half-BPS fashion. In terms of branes, this is the boundary condition for a family of D3-branes ending on a single NS5-brane in the absence of a gauge theory θ-angle. Upon compactification on C and reduction to two dimensions, this boundary condition gives a brane B NS5 on M H corresponding to a trivial flat line bundle over M H . In other words, the support of the brane B NS5 is all of M H , and its Chan-Paton connection is trivial. The brane B NS5 is of type (B, B, B), meaning that it is a B-brane in every complex structure. This reflects the fact that the trivial bundle over M H is holomorphic in every complex structure. Under S-duality or electric-magnetic duality, the D3-NS5 boundary condition is converted to a D3-D5 boundary condition, still with θ = 0. On the other hand, S-duality acts in the dimensionally reduced theory as T -duality on the fibers of the Hitchin fibration [60,61]. Hence the brane B NS5 must be mapped by S-duality to a brane B D5 supported on a section of the Hitchin fibration. Moreover, the S-dual of a brane of type (B, B, B) is of type (B, A, A) (this is shown in [15]). For a middle-dimensional brane to be an A-brane, its support must be a Lagrangian submanifold, and its Chan-Paton bundle must be flat. So the section of the Hitchin fibration on which B D5 is supported must be complex Lagrangian from the point of view of complex structure I. On the other hand, concretely the D3-D5 boundary condition, for the case of a single D5-brane, is described by the Nahm pole [19] at t = 1. (This value of t corresponds to unbroken supersymmetry of type (B, A, A).) Our analysis above determines the section of the Hitchin fibration that corresponds to the Nahm pole; it corresponds to the family of Higgs bundles described in (8.2) and (8.3). The D3-NS5 boundary condition can be deformed by turning on θ and more generally by turning on a U (1) gauge field on the NS5-brane. These deformations, which are described in [19], preserve the half-BPS nature of the boundary condition but rotate the unbroken supersymmetry. A particular deformation in this family, described in section 12 of [15], gives a brane -the canonical coisotropic brane B cc -that is important in the gauge theory approach to the geometric Langlands correspondence. B cc is a rank one brane supported on all of M H , with a Chan-Paton bundle whose curvature is a linear combination of the Kahler forms of M H . The precise combination depends on a parameter analogous to our ζ. With a choice that is convenient for geometric Langlands, the curvature of the Chan-Paton bundle is a multiple of ω J (the Kahler form for complex structure J) and then B cc is of type (A, B, A). The S-dual of the deformation of B NS5 that gives B cc is a deformation of B D5 that is obtained by rotating the Nahm pole in the space of fields A and φ. This is analyzed in [19], and the appropriate type of "rotation" was briefly described in section 3.1. As we have seen, the rotated Nahm pole boundary condition leads to a brane B oper that is supported on the variety of opers, that is on V ζ for some ζ. If B cc is defined in the standard fashion as a brane of type (A, B, A), then its S-dual must have the same supersymmetry. In that case, B oper is supported on the ordinary variety of opers, with ζ = i. (In the present paper, it is more natural for ζ to be real.) That the S-dual of the brane B cc is the brane B oper supported on the variety of opers is important in mathematical treatments of the geometric Langlands correspondence. See for example [62] for an explanation of the role of opers in the geometric Langlands correspondence. The facts that we have just described give a gauge theory way to understand the S-duality between B cc and B oper . It has been argued [63] that the S-duality between these two branes is important in understanding the AGT correspondence [64] as well as recent developments relating supersymmetric gauge theory and integrable systems [65]. The role of the S-duality between B cc and B oper is more explicit in [66]. Monodromy Defects The concept of a Higgs bundle can be generalized by allowing singularities at isolated points p i ∈ C. For what follows, the case of interest will be a regular singularity. To introduce a regular singularity near a point p in C, we pick a local complex coordinate z near C and then we introduce polar coordinates r, θ with z = re iθ . We select elements α, β, γ in the Lie algebra t of a maximal torus T ⊂ G, and look for solutions of Hitchin's equations with a singularity at p of the form A = α dθ + . . . φ = β dr r − γ dθ + . . . (8.8) where the ellipses refer to additional terms that are less singular than 1/r. We call this sort of codimension two singularity a monodromy defect. The general theory of Hitchin's equations adapts well to this situation [67] and one can define a moduli space M H (C; p, α, β, γ) of solutions that is a hyper-Kahler manifold with properties rather similar to what one has in the absence of the monodromy defect. Everything we will say generalizes in an obvious way to the case of any number of monodromy defects. Once we introduce monodromy defects, the limitation of some of the above statements to the case that the genus of C is at least 2 can be dropped. All above statements hold for C of any genus in the presence of a sufficient number of monodromy defects (for G = SU (2), the required number is 3 if C has genus 0, and is 1 if C has genus 1). In the context of N = 4 super Yang-Mills on Σ × C, where Σ is another two-manifold, one can consider a monodromy defect supported on Σ × p, with p ∈ C. The singular solution (8.8) of Hitchin's equations embeds naturally as a solution of the four-dimensional equations (1.1). In the limit that C is small compared to Σ, the N = 4 theory on Σ × C reduces to a sigma-model on Σ with target M H (C; p, α, β, γ). In this description, there is an additional parameter η that arises [68] as a theta-angle for the abelian subgroup of G that is unbroken along Σ × p. So quantum mechanically, a monodromy defect really has four parameters α, β, γ, η. Under S-duality, a monodromy defect of the above-described type in G gauge theory is mapped to a similar monodromy defect in G ∨ gauge theory. The transformation of the parameters under S-duality is (α, η) → (η, −α) while β and γ are rescaled (for more detail see section 2.4 of [68]). If we drop the subleading terms represented by the ellipses in (8.8), we find that the monodromy of the complex flat connection A = A + iφ is U = exp(−2π(α − iγ)). The subleading terms do not modify the monodromy as long as U is regular -meaning that the subgroup of G that commutes with U has dimension equal to r, the rank of G. If U is not regular, there is an important subtlety, explained in detail in [68], section 3.3. For brevity, we will here consider only the case that G = SU (2), so that the nonregular values of U are only ±1. If U = ±1, the monodromy V of a connection of the form (8.8) is not necessarily conjugate to U ; on the contrary, generically it is in the "unipotent" conjugacy class containing the element U ′ = ± 1 1 0 1 . (8.9) A general element V of this conjugacy class is ±1 plus an arbitrary nilpotent matrix: V = ±1 + x y z −x , x 2 + yz = 0. (8.10) The equation x 2 + yz = 0 describes an A 1 singularity C 2 /Z 2 . The singular point is located at x = y = z = 0 where the monodromy V is precisely ±1; in other words, this is the case that the subleading terms in (8.8) do not correct the monodromy. In setting α and γ to special values at which U = ±1, we will assume that β remains generic. In this case, even if U = ±1, the solution has a "symmetry breaking direction" built in, given by the singular term in the connection proportional to β. The effect of this is to blow up the A 1 singularity, replacing C 2 /Z 2 with T * CP 1 . This important fact is established in [67]. The precise meaning of this T * CP 1 is that if U = ±1 and β = 0, then M H (C; α, β, γ) has a locus of A 1 singularities, which parametrizes Higgs bundles for which the monodromy around p is precisely ±1. But if β = 0 with U still equal to ±1, then this singular locus is blown up, replacing the singularities by a family of CP 1 's. The moduli space M H (C; p, α, β, γ) is invariant under shifting α by a cocharacterfor G = SU (2), this means that it is invariant under α → α + diag(i, −i), which is a shift that can be induced by a gauge transformation that has a singularity at p. However, we will be interested in brane constructions that are not invariant under such shifts of α, and for this reason, it will be best for our purposes not to view α as a periodic variable. The first brane that we want to consider is the oper brane. It is defined as usual by the Nahm pole boundary condition. For this, we take Σ to be R × R + , where R + is the usual half-line y ≥ 0, and we impose the Nahm pole boundary condition at y = 0. Suppressing the R or time direction, the picture on C × R + is sketched in fig. 26: there are monodromy defects supported on p i × R + , with respective parameters (α i , β i , γ i , η i ), and a Nahm pole boundary condition at y = 0. Of course, we need to explain what sort of singularity we want where the monodromy defect ends on a boundary with the Nahm pole. As usual, this kind of question is answered by finding a model solution with the desired singularity. For the present case, this has been done in section 3.6 of [14], and in greater generality in [69]. Let us set α − iγ = λ diag(i, −i), for a complex parameter λ, and consider a Higgs bundle E with a singularity of this type at, say, z = 0. If E (viewed in complex structure Figure 26. Monodromy defects in C × R + . supported on p i × R + where the p i are points in C. C is represented by the rectangle. We assume a Nahm pole boundary condition at y = 0. I ζ ) is also an oper, then it can be described by the classical stress tensor t = − λ(λ + 1) z 2 + . . . ,(8.11) where we have omitted less singular terms. This formula is just like (3.48), with j a = k a /2 replaced by λ. Flat sections of E correspond to holomorphic solutions of the differential equation ∂ 2 ∂z 2 + t ψ = 0. (8.12) For generic λ, one can find two linearly independent solutions with ψ 1 = z −λ 1 + ∞ i=1 c i z i , ψ 2 = z λ+1 1 + ∞ i=1 c i z i . This means that, as expected, the monodromy is U = diag(exp(−2πiλ), exp(2πiλ)). (8.13) What happens if instead λ = k/2 with k ∈ Z? There is always a solution ψ 2 = z k/2+1 (1 + ∞ i=1 c i z i ), but if we look for a solution with ψ 1 = z −k/2 (1 + c 1 z + . . . ), we find that generically when we carry this expansion to order z k/2+1 , we need logarithmic terms of order z k/2+1 log z + . . . . The logarithmic terms are simply a multiple of (log z)ψ 2 . Accordingly, the monodromy around z = 0 is actually generically of the unipotent form ψ 1 ψ 2 → (−1) k 1 s 0 1 ψ 1 ψ 2 (8.14) for some complex constant s. So if we want the monodromy around z = 0 to be trivial, we need to impose one condition on the subleading coefficients in the stress tensor (8.11), so as to get s = 0. This means that having trivial monodromy around z = 0 is a middle-dimensional condition. Indeed, without this condition, a monodromy defect for G = SU (2) increases the complex dimension of M H by 2, but the trivial monodromy condition fixes 1 of the 2 parameters. As this point is important, we will dwell on it a bit. Generically, the monodromy around the defect is an element of SL(2, C) (complex dimension 3) that obeys 1 constraint specifying its conjugacy class, leaving 2 complex parameters. For example, when α = γ = 0, the conjugacy class is two-dimensional, as exhibited explicitly in (8.10). The condition of trivial monodromy (which is defined only when U = exp(−2π(α − iγ)) equals ±1, and has no analog for other values) fixes 1 of the 2 parameters associated to the defect, so it leaves 1 parameter. One can think of this 1 parameter as the direction of symmetry breaking associated to the term β dr/r in eqn. (8.8). The choice of a symmetry-breaking direction determines a point in a copy of CP 1 ; this CP 1 is the projectivization of the fiber of E at the point p ∈ C where the monodromy defect lives. A more detailed explanation of the origin of this CP 1 is as follows. First of all, because of the equation x 2 + yz = 0, the unipotent conjugacy class described in eqn. (8.10) is explicitly isomorphic as a complex manifold to C 2 /Z 2 , with an A 1 singularity at x = y = z = 0. The singularity is precisely the point at which the group element V in (8.10) equals ±1. In the context of the construction of M H as a hyper-Kahler manifold, the β parameter is a Kahler parameter that blows up the A 1 singularity, replacing the conjugacy class C 2 /Z 2 by its resolution, the Eguchi-Hansen manifold T * CP 1 . In the blowup, the singular point at the origin is replaced by a copy of CP 1 . See [67] for the interpretation of β as a blowup parameter, and [68] for a leisurely explanation of some of these matters. We have essentially already run into the fact that in this situation, vanishing monodromy is a middle-dimensional condition. Let us specialize to the case that C = CP 1 (we could similarly treat the case that C = C with an irregular singularity at infinity). We know from section 3.4 that for a given set of singular points z a and charges k a , a = 1, . . . , d, there are finitely many opers with monodromy-free singularities. The condition that a flat G C bundle should be an oper is a middle-dimensional condition. To reduce to a finite set of opers with monodromy-free singularity, the condition of vanishing monodromy must also be middle-dimensional. (This assertion tacitly assumes that the two conditions intersect in a transverse fashion, which is in fact the case.) In fact, dropping the oper condition, we can explicitly describe the moduli space of solutions of Hitchin's equations on C, with monodromy defects characterized by λ a = k a /2, for which the complex connection A has trivial monodromy around those points. As C is simply connected, a flat bundle on C with no monodromy around the points p a is completely trivial as a flat bundle. The only possible moduli arise because the symmetry breaking associated to the parameters β a (which we assume to be all nonzero) generates a copy of CP 1 at each singular point p a . To get the moduli space, we must divide the product of these CP 1 's by the automorphism group of the trivial flat bundle E; this is a copy of SL(2, C). So finally the locus U of solutions of Hitchin's equations corresponding to flat bundles with trivial monodromy at each singular point is isomorphic to (CP 1 ) d /SL(2, C). This is a complex submanifold of M H in complex structure J (it is defined by a condition on the monodromies, which are holomorphic in that complex structure). Its dimension is d − 3, which is one-half the dimension of M H . In fact, U is complex Lagrangian from the point of view of complex structure J; this is true roughly because each CP 1 is complex Lagrangian in T * CP 1 . So the brane B triv supported on U with trivial Chan-Paton bundle This gives us a new way to think about opers of trivial monodromy. They are intersection points of two Lagrangian submanifolds of type (A, B, A) -one is the variety of opers and one parametrizes bundles with trivial monodromy. So the opers of trivial monodromy give a basis for the space of supersymmetric open strings stretching between the brane B oper and the brane B triv . We call this the space of (B oper , B triv ) strings. Technically here we want the space of (B oper , B triv ) strings in the B-model of type J. We can study this space of supersymmetric string states using S-duality, which converts the B-model of type J to the A-model of type ω K . S-duality converts the brane B oper to the canonical coisotropic brane B cc , as we learned in section 8.3. It turns out that, as we describe shortly, B triv is mapped to itself by S-duality (with the usual transformation of the monodromy defect parameters (α a , β a , γ a , η a )). So the S-dual of the space of (B oper , B triv ) strings is the space of (B cc , B triv ) strings, now viewed in the A-model of type ω K . The key aspect of this problem is that although the support of B triv is Lagrangian for ω K , it is actually symplectic for ω J -indeed, the support of B triv is a complex submanifold in complex structure J, and accordingly has ω J as a Kahler form. This being the case, the problem of describing the space of (B cc , B triv ) strings is governed by the analysis of quantization and branes in [70]. The space of (B cc , B triv ) strings is obtained by quantizing the support U of B triv ; here U is viewed as a symplectic manifold with symplectic structure ω J . A Selfdual Brane There is a simple gauge theory explanation of why B triv is selfdual. Forgetting about supersymmetry for a moment, we can think of a monodromy defect line as the Dirac string associated to a magnetic monopole that may have been improperly quantized. Hence a monodromy defect line can end on a singular magnetic monopole ( fig. 27). Since monodromy defects are mapped to themselves by S-duality (with some transformation of the parameters), pictures in which the monodromy defects end on singular monopoles are similarly mapped to themselves by duality. Supersymmetry imposes some constraints on the values of the parameters at which such pictures exist. In the context of the B-model of type J, the monodromy around a given defect line must be trivial if the defect line is going to end. This means that, in this B-model, the picture of fig. 27 only exists if α = γ = 0 (here we will view α and η as periodic variables). Of course, that is anyway the only case that the brane B triv can be defined. Dually, in the A-model of type ω K , a picture like that of fig. 27 only exists if γ = η = 0. (For example, η must vanish because the worldsheet theta-angle η fails to preserve the topological supersymmetry of the A-model if the support of the monodromy defect ends at a place where the U (1) bundle along the monodromy defect is not trivialized.) In the context of the present paper, opers with trivial monodromy arise most directly from singular monopoles at y = 0. However, without changing anything essential, we can move the singular monopoles away from the boundary as long as we connect them to the boundary via monodromy defects, as in fig. 27. This has the advantage of making it obvious that opers with trivial monodromy are intersection points of two branes, and also making clear the selfduality of one of these branes. In the general context of a defect line ending on a singular monopole, the monopole may be incorrectly quantized. However, for λ = k/2, which is equivalent to γ = 0, α = (k/2)diag(i, −i), the monopole at the end of the string obeys Dirac quantization, but the string is observable because we assume β = 0. Application To The Gaudin Model The selfduality of the brane B triv provides a gauge theory explanation of the main result of [22,25]: opers on CP 1 with trivial monodromy correspond to simultaneous eigenvectors of the commuting Hamiltonians of the Gaudin model. Let us consider the duality between the space of (B cc , B triv ) strings and the space of (B oper , B triv ) strings. The following discussion assumes familiarity with the framework of [70]. To construct the space of (B cc , B triv ) strings, we have to quantize a moduli space ( d a=1 CP 1 a )/SL(2, C), where CP 1 a is a copy of CP 1 attached to the monodromy defect at z = z a . Quantization of CP 1 a gives an irreducible representation R a of SU (2) of spin j a = k a /2, and quantization of ( d a=1 CP 1 a )/SL(2, C) gives a quantum Hilbert space H that is the SU (2)-invariant part of ⊗ a R a , H = (⊗ a R a ) SU (2) . (8.15) The classical commuting Hamiltonians of Hitchin's integrable systems can be interpreted (in the A-model of type ω K ) as (B cc , B cc ) strings. So they act on the space H of (B cc , B triv ) strings. In fact, the Hitchin Hamiltonians become the commuting Hamiltonians of the Gaudin model. To demonstrate the last statement, one interprets the generators of the SU (2) action on R a as arising from first order differential operators on CP 1 a , whence the Gaudin Hamiltonians (3.23) become second order differential operators. The "symbols" (or coefficients of the leading terms) of these operators are functions on the base of the Hitchin fibration that are precisely the Hitchin Hamiltonians. So, reading this in reverse, the Gaudin Hamiltonians represent a quantization of the Hitchin Hamiltonians (and this quantization is unique, given the commutativity of the Hitchin Hamiltonians, modulo the possibility of adding c-numbers). To understand the eigenvectors and eigenvalues of the commuting Hamiltonians, we use the equivalence of (B cc , B triv ) strings in the A-model of type ω K to (B oper , B triv ) strings in the B-model of type J. The latter strings simply correspond to intersection points of the classical branes B cc and B oper . So opers with trivial monodromy give a basis for the quantum Hilbert space H of the Gaudin model. In the B-model description, the commuting Hamiltonians simply become functions on the variety V of opers, which is the support of the brane B oper . Hitchin's classical Hamiltonians are holomorphic functions on the space of quadratic differentials on C (with poles of prescribed type at the positions z a of the monodromy defects). The support of V is the space of stress tensors on C (with prescribed poles at the z a ). The space of stress tensors differs from the space of quadratic differentials only because of the c-number conformal anomaly. This matches the additive c-number ambiguity in the quantization of the Hitchin Hamiltonians. The eigenvalues of the quantized Hamiltonians corresponding to a given oper are simply given by the stress tensor associated to that oper. Recently [66], a "noncompact" version of the Gaudin model has been described in which the finite-dimensional representations R a are replaced by infinite-dimensional ones. The eigenvectors of the commuting Hamiltonians are again expressed as opers, now with certain conditions on their monodromies. It is natural to suspect that this construction again reflects the existence of a selfdual brane. There actually is a good candidate -a selfdual brane that is constructed by replacing ends of monodromy defects, as in fig. 27, by junctions of such defects, as in fig. 28. Such a junction is defined by a solution of Hitchin's equations on a small two-sphere S linking the junction with singularities (of a type depending on the parameters α a , β a , γ a , η a ) at the intersection points of S with the monodromy defects. D Dx 3 D 2 = [φ 2 − iφ 3 , · ] D 3 = D Dy − i[φ 1 , · ] (A.2) Or, in a complex notation, D 1 = 2 D Dz D 2 = 2[ϕ, · ] D 3 = D Dy . (A.3) This system of equations can be generalized to a one-real-parameter family of 3d BPS equations, which can be written as in A.1, but with a different choice of operators D i : D 1 = 2 D Dz + 2ζ[ϕ, · ] D 2 = −2ζ D Dz + 2[ϕ, · ] D 3 = D Dy . (A.4) The generalization was studied in section 3. This family of 3d equations can be usefully derived from six dimensions. We start in R 6 with coordinates x a , x a+3 , a = 1, 2, 3. Then we constrain a gauge field by requiring that the field strength, seen as an element of the SO(6) Lie algebra, lies in a specified SU (3) subgroup. As long as one is in six dimensions, the choice of a subgroup does not matter; it just amounts to the choice of an identification of R 6 with C 3 . But if we require that the fields actually only depend on the first three coordinates x a , and are invariant under constant shifts of x a+3 , then the choice of an SU (3) subgroup does matter. So after dimensional reduction to three dimensions, one can obtain a family of inequivalent three-dimensional equations depending on a parameter. A simple way to show that the family is of real dimension one, modulo equivalences, is as follows. First, a choice of embedding of SU (3) in SO (6), parametrized by SO (6) (4), and its real and imaginary parts break SO(4) to SO (2). Hence the family of 3d equations obtained as dimensional reduction of the 6d equations is of real dimension one, and (A.4) is a generic representative. A consequence of this picture is that we can change ζ by an SO(6) rotation. Indeed, we can change ζ as desired by acting with an appropriate element of a group that we will call SO(2) ζ , which rotates D/Dx 2 and [φ 2 , · ] into each other, and also rotates D/Dx 3 and [φ 3 , · ] into each other. To be precise, the SO(2) ζ rotation acting on the D i defined at ζ = 0 will give a slightly rescaled version of the D i , with a prefactor (1 + ζ 2 ) −1/2 . This prefactor can be absorbed by a simple rescaling of the z coordinate. Now let us discuss the Nahm pole boundary condition that has been so important in the present paper. If we assume a dependence on y only, and further assume that A z = 0 (so that we can disregard D 1 ), the equations (A.1) with the D i defined as in (A.3) reduce to Nahm's equations. The Nahm pole boundary condition is defined by requiring that for y → 0, the fields can be approximated by a certain singular solution of Nahm's equations. There is a similar boundary condition for the 3d BPS equations at generic ζ. Indeed, SO(2) ζ maps a solution of (A.1) which only depends on y to a solution of (A.4) which only depends on y. More explicitly, taking the general form of the D i in (A.4), we can look for solutions which depend on y only, and such that D 1 reduces to 2∂/∂z, i.e. A z = −ζϕ. Then D 2 = 2(1 + ζ 2 )[ϕ, · ] and hence we can embed solutions of the Nahm equations as solutions of the general 3d BPS equations, at the price of a rescaling of the complex scalar ϕ by 1 + ζ 2 . This leads to the rotated Nahm pole boundary condition which we found useful in this paper. Of course, what we have just described is not the only embedding of the Nahm pole which would be compatible with the general 3d BPS equations. For example, at ζ = 0, we could have chosen to look for an embedding in which D 2 rather than D 1 is trivial; this would lead to what we might call anti-opers -flat bundles with an oper-like constraint on their antiholomorphic structure, rather than on their holomorphic structure. Any rotation of our choice of Nahm pole by the U (2) subgroup of SU (3) which preserves D 3 would produce a possible boundary condition, but we will generally stick to the "oper" Nahm pole. We will conclude with an alternative explanation of the meaning of the parameter ζ. For finite, non-zero ζ, the D i can be rescaled and interpreted as a generic complex 3d connection. In Cartesian coordinates, we can denote the components of the connection as .4), we have a useful relation: ω zz /ω zz = ζ 2 . If ζ 2 = 1, ω ab is symmetric, but in general that is not so. Generically, under linear coordinate redefinitions, there is a one-dimensional parameter space of possible ω ab . For example, if the symmetric part of ω ab is positive definite, as it is for (A.4), we can make it into the identity matrix δ ab . Then the antisymmetric part B ab can be rotated to live in the z, z plane, and its magnitude is controlled by a single real parameter, which we can identify with ζ. B Three-Dimensional BPS Equations From Four And Eight Dimensions In this appendix, we will discuss how the 3d BPS equations of parameter ζ can arise from time-independent solutions of the four-dimensional BPS equations (1.1). We will generalize the statement that the ζ = 0 equations in three dimensions arise from the 4d equations at t = 1 if we drop the dependence on one coordinate, say x 1 , and also set A 1 = φ y = 0. First we will show that this is not a feature of a specific choice. We can start with any choice of t, set d/dx 1 = 0 and set A 1 and φ y to any two linear combinations of the other three components of φ, and the resulting 3d equations will be equivalent to the 3d BPS equations discussed in the last appendix for some value of the parameter ζ in (A.4). For that purpose, it is rather convenient to rewrite the 4d BPS equations in a compact form, as a dimensional reduction of BPS equations in eight-dimensional Yang-Mills theory. A succinct way to describe the desired eight-dimensional equations is to pick a Spin (8) spinor ǫ of definite chirality and require F IJ Γ IJ ǫ = 0. (B.1) If the curvature F IJ is understood as an element of the Lie algebra of SO (8), then the equations restrict the curvature to a Spin(7) subalgebra of SO (8). These are really 7 equations, because of the obvious relation ǫ T F IJ Γ IJ ǫ = 0, (B.2) as Γ IJ are antisymmetric. Dimensional reduction to four dimensions breaks SO(8) to a subgroup that we will call SO(4) s × SO(4) R , acting respectively on the first four and last four coordinates. The spinor ǫ decomposes into a piece ǫ L which is left chiral under both SO(4) s and SO(4) R , and a piece ǫ R which is right chiral under both SO(4) s and SO(4) R . If both ǫ L and ǫ R are non-zero, they fix a choice of a twisted SO(4) ′ s diagonally embedded in SO(4) s × SO(4) R , such that ǫ L and ǫ R are SO(4) ′ s scalars. Then the 7 equations decompose under SO(4) ′ s into a triplet of self-dual two-forms, a triplet of anti-self-dual forms and a scalar equation. This is the form familiar from (1.1). We write a ′ as an abbreviation for a + 4 and adopt a complex notation with a as an abbreviation for a + ia ′ and a as an abbreviation for a − ia ′ . In order to bring the 8d equations explicitly to the form (1.1), it is useful to combine the Γ matrices to raising operators γ a = Γ a + iΓ a+4 (B .3) and lowering operators γ a = Γ a − iΓ a+4 (B.4) with a = 1, . . . , 4. We write |Ω for a state annihilated by the lowering operators, and |℧ for its complex conjugate, a state annihilated by the raising operators. Being invariant under SO(4) ′ s , ǫ is a linear combination of |Ω and |℧ ; being real, it is actually ǫ = e −iα |Ω + e iα |℧ , for some real α. Then the 8d equations can be written in terms of the (2, 0), (1, 1) and (0, 2) components of the curvature F ab , F ab and F ab : e −iα F ab + e iα 1 2 ǫ ab cd F cd = 0 a F aa = 0. (B.5) When we reduce to 4d, the first equation tells us that the selfdual part of Re (e −iα F ab ) vanishes, as does the anti-selfdual part of Im (e −iα F ab ). With φ = a A a+4 dx a , we recover the familiar 4d equations (F − φ ∧ φ + t d A φ) + = 0 (F − φ ∧ φ − t −1 d A φ) − = 0 d A ⋆ φ = 0, (B.6) with t = tan α. If we start from the 8d form of the equations, it is clear that solutions which are independent of some of the eight directions preserve additional supersymmetry. For example, any solution such that F I8 = 0 for some I also satisfies F IJ Γ IJ Γ 8 ǫ = 0 (B.7) and hence preserves the supersymmetry generated by the real anti-chiral spinor Γ 8 ǫ of SO(8). The 7 equations remain independent, and describe a reduction of SO(7) to G 2 preserving a 7d spinor ǫ 7 . Solutions that satisfy F I8 = 0 and F I7 = 0 preserve generically four spinors: ǫ, Γ 7 ǫ, Γ 8 ǫ, Γ 78 ǫ. The 7 equations then describe the reduction of SO(6) to SU (3) preserving the supersymmetries generated by a 6d complex spinor ǫ 6 and its complex conjugate. They decompose into 3 complex equations and a real moment map condition [D i , D j ] = 0 i [D i , D † i ] = 0, (B.8) as discussed in Appendix A. This is exactly the situation we are in whenever in the four-dimensional equations (B.6), for any value of t, we set d/dx 1 = 0 and set A 1 , φ y to any two linear combinations of the remaining three scalar fields φ in φ. Any such choices will produce a 3d reduction of the 6d BPS equations, and hence, according to the analysis in Appendix A, will be equivalent to the standard 3d BPS equations for some ζ. The 3d BPS equations admit the oper-Nahm pole boundary condition. This will induce a boundary condition in the original 4d BPS equations, which will be some deformation of the standard Nahm pole boundary condition. Vice-versa, with this boundary condition, the usual vanishing theorems will guarantee that time-independent solutions arise from solutions of the corresponding 3d BPS equations. Finally, we will describe a simple explicit choice of reduction from 4d to 3d which gives whatever ζ we wish. Starting from α = 0 and the standard reduction with A 1 = φ y = 0, we make simultaneous SO(2) rotations in the (a, a + 4) planes for a = 1, 2, 3, i.e. rotations of D/Dx a and [φ a , · ] into each other by angles θ a . (We do not make such a rotation for x 4 = y, as this would not behave well when we introduce a boundary at y = 0.) The rotation multiplies the creation and destruction operators by phases e ±iθa/2 , and hence the vacuum |Ω by the phase e −i a θa/2 . Hence it shifts the angle α by a θ a /2, and acts correspondingly on the t parameter. In order to preserve the SO(2) symmetry that rotates x 2 and x 3 , it is natural to keep θ 2 = θ 3 . Given how the rotation transforms D 2 and D 3 in (A.3), we will then have clearly ζ = tan θ 2 . Concerning the relation between θ 1 and θ 2 , there are two particularly natural choices. If we want to keep three-dimensional topological symmetry along the boundary, we should keep θ 1 = θ 2 = θ 3 = θ. A rotation by these angles will change t to tan(3θ/2 + π/4), and set ζ to tan θ. On the other hand, if we content ourselves with two-dimensional symmetry, we can keep t = 1, by setting θ 1 = −2θ 2 = −2θ 3 = −2θ. Again, ζ will be tan θ. With this second choice, we deform only the Nahm pole boundary condition, and not the fourdimensional equations. C On Boundary Conditions And A Special Solution Of The BPS Equations Here we will describe the Nahm pole boundary condition for the 3d BPS equations with generic ζ, allowing for singular monopoles on the boundary, and describe explicitly the model solution for the case of just one singular monopole. We work throughout on R 2 × R (the generalization to C × R + is straightforward). We will write the BPS equations simply as a flatness condition This affects the relative normalization of the [ D i , D i † ] terms in the moment map constraint. Of course, we can always rescale the y coordinate with respect to z, z. If we write the moment map constraint as i,j ω ij [ D i , D j † ] = 0, (C.2) for a constant diagonal matrix ω ij , the statement invariant under scaling is that ω 22 = ζ 2 ω 11 . We will find it convenient to set ω 11 = ζ −2 , ω 22 = 1, ω 33 = 1. If ζ 2 = 1, then ω ij ∂ i ∂ † j = ∂ 2 y + 2∂ z ∂ z (C.3) is the Laplace operator for a Euclidean metric on the half-space R 2 × R + that is normalized in a slightly unconventional way ds 2 = dy 2 + 2|dz| 2 . (C.4) This normalization will be useful later. The flatness condition (C.1) tells us that D i = g∂ i g −1 for a complex gauge transformation g, that is, a map from R 2 × R + to G C . The moment map condition (C.2) is invariant under unitary (G-valued) gauge transformations g → U g. We can eliminate the gauge-invariance by introducing the gauge-invariant hermitean matrix h = g † g. Then the moment map equation can be conjugated to ω ij ∂ i (h −1 ∂ † j h) = 0 (C.5) or ω ij ∂ i ∂ † j h = ω ij (∂ i h)h −1 (∂ † j h) (C.6) When ζ 2 = 1, this equation says that the map h from the half-space R 2 ×R + to the quotient space G\G C (endowed with its natural G C -invariant metric Tr (h −1 dh) 2 /2) is harmonic. Problems of this type are much-studied, but usually (for example, see [71]) in the context of a hyperbolic metric on R 2 × R + , rather than a Euclidean metric, as in our case. For simplicity, we will specialize to the case G = SU (2), so that G\G C is a copy of hyperbolic threespace H 3 or AdS 3 . We can write g = Y −1/2 0 0 Y 1/2 1 −Σ 0 1 (C.7) for a real function Y and a complex function Σ. This is a general parametrization, in the sense that every g ∈ SL(2, C) can be uniquely written in this form, modulo a unitary gauge transformation g → U g. With this parametrization, we have h = Y −1 −ΣY −1 −ΣY −1 |Σ| 2 Y −1 + Y (C.8) In these coordinates, the natural metric on H 3 takes a familiar form 1 2 Tr h −1 dh 2 = dY 2 + dΣ dΣ Y 2 . (C.9) In general, in terms of the variables Y and Σ, the equations for h become ω ij ∂ i Y −1 ∂ † j Y + Y −2 ∂ i Σ∂ † j Σ = 0 ω ij ∂ i Y −2 ∂ † j Σ = 0. (C.10) In the framework of section 3.2, we want a boundary condition that is determined by the properties of the "small section." If we write s for the small section in the complex gauge A i = 0, then in the unitary gauge with D i = g∂ i g −1 , the small section becomes gs. We must require gs to go as y 1/2 as y → 0, while g itself diverges as y −1/2 . This means that h diverges as y −1 while hs and s † h are finite and s † hs goes as y. The standard Nahm pole solution corresponds to s = z 1 and g = y −1/2 0 0 y 1/2 1 −z 0 1 . (C.11) This formula, which is familiar from eqn. (3.8), is equivalent to h = y −1 −zy −1 −zy −1 |z| 2 y −1 + y . (C.12) Comparing to the general parametrization (C.8), we see that the standard Nahm pole solution is Y = y, Σ = z. (The normalization ω 11 = ζ −2 , ω 22 = 1, ω 33 = 1 was chosen to ensure that this is a solution for all ζ.) In other words, this solution is the "identity" map from the half-space R 2 × R + endowed with the Euclidean metric (C.4) to the halfspace endowed with the hyperbolic metric (C.9). For ζ 2 = 1, the assertion that this gives a solution is simply the statement that the "identity" map between half-spaces endowed with these two metrics is harmonic. In general, if s = P Q , we want to require that Y ∼ y and P − QΣ ∼ y as y → 0. The last statement means that if we set σ(z) = P/Q, then Σ = σ at y = 0. The fact that Y → 0 for y → 0 means that the boundary y = 0 of the half-space R 2 × R + is mapped to the conformal boundary at infinity of the hyperbolic space H 3 . That conformal boundary is a copy of CP 1 . By adjoing CP 1 to H 3 , one makes the usual conformal compactification H 3 of H 3 . The choice of an oper without monodromy determines a holomorphic map σ(z) from R 2 ∼ = C to CP 1 , and the condition Σ| y=0 = σ means that, as a map of the boundary of the half-space to CP 1 , h coincides with σ. So our problem is this: given a holomorphic map σ from the boundary of the half-space to the conformal boundary of the hyperbolic space, we want to extend σ to a map h : R 2 × R + → H 3 that obeys (C.5) when restricted to y > 0. For ζ 2 = 1, we are simply trying to extend the given map σ to a harmonic map from the half-space R 2 × R + to H 3 . (Technically, we assume that the map σ has only polynomial growth so that it extends to a holomorphic map from the one-point compactification of C to CP 1 , and we similarly require that h extends to a continuous map from the one-point compactification of the half-space to H 3 .) For any σ(z), at least away from the branch points of the map σ -in other words, the zeroes of dσ/dz -it is not difficult to write a systematic expansion of Y and Z for y → 0, involving powers of y and powers of log y. The expansion roughly starts with Y = y|σ ′ (z)| + · · · and Σ = σ(z) + · · · , and the coefficients are rational functions in derivatives of σ(z), and of three undetermined real functions of z and z. The denominators of these rational functions are powers of σ ′ (z) and its complex conjugate. So, away from the zeroes of σ ′ (z), boundary condition behaves well, and cuts in half the degrees of freedom of a solution. The branch points are precisely the points with P Q ′ − QP ′ = 0 -in other words, the points at which there are singular monopoles. We still need to show that it is possible for a solution to be smooth away from the boundary in the presence of branch points or in other words singular monopoles on the boundary. The basic problem is to find a model solution in the presence of just one singular monopole; we then ask for the behavior near every singular monopole to match the model solution. In order to describe a singular monopole of charge k, we consider the special case s = z k+1 /(k + 1) 1 , or in other words σ(z) = z k+1 /(k + 1). We also make use of the invariances of the BPS equations. The equations (C.10) are invariant under scale transformations y → λy, z → λz with real λ, and under rotations z → e iθ z. They are also invariant under reflections z → z of R 2 , accompanied, if ζ 2 = 1, by Σ → Σ. The boundary conditions Y ∼ 1/y for y → 0, Σ| y=0 = z k+1 /(k + 1) are invariant under all these symmetries, accompanied by obvious rescalings of Y and Σ (which correspond to SL(2, C) transformations of the hyperbolic space). We expect the solution of the moment map condition that obeys the boundary condition to be unique, so it must be invariant under all these symmetries. Hence we require Y to be of the form y|z| k e u(ρ) and Σ to be of the form z k+1 e u(ρ) v(ρ)/(k + 1), for real functions u and v of ρ = y/|z|. Then the equations for h turn into two unfortunately rather complicated-looking nonlinear PDEs: v(ρ) ζ 2 (4k + 3)ρ 2 − 8 − ρ 2 u ′ (ρ) − 4k(k + 1)ρζ 2 v(ρ) +ρ ρ 2 + ρ 2 + 4 ζ 2 v(ρ)u ′′ (ρ) − ρ 3 + ρ 2 + 4 ρζ 2 v(ρ)u ′ (ρ) 2 +ρ ρ 2 + ρ 2 + 4 ζ 2 v ′′ (ρ) + −ρ 2 − ρ 2 + 8 ζ 2 v ′ (ρ) = 0 (k + 1) 2 ρ 2 ρ 2 + ρ 2 + 4 ζ 2 u ′′ (ρ) +v ′ (ρ) 2 ρ 2 + ρ 2 + 4 ζ 2 v(ρ)u ′ (ρ) − 4(k + 1)ρζ 2 v(ρ) +(k + 1)ρu ′ (ρ) (k + 1)ρ 2 ζ 2 + 1 − 4ζ 2 v(ρ) 2 + 4(k + 1) 2 ζ 2 v(ρ) 2 − 1 + ρ 2 + ρ 2 + 4 ζ 2 v(ρ) 2 u ′ (ρ) 2 + ρ 2 + ρ 2 + 4 ζ 2 v ′ (ρ) 2 = 0. (C.13) These equations involve v and the first two derivatives of u and v, but not u itself. Indeed, a constant shift of u is a symmetry of the equations, though not of the desired boundary conditions for y → 0. As PDEs for v and the derivative u ′ , these equations have a space of solutions which is locally three-dimensional. The requirement that the solution should be smooth as z → 0 poses two constraints. It turns out that at large ρ, u behaves as k log ρ, so that Y ∼ y k+1 , while v(ρ) scales as ρ −k , so that Σ ∼ δz k+1 for some constant δ. The solution admits for large ρ a convergent power series expansion in 1/ρ, which depends on δ. On the other hand, the boundary condition at ρ → 0 is more forgiving, and only imposes a single further constraint on the solution, which basically reduces to the requirement that v → 1 as ρ → 0. It is not difficult to check numerically that δ can be tuned so that the solution satisfies the constraint, and it is hopefully possible to prove this rigorously for any non-zero finite ζ. As δ is tuned, given the behavior for large ρ imposed in the last paragraph, there are two possible behaviors for v(ρ) as ρ becomes small. If δ is small, v(ρ) does not reach 1, and goes to zero as ρ → 0. If δ is large, it crosses 1 at some finite ρ, and then blows up before reaching ρ = 0. The solution we are after corresponds to the critical value of δ which separates these two behaviors. ( We write here φ z and φ z instead of ϕ and ϕ.) For any ζ, to two real linear combinations of the Hitchin equations (3.1). The possible complex structures on M H (G, C) are parametrized by ζ, where we add a point at infinity to the ζ plane to make CP 1 . For every ζ, one defines a complex structure I ζ in which the equation (3.4) is regarded as an equation governing holomorphic data; in this complex structure, the holomorphic variables are A ζ z = A z − ζ −1 φ z and A ζ z = A z + ζφ, and the equation (3.4) is holomorphic in those variables. The third linear combination of the equations is regarded in complex structure I ζ as a moment map condition. Figure 6 . 6A snapshot at fixed time of a time-independent situation. In the three-manifold M 3 = C × R + , knots are present at boundary points z 1 , . . . , z 4 . of this relation is that N J I is what we want, the counting of time-dependent solutions, while n J I is more easily computed, since this requires only the study of timeindependent problems.One explanation of (4.16) is as follows. By definition, J I [ z i ] is the set of points in U which can be reached by flowsdw i dt = −e iα g ij ∂W(w, z i ) ∂w j (4.17)which asymptote to I in the past. Here the z i are regarded as constants. Now consider the equation(4.14) for forced gradient flow on the semi-infinite interval (−∞, t 0 ]. For t 0 ≤ −T , the values at t 0 of a solution of this equation parametrize J I [ z i ], but for t 0 > −T , they parametrize a t 0 -dependent continuous deformation of this space that we will call J I [ z i ; t 0 ]. We saw in section 4.1.3 that that for any given cycle Γ in H m (U , U < ) z f , the coefficients c J in the expansion Γ = J c J J J [ z f ] (4.18) count, in the sense of an index, the number of flows 11) with k the Chern-Simons level, then the stress-tensor has central charge c = 1 + 6(b + b −1 ) 2 and in the semiclassical limit,b 2 T (z) → t(z)[A].If we combine (5.11) with (5.7) and (5.4), we find the relationship between the variable q usually used in describing the Jones polynomial and the parameter b used in describing Virasoro conformal blocks: q = exp(−2πi/b 2 ).(5.12) Figure 7 .Figure 8 . 78The thimble C 12 (top) compared to a closed integration contour (bottom), which is equivalent to (1 − q −1 )C 12 . This ray parallel to the real axis is the Lefschetz thimble for the case of one primary field of minimal charge with symmetry breaking. Figure 13 .Figure 14 . 1314The pattern of Stokes walls and the bases of thimbles in the c(z 1 − z The two Morse flows from the critical point (empty dot) near z 2 to the critical point (empty dot) near z 1 . The flows occur at slightly different values of Im Z. Figure 15 . 15The pattern of Stokes walls in the space of shapes of the triangle with vertices z 1 , z 2 , z 3 . approximated by the Bethe equations for that number w's in the absence of symmetry breaking. On the other hand, the Bethe equations for the w (∞) j are well approximated by the Bethe equations for that number of w's, in the presence of a single z of charge k eff = a k a − 2q 0 . Eqn. (3.22) ensures that k eff is non-negative, so in fact 0 ≤ k eff ≤ a k a . (6.19) As we have discussed in section 6.5.1, the Bethe equations for the w (∞) i have a single solution if k eff ≥ q ∞ ; otherwise, they have no solutions. Summing over all decompositions q = q 0 + q ∞ and all solutions of the Bethe equations for the w (0) i , and finally over all possible values of q, one gets the expected number d a=1 (k a + 1) of solutions of the Bethe equations. Figure 20 . 20The vertex model assigns the indicated factors to every crossing of two strands. The + and − signs labeling the strands express upward flow of magnetic charge +1 or −1; if one turns the picture upside down, the weights remain unchanged, provided one exchanges all + and − labels. Charge can be exchanged between strands, but only when a positive charge flows in from the bottom above a negative charge. If one reflects the picture from left to right, while also replacing q with q −1 , the weights remain invariant; this reflects the behavior of Chern-Simons theory under reversal of orientation. by its magnetic charge 1 or −1. The charges are unchanged at crossings, unless a strand of positive charge passes over a strand of negative charge, in which case a charge exchange process (corresponding in Morse theory to a non-trivial gradient flow) is possible. Figure 21 . 21The weights of the vertex model for creation or annihilation of a pair of strands. As in fig. 20, the weights are invariant under rotating the picture upside down if one exchanges + and − labels, or under a reflection from left to right if one replaces q by q −1 . Figure 23 . 23Use of the vertex model to compare two different projections of a single strand to the x 1 − x 3 plane -with a twist (left) or no twist (right). For either sign of the charge carried by the strand, the twist introduces a factor of −q 3/4 . The computation is quite different for the two possible values of the charge carried by the strand; for charge −1, as shown in (a), the vertex model sum has only one nonzero contribution, corresponding to the indicated labeling, but for charge +1, there are two possible contributions, shown in (b); they add to the same result. In (b), the second contribution involves a charge exchange process in which upper and lower strands exchange charge where they cross. A similar twist with undercrossing instead of overcrossing (or with the whole picture replaced by a mirror image) leads instead to a factor of −q −3/4 , as the reader can verify. Figure 27 . 27This figure differs from fig. 26 only in that each monodromy defect line ends on a singular monopole, indicated by a black dot on the right. Since the defect lines themselves are selfdual (with a suitable transformation of their parameters), the brane defined by ending of the defect lines is also selfdual. is a half-BPS brane of type (A, B, A). Figure 28 . 28Another selfdual brane can be constructed by replacing the ends of monodromy defects, which we used infig. 27, with junctions of monodromy defects, as depicted here. ∼ CP 3 , is equivalent to the choice of a complex line in the space of 6d spinors of positive chirality. After dimensional reduction to 3d, the inequivalent sets of 3d equations are parametrized by such a choice modulo the SO(3) × SO(3) group of space rotations and rotations of the three scalars φ i . Although this group is six-dimensional, just like CP 3 , it does not act freely on CP 3 ; rather, a generic point in CP 3 preserves an SO(2) subgroup of SO(3) × SO(3). For example, all the 3d equations parametrized by ζ are invariant under a simultaneous phase rotation of D/Dz and φ z . In general, SO(3) × SO(3) acts as SO(4) ⊂ SU (4) on the space of 6d spinors. So a complex spinor of SU (4) is a complex vector of SO D a . The complex equations tell us that the connection is flat. Then we have a moment map constraint, which set to zero a certain constant linear combination of the commutators [ D a , D † b ]. From this point of view, ζ only appears in the choice of moment map equation. A generic linear combination of the commutators is described by a 3 × 3 matrix of coefficients ω ab , [ D i , D j ] = 0 (C.1) for a complex 3d connection D i = d i + [A i , · ] together with a moment map constraint. Just as in eqn. (A.4), the indices i = 1, 2, 3 refer to z, z, and y. The definition of the D i differs from eqn. (A.4) by a rescaling of D 2 . Figure 29 . 29The numerical solutions as δ is varied across the critical value, for k = 1. 6 ) 6with F yz = [D y , D z ]. To be more exact, in varying W to derive the conditions [D i , D j ] = 0, one should require the variation of A at y = 0 to vanish. Otherwise, the variation of W contains additional delta function terms at y = 0. The equations [D i , D j ] = i [D i , D i † ] = 0 have been called the extended Bogomolny equations in and its first r a − 1 derivatives are invariants at each point z = z a . So there are r a moduli associated to each z a , where the possible values of r a are {0, 1, . . . , [k a /2]}. We will argue beginning in section 2.4 and in most detail in section 7 that the moduli represent the positions in the y direction of some smooth BPS monopoles, together with some conjugate angles.22) with m a + 2r a = k a . (2.23) For Q as in (2.22), we cannot set P to 1 by a transformation (2.20); rather, the values of P We reverse notation from[14], writing G ∨ for the gauge group in the Chern-Simons description and G for the gauge group in the dual "magnetic" description, on which we focus in this paper. For general W and a general choice of the boundary condition at y = ∞, P takes values not in Z but in a certain coset of Z in Q. This equation has also been formulated and some basic properties described in[17]. Later on, in the presence of symmetry breaking, we will have to relax this condition. We explain in section 4.1 why time-dependent instanton corrections in the G gauge theory do not affect this counting of states. For a relatively accessible introduction to the relevant aspects of Morse theory, see[39]. For the stated asymptotics to break down, the first step is to cross a Stokes wall, so that the thimble we started with evolves into a linear combination of thimbles with at least two terms. Initially, the asymptotics (5.3) remain valid, as any extra thimbles that appear at the Stokes wall initially make exponentially small contributions. If one varies α further, one of the extra thimbles may eventually become dominant. The combined process always involves varying α by an angle strictly greater than π/2 from its initial value. To show this, one just compares the values of W at the two critical points; these values have equal imaginary parts at the Stokes wall, and equal real parts when the two critical points exchange dominance. Acknowledgments We thank D. Bar-Natan, L. Kauffman, S. Lewallen, P. Li, R. Mazzeo, G. Moore, R. Schoen, P. Seidel, L.-F. Tam, and V. Toledano-Laredo for discussions and comments. We also thank M. Turansick for assistance with the figures.A Three-Dimensional BPS Equations From Six DimensionsThe time-independent configurations we consider in section 2 are solutions of the 3d BPS equationsfor a 3d connection together with three adjoint scalar fields, packaged together in the operators D i as A Polynomial Invariant For Links Via Von Neumann Algebras. V F R Jones, Bull. AMS. 12103V. F. R. Jones, "A Polynomial Invariant For Links Via Von Neumann Algebras," Bull. AMS 12 (1985) 103. A Categorification Of The Jones Polynomial. M Khovanov, Duke. Math. J. 101M. Khovanov, "A Categorification Of The Jones Polynomial," Duke. Math. J. 101 (2000) 359-426. Khovanov-Rozansky Homology And Topological Strings. S Gukov, A S Schwarz, C Vafa, hep-th/0412243Lett. Math. Phys. 74S. Gukov, A. S. Schwarz, and C. Vafa, "Khovanov-Rozansky Homology And Topological Strings," Lett. Math. Phys. 74 (2005) 53-74, hep-th/0412243. Knot Invariants And Topological Strings. H Ooguri, C Vafa, hep-th/9912123Nucl. Phys. 577H. Ooguri and C. Vafa, "Knot Invariants And Topological Strings," Nucl. Phys. B577 (2000) 419, hep-th/9912123. Knots, Links, and Branes at Large N. J M F Labastida, M Mariño, C Vafa, hep-th/0010102JHEP. 00117J. M. F. Labastida, M. Mariño and C. Vafa, "Knots, Links, and Branes at Large N ,"' JHEP 0011(2000) 007, hep-th/0010102. Topological Strings And Their Physical Applications. A Neitzke, C Vafa, hep-th/0410178A. Neitzke and C. Vafa, "Topological Strings And Their Physical Applications," hep-th/0410178. M Marinõ, Chern-Simons Theory, Matrix Models, And Topological Strings. Oxford University PressM. Marinõ, Chern-Simons Theory, Matrix Models, And Topological Strings (Oxford University Press, 2005). The Superpotential For Knot Homologies. N M Dunfield, S Gukov, J Rasmussen, arXiv:math/0505662Experiment. Math. 15129N. M. Dunfield, S. Gukov, and J. Rasmussen, "The Superpotential For Knot Homologies," Experiment. Math. 15 (2006) 129, arXiv:math/0505662. M -Theory And A Topological String Duality. R Dijkgraaf, C Vafa, E Verlinde, hep-th/0602087R. Dijkgraaf, C. Vafa, and E. Verlinde, "M -Theory And A Topological String Duality," hep-th/0602087. M Aganagic, M Yamazaki, Open BPS Wall Crossing And M -Theory. 834M. Aganagic and M. Yamazaki, "Open BPS Wall Crossing And M -Theory," Nucl. Phys. B834 (2010) 258-272. S Cecotti, A Neitzke, C Vafa, arXiv:1006:3435R-Twisting And 4d/2d Corresondence. S. Cecotti, A. Neitzke, and C. Vafa, "R-Twisting And 4d/2d Corresondence," arXiv:1006:3435. M Aganagic, M C N Cheng, R Dijkgraaf, D Krefl, C Vafa, arXiv:1105.0630Quantum Geometry Of Refined Topological Strings. M. Aganagic, M. C. N. Cheng, R. Dijkgraaf, D. Krefl, and C. Vafa, "Quantum Geometry Of Refined Topological Strings," arXiv:1105.0630. Knot Homology From Refined Chern-Simons Theory. M Aganagic, S Shakirov, to appearM. Aganagic and S. Shakirov, "Knot Homology From Refined Chern-Simons Theory," to appear. . E Witten, arXiv:1101.3216Fivebranes And Knots. E. Witten, "Fivebranes And Knots," arXiv:1101.3216. Electric-Magnetic Duality And The Geometric Langlands Program. A Kapustin, E Witten, hep-th/0604151Commun. Numb. Th. Phys. 1A. Kapustin and E. Witten, "Electric-Magnetic Duality And The Geometric Langlands Program," Commun. Numb. Th. Phys. 1 (2007) 1-236, hep-th/0604151. Instantons And The Geometry Of The Nilpotent Variety. P B Kronheimer, J. Diff. Geom. 32P. B. Kronheimer, "Instantons And The Geometry Of The Nilpotent Variety," J. Diff. Geom. 32 (1990) 473-490. A Haydys, arXiv:1010.2353Fukaya-Seidel Category And Gauge Theory. A. Haydys, "Fukaya-Seidel Category And Gauge Theory," arXiv:1010.2353. . P B Kronheimer, Oxford UniversityMSc. thesisunpublishedP. B. Kronheimer, MSc. thesis (Oxford University, 1986), unpublished. D Gaiotto, E Witten, arXiv:08042907Supersymmetric Boundary Conditions In N = 4 Super Yang-Mills Theory. D. Gaiotto and E. Witten, "Supersymmetric Boundary Conditions In N = 4 Super Yang-Mills Theory," arXiv:08042907. The Self-Duality Equations On A Riemann Surface. N Hitchin, Proc. London Math. Soc. 3N. Hitchin, "The Self-Duality Equations On A Riemann Surface," Proc. London Math. Soc. (3) 55 (1987) 59-126. Flat G-Bundles With Canonical Metrics. K Corlette, J. Diff. Geom. 28K. Corlette, "Flat G-Bundles With Canonical Metrics," J. Diff. Geom. 28 (1988) 361-382. E Frenkel, arXiv:math/0407524Gaudin Model And Opers. E. Frenkel, "Gaudin Model And Opers," arXiv:math/0407524. Quantization Of The Hitchin Moduli Spaces, Liouville Theory, And The Geometric Langlands Correspondence I. J Teschner, arXiv:1005:2846J. Teschner, "Quantization Of The Hitchin Moduli Spaces, Liouville Theory, And The Geometric Langlands Correspondence I," arXiv:1005:2846. Off-Shell Bethe Ansatz Equation For Gaudin Magnets and Solutions Of Knizhnik-Zamolodchikov Equations. H M Babujian, R Flume, Mod. Phys. Lett. 9H. M. Babujian and R. Flume, "Off-Shell Bethe Ansatz Equation For Gaudin Magnets and Solutions Of Knizhnik-Zamolodchikov Equations," Mod. Phys. Lett. A9 (1994) 2029-2039. B Feigin, E Frenkel, N Reshetikin, Gaudin Model, Bethe Ansatz, and Critical Level. 166B. Feigin, E. Frenkel, and N. Reshetikin, "Gaudin Model, Bethe Ansatz, and Critical Level," Commun. Math. Phys. 166 (1994) 27-62. Vertex Operators In Conformal Field Theory On P 1 And Monodromy Representations of Braid Group. A Tsuchiya, Y Kanie, Adv. Stud. Pure Math. 16A. Tsuchiya and Y. Kanie, "Vertex Operators In Conformal Field Theory On P 1 And Monodromy Representations of Braid Group," Adv. Stud. Pure Math. 16 297-372. Current Algebra And Wess-Zumino Model In Two-Dimensions. V G Knizhnik, A B Zamolodchikov, Nucl. Phys. 247V. G. Knizhnik and A. B. Zamolodchikov, "Current Algebra And Wess-Zumino Model In Two-Dimensions," Nucl. Phys. B247 83-103. Argument Shift Method And Gaudin Model. L G Rybnikov, math.RT/0606380Func. Anal. Appl. 403L. G. Rybnikov, "Argument Shift Method And Gaudin Model," Func. Anal. Appl. 40 (2006) No. 3, math.RT/0606380. B Feigin, E Frenkel, V Toledano-Laredo, arXiv:math/0612798Gaudin Models With Irregular Singularities. and to appearB. Feigin, E. Frenkel, and V. Toledano-Laredo, "Gaudin Models With Irregular Singularities," arXiv:math/0612798, and to appear. Opers With Irregular Singularity And Spectra Of The Shift of Argument Subalgebra. B Feigin, E Frenkel, L Rybnikov, arXiv:0712.1183B. Feigin, E. Frenkel, and L. Rybnikov, "Opers With Irregular Singularity And Spectra Of The Shift of Argument Subalgebra," arXiv:0712.1183. Infinite Conformal Symmetry In Two-Dimensional Quantum Field Theory. A A Belavin, A M Polyakov, A B Zamolodchikov, Nucl. Phys. 241A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, "Infinite Conformal Symmetry In Two-Dimensional Quantum Field Theory," Nucl. Phys. B241 (1984) 333-380. Conformal Algebra And Multipoint Correlation Functions in 2D Statistical Models. S Vl, V A Dotsenko, Fateev, Nucl. Phys. 240Vl. S. Dotsenko and V. A. Fateev, "Conformal Algebra And Multipoint Correlation Functions in 2D Statistical Models," Nucl. Phys. B240 (1984) 312-348. BRST Approach To Minimal Models. G Felder, Nucl. Phys. 317215G. Felder, "BRST Approach To Minimal Models," Nucl. Phys. B317 (1989) 215. Homological Representations Of The Hecke Algebra. R J Lawrence, Commun. Math. Phys. 135R. J. Lawrence, "Homological Representations Of The Hecke Algebra," Commun. Math. Phys. 135 (1990) 141-191. Arrangements of Hyperplanes And Lie Algebra Homology. V V Schechtman, A N Varchenko, Invent. Math. 106139V. V. Schechtman and A. N. Varchenko, "Arrangements of Hyperplanes And Lie Algebra Homology," Invent. Math. 106 (1991) 139. Asymptotically Free N = 2 Theories And Irregular Conformal Blocks. D Gaiotto, arXiv:0908.0307D.Gaiotto, "Asymptotically Free N = 2 Theories And Irregular Conformal Blocks," arXiv:0908.0307. Clifford Operators And Riemann's Monodormy Problem. T Miwa, Publ. Res. Inst. Math. Sci. Kyoto. 17665T. Miwa, "Clifford Operators And Riemann's Monodormy Problem," Publ. Res. Inst. Math. Sci. Kyoto 17 (1981) 665. Matrix Models Of 2-D Gravity And Isomonodromic Deformation. G W Moore, Prog. Theor. Phys. Suppl. 102G. W. Moore, "Matrix Models Of 2-D Gravity And Isomonodromic Deformation," Prog. Theor. Phys. Suppl. 102 (1990) 255-286. Lecture Notes On Morse Homology (with an eye towards Floer theory and pseudoholomorphic curves)," available at math. M Hutchings, M. Hutchings, "Lecture Notes On Morse Homology (with an eye towards Floer theory and pseudoholomorphic curves)," available at math.berkeley.edu/~hutching/teach/276-2010/mfp.ps. Analytic Continuation Of Chern-Simons Theory. E Witten, arXiv:1001.2933E. Witten, "Analytic Continuation Of Chern-Simons Theory," arXiv:1001.2933. A New Look At The Path Integral Of Quantum Mechanics. E Witten, arXiv:1009:6032E. Witten, "A New Look At The Path Integral Of Quantum Mechanics," arXiv:1009:6032. Supersymmetry And Morse Theory. E Witten, J. Diff. Geom. 17E. Witten, "Supersymmetry And Morse Theory," J. Diff. Geom. 17 (1982) 661-692. Instantons Beyond Topological Theory, I. E Frenkel, A Losev, N Nekrasov, hep-th/0610149E. Frenkel, A. Losev, and N. Nekrasov, "Instantons Beyond Topological Theory, I," hep-th/0610149. Quantum Field Theory And The Jones Polynomial. E Witten, Commun. Math. Phys. 121E. Witten, "Quantum Field Theory And The Jones Polynomial," Commun. Math. Phys. 121 (1989) 351-399. Conformal Field Theory, Two-Dimensional Gravity, and Quantization Of Teichmuller Space. H Verlinde, Nucl. Phys. 337H. Verlinde, "Conformal Field Theory, Two-Dimensional Gravity, and Quantization Of Teichmuller Space," Nucl. Phys. B337 (1990) 652-680. W -Algebras From Chern-Simons Theory. A , Phys. Lett. 267A. Bilal, "W -Algebras From Chern-Simons Theory," Phys. Lett. B267 (1991) 487-496. W Gravity From Chern-Simons Theory. J Boer, J Goeree, Nucl. Phys. 381J. de Boer and J. Goeree, "W Gravity From Chern-Simons Theory," Nucl. Phys. B381 (1991) 329-359. The Homological Approach To Higher Representations. R Lawrence, Harvard preprintR. Lawrence, "The Homological Approach To Higher Representations" (Harvard preprint, 1990), available at http://www.ma.huji.ac.il/~ruthel/. L H Kauffman, Knots And Physics. World ScientificL. H. Kauffman, Knots And Physics (World Scientific, 1991). Map Coloring, q-Deformed Spin Networks, and Turaev-Viro Invariants For 3-Manifolds. L H Kauffman, Int. J. Mod. Phys. 6L. H. Kauffman, "Map Coloring, q-Deformed Spin Networks, and Turaev-Viro Invariants For 3-Manifolds," Int. J. Mod. Phys. B6 (1992) 1765-1794. Gauge Theories, Vertex Models, and Quantum Groups. E Witten, Nucl. Phys. 330E. Witten, "Gauge Theories, Vertex Models, and Quantum Groups," Nucl. Phys. B330 (1990) 285-346. A Mathematical Construction Of The Non-Abelian Chern-Simons Functional Integral. S Albeverio, A N Sengupta, Commun. Math. Phys. 186S. Albeverio and A. N. Sengupta, "A Mathematical Construction Of The Non-Abelian Chern-Simons Functional Integral," Commun. Math. Phys. 186 (1997) 563-579. The Wilson Loop Observables Of Chern-Simons On R 3 In Axial Gauge. A Hahn, Commun. Math. Phys. 248A. Hahn, "The Wilson Loop Observables Of Chern-Simons On R 3 In Axial Gauge," Commun. Math. Phys. 248 (2004) 467-499. Knot Invariants And Universal R-Matrices From Perturbative Chern-Simons Theory In The Almost Axial Gauge. J F W H Van De Wetering, Nucl. Phys. 379J. F. W. H. van de Wetering, "Knot Invariants And Universal R-Matrices From Perturbative Chern-Simons Theory In The Almost Axial Gauge," Nucl. Phys. B379 (1992) 172-198. The Chern-Simons Theory And Knot Polynomials. J Fröhlich, C King, Commun. Math. Phys. 126J. Fröhlich and C. King, "The Chern-Simons Theory And Knot Polynomials," Commun. Math. Phys. 126 (1989) 167-199. M F Atiyah, N Hitchin, The Geometry And Dynamics Of Magnetic Monopoles. Princeton University PressM. F. Atiyah and N. Hitchin, The Geometry And Dynamics Of Magnetic Monopoles (Princeton University Press, 1988). Monopoles And Rational Maps: A Note On A Theorem of Donaldson. J Hurtubise, Commun. Math. Phys. 100J. Hurtubise, "Monopoles And Rational Maps: A Note On A Theorem of Donaldson," Commun. Math. Phys. 100 (1985) 191-196. Twisted Multiplets And New Supersymmetric Nonlinear Sigma Models. S J Gates, C M Hull, M Rocek, Nucl. Phys. 248157S. J. Gates, C. M. Hull and M. Rocek, "Twisted Multiplets And New Supersymmetric Nonlinear Sigma Models," Nucl. Phys. B248 (1984) 157. D-Branes, Monopoles and Nahm Equations. D E Diaconescu, arXiv:hep-th/9608163Nucl. Phys. B. 503220D. E. Diaconescu, "D-Branes, Monopoles and Nahm Equations," Nucl. Phys. B 503 (1997) 220, arXiv:hep-th/9608163. Topological Reduction of 4-d SYM to 2-d Sigma Models. M Bershadsky, A Johansen, V Sadov, C Vafa, hep-th/9501096Nucl. Phys. 448M. Bershadsky, A. Johansen, V. Sadov, C. Vafa, "Topological Reduction of 4-d SYM to 2-d Sigma Models," Nucl. Phys. B448 (1995) 166-186, hep-th/9501096. Reducing S Duality to T Duality. J A Harvey, G W Moore, A Strominger, hep-th/9501022Phys. Rev. 52J. A. Harvey, G. W. Moore, A. Strominger, "Reducing S Duality to T Duality, Phys. Rev. D52 (1995) 7161-7167 hep-th/9501022. E Frenkel, hep-th/0512172Lectures on the Langlands Program and Conformal Field Theory. E. Frenkel, "Lectures on the Langlands Program and Conformal Field Theory," hep-th/0512172. The Omega Deformation, Branes, Integrability, and Liouville Theory. N Nekrasov, E Witten, arXiv:1002.0888JHEP. 100992hep-thN. Nekrasov and E. Witten, "The Omega Deformation, Branes, Integrability, and Liouville Theory," JHEP 1009 (2010) 092, arXiv:1002.0888 [hep-th]. Liouville Correlation Functions From Four-Dimensional Gauge Theories. L F Alday, D Gaiotto, Y Tachikawa, arXiv:0906.3219L. F. Alday, D. Gaiotto, and Y. Tachikawa, "Liouville Correlation Functions From Four-Dimensional Gauge Theories," arXiv:0906.3219. Quantum Integrability and Supersymmetric Vacua. N A Nekrasov, S L Shatashvili, arXiv:0901.4748Prog. Theor. Phys. Suppl. 177hep-thN. A. Nekrasov and S. L. Shatashvili, "Quantum Integrability and Supersymmetric Vacua," Prog. Theor. Phys. Suppl. 177 (2009) 105-119, arXiv:0901.4748 [hep-th]. N Nekrasov, A Rosly, S Shatashvili, arXiv:1103.3919Darboux Coordinates, Yang-Yang Functional, and Gauge Theory. hep-thN. Nekrasov, A. Rosly, and S. Shatashvili, "Darboux Coordinates, Yang-Yang Functional, and Gauge Theory," arXiv:1103.3919 [hep-th]. Harmonic Bundles On Noncompact Curves. C Simpson, J. Am. Math. Soc. 3C. Simpson, "Harmonic Bundles On Noncompact Curves," J. Am. Math. Soc. 3 (1990) 713-770. Gauge Theory, Ramification, And The Geometric Langlands Program. S Gukov, E Witten, hep-th/0612073S. Gukov and E. Witten, "Gauge Theory, Ramification, And The Geometric Langlands Program," hep-th/0612073. . V Mikhaylov, to appearV. Mikhaylov, to appear. S Gukov, E Witten, arXiv:0809.0305Branes And Quantization. S. Gukov and E. Witten, "Branes And Quantization," arXiv:0809.0305. The Heat Equation And Harmonic Maps Of Complete Manifolds. P Li, L.-F Tam, Invent. Math. 105P. Li and L.-F. Tam, "The Heat Equation And Harmonic Maps Of Complete Manifolds," Invent. Math. 105 (1991) 1-46.
{'fraction_non_alphanumeric': 0.04845409260198446, 'fraction_numerical': 0.0199721839345109, 'mean_word_length': 3.7154154256957996, 'pattern_counts': {'":': 0, '<': 37, '<?xml version=': 0, '>': 38, 'https://': 0, 'lorem ipsum': 0, 'www.': 1, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 207, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'It has been argued based on electric-magnetic duality and other ingredients that the Jones polynomial of a knot in three dimensions can be computed by counting the solutions of certain gauge theory equations in four dimensions. Here, we attempt to verify this directly by analyzing the equations and counting their solutions, without reference to any quantum dualities. After suitably perturbing the equations to make their behavior more generic, we are able to get a fairly clear understanding of how the Jones polynomial emerges. The main ingredient in the argument is a link between the four-dimensional gauge theory equations in question and conformal blocks for degenerate representations of the Virasoro algebra in two dimensions. Along the way we get a better understanding of how our subject is related to a variety of new and old topics in mathematical physics, ranging from the Bethe ansatz for the Gaudin spin chain to the M -theory description of BPS monopoles and the relation between Chern-Simons gauge theory and Virasoro conformal blocks. 7 An Effective Superpotential For Monopoles 84 7.1 Overview Of Results 84 7.2 Coordinates for Monopoles 88 7.3 Realization Via M -Theory And Branes 90 7.3.1 M -Theory Preliminaries 91 7.3.2 Reduction To Gauge Theory 92 7.3.3 Reducing On A Half Space 93 7.3.4 The Instanton 93 8 Opers And Branes 95 8.1 Back to t = 1 95 8.2 General t 96 8.3 S-Duality 98 -ii -8.4 Monodromy Defects 99 8.5 A Selfdual Brane 103 8.6 Application To The Gaudin Model 104Hence the number of four-dimensional solutions which flow from J I [ z i ; t 0 ] at some given time t 0 after the braiding occurs to the critical point J in the future are the coefficients N J I in the expansion(4.20)But since J I [ z i ; t 0 ] parametrizes flows on the interval (−∞, t 0 ] that start at I, a flow from J I [ z i ; t 0 ] to J on the interval [t 0 , ∞) is equivalent to a flow from I to J defined on the whole real line. So the N J I are the same as the desired invariants N J I :', 'arxivid': '1106.4789', 'author': ['Davide Gaiotto \nSchool of Natural Sciences\nInstitute for Advanced Study\n1 Einstein Drive08540PrincetonNJUSA\n', 'Edward Witten \nSchool of Natural Sciences\nInstitute for Advanced Study\n1 Einstein Drive08540PrincetonNJUSA\n\nDepartment of Physics\nStanford University\n94305Palo AltoCA\n'], 'authoraffiliation': ['School of Natural Sciences\nInstitute for Advanced Study\n1 Einstein Drive08540PrincetonNJUSA', 'School of Natural Sciences\nInstitute for Advanced Study\n1 Einstein Drive08540PrincetonNJUSA', 'Department of Physics\nStanford University\n94305Palo AltoCA'], 'corpusid': 119612030, 'doi': '10.4310/atmp.2012.v16.n3.a5', 'github_urls': [], 'n_tokens_mistral': 92521, 'n_tokens_neox': 83227, 'n_words': 59750, 'pdfsha': 'db11924db94a8845e4ecc79e5dbbb97fe2e266e8', 'pdfurls': ['https://arxiv.org/pdf/1106.4789v1.pdf'], 'title': ['Knot Invariants from Four-Dimensional Gauge Theory', 'Knot Invariants from Four-Dimensional Gauge Theory'], 'venue': []}
arxiv
Quantum Quench dynamics in Non-local Luttinger Model: Rigorous Results 20 Nov 2017 Zhituo Wang [email protected] Institute for Advanced Study in Mathematics Research Center for Operator Algebras Harbin Institute of Technology 150006HarbinChina East China Normal University Quantum Quench dynamics in Non-local Luttinger Model: Rigorous Results 20 Nov 2017 We investigate, in the Luttinger model with fixed box potential, the time evolution of an inhomogeneous state prepared as a localized fermion added to the noninteracting ground state. We proved that, if the state is evolved with the interacting Hamiltonian, the averaged density has two peaks moving in opposite directions, with a constant but renormalized velocity. We also proved that a dynamical 'Landau quasi-particle weight' appears in the oscillating part of the averaged density, asymptotically vanishing with large time. The results are proved with the Mattis-Lieb diagonalization method. A simpler proof with the exact Bosonization formulas is also provided. INTRODUCTION Recent experiments on cold atoms [1] have motivated increasing interest in the dynamical properties of many body quantum systems which are closed and isolated from any reservoir or environment [2]. Nonequilibrium properties can be investigated by quantum quenches, in which the system is prepared in an eigenstate of the non-interacting Hamiltonian and its subsequent time evolution driven by an interacting many-body Hamiltonian is observed. As the resulting dynamical behavior is the cumulative effect of the interactions between an infinite or very large number of particles, the computation of local observables averaged over time-evolved states poses typically great analytical difficulties; therefore, apart for some analysis in two dimensions (see, for instance [3,4]), the problem is mainly studied in one dimension [5]- [30]. A major difference with respect to the equilib-rium case relies on the fact that in such a case a form of universality holds, ensuring that a number of properties are essentially insensitive to the model details. At non-equilibrium the behavior depends instead on model details; for instance integrability in spin chains dramatically affects the non equilibrium behavior [13], [40], [41] while it does not alter the T = 0 equilibrium properties [43]. This extreme sensitivity to the details or approximations asks for a certain number of analytical exact results at non-equilibrium, to provide a benchmark for experiments or approximate computations. One of the interacting Fermionic system where non-equilibrium properties can be investigated is the Luttinger model [32,33] (see also [34][35][36]), which provides a great number of information in the equilibrium case. In the Luttinger model model the quadratic dispersion relation of the non relativistic fermions is replaced with a linear dispersion relation, leading to the "anomaly" in the distribution of the ground states density. This anomaly is proved to be universal for a large class of one dimensional Fermionic system, called the Luttinger liquid [31]. Luttinger model became of great interest in mathematical physics ever since the exact solutions founded by Mattis-Lieb [34] and is a key to investigate the mathematical properties of condensed matter physics. It is important to stress that there exist two versions of this model, the local Luttinger model (LLN) and the non local Luttinger model (NLLM); in the former a local delta-like interaction is present while in the latter the interaction is short ranged but non local. The finite range of the interaction plays as an ultraviolet cut-off. At equilibrium such two models are often confused as they have similar behavior, due to the above mentioned insensitivity to model details; there is however no reason to expect that this is true also at non equilibrium. It should be also stressed that the LLM is plagued by ultraviolet divergences typical of a QFT and an ad-hoc regularization is necessary to get physical predictions; the short time or distance behavior depends on the chosen regularization. In this paper we study the evolution of inhomogeneous states in the non-local Luttinger Model with a fixed box potential, with the Mattis-Lieb diagonalization method, which was proved to be mathematically rigorous ( [35,36]). Then we perform rigorous analysis of the asymptotic behavior in the infinite volume limit. The main result shows that (see Theorem 2.2), when the interaction is turned on, the dynamics is ballistic with a constant but renormalized velocity, and the interaction produces a dynamical 'Landau quasi-particle weight' in the oscillating part, asymptotically vanishing with time. The expressions we get do not require any ultraviolet regularization, and correctly capture also the short time dynamics. We also invite the physically oriented reader to read this article along with a short letter [18], in which we studied the quench dynamics of non-local Luttinger model but without giving full details of the proof. In the current article we put full details of the proof and specialize to the box potential, for which the change of velocity due to the many-body interaction is more transparent; we provide also a simpler proof of the main theorem with the exact Bosonization formulas. The quantum quench of homogeneous states in the NLLM was derived in [20], [21], in which steady states were found. However mathematical rigor is lacking in these work. The quenched evolution of the NLLM prepared in domain wall initial state was studied in [42] and the universality of the quantum Landauer conductance for the final states was proved, in a mathematically rigorous way. The plan of the paper is the following. We introduce the NLLM with box potential in §II. In §III we prove Theorem 2.2 with the Mattis-Lieb diagonalization method. Some details of the proof are presented in the Appendix. The proof of Theorem 2.2 based on the Bosonization method is given in §IV. THE LUTTINGER MODEL AND MAIN RESULTS A. The Luttinger model with box potential The non-local Luttinger model (NLLM) is defined by the Hamiltonian: H λ = L/2 −L/2 dx i v F (: ψ + x,1 ∂ x ψ − x,1 : − : ψ + x,2 ∂ x ψ − x,2 :) +λ L 2 − L 2 dxdy v(x − y) : ψ + x,1 ψ − x,1 :: ψ + y,2 ψ − y,2 : (2.1) where ψ ± x,ω = 1 √ L k a k ,ω e ±ikx , ω = 1, 2, k = 2πn L , n ∈ N are fermionic creation or annihilation operators, :: denotes Wick ordering and v F is the Fermi velocity. We are choosing units so that v F = 1. The two-body interaction potential v(x − y) is given by: v(x − y) = sin(x − y) x − y ,(2.2) whose fourier transform reads: v(p) = v 0 f or p ≤ 1, 0 f or p > 1. (2.3) The potential v(x) or v(p) is also called the box potential and v 0 is called the strength of v(p). Equilibrium Luttinger model with box potential was first considered in [44]. In the Fourier space the Luttinger Hamiltonian can be written as H = H 0 + V = k>0 k[(a + k,1 a − k,1 + a − −k,1 a + −k,1 ) + (a + −k,2 a − −k,2 + a − k,2 a + k,2 ) + λ L p>0 v(p)[ρ 1 (p)ρ 2 (−p) + ρ 1 (−p)ρ 2 (p)] + λ L v(0) N 1 N 2 (2.4) where, for p > 0, ρ ω (p) = k a + k+p,ω a − k,ω , N ω = k>0 (a + k,ω a − k,ω − a − −k,ω a + −k,ω ). (2.5) It is well known that Fock space canonical commutation relations don't have a unique representation in a system with infinite degree of freedom. So one has to introduce a cutoff function χ Λ (k) with Λ a large positive number such that χ Λ (k) = 1 for |k| ≤ Λ and equals 0 otherwise and the regularized operators ρ ω (p) must be thought as lim Λ→∞ k χ Λ (k)χ Λ (k+ p)a + k+p,ω a − k,ω . The Hamiltonian H as well as ρ ω (p) can be regarded as operators acting on the Hilbert space H constructed as follows. Let H 0 be the linear span of vectors obtained by applying finitely many times creation or annihilation operators on |0 >= k≤0 a + k,1 a + −k,2 |vac > . The basic property of the Luttinger model is the validity of the following anomalous commutation relations, first proved in [34], for p, p ′ > 0 [ρ 1 (−p), ρ 1 (p ′ )] = [ρ 2 (p), ρ 2 (−p ′ )] = pL 2π δ p,p ′ . (2.7) Remark that this commutator acting on the Fock space is not precise due to the infinitely many degrees of freedom of the system. So one should introduce a cutoff Λ so that the commutator: − Λ k=Λ+p a + k,ω a − k,ω + Λ−p k=−Λ a + k,ω a − k,ω = −Λ+p k=−Λ a + k,ω a − k,ω − Λ k=Λ−p a + k,ω a − k,ω . (2.8) on any state of H is equal, in the limit Λ → ∞, to pL 2π . Moreover one can verify that ρ 2 (p)|0 >= 0 , ρ 1 (−p)|0 >= 0 . (2.9) Other important commutation relations (see [34,45] for proofs) are as follows: [H 0 , ρ ω (±p)] = ±ε ω pρ p (±p), [ρ ω , ψ ± ω,x ] = e ipx ψ ± ω,x (2.10) where ω = 1, 2; ε ω = 1 for ω = 1 and ε ω = −1 for ω = 2. B. The Mattis-Lieb diagonalization The Hamiltonian (2.4) can be diagonalized with the method of Lieb-Mattis [34], as follows. First of all we introduce an operator T = 1 L p>0 [ρ 1 (p)ρ 1 (−p) + ρ 2 (−p)ρ 2 (p)] (2.11) and write H = (H 0 − T ) + (V + T ) = H 1 + H 2 . Note that H 1 is already diagonalized in that it commutes with ρ ω . The key for the diagonalization of H 2 is the introduction of a bounded operator S acting on the Hilbert space H: S = 2π L p =0 φ(p)p −1 ρ 1 (p)ρ 2 (−p), tanhφ(p) = − λv(p) 2π . (2.12) Using the following Bogolyubov transformations for the operators ρ ω (±p): e iS ρ 1,2 (±p)e −iS = ρ 1,2 (±p) cosh φ(p) + ρ 2,1 (±p) sinh φ,(2.13) we can easily prove that H 2 can be written in diagonal form: e iS H 2 e −iS =H 2 := 2π L p sech2φ(p)[ρ 1 (p)ρ 1 (−p) + ρ 2 (−p)ρ 2 (p)] + E 0 . (2.14) By Formula (2.12) we can easily find that the operator S hence the transformation in (2.14) is well defined only for |λv(p)| < 2π; The model is instable for |λv(p)| > 2π. Define D =H 2 − T = 2π L p σ(p)[ρ 1 (p)ρ 1 (−p) + ρ 2 (−p)ρ 2 (p)] + E 0 , (2.15) we have [H 0 , D] = 0. The diagonalization formula for the Hamiltonian reads: e iS e iHt e −iS = e i(H 0 +D)t . (2.16) C. The time evolution of the one particle state and the main theorem Define ψ ± x,δ = e iH 0 t ψ ± ω,x e −iH 0 t = 1 √ L k a ± ω,k e ±i(kx−εωkt)−δ|k| ,(2.17) where δ → 0 + , ε 1 = +, ε 2 = −. By direct calculation we find that: < 0|ψ εω ω,x,δ ψ −εω ω,y,δ |0 >= (2π) −1 iε ω (x − y) − i(t − s) + δ . (2.18) The relation between the creation or annihilation Fermionic operators and the quasiparticle operators is ψ x = e ip F x ψ x,1 + e −ip F x ψ x,2 , (2.19) where p F is the Fermi momentum and we call e ip F x ψ x,1 =ψ x,1 and e −ip F x ψ x,−1 =ψ x,2 . In momentum space this simply means that the momentum k is measured from the Fermi points, that is c k,ω =c k+εωp F ,ω . The ground state of H is |GS >= e iS |0 >, where |0 > is the ground state of H 0 and the inhomogeneous one particle initial state is given by: |I t >= e iH λ t (ψ + 1,x +ψ + 2,x )|0 > . (2.20) Let n(z) be the density operator, which is defined as the limit δ → 0, ε → 0 of the following expression: 1 2 ρ=± (ψ + 1,z+ρεψ − 2,z, +ψ + 2,z+ρε ψ − 1,z +ψ + 2,z+ρεψ − 2,z (2.21) +ψ + 1,z+ρεψ − 1,z +ψ + 1,z+ρεψ − 1,z +ψ + 2,z+ρεψ − 2,z ). Note that summing over ρ = ± is the point spitting regularization, which plays the same role as the Wick ordering for avoiding divergences. We are interested in the average value of the density operator w. r. t. the 1-particle initial state (2.20), formally defined by: G(x, z, t, δ) :=< I t |n(z)|I t > (2.22) := ω,ω ′ =1,2 0|ψ − ω,x e iHtψ+ ω,z+ρεψ − ω ′ ,z e −iHtψ+ ω ′ ,x |0 + 0|ψ − ω,x e iHtψ+ ω ′ ,z+ρεψ − ω ′ ,z e −iHtψ+ ω,x |0 As a first step we consider the non-interacting case. Let |I 0,t >:= e iH 0 t (ψ + 1,x +ψ + 2,x )|0 >, we have: Theorem 2.1 When λ = 0, H = H 0 , we have lim L→∞ < I 0,t |n(z)|I 0,t > (2.23) = 1 2π 2 cos 2p F (x − y) (x − z) 2 − t 2 + 1 4π 2 [ 1 ((x − z) − t) 2 + 1 ((x − z) + t) 2 ]. Proof 2.1 We consider first the term with ω = 1, ω ′ = 2. Using the explicit expressions of the Fermionic operators and taking the limit ε → 0, we can easily find that this term is equal to e 2ip F (x−y) (4π 2 ) −1 [(x − z) 2 − t 2 ] −1 ; a similar result is found for the second term. The third and fourth terms are vanishing as ρ 1 ρε = 0; similarly the last two term give (4π 2 ) −1 [(x − z) ± t] −2 . Combine all these terms we can derive Formula (2.23), hence proved this theorem. Remark 2.1 The physical meaning of Theorem 2.1 is quite clear: when the interaction is turned off, the average of the density is sum of two terms, an oscillating and a non oscillating part (when the particle is added to the vacuum there are no oscillations p F = 0). At t = 0 the density is peaked at z = x, where the average is singular. With the time increasing the particle peaks move in the left and right directions with constant velocity v F = 1 (ballistic motion); that is, the average of the density is singular at z = x ± t and a "light cone dynamics" is found. When we turn on the interaction and let the system driven by the full interacting Hamiltonian, the ground states and the dynamics will be significantly changed. The explicit expression of (2.22) can be derived with the Mattis-Lieb diagonalization method followed by a rigorous analysis of the asymptotic behavior for L → ∞ and large t. We have Theorem 2.2 Let the interacting box potential (see (2.3)) be turned on in the Hamilto- nian, let γ 0 = v 0 2 and ω 0 = 1 − v 0 2π 2 . The average of the density operator with respect to the one particle initial state |I λ,t > in the limit L → ∞ reads: lim L→∞ < I λ,t |n(z)|I λ,t > = 1 4π 2 [ 1 ((x − z) − t) 2 + 1 ((x − z) + t) 2 ] + 1 2π 2 cos 2p F (x − z) e Z(t) (x − z) 2 − (ω 0 t) 2 . (2.24) where Z(t) = γ 0 1 0 dp p (cos 2ω 0 pt − 1) (2.25) is the Landau quasi particle factor, such that Z(0) = 1 and exp Z(t) ∼ cst( 1 2ω 0 t ) γ 0 ,(2. 26) for t ≥ 1. PROOF OF THEOREM 2.2 We consider first the term: 0| ψ − 1,x e iHt ψ + 1,z ψ − 2,z e −iHt ψ + 2,x |0 ,(3.27) and forget the phase factor e ±ip F x for the moment for simplicity; these factors are very easy to restore. The rest of this subsection is devoted to the calculation of (3.27). Let I be an identity operator in H. Using the fact that e −iεS e iεS = I and e −iHt e iHs | t=s = I, we can write (3.27) as 0| ψ − 1,x e −iεS (e iεS e iHt e −iεS )(e iεS ψ + 1,z e −iεS ) · (3.28) ·(e iεS ψ − 2,z e −iεS )(e iεS e −iHs e −iεS )e iεS ψ + 2,x |0 | ε=1,s=t Lemma 3.1 LetÎ 1 be an operator valued function of ρ 1 (±p) and ψ ± 1 andÎ 2 be an operator valued function of ρ 2 (±p) and ψ ± 2 , then we have the following factorization Formula for (3.27): G 1 = I 1 I 2 ,(3. 29) where I 1 = 0|Î 1 |0 and I 2 = 0|Î 2 |0 . Proof 3.1 We shall prove this lemma by deriving the explicit expressions ofÎ 1 andÎ 2 . Using the diagonalization formula (2.16), formula (3.28) can be written as: 0| ψ − 1,x e −iεS e i(H 0 +D)t e iεS ψ + 1,z e −iεS e −i(H 0 +D)t e −iεS · (3.30) ·e iεS e i(H 0 +D)s e iS ψ − 2,z e −iS e −i(H 0 +D)s e iεS ψ + 2,x |0 | ε=1, s=t . Now we consider the term of e iεS ψ + 1,z e −iεS . It is a well known result [34] that: e iεS ψ ∓ 1,z e −iεS = ψ ∓ 1,z W ± 1,z R ± 1,z , (3.31) where W ± 1,z = exp{∓ 2π L p>0 1 p [ρ 1 (p)e −ipz − ρ 1 (−p)e ipz ](cosh εφ − 1)} R ± 1,z = exp{± 2π L p>0 1 p [ρ 2 (p)e −ipz − ρ 2 (−p)e ipz ] sinh εφ}. (3.32) Similarly one has e iεS ψ ∓ 2,z e −iεS = ψ ∓ 2,z W ± 2,z R ± 2,z (3.33) where W ± 2,z = exp{∓ 2π L p>0 1 p [ρ 1 (p)e −ipz − ρ 1 (−p)e ipz ] sinh εφ} R ± 2,z = exp{± 2π L p>0 1 p [ρ 2 (p)e −ipz − ρ 2 (−p)e ipz ](cosh εφ − 1)}. (3.34) Then we consider the term Combining the above formula with (2.13) and (2.17) we find that (3.36) can be written as a product of e −iεS e i(H 0 +D)t W − 1,z R − 1,z e −i(H 0 +D)t e iεS ,(3.[e −iεS e i(H 0 +D)t W − 1,z e −i(H 0 +D)t e iεS ] · [e −iεS e i(H 0 +D)t R − 1,z e −i(H 0 +D)t e iεS ].e −iεS e i(H 0 +D)t W ± 1,z e −i(H 0 +D)t e iεS = exp ± 2π L p (cosh φ − 1) p [ (ρ 1 (−p) cosh εφ − ρ 2 (−p) sinh εφ)e ipx−ipt(σ+1) − (ρ 1 (p) cosh εφ − ρ 2 (p) sinh εφ)e −ipx+ipt(σ+1) ] :=W ± 1,z . (3.39) and e −iεS e i(H 0 +D)t R − 1,z e −i(H 0 +D)t e iεS = exp ± 2π L p sinh φ p [ (ρ 2 (−p) cosh εφ − ρ 1 (−p) sinh εφ)e ipy+ips(σ+1) − (ρ 2 (p) cosh εφ − ρ 1 (p) sinh εφ)e −ipy−ips(σ+1) ] :=R ± 1,z . (3.40) Using again (3.31), (3.33) and (3.46), we have: whereW −1 1,2,t,ε ,R −1 1,2,t,ε andŴ −1 1,2,t,ε ,R −1 1,2,t,ε are operators depending on ρ 1,2 (±p), respectively and z a , z b are functions of p. The explicit expressions of the above factors are given in the Appendix. e −iεS e i(H 0 +D)t e iS ψ + 1,z e −iS e −i(H 0 +D)t e iεS = z a A 1+ A 1− A 2+ A 2− ψ + 1,zt,δW −1 1tR −1 1t W −1 1tε R −1 1tεŴ −1 1tεR −1 1tε ,(3. Then we can easily find that the terms depending on ρ 1 (±p) and ψ ± 1 are factorized with respect to the terms depending on ρ 2 (±p) and ψ ± 2 . Let I 1 := 0|Î 1 |0 := 0|ψ 1x A 1+ A 1− ψ + 1,ztW −1 1W −1 1tŴ −1 1tW 2tW2tŴ2t B 1+ B 1− |0 ,(3. 43) and I 2 := 0|Î 2 |0 := 0|A 2+ A 2−R −1 1R −1 1R −1 1R 2R2R2 ψ 2,zt B 2+ B 2− ψ † 2x |0 ,(3. 44) and using the fact that z a = z −1 b we have G 1 = I 1 I 2 ,(3. 45) So we proved Lemma 3.1. A. Calculation of I 1 and I 2 In this part we derive the explicit expressions for I 1 and I 2 . It is also useful to introduce the following proposition, which can be easily proved using (2.10): Proposition 3.1 Let f (p, t) is an arbitrary regular function. Then we have: e iH 0 t e f (p,t) ρω(±p) e −iH 0 t = e f (p,t) e ±εω i(σ+1)pt ρω(±p) , ω = 1, 2; ε 1 = +, ε 2 = −,(3.46) The basic idea to calculate I 1 and I 2 is to use repeatedly the Hausdorff to move the operators ρ 1 (−p) and ρ 2 (p) to the right most of the expressions in (3.43) and (3.44), and move ρ 1 (p), ρ 2 (−p) to the left most of the above expressions. By formula (2.9) and its adjoint form we know that these operator annihilate |0 and 0|, respectively; the survived terms are those independent of ρ 1,2 (±p). Setting ε = 1, we have: I 1 = exp{ 2π L p 1 p [(e −ip(σ+1)(t+s) − 1)(2 cosh 2 φ sinh 2 φ + cosh 3 φ sinh φ) + (e ip(σ+1)(t+s) − 1) cosh φ sinh 3 φ + e −ipσt (− cosh 2 φ − sinh 2 φ) + e ip(x−z)+ip(σ+1)s (cosh φ sinh φ + cosh 2 φ) − e ip(x−z)+ips + e ip(x−z)−ip(σ+1)t (− sinh φ − sinh 2 φ) ]} 0|ψ 1x ψ + 1,z,t,δ |0 ,(3.47) and I 2 = 0|ψ + 2,z,t,δ ψ 2x |0 exp{ 2π L p 1 p [(e −ip(σ+1)(t+s) − 1) cosh φ sinh 3 φ + (e ip(σ+1)(t+s) − 1)(cosh 3 φ sinh φ + 2 cosh 2 φ sinh 2 φ) + e −ipσt (cosh 2 φ + sinh 2 φ) − e ip(x−z)−ipt + e ip(x−z)+ip(σ+1)s (− cosh φ sinh φ − sinh 2 φ) + e ip(x−z)−ip(σ+1)t (sinh φ + cosh 2 φ) ]}. (3.48) Combining (3.47) with (3.48) and setting s = t, we get: 0| ψ − 1,x e iHt ψ + 1,z ψ − 2,z e −iHt ψ + 2,x |0 (3.49) = 0|ψ 1x ψ + 1,z,t,δ |0 0|ψ + 2,z,t,δ ψ 2x |0 × exp p 1 p (e ip(x−z)+ip(σ+1)t − e ip(x−z)+ipt ) +(e ip(x−z)−ip(σ+1)t − e ip(x−z)−ipt ) +2 sinh φ cosh φ(sinh φ + cosh φ) 2 (cos 2p(σ + 1)t − 1) . It is useful to derive the asymptotic behavior for the second line in (3.49) and we have: lim δ→0 lim L→∞ 0|ψ 1x ψ + 1,zt,δ |0 0|ψ + 2,ztδ ψ 2x |0 = 1 4π 2 1 (x − z) 2 − t 2 (3.50) With the same method we can derive the explicit expression for the other terms in (2.22). Restoring the phase factor e ±ip F (x−z) and combine all the terms of (2.22), we obtain the following desired result: < I λ,t |n(z)|I λ,t >= 1 4π 2 [ 1 ((x − z) − t) 2 + 1 ((x − z) + t) 2 ] (3.51) + 1 4π 2 e Z(t) (x − z) 2 − t 2 e 2ip F (x−z) e Qa(x,z,t) + e −2ip F (x−z) e Q b (x,z,t) , where Z(t) = p 2 p sinh φ cosh φ(sinh φ + cosh φ) 2 (cos 2p(σ + 1)t − 1), (3.52) Q a = p 1 p [(e ip(x−z)+ip(σp+1)t − e ip(x−z)+ipt ) + (e ip(x−z)−ip(σp+1)t − e ip(x−z)−ipt )], Q b = p 1 p [(e −ip(x−z)+ip(σp+1)t − e −ip(x−z)+ipt ) + (e −ip(x−z)−ip(σp+1)t − e −ip(x−z)−ipt )] .1 + v(p) 4π 1 + v(p) 2π − 1 , cosh φ = 1 2 1 + v(p) 4π 1 + v(p) 2π + 1 , (3.54) where v(p) is the box potential with strength v 0 (see Formula (2.3)), we have the following expression for the critical exponent: γ(p) = 2 sinh φ(p) cosh φ(p)(sinh φ(p) + cosh φ(p)) 2 = v(p) 4π . (3.55) Taking the limit L → ∞ means that we should consider the discrete sum over p as integral over continuous variables. We have: There are three cases to be considered, depending on the range of t: Z(t) = ∞ 0 γ(p)dp p (cos 2ω 0 pt − 1) = γ 0 1 0 dp p (cos 2ω 0 pt − 1) = γ 0 2ω 0 t 0 d(2ω 0 p) 2ω 0 p (cos 2ω 0 pt − 1),(3. • when t ≪ 1, which corresponds to the short time behavior and implies that y ≪ 1 and w ≪ 1 (to remember that the v(p) is vanishing for p > 1); In this case we have Z(t) = γ 0 w 0 dy y (cos y − 1) ∼ γ 0 w 0 dy(− y 2 + O(y 3 )) ≪ 1. (3.58) So that Z(t) is well defined for y ≪ 1. Furthermore, it is vanishing as y → 0 + and we have e Z(t) | t→0 + → 1. • when t ∈ (0, 1]; In this case we can repeat the analysis as above and easily prove that Z(t) is a bounded function. • when t ∈ [1, ∞]; let p 0 > 0 be the minimal value of p and u = 2ω 0 p 0 t, we have where C = 0.577215 · · · is the Euler constant and u 0 cos y−1 y dy is a bounded function. Z(t) = γ 0 Remark that (3.59) is well defined for u → 0, due to the cancellation of ln u. So we have e Z(t) ∼ cst · [ 1 2ω 0 t ] γ 0 , f or t ≥ 1. (3.61) Now we derive the asymptotic formula for Q a and Q b . Replacing the discrete sum over p in (3.53) by integrals and performing the integrations, we can easily find that: Q a = Q b = ln (x − z) 2 − t 2 (x − z) 2 − ω 2 0 t 2 . (3.62) Collecting all the above terms we have: Bosonization formulas was given very recently in a paper by Langmann and Moosavi [45]. In this section we shall prove Theorem 2.2 with the exact Bosonization formulas in [45]. This can be considered as a verification of the use of Bosonization formula in the non-equilibrium setting. lim L→∞ < I λ,t |n(z)|I λ,t > = 1 4π 2 [ 1 ((x − z) − t) 2 + 1 ((x − z) + t) 2 ] + 1 2π 2 cos 2p F (x − z) e Z(t) (x − z) 2 − (ω 0 t) 2 . First of all we shall derive Formula (3.51). Following the notations in [45] we have Proposition 4.1 Let ρ ω be the Bosonic operators introduced before and let R εω ω be the Klein factor, then we can express the Fermionic operators ψ − in terms of the Bosonic operators and the Klein factor as follows: ψ − ω (x, δ) = : N δ e iπεωxQω/L R −εω ω e iπεωxQω/L × (4.64) exp ε ω p>0 2π Lp [ρ ω (p)e −ipx−δ|p| − ρ ω (−p)e ipx−δ|p| , where ω, ω ′ = 1, 2, ε 1 = +, ε 2 = −, Q ω = ρ ω (0) and N δ = 1 L(1−e −2πδ/L ) 1/2 is the normalization factor. R ± ω is the Klein factor such that R − ω = (R + ω ) † . They obey the following commutation relation (see [45] for the detailed derivation): [ρ ω (p), R ω ′ ] = ε ω δ ω,ω ′ δ p,0 R ω , [H 0 , R ω ] = ε ω π L ρ ω (0), R ω , (4.65) 0|R q 1 ω R q 2 ω ′ |0 = δ ω,ω ′ δ q 1 ,0 δ q 2 ,0 , R q 1 1 R q 2 2 = (−1) q 1 q 2 R q 2 2 R q 1 1 , [Q ω , R q 1 1 R q 2 2 ] = q ω R q 1 1 R q 2 2 , q ω ∈ Z . We shall not repeat the proof here and the interested reader is invited to look at [45] for details. LetẐ − ω = e iπεωxQω/L R −εω ω e iπεωxQω/L andẐ + be its adjoint, we can write the Fermionic operators as: ψ ± ω (x, δ) = N δẐ ± ω e ∓εω p>0 2π Lp [ρω(p)e −ipx−δ|p| −ρω(−p)e ipx−δ|p| . (4.66) We calculate first the term 0| ψ − 1,x e iHt ψ + 1,z ψ − 2,z e −iHt ψ + 2,x |0 in(2.22) forget the phase factor e ip F (x−z) for the moment. Inserting the identity operators I = e iHt e −iHt and I = e iS e −iS we derived Formula (3.28), which is the starting point of our analysis. First of all, it is easily to find that Using the fact that: e iSẐ ± ω e −iS =Ẑ ± ω .[H 0 + D, R ± ω ] = ± 2π(σ(0) + 1) L R ± ω (2ε ω ρ ω (0) + 1),(4.69) and e i(H 0 +D)t R ± ω e −i(H 0 +D)t = R ± ω exp [± 2π(σ(0) + 1) L (2ε ω ρ ω (0) + 1)t ],(4.70) we have: where e −iS e i(H 0 +D)t e iS ψ + 1,z e −iS e i(H 0 +D)t e iS (4.71) = N δẐ1 (t) exp 2π L p>0 1 p e −δp [A 1 ρ 1 (p) + A −1 ρ 1 (−p) + A 2 ρ 2 (p) + A −2 ρ 2 (−p)] ,A ±1 = ±e ∓ip[z+(σ+1)t] sinh 2 φ ∓ e −ip[z−(σ+1)t] cosh 2 φ , A ±2 = ±e ∓ip[z−(σ+1)t] sinh φ cosh φ ∓ e ∓ip[z+(σ+1)t] cosh φ sinh φ, B ±1 = ±e ∓ip[z+(σ+1)t] sinh φ cosh φ ∓ e ∓ip[z−(σ+1)t] cosh φ sinh φ, B ±2 = ±e ∓ip[z−(σ+1)t] sinh 2 φ ∓ e ∓ip[z+(σ+1)t] cosh 2 φ Z 1 (t) = e iπxρ 1 (0)/L exp[− 2π(σ(0) + 1) L (2ρ 1 (0) + 1) t]R −1 1 e iπxρ 1 (0)/L , Z 2 (t) = e iπxρ 2 (0)/L exp{ 2π(σ(0) + 1) L (−2ρ 2 (0) + 1)t}R 2 e iπxρ 2 (0)/L . (4.73) When p = 0, by using the fact that ρ ω (0)|0 = 0 and 0|R q 1 ω R q 2 ω ′ |0 = δ ω,ω ′ δ q 1 ,0 δ q 2 ,0 , we have 0|Ẑ 1Ẑ + 1 (t)Ẑ 2 (t)Ẑ † 2 |0 = 1. (4.74) So the nontrivial contributions come from the p > 0 part. Using repeatedly the Hausdorff formula we can factorize the terms depending on ρ 1 (±p) and ρ 2 (±p): 0| N δ exp 2π L p>0 1 p [e −δp e −ipx ρ 1 (p) − e −δp e ipx ρ 1 (−p)] ×N δ exp 2π L p>0 1 p e −δp [A +1 ρ 1 (p) + A −1 ρ 1 (−p) + A +2 ρ 2 (p) + A −2 ρ 2 (−p)] ×N δ exp 2π L p>0 1 p e −δp [B +1 ρ 1 (p) + B −1 ρ 1 (−p) + B +2 ρ 2 (p) + B −2 ρ 2 (−p)] ×N δ e 2π L p>0 1 p [e −δp e ipx ρ 2 (p)−e −δp e −ipx ρ 2 (−p)] |0 =: N 4 δ I 1 I 2 ,(4.75) where I 1 = 0|e 2π L p>0 1 p e −δp [e −ipx ρ 1 (p)−e ipx ρ 1 (−p)] e 2π L p>0 1 p e −δp [A +1 ρ 1 (p)+A −1 ρ 1 (−p)] × e 2π L p>0 1 p e −δp [B +1 ρ 1 (p)+B −1 ρ 1 (−p)] |0 ,(4.I 2 = 0|e 2π L p>0 1 p e −δp [A +2 ρ 2 (p)+A −2 ρ 2 (−p)] e 2π L p>0 1 p e −δp [B +2 ρ 2 (p)+B −2 ρ 2 (−p)] × e 2π L p>0 1 p e −δp [e −ipx ρ 2 (p)−e ipx ρ 2 (−p)] |0 . (4.77) Following exactly the same procedure as section 3 A, namely using repeatedly the Hausdorff formula and the annihilation formulas we have: I 1 I 2 = exp 2π L p>0 1 p e −2δp [(e ip(x−z)+ip(σ+1)t − 1) + (e ip(x−z)−ip(σ+1)t − 1) + 2 sinh φ cosh φ(sinh φ + cosh φ) 2 (cos 2p(σ + 1)t − 1)]. (4.78) In order to reproduce the expressions in (3.51) we need to extract from the above formula the noninteracting 2-point correlation function (see [45]), as follows. We write the terms e ±ip(x−z)±ip(σ+1)t − 1 in the above formula as (e ip(x−z)±ip(σ+1)t − e ip(x−z)±ip(σ+1)t ) + (e ip(x−z)±ip(σ+1)t − 1), while the first term gives the factors Q, the second term contributes to the non-interacting correlation function: A 2± = exp ± 2π L p 1 p ρ 2 (±p) sinh εφ(e ∓ipx±ipt − e ∓ipx±ipt(σ+1) ). N 4 δ exp 2π L p>0z b = exp 2π L p 1 p (1 − e −ipσt ) = z −1 a ,(5.W 2tε = exp − 2π L p 1 p (cosh φ − 1) sinh εφ[ρ 1 (p)e −ipz−ipt(σ+1) − ρ 1 (−p)e ipz+ipt(σ+1) ], R 2tε = exp − 2π L p sinh φ sinh εφ p [ ρ 2 (p)e −ipz+ipt(σ+1) − ρ 2 (−p)e ipz−ipt(σ+1) ], R 2tε = exp 2π L p 1 p (cosh φ − 1) cosh εφ [ ρ 2 (p)e −ipz−ipt(σ+1) − ρ 2 (−p)e ipz+ipt(σ+1) ]. way we get an abstract linear space to which we introduced scalar products between any pair of vectors. H is defined as the completion of H 0 in the scalar product just introduced. Moreover the operators H and ρ ω (p), regarded as operators on H with domain H 0 , are self adjoint. 35) which, after inserting the identity operator I = e iεS e −iεS and I = e −i(H 0 +D)t e i(H 0 +D)t , is equal to ( 3 . 36 ) 336Let f (p, t) be an arbitrary regular function, define σ(p) = sech2φ − 1 and ω(p) = σ(p) + 1 = sech2φ, we have the following commutation relation [H 0 + D, ρ ω (±p)] = ±ε ω p(σ(p) + 1)ρ ω (±p), ω = 1, 2, ε 1 = +, ε 2 = − ,(3.37) which implies that e i(H 0 +D)t e f (p,t)ρω (±p) e −i(H 0 +D)t = e e ±εω i(σ+1)pt f (p,t)ρω (±p) .(3.38) 41) and e −iS e iS e iHs e −iS e iS ψ − 2,z e −iS e iS e −iHs e −iS e iS = z bW2sεR2sεŴ2sεR2sεW2R2 ψ 2,z,s,δ B 1− B 1+ B 2− B 2+ , (3.42) The asymptotic behavior for L → ∞ In this section we shall derive the asymptotic behavior of Formula (3.51) in the limit L → ∞. Using definitions of the hyper-geometric functions we find that sechφ(p) = 1 2 56) where γ 0 := v 0 4π and ω 0 := 1 − v 0 2π 2 . The second line is true is due to the fact that γ(p) = 0 for p ∈ (1, ∞]. Let y = 2ω 0 pt and w = 2ω 0 t, Z(t) can be written as: − [ln 2ω 0 t − ln u] = γ 0 (− ln 2ω 0 t − C − Lieb-Mattis method for solving Luttinger model is mathematically rigorous, technically it is very complicated. There exist another very popular method for studying the one dimensional interacting Fermions models, called the Bosonizations, which states that certain two dimensional models of fermions are equivalent to the corresponding Bosonic models: the corresponding Fermionic Hilbert space and the Bosonic one are isomorphic and the the Fermionic operator can be expressed in terms of the Bosonic operators. While the Bosonization method can reduce significantly the difficulty for the calculation, it has the reputation of not mathematically rigorous. A Rigorous proof of − e −δp e −ipz [cosh φρ 1 (p) + sinh φρ 2 (p)] +e −δp e ipz [cosh φρ 1 (−p) + sinh φρ 2 (−p)] , e −δp e −ipz [cosh φρ 2 (p) + sinh φρ 1 (p)] +e −δp e ipz [cosh φρ 2 (−p) + sinh φρ 1 (−p)] .(4.68) and e −iS e i(H 0 +D)t e iS ψ − 2,z e −iS e i(H 0 +D)t e iS (4.72) = N δẐ2 (t) exp 2π L p>0 1 p e −δp [B 1 ρ 1 (p) + B −1 ρ 1 (−p) + B 2 ρ 2 (p) + B −2 ρ 2 (−p)] , 2δp [(e ip(x−z)+ipt − 1) + (e −ip(x−z)+ipt − 1)]. (4.79)Now we derive the asymptotic formula for (4.79). Using the Poisson summation for− z) 2 − t 2 . same procedure we can calculate all the other terms in (2.22) and derive Formula (3.51). The asymptotic expressions for the terms in the exponential can be derived with the same procedure as in the last section and we shall not repeat it here. So we proved Theorem (2.2) with the exact Bosonization formulas. expressions of the factors in Formulas (3.43) and (3.44) With some very long but elementary calculation we find that the expressions of the terms in formula (3.43) and (3.44) read: (±p) cosh εφ(∓e ∓ipx±ipt ± e ∓ipx±ipt(σ+1) ) (±p) sinh εφ(∓e ∓ipz∓ipt ± e ∓ipz∓ipt(σ+1) ) (±p) cosh εφ(±e ∓ipz∓ipt(σ+1) ∓ e ∓ipz∓ipt ) .cosh εφ − 1)e −ipz+ipt ρ 1 (p) − (cosh εφ − 1)e ipz−ipt ρ 1 (−p) ] εφe −ipz+ipt ρ 2 (p) − sinh εφe ipz−ipt ρ 2 (−p) ]. cosh φ − 1) cosh εφe −ipz+ip(σ+1)t ρ 1 (p) (5.86) − (cosh φ − 1) cosh εφe ipz−ip(σ+1)t ρ 1 (−p) ] cosh φ − 1) sinh εφe −ipz+ip(σ+1)t ρ 2 (p) − (cosh φ − 1) sinh εφe ipz−ip(σ+1)t ρ 2 (−p) ] 1 (p) sinh εφe −ipx−ipt(σ+1) − ρ 1 (−p) sinh εφe ipx+ipt(σ+1) ], ρ 2 (−p) cosh εφe ipx+ipt(σ+1) − ρ 2 (p) cosh εφe −ipx−ipt(σ+1) ] . εφ − 1)(e −ipz−ipt ρ 2 (p) − e ipz+ipt ρ 2 (−p)). sinh φ cosh εφ [e −ipz+ip(σ+1)t ρ 1 (p) (5.88) − e ipz−ip(σ+1)t ρ 1 (−p) ], 76) . I Bloch, J Dalibard, W Zwerger, Rev. Mod. Phys. 80885I. Bloch, J. Dalibard and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008). . A Polkovnikov, K Sengupta, Silva A Vengalattore, M , Rev. Mod. Phys. 83863Polkovnikov A, Sengupta, K, Silva A and Vengalattore M Rev. Mod. Phys. 83 863 (2011) . M Rigol, V Dunjko, M Olshanii, Nature. 452854M. Rigol, V. Dunjko, and M. Olshanii, Nature 452, 854 (2008). . N Nessi, A Iucci, A M Cazalilla, Phys. Rev, Lett. 113210402N. Nessi, A. Iucci, A.M. Cazalilla, Phys. Rev, Lett. 113, 21, 210402 (2014) . T Antal, Z Racz, G M Schutz, Phys. Rev. E. 594912T. Antal, Z. Racz, G.M. Schutz, Phys. Rev. E 59, 5 4912 (1999) . M Rigol, A Muramatsu, Phys. Rev. Lett. 93230404M. Rigol and A. Muramatsu, Phys. Rev. Lett. 93, 230404 (2004). . M Rigol, A Muramatsu, Phys. Rev. Lett. 94240403M. Rigol and A. Muramatsu, Phys. Rev. Lett. 94, 240403 (2005) . M Rigol, A Muramatsu, M Olshanii, Phys. Rev. A. 7453616M. Rigol, A. Muramatsu, and M. Olshanii,Phys. Rev. A 74, 053616 (2006) . M Cazalilla, Phys. Rev. Lett. 97156403M Cazalilla Phys. Rev. Lett. 97 156403 (2006) . A Iucci, A , M , Cazalilla Phys. Rev. A. 8063619A. Iucci A, M. Cazalilla Phys. Rev. A 80 063619 (2009) . S R Manmana, S Wessel, R M Noack, A Muramatsu, Phys. Rev. Lett. 98210405S.R. Manmana, S. Wessel, R.M. Noack, A. Muramatsu Phys. Rev. Lett.98.210405 (2006) . P Calabrese, J Cardy, Phys.Rev.Lett. 96136801P. Calabrese, J. Cardy Phys.Rev.Lett. 96, 136801 (2006) . S Langer, F Heidrich-Meisner, J Gemmer, I P Mccul-Loch, U Schollwock, Phys. Rev. B. 79214409S. Langer, F. Heidrich-Meisner, J. Gemmer, I.P. McCul-loch, and U. Schollwock, Phys. Rev. B 79, 214409 (2009) . D Fioretto, M , Mussardo New J.Phys. 1255015D.Fioretto, M. Mussardo New J.Phys.12:055015,2010 . J Lancaster, A Mitra, Phys. Rev. E. 8161134J. Lancaster A. Mitra Phys. Rev. E 81, 061134 (2010) . A Mitra, T Giamarchi, Phys. Rev. Lett. 107150602A. Mitra and T. Giamarchi, Phys. Rev. Lett. 107, 150602 (2011). . J Lancaster, T Giamarchi, A Mitra, Phys. Rev. B. 8475143J. Lancaster, T. Giamarchi, A. Mitra Phys. Rev. B 84, 075143 (2011) . Vieri Mastropietro, Zhituo Wang, Phys. Rev. B. 9185123Vieri Mastropietro, Zhituo Wang, Phys. Rev. B 91, 085123 (2015) . B Dra, M Haque, G Zarnd, Phys. Rev. Lett. 106156406B. Dra, M. Haque, G. Zarnd Phys. Rev. Lett. 106, 156406 (2011) . J Rentrop, D Schuricht, V , Meden New J. Phys. 1475001J. Rentrop, D. Schuricht, V. Meden New J. Phys. 14, 075001 (2012) . C Karrasch, J Rentrop, D Schuricht, V Meden, Phys. Rev. Lett. 109126406C. Karrasch, J. Rentrop, D. Schuricht, V. Meden Phys. Rev. Lett. 109, 126406 (2012) . S A Hamerla, G Uhrig, New J. Phys. 1573012S. A. Hamerla, G. S Uhrig New J. Phys. 15 073012 (2013) . D M Kennes, V Meden, Phys. Rev. B. 88165131D. M. Kennes and V. Meden Phys. Rev. B 88, 165131 (2013) . T Sabetta, G Misguich, Phys. Rev. B. 88245114T. Sabetta, G. Misguich Phys. Rev. B 88, 245114 (2013) . L Bonnes, F H L Essler, A M Luchli, Phys. Rev. Lett. 113187203L. Bonnes, F. H. L. Essler, A. M. Luchli Phys. Rev. Lett. 113, 187203 (2014) . V Alba, F Heidrich-Meisner, Phys. Rev. B. 9075144V. Alba, F. Heidrich-Meisner Phys. Rev. B 90, 075144 (2014) . R Sachdeva, Nag, A Agarwal, Dutta, Phys. Rev. B. 9045421R Sachdeva, T Nag, A Agarwal, A. Dutta Phys. Rev. B 90, 045421 (2014) . W Liu, N Andrei, Phys. Rev. Lett. 112257204W,Liu, N. Andrei Phys. Rev. Lett. 112, 257204 (2014) . C Karrasch, J E Moore, F Heidrich-Meisner, Phys. Rev. B. 8975139C. Karrasch, J. E. Moore, F. Heidrich-Meisner Phys. Rev. B 89, 075139 (2014) . D M Kennes, C Klckner, V Meden, Phys. Rev. Lett. 113116401D.M. Kennes, C. Klckner, V. Meden Phys. Rev. Lett. 113, 116401 (2014) . D Haldane, Phys. Rev. Lett. 451358D. Haldane. Phys. Rev. Lett. 45, 1358 (1980) . J M Luttinger, J. Math. Phys. 41154J. M. Luttinger, J. Math. Phys. 4 (1963) 1154. . S Tomonaga, Prog. Theor. Phys. 5544S. Tomonaga,Prog. Theor. Phys. 5, 544 (1950) . D C Mattis, E H Lieb, J. Math. Phys. 62304D. C. Mattis and E. H. Lieb J. Math. Phys. 6, 2304 (1965) . V Mastropietro, Il Nuovo Cimento. 1091V. Mastropietro Il Nuovo Cimento 109, 1 (1994) V , Mastropietro D Mattis, Luttinger Model. World ScientificV. Mastropietro D. Mattis (Editors), Luttinger Model. World Scientific 2014 . P Calabrese, J Cardy, J. Stat. Mech. 10004P. Calabrese and J. Cardy, J. Stat. Mech. 10004 (2007) . M Ganahl, E Rabel, F H L Essler, H G Evertz, Phys. Rev. Lett. 10877206M. Ganahl, E. Rabel, F.H.L. Essler, and H.G. Evertz, Phys. Rev. Lett. 108, 077206 (2012). . G Benfatto, P Falco, V Mastropietro, Phys. Rev. Lett. 10475701G.Benfatto, P. Falco V. Mastropietro. Phys. Rev. Lett. 104 075701 (2010). . J Sirker, R G Pereira, I , Affleck Phys. Rev. B. 8335115J. Sirker, R.G. Pereira, I. Affleck Phys. Rev. B 83, 035115 (2011) . T Prosen, Phys. Rev. Lett. 106217206T. Prosen Phys. Rev. Lett. 106, 217206 (2011) . E Langmann, J Lebowitz, V Mastropietro, P Moosavi, Comm. Math. Phys. 349551582E. Langmann, J. Lebowitz, V. Mastropietro, P. Moosavi, Comm. Math. Phys. 349, 2, 551582, (2017) . V Mastropietro, Phys. Rev. E. 8742121V. Mastropietro. Phys. Rev. E 87, 042121 (2013) . A Theumann, J. Math. Phys. 82460A. Theumann J. Math. Phys. 8, 2460 (1967) . E Langmann, P Moosavi, J. Math. Phys. 5691902E. Langmann, P. Moosavi, J. Math. Phys., 56, 091902 (2015)
{'fraction_non_alphanumeric': 0.12427532113220416, 'fraction_numerical': 0.06644310560418325, 'mean_word_length': 3.114710009354537, 'pattern_counts': {'":': 0, '<': 7, '<?xml version=': 0, '>': 49, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 16, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': "We investigate, in the Luttinger model with fixed box potential, the time evolution of an inhomogeneous state prepared as a localized fermion added to the noninteracting ground state. We proved that, if the state is evolved with the interacting Hamiltonian, the averaged density has two peaks moving in opposite directions, with a constant but renormalized velocity. We also proved that a dynamical 'Landau quasi-particle weight' appears in the oscillating part of the averaged density, asymptotically vanishing with large time. The results are proved with the Mattis-Lieb diagonalization method. A simpler proof with the exact Bosonization formulas is also provided.", 'arxivid': '1711.07507', 'author': ['Zhituo Wang [email protected] \nInstitute for Advanced Study in Mathematics\nResearch Center for Operator Algebras\nHarbin Institute of Technology\n150006HarbinChina\n\nEast China Normal University\n\n'], 'authoraffiliation': ['Institute for Advanced Study in Mathematics\nResearch Center for Operator Algebras\nHarbin Institute of Technology\n150006HarbinChina', 'East China Normal University\n'], 'corpusid': 56319085, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 16368, 'n_tokens_neox': 13915, 'n_words': 7098, 'pdfsha': '62c41cab884f2b2547d101e9450885a281dae505', 'pdfurls': ['https://arxiv.org/pdf/1711.07507v1.pdf'], 'title': ['Quantum Quench dynamics in Non-local Luttinger Model: Rigorous Results', 'Quantum Quench dynamics in Non-local Luttinger Model: Rigorous Results'], 'venue': []}
arxiv
Comment on "Ehrenfest times for classically chaotic systems" 29 May 2002 Steven Tomsovic Department of Physics Washington State University 99164-2814PullmanWA Eric J Heller Department of Physics Department of Chemistry and Chemical Biology Harvard University 02138CambridgeMA Comment on "Ehrenfest times for classically chaotic systems" 29 May 2002(November 21, 2018)arXiv:nlin/0205065v1 [nlin.CD] In a recent Rapid Communication [1], the authors, Silvestrov and Beenakker, introduce a way to lengthen the Ehrenfest time, τ , for fully chaotic systems. We disagree with several statements made in their paper, and address the following points essential to their conclusions: 1) it is not true that all semiclassical approximations for chaotic systems fail at a so-called 'logtime', τ ∝ − ln(h), differing only by a numerical coefficient; and 2) the limitation of the semiclassical approximation as expressed in the authors' Eq.(8)is not limited by their argument leading to Eq. (12).It is important to distinguish between the correspondence of quantum and classical dynamical propagations, and the faithfulness of semiclassical approximations. If one takes the Ehrenfest time, τ , to be the upper limit for which a quantum mechanical wave packet is described by solving classical equations of motion without invoking a semiclassical construction of the wave packet, then the Ehrenfest time increases logarithmically slowly for chaotic systems as τ ∝ λ −1 ln(S/h)[2,3]; there is no controversy on this point. In this expression λ is sum of the positive Lyapunov exponents, and S is some characteristic classical action such as that of the shortest periodic orbit. If, instead, one defines τ as the time scale beyond which the semiclassical approximation no longer faithfully reproduces the quantum propagation of a wave packet, then τ is not a so-called "logtime", but is proportional to inverse algebraic powers of h [4-6].The precise exponent in the breakdown time scale has been shown to depend on a few basic features of the chaotic dynamical system being considered. We mention work on three separate paradigms of chaos. It was shown in the stadium billiard[4], that τ ∝h −1/2 ln S/h (essentially 1 In a recent Rapid Communication [1], the authors, Silvestrov and Beenakker, introduce a way to lengthen the Ehrenfest time, τ , for fully chaotic systems. We disagree with several statements made in their paper, and address the following points essential to their conclusions: 1) it is not true that all semiclassical approximations for chaotic systems fail at a so-called 'logtime', τ ∝ − ln(h), differing only by a numerical coefficient; and 2) the limitation of the semiclassical approximation as expressed in the authors' Eq. (8) is not limited by their argument leading to Eq. (12). It is important to distinguish between the correspondence of quantum and classical dynamical propagations, and the faithfulness of semiclassical approximations. If one takes the Ehrenfest time, τ , to be the upper limit for which a quantum mechanical wave packet is described by solving classical equations of motion without invoking a semiclassical construction of the wave packet, then the Ehrenfest time increases logarithmically slowly for chaotic systems as τ ∝ λ −1 ln(S/h) [2,3]; there is no controversy on this point. In this expression λ is sum of the positive Lyapunov exponents, and S is some characteristic classical action such as that of the shortest periodic orbit. If, instead, one defines τ as the time scale beyond which the semiclassical approximation no longer faithfully reproduces the quantum propagation of a wave packet, then τ is not a so-called "logtime", but is proportional to inverse algebraic powers of h [4][5][6]. The precise exponent in the breakdown time scale has been shown to depend on a few basic features of the chaotic dynamical system being considered. We mention work on three separate paradigms of chaos. It was shown in the stadium billiard [4], that τ ∝h −1/2 ln S/h (essentially 1 h −1/2 ). Theh −1/2 behavior was linked to the fact that the stable and unstable manifolds associated with trajectories in the stadium have discontinuities in their slopes where they fold over upon themselves. The ln S/h part of the expression is due to the 'stickiness' of phase space in the neighborhood of the marginally stable bouncing ball trajectories. In contrast, a general dynamical system possessing stable and unstable manifolds that are continuous in their slopes gives τ ∝h −1/3 [5]; this was illustrated with the kicked rotor. A third example that has been studied extensively is the quantum bakers map. There it was shown that for some quantities, the breakdown time scale could be as great as τ ∝h −1 [6], althoughh −1/2 was typical [7]. Note that the semiclassical approximations in Refs. [4][5][6][7] involve no uniformizations or caustic corrections. They are, in fact, either exactly or poor man's versions of the standard WKB method, and developed specifically for chaotic systems. For wave packets, the standard timedependent WKB method involves sets of complex trajectories [8]. Nevertheless, in the above cited work on τ , no classically non-allowed processes are taken into account. One essential ingredient relied upon in these works, the 'area-h rule' is contained in Ref. [3]. This rule is in contradiction with the argument of Ref. [1] leading to Eq. (12) which contains the relation h 7/6−c << 1, c being the coefficient of proportionality in the logtime scale relation. The consequences of the area-h rule carefully considered in conjunction with the geometrical properties of evolving stable and unstable manifolds give a precise formulation of the semiclassical breakdown due to caustics and the resultant algebraic time scales [4][5][6]. The crucial point is that the distance between local classical manifolds (the criterion used by Silvestrov and Beenakker) is actually of no importance -what matters is the area enclosed by following the manifold from one branch to the next in a given locality. To miss this point unfortunately leads to a qualitatively different and incorrect result. Finally, we do agree with the authors that there should be important or morphological distinctions in the nature of evolving wave packets as they surpass each relevant time scale. Examples include interference phenomena necessarily arising beyond the logtime, and localization effects sometimes supressing classical diffusion beyond algebraic time scales [9]. 2 . P G Silvestrov, C W J Beenakker, Phys. Rev. E. 6535208P. G. Silvestrov and C. W. J. Beenakker, Phys. Rev. E 65, 035208(R) (2002). . G P Berman, G M Zaslavsky, Physica A. 91450G. P. Berman and G. M. Zaslavsky, Physica A 91, 450 (1978); . G M Zaslavsky, Phy. Rep. 80157G. M. Zaslavsky, Phy. Rep. 80 (1981) 157. . M V Berry, N L Balazs, J. Phys.A. 12625M. V. Berry and N. L. Balazs, J. Phys.A 12, 625 (1979); . M V Berry, N L Balazs, M Tabor, A Voros, Ann. Phys. (N. Y.). 12226M. V. Berry, N. L. Balazs, M. Tabor, and A. Voros, Ann. Phys. (N. Y.) 122, 26 (1979). . S Tomsovic, E J Heller, Phys. Rev. Lett. 67664S. Tomsovic and E. J. Heller, Phys. Rev. Lett. 67, 664 (1991); . Phys. Rev. E. 47282Phys. Rev. E 47, 282 (1993); . Phys. Rev. Lett. 701405Phys. Rev. Lett. 70, 1405 (1993); . E J Heller, S Tomsovic, Physics Today. 46738E. J. Heller and S. Tomsovic, Physics Today 46 (7), 38 (1993). . M.-A Sepúlveda, S Tomsovic, E J Heller, Phys. Rev. Lett. 69402M.-A. Sepúlveda, S. Tomsovic and E. J. Heller, Phys. Rev. Lett. 69, 402 (1992). . P W O&apos;connor, S Tomsovic, E J Heller, Physica D. 55340P. W. O'Connor, S. Tomsovic, and E. J. Heller, Physica D 55, 340 (1992); . J. Stat. Phys. 68131J. Stat. Phys. 68, 131 (1992); . P W O&apos;connor, University of WashingtonDoctoral dissertationP. W. O'Connor, Doctoral dissertation, University of Washington, (1991). . F.-M Dittes, E Doron, U Smilansky, Phys. Rev. E. 52963F.-M. Dittes, E. Doron, and U. Smilansky, Phys. Rev. E 52, R963 (1994). . D Huber, E J Heller, R G Littlejohn, J. Chem. Phys. 8942003D. Huber, E. J. Heller and R. G. Littlejohn, J. Chem. Phys. 89 (4), 2003 (1988). . S Fishman, D R Grempel, R E Prange, Phys. Rev. Lett. 49509S. Fishman, D. R. Grempel, and R. E. Prange, Phys. Rev. Lett. 49, 509 (1982); . Phys. Rev. A. 291639Phys. Rev. A 29, 1639 (1984).
{'fraction_non_alphanumeric': 0.06427289048473968, 'fraction_numerical': 0.040813883901855176, 'mean_word_length': 4.167594310451453, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'In a recent Rapid Communication [1], the authors, Silvestrov and Beenakker, introduce a way to lengthen the Ehrenfest time, τ , for fully chaotic systems. We disagree with several statements made in their paper, and address the following points essential to their conclusions: 1) it is not true that all semiclassical approximations for chaotic systems fail at a so-called \'logtime\', τ ∝ − ln(h), differing only by a numerical coefficient; and 2) the limitation of the semiclassical approximation as expressed in the authors\' Eq.(8)is not limited by their argument leading to Eq. (12).It is important to distinguish between the correspondence of quantum and classical dynamical propagations, and the faithfulness of semiclassical approximations. If one takes the Ehrenfest time, τ , to be the upper limit for which a quantum mechanical wave packet is described by solving classical equations of motion without invoking a semiclassical construction of the wave packet, then the Ehrenfest time increases logarithmically slowly for chaotic systems as τ ∝ λ −1 ln(S/h)[2,3]; there is no controversy on this point. In this expression λ is sum of the positive Lyapunov exponents, and S is some characteristic classical action such as that of the shortest periodic orbit. If, instead, one defines τ as the time scale beyond which the semiclassical approximation no longer faithfully reproduces the quantum propagation of a wave packet, then τ is not a so-called "logtime", but is proportional to inverse algebraic powers of h [4-6].The precise exponent in the breakdown time scale has been shown to depend on a few basic features of the chaotic dynamical system being considered. We mention work on three separate paradigms of chaos. It was shown in the stadium billiard[4], that τ ∝h −1/2 ln S/h (essentially 1', 'arxivid': 'nlin/0205065', 'author': ['Steven Tomsovic \nDepartment of Physics\nWashington State University\n99164-2814PullmanWA\n', 'Eric J Heller \nDepartment of Physics\nDepartment of Chemistry and Chemical Biology\nHarvard University\n02138CambridgeMA\n'], 'authoraffiliation': ['Department of Physics\nWashington State University\n99164-2814PullmanWA', 'Department of Physics\nDepartment of Chemistry and Chemical Biology\nHarvard University\n02138CambridgeMA'], 'corpusid': 8274613, 'doi': '10.1103/physreve.68.038201', 'github_urls': [], 'n_tokens_mistral': 2665, 'n_tokens_neox': 2213, 'n_words': 1334, 'pdfsha': '21e23c8ec76d0e78d623084eeed0fc5f29c35877', 'pdfurls': ['https://arxiv.org/pdf/nlin/0205065v1.pdf'], 'title': ['Comment on "Ehrenfest times for classically chaotic systems"', 'Comment on "Ehrenfest times for classically chaotic systems"'], 'venue': []}
arxiv
On Random Construction of a Bipolar Sensing Matrix with Compact Representation 1 Aug 2009 Tadashi Wadayama [email protected] Nagoya Institute of Technology Nagoya AichiJAPAN On Random Construction of a Bipolar Sensing Matrix with Compact Representation 1 Aug 2009 A random construction of bipolar sensing matrices based on binary linear codes is introduced and its RIP (Restricted Isometry Property) is analyzed based on an argument on the ensemble average of the weight distribution of binary linear codes. I. INTRODUCTION Research in compressed sensing [2] [3] is expanding rapidly. The sufficient condition for ℓ 1 -recovery based on the Restricted Isometry Property (RIP) [3] [4] is one of the celebrated results in this field. The design of sensing matrices with small RIP constants is a theoretically interesting and challenging problem. Currently, random constructions provide the strongest results, and the analysis of random constructions is based on large deviations of maximum and minimum singular values of random matrices [5] [3]. In the present paper, a random construction of bipolar sensing matrices based on binary linear codes is introduced and its RIP is analyzed. The column vectors of the proposed sensing matrix are nonzero codewords of a randomly chosen binary linear code. Using a generator matrix, a p × m sensing matrix can be represented by O(p log 2 m)-bits. The existence of sensing matrices with the RIP is shown based on an argument on the ensemble average of the weight distribution of binary linear codes. II. PRELIMINARIES A. Notation The symbols R and F 2 represent the field of real numbers and the finite field with two elements {0, 1}, respectively. The set of all p × m real matrices is denoted by R p×m . In the present paper, the notation x ∈ R p indicates that x is a column vector of length p. The notation || · || p denotes ℓ p -norm (1 ≤ p < ∞) defined by ||x|| p △ = p i=1 |x i | p 1/p .(1) The ℓ 0 -norm is defined by ||x|| 0 △ = |supp(x)|,(2) where supp(x) denotes the index set of nonzero components of x. The functions w h (·) and d h (·, ·) are the Hamming weight and Hamming distance functions, respectively. B. Restricted isometry property (RIP) Let Φ △ = {φ 1 , . . . , φ m } ∈ R p×m be a p × m real matrix, where the ℓ 2 -norm of the j-th (j ∈ [1, m]) column vector φ j is normalized to one, namely, ||φ i || 2 = 1. The notation [a, b] represents the set of consecutive integers from a to b. The restricted isometry property of Φ introduced by Candes and Tao [3] plays a key role in a sufficient condition of ℓ 1recovery. Definition 1: A vector x ∈ R m is called an S-sparse (S ∈ [1, m]) vector if ||x|| 0 ≤ S. If there exists a real number δ(0 ≤ δ < 1) satisfying (1 − δ)||x|| 2 2 ≤ ||Φx|| 2 2 ≤ (1 + δ)||x|| 2 2(3) for any S-sparse vector x ∈ R m , then we say that Φ has the RIP of order S. If Φ has the RIP of order S, then the smallest constant satisfying (3) is called the RIP constant of Φ, which is denoted by δ S . Assume that Φ has the RIP with small δ S . In such a case, any sub-matrix composed from Q-columns (1 ≤ Q ≤ S) of Φ is nearly orthonormal. Recently, Candes [4] reported the relation between the RIP and the ℓ 1 -recovery property. A portion of the main results of [4] is summarized as follows. Let S ∈ [1, m], and assume that Φ has the RIP with δ 2S ≤ √ 2 − 1.(4) For any S-sparse vector e ∈ R m (i.e., ||e|| 0 ≤ S), the solution of the following ℓ 1 -minimization problem minimize||d|| 1 subject to Φd = s(5) coincides exactly with e, where s = Φe. Note that [4] considers stronger reconstruction results (i.e., robust reconstruction). The matrix Φ in (5) is called a sensing matrix. C. Relation between incoherence and the RIP The incoherence of Φ defined below and the RIP constant are closely related. Definition 2: The incoherence of Φ is defined by µ(Φ) △ = max i,j∈[1,m],i =j |φ T i φ j |.(6) The following lemma shows the relation between the incoherence and the RIP constant. Similar bounds are well known (e.g., [9]). Lemma 1: Assume that Φ ∈ R p×m is given. For any S ∈ [1, m], δ S is upper bounded by δ S < µ(Φ)S.(7) An elementary proof (different from that in [9]) is presented in Appendix. III. CONSTRUCTION OF SENSING MATRICES BASED ON BINARY LINEAR CODES In this section, we present a construction method for sensing matrices based on binary linear codes. A sensing matrix obtained from this construction has a concise description. A sensor can store a generator matrix of a binary linear code, instead of the entire sensing matrix. A. Binary to bipolar conversion function The function β p : F p 2 → R p is called a binary to bipolar conversion function defined by β : x ∈ F p 2 → 1 √ p (e − 2x) ∈ R p ,(8) where e is an all-one column vector of length p. Namely, using the binary to bipolar conversion function, a binary sequence is converted to a {+1/ √ p, −1/ √ p}-sequence. The following lemma demonstrates that the inner product of two bipolar sequences β p (a) and β p (b) is determined from the Hamming distance between the binary sequences a and b. Lemma 2: For any a, b ∈ F p 2 , the inner product of β p (a) and β p (b) is given by (9) is derived as follows: β p (a) T β p (b) = 1 − 2d h (a, b) p . (9) (Proof) Let β p (a) = (a 1 , . . . , a p ) T and β p (b) = (b 1 , . . . , b p ) T . Define Y 1 and Y 2 by Y 1 △ = {i ∈ [1, p] : a i = b i }, Y 2 △ = {i ∈ [1, p] : a i = b i },(10)where |Y 1 | = p − d h (a, b) and |Y 2 | = d h (a, b). Equationβ p (a) T β p (b) = p i=1 a i b i = p i∈Y1 a i b i + p i∈Y2 a i b i = p i∈Y1 1 p + p i∈Y2 − 1 p = (p − d h (a, b)) 1 p + d h (a, b) − 1 p = 1 − 2d h (a, b) p .(11) It is easy to confirm that β p (a) is normalized, i.e., ||β p (a)|| 2 = 1, for any a ∈ F p 2 . B. Construction of the sensing matrix Let H ∈ F r×p 2 (p > r) be a binary r×p parity check matrix where 2 p−r ≥ p holds. The binary linear coded C(H) defined by H is given by C(H) △ = {x ∈ F p 2 : Hx = 0 r },(12) where 0 r is a zero-column vector of length r. The following definition gives the construction of sensing matrices. − 1 ≥ 2 p−r − 1. The sensing matrix Φ(H) ∈ R p×m is defined by Φ(H) △ = (β p (c 1 ), β p (c 2 ), . . . , β p (c m )) ,(13) where m = 2 p−r − 1. If Φ(H) has the RIP of order S, the RIP constant corresponding to Φ(H) is denoted by δ S (H). Since the order of the columns is unimportant, we do not distinguish between sensing matrices of different column order (or choice of codewords from C(H)). If the weights of all nonzero codewords of C(H) are very close to p/2, then the incoherence of Φ(H) becomes small, as described in detail in the following lemma. Lemma 3: Assume that ǫ(0 < ǫ < 1) is given and that 1 − ǫ 2 p ≤ w h (c) ≤ 1 + ǫ 2 p(14) holds for any c ∈ C(H)\0 p . In such a case, the incoherence Φ(H) is upper bounded by µ(Φ(H)) ≤ ǫ.(15) (Proof) For any pair of codewords a, b(a = b) ∈ C(H), the Hamming weight of a + b is in the range: 1 − ǫ 2 p ≤ w h (a + b) ≤ 1 + ǫ 2 p.(16) due to the linearity of C(H). This means that 1 − ǫ 2 p ≤ d h (a, b) ≤ 1 + ǫ 2 p(17) holds for any a, b ∈ C(H)(a = b). Using Lemma 2, we immediately obtain ∀i, j(i = j) ∈ [1, m], −ǫ ≤ β p (c i ) T β p (c j ) ≤ ǫ,(18) where Φ(H) = (β p (c 1 ), β p (c 2 ), . . . , β p (c m )) . The definition of incoherence and the above inequalities lead to an upper bound on the incoherence: µ(Φ(H)) ≤ ǫ.(20) C. Analysis based on ensemble average of weight distribution We here consider binary linear codes whose weight distribution is tightly concentrated around the Hamming weight p/2. Before starting the analysis, we introduce the weight distribution {A w (H)} w∈ [1,n] , which is defined by A w (H) △ = |{c : c ∈ C(H), w h (c) = w}|.(21) In the present paper, we consider an ensemble of binary parity check matrices, which is referred to herein as the random ensemble. The random ensemble R r,p contains all binary r ×p matrices. Equal probability P (H) = 1/2 rp is assigned to each matrix in R r,p . Let f be a real-valued function defined on R r,p , which can be considered as a random variable defined over the ensemble R r,p . The expectation of f with respect to the ensemble R r,p is defined by E Rr,p [f ] △ = H∈Rr,p P (H)f (H).(22) The expectation of weight distributions with respect to the random ensemble has been reported [8] to be E Rr,p [A w (H)] = p w 2 −r .(23) In the following, a combination of average weight distribution and Markov inequality is used to show that the RIP holds for Φ(H) with overwhelmingly high probability. Lemma 4: Assume that we draw a parity check matrix from R r,p . The probability of selecting H that satisfies µ(Φ(H)) ≤ ǫ is lower bounded by 1 − 2 1−r ⌊( 1−ǫ 2 )p⌋ w=0 p w .(24) (Proof) Let us define K ǫ (H) as K ǫ (H) △ = ⌊( 1−ǫ 2 )p⌋ w=1 A w (H) + p w=⌈( 1+ǫ 2 )p⌉ A w (H)(25) for H ∈ R r,p . The condition K ǫ (H) = 0 implies that 1 − ǫ 2 p ≤ w h (c) ≤ 1 + ǫ 2 p(26) for any c ∈ C(H)\0 p . Namely, if K ǫ (H) = 0 holds, then µ(Φ(H)) is proven to be smaller than or equal to ǫ by Lemma 3. Next, we evaluate the ensemble expectation of K ǫ (H): E Rr,p [K ǫ (H)] = ⌊( 1−ǫ 2 )p⌋ w=1 E Rr,p [A w (H)] + p w=⌈( 1+ǫ 2 )p⌉ E Rr,p [A w (H)] = ⌊( 1−ǫ 2 )p⌋ w=1 2 −r p w + p w=⌈( 1+ǫ 2 )p⌉ 2 −r p w < 2 1−r ⌊( 1−ǫ 2 )p⌋ w=0 p w .(27) The final inequality is due to the following identity on the binomial coefficients: ∀w ∈ [0, p], p w = p p − w .(28) Using the Markov inequality, we obtain the following upper bound on the probability of the event K ǫ (H) ≥ 1: P rob[K ǫ (H) ≥ 1] ≤ E Rr,p [K ǫ (H)] < 2 1−r ⌊( 1−ǫ 2 )p⌋ w=0 p w .(29) Since K ǫ (H) takes a non-negative integer-value, we have P rob[K ǫ (H) = 0] > 1 − 2 1−r ⌊( 1−ǫ 2 )p⌋ w=0 p w .(30) This completes the proof. The following theorem is the main contribution of the present paper. Theorem 1: Assume that H is chosen randomly according to the probability assignment of R r,p . If S < Z p log 2 m ,(31) holds, then δ 2S (H) < √ 2 − 1 holds with probability greater than 1 − 2 1−p+r ,(32) where m = 2 p−r − 1. The constant Z is given by Z △ = √ 2 − 1 2 √ 6 .(33) (Proof) A simpler upper bound on 2 1−r ⌊( 1−ǫ 2 )p⌋ w=0 p w(34) is required. Using the inequality on binomial coefficients [6]: p w ≤ 2 pH(w/p) ,(35) we have 2 1−r ⌊( 1−ǫ 2 )p⌋ w=0 p w ≤ 2 1−r ⌊( 1−ǫ 2 )p⌋ w=0 2 pH(w/p) < 2 1−r × p × 2 pH( 1−ǫ 2 ) = 2 1−r+log 2 p+pH( 1−ǫ 2 ) ,(36) where H(x) is the binary entropy function defined by H(x) △ = −x log 2 x − (1 − x) log 2 (1 − x).(37) In order to consider the exponent of an upper bound, we take the logarithm of (34) and obtain an upper bound of the exponent: log 2   2 1−r ⌊( 1−ǫ 2 )p⌋ w=0 p w   < 1 + log 2 (m + 1) − p + log 2 p + pH 1 − ǫ 2 (38) < 1 + 2 log 2 (m + 1) − 1 2 pǫ 2 .(39) In the above derivation, we used the relation r = p − log 2 (m + 1)(40) and the assumption 2 p−r ≥ p. A quadratic upper bound on the binary entropy function (Lemma 6 in Appendix) was also exploited to bound the entropy term. Letting ǫ △ = 6 log 2 (m + 1) p ,(41) we have 1 + 2 log 2 (m + 1) − 1 2 pǫ 2 = 1 − log 2 (m + 1) = 1 − p + r.(42) Lemma 1 and Lemma 4 imply that, in this case, δ S (H) < ǫS holds with probability greater than 1−2 1−p+r . Due to Lemma 1, the ℓ 1 -recovery condition (4) can be written as δ 2S < 2 6 log 2 (m + 1) p S < √ 2 − 1.(43) From this inequality, we have S < Z p log 2 (m + 1) < Z p log 2 m ,(44) which proves the claim of the theorem. D. Asymptotic analysis In this subsection, the asymptotic properties of the proposed construction are given. Lemma 5: Assume that we draw a parity check matrix from R r,p . The probability of selecting H that satisfies µ(Φ(H)) ≤ ǫ is upper bounded by (1 − 2 −r )2 1+r ⌊( 1−ǫ 2 )p⌋ w=0 p w 2 ⌊( 1−ǫ 2 )p⌋ w=0 p w − 1 2 . (45) (Proof) Here, we use a variant of Chebyschev's inequality [1]: P rob[K ǫ (H) = 0] ≤ V AR Rr,p (K ǫ (H)) E Rr,p [K ǫ (H)] 2 ,(46) where V AR Rr,p (·) denotes the variance with respect to R r,p . The variance V AR Rr,p (K ǫ (H)) is given by V AR Rr,p (K ǫ (H)) = A w1=1 A w2=1 Cov(w 1 , w 2 ) + Definition 3 : 3Assume that all of the nonzero codewords of C(H) are denoted by c 1 , c 2 , . . . , c M (based on any predefined order), where M = 2 p−rank(H) w 1 , w 2 ),(47)where A = ⌊(1−ǫ)p/2⌋ and B = ⌈(1+ǫ)p/2⌉. The covariance of weight distributions denoted by Cov(w 1 , w 2 ) is defined as follows:Cov(w 1 , w 2 ) △ = E Rr,p [A w1 (H)A w2 (H)] − E Rr,p [A w1 (H)]E Rr,p [A w2 (H)] (48)for w 1 , w 2 ∈ [1, n]. The covariance for the random ensemble has the following closed formula[10]:Cov(w 1 , w 2 ) = (1 − 2 −r )2 −r p w w 1 = w 2 = w 0 w 1 = w 2 (49) for w 1 , w 2 ∈ [1, n].Applying the covariance formula to (47), we have V AR Rr,p (K ǫ (Hupper bound on the variance (50) into (46) proves the lemma. The asymptotic behavior of P rob[K ǫ (H) = 0] and P rob[K ǫ (H) = 0] is summarized in the following theorem.Theorem 2: Assume that α = r/p is fixed (0 < α < 1). LetThe following inequalities give upper bounds on f 1 (ǫ) and f 2 (ǫ), respectively:(Proof) We first discuss (54). LetUsing the inequality on the binomial coefficientsX can be bounded from below:The inequality (45) can be simplified asfor sufficiently large X. The right-hand side of the above inequality can be bounded from above using (58):We are now able to derive the inequality given in (54) as follows:The inequality given in (55) is readily obtained from (38). Theorem 2 implies a sharp threshold behavior in the asymptotic regime. Let α * (ǫ) bewhich is referred to as the critical exponent. If α < α * (ǫ), (54) means that the probability to draw a p × r matrix with µ(Φ(H)) ≤ ǫ decreases exponentially as p goes to infinity. On the other hand, (55) indicates that the probability not to select a matrix with µ(Φ(H)) ≤ ǫ decreases exponentially if α > α * (ǫ).IV. CONCLUDING REMARKSIn the present paper, a construction of a bipolar sensing matrix is introduced and its RIP is analyzed. The existence of sensing matrices with the RIP has been shown based on a probabilistic argument. An advantage of this type of sensing matrix is its compactness. A sensor requires O(pm)-bits in order to store a truly random p × m bipolar matrix. On the other hand, we need only O(p log 2 m)-bits to store Φ(H) because we can use a generator matrix of C(H) as a compact representation of C(H). However, this limited randomness of matrices results in a penalty on the RIP constant. Although the present construction is based on a probabilistic construction, the results shown in Theorem 1 are weaker than the ℓ 1recovery condition O(S log e (m/S)) < p for the truly random p × m bipolar matrix ensemble shown in[5]. The condition shown in Theorem 1 can be written as O(S log 2 m) < √ p and is more similar to the conditions of deterministic constructions, such as that given in[7]. Lemma 3 may be useful for evaluating the goodness of a randomly generated instance. The weight distribution of C(H) can be evaluated with time complexity O(mp), and an upper bound on the RIP constant can be obtained using Lemma 3.APPENDIXLemma 6:The following inequality 1the domain of which is 0 < x < 1. The first and second derivatives of f (x) are given byandrespectively. It is easy to verify that f ′′ (x) > 0 for 0 < x < 1, which indicates that f (x) is convex. Thus, we can obtain the global minimum of f (x) by solving f ′ (x) = 0, and we have f ′ (1/2) = 0 and f (1/2) = 0.1 This bound becomes tighter as x approaches to 1/2.Proof of Lemma 1Let Q be an index set satisfying Q ⊂ {1, . . . , m}, |Q| ≤ S. For any c = (c i ) i∈Q ∈ R |Q| , we havewhere Φ Q is a sub-matrix of φ composed from the columns corresponding to the index set Q. For any a, b ∈ R,holds since (|a| − |b|) 2 = a 2 + b 2 − 2|ab| ≥ 0. We use this inequality to bound |c i c j | in (67) and obtainSimilarly, ||Φ Q c|| 2 2 can be lower bounded by ||Φ Q c|| 2 2 ≥ ||c|| 2 2 (1 − µ(Φ)S). From the definition of δ S , the lemma is proven.ACKNOWLEDGMENTThe author would like to thank the anonymous reviewers of IEEE Information Theory Workshop 2009 for their constructive comments. The present study was supported in part by the Ministry of Education, Science, Sports and Culture of Japan through a Grant-in-Aid for Scientific Research on Priority Areas (Deepening and Expansion of Statistical Informatics) 180790091. The probabilistic method. N Alon, J H Spencer, John Wiley & Sons2nd ed.N.Alon and J.H.Spencer, "The probabilistic method, " 2nd ed., John Wiley & Sons, 2000. Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. E Candes, J Romberg, T Tao, IEEE Trans. on Information Theory. 522E.Candes, J. Romberg and T.Tao, "Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information," IEEE Trans. on Information Theory, vol.52(2), pp. 489 -509, 2006. Decoding by linear programming. E Candes, T Tao, IEEE Trans. on Information Theory. 5112E.Candes and T.Tao, "Decoding by linear programming, " IEEE Trans. on Information Theory, vol.51(12), pp. 4203 -4215, 2005. The restricted isometry property and its implications for compressed sensing. E Candes, Compte Rendus de l'Academie des Sciences. Paris, Serie IE.Candes, "The restricted isometry property and its implications for compressed sensing," Compte Rendus de l'Academie des Sciences, Paris, Serie I, pp. 589-592. Near optimal signal recovery from random projections: universal encoding strategies?. E Candes, T Tao, IEEE Trans. on Information Theory. 5212E.Candes and T.Tao, "Near optimal signal recovery from random pro- jections: universal encoding strategies?, " IEEE Trans. on Information Theory, vol.52(12), pp. 5406 -5425, 2006. Elements of Information Theory. T M Cover, J A Thomas, 2nd ed. Wiley-InterscienceT.M.Cover and J.A.Thomas, "Elements of Information Theory", 2nd ed. Wiley-Interscience 2006. Deterministic constructions of compressed sensing matrices. R A Devore, J. of Complexity. 23R.A. DeVore, "Deterministic constructions of compressed sensing ma- trices, " J. of Complexity, 23, pp. 918 -925, 2007. Low Density Parity Check Codes. R G Gallager, MIT PressCambridge, MAR.G.Gallager, "Low Density Parity Check Codes". Cambridge, MA:MIT Press 1963. Compressed sensing and redundant dictionaries. H Rauhut, K Schass, P Vandergheynst, IEEE Trans. on Information Theory. 545H.Rauhut, K.Schass, and P.Vandergheynst, "Compressed sensing and redundant dictionaries," IEEE Trans. on Information Theory, vol.54(5), pp. 2210 -2219, 2008. On undetected error probability of binary matrix ensembles. T Wadayama, arXiv:0705.3995Proceedings of IEEE International Symposium on Information Theory (ISIT2008). IEEE International Symposium on Information Theory (ISIT2008)Trontorelated preprintT.Wadayama, "On undetected error probability of binary matrix ensem- bles", in Proceedings of IEEE International Symposium on Information Theory (ISIT2008), Tronto (2008) (related preprint: arXiv:0705.3995).
{'fraction_non_alphanumeric': 0.09538216560509555, 'fraction_numerical': 0.03550955414012739, 'mean_word_length': 3.485952380952381, 'pattern_counts': {'":': 0, '<': 26, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 12, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'A random construction of bipolar sensing matrices based on binary linear codes is introduced and its RIP (Restricted Isometry Property) is analyzed based on an argument on the ensemble average of the weight distribution of binary linear codes.', 'arxivid': '0908.0045', 'author': ['Tadashi Wadayama [email protected] \nNagoya Institute of Technology Nagoya\nAichiJAPAN\n'], 'authoraffiliation': ['Nagoya Institute of Technology Nagoya\nAichiJAPAN'], 'corpusid': 9610512, 'doi': '10.1109/itw.2009.5351188', 'github_urls': [], 'n_tokens_mistral': 6985, 'n_tokens_neox': 6181, 'n_words': 3602, 'pdfsha': '88a9156b8f0fd9dc410d5bc26e27f2ed2b058030', 'pdfurls': ['https://arxiv.org/pdf/0908.0045v1.pdf'], 'title': ['On Random Construction of a Bipolar Sensing Matrix with Compact Representation', 'On Random Construction of a Bipolar Sensing Matrix with Compact Representation'], 'venue': []}
arxiv
ANACONDA: An Improved Dynamic Regret Algorithm for Adaptive Non-Stationary Dueling Bandits 25 Oct 2022 Thomas Kleine Buening Aadirupa Saha ANACONDA: An Improved Dynamic Regret Algorithm for Adaptive Non-Stationary Dueling Bandits 25 Oct 2022 We study the problem of non-stationary dueling bandits and provide the first adaptive dynamic regret algorithm for this problem. The only two existing attempts in this line of work fall short across multiple dimensions, including pessimistic measures of non-stationary complexity and non-adaptive parameter tuning that requires knowledge of the number of preference changes. We develop an elimination-based rescheduling algorithm to overcome these shortcomings and show a near-optimalÕ( √ S CW T ) dynamic regret bound, where S CW is the number of times the Condorcet winner changes in T rounds. This yields the first near-optimal dynamic regret algorithm for unknown S CW . We further study other related notions of non-stationarity for which we also prove near-optimal dynamic regret guarantees under additional assumptions on the underlying preference model. Introduction Multi-Armed Bandits (MAB) [42,26,23] are a well-studied online learning framework, which can be used to model online decision-making under uncertainty. Due to its exploration-exploitation tradeoff, the MAB framework is able to model situations such as clinical trials or job scheduling, where the goal is to keep selecting the 'best item' in hindsight through sequentially querying one item at a time and subsequently observing a noisy reward feedback for the queried item [6,5,2,12]. The MAB framework has been studied and generalized to different settings, among which a popular variant is known as Dueling Bandits (DB) which has gained much attention in the machine learning community over the last two decades [47,49,50,43]. DB are a preference-based variant of MAB in which every round the learner selects a pair of items (or arms) whereupon a noisy preference between the two items is observed. Such a model is particularly useful in applications, where direct numerical feedback is unavailable, but observed feedback or behavior implies a preference of one item over the other. For instance, the DB framework can be used for search engine optimization through interleaved comparisons [25,20]. In the classical stochastic dueling bandit problem, it is assumed that the underlying preferences between items remain fixed over time. However, this assumed stationarity of preferences is likely to be violated in many applications. For example, preferences over movies may change depending on the season or other external influences. Despite its strong practical motivation, regret minimization in non-stationary dueling bandits has only recently been studied for the fist time [19,21]. In contrast to the classical stochastic [47,48,8] and adversarial [16,37,30] dueling bandit problem, which measures performance in terms of static regret w.r.t. a fixed benchmark (or best item in hindsight), in non-stationary dueling bandits we consider the stronger dynamic regret, which compares the algorithm's selection against a dynamic benchmark every round. In general, the achievable dynamic regret depends on the amount of non-stationarity in the environment. Here, prior work [19,21] studied the number of changes in the preference matrix as a measure non-stationary complexity. While the number of such preference switches indeed relates to the hardness of the problem, it is, however, a pessimistic measure of non-stationarity. For example, a change in the preference between two widely suboptimal arms or a minor change in the preference matrix under which the optimal arm remains optimal should not significantly impact our ability to achieve low dynamic regret. To this end, one question that we aim to address in this paper for the paradigm of non-stationary dueling bandits is: Q.1: Can we guarantee low dynamic regret for stronger and more meaningful notions of non-stationarity? Moreover, prior work in non-stationary dueling bandits [19,21] assumes knowledge of the nonstationary complexity, i.e. prior knowledge of the total number of preference switches (or total variation), which is a highly impractical assumption. The second question we thus address is: Q.2: Can we achieve near-optimal dynamic regret in non-stationary dueling bandits adaptively, without the knowledge of the underlying non-stationary complexity? Our Contributions We answer these two questions affirmatively. Our main contribution is a new algorithm ANACONDA that adaptively achieves near-optimal regret with respect to the number of 'best arm' switchesa measure that is sensitive only to the variations of the best arms in the preference sequence and indifferent to any other 'background noise' due to suboptimal arms. More precisely, our contributions can be listed as follows: • Connecting Different Notions of Non-Stationary Complexity in DB. We first give an overview over different notions of non-stationarity measures for dueling bandits and analyze their interdependencies towards a better understanding of the implications of one to another (Section 2.2). • Proposing Tighter Notions of Non-Stationarity (towards Q.1). We propose three new notions of non-stationary complexity for dueling bandits: (i) S CW which measures the number of Condorcet Winner Switches in the preference sequence, (ii)Ṽ which measures the preference variation of the Condorcet arms, and (iii)S CW that counts only the 'significant variations' in the Condorcet arms (Section 2.2). The novelty of our proposed non-stationarity measures lies in capturing only the non-stationarity observed for the 'best arms' of the preference sequences. They remain unaffected by any changes in the suboptimal arms, which of course captures a stronger notion of non-stationarity than simply counting the number of preference shifts S P , or total variation V , of the preference sequence {P t } t∈ [T ] , as studied in prior work [19,21]. In particular, we show thatS CW ≤ S CW ≤ S P andṼ ≤ V justifying the strength of our proposed non-stationarity measures. • Adaptive Algorithm (towards Q.2). Besides using weaker notions of non-stationary complexity, another drawback of existing work on non-stationary dueling bandit is that, in order to optimize dynamic regret, their algorithms require exact knowledge of the non-stationary complexity (e.g. S P or V ), which is in practice of course expected to be unknown to the system/algorithm designed ahead of time. Our next main contribution lies in designing an adaptive algorithm (ANACONDA, Algorithm 1) that does not require knowledge of any underlying non-stationary complexity-it can adapt to any unknown number of best arm switches S CW and yields a near-optimal regret bound of O √ S CW T (Theorem 3.1, Section 3). 1 • Improved and (Near-)Optimal Dynamic Regret Bounds. Owing to the fact that S CW ≤ S P , our dynamic regret bounds can be much tighter compared to the previous results by [19,21] which can only give a regret guarantee ofÕ √ S P T (Remark 2.1). Further our regret bound is also provably order optimal in T and S CW as justified in Remark 3.1. • Better Guarantees for Structured Preferences. Moreover, in Section 5 we discover a special class of preference matrices, those that respect a type of transitive property, for which we can prove even stronger dynamic regret guarantees ofÕ √S CW T in terms of Significant CW SwitchesS CW andÕ Ṽ 1 /3 T 2 /3 in terms of Condorcet Winner VariationṼ . The optimality of these bounds is discussed in Remark 5.2 and Remark 5.3. Related Works The non-stationary MAB problem has been extensively studied for various non-stationarity measures, such as total variation [9,10], distribution switches [17,4,7], or best arm switches [1,41]. Moreover, its study has been extended to more complex setups including linear bandits [27,28] and contextual MAB [24,13,44]. We will particularly take inspiration from the recent advances of [7,1,41] that were able to achieve near-optimal dynamic regret rates without knowledge of the number of distribution (or best arm) changes. While the non-stationary MAB problem has seen much attention in recent years, its DB counterpart remains widely unexplored. The only two earlier works that address the non-stationary dueling bandit problem are [19] and [21]. However, these works are limited in a) the weakness of the analyzed non-stationarity measures, namely, general preference switches or total variation (see Section 2.2), and b) in the fact that their algorithms require knowledge of the total amount of non-stationarity in advance, an unrealistic assumption. Here, we improve upon prior work by designing an adaptive algorithm ANACONDA that does not require knowledge of the amount of nonstationarity in the environment and achieves near-optimal dynamic regret w.r.t. the number of Condorcet winner switches, a stronger notion of non-stationarity than general preference switches. A more detailed review of previous work that is related to the non-stationary MAB and DB problem is provided in Appendix C. Problem Setting We consider preference matrices P ∈ [0, 1] K×K such that P (a, b) indicates the probability of arm a being preferred over arm b. Here, P satisfies P (a, b) = 1−P (a, b) and P (a, a) = 0.5 for all a, b ∈ [K]. We say that a dominates b and write a ≻ b if P (a, b) > 0.5, i.e. arm a has a higher chance of winning over arm b in a duel (a, b). A well-studied concept of a good benchmark arm in dueling bandits is the Condorcet Winner (CW): Given any preference matrix P ∈ [0, 1] K×K , an arm a * ∈ [K] is called a Condorcet winner of P if P (a * , b) > 0.5 for all b ∈ [K] \ {a * } [48,22,43,8,30]. Note that any preference matrix with a total ordering over arms invariably has a Condorcet winner. For example, assuming a total ordering 1 ≻ 2 ≻ . . . ≻ K implies that the Condorcet winner is arm 1. Any RUM-based preference matrix [32,34,38], or more generally any P that satisfies stochastic transitivity [45], always respects a total ordering. However, note that CW-based preference matrices consider a much bigger class of pairwise relations than total ordering. Despite this, in general a preference matrix might not have a Condorcet winner, which led to more general notions of benchmark arms in DB, such as the Borda winner [37], the Copeland winner [50] or the von Neumann winner [14,36]. Non-Stationary Dueling Bandits (NSt-DB) We consider a decision space of K arms denoted by [K]. At each round t ∈ [T ], the task of the learner is to select a pair of actions (a t , b t ) ∈ [K] × [K], upon which a preference feedback o t (a t , b t ) ∼ Ber(P t (a t , b t ) ) is revealed to the learner according to the underlying preference matrix P t ∈ [0, 1] K×K . The sequence of preferences P 1 , P 2 , . . . , P T is generated adversarially and for any such preference matrix P t we define δ t (a, b) := P t (a, b) − 1/2 as the gap or preference-strength of arm a over arm b in round t. We here assume that every preference matrix P t has a Condorcet winner, which we denote by a * t . Static Regret in Dueling Bandits. In classical (stochastic) dueling bandits, where it is assumed that P 1 = . . . = P T = P for some fixed preference matrix P, the performance of the learner is often measured w.r.t. the CW of P, described by the static regret R(T ) := T t=1 δ t (a * , a t ) + δ t (a * , b t ) 2 , where a * is the CW of P [45,40,8,29]. Note that here δ t (a * , a) = P t (a * , a) − 1/2 essentially quantifies the net loss of arm a against the fixed benchmark arm a * . However, regret with respect to any fixed benchmark (comparator arm) soon becomes meaningless when the underlying preference matrices are changing over time, since no single fixed arm may represent a reasonably good benchmark over T rounds. Consider the following simple motivating example: Example 2.1. Let K = 2 and define P 1 = 0.5 1 0 0.5 , P 2 = 0.5 0 1 0.5 . Now, assume a preference sequence such that P t = P 1 for the first ⌊T /2⌋ rounds and P t = P 2 for the last ⌈T /2⌉ rounds. We see that a policy that plays any of the two arms all T rounds, e.g. π t = 1 for all t ∈ [T ], has regret O(1) against any fixed benchmark arm, since δ t (1, 2) = 1/2 for the first T /2 rounds and δ t (1, 2) = −1/2 for last T /2 rounds. However, against a dynamic benchmark, e.g. arm 1 for t < T /2 and arm 2 for t ≥ T /2, any policy that plays a fixed arm all T rounds suffers O(T /2) regret (while suffering only constant regret against any fixed benchmark). Dynamic Regret in Dueling Bandits. Drawing motivation from the above, we seek to formulate a stronger and more meaningful notion of dueling bandit regret, where the benchmark in every round is chosen dynamically based on P t . More precisely, letting a * t be the CW of P t , we define dynamic regret as DR(T ) := T t=1 δ t (a * t , a t ) + δ t (a * t , b t ) 2 . Measures of Non-Stationarity Clearly, without any control over the amount of non-stationarity in the sequence {P t } t∈ [T ] , it is impossible for any learner to achieve sublinear o(T ) dynamic regret in the worst case. To see this, consider the matrices from Example 2.1 and note that for any choice of arms (a t , b t ), the adversary can choose a matrix so as to guarantee instantaneous regret of at least 1/2. This consequently leads to linear regret for the learner, implying that to achieve sublinear dynamic regret, we need to restrict the adversary in terms of the total amount of non-stationarity it can induce in the sequence P 1 , . . . , P T . But what could be a good measure of non-stationarity? In this paper, we study several of these measures, which we will now formally introduce and put in relation to one another. Preference Switches. A non-stationarity measure that has been studied in the previous work on NSt-DB is the number of times P t changes [21,19]: S P := T t=2 1{P t = P t−1 }. However, S P can be a quite pessimistic measure of non-stationarity, as changes in the preference between two suboptimal arms or minor preference shifts that do not change the CW are counted toward S P , whereas they should not significantly affect the performance of a good learning algorithm. Condorcet Winner Switches. A naturally stronger measure of non-stationarity is the total number of Condorcet Winner Switches, i.e. the number of times the identity of a * t changes: and a preference sequence such that P t = P 1 when t is odd and P t = P 2 otherwise. We then find that S CW = 0 (since 1 is the CW in all rounds t), whereas S P = T . S CW := T t=2 1{a * t = a * t−1 }. Significant Condorcet Winner Switches. Recently, [41] proposed a new (and strong) notion of non-stationarity for multi-armed bandits, called Significant Shifts, that aims to account only for severe distribution shifts and comprises previous complexity measures. We can define a similar concept for dueling bandits: Let ν 0 := 1 and define ν i+1 recursively as the first round in [ν i , T ) such that for all arms a ∈ [K] there exist rounds ν i ≤ s 1 < s 2 < ν i+1 such that s 2 t=s 1 δ t (a * t , a) ≥ K(s 2 − s 1 ). LetS CW denote the number of such Significant CW Switches ν 1 , . . . , νS CW . We immediately see that we haveS CW ≤ S CW , since not all CW Switches are also Significant CW Switches. For example, a 'non-severe' and quickly reverted change of the Condorcet winner may not be counted towardsS CW . Total Variation. Another common notion of non-stationarity studied in the multi-armed bandits literature is the total variation in the rewards [9,24]. Its analogue in dueling bandits can be defined as V := T t=2 max a,b∈[K] |P t (a, b) − P t−1 (a, b)| , which has been previously studied in [19]. However, V can also be a pessimistic measure of complexity, as it can be of order O(T ) even though the Condorcet winner remains fixed throughout all rounds. Condorcet Winner Variation. We can then formulate a more refined version of total variation by accounting only for the maximal drift in the winning probabilities of the current Condorcet winner:Ṽ := T t=2 max a∈[K] |P t (a * t , a) − P t−1 (a * t , a)| . Remark 2.2 (V vsṼ Proposed Algorithm: ANACONDA Following recent advances in non-stationary multi-armed bandits [7,13,1] and especially [41], we construct an episode-based algorithm with a carefully chosen replay schedule, called ANACONDA. Recall that our goal is to minimize dynamic regret w.r.t. a changing benchmark a * t . However, we quickly notice that we cannot reliably track the dynamic regret of some arm a ∈ [K], i.e. Algorithm 1 ANACONDA: Adaptive Non-stationAry CONdorcet Dueling Algorithm 1: input: horizon T 2: t ← 1 3: while t ≤ T do 4: t ℓ ← t start of the ℓ-th episode 5: A good ← [K] 6: for m ∈ {2, . . . , 2 ⌈log(T )⌉ } and s ∈ {t ℓ + 1, . . . , T } do set replay schedule 7: Sample B s,m ∼ Bern 1 √ m(s−t ℓ ) 8: Run CondaLet(t ℓ , T + 1 − t ℓ ) root replay in ℓ-th episode Algorithm 2 CondaLet(t 0 , m 0 ) 1: input: scheduled time t 0 and duration m 0 2: initialize: t ← t 0 , A t ← [K] 3: while t ≤ T and t ≤ t 0 + m 0 and A good = ∅ do restart if no good arms are left 4: Play arm-pair (a t , b t ) ∈ A t with each arm being selected with probability 1/|A t | 5: A good ← A good \{a ∈ [K] : ∃[s 1 , s 2 ] ⊆ [t ℓ , t) s.t. (2) holds} eliminate bad arms from A good 6: A local ← A t save active set of arms locally 7: t ← t + 1 8: if ∃m such that B t,m = 1 then check for scheduled child replays Run CondaLet(t, m) with m = max{m ∈ {2, . . . , 2 ⌈log(T )⌉ } : B t,m = 1} 10: A t ← A local \ {a ∈ [K] : ∃[s 1 , s 2 ] ⊆ [t 0 , t) s.t. (2) holds} eliminate bad arms from A t max a ′ ∈[K] s 2 t=s 1 δ t (a ′ , a) instead. It will be the main challenge of our analysis to ensure that properly timed replays will occur (and not too many of these) so that it is in fact sufficient to track the static regret to guarantee low dynamic regret. In the following, we explain our algorithmic approach in more detail. The algorithm is organized in episodes, denoted ℓ. Similar to recent approaches to non-stationary multi-armed bandits [7,1,41], the algorithm maintains a set of good arms, A good , and a replay schedule, {B s,m } s,m , within each episode. When no good arms are left in A good , a new episode begins and the set of good arms and the replay schedule are being reset. Here, ANACONDA (Algorithm 1) is the meta procedure that initializes each episode by resetting the set of good arms to [K], sampling a new replay schedule, and triggering the root call of CondaLet(t ℓ , T + 1 − t ℓ ). When active in round t, a run of CondaLet(t 0 , m 0 ) (Algorithm 2) samples two arms uniformly at random from the active set of arms at round t, denoted A t . The set A t is globally maintained by all calls of CondaLet and reset to [K] at the beginning of each replay, i.e. call of CondaLet. When a child replay CondaLet(t, m) is scheduled in round t, i.e. B t,m = 1 for some m, the parent algorithm, say CondaLet(t 0 , m 0 ), is interrupted (before eventually resuming if t ≤ t 0 + m 0 and A good = ∅). To not overwrite arm eliminations of a parent by resetting A t to [K] in interrupting calls of CondaLet, each version of CondaLet saves a local set of arms, A local , before checking for children. Gap Estimates. Recall the definition of the gap between two arms as δ t (a, b) = P t (a, b) − 1/2. Based on observed outcomes of duels, ANACONDA maintains the following importance weighted estimates of δ t (a, b):δ t (a, b) = |A t | 2 1 {at=a,bt=b} o t (a, b) − 1/2.(1) Wee see that whenever a, b ∈ A t , i.e. both arms are in the active set in round t, the estimator δ t (a, b) is an unbiased estimate of δ t (a, b), as we select a pair of arms uniformly at random from A t every round (see Line 4 in Algorithm 2). Elimination Rule. In Line 5 and Line 10 of Algorithm 2, we eliminate an arm a ∈ [K] in round t if there exist rounds 0 ≤ s 1 < s 2 ≤ t such that max a ′ ∈[K] s 2 t=s 1δ t (a ′ , a) > C log(T )K (s 2 − s 1 ) ∨ K 2 ,(2) where C > 0 is some universal constant that does not depend on T , K, or S CW , and can be derived from the regret analysis. Main Result The main result of this paper is aÕ( √ S CW T ) dynamic regret bound of ANACONDA without knowledge of the number of CW Switches S CW . When S CW ≪ S P , this bound substantially improves upon the non-adaptiveÕ( √ S P T ) rates in [19] and [21]. In particular, as previously mentioned, the number of preference switches S P can be a very pessimistic measure of complexity. For example, a change in the preference between two suboptimal arms, or a minor change of the winning probabilities of the Condorcet winner under which it remains optimal, should not substantially affect our performance (see Remark 2.1). DR(T ) ≤ c log 3 (T )K S CW i=0 √ τ i+1 − τ i . An application of Jensen's inequality shows that this implies a dynamic regret bound of order O(K √ S CW T ), stated in the following corollary. Corollary 3.1 (Dynamic Regret w.r.t. S CW ). For some constant c > 0, the dynamic regret of ANACONDA is bounded as DR(T ) ≤ c log 3 (T )K (S CW + 1)T . Remark 3.1 (Regret Lower Bound and Tightness of Theorem 3.1). Note that a lower bound of Ω( √ KS P T ) has recently been shown by [19], which can also be seen to give a lower bound Ω( √ KS CW T ) in terms of CW Switches S CW as S CW ≤ S P (in particular, the lower bound problem instance used in [19] is precisely such that S CW = S P ). As a result, we find that the above bound is optimal up to logarithmic factors in its dependence on S CW and T , whereas its dependence on K may not be tight. Regret Analysis of ANACONDA We build on recent advances in non-stationary multi-armed bandits, which are able to achieve near-optimal dynamic guarantees [7,1,41] without knowledge of the non-stationary complexity. A common basis of the regret analysis in these works is a decomposition of the dynamic regret using the notion of good arms. Challenges in the Dueling Setting. More precisely, within each episode ℓ, prior work in multiarmed bandits [7,1,41] decomposes the regret of their algorithm's selection, say, a t into its relative regret against the last good arm a g ℓ ∈ A good , and the relative regret of a g ℓ against the best arm, say, a * t . A key advantage of this decomposition is that estimating the relative regret of some arm a w.r.t. a g ℓ instead of a * t is much easier. In particular, since a g ℓ is by definition considered good throughout the episode, it is always actively played, which guarantees unbiased estimates of the difference in rewards between any played arm a and the last good arm a g ℓ . However, pairwise preferences are generally not transitive, let alone linear, so that a triangle inequality does not hold , i.e. δ t (a * t , a) ≤ δ t (a * t , a g ℓ ) + δ t (a g ℓ , a). In NSt-DB, we can thus generally not utilize a g ℓ , or any other temporarily fixed arm, as a benchmark to detect large regret. Instead, in contrast to prior work in multi-armed bandits, we face the difficulty of having to argue directly that we can guarantee low dynamic regret t δ t (a * t , a) without a proxy benchmark such as a g ℓ . Key Ideas to Overcome these Challenges. To overcome these challenges, we consider every fixed arm a ∈ [K] in isolation and split each episode ℓ into the rounds before arm a gets eliminated from A good and the rounds after it gets eliminated from A good . Letting t a ℓ be the elimination round of arm a, we will then argue that t a ℓ will occur sufficiently early to guarantee low regret (in episode ℓ) before round t a ℓ . For the rounds after elimination from A good , it will be key to dissect each possible replay of the eliminated arm and obtain replay-specific regret bounds, where we distinguish between 'confined' and 'unconfined' replays of arms. We now give an outline of our regret analysis. Proof Sketch of Theorem 3.1 In the following, we letc > 0 denote a positive constant that does not dependent on T , K, or S CW , but may change from line to line. To begin our analysis, we state a concentration bound on the martingale difference sequenceδ t (a, b) − E[δ t (a, b) | F t−1 ] as it can be found in similar form in [11] and [41]. s 2 t=s 1δ t (a, b) − s 2 t=s 1 E δ t (a, b) | F t−1 ≤c log(T ) K (s 2 − s 1 ) + K 2 (3) for a sufficiently large constantc > 0 and where F = {F t } t∈N 0 denotes the canonical filtration. Then, event E occurs with probability at least 1 − 1/T 2 . Note that our elimination rule (2) has been chosen in accordance with the above concentration bound. In particular, let t a ℓ denote the round in episode ℓ in which arm a is eliminated from A good . Then, on the concentration event E, if a ′ ∈ A good for all t ℓ ≤ t < t a ℓ , we must have t a ℓ −1 t=t ℓ δ t (a ′ , a) = t a ℓ −1 t=t ℓ E δ t (a ′ , a) | F t−1 ≤c log(T )K (t a ℓ − t ℓ ) ∨ K 2 , where the initial identity holds asδ t (a ′ , a) is unbiased when a, a ′ ∈ A t and the inequality follows from the elimination rule (2) and the concentration bound (3). However, note that the above crucially used that both a and a ′ are actively played throughout the interval [t ℓ , t a ℓ ), as we are otherwise not able to accurately estimate t δ t (a ′ , a). It will be the primary challenge of our analysis to ensure that through properly timed replays, i.e. calls of CondaLet, we can obtain unbiased estimates w.r.t. the changing CW that allow us to eliminate bad arms before they amass large regret. Bounding Regret Within Episodes. We proceed by bounding regret within each episode separately. Recall that we let τ 1 < . . . < τ S CW denote the (unknown) rounds in which the Condorcet winner changes. We then refer to the interval [τ i , τ i+1 ) as the i-th phase, i.e. the interval for which a * t = a * τ i for all t ∈ [τ i , τ i+1 ). Let Phases(t 1 , t 2 ) = {i : [τ i , τ i+1 ) ∩ [t 1 , t 2 ) = ∅} be the set of phases i such that [τ i , τ i+1 ) intersects with the interval [t 1 , t 2 ) . Our main claim is the following upper bound on the dynamic regret within each episode: E   t ℓ+1 −1 t=t ℓ δ t (a * t , a t ) + δ t (a * t , b t ) 2   ≤cK log 3 (T ) E   i∈Phases(t ℓ ,t ℓ+1 ) √ τ i+1 − τ i   .(4) By conditioning on t ℓ and carefully applying the tower property, we can rewrite the expected dynamic regret within an episode in terms of fixed arms a ∈ [K]: Lemma 4.2. We have E   t ℓ+1 −1 t=t ℓ δ t (a * t , a t ) + δ t (a * t , b t ) 2   = E   K a=1 t ℓ+1 −1 t=t ℓ δ t (a * t , a) |A t | 1 {a∈At}   . In a next step, we split the RHS into the rounds before a fixed arm a ∈ [K] has been eliminated from the good set, and the rounds after its elimination. Recall t a ℓ to be the round in episode ℓ in which arm a is eliminated from A good and consider E K a=1 t a ℓ −1 t=t ℓ δ t (a * t , a) |A t | R 1 (ℓ) + E K a=1 t ℓ+1 −1 t=t a ℓ δ t (a * t , a) |A t | 1 {a∈At} R 2 (ℓ) , where we could drop the indicator in R 1 (ℓ), since A good ⊆ A t by construction of these sets. The remainder of our analysis is mostly concerned with showing that both, R 1 (ℓ) and R 2 (ℓ), are upper bounded by the RHS in (4). Regret Before Elimination. The main difficulty in bounding R 1 (ℓ) lies in the fact that some arm could have been eliminated due to being suboptimal, only to become the Condorcet winner shortly after. As a result, large regret could go undetected, as the current Condorcet winner is not being actively played anymore. To this end, we have to argue that with high probability there will always be a replay scheduled that eliminates any bad arm from A good in a timely manner, thereby eventually triggering a restart. Here, we specifically consider calls of CondaLet(s, m) that provably eliminate bad arms from A good . Importantly, by construction of our elimination rule (2), we can guarantee on the concentration event E that any run of CondaLet that is scheduled within some phase i will actively play the Condorcet winner of said phase. Lemma 4.3. On event E, no call of CondaLet(s, m) with τ i ≤ s < τ i+1 eliminates arm a * i before round τ i+1 . Roughly speaking, we can then argue that a replay that eliminates arm a will be scheduled with high probability before the smallest round s(a) > t ℓ such that s(a) t=t ℓ δ t (a * t , a) s(a) − t ℓ . In other words, arm a is going to be eliminated from A good before it suffers too much regret. Since t a ℓ is defined as the round in episode ℓ in which a is eliminated from A good , we must have t a ℓ < s(a), which implies that the inner sum in R 1 (ℓ) is at most of order t a ℓ − t ℓ for every fixed arm a ∈ [K]. Finally, using that t a ℓ − t ℓ ≤ i∈Phases(t ℓ ,t a ℓ ) √ τ i+1 − τ i and summing over all arms, we obtain the desired bound of (4). Note that here summing over arms can be seen to account for a log(K) factor which we coarsely upper bound by log(T ). Regret After Elimination. R 2 (ℓ) can be viewed as the regret due to replaying arms after they have been eliminated from the good set A good . We here distinguish between two types of replays, i.e. calls of CondaLet: To bound the regret within a confined replay, we recall that according to Lemma 4.3, on the concentration event E, no call of CondaLet will eliminate the Condorcet winner within the phase it is scheduled in. Thus, whenever some arm a is being played by a confined replay, we obtain unbiased estimates of δ t (a * t , a). It is then straightforward to show that for any confined CondaLet(s, m), we have that s+m t=s δ t (a * t , a) is at most of order √ m. A similar line of argument does not work for unconfined replays, as they intersect with several phases. We then face a similar difficulty as when bounding R 1 (ℓ), where the Condorcet winner of the current phase could have been eliminated (from the replay) in an earlier phase. Using similar arguments than for bounding R 1 (ℓ), we show that for any unconfined CondaLet(s, m), we have that s+m t=s δ t (a * t , a) is at most of order √ s − t ℓ + √ m. Lastly, recall that in episode ℓ a replay CondaLet(s, m) is scheduled with probability 1/ m(s − t ℓ ). Crucially, any unconfined CondaLet scheduled in [τ i , τ i+1 ) must have duration at least m ≥ τ i+1 − s (otherwise it is not unconfined). Careful summation over confined and unconfined CondaLet then yields the desired upper bound (4). Counting Episodes. Lastly, we show that ANACONDA only restarts if there has been a CW switch. Lemma 4.4. On event E, for all episodes ℓ but the last there exists a change of the CW t ℓ ≤ τ i < t ℓ+1 . This follows directly from the fact that on the concentration event within a single phase the CW will never be eliminated from A good . Thus, if there is a restart, i.e. every arm has been eliminated from A good , there must have been a change of CW. Lemma 4.4 thus tells us that any phase intersects with at most two episodes. Summing the RHS of (4) over episodes then gives the claimed upper bound of E T t=1 δ t (a * t , a t ) + δ t (a * t , b t ) 2 ≤ 2cK log 3 (T )E   S CW i=1 √ τ i+1 − τ i   . A detailed proof of Theorem 3.1 is given in Appendix A. Tighter Bounds Under SST and STI We show that ANACONDA can in fact yield a stronger regret guarantee in terms of a more refined notion of non-stationarity, Significant Condorcet Winner Switches (see Section 2.2), under additional assumptions on the preference sequence P 1 , . . . , P T : Strong Stochastic Transitivity (SST) and Stochastic Triangle Inequality (STI) [45,47,46]. Let a, b, c ∈ [K] and let a ≻ t b denote that a is preferred over b in round t. Assumption 1 (Strong Stochastic Transitivity). Every preference matrix P t satisfies that if a ≻ t b ≻ t c, we have δ t (a, c) ≥ δ t (a, b) ∨ δ t (b, c). Assumption 2 (Stochastic Triangle Inequality). Every preference matrix P t satisfies that if a ≻ t b ≻ t c, we have δ t (a, c) ≤ δ t (a, b) + δ t (b, c). Remark 5.1 (Example of SST & STI) . Among the preference models that satisfy Assumption 1 and Assumption 2, are utility-based models with a symmetric and monotonically increasing link function σ. In these models, every arm a has an associated (time-dependent) utility u t (a) and the probability of arm a winning a duel against arm b is given by P t (a ≻ b) = σ(u t (a) − u t (b)), where σ is an increasing function satisfying σ(x) = 1 − σ(−x) and σ(0) = 1/2 that maps utility differences to probabilities [47,8]. Improved Dynamic Regret Analysis We now show that ANACONDA achieves strong regret guarantees in terms of Significant CW Switches and CW Variation under SST and STI. Significant Condorcet Winner Switches. Under Assumption 1 and Assumption 2, we are able to obtain the following adaptive dynamic regret bound in terms ofS CW . Proof Overview. With some additional effort, Assumption 1 and Assumption 2 allow us to utilize a dynamic regret decomposition similar to prior work in non-stationary multi-armed bandits [7,41,1]. Roughly speaking, this allows us to reuse the regret analysis for CW Switches (Theorem 3.1) in the analysis under Significant CW Switches. We want to give a brief intuition about why additional assumptions are necessary when bounding dynamic regret w.r.t. Significant CW SwitchesS CW opposed to CW Switches S CW . 2 Consider a phase [ν i , ν i+1 ) in the sense of Significant CW Switches as defined in Section 2.2. As previously mentioned, the definition of a Significant CW Switch allows for several (non-severe) CW changes within each phase [ν i , ν i+1 ). As a result, we cannot guarantee that there will be any intervals during which the CW remains fixed, which would enable us to accurately estimate the relative regret t δ t (a * t , a) so as to eliminate bad arms. Broadly speaking, assuming a sort of transitivity (i.e. SST and STI) enables us to identify bad arms based on knowledge of t δ t (a ′ , a) for some temporarily fixed benchmark a ′ . More details and a complete proof can be found in Appendix B. Condorcet Winner Variation. Recall the definition of the Condorcet Winner VariationṼ from Section 2.2. As a consequence of Theorem 5.1, we can show that ANACONDA also achieves near-optimal dynamic regret w.r.t.Ṽ . Remark 5.3. By definition, we haveṼ ≤ V , which means that Corollary 5.1 may yield a tighter dynamic regret bound than the (non-adaptive)Õ (KV ) 1 /3 T 2 /3 guarantee in [19]. In view of the lower bound of Ω (KV ) 1 /3 T 2 /3 shown in [19], the regret guarantee of ANACONDA is also tight up to logarithmic factors and a factor of K 1 /3 . Note again that the lower bound in [19] is not violated as their lower bound uses a worst-case preference sequence P 1 , . . . , P T whereṼ = V . Discussion We studied the problem of dynamic regret minimization in non-stationary dueling bandits and proposed an adaptive algorithm that yields provably optimal regret guarantees in terms of strong notions of non-stationary complexity. Our proposed algorithm is the first to achieve optimal dynamic dueling bandit regret without prior knowledge of the underlying non-stationary complexity. While our results certainly close some of the practical open problems in preference elicitation in time-varying preference models, it also leads to plethora of new questions along the line. We provide an outlook to future directions and open problems in the supplementary material. Future Work. While our results certainly address some of the practical open problems for preference elicitation in time-varying preference models, it also leads to plethora of new questions along the line. In particular, as an extension to this work, one obvious question would be to understand non-stationary dueling bandits for more general preference matrices: What happens if the preference sequences do not have a Condorcet winner in each round? What could be a good dynamic benchmark in that case? Hereto related, another open question is whether it is possible to obtain dynamic regret bounds in terms of Significant CW Switches (S CW ) for general preference sequences (without transitivity assumptions). Extending the considered pairwise preference setting to more general subsetwise feedback [31,33,18,35] would be another interesting direction from a practical point of view. Supplementary: ANACONDA: An Improved Dynamic Regret Algorithm for Adaptive Non-Stationary Dueling Bandits A Proof of Theorem 3.1 We organize the proof of Theorem 3.1 as follows. Section A.1 contains basic preliminary facts that will be the foundation of the upcoming proof. Section A.2 then bounds the regret any fixed arm suffers within each episode before being eliminated from the good set. Complementary to this, Section A.3 then deals with the regret an arm suffers after being eliminated. A.1 Preliminaries In this preliminary section, we introduce a concentration bound on the sum of our estimatesδ t in Section A.1.1. We then show in Section A.1.2 that the beginning of a new episode implies that the Condorcet winner has changed (on the concentration event), which will be useful later. Finally, Section A.1.3 decomposes the regret in terms episodes, arms, and rounds, which will form the basis of our analysis. A.1.1 Martingale Concentration Bound We will rely on a similar martingale tail bound as [11] and [41], which is based on a version of Freedman's inequality given below. Lemma A.1 (Theorem 1 in [11]). Let (X t ) t∈N be a martingale difference sequence w.r.t. some filtration (F t ) t∈N 0 . Assume that is X t is almost surely uniformly bounded, i.e. X t ≤ R a.s. for some constant R. Moreover, suppose that t s=1 E[X 2 s | F s−1 ] ≤ V t a. s. for some sequence of constants (V t ) t∈N . Then, for any δ ∈ (0, 1), with probability at least 1 − δ, we have t s=1 X s ≤ (e − 1) V t log(1/δ) + R log(1/δ) .(5) Proof. See Theorem 1 in [11] and Lemma 1 in [41]. We now apply the above concentration bound to the martingale difference sequenceδ t (a, b) − E[δ t (a, b) | F t−1 ]. Lemma A.2. Let E be the event that for all rounds s 1 < s 2 and all arms a, b ∈ [K]: s 2 t=s 1δ t (a, b) − s 2 t=s 1 E δ t (a, b) | F t−1 ≤ c 1 log(T ) K (s 2 − s 1 ) + K 2 (6) for an appropriately large constant c 1 > 0 and where F = {F t } t∈N 0 is the canonical filtration generated by observations in past rounds. Then, event E occurs with probability at least 1 − 1/T 2 . Proof. Note thatδ t (a, b) − E[δ t (a, b) | F t−1 ] is naturally a martingale difference, since E δ t (a, b) − E[δ t (a, b) | F t−1 ] | F t−1 = 0 a.s. Using that |A t | ≤ K, we have that X t ≤ 2K 2 a. s. for all rounds t. Moreover, we get that s 2 t=s 1 E δ 2 t (a, b) | F t−1 ≤ s 2 t=s 1 |A t | 4 E 1 {at=a,bt=b} | F t−1 = s 2 t=s 1 |A t | 2 ≤ K 2 (s 2 − s 1 ). We can thus apply Lemma A.1 with R = K 2 and V t = 2K 2 t. Using |x − y| ≤ |x| + |y| and taking union bounds over a, b and s 1 , s 2 , we then obtain Lemma A.2. A.1.2 Episodes and Condorcet Winner Switches Lemma A.3. On event E, for each episode [t ℓ , t ℓ+1 ) with t ℓ+1 ≤ T , there exists a change of the CW τ i ∈ [t ℓ , t ℓ+1 ). This implies that any phase [τ i , τ i+1 ) will intersect with at most two episodes. Proof. The start of a new episode means that every arm a ∈ [K] has been eliminated from A good at some round in t a ℓ ∈ [t ℓ , t ℓ+1 ). As a result, there must exist an interval [s 1 , s 2 ] ⊆ [t ℓ , t a ℓ ) and some arm a ′ ∈ [K] so that the elimination rule (2) holds. Using Lemma A.2, we then find that for some constant c 2 > 0: s 2 t=s 1 E δ t (a ′ , a) | F t−1 > c 2 log(T )K (s 2 − s 1 ) ∨ K 2 .(7) Note that by construction ofδ t (a ′ , a), we always have δ t (a ′ , a) ≥ E[δ t (a ′ , a) | F t−1 ] since E[δ t (a ′ , a) | F t−1 ] = δ t (a ′ , a) a ′ , a ∈ A t −1/2 otherwise.(8) Thus, in view of inequality (7), there exists no arm a ∈ [K] such that max a ′ δ t (a ′ , a) = 0 for all t ∈ [t ℓ , t ℓ+1 ), i.e. no fixed arm is optimal throughout the episode and there must have been a change of Condorcet winner. A.1.3 Decomposing Regret across Episodes and Arms We will bound regret of the algorithm withing each episode separately, i.e. we consider E   t ℓ+1 −1 t=t ℓ δ t (a * t , a t ) + δ t (a * t , b t ) 2   ,(9) where t ℓ is the first round in episode ℓ and a * t is the Condorcet winner in round t ∈ [T ]. Recall that, every round t ∈ [T ], the algorithm selects an arm a uniformly at random from the active set A t . It will then be useful to rewrite (11) in terms of fixed arms a ∈ [K]. Lemma A.4. We can write (11) in terms of the regret suffered by fixed arms: E   t ℓ+1 −1 t=t ℓ δ t (a * t , a t ) + δ t (a * t , b t ) 2   = E   K a=1 t ℓ+1 t=t ℓ δ t (a * t , a) |A t | 1 {a∈At}   (10) Proof. As the algorithm independently and symmetrically selects two arms (a t , b t ) in each round (Line 4 in Algorithm 2), we can focus on bounding regret for one of the two arms, say a t , by writing E   t ℓ+1 −1 t=t ℓ δ t (a * t , a t ) + δ t (a * t , b t ) 2   = E   t ℓ+1 −1 t=t ℓ δ t (a * t , a t )   .(11) Conditioning on t ℓ and using the tower property, we then further find that E   t ℓ+1 t=t ℓ δ t (a * t , a t )   = E   E   t ℓ+1 t=t ℓ δ t (a * t , a t ) | t ℓ     = E   T t=t ℓ E 1 {t<t ℓ+1 } E [δ t (a * t , a t ) | F t−1 ] | t ℓ   = E   T t=t ℓ a∈At E 1 {t<t ℓ+1 } | t ℓ δ t (a * t , a) |A t |   = E   t ℓ+1 t=t ℓ a∈At δ t (a * t , a) |A t |   , where we used that 1 {t<t ℓ+1 } is F t−1 -measurable and E [δ t (a * t , a t ) | F t−1 ] = a∈At δ t (a * t , a) |A t | . Lastly, Lemma A.4 then follows from rewriting the sum over a ∈ A t using the indicator 1 {a∈At} and swapping the order of the sums. In an important next step, we split the dynamic regret for each fixed arm a ∈ [K] into: (i) the regret we suffer from playing arm a in the ℓ-th episode before its elimination from A good , (ii) the regret we suffer from (re)playing arm a in the ℓ-th episode after its elimination from A good . Recall that t a ℓ ∈ [t ℓ , t ℓ+1 ) denotes the time that arm a is eliminated from A good in episode ℓ. Using Lemma A.4, we then decompose the dynamic regret in episode ℓ as E   t ℓ+1 −1 t=t ℓ δ t (a * t , a t ) + δ t (a * t , b t ) 2   = E   K a=1 t a ℓ −1 t=t ℓ δ t (a * t , a) |A t |   R 1 (ℓ) + E   K a=1 t ℓ+1 −1 t=t a ℓ δ t (a * t , a) |A t | 1 {a∈At}   R 2 (ℓ) ,(12) where for R 1 (ℓ) we used that a ∈ A good implies a ∈ A t by construction of these sets. For every fixed arm, R 1 (ℓ) corresponds to the regret suffered before said arm is eliminated from the master set. Accordingly, R 2 (ℓ) is the regret due to replaying an arm after its elimination from the master set. The remainder of the proof is mainly concerned with bounding R 1 (ℓ) and R 2 (ℓ) appropriately. A.2 Bounding R 1 (ℓ): Regret Before Elimination We begin by assuming w.l.o.g. that t 1 ℓ ≤ · · · ≤ t K ℓ so that for each round t < t a ℓ all arms a ′ ≥ a are element in A good ⊆ A t . As a result, we have |A t | ≥ K + 1 − a for all t ≤ t a ℓ , and thus E   K a=1 t a ℓ −1 t=t ℓ δ t (a * t , a) |A t |   ≤ E   K a=1 t a ℓ −1 t=t ℓ δ t (a * t , a) K + 1 − a   .(13) As we can see, the denominator will eventually account for a factor of log(K) ≈ K a=1 1/a. We now concentrate on bounding the inner sum in (13), i.e. the regret of any fixed arm before being eliminated in the ℓ-th episode. A.2.1 Bounding E[ t a ℓ −1 t=t ℓ δ t (a * t , a)] for any fixed arm a ∈ [K] This section is devoted to proving the following upper bound. Lemma A.5. For some constant c > 0: E   t a ℓ −1 t=t ℓ δ t (a * t , a)   ≤ c log 2 (T )K E   i∈Phases(t ℓ ,t a ℓ ) √ τ i+1 − τ i   + K T 2 + 1 T .(14) To prove Lemma A.5, we will divide the interval [t ℓ , t a ℓ ) into segments over the course of which arm a suffers large regret and show that not too many of such segments will occur in interval [t ℓ , t a ℓ ), i.e. until arm a is being eliminated from A good . The definition of such bad segments is analogous to their construction in [1] and [41]. Whereas prior work utilizes such segments to bound the regret of the last arm considered good in an episode, i.e. the last arm in A good , we will instead derive a regret bound for any fixed arm a. While the according regret bound will be in some sense weaker, it will still be sufficiently tight for our purposes. We here follow the notation in [41]. Definition A.1 (Bad Segments). Fix t ℓ and let [τ i , τ i+1 ) be any phase intersecting [t ℓ , T ). For an arm a, define rounds s i,j (a) ∈ [t ℓ ∨ τ i , τ i+1 ) recursively as follows: let s i,0 (a) = t ℓ ∨ τ i and define s i,j+1 (a) as the smallest round in (s i,j (a), τ i+1 ) such that arm a satisfies for some constant c 3 > 0: s i,j+1 (a) t=s i,j (a) δ t (a * i , a) > c 3 log(T )K s i,j+1 (a) − s i,j (a),(15) if such round s i,j+1 (a) exists. Otherwise, we let s i,j+1 (a) = τ i+1 − 1. We refer to the intervals [s i,j , s i,j+1 ) as bad segments if (15) is satisfied. If a segment does not satisfy (15), we refer to them as non-bad segments. 3 Note that the concept of bad segments will become useful later as, for a fixed t ℓ , by definition of the bad segments, we can always upper bound the dynamic regret on an interval [s i,j (a), s i,j+1 (a)) by s i,j+1 (a)−1 t=s i,j (a) δ t (a * t , a) ≤ c 3 log(T )K s i,j+1 (a) − s i,j (a).(16) We now define the bad round for an arm a as the smallest round when the aggregated regret of bad segments exceeds √ interval length regret. Definition A.2 (Bad Round). Fix t ℓ and some arm a. The bad round s(a) > t ℓ is defined as the smallest round which satisfies for some universally fixed constant c 4 > 0: (i,j) : s i,j+1 (a)<s(a) s i,j+1 (a) − s i,j (a) > c 4 log(T ) s(a) − t ℓ ,(17) where the sum is over all bad segments with s i,j+1 (a) < s(a). For a given episode ℓ, we will show that arm a is eliminated with high probability by the time the bad round s(a) occurs. To this end, we will introduce perfect replays, i.e. those runs of CondaLet which are properly timed and eliminate arm a before it aggregates large regret. A.2.2 Perfect Replays The following result will become very useful and makes the intuition precise that on the concentration event the Condorcet winner will not be eliminated. More precisely, any run of CondaLet(s, m) scheduled in phase i will never eliminate a * i inside phase i as long as our concentration bound holds. Lemma A.6. On event E, no run of CondaLet(s, m) with s ∈ [τ i , τ i+1 ) ever eliminates arm a * i before round τ i+1 . Proof. Suppose the contrary that some CondaLet(s, m) with s ∈ [τ i , τ i+1 ) eliminates arm a * i before round τ i+1 . Then, we must have for some arm a ∈ [K] and interval [s 1 , s 2 ] ⊆ [s, τ i+1 ) that C log(T )K (s 2 − s 1 ) ∨ K 2 < s 2 t=s 1δ t (a, a * i ),(18) which using the concentration bound (6) implies on event E that c 2 log(T )K (s 2 − s 1 ) ∨ K 2 < s 2 t=s 1 E δ t (a, a * i ) | F t−1 ≤ s 2 t=s 1 δ t (a, a * i ),(19) where the last inequality holds by merit of (8). Now, by the definition of arm a * i as the Condorcet winner in phase i, we must have δ t (a, a * i ) ≤ 0 for all t ∈ [τ i , τ i+1 ) and all a ∈ [K]. Lemma A.6 then follows from contradiction. Using that , a) is an unbiased estimate of , a) and applying the concentration bound (6), this shows that arm a satisfies the elimination rule (2) over interval [s i,j (a), s i,j+1 (a)] and will thus be eliminated by CondaLet(s, m). s i,j+1 (a) t=s i,j (a)δ t (a * is i,j+1 (a) t=s i,j (a) δ t (a * i A.2.3 Perfect replays are scheduled w.h.p. Following [41], we will now show that a perfect replay that eliminates arm a is scheduled before 2 . We will obtain the high probability guarantee via concentration on the sum X(t ℓ , s(a)) = (i,j) : s i,j+1 (a)<s(a)s i,j (a) s=s i,j (a) B s,m i,j .(20) Lemma A.7. Let E ′ (t ℓ ) denote the event that X(t ℓ , s(a)) ≥ 1 for all arms a, i.e. a perfect replay is scheduled before round s(a). We have P(E ′ (t ℓ ) | t ℓ ) ≥ 1 − K/T 3 . Proof. Recalling that B s,m | t ℓ ∼ Bernoulli 1 √ m(s−t ℓ ) , we find that For c 4 sufficiently large the standard Chernoff bound tells us that E[X(t ℓ , s(a)) | t ℓ ] ≥ 1 √ 2 (i,j) : s i,j+1 (a)<s(a)s i,j (a) − s i,j (a) s i,j+1 (a) − s i,j (a) s(a) − t ℓ ≥ 1 4 (i,j) : s i,P X(t ℓ , s(a)) ≤ E[X(t ℓ , s(a)) | t ℓ ] 2 | t ℓ ≤ exp − E[X(t ℓ , s(a)) | t ℓ ] 8 ≤ 1 T 3 . The desired bound then follows from taking a union bound over all arms in [K]. Now, on event E ∩ E ′ (t ℓ ) , it must hold that t a ℓ < s(a) for all arms a ∈ [K], since otherwise a would have been eliminated by some perfect replay before round t a ℓ (by definition of event E ′ (t ℓ )). As the bad round s(a) is defined as the smallest round satisfying (17), we then have (i,j) : s i,j+1 (a)<t a ℓ s i,j+1 (a) − s i,j (a) ≤ c 4 log(T )K t a ℓ − t ℓ .(21) Hence, in view of equation (16), over the bad segments, the regret of arm a is at most of order log 2 (T ) t a ℓ − t ℓ . Moreover, for every last segment in some phase i, [s i,j , s i,j+1 (a)), as well as the final segment [s i,j (a), t a ℓ ), we know that the regret suffered from playing a is upper bounded by c 3 log(T ) √ τ i+1 − τ i by definition of non-bad segments (Definition A.1). Therefore, on event E ∩ E ′ (t ℓ ), it follows from equation (21) and the above that t a ℓ −1 t=t ℓ δ t (a * t , a) ≤ c 5 K log 2 (T ) i∈Phases(t ℓ ,t a ℓ ) √ τ i+1 − τ i ,(22) where we used that t a ℓ − t ℓ ≤ i∈Phases(t ℓ ,t a ℓ ) √ τ i+1 − τ i . Finally, we obtain Lemma A.5 by taking expectation and using that E ∩ E ′ (t ℓ ) holds with high probability, E   t a ℓ −1 t=t ℓ δ t (a * t , a)   ≤ E   1 {E∩E ′ (t ℓ )} t a ℓ −1 t=t ℓ δ t (a * t , a) | t ℓ   + T P(E c ) + P(E ′ (t ℓ ) c | t ℓ ) ≤ c 5 K log 2 (T ) E   i∈Phases(t ℓ ,t a ℓ ) √ τ i+1 − τ i   + 1 T + K T 2 . A.2.4 Summing Over Arms Note that t a ℓ ≤ t ℓ+1 for all a ∈ [K] by definition of t a ℓ . Then, summing over all arms, it follows from Lemma A.5 and (13) that for some constant c 6 > 0: E   K a=1 t a ℓ −1 t=t ℓ δ t (a * t , a) |A t |   ≤ c 6 K log 3 (T ) E   i∈Phases(t ℓ ,t ℓ+1 ) √ τ i+1 − τ i   ,(23) where we loosely upper bound log(K) by log(T ). A.3 Bounding R 2 (ℓ): Regret After Elimination Before we can begin, we will have to lay some groundwork to simplify the analysis in later steps. Recall the definition of bad segments from Section A.2 and define for every phase [τ i , τ i+1 ) intersecting with [t a ℓ , t ℓ+1 ), i.e. i ∈ Phases(t a ℓ , t ℓ+1 ), the segments [s i,j (a), s i,j+1 ) as in Definition A.1. We will split the regret due to bad segments, i.e. those that satisfy (15), from the regret due to non-bad segments, i.e. the last segments in a phase that do no satisfy (15). For a fixed arm a ∈ [K], we let bad(a) denote the rounds t ∈ [t ℓ , t ℓ+1 ) such that t ∈ [s i,j (a), s i,j+1 (a)) for any bad segment [s i,j (a), s i,j+1 (a)). By the definition of a non-bad segment (w.r.t. arm a), we know that that there is at most one such segment in every phase and that the regret of arm a in each segment is upper bounded by c 3 log(T ) √ τ i+1 − τ i , where [τ i , τ i+1 ) is the phase that contains the segment. To take care of the denominator |A t |, assume w.l.o.g. that there is a run of CondaLet(t a ℓ , m) that remains active and uninterrupted until the final round T . 4 We can then reorder arms a ∈ [K] according to the round that they are being eliminated by CondaLet(t a ℓ , m), which gives |A t | ≥ K + 1 − a whenever a ∈ A t . As before, this yields a factor of log(K) when summing over all arms. We then bound R 2 (ℓ) over non-bad segments as (s ′ , m ′ , a). When a replay CondaLet(s ′ , m ′ ) intersects with more than one phase, the CW in the next phase [τ i , τ i+1 ), denoted a * i+1 , could be evicted before the beginning of that phase, i.e. in the interval [s ′ , τ i ). E   K a=1 t ℓ+1 −1 t=t a ℓ δ t (a * t , a) |A t | 1 {a∈At,t ∈bad(a)}   ≤ c 3 K log(K) log(T )E   i∈Phases(t a ℓ ,t ℓ+1 ) √ τ i+1 − τ i   . (24) t ℓ t a ℓ t ℓ+1 s s + m confined replay M s ′ s ′ + m ′ unconfined replay M ′ τ i a * i a * i+1 The more challenging task is now to bound R 2 (ℓ) for rounds in bad segments. Recall that, for a fixed arm a ∈ [K], the sum in question relates to the expected regret suffered within an episode from replaying arm a after it has been eliminated from A good , i.e. after time t a ℓ . We begin by a straightforward upper bound. To this end, for a given replay CondaLet(s, m), let M (s, m, a) be the last round in [s, s + m], where arm a is active in CondaLet(s, m) and all of its children. Then, E   K a=1 t ℓ+1 −1 t=t a ℓ δ t (a * t , a) |A t | 1 {a∈At,t∈bad(a)}   ≤ E   K a=1 t ℓ+1 −1 s=t ℓ +1 m 1 {Bs,m=1} M (s,m,a) t=s∨t a ℓ δ t (a * t , a) |A t | 1 {t∈bad(a)}   ,(25) where the most inner sum on the right hand side is for m ∈ {2, . . . , 2 ⌈log(T )⌉ }. We will keep the convention that whenever a sum over m is not further specified, it will be over the above set. Note that (25) is a loose upper bound. While of course only a single call of CondaLet can be active at any point in time, we here sum over every possible replay and ignore the potential nesting and interleaving of replays. In particular, this upper bound is justified as each δ t (a * t , a) is non-negative by definition of the CW a * t . The looseness of (25) will pose no obstacle, as the remainder of our upper bounds will be sufficiently tight as we will see. Again, we first take care of the dependence on K due to the denominator on the right hand side of (25). Note that for a fixed CondaLet(s, m) if a k is the k-th arm to be eliminated by CondaLet(s, m), then min t∈[s,M (s,m,a k )] |A t | ≥ K + 1 − k. Similarly to before, this will result in a multiplicative log(K) term when eventually switching the order of the sums and summing over all arms. For now, we therefore focus on the expression E   t ℓ+1 −1 s=t ℓ +1 m 1 {Bs,m=1} M (s,m,a) t=s∨t a ℓ δ t (a * t , a)1 {t∈bad(a)}  (26) for any fixed arm a ∈ [K]. To deal with this quantity, it will be helpful to distinguish between two types of replays, i.e. calls of CondaLet, which we refer to as confined and unconfined replays. τ i −1 t=s∨t a ℓ δ t (a * i , a) ≤ c 2 log(T )K √ m. The second sum cannot be bounded in a similar way, as we cannot guarantee that the Condorcet winner in some round t ∈ [τ i , M (s, m, a)] has not been eliminated in prior rounds [s ∨ t a ℓ , τ i ). For example in Figure 1, the unconfined replay CondaLet(s ′ , m ′ ) could have eliminated a * i+1 on interval [s ′ , τ i ) before it became the Condorcet winner. We may therefore fail to detect that a suffers large regret without additional replays. To resolve this difficulty, we can reuse part of the arguments from Section A.2. Define the bad segments [s k,j (a), s k,j+1 (a)) for k ≥ i as in Definition A.1. Similarly to before, we now define the bad round s ′ (a) as the smallest round s ′ (a) > τ i such that for the same constant c 4 > 0 as in (17) (k,j) : s k,j+1 (a)<s ′ (a) s k,j+1 (a) − s k,j (a) > c 4 log(T ) s ′ (a) − t ℓ ,(28) where the sum is over all bad segments with k ≥ i and s k,j+1 (a) < s ′ (a). Importantly, for this definition of s ′ (a) and with the sum X(t ℓ , s ′ (a)) defined accordingly, the high probability guarantee of Lemma A.7 still holds. This implies that a perfect replay (see Proposition A.1) that eliminates arm a (from the unconfined replay CondaLet(s, m)) is scheduled w.h.p. before the bad round s ′ (a) occurs. Let the corresponding event denote E ′′ (t ℓ ) as in Lemma A.7. The round M (s, m, a) was defined as the last round for which a is retained in CondaLet(s, m) and all of its children. Hence, on event E ∩ E ′′ (t ℓ ), we must have M (s, m, a) < s ′ (a) as otherwise a would have been eliminated from CondaLet(s, m) (or one of its children) before round M (s, m, a), a contradiction. By merit of (16), this yields Further note that, as explained before, the denominator |A t | can be seen to account for a factor of log(K), which we loosely upper bounded by log(T ). Together with (24), we then obtain for some constant c 8 > 0 the desired bound of E   K a=1 t ℓ+1 −1 t=t a ℓ δ t (a * t , a) |A t | 1 {a∈At}   ≤ c 8 K log 3 (T ) E   i∈Phases(t ℓ ,t ℓ+1 ) √ τ i+1 − τ i   .(29) A.4 Summing Over Episodes In Section A.2 and Section A.3, we bounded the regret of arms within an episode before and after their elimination, respectively. Combining (23) and (29), and summing over episodes, we then obtain E T t=1 δ t (a * t , a t ) + δ t (a * t , b t ) 2 ≤ c 9 K log 3 (T ) E   1 {E} L ℓ=1 i∈Phases(t ℓ ,t ℓ+1 ) √ τ i+1 − τ i   + 1 T . Now, on the concentration event E, Lemma A.3 tells us that any phase [τ i , τ i+1 ) intersects with at most two episodes. Recall that τ 0 := 1 and τ S CW +1 := T . It then follows from the above that E T t=1 δ t (a * t , a t ) + δ t (a * t , b t ) 2 ≤ 2c 9 K log 3 (T ) S CW i=0 √ τ i+1 − τ i + 1 T . B Missing Details from Section 5 B.1 Significant CW Switches Let us first recall the definition of Significant Condorcet Winner Switches from Section 2.2. Let ν 0 := 1 and define ν i+1 recursively as the first round in [ν i , T ) such that for all arms a ∈ [K] there exist rounds ν i ≤ s 1 < s 2 < ν i+1 such that LetS CW denote the number of such Significant CW Switches ν 1 , . . . , νS CW . The key idea of [41] when developing this notion of non-stationarity (for multi-armed bandits) is that a restart in exploration is only warranted if there are no safe arms left to play, i.e. there is no arm left that does not suffer regret (30) on some interval [s 1 , s 2 ]. For every phase [ν i , ν i+1 ), we denote by a s i the last safe arm in phase i, i.e. the last arm to satisfy (30) in phase i. Moreover. we define the sequence of safe arms as a s t = a s i for t ∈ [ν i , ν i+1 ). Significant CW Switches are able to reconcile switch-based non-stationarity measures such as CW Switches S CW and variation-based non-stationarity measures such as the CW VariationṼ . More specifically, it naturally holds thatS CW ≤ S CW and Corollary 5.1 shows that near-optimal dynamic regret w.r.t.S CW also implies near-optimal dynamic regret w.r.t.Ṽ . We can then sum over all phases i ∈ [S CW ] to obtain T t=1 δ t (a * t , a s t ) ≤S C More Related Work Related to the non-stationary dueling bandit problem studied in this paper are adversarial dueling bandits [3,16,37,39]. Here, [3] was the first to study the dueling bandit problem in an adversarial setup and introduced a popular sparring idea, which has been picked up by many follow-up works [16,15,37,19]. The settings in [3] and [16] are restricted to utility-based preference models, where each arm is has an associated utility in each round. This entails a complete ordering over the arms in each round, which only covers a small subclass of [K] × [K] preference matrices. Moreover, [16] assume that the feedback includes not only the winner but also the difference in the utilities between the winning and losing arm, which is more similar to MAB feedback and than the 0/1 one bit preference feedback considered by us. [37] consider the dueling bandit setup for general adversarial preferences, but they measure performance in terms of (static) regret w.r.t. Borda-scores. This measure of regret is very different from our preference-based regret objective. In general, the adversarial dueling bandit problem aims to minimize static regret w.r.t. some fixed benchmark a * , whereas we study dynamic regret w.r.t. a time-varying benchmark a * t . As discussed in Section 2, static regret can be an undesirable measure of performance when no single fixed arm represents a reasonably good benchmark over all rounds (see Example 2.1). Another somewhat related line of work considers the sleeping dueling bandit problem, where the action space is non-stationary (as opposed to the preference sequence). The objective here is to be competitive w.r.t. the best active arm at each round. [29] studies the setup for adversarial sleeping but assumes a fixed preference matrix across all rounds. Remark 2. 1 1(S P vs S CW ). Of course, we always have S CW ≤ S P . In fact, it is easy to construct a simple scenario where S CW ≪ S P : Assume K = 3 and consider the following two preference matrices Theorem 3. 1 ( 1Dynamic Regret of ANACONDA). Let S CW denote the unknown number of Condorcet Winner Switches. Let τ 1 , . . . , τ S CW be the unknown times of these switches and let τ 0 := 1 and τ S CW +1 := T . For some constant c > 0, the dynamic regret of ANACONDA is bounded as Lemma 4. 1 . 1Let E be the event that for all rounds 1 ≤ s 1 < s 2 ≤ T and all arms a, b ∈ [K]: Corollary 5. 1 . 1LetṼ be the unknown Condorcet Winner Variation. Under Assumption 1 and Assumption 2, ANACONDA has dynamic regretÕ K √ T +Ṽ 1 /3 (KT ) 2 /3 . 2 Note that this is a limitation of our regret analysis. It is an open question whether it is possible to achieve O( √S CW T ) dynamic regret in NSt-DB with general preference models. round s(a) with high probability. A replay CondaLet(s, m) is scheduled if B s,m = 1 and the random variables B s,m with s ≥ t ℓ are conditionally independent on t ℓ (see Line 7 in Algorithm 1). We are thus interested in perfect replays CondaLet(s, m) such that for any bad segment [s i,j (a), s i,j+1 (a)) with s i,j+1 (a) < s(a), we have s ∈ [s i,j (a),s i,j (a)] and m ≥ s i,j+1 (a) − s i,j (a). Moreover, we define m i,j as the smallest element in {2, . . . , 2 ⌈log(T )⌉ } such that m i,j ≥ s i,j+1 (a) − s i,j (a), which implies that s i,j+1 (a) − s i,j (a) ≥ m i,j j+1 (a)<s(a) s i,j+1 (a) − s i,j (a) s(a) − t ℓ Figure 1 : 1For some episode [t ℓ , t ℓ+1 ) and arm a ∈ [K], an example of a confined replay and a unconfined replay, where M = M (s, m, a) and M ′ = M (k,j) : s k,j+1 (a)<M (s,m,a) s k,j+1 (a) − s k,j (a) ≤ c 4 log(T )K M (s, m, a) − t ℓ The regret on the final segment [s k,j (a), M (s, m, a)] can trivially be bounded by c 3 log(T )K √ m, as it must be a non-bad segment and M (s, m, a) − s k,j (a) ≤ m. Finally, in view of (16), it follows that M (s,m,a) t=s∨t a ℓ δ t (a * t , a)1 {t∈bad(a)} ≤ c 5 log 2 (T )K( M (s, m, a) − t ℓ + √ m)≤ c 5 log 2 (T )K( √ s − t ℓ + 2 √ m),where the second inequality uses M (s, m, a) − t ℓ ≤ √ s − t ℓ + √ m, since M (s, m, a) ≤ s + m and s ≥ t ℓ . t (a * t , a) ≥ K(s 2 − s 1 ). Definition 4.1. We call CondaLet(s, m) confined if there exists i ∈ Phases(t ℓ , T ) s.t. [s, s + m] ⊆ [τ i , τ i+1 ). In turn, we say that CondaLet(s, m) is unconfined if for all i ∈ Phases(t ℓ , T ), we have [s, s + m] ⊆ [τ i , τ i+1 ). CW ≤ S CW (as not all CW Switches are also Significant CW Switches), Theorem 5.1 gives a tighter dynamic regret guarantee for the class of non-stationary preference sequences with SST and STI. Also note that this bound does not violate the Ω( √ KS P T ) lower bound from 3.1, as the lower bound is shown for a worst-case preference sequence P 1 , . . . , P T whereS CW = S CW = S P .Theorem 5.1. LetS CW be the unknown number of Significant Condorcet Winner Switches. Under Assumption 1 and Assumption 2, ANACONDA has dynamic regretÕ K √S CW T . Remark 5.2. Recall from Section 2.2, sinceS Table of Contents ofPreliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 A.2 Bounding R 1 (ℓ): Regret Before Elimination . . . . . . . . . . . . . . . . . . . . . . 21 A.3 Bounding R 2 (ℓ): Regret After Elimination . . . . . . . . . . . . . . . . . . . . . . . 25 A.4 Summing Over Episodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 B.1 Significant CW Switches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 B.2 Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 B.3 Proof of Corollary 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Round in the ℓ-th episode in which a is eliminated from A good S CW Number of Condorcet Winner Switches τ 1 , . . . , τ S CW Rounds in which the Condorcet winner changes a * Condorcet winner in phase i ∈ [S CW ], i.e. a * Last safe arm in phase [ν i , ν i+1 ), i.e. last arm to satisfy (30) V Condorcet Winner VariationA Proof of Theorem 3.1 19 A.1 B Missing Details from Section 5 30 C More Related Work 35 Notation a t , b t Arms selected by the algorithm in round t a, a ′ , b Generic fixed arms in [K] δ t (a, b) Gap between arm a and arm b δ t (a, b) Importance weighted gap estimate a * t Condorcet winner in round t t ℓ First round in the ℓ-th episode t a ℓ i t = a * i for t ∈ [τ i , τ i+1 ) S CW Number of Significant Condorcet Winner Switches ν 1 , . . . , νS CW Rounds of Significant CW Switches a s i Definition A.3 (Confined and Unconfined Replays). For a fixed t ℓ , we call CondaLet(s, m) confined if there exists i ∈ Phases(t ℓ , T ) such that [s, s + m] ⊆ [τ i , τ i+1 ), i.e. the replay intersects with a single phase only. In turn, we say that CondaLet(s, m) is unconfined if for all i ∈ Phases(t ℓ , T ), we have [s, s + m] ⊆ [τ i , τ i+1 ). Note that the interval [τ i ,M (s, m, a)] can itself span over several phases. The first sum on the right hand side can be bounded as in Lemma A.8. Using Lemma A.6, the elimination rule, and the concentration bound, we get Here,Õ notation hides logarithmic dependencies. t δ t (a * t, a), as the identity of the benchmark, a * t , changes at unknown times. As a resolution to this, we aim to detect relevant changes in the preference matrix by tracking the static regret Note that by definition every segment but the last segment in a given phase must always satisfy(15) Note that this is w.l.o.g. when bounding 1/|At| as any interrupting call of CondaLet would only increase |At| by resetting it to[K]. AcknowledgementThomas Kleine Buening was supported by the Research Council of Norway, Grant No. 302203.This leads to the following important property of CondaLet that states that properly timed replays of sufficient length will eliminate arms from A good in the course of their bad segments. We call such calls of CondaLet perfect replays. Proof. Let CondaLet(s, m) be a replay such that s ∈ [s i,j (a),s i,j (a)] and m ≥ s i,j+1 (a) − s i,j (a). As any bad segment is by definition contained inside a phase, Lemma A.6 tells us that a * i ∈ A t for all t ∈ [s i,j (a), s i,j+1 (a)]. Recall that the estimatesδ t (a * i , a) are unbiased if a, a * i ∈ A t and we are thus able to obtain unbiased estimates of. What is left to show is that in fact arm a suffers sufficiently large regret to cause its elimination on this interval. To this end, by definition of the bad segments and basic algebraic manipulation, we find thatAn illustration of confined and unconfined replays is given inFigure 1. We proceed by upper bounding the inner sum M (s,m,a) t=s∨t a ℓ δ t (a * t , a)1 {t∈bad(a)} for confined and unconfined replays separately. The bound for confined replays comes with no major intricacies, whereas bounding the regret due to unconfined replays is slightly more involved.A.3.1 Bounding Regret for Confined ReplaysWe begin by bounding, the inner sumwhere the last inequality uses that M (s, m, a) ≤ s + m.A.3.2 Bounding Regret for Unconfined ReplaysLemma A.9. On event E ∩ E ′′ (t ℓ ), for any fixed arm a and unconfined replay (s, m), it holds thatHere, the event E ′′ (t ℓ ) is a concentration event similar to that in Lemma A.7 and will be defined in the following.Proof of Lemma A.9. Consider any unconfined replay CondaLet(s, m) with s ∈ [t ℓ , t ℓ+1 ). Let i be the phase so that s ∈ [τ i−1 , τ i ). We can then split the sum over t ∈ [s ∨ t a ℓ , M (s, m, a)] into the rounds before the Condorcet winner changes for the first time within [s, s + m] and the remaining rounds, i.e.A.3.3 Combining Confined and Unconfined ReplaysWe will now conclude the bound on R 2 (ℓ). To this end, recall that the replay schedule is chosenMoreover, note that we can rewrite a sum over s ∈ [t ℓ + 1, t ℓ+1 ) as a double sum over i ∈ Phases(t ℓ , t ℓ+1 ) and s ∈ [τ i ∨ (t ℓ + 1), τ i+1 ∧ t ℓ+1 ). For unconfined replays, we notice that whenNow, combining Lemma A.8 and Lemma A.9, we obtainWe here repeatedly used that n k=1 1/ √ k ≤ 2 √ n in the third and fourth inequality. In particular, the fourth inequality holds asWe see that together Assumption 1 and Assumption 2 imply a more general type of triangle inequality for any triplet a, b, c ∈ [K] with a ≻ b and a ≻ c.Lemma B.1.Under Assumption 1 and Assumption 2, for any triplet a,Proof. Suppose that b ≻ t c. Then, the claim follows directly from the stochastic triangle inequality,a, b). By definition of the gaps, this alsoAs briefly discussed in Section 5, these assumptions on the preference sequence P 1 , . . . , P T allow us to decompose the dynamic regret so that we can compare against a temporarily fixed benchmark.We can w.l.o.g. assume that a * t ≻ t a t and a * t ≻ t a s t . To see that this assumption is valid, note that a * t is the Condorcet winner in round t and it is then easy to verify that Lemma B.1 also holds if a * t equals one (or both) of a t and a s t . Applying Lemma B.1 to a * t , a s t and a t , we have δ t (a * t , a t ) ≤ 2δ t (a * t , a s t ) + δ t (a s t , a t ).Recalling equation(11)from Section A, we then get the following decomposition of the dynamic regret within each episode as.B.2.1 BoundingR 1 (ℓ)We can boundR 1 (ℓ) directly using the definition of Significant CW Switches. By definition of a s i as the last safe arm in phase [ν i , ν i+1 ), i.e. the last arm to satisfy (30) for some interval [s 1 ,B.2.2 BoundingR 2 (ℓ)As briefly mentioned in the main text, the difficulty in bounding t ℓ+1 −1 t=t ℓ δ t (a * t , a t ) for Significant CW Switches is that the identity of the Condorcet winner, i.e. a * t , may change several times within each significant phase i ∈ [S CW ]. This makes accurately tracking δ t (a * t , a) (nearly) impossible even across small intervals and the arguments that we used to prove Theorem 3.1 fail.In contrast, when we consider the relative regret of a t against the last safe arm a s t (or sequence thereof), this difficulty can be resolved. Considering a s t (instead of a * t ) as a benchmark ensures that within each phase i ∈ [S CW ] the comparator arm is fixed, since a s t = a s i for all t ∈ [ν i , ν i+1 ). Hence, the relative regret w.r.t. a s t can still be dealt with. In particular, the proof of Theorem 3.1 from Section A can be seen to hold with minor changes when swapping a * t for a s t and considering significant phases ν 1 , . . . , νS CW . For completeness, we reformulate and prove two important lemmas from Section A that relied on properties of a * t and τ 1 , . . . , τ S CW . We want to emphasise that we here again rely on Assumption 1 and Assumption 2.The following lemma shows that the beginning of a new episode implies a Significant CW Switch, i.e. every arm suffers at least (30) much regret over some interval within the episode.Proof. The start of a new episode means that every arm a ∈ [K] has been eliminated from A good at some round in t a ℓ ∈ [t ℓ , t ℓ+1 ). As a result, there must exist an interval [s 1 , s 2 ] ⊆ [t ℓ , t a ℓ ) and some arm a ′ ∈ [K] so that the elimination rule (2) holds. Using Lemma A.2, we then find that for some constant c 2 > 0:Note that by construction ofδ t (a ′ , a), we always haveApplying Lemma B.1 to the triplet (a * t , a ′ , a), we get that δ t (a * t , a) ≥ 2δ t (a * t , a ′ )+δ t (a ′ , a) ≥ δ t (a ′ , a). Thus, (31) tells us that there exists no arm a ∈ [K] such that for all [s 1 ,In other words, there is no arm that remains safe to play throughout the episode and there must have been a Significant CW Switch ν i ∈ [t ℓ , t ℓ+1 ).The following lemma ensures that the last safe arm a s i within phase i is not being eliminated before round ν i+1 by any replay CondaLet(s, m) that is scheduled in said phase.Lemma B.3 (Lemma A.6 for a s t ).On event E, no run of CondaLet(s, m) with s ∈ [ν i , ν i+1 ) ever eliminates arm a s i before round ν i+1 .Proof. Suppose on the contrary that some CondaLet(s, m) with s ∈ [ν i , ν i+1 ) eliminates arm a s i before round ν i+1 . Then, we must have for some arm a ∈ [K] and interval [s 1 ,In view of the concentration bound(6), this implies on event E thatwhere the last inequality holds by merit of(32). Now, by the definition of a s i as the last safe arm in phase i, it must hold that δ t (a, a s i ) < K(s 2 − s 1 ) for all t ∈ [ν i , ν i+1 ) and all a ∈ [K]. This stands in contradiction to the above which proves Lemma B.3. Now, following the same steps as in the proof of Theorem 3.1 in Section A, we obtain for some constantc > 0RAn application of Jensen's inequality shows that DR(T ) ≤Õ(K √S CW T ).B.3 Proof of Corollary 5.1Recall the definition of the Condorcet Winner Variation from Section 2.2:We define the CW Variation over phaseNote that in view of the bound in(35), it suffices to show thatConsider a phase [ν i , ν i+1 ) with 0 ≤ i <S CW . By definition of Significant CW Switches, every arm a ∈ [K] must satisfy on some interval [s 1 ,In particular, this is also the case for the Condorcet winner a * ν i+1 in round ν i+1 . Then, since √ s 2 − s 1 > s 2where we used that δ ν i+1 (a * ν i+1 , a * t ) ≥ 0 and δ ν i+1 (a * ν i+1 , a * t ) = −δ t (a * t , a * ν i+1 ) in the second and third inequality, respectively. We can now apply Hölder's inequality to obtaiñThe above dependence on K can be improved to K 4/9 (which is even smaller than the K 2/3 dependence in Corollary 5.1) by modifying the definition of Significant CW Switches so that ν i+1 is the first round in [ν i , T ) such that for all arms a ∈ [K] there exist rounds ν i ≤ s 1 < s 2 < ν i+1 withIt is straightforward to verify that Theorem 5.1 also holds true for this definition of Significant CW Switches. A new look at dynamic regret for non-stationary stochastic bandits. Yasin Abbasi-Yadkori, Andras Gyorgy, Nevena Lazic, arXiv:2201.06532arXiv preprintYasin Abbasi-Yadkori, Andras Gyorgy, and Nevena Lazic. A new look at dynamic regret for non-stationary stochastic bandits. arXiv preprint arXiv:2201.06532, 2022. Analysis of Thompson sampling for the multi-armed bandit problem. Shipra Agrawal, Navin Goyal, Conference on Learning Theory. Shipra Agrawal and Navin Goyal. Analysis of Thompson sampling for the multi-armed bandit problem. In Conference on Learning Theory, pages 39-1, 2012. Reducing dueling bandits to cardinal bandits. Nir Ailon, Zohar Shay Karnin, Thorsten Joachims, ICML. 32Nir Ailon, Zohar Shay Karnin, and Thorsten Joachims. Reducing dueling bandits to cardinal bandits. In ICML, volume 32, pages 856-864, 2014. The non-stationary stochastic multi-armed bandit problem. Robin Allesiardo, Raphaël Féraud, Odalric-Ambrym Maillard, International Journal of Data Science and Analytics. 34Robin Allesiardo, Raphaël Féraud, and Odalric-Ambrym Maillard. The non-stationary stochas- tic multi-armed bandit problem. International Journal of Data Science and Analytics, 3(4): 267-283, 2017. Best arm identification in multi-armed bandits. Jean- , Yves Audibert, Sébastien Bubeck, COLT-23th Conference on Learning Theory-2010. 13Jean-Yves Audibert and Sébastien Bubeck. Best arm identification in multi-armed bandits. In COLT-23th Conference on Learning Theory-2010, pages 13-p, 2010. Finite-time analysis of the multiarmed bandit problem. Peter Auer, Nicolo Cesa-Bianchi, Paul Fischer, Machine learning. 472-3Peter Auer, Nicolo Cesa-Bianchi, and Paul Fischer. Finite-time analysis of the multiarmed bandit problem. Machine learning, 47(2-3):235-256, 2002. Adaptively tracking the best bandit arm with an unknown number of distribution changes. Peter Auer, Prateek Gajane, Ronald Ortner, Proceedings of the 32nd International Conference on Learning Theory. the 32nd International Conference on Learning Theory99Peter Auer, Prateek Gajane, and Ronald Ortner. Adaptively tracking the best bandit arm with an unknown number of distribution changes. In Proceedings of the 32nd International Conference on Learning Theory, 99:138-158, 2019. Preferencebased online learning with dueling bandits: A survey. Viktor Bengs, Róbert Busa-Fekete, Adil El Mesaoudi-Paul, Eyke Hüllermeier, Journal of Machine Learning Research. Viktor Bengs, Róbert Busa-Fekete, Adil El Mesaoudi-Paul, and Eyke Hüllermeier. Preference- based online learning with dueling bandits: A survey. Journal of Machine Learning Research, 2021. Stochastic multi-armed-bandit problem with nonstationary rewards. Omar Besbes, Yonatan Gur, Assaf Zeevi, Advances in Neural Information Processing Systems. 27Omar Besbes, Yonatan Gur, and Assaf Zeevi. Stochastic multi-armed-bandit problem with non- stationary rewards. Advances in Neural Information Processing Systems, 27:199-207, 2014. Non-stationary stochastic optimization. Operations research. Omar Besbes, Yonatan Gur, Assaf Zeevi, 63Omar Besbes, Yonatan Gur, and Assaf Zeevi. Non-stationary stochastic optimization. Opera- tions research, 63(5):1227-1244, 2015. Contextual bandit algorithms with supervised learning guarantees. Alina Beygelzimer, John Langford, Lihong Li, Lev Reyzin, Robert Schapire, Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics. the Fourteenth International Conference on Artificial Intelligence and StatisticsJMLR Workshop and Conference ProceedingsAlina Beygelzimer, John Langford, Lihong Li, Lev Reyzin, and Robert Schapire. Contextual bandit algorithms with supervised learning guarantees. In Proceedings of the Fourteenth In- ternational Conference on Artificial Intelligence and Statistics, pages 19-26. JMLR Workshop and Conference Proceedings, 2011. Regret analysis of stochastic and nonstochastic multi-armed bandit problems. Foundations and Trends® in Machine Learning. Sébastien Bubeck, Nicolo Cesa-Bianchi, 5Sébastien Bubeck, Nicolo Cesa-Bianchi, et al. Regret analysis of stochastic and nonstochastic multi-armed bandit problems. Foundations and Trends® in Machine Learning, 5(1):1-122, 2012. A new algorithm for nonstationary contextual bandits: Efficient, optimal, and parameter-free. Yifang Chen, Chung-Wei Lee, Haipeng Luo, Chen-Yu Wei, Proceedings of the 32nd Conference on Learning Theory. the 32nd Conference on Learning Theory99Yifang Chen, Chung-Wei Lee, Haipeng Luo, and Chen-Yu Wei. A new algorithm for non- stationary contextual bandits: Efficient, optimal, and parameter-free. In Proceedings of the 32nd Conference on Learning Theory, 99:1-30, 2019. Contextual dueling bandits. Miroslav Dudík, Katja Hofmann, Robert E Schapire, Aleksandrs Slivkins, Masrour Zoghi, Conference on Learning Theory. Miroslav Dudík, Katja Hofmann, Robert E Schapire, Aleksandrs Slivkins, and Masrour Zoghi. Contextual dueling bandits. In Conference on Learning Theory, pages 563-587, 2015. Miroslav Dudik, Katja Hofmann, Robert E Schapire, Aleksandrs Slivkins, Masrour Zoghi, Contextual dueling bandits. Conference on Learning Theory. Miroslav Dudik, Katja Hofmann, Robert E Schapire, Aleksandrs Slivkins, and Masrour Zoghi. Contextual dueling bandits. Conference on Learning Theory, pages 563-587, 2015. A relative exponential weighing algorithm for adversarial utility-based dueling bandits. Pratik Gajane, Tanguy Urvoy, Fabrice Clérot, Proceedings of the 32nd International Conference on Machine Learning. the 32nd International Conference on Machine LearningPratik Gajane, Tanguy Urvoy, and Fabrice Clérot. A relative exponential weighing algorithm for adversarial utility-based dueling bandits. In Proceedings of the 32nd International Confer- ence on Machine Learning, pages 218-227, 2015. On upper-confidence bound policies for switching bandit problems. Aurélien Garivier, Eric Moulines, International Conference on Algorithmic Learning Theory. Aurélien Garivier and Eric Moulines. On upper-confidence bound policies for switching ban- dit problems. In International Conference on Algorithmic Learning Theory, pages 174-188. . Springer, Springer, 2011. Exploiting correlation to achieve faster learning rates in low-rank preference bandits. Suprovat Ghoshal, Aadirupa Saha, International Conference on Artificial Intelligence and Statistics. PMLRSuprovat Ghoshal and Aadirupa Saha. Exploiting correlation to achieve faster learning rates in low-rank preference bandits. In International Conference on Artificial Intelligence and Statistics, pages 456-482. PMLR, 2022. Optimal and efficient dynamic regret algorithms for nonstationary dueling bandits. Shubham Gupta, Aadirupa Saha, International Conference on Machine Learning. PMLRShubham Gupta and Aadirupa Saha. Optimal and efficient dynamic regret algorithms for non- stationary dueling bandits. In International Conference on Machine Learning, pages 19027- 19049. PMLR, 2022. A probabilistic method for inferring preferences from clicks. Katja Hofmann, Shimon Whiteson, Maarten De Rijke, Proceedings of the 20th ACM international conference on Information and knowledge management. the 20th ACM international conference on Information and knowledge managementKatja Hofmann, Shimon Whiteson, and Maarten De Rijke. A probabilistic method for in- ferring preferences from clicks. In Proceedings of the 20th ACM international conference on Information and knowledge management, pages 249-258, 2011. Patrick Kolpaczki, arXiv:2202.00935Viktor Bengs, and Eyke Hüllermeier. Non-stationary dueling bandits. arXiv preprintPatrick Kolpaczki, Viktor Bengs, and Eyke Hüllermeier. Non-stationary dueling bandits. arXiv preprint arXiv:2202.00935, 2022. Regret lower bound and optimal algorithm in dueling bandit problem. Junpei Komiyama, Junya Honda, Hisashi Kashima, Hiroshi Nakagawa, COLT. Junpei Komiyama, Junya Honda, Hisashi Kashima, and Hiroshi Nakagawa. Regret lower bound and optimal algorithm in dueling bandit problem. In COLT, pages 1141-1154, 2015. . Tor Lattimore, Csaba Szepesvári, Bandit algorithms. preprintTor Lattimore and Csaba Szepesvári. Bandit algorithms. preprint, 2018. Efficient contextual bandits in non-stationary worlds. Haipeng Luo, Chen-Yu Wei, Alekh Agarwal, John Langford, Proceedings of the 31st Conference On Learning Theory. the 31st Conference On Learning Theory75Haipeng Luo, Chen-Yu Wei, Alekh Agarwal, and John Langford. Efficient contextual bandits in non-stationary worlds. In Proceedings of the 31st Conference On Learning Theory, 75: 1739-1776, 2018. Optimized interleaving for online retrieval evaluation. Filip Radlinski, Nick Craswell, Proceedings of the sixth ACM international conference on Web search and data mining. the sixth ACM international conference on Web search and data miningFilip Radlinski and Nick Craswell. Optimized interleaving for online retrieval evaluation. In Proceedings of the sixth ACM international conference on Web search and data mining, pages 245-254, 2013. Some aspects of the sequential design of experiments. Herbert Robbins, Bulletin of the American Mathematical Society. 585Herbert Robbins. Some aspects of the sequential design of experiments. Bulletin of the American Mathematical Society, 58(5):527-535, 1952. Weighted linear bandits for non-stationary environments. Yoan Russac, Claire Vernade, Olivier Cappé, Advances in Neural Information Processing Systems. 32Yoan Russac, Claire Vernade, and Olivier Cappé. Weighted linear bandits for non-stationary environments. Advances in Neural Information Processing Systems, 32, 2019. Algorithms for non-stationary generalized linear bandits. Yoan Russac, Olivier Cappé, Aurélien Garivier, arXiv:2003.10113arXiv preprintYoan Russac, Olivier Cappé, and Aurélien Garivier. Algorithms for non-stationary generalized linear bandits. arXiv preprint arXiv:2003.10113, 2020. Dueling bandits with adversarial sleeping. Aadirupa Saha, Pierre Gaillard, Advances in Neural Information Processing Systems. 34Aadirupa Saha and Pierre Gaillard. Dueling bandits with adversarial sleeping. Advances in Neural Information Processing Systems, 34:27761-27771, 2021. Versatile dueling bandits: Best-of-both-world analyses for online learning from preferences. Aadirupa Saha, Pierre Gaillard, PMLRInternational Conference on Machine Learning. Aadirupa Saha and Pierre Gaillard. Versatile dueling bandits: Best-of-both-world analyses for online learning from preferences. In International Conference on Machine Learning. PMLR, 2022. Battle of bandits. Aadirupa Saha, Aditya Gopalan, In Uncertainty in Artificial Intelligence. Aadirupa Saha and Aditya Gopalan. Battle of bandits. In Uncertainty in Artificial Intelligence, 2018. Combinatorial bandits with relative feedback. Aadirupa Saha, Aditya Gopalan, Advances in Neural Information Processing Systems. Aadirupa Saha and Aditya Gopalan. Combinatorial bandits with relative feedback. In Advances in Neural Information Processing Systems, 2019. PAC Battling Bandits in the Plackett-Luce Model. Aadirupa Saha, Aditya Gopalan, Algorithmic Learning Theory. Aadirupa Saha and Aditya Gopalan. PAC Battling Bandits in the Plackett-Luce Model. In Algorithmic Learning Theory, pages 700-737, 2019. Best-item learning in random utility models with subset choices. Aadirupa Saha, Aditya Gopalan, International Conference on Artificial Intelligence and Statistics. PMLRAadirupa Saha and Aditya Gopalan. Best-item learning in random utility models with subset choices. In International Conference on Artificial Intelligence and Statistics, pages 4281-4291. PMLR, 2020. From pac to instance-optimal sample complexity in the plackett-luce model. Aadirupa Saha, Aditya Gopalan, International Conference on Machine Learning. PMLRAadirupa Saha and Aditya Gopalan. From pac to instance-optimal sample complexity in the plackett-luce model. In International Conference on Machine Learning, pages 8367-8376. PMLR, 2020. Efficient and optimal algorithms for contextual dueling bandits under realizability. Aadirupa Saha, Akshay Krishnamurthy, International Conference on Algorithmic Learning Theory. PMLRAadirupa Saha and Akshay Krishnamurthy. Efficient and optimal algorithms for contextual du- eling bandits under realizability. In International Conference on Algorithmic Learning Theory, pages 968-994. PMLR, 2022. Adversarial dueling bandits. Aadirupa Saha, Tomer Koren, Yishay Mansour, International Conference on Machine Learning. PMLRAadirupa Saha, Tomer Koren, and Yishay Mansour. Adversarial dueling bandits. In Interna- tional Conference on Machine Learning, pages 9235-9244. PMLR, 2021. Preference elicitation for general random utility models. Hossein Azari Soufiani, C David, Lirong Parkes, Xia, Uncertainty in Artificial Intelligence. Citeseer596Hossein Azari Soufiani, David C Parkes, and Lirong Xia. Preference elicitation for general random utility models. In Uncertainty in Artificial Intelligence, page 596. Citeseer, 2013. Multi-dueling bandits with dependent arms. Yanan Sui, Vincent Zhuang, Joel Burdick, Yisong Yue, Conference on Uncertainty in Artificial Intelligence, UAI'17. Yanan Sui, Vincent Zhuang, Joel Burdick, and Yisong Yue. Multi-dueling bandits with depen- dent arms. In Conference on Uncertainty in Artificial Intelligence, UAI'17, 2017. Advancements in dueling bandits. Yanan Sui, Masrour Zoghi, Katja Hofmann, Yisong Yue, IJCAI. Yanan Sui, Masrour Zoghi, Katja Hofmann, and Yisong Yue. Advancements in dueling bandits. In IJCAI, pages 5502-5510, 2018. Tracking most significant arm switches in bandits. Joe Suk, Samory Kpotufe, Conference on Learning Theory. PMLRJoe Suk and Samory Kpotufe. Tracking most significant arm switches in bandits. In Conference on Learning Theory, pages 2160-2182. PMLR, 2022. On the likelihood that one unknown probability exceeds another in view of the evidence of two samples. William R Thompson, Biometrika. 253-4William R Thompson. On the likelihood that one unknown probability exceeds another in view of the evidence of two samples. Biometrika, 25(3-4):285-294, 1933. Double Thompson sampling for dueling bandits. Huasen Wu, Xin Liu, Advances in Neural Information Processing Systems. Huasen Wu and Xin Liu. Double Thompson sampling for dueling bandits. In Advances in Neural Information Processing Systems, pages 649-657, 2016. Learning contextual bandits in a nonstationary environment. Qingyun Wu, Naveen Iyer, Hongning Wang, Proceedings of the 41st International ACM SIGIR Conference on Research & Development in Information Retrieval. the 41st International ACM SIGIR Conference on Research & Development in Information RetrievalQingyun Wu, Naveen Iyer, and Hongning Wang. Learning contextual bandits in a non- stationary environment. In Proceedings of the 41st International ACM SIGIR Conference on Research & Development in Information Retrieval, pages 495-504, 2018. Interactively optimizing information retrieval systems as a dueling bandits problem. Yisong Yue, Thorsten Joachims, Proceedings of the 26th Annual International Conference on Machine Learning. the 26th Annual International Conference on Machine LearningACMYisong Yue and Thorsten Joachims. Interactively optimizing information retrieval systems as a dueling bandits problem. In Proceedings of the 26th Annual International Conference on Machine Learning, pages 1201-1208. ACM, 2009. Beat the mean bandit. Yisong Yue, Thorsten Joachims, Proceedings of the 28th International Conference on Machine Learning (ICML-11). the 28th International Conference on Machine Learning (ICML-11)Yisong Yue and Thorsten Joachims. Beat the mean bandit. In Proceedings of the 28th Inter- national Conference on Machine Learning (ICML-11), pages 241-248, 2011. The k-armed dueling bandits problem. Yisong Yue, Josef Broder, Robert Kleinberg, Thorsten Joachims, Journal of Computer and System Sciences. 785Yisong Yue, Josef Broder, Robert Kleinberg, and Thorsten Joachims. The k-armed dueling bandits problem. Journal of Computer and System Sciences, 78(5):1538-1556, 2012. Relative upper confidence bound for the k-armed dueling bandit problem. Masrour Zoghi, Shimon Whiteson, Remi Munos, Maarten De Rijke, JMLR Workshop and Conference Proceedings. JMLR32Masrour Zoghi, Shimon Whiteson, Remi Munos, Maarten de Rijke, et al. Relative upper confidence bound for the k-armed dueling bandit problem. In JMLR Workshop and Conference Proceedings, number 32, pages 10-18. JMLR, 2014. Relative confidence sampling for efficient on-line ranker evaluation. Masrour Zoghi, A Shimon, Maarten Whiteson, Remi De Rijke, Munos, Proceedings of the 7th ACM international conference on Web search and data mining. the 7th ACM international conference on Web search and data miningACMMasrour Zoghi, Shimon A Whiteson, Maarten De Rijke, and Remi Munos. Relative confidence sampling for efficient on-line ranker evaluation. In Proceedings of the 7th ACM international conference on Web search and data mining, pages 73-82. ACM, 2014. Copeland dueling bandits. Masrour Zoghi, Shimon Zohar S Karnin, Maarten Whiteson, De Rijke, Advances in Neural Information Processing Systems. Masrour Zoghi, Zohar S Karnin, Shimon Whiteson, and Maarten De Rijke. Copeland dueling bandits. In Advances in Neural Information Processing Systems, pages 307-315, 2015. Every preference matrix P t satisfies that if a ≻ t b ≻ t c. Assumption 1 (Strong Stochastic Transitivity). we have δ t (a, c) ≥ δ t (a, b) ∨ δ t (b, cAssumption 1 (Strong Stochastic Transitivity). Every preference matrix P t satisfies that if a ≻ t b ≻ t c, we have δ t (a, c) ≥ δ t (a, b) ∨ δ t (b, c).
{'fraction_non_alphanumeric': 0.0727063339731286, 'fraction_numerical': 0.022067452700850015, 'mean_word_length': 3.592092671871065, 'pattern_counts': {'":': 0, '<': 36, '<?xml version=': 0, '>': 26, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 133, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'We study the problem of non-stationary dueling bandits and provide the first adaptive dynamic regret algorithm for this problem. The only two existing attempts in this line of work fall short across multiple dimensions, including pessimistic measures of non-stationary complexity and non-adaptive parameter tuning that requires knowledge of the number of preference changes. We develop an elimination-based rescheduling algorithm to overcome these shortcomings and show a near-optimalÕ( √ S CW T ) dynamic regret bound, where S CW is the number of times the Condorcet winner changes in T rounds. This yields the first near-optimal dynamic regret algorithm for unknown S CW . We further study other related notions of non-stationarity for which we also prove near-optimal dynamic regret guarantees under additional assumptions on the underlying preference model.', 'arxivid': '2210.14322', 'author': ['Thomas Kleine Buening ', 'Aadirupa Saha '], 'authoraffiliation': [], 'corpusid': 253116915, 'doi': '10.48550/arxiv.2210.14322', 'github_urls': [], 'n_tokens_mistral': 30660, 'n_tokens_neox': 27895, 'n_words': 16556, 'pdfsha': '351afaa6ee3fa9f7ff5a5ff3418b1181aa825416', 'pdfurls': ['https://export.arxiv.org/pdf/2210.14322v1.pdf'], 'title': ['ANACONDA: An Improved Dynamic Regret Algorithm for Adaptive Non-Stationary Dueling Bandits', 'ANACONDA: An Improved Dynamic Regret Algorithm for Adaptive Non-Stationary Dueling Bandits'], 'venue': []}
arxiv
A NOTE ON THE GEOMETRY OF CERTAIN CLASSES OF LINEAR OPERATORS Diogo Diniz Anselmo Raposo J R A NOTE ON THE GEOMETRY OF CERTAIN CLASSES OF LINEAR OPERATORS In this note we introduce a new technique to answer an issue posed in[7]concerning geometric properties of the set of non-surjective linear operators. We also extend and improve a related result from the same paper.1991 Mathematics Subject Classification. 46B87, 15A03, 47B37, 47L05. Introduction In 1872 K. Weierstrass constructed an example af a nowhere differentiable continuous function from [0, 1] on R. This non intuitive result, now known as Weierstrass Monster, was pushed further in 1966, when V. Gurariy constructed an infinite-dimensional subspace formed, except for the null vector, by continuous nowhere differentiable functions. In 2004, Aron, Gurariy and Seoane [1] investigated similar problems in other settings, initiating the field of research known as "lineability": the idea is to look for linear structures inside exotic subsets of vector spaces. If V is a vector space and α is a cardinal number, a subset A of V is called α-lineable in V if A ∪ {0} contains an α-dimensional linear subspace W of V . When V has a topology and the subspace W can be chosen to be closed, we say that A is spaceable. We refer to the book [2] for a general panorama of the subject. As a matter of fact, with the development of the theory, it was observed that positive results of lineability were quite common, although general techniques are, in general, not available. Towards a more demanding notion of linearity, Fávaro, Pellegrino and Tomaz introduced a more involved geometric concept: let α, β and λ be cardinal numbers, with α < β ≤ λ, and let V be a vector space such that dim V = λ. A subset A of V is called (α, β)-lineable if, for every subspace W α ⊂ V such that dim W α = α and W α ⊂ A ∪ {0} there is a subspace W β ⊂ V with dim W β = β and W α ⊂ W β ⊂ A ∪ {0}. When V is a topological vector space, we shall say that A is (α, β)-spaceable when the subspace W β can be chosen to be closed . A well-known technique in lineability is known as "mother vector technique": it consists of choosing a vector v in the set A and generating a subspace W ⊂ A ∪ {0} containing "copies" of v. However, in general, the vector v does not belong to the generated subspace (see, for instance, [8]). Constructing a vector space of prescribed dimension and containing an arbitrary given vector is a rather more involving problem, which is probably another motivation of this more strict approach to lineability. Under this new perspective, several simple problems, from the point of view of ordinary lineability, gain more subtle contours. For instance, it is obvious that the set of the continuous linear operators u : p → q that are non-surjective is c-spaceable (here and henceforth c denotes the continuum). In fact, if π 1 : q → K is the projection at the first coordinate, just consider the colection of the continuous and non-surjective linear operators for which π 1 • u ≡ 0. Therefore, only (α, β)-lineability matters in this framework. In this note we answer a question posed in [7] on the (1, c)-lineability of a certain set of non surjective functions. Our solution uses a technique that, to the best of the authors knowledge, is new. Lineability vs injectivity and surjectivity Lineability properties of the sets of injective and surjective continuous linear operators between classical sequence spaces were recently investigated by [3] and [5]. In In this section we shall show that D is (1, c)-lineable and, as a matter of fact, our technique works in a more general environment of sequence spaces. We shall say that a Banach sequence space E of X-valued sequences where X is a Banach space is reasonable if c 00 (X) ⊂ E and, for all x = (x j ) ∞ j=1 ∈ E and (α j ) ∞ j=1 ∈ ∞ , we have (α j x j ) ∞ j=1 ∈ E with (2.3) (α j x j ) ∞ j=1 ≤ (α j ) ∞ j=1 ∞ (x j ) ∞ j=1 . The class of reasonable sequence spaces includes various classical sequence spaces. For instance, for 1 < p < ∞, the p (X) spaces of p-summable sequences, the w p (X) spaces of weakly p-summable sequences and the u p (X) spaces of unconditionally psummable sequences are reasonable sequence spaces. The spaces ∞ (X), c 0 (X), of bounded and null sequences, respectively and the Lorentz spaces (w,p) (X) are also reasonable sequence spaces. Our result reads as follows: The proof Fixed v ∈ D V,E \ {0}, let N v = {k ∈ N : π k • v ≡ 0} , where π k : E → X (x j ) ∞ j=1 → x k is the k-th projection over X. If N v is a proper subset of N, the proof is simple. In fact, if j 0 ∈ N\N v , since c 00 (X) ⊂ E, it is obvious that the subspace N := {u : V → E : u is linear, continuous and π j 0 • u ≡ 0} is contained in D V,E and it is also plain that v ∈ N . We will prove that dim N ≥ c. Since v is not identically zero, there exists x 0 ∈ V such that v (x 0 ) = w 0 = 0. By the Hahn-Banach Theorem, there is a continuous linear functional ϕ : E → K, such that ϕ (w 0 ) = 1. Fixing a ∈ X\ {0}, for each k ∈ N, let us define w k = (0, . . . , 0 j 0 +k−1 , a, 0, 0, . . .) ∈ E and consider the linear operators T k : E → E given by T k (w) = ϕ (w) w k w k . Obviously, the operators T k are continuous and T k = sup w ≤1 ϕ (w) w k w k = sup w ≤1 |ϕ (w)| = ϕ . Hence, the operators S k = T k • v are continuous too and S k = T k • v ≤ T k v = ϕ v . Notice that S k (x 0 ) = w k / w k and, consequently, π j 0 +k • S k ≡ 0. Hence, S k ∈ N for each k ∈ N. It is obvious that the set {S k : k ∈ N} ⊂ N is linearly independent. Define Ψ : 1 → L (V ; E) (a n ) ∞ n=1 → ∞ n=1 a n S n . Since Ψ is well-defined, linear and injective we have dim Ψ( 1 ) = c and since Ψ( 1 ) ⊂ N the proof of the case N v = N is done. Now, let us suppose that N v = N. By (2.3) we know that, for each (α n ) ∞ n=1 ∈ ∞ , S v (αn) ∞ n=1 : V → E x → (α n (v (x)) n ) ∞ n=1 is a well-defined continuous linear operator. It is plain that S v (αn) ∞ n=1 ∈ D V,E whenever (α n ) ∞ n=1 is a sequence in ∞ having some null entry (because, if α i = 0, then the i-th coordinate of S v (αn) ∞ n=1 (x) is zero for all x ∈ V ). Let us consider, therefore, (α n ) ∞ n=1 ∈ ∞ such that α n = 0 for all n ∈ N and fix w = (w n ) ∞ n=1 ∈ E\v (V ). Since (α n w n ) ∞ n=1 ∈ E, we have (α n w n ) ∞ n=1 ∈ E\S v (αn) ∞ n=1 (V ) . In fact, if there was x ∈ V such that S v (αn) ∞ n=1 (x) = (α n w n ) ∞ n=1 , we would have (α n w n ) ∞ n=1 = (α n (v (x)) n ) ∞ n=1 and, since α n = 0 for all n ∈ N, we would have v (x) = w, which is impossible. Hence, S v (αn) ∞ n=1 ∈ D V,E . Now consider the linear map Λ : ∞ → L (V ; E) Λ ((µ n ) ∞ n=1 ) = S v (µn) ∞ n=1 . We have just proved that S v (αn) ∞ n=1 ∈ D V,E for all (α n ) ∞ n=1 ∈ ∞ ; thus Λ ( ∞ ) ⊂ D V,E . Note that Λ is injective. In fact, if (µ n ) ∞ n=1 ∈ ∞ and Λ ((µ n ) ∞ n=1 ) = 0, since N v = N, it follows that, for all k ∈ N, there is x (k) ∈ V such that v x (k) k = 0. However, the k-th coordinate of S v (µn) ∞ n=1 x (k) is µ k v x (k) k , which must be null and, consequently, µ k = 0 for all k and (µ n ) ∞ n=1 = 0. Observe that, choosing (λ n ) ∞ n=1 = (1, 1, 1, . . .) ∈ ∞ , then v = Λ ((λ n ) ∞ n=1 ) ∈ Λ ( ∞ ). Since Λ is injective, we have dim (Λ ( ∞ )) = dim ( ∞ ) = c, and the proof is done. Final remarks Proof. Fixed v ∈ A V,E \ {0}, let x 0 , y 0 ∈ V , with x 0 = y 0 , be such that v(x 0 ) = ((v (x 0 )) n ) ∞ n=1 = ((v (y 0 )) n ) ∞ n=1 = v(y 0 ) . It is obvious that the subspace M := {u : V → E : u is linear, continuous and u (x 0 ) = u (y 0 )} is contained in A V,E and v ∈ M . It is sufficient to show that M is closed and dim M ≥ β. Since v is not identically zero, there exists ξ 0 ∈ V such that v (ξ 0 ) = w 0 = 0. By the Hahn-Banach Theorem, there is a continuous linear functional ϕ : E → K, such that ϕ (w 0 ) = 1. Let {a γ : γ ∈ Γ} be a Hamel basis of V . For each γ ∈ Γ and each k ∈ N, let us define w γ k = (0, . . . , 0 k−1 , a γ , 0, 0, . . .) ∈ E and consider the linear operators T γ k : E → E given by T γ k (w) = ϕ (w) w γ k w γ k . Obviously, the operators T γ k are continuous and, thus, the operators R γ k = T γ k • v are continuous too. Notice that R γ k (ξ 0 ) = w γ k / w γ k and, consequently, π k • R γ k ≡ 0 and thus, R γ k ∈ M for each k ∈ N. Let us see that {R γ k : k ∈ N, γ ∈ Γ} ⊂ M is linearly independent. In fact, let (k 1 , γ 1 ) , . . . , (k n , γ n ) ∈ N × Γ be pairwise distinct and let λ 1 , . . . , λ n ∈ K such that λ 1 R γ 1 k 1 + · · · + λ n R γn kn = 0. Then 0 = λ 1 R γ 1 k 1 (ξ 0 ) + · · · + λ n R γn kn (ξ 0 ) = λ 1 w γ 1 k 1 w γ 1 k 1 + · · · + λ n w γn kn w γn kn and it is plain that λ 1 = · · · = λ n = 0 if k i = k j whenever i = j, with i, j ∈ {1, . . . , n}. With no lost of generality, assuming that k = k 1 = · · · = k p 1 , p 1 ≤ n, and k i = k if i > p 1 , we have that the k-th coordinate of λ 1 w γ 1 k 1 w γ 1 k 1 + · · · + λ n w γn kn w γn kn is 0 = λ 1 w γ 1 k 1 a γ 1 + · · · + λ p 1 w γp 1 kp 1 a γp 1 . By hypothesis, k 1 = · · · = k p 1 implies γ 1 , . . . , γ p 1 pairwise distinct and, therefore, a γ 1 , . . . , a γp 1 are linearly independent. Hence, λ 1 = · · · = λ p 1 = 0 and 0 = λ p 1 +1 w γ p 1 +1 k p 1 +1 w γ p 1 +1 k p 1 +1 + · · · + λ n w γn kn w γn kn . [ 7 , 7Theorem 3.1] the authors investigated more subtle geometric properties in the setting of non injective continuous linear operators by proving that if p, q ≥ 1 and (2.1) A := {u : p → q : u is linear, continuous and non injective} , then A is (1, c)-lineable. In the same paper the authors pose a question on the (1, c)lineability of the set (2.2) D := {u : p → q : u is linear, continuous and non-surjective} . Theorem 2. 1 . 1Let V = {0} be a normed vector space and X = {0} be a Banach space. Let E be a reasonable sequence space of X-valued sequences. The set D V,E = {u : V → E : u is linear, continuous and non-surjective} is (1, c)-lineable. We finish this note by showing that the previous technique gives us the following improvement of [7, Theorem 3.1]: Theorem 4.1. Let V = {0} be a normed vector space and X = {0} be a Banach space. Let E be a Banach sequence space such that c 00 (X) ⊂ E. If A V,E := {u : V → E : u is linear, continuous and non-injective} = {0} then A V,E is (1, β)-spaceable, where β = max {c, dim X}. We recall that, as commented in[7]it is not true that (1, c)-lineability is inherited by inclusions. So, the following result, which is proved by combinations of the previous techniques, shall be noticed.Again, with no lost of generality, assuming that k = k p 1 +1 = · · · = k p 2 , p 2 ≤ n, and k i = k if i > p 2 , we have that the k-th coordinate ofBy hypothesis, k p 1 +1 = · · · = k p 2 implies γ p 1 +1 , . . . , γ p 2 pairwise distinct and, therefore, a γ p 1 +1 , . . . , a γp 2 are linearly independent. Hence, λ p 1 +1 = · · · = λ p 2 = 0. Proceeding in this way, after finitely many steps, or we get λ 1 = · · · = λ n = 0 or we obtain m < n such that λ 1 = · · · = λ m = 0 and k m+1 , . . . , k n are pairwise distinct. So we have+ · · · + λ n w γn kn w γn kn and, as we know, this implies λ m+1 = · · · = λ n = 0. Notice that, at this moment, we have shown thatSupposing that u lies in the closure of M , let (u n ) ∞ n=1 be a sequence in M such that lim Seoane-Sepúlveda, Lineability and spaceability of sets of functions on R. R M Aron, V I Gurariy, J B , Proc. Am. Math. Soc. 133R.M. Aron, V.I. Gurariy, J.B. Seoane-Sepúlveda, Lineability and spaceability of sets of functions on R, Proc. Am. Math. Soc. 133 (2005), 795-803. R M Aron, L Bernal-González, D Pellegrino, J B Seoane-Sepúlveda, Lineability , The Search for Linearity in Mathematics. Monographs and Research Notes in Mathematics. Boca RatonCRC PressR.M. Aron, L. Bernal-González, D. Pellegrino, J.B. Seoane-Sepúlveda, Lineability: The Search for Linearity in Mathematics. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton (2016). Seoane-Sepúlveda, On the size of special families of linear operators. R M Aron, L Bernal-González, P Jiménez-Rodríguez, G Muñoz-Fernández, J B , Linear Algebra Appl. 544R. M. Aron, L. Bernal-González, P. Jiménez-Rodríguez, G. Muñoz-Fernández, J. B. Seoane- Sepúlveda, On the size of special families of linear operators, Linear Algebra Appl. 544 (2018), 186-205. Seoane-Sepúlveda, Linear subsets of nonlinear sets of topological vector spaces. L Bernal-González, D Pellegrino, J B , Bull. Amer. Math. Soc. 51L. Bernal-González, D. Pellegrino, J.B. Seoane-Sepúlveda, Linear subsets of nonlinear sets of topo- logical vector spaces, Bull. Amer. Math. Soc. 51 (2014) 71-130. Spaceability of the sets of surjective and injective operators between sequence spaces. D Diniz, V V Fávaro, D Pellegrino, A RaposoJr, to appear in RACSAMD. Diniz, V.V. Fávaro, D. Pellegrino, A. Raposo Jr, Spaceability of the sets of surjective and injective operators between sequence spaces, to appear in RACSAM. On the size of the set of unbounded multilinear operators between Banach sequence spaces. V V Fávaro, D Pellegrino, P Rueda, Linear Algebra Appl. 606V.V. Fávaro, D. Pellegrino, P. Rueda, On the size of the set of unbounded multilinear operators between Banach sequence spaces, Linear Algebra Appl. 606 (2020), 144-158. Lineability and spaceability: a new approach. V V Fávaro, D Pellegrino, D Tomaz, Bull. Braz. Math. Soc. 51V.V. Fávaro, D. Pellegrino, D. Tomaz, Lineability and spaceability: a new approach, Bull. Braz. Math. Soc. 51 (2020), 27-46. Norm optimization problem for linear operators in classical Banach spaces. D Pellegrino, E Teixeira, Bull. Braz. Math. Soc. 40D. Pellegrino, E. Teixeira, Norm optimization problem for linear operators in classical Banach spaces. Bull. Braz. Math. Soc. 40 (2009), 417-431. 58109-970 -Campina Grande, Brazil. E-mail address: [email protected] and [email protected] Departamento de Matemática. São Luís, BrazilUnidade Acadêmica de Matemática e Estatística, Universidade Federal de Campina Grande ; Universidade Federal do MaranhãoUnidade Acadêmica de Matemática e Estatística, Universidade Federal de Campina Grande, 58109-970 -Campina Grande, Brazil. E-mail address: [email protected] and [email protected] Departamento de Matemática, Universidade Federal do Maranhão, 65085-580 -São Luís, Brazil. E-mail address: [email protected]
{'fraction_non_alphanumeric': 0.09559492961691995, 'fraction_numerical': 0.027102738286994887, 'mean_word_length': 3.0123630233211576, 'pattern_counts': {'":': 2, '<': 4, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 5, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 40, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'In this note we introduce a new technique to answer an issue posed in[7]concerning geometric properties of the set of non-surjective linear operators. We also extend and improve a related result from the same paper.1991 Mathematics Subject Classification. 46B87, 15A03, 47B37, 47L05.', 'arxivid': '2009.02407', 'author': ['Diogo Diniz ', 'Anselmo Raposo ', 'J R '], 'authoraffiliation': [], 'corpusid': 221516376, 'doi': '10.1007/s00574-021-00246-9', 'github_urls': [], 'n_tokens_mistral': 5388, 'n_tokens_neox': 4825, 'n_words': 2885, 'pdfsha': '40f006a1318d8ab41890d8095f84c00bb7d23f1c', 'pdfurls': ['https://arxiv.org/pdf/2009.02407v1.pdf'], 'title': ['A NOTE ON THE GEOMETRY OF CERTAIN CLASSES OF LINEAR OPERATORS', 'A NOTE ON THE GEOMETRY OF CERTAIN CLASSES OF LINEAR OPERATORS'], 'venue': []}
arxiv
(2,3)-Cordial Trees and Paths * † Manuel Santana [email protected] Jonathan Mousley [email protected] Dave Brown [email protected] Leroy B Beasley [email protected] Department of Mathematics and Statistics Department of Mathematics and Statistics Utah State University Logan 84322-3900UtahU.S.A Department of Mathematics and Statistics Utah State University Logan 84322-3900UtahU.S.A Department of Mathematics and Statistics Utah State University Logan 84322-3900UtahU.S.A Utah State University Logan 84322-3900UtahU.S.A (2,3)-Cordial Trees and Paths * † AMS Classification number: 05C2005C3805C78 † Key words and phrases: orientation of an undirected graphgraph labelingcordial labeling(23)-cordial digraph ‡ corresponding Author Recently L. B. Beasley introduced (2, 3)-cordial labelings of di- * Introduction Let G = (V, E) be an undirected graph with vertex set V and edge set E, a convention we will use throughout this paper. A (0, 1)-labelling of Applications of balanced graph labelings can be found in the introduction of [7]. In this paper we will formally define (2, 3)-cordiality starting from the view of quasi-groups, resolve two conjecture posed in [1], both in the negative, and discuss the (2, 3) cordiality of the Petersen graph and complete graphs. Preliminaries In [4] Hovey introduced A cordial graphs, with vertex labeling of an abelian group A. We use the short hand that for any a ∈ A that V a , E a is the number of vertices or edges labeled a respectively We repeat the definition here, and then proceed to generalize it to the directed graph case. f , ||f −1 (a)| − |f −1 (b)|| ≤ 1. Definition 2.2. Let A be an abelian group. A graph G is is A−cordial if there is a balanced labeling f : V → A that induces an edge labeling f (a, b) = f (a) + f (b) such that the vertex labeling and edge labeling are balanced. In [8] Penchenik and Wise generalized this idea by introducing quasigroup cordiality. A quasi-group Q is a set with binary operation · such that for all a, b ∈ Q there exist unique c, d ∈ Q such that a · c = b and d · a = b. Particularly all (non-abelian) groups are quasi groups. They offer the following definition for quasigroup cordiality. We extend this now to a different form of quasi-group Cordiality defined as thus. In this paper consider only the simplest case (Z 2 , Z − 3 ) − cordial defined by Beasly as (2, 3)−cordial in [1]. We will offer the formal definition of (2, 3)−cordial here. Definition 2.5. A labeling f : V → {0, 1} is said to be friendly if it is balanced. Definition 2.6. Let D n be the set of all digraphs on n vertices. We will define T n as the subset of D n that consists of all digon-free digraphs, where a digon is a two cycle on a digraph. Definition 2.7. Let D ∈ T n with D = (V, A). Let f : V → {0, 1} be a friendly labeling of the vertex set V of D. Let g : A → {−1, 0, 1} be an induced labeling of the arcs of D such that for any i, j ∈ {−1, 0, 1}, −1 ≤ |g −1 (i)| − |g −1 (j)| ≤ 1. Such a labeling is called a (2, 3)-cordial labeling, and a digraph D ∈ T n that can possess a (2, 3)-cordial labeling will be called a (2, 3)-cordial digraph. Definition 2.8. Let D = (V, A) be a digraph with vertex labelling f : V → {0, 1} and with induced arc labelling g : A → {−1, 0, 1, }. Define Γ f,g to be the real triple Γ f,g (D) = (α, β, γ) where α = |g −1 (1)|, β = |g −1 (−1)|, and γ = |g −1 (0)|. Let D ∈ T n and let D r be the digraph such that every arc of D is Let g be the corresponding induced arc labeling of D, g( − → uv) = f (v) − f (u). Lemma 2.1. Let D ∈ T n with vertex labeling f and induced arc labeling g. Let Γ f,g (D) = (α, β, γ). Then 1. Γ f,g (D r ) = (β, α, γ). 2. Γ f ,g (D) = (β, α, γ), and 3. Γ f ,g (D R ) = Γ f,g (D). Proof. If an arc is labeled 1, -1, 0 respectively then reversing the labeling of the incident vertices gives a labeling of -1, 1, 0 respectively, If an arc − → uv is labeled 1, -1, 0 respectively, then − → vu would be labeled -1, 1, 0 respectively. Also in this article we will study when undirected graphs can have their arcs given an orientation such that the resulting graph is (2, 3)−cordial. We finish this section with a couple of definitions for that. Definition 2.9. Define G n to be the set of all simple, undirected, connected graphs. We say G ∈ G n has vertex set V and edge set E and denote it by G = (V, E). Definition 2.10. Let G ∈ G n . An orientation of G is a digraph D(G) whose vertex set is the same as the vertex set of G and whose arc set Resolution of Two Conjectures We begin with the first conjecture. The orientation in Figure 1 of the ten path has no cordial labeling. We used the following brute force algorithm to test to see if a certain arc orientation is cordial on a friendly labeling of a graph. Data: Arcs, Verticies Result: Determine if an orientation is cordial on a path for arc in Arcs do current = first vertex next = second vertex if arc is left then edgeLabel = current -next end else edgeLabel = current -next end store edge label current = next next = next vertex end if edge labels are cordial then return it is cordial end In investigating the conjecture we had to test every possible friendly labeling and arc orientation on ten vertices. If we let n denote the number of vertices on the path, then checking every possible friendly labeling against every arc set has complexity of O(2 k ). As a slight optimization by Lemma 2.1 with out loss of generality we can fix the first arc and the first label and still account for all cases up to isomorphism. That means there will be 2 n−2 arc orientations to test. In calculating all arc orientations of the ten path we found the only orientation that is not (2, 3)−coridal is the one in a similar argument as on the one above, we will prove by cases. Proof. First note that the value of γ in a vertex labeling of the graph does not depend on arc orientation. Thus we show that there does not exist a friendly labeling on the vertex set of the graph that induces an edge labeling of γ = 3, by only considering vertex labelings on the undirected graph. We will separate our graph into a right and left subgraph and then connect them The structure of the proof leads to the following theorem. 3 of the edges are connected by vertices of different labels, and therefore arcs may be assigned such that G is (2, 3)−cordial. If G is (2, 3)−cordial, then clearly G has a friendly labeling that satisfies the above conditions. Though fairly intuitive we will now use theorem 3.1 to show two more results that stem from it in the following section. There are two cases as shown in Figure 9, up to isomorphism. γ = 2 is not possible since there is no way to label two 0's and two 1's without the third 1 connecting to another vertex labeled one. This means we need to connect the star sub graph such exactly two or exactly four more edges Therefore by Theorem 4.1 the Petersen graph is not (2, 3) cordial. One more example shows how this theorem proved useful in proving an upper bound on the size of the edge set for any graph to be (2, 3)-orientable. For work on the upper bounds for other graph labelings see [6]. (2) Proof. It can be shown, see [2], that any tournament with n ≤ 5 vertices is (2, 3)-cordial, save for the case when n = 4. Thus we begin with a complete graph with n ≥ 6. Recall that the number of vertices on a complete graph is n 2 . Thus Z is the number of edges in two cliques comprised of the the subgraph of the vertices labeled 0 and the vertices labeled 1. In this way Z counts the number of edges labeled zero regardless of arc orientation. If n is even that will mean that Z = 2 n 2 2 . If n is odd then Z is as above. This also implies there are n 2 − Z edges that cannot be labeled zero. For every complete graph with n ≥ 6 vertices Z > 1/3 n 2 . By Theorem 3.1 this is too many edges to be (2, 3)−orientable. The number edges labeled 0 must be balanced with the number of edges labeled 1 and −1, and the number of edges labeled 1 or −1 is 1/2( n 2 − Z). Thus the upper bound on the number of edges a graph on n vertices can have and be (2, 3)−orientable is as in equation 2. the vertex set is a mapping f : V → {0, 1} and is said to be friendly if approximately one half of the vertices are labelled 0 and the others labelled 1. An induced labelling of the edge set is a mapping g : E → {0, 1} where for an edge uv, g(uv) =ĝ(f (u), f (v)) for someĝ : {0, 1} × {0, 1} → {0, 1} and is said to be cordial if f is friendly and about one half the edges of G are labelled 0. A graph, G, is called cordial if there exists a cordial induced labelling of the edge set of G. In this article we investigate a cordial labelling of directed graphs that is not merely a cordial labelling of the underlying undirected graph. This labeling was introduced by L. B. Beasley in [1]. Let D = (V, A) be a directed graph with vertex set V and arc set A, with a (0, 1) vertex set mapping f : V → {0, 1}. Let g : A → {−1, 0, 1} be the induced labeling of the arcs of D such that for any − → uv, by which we mean an arc going from u to v, g( − → uv) = f (v) − f (u). The digraph D is said to be (2, 3)-cordial if there exists a friendly labeling on D with this induced labeling on the arc set such that approximately one third of the arcs receive each labeling. Definition 2. 3 . 3Let Q be a quasi-group and G a directed graph. A labeling f : V → Q induces a labeling of the arcs in the following way. If (a, b) is an arc with head a, then f (a, b) = f (a) · f (b). If there is a balanced vertex labeling of G that induces a balanced edge labeling of G, then we say that G is Q − cordial. Definition 2. 4 . 4Let Q be a quasi-group with subset Q and G a directed graph. A labeling f : V → Q induces an arc labeling as in definition 2.3. If there is a balanced vertex labeling of G that induces a balanced arc labeling of G, then we say G is (Q, Q) − cordial. reversed, so that − → uv is an arc in D r if and only if − → vu is an arc in D. Let f be a (0, 1)-labeling of the vertices of D and let g( − → uv) = f (v) − f (u) so that g is a (−1, 0, 1)-labeling of the arcs of D. Let f be the complementary (0, 1)-labeling of the vertices of D, so that f (v) = 0 if and only if f (v) = 1. consists of the same number of arcs as the number of edges of G such that given an edge {u, v} of G, either − → uv or − → vu is an arc of D(G) but not both, so that D(G) is digon free. A graph G is said to be (2, 3)-orientable if there exists and orientation of G, D(G), that is (2, 3)-cordial. Conjecture 1[ 1 , 1Conjecture 4.1]: Every orientation of every path is (2,3) cordial except for a path with four vertices. Figure 1 : 1Figure 1: figure 3 . 3The next known case of a non (2, 3)−orientable path is one on 22 vertices with the same alternating arc structure. Conjecture 2[1, Conjecture2.3]: Every tree of max degree 3 is (2, 3)−orientable. Figure 2 2is a counter example. Though easily proved with a computer by Figure 2 : 2Figure 2: Figure 3 :Figure 4 :Figure 5 : 345Left and Right SubgraphLet x be the number of vertices labeled 1 on one subgraph in a friendly labeling of the whole graph. In order for the graph to be cordial x cannot be 4 or 5 since this would make γ > 3 on the entire graph. Therefore x also cannot be 0 or 1 since that would mean x would be too large on the other sub graph. Thus we must have x = 2 on one subgraph x = 3 on the other subgraph.Given these constraints it is not possible to have a sub graph such that γ = 0, since there is no way to label two of the vertices on a subgraph without having at least two vertices of the other label connected. This means we do not need to account for the case when γ = 3 on a subgraph, since the other subgraph cannot have γ = 0. Now we will consider two cases.Case 1. With out loss of generality let the left subgraph have γ All γ = 2 subgraph labelings x = 2 and the right subgraph have γ = 2, x = 3. This would mean we would need to connect the subgraphs such that the connecting edge All γ = 1 subgraph labelings be labeled 0. Considering all cases we see that there is no way to connect the two subgraphs to make the full graph without γ = 4 on the full graph. Case 2. With out loss of generality let the left subgraph have γ = 1, x = 2, and the right subgraph have γ = 1, x = 3. This would mean we need to find a way to connect the two subgraphs such that the connecting edge is labeled 0. Considering all cases shows that this is not possible. Theorem 3. 1 . 1Let G ∈ G n . We define Λ(G) to be the number of edges, uv such that vertices u and v have the same label for a given friendly labeling on G. G is (2, 3)-orientable if and only if there exsits a friendly vertex labling on G such that Λ(G) = 1 3 |E| or Λ(G) = 1 3 |E| , where |E| is the cardinality of the edge set of G Proof. Suppose G satisfies Λ(G) = 1 3 |E| or Λ(G) = 1 3 |E| . This would mean about 2 Theorem 4 . 1 . 41The Petersen Graph,Figure 6, is not (2,3) -orientable.Proof. By Theorem 3.1 only need to show that there is no friendly labeling such that one third of the edges are connected with the same label of the vertex. Again let x be the number of vertices labeled 1. We will start by dividing the Petersen graph into two subgraphs and then connect them. Figure 6 :Figure 7 : 67The Two Subgraphs of the Petersen Graph With out loss of generality the only way for our friendly label to be cordial we must have one sub graph have x = 2, and have x = 3 on the other. The case when either sub graph has x ≥ 4 of a certain label would result in γ being too large. Let us first consider the outer subgraph letting x = 3 for this subgraph. Figure 8: γ = 1, γ = 3 Figure 8 Figure 9 : 89γ = 1, γ = 3 with γ = 1 or γ = 3 possible on the subgraph. Upon inspection there is no possible way to connect any rotation of the inner sub graph with either of the outer sub graphs such that γ = 5 for the resulting Petersen graph. Theorem 4. 2 . 2Given a directed graph G = (V, E) with vertex set V and n = |V | with n ≥ 6, and edge set E. The maximum size of E such that G is (2, 3) orientable for any given n Definition 2.1. A labeling function f is said to be balanced if it is surjective, and if for all a and b in the image of . L B Beasley, Cordial Digraphs, In PressL. B. Beasley, Cordial Digraphs, In Press. L B Beasley, M A Santana, J M Mousley, D E Brown, Cordial Digraphs. PreprintL. B. Beasley, M. A. Santana, J. M. Mousley and D.E. Brown, (2,3)- Cordial Digraphs, Preprint. Cordial graphs: A weaker version of graceful and harmonious graphs. I Cahit, Ars Comb. 23I. Cahit, Cordial graphs: A weaker version of graceful and harmonious graphs, Ars Comb. 23(1987) 201-208. A-cordial graphs. M Hovey, Discrete Math. 93M. Hovey, A-cordial graphs, Discrete Math. 93 (1991) 183-194. PC-labelling of a graph and its PC-set. E Salehi, Bull. Ins.t Comb. Appl. 58E Salehi, PC-labelling of a graph and its PC-set, Bull. Ins.t Comb. Appl., 58(2010) 112-121. Upper Bounds of Four Types of Graph Labelings. M A Seoud, M A Salim, Ars Combinatoria. 127M. A. Seoud and M. A. Salim, Upper Bounds of Four Types of Graph Labelings, Ars Combinatoria, 127(2016) 271-278 Full friendly index set -I. Sinah Deepa, Kuar Jaspreet, Discrete Applied Mathematics. 161Sinah Deepa, Kuar Jaspreet, Full friendly index set -I Discrete Applied Mathematics 161 (2013) 1262-1274. . O Penchenik, J Wise, Generalized Graph Cordiality Discussiones Mathematicae Graph Theory. 32O. Penchenik, J. Wise, Generalized Graph Cordiality Discussiones Mathematicae Graph Theory 32 (2012) 557-567
{'fraction_non_alphanumeric': 0.06156213928434013, 'fraction_numerical': 0.02879312556111325, 'mean_word_length': 3.5020207852193996, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 4, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'Recently L. B. Beasley introduced (2, 3)-cordial labelings of di- *', 'arxivid': '2012.10591', 'author': ['Manuel Santana [email protected] ', 'Jonathan Mousley [email protected] ', 'Dave Brown [email protected] ', 'Leroy B Beasley [email protected] ', '\nDepartment of Mathematics and Statistics\nDepartment of Mathematics and Statistics\nUtah State University Logan\n84322-3900UtahU.S.A\n', '\nDepartment of Mathematics and Statistics\nUtah State University Logan\n84322-3900UtahU.S.A\n', '\nDepartment of Mathematics and Statistics\nUtah State University Logan\n84322-3900UtahU.S.A\n', '\nUtah State University Logan\n84322-3900UtahU.S.A\n'], 'authoraffiliation': ['Department of Mathematics and Statistics\nDepartment of Mathematics and Statistics\nUtah State University Logan\n84322-3900UtahU.S.A', 'Department of Mathematics and Statistics\nUtah State University Logan\n84322-3900UtahU.S.A', 'Department of Mathematics and Statistics\nUtah State University Logan\n84322-3900UtahU.S.A', 'Utah State University Logan\n84322-3900UtahU.S.A'], 'corpusid': 229339994, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 5005, 'n_tokens_neox': 4461, 'n_words': 2995, 'pdfsha': '39b33330660b7918c54ce4f13b4ee571e8f1f66e', 'pdfurls': ['https://arxiv.org/pdf/2012.10591v2.pdf'], 'title': ['(2,3)-Cordial Trees and Paths * †', '(2,3)-Cordial Trees and Paths * †'], 'venue': []}
arxiv
WIMP Dark Matter in Composite Higgs Models and the Dilaton Portal Manki Kim Seung J Lee Alberto Parolini Department of Physics Department of Physics Korea Advanced Institute of Science and Technology 335 Gwahak-ro, Yuseong-gu305-701DaejeonKorea School of Physics Institute for Advanced Study Korea University 136-713, 130-722Seoul, SeoulKorea, Korea, Korea Quantum Universe Center Institute for Advanced Study 130-722SeoulKorea, Korea WIMP Dark Matter in Composite Higgs Models and the Dilaton Portal We study under which conditions a scalar particle is a viable WIMP Dark Matter candidate with Higgs and dilaton interactions. The theory is a composite Higgs model with top partial compositeness where both the Higgs and the Dark Matter candidate arise as pseudo Goldstone boson of the coset SO(6)/SO(5) from a new physics sector. We highlight the role of the dilaton in direct and indirect searches. We find that a Dark Matter particle with a mass around 200-400 GeV and a relatively light dilaton are a fair prediction of the model. I. INTRODUCTION Light scalars are believed to be unlikely in Nature, unless there is a fine tuning or there exists an underlying dynamics screening the quadratic ultraviolet sensitivity. Indeed the Standard Model (SM) suffers from the hierarchy problem because of the Higgs boson: an interesting possibility is that the Higgs boson, rather than an elementary particle, is a composite object, a bound state of a new, yet undiscovered, interacting theory which gets strong at the TeV scale. In particular the idea that the Higgs is not only a composite object but a pseudo Nambu Goldstone boson (pNGB), like pions in QCD, is especially appealing, because of the approximate built in shift symmetry. From a different perspective, also the Dark Matter (DM) density in the Universe could be accounted for by a scalar particle, again subject to the same naturalness issue, and if it is a weakly interacting massive particle (WIMP), its mass should be broadly in the TeV range. Therefore a very compelling picture emerges if a single new strongly interacting sector is responsible for both the Higgs and the DM. We pursue this approach in a next to minimal pNGB Composite Higgs Model (CHM), based on the symmetry breaking coset SO(6)/SO (5): it includes a custodial SO(4) and it is exactly described by five Goldstone modes, a bidoublet H and a singlet η. This coset, or the isomorphic SU(4)/Sp(4), can be formulated in an underlying theory of fundamental techni-quarks and it has already received some attention [1][2][3][4][5][6]. If η is sufficiently stable it is a perfect DM candidate: this is achieved if the * Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] theory respects a global Z 2 symmetry under which η is odd. The main difference with the case of elementary scalars is in the form of the interactions. This very predictive setup has already been explored [7,8]. We want to extend the analysis assuming that the strong sector provides a second DM portal to SM particles: on top of Higgs exchange the dilaton could play an important role, if the strong sector is an approximate Conformal Field Theory (CFT) and it features a light dilaton. A light dilaton is also a rare phenomenon in spontaneously broken CFTs in the sense that it requires fine tuning, [9][10][11][12][13], but if present it affects the DM phenomenology, if it is a different state than the Higgs scalar. We will show how in our model the light dilaton affects the DM phenomenology, mainly fixing a lighter DM mass; moreover it gives the dominant contribution to Sommerfeld enhanced processes. The dilaton portal in composite DM models has been studied in [14], but neglecting Higgs effects. A complete picture including both is the main object of our present work. In [15] a similar interplay was studied, but without the pNGB structure. The rest of the paper is organized as follows. After defining an effective Lagrangian in section II, including the other composite resonances typically considered in CHM, we introduce the dilaton field σ and we derive the interactions between the light scalars, h, η and σ, and the SM fermions and vectors in section III. We move to DM properties, starting from the computation of the relic density, section IV, to direct and indirect constraints, in section V and VI respectively. We take into account collider constraints in section VII. Finally we summarize and we draw our conclusions in section VIII. II. THE SO(6)/SO(5) MODEL A. Scalar Sector The new physics sector, behaving as a CFT, is perturbed by a deformation, which becomes strong at an energy scale around the TeV. It possesses, in isolation, an approximate global SO(6) symmetry, spontaneously broken to SO (5). As a result five pseudo Goldstone bosons arise, a complex doublet H and a singlet η. H transforms as a bi-doublet under the custodial SU(2) L × SU(2) R ⊆ SO(5), and η is a singlet. According to the Callan Coleman Wess Zumino (CCWZ) formalism [16,17] the Lagrangian for the Goldstone bosons, in the unitary gauge, is written as L = f 2 4 Tr[d µ d µ ] = f 2 2 (D µ Σ) T D µ Σ (1) = 1 2 ∂ µ h∂ µ h + ∂ µ η∂ µ η + (h∂ µ h + η∂ µ η) 2 f 2 − h 2 − η 2 + h 2 8 (g 2 0 ((W 1 µ ) 2 + (W 2 µ ) 2 ) + (g 0 B µ − g 0 W 3 µ ) 2 ) . where U (x) and Σ are defined in terms of the broken SO(6) generators Tâ as U (x) =e (i √ 2θâTâ) , Σ(x) = U (x) (0 0 0 0 0 1) T ,(2) and in the unitary gauge Σ = 1 f 0, 0, h, 0, η, f 2 − h 2 − η 2 T .(3) dâ µ is defined as iTr(U † ∂ µ U Tâ). The scalar potential is radiatively generated once SO(6) breaking effects are included, namely once the strong sector is coupled to the SM, and it depends on the details of the composite sector and of the mixings, therefore it is model dependent. Nonetheless it can be parametrized in the following way: V f (h, η, χ) = µ 2 h,f 2 h 2 + µ 2 η,f 2 η 2 + λ h,f 4 h 4 + λ η,f 4 η 4 + λ hη,f 4 η 2 h 2 .(4) We limit to models in whose vacuum the ElectroWeak (EW) symmetry is broken h = h + 1 − ξh phys , η = 0 ,(5) where h = v = f √ ξ 246 GeV and we work in the assumption of v f . B. Composite Resonances Fermion Resonances In order to generate fermion Yukawa couplings and the effective potential of the composite Higgs and the composite DM, we adopt the partial compositeness scenario [18]. Additionally, when we formally embed the SM fermions in SO (6) representations, the embedding should preserve the Z 2 symmetry stabilizing the DM. According to [7,8], we embed the left and right handed fermions in the fundamental representation of SO (6): ξ u L = 1 √ 2 b L −ib L t L it L 0 0 T 2/3 , ξ u R = 0 0 0 0 0 t R T 2/3 ,(6) where we focus on the top quark and the subscript is the X charge assignment necessary to reproduce the top hypercharge. Other quarks and leptons can be embedded in a similar way, or could receive their mass from a different mechanism, as bilinear Yukawalike interactions [19][20][21]. Partial compositeness is introduced as L ψ SM O ψ + h.c .(7) According to the CCWZ formalism, at low energy, O ψ can be represented as a function of U (x) and Ψ, where U is the NGB matrix and Ψ is a collection of SO (5) fields. We focus for definiteness and for simplicity on cases of Ψ resonances S i and F j transforming in the trivial and in the fundamental representation of SO (5). Details on the Lagrangian can be find in Appendix A, where we also show how the effects of the heavy resonances can be encoded in form factors. Vector Resonances Vector resonances are generically expected as well as fermion resonances. For simplicity we present one adjoint vector resonance ρ µ and one fundamental vector resonance a µ , introduced following [22]: again we refer to Appendix A for detailed expressions. III. DILATON EXTENSION OF THE COMPOSITE HIGGS MODEL As we previously stated the strong sector in isolation is a CFT enjoying a global SO(6) symmetry. In the vacuum both the conformal and the global symmetry are spontaneously broken. In this section we want to specify the general relations given in section II including the dilaton field. The dilaton dependence is introduced promoting f to be a dynamical field χ = f e σ/f and dressing composite fields with the appropriate powers of χ/f . Notice that for simplicity we identify the scale associated to the dilaton f σ with f . The Goldstone kinetic term becomes L ⊇ χ 2 4 Tr[d µ d µ ] .(8) In a similar manner the fermionic and vector Lagrangian are modified by the presence of the dilaton χ/f . At energies below the masses of the resonances the effective Lagrangian is L ef f ⊇ Π t Lt L/ pt L + Π t Rt R/ pt R − (Π t L t Rt L t R + h.c) + P µν T 2 (Π 0 W a µ W a ν + Π 1 h 2 4f 2 (W 1 µ W 1 ν + W 2 µ W 2 ν )) + P µν T 2 (Π B B µ B ν + Π 1 h 2 4f 2 cos 2 θ w Z µ Z ν )(9) where the form factors are modified by the presence of the dilaton. The scalar potential V (h, η, χ) is obtained integrating out the SM top and vector bosons with a standard one loop computation. We briefly review the results of this computation in the following. A. The Scalar Potential The gauge contribution to the scalar effective potential V g (h, η, χ) is V g = 3 2 d 4 p E (2π) 4 (2 log[Π W W (−p 2 E )](10)+ log[Π BB (−p 2 E )Π W W (−p 2 E ) − Π W3B (−p 2 E )]) . Notice that as a result no potential is generated for η. Fermion loops generate in principle all the possible terms containing Higgs and η fields, but the case N F = N S = 1 leads to the unsatisfactory prediction µ η = λ η = 0. Therefore we move to the next to minimal case, namely N F = 1, N S = 2. The fermion contribution to the effective potential V f (h, η, χ) is computed from V f = −2N c d 4 p E (2π) 4 log(p 2 E Π t L Π t R + Π 2 t L t R ) .(11) We impose the generalized Weinberg sum rules [23] and in order to get unsuppressed µ η and λ, we assume m 2S >> m F >> m 1S ∼ f . There is one subtlety: loops of top quarks, due to the large top Yukawa, induce a mixing between the Higgs and the dilaton field. Indeed the most general Lagrangian takes the form V f (h, η, χ) = χ 4 f 4 i+j<3 κ i,j χ 2γ(i+j) f 2γ(i+j) h 2i η 2j(12) where γ is the top anomalous dimension [13]. Therefore < ∂ χ ∂ h V > < ∂ 2 h V > γv f(13) and we get that the mixing is proportional to the top anomalous dimension: since γ 0 we safely neglect it. Similarly the Higgs radion mixing has been studied in a warped extra dimensional background and argued to be small for a pNGB Higgs [24]. We refer to Appendix A 3 for a discussion on the dilaton potential. In the following we are going to treat the dilaton mass as a free parameter of the model, given its unpredictability in an effective description. B. Interactions with Massless Gauge Bosons The precise determination of interaction couplings between scalars such as dilaton, DM, and Higgs and gauge bosons is of primary importance in order to study LHC phenomenology and various aspects of DM detection. We therefore proceed in analyzing them. First, we study the dilaton. It couples to gauge bosons via trace anomaly terms, which depend on the beta functions of the theory, and via triangle diagrams generated by loops of charged fields [9,14,[25][26][27][28]: L ⊇ α s 8π (b 3 IR − b 3 U V + 1 2 F 1/2 (x t )) σ f G a µν G aµν(14)+ α em 8π (b em IR − b em U V + 4 3 F 1/2 (x t ) − F 1 (x W )) σ f F µν F µν where x i = 4m 2 i /m 2 σ . F 1/2 and F 1 are loop functions defined as F 1/2 (x) = 2x(1 + (1 − x)f (x)),(15)F 1 (x) = 2 + 3x + 3x(2 − x)f (x), f (x) = arcsin 2 (1/ √ x) if x ≥ 1 − 1 4 (log( 1+ √ x−1 1− √ x−1 ) − iπ) 2 if x < 1 . The loops of heavy top partners cancel with the IR beta function of the same in the limit of masses larger than m σ /2, as we discuss in Appendix B. Therefore the top partners decouple and the only effects from the IR are from the light degrees of freedom. Among the light composite states we count the Higgs boson doublet, which enters the beta function coefficients with b 2 IR = − 1 6 , b 1 IR = − 1 6 .(16) In case the right handed top is fully composite then b 3 IR = − 1 3 , b 1 IR = − 8 27 N c ,(17) while it does not contribute to the composite beta functions if it is elementary. As a result the IR beta function coefficients are b 3 IR 0 , b em IR = − 1 3 ,(18)or b 3 IR = − 1 3 , b em IR = − 11 9(19) if also t R belongs to the composite fields. The UV coefficients b 3,em U V are model dependent and we cannot specify them in our effective construction. Since they enter the couplings of the dilaton in the following discussion we will focus on simple benchmark values. We now turn to Higgs couplings. According to [28] the effect of composite fermion loops is expected to be negligible and the main contribution is given by top loops, closely resembling the SM result: L ⊇ α em 8π ( 1 − 2ξ √ 1 − ξ 4q 2 t − 1 − ξF 1 ( 4m 2 W m 2 h )) h v F µν F µν + α s 12π 1 − 2ξ √ 1 − ξ h v G a µν G aµν .(20) Similarly, since DM couples at tree level to SM fermions, we have DM to gauge bosons interactions at one loop. Given the coupling of η to fermions [65] L ⊇ ξ 2(1 − ξ) m ψψ ψ η 2 v 2(21) we easily read the couplings to gauge bosons L ⊇ − α s 32π F 1/2 ( 4m 2 t m 2 η ) ξ 1 − ξ η 2 v 2 G a µν G aµν − 3α em 16π ξ 1 − ξ η 2 v 2 q 2 t F 1/2 ( 4m 2 t m 2 η )F µν F µν . (22) We neglect possible couplings of η to pair of gauge bosons arising from the Wess-Zumino-Witten term, they could be computed in principle given the details of the fundamental underlying theory, as done in [29]. C. Effective Lagrangian An effective Lagrangian for the SM fields, the DM candidate η and the dilaton σ is obtained, expanding the scalars around their VEV h = v + 1 − ξh phys , η = η phys , σ = σ phys . (23) The resulting Lagrangian, the starting point of our phenomenological analysis, has the following form: L ⊇ + 1 2 (∂ µ h) 2 (1 + a hh h v + b hh h 2 v 2 + b hη η 2 v 2 )e 2σ/f + 1 2 (∂ µ η) 2 (1 + b ση η 2 v )e 2σ/f + (∂ µ h∂ µ η)(c η η v + d ηh ηh v 2 )e 2σ/f + V m 2 V 2 V µ V µ (1 + 1 − ξ h v ) 2 e 2σ/f − i m ψψi ψ(1 + a ψh h v + b ψh h 2 v 2 )e σ/f + α s 8π (b 3 IR − b 3 U V + 1 2 F 1/2 (x t )) σ f G a µν G aµν + α em 8π (b em IR − b em U V + 4 3 F 1/2 (x t ) − F 1 (x W )) σ f F µν F µν + α em 8π ( 1 − 2ξ √ 1 − ξ 4q 2 t − 1 − ξF 1 ( 4m 2 W m 2 h )) h v F µν F µν + α s 12π 1 − 2ξ √ 1 − ξ h v G a µν G aµν − α s 32π F 1/2 ( 4m 2 t m 2 η ) ξ 1 − ξ η 2 v 2 G a µν G aµν − 3α em 16π ξ 1 − ξ η 2 v 2 q 2 t F 1/2 ( 4m 2 t m 2 η )F µν F µν − V ef f (h, η, χ) .(24) with V ef f (h, η, χ) given in (A19). Concerning the dilaton mass we are mostly interested in m σ > 0.1f [9][10][11], because a too light dilaton requires too much fine tuning, and m σ ≤ 4πf because of NDA [30]. IV. RELIC ABUNDANCE A. Introduction to WIMPs WIMP is one of the most compelling paradigm for DM. In case of scalar DM fundamental and composite singlet scalar WIMPs have been extensively studied, see e.g. [7,8,36,37]. In order to implement the WIMP scenario, we need to assume that the DM candidate is in thermal equilibrium since the very early universe. In case of composite DM there exists an energy threshold above which DM particles are resolved in their constituents. Since we have f v we can safely assume thermal equilibrium; moreover heavy degrees of freedom of the strong theory are irrelevant being, indeed, heavy. As a result we can use the standard picture of WIMPs [38]. We recall that the measured DM relic density is Ωh 2 = 0.1199 ± 0.002 [39]. The current relic density is predicted using the Weinberg-Lee equation [38] dn dt + 3Hn =< σv > (n 2 eq − n 2 )(25) where σv is the thermal average of cross sections times relative speed, and H is the Hubble constant. Expanding σv for small velocities as σv = a + bv 2 we get < σv >= a + 6b/x, where x = m/T . We use this expansion because s-wave processes are dominant in our model. By solving the above equation, we get the freeze out temperature x F = ln 5 4 45 8 g 2π 3 M pl m η (a + 6b/x F ) √ g * x F ,(26) where g is the number of degrees of freedom of the DM and g * is the effective relativistic degrees of freedom in thermal equilibrium. As a result, the DM relic abundance is given by Ωh 2 1.07 × 10 9 GeV M pl √ g * x F a + 3(b − a/4)/x F .(27) B. Annihilation Cross Sections In our model the DM candidate is the fifth pseudo Goldstone boson of the coset SO(6)/SO(5), η. Its effective potential is determined by the underlying theory and can be reliably computed using an effective IR Lagrangian, as we outlined before: the form of this Lagrangian depends on the details of the theory, as the number of top partners N F and N S . If N F = N S = 1 the mass is fixed to be m η m h /2 and the predicted relic density is too small to be a viable option. Therefore we focus on the next to minimal case N F = 1, N S = 2, where the η mass varies as a free parameters over an interval. We fix the portal coupling λ hη 0.13, following [8]. We computed the annihilation channels including ηη → W W, ZZ, hh, hσ, σσ, AA, GG, andψψ, where ψ runs over the SM fermions. Note that the above processes are dominated by s-wave exchange since p and higher order terms are suppressed by v 2 . Full expressions are reported in Appendix D. We present here asymptotic forms valid in certain limits. We focus on m σ , m η m Z : as a result ηη → V V dominates the annihilation cross section. First we take m η m σ . If this is the case we obtain σv AA m 2 η c AA πf 4 ,(28) where c ZZ = 16, c W W = 8, c σσ = 4 and c hh = 16. The ηη → hσ process is controlled by η∂ µ h∂ µ η and suppressed by ξ 3 /(1 − ξ). The total thermally averaged cross section is then σv m 2 η 2πf 4 3 × 10 −26 8.5 TeV m η f 2 2 cm 3 /s .(29) Note that σv should be equal to or larger than 3 × 10 −26 cm 3 /s in order to reproduce a relic density equal to or smaller than the observed one. In the massive dilaton limit, m σ m η , the dilaton exchanging processes are suppressed by m 2 η /m 2 σ and Higgs exchanging processes have a similar asymptotic form as before. Consequently we get larger annihilation cross section, parametrized as in (28) where now c ZZ = 4, c W W = 2, and c hh = 4. As a result, the total thermally averaged cross section is σv m 2 η πf 4 3 × 10 −26 12 TeV m η f 2 2 cm 3 /s . (30) In Fig. 1 we present the predicted relic density of DM particles in the m σ −m η plane. We clearly distinguish a depletion of Ωh 2 in correspondence of the points with m η = m h /2 63 GeV and m η = m σ /2. If Fig. 2 we present the value of the scale f which is necessary to reproduce the observed relic density, in the same plane. V. DIRECT DETECTION Null results from direct detection experiments, as LUX [40,41], put limits on the nucleon-DM scattering cross section. The interactions in (24) relevant in this regard are the vertices between the scalars h, η and σ with the fermion bilinearsψψ and the field strength operator G µν G µν of colored interactions. From those we derive an effective theory for nucleons L ⊇ − i=n,pψ i ψ i (y s,i σ + y h,i h + y η,i η 2 ) ,(31) where y σ,i = ψ 1 f i|m ψψ ψ|i − C s 8πf i|α s G a µν G aµν |i , y h,i = 1 v 1 − 2ξ √ 1 − ξ ( ψ i|m ψψ ψ|i − 1 12π i|α s G a µν G aµν |i ) , y η,i = − 1 2v 2 ξ 1 − ξ ψ i|m ψψ ψ|i(32)+ 1 32π 2 v 2 ξ 1 − ξ F 1/2 ( 4m 2 t m 2 η ) i|α s G a µν G aµν |i , where i stands for neutron and proton and ψ stands for SM quarks [66]. Integrating out the dilaton and the Higgs we obtain L ef f ⊇ −a nn nη 2 − a pp pη 2(33) where a i y η,i − 2m 2 η y σ,i f m 2 σ − λ hη v √ 1 − ξy h,i 2m 2 h .(34) For the matrix elements, we take the values for u and d quarks from [42], and for s, c, b, and t quarks from [43]: f i ψ = i|ψψ|i m ψ m i f n u 0.016 , f p u 0.018 , f n d 0.038 , f p d 0.034 , f n s f p s 0.043 , f c 0.0814 , f b 0.0785 , f t 0.0820 , α s n|G a µν G aµν |n −2.4 GeV .(35) We then derive the nucleon-DM cross section σ η,i a 2 i m 2 i πm 2 η .(36) By comparing with the LUX data we get the allowed parameter region, shown in Fig. 3. For the points for which the model predicts a relic density lower than the observed one we rescale the bound. VI. INDIRECT DETECTION A. Sommerfeld Enhancement To correctly evaluate the signals searched by indirect detection experiments we take into account Sommerfeld enhancement, following [44,45]. To this end we need the three fields interaction vertices of DM η with dilaton and Higgs, which are respectively of the form 1 2 (∂η) 2 2 σ f − 1 2 m 2 η η 2 4 σ f(37) and where λ hη = 0.013. These lead to interaction potential, in momentum space, of the form ξ 1 − ξ ∂ µ η∂ µ h η v − v 1 − ξ λ hη 2 η 2 h(38)V (p − q) = 1 4m 2 η ( 2 f ) 2 Π i (p iµ (p 0 + (−1) i+1 q) µ + 2m 2 η ) (p − q) 2 − m 2 σ − Π σ (4m 2 η ))(39) and V (p − q) = 1 4m 2 η Π 2 i=1 ( (−1) i ξ v(1−ξ) p i (p − q) − v √ 1 − ξ λ hη 2 ) (p − q) 2 − m 2 h(40) for dilaton and Higgs respectively, where p 1 and p 2 are the momenta of the incoming particles and p = (p 1 − p 2 )/2. In the non-relativistic limit, in the instant interaction limit and in the CM frame the above expressions reduce to V (p − q) = − 1 4m 2 η ( 2 f ) 2 m 4 η ( p − q) 2 + m 2 σ(41) and V (p − q) = − 1 4m 2 η v 2 (1 − ξ)λ 2 hη 4( p − q) 2 + 4m 2 h .(42) As a result, the following Yukawa potential arises V (r) = − α σ r e −mσr − α h r e −m h r(43) where α σ = 9m 2 η 4πf 2 and α h = (1−ξ)v 2 λ 2 hη 16πm 2 η . Notice that α σ α h , and DM is in non relativistic regime, thus Sommerfeld enhancement is dilaton dominated. According to [45][46][47], an analytic approximate formula for dilaton mediated Sommerfeld enhancement is S = π v sinh( 12 v π σ ) cosh( 12 v π σ ) − cos(2π 6 π 2 σ − ( 6 v π 2 σ ) 2 )(44) where v = v/α σ and σ = m σ /(α σ m η ). B. Antiproton Flux DM annihilation can produce antiprotons in various ways and we take into account the AMS-02 [48,49] measure to constraint the parameter space of the model, demanding that the predicted antiproton flux does not exceed the observed one. Following [14,50], we derive a bound on the antiproton flux produced by DM annihilation by imposing that the amount of antiprotons produced by the DM annihilation in the Galactic disk is smaller than the antiproton flux due to primary cosmic rays colliding with interstellar medium in the disc [51]. We followed [52,53] to compute antiproton spectrum, and [54] to evaluate cascade annihilation processes initiated by ηη → σσ , hh, including the Sommerfeld enhancement (44). The injection rate density of antiprotons produced by DM annihilation is Qp(E) = 1 2 n 2 η σv dNp dE (45) 5 × 10 −36 cm −3 s −1 GeV −1 ρ η 0.4 GeVcm −3 2 σv 3 × 10 −26 cm 3 s −1 m η 1 TeV −3 m η dNp dE where ρ η = m η n η and dNp/dE is the differential antiproton spectrum per annihilation event. According to [54] dilaton and higgs contributions to antiproton flux is given by dNp dx = 2 tmax tmin dx 0 x 0 β σ dNp ,S dx 0(46) where S = h, σ, β σ = 1 − γ −2 σ , x = E/m η , t min = 2xγ 2 σ (1 − β σ ), t max = min[1, 2xγ 2 σ (1 + β σ )] and γ σ = m η /m σ . By including cascade effects, we obtain the full differential antiproton spectrum, follow-ing [53]. Fig. 5 shows a typical spectrum at f = 1500 GeV and for m η = 300 GeV and m σ = 1000 GeV. In order to impose our condition we use a propagation model independent injection rate [51] given by Qp ,CR (E) 8.4 × 10 −33 cm −3 s −1 GeV −1 E 100 GeV −2.8 1 − 0.22 log 2 10 E 500 GeV J p (1 TeV) J p,0 (1 TeV)(47) where J p (1 TeV) is the local proton flux at E = 1 TeV and scaled to measured value J p,0 (1 TeV) 8 × 10 −9 GeV −1 cm −2 s −1 sr −1 . Due to uncertainty in the derivation of the injection rate, it varies within a factor of 2 [51]. The results of our analysis are shown in Fig. 4. Also in this case for the points predicting a too low relic density we assume that our DM candidate is the only source of antiprotons. Furthermore, by adopting the Cosmic Rays (CR) grammage given in [51], we compute the antiproton flux and compare antiproton to proton flux to measuredp/p data reported by AMS-02 [48,49,55]. Fig. 6 presents the allowed region by imposing that the computedp/p ratio does not exceed thep/p measured by AMS-02. We found that the points reproducing a nearly exact DM relic density do not give significant antiproton flux, and points fitting thep/p flux predict a too low relic density. Note that the allowed region can be significantly changed by precise determination of CR grammage and proper knowledge on spallation loss, propagation and solar modulation. In addition, we find that parameter points which generate resonant Sommerfeld enhancement factor are excluded by AMS-02 data. For sake of illustration we provide thep/p flux spectra for two points in the Sommerfeld enhanced region in Fig. 7: data points are the measuredp/p flux ratio reported by [55], the red line is the secondary prediction as given by [49], the blue area is the deviation of the secondary prediction due to uncertainities. Model predictions are computed at two parameter points, where (1) is f = 1500 GeV, m η = 1866 GeV and m σ = 1303 GeV, and (2) is f = 1000 GeV, m η = 1183 GeV and m σ = 746 GeV. C. Gamma Ray Flux As well known, see for instance [56], gamma ray excesses can be a good probe of DM. Since, in our model, DM annihilation produces gamma ray via direct annihilation and Higgs and dilaton mediation, we check whether our model fits the experimental data. Because of the fact that the dwarf spheroidal satellite galaxies (dSphs) of the Milky Way are expected to contain considerable DM amount [57] and have ignorable noise of non-thermal astrophysical gamma ray production, we use the limit on thermally averaged scattering cross sections observed by the Fermi-LAT Collaboration [58] to constrain our model. Note that the analysis is relatively insensitive to the detailed DM distribution inside the dSphs. Following [53,54] we compute the gamma ray spectrum per annihilation, and we compare with the SM channels, which we find in [53]. Fig. 9 shows the ratio of gamma ray spectrum at each energy. The spectrum generated by DM annihilation of our model is within a factor of 2 or 3 with respect to the gamma ray spectrum generated by pure ηη →bb channel and ηη → W W channel, thus we assume that the constraints given by [58] is applicable to our model. In many points of the parameter space the correct relic density of DM is not reproduced, as we discussed above and we showed in Fig. 1. For those points we assume that η only partially accounts for the DM density around the dSphs and the additional DM does not contribute to the CR production. Under such assumptions the resulting effective J factor contributing to the gamma ray flux is J ef f = ( Ω η Ω DM ) 2 J(48) where Ω DM h 2 0.12 and Ω η is the relic density for η DM. Consequently we derive a cross section bound much weaker the bound given by [58]. Fixing m σ = 1000 GeV we present thermally averaged cross section and bounds given by the Fermi-LAT Collaboration in Fig. 10. Fig. 8 shows the allowed parameter region imposing the constraints from the Fermi-LAT experiment at 95% confidence level. We do not observe any peak in the gamma ray spectrum because σv ηη→γγ /σv tot is negligible in our model. In the high DM mass region, where m η ≥ 1 TeV, experimental constraints given by the H.E.S.S Collaboration [59] provide tighter bound though we have more dependence on the propagation model. By assuming that DM distribution follows a cusp distribution such as the Navarro-Frenk-White [60], we could superimpose this additional bound on the constraints given by Fermi-LAT, but that region is already ruled out and this procedure does not provide additional information. VII. COLLIDER CONSTRAINTS A. Higgs Measurements We consider the impact of the measurements of signal strengths reported in [31,32] on the allowed parameter space of the theory, namely on ξ or equivalently on f . We perform a χ 2 analysis using the following channels µ V /µ F = 1. and the result is shown in Fig. 13, from which we read that at 95% CL f larger than 960 GeV is still allowed. B. Heavy Scalar Searches Since the dilaton has couplings to SM particles similar to the Higgs' ones its parameter space is constrained by searches for heavy Higgses [33][34][35]. A dilaton whose mass lies between 200 and 1000 GeV is probed by such searches, and the experimental measures convert to a lower bound on f . In Fig. 12 we report the allowed minimum value for f at 95% CL for each choice of dilaton mass, focusing for definiteness on specific values for the UV beta functions b 3,em U V , chosen as representative. C. Precision Tests We proceed inspecting the contribution of new physics to the EW precision parameters measured by LEP [62]. The presence of composite resonances is expected to have an impact on EW precision tests .At tree level vector resonances give, imposing the generalized Weinberg sum rules as in [23], δS = 8 sin 2 θ w m 2 W αm 2 ρ 1 − f 2 4f 2 ρ(50) which in turn implies for instance m ρ > 2 TeV if f ρ = f . Also modification of Higgs couplings play a role in enhancing EW precision parameters: interestingly enough once we include the dilaton we get vanishing T corrections due to the fact that c 2 V,h +c 2 V,σ = 1. Furthermore, for the same reason, S correction are also suppressed. Higgs and dilaton loops are computed following [63]. From the Lagrangian L ⊇ 2m 2 W W + µ W −µ + m 2 Z Z µ Z µ c V,h h v + c V,σ σ v − 1 4 σ v (2c Zγ F µν Z µν + c γγ F µν F µν )(51) we easily read α∆T − 3g 2 Y 32π 2 (1 − c 2 V,h − c 2 V,σ ) log(Λ/m Z ) = 0 , α∆S g L g Y log(Λ/m Z ) 48π 2 (g 2 L + g 2 Y ) (2g L g Y (1 − c 2 V,h − c 2 V,σ ) + 6c V,σ (2g L g Y c γγ + c Zγ (g 2 L − g 2 Y )) + 3(g L g Y (c 2 Z,γ − c 2 γγ ) − (g 2 L − g 2 Y )c γγ c Zγ )) , α∆W g 2 L 192π 2 c γγ + g L g Y c Zγ 2 log(Λ/m Z ) , α∆Y g 2 L 192π 2 c γγ − g L g Y c Zγ 2 log(Λ/m Z ) ,(52) where Λ 4πf and in our model c V,h = 1 − ξ , c V,σ = ξ ,(53)c γγ = − α em 2π (b em IR − b em U V + 4 3 F 1/2 (x t ) − F 1 (x W )) ξ , c Z,γ = − α em 2π tan θ W (b 2 IR − b 2 U V − t 2 W (b 1 IR − b 1 U V )) ξ + eg L 8π 2 (A Z 1 (τ W , λ W ) + f N f q f g f A Z 1/2 (τ f , λ f )) ξ , with A 1 and A 1/2 given in [64]. As a result EW precision tests do not significantly constraint the model for f ≥ 900 GeV. Finally note that typical values of αW and αY are ∼ 10 −7 . Fermionic resonances are expected to affect EW parameters as well but in a model dependent way: we rely on the fact that this effect is well studied and understood in the literature and it is shown to be compatible with observations for large regions in parameters space in similar models. VIII. SUMMARY AND CONCLUSIONS The presence of additional light scalars, beyond the Higgs, is an expected feature of CHM. We have considered a candidate DM scalar particle in a specific CHM based on the coset SO(6)/SO(5), enlightening the possible role of a light dilaton as a mediator of DM interactions with the SM. To summarize our analysis we combine results from collider constraints, direct and indirect searches discussed in the previous sections. Fig. 11 shows the predicted density for two given symetry breaking scales f = 1000 GeV and f = 1500 GeV. For these plots we use benchmark UV beta functions b 3 U V = b em U V = 0. While for f = 1000 GeV the available parameter space, in which our candidate DM scalar entirely accounts for the observed density, shrinks to zero, if we allow for f = 1500 GeV we have a region in parameter space starting with m η 200 GeV and m σ 500 GeV; a heavier dilaton requires a heavier DM particle and an asymptotic value of m η 300 GeV is reached at m σ 1500 GeV. Interestingly, according to the scan performed in [8], η mass can vary between 100 and 700 GeV for f = 800 − 1100 GeV. Notice that f = 1000 GeV returns to be a viable option if a fraction of the DM relic density is accounted for by a different particle, as for instance an axion. + iS i (i / ∇ − m iS )S i + jF j (i / ∇ − m jF )F j + i ( i tSξR P L U S i + i qSξL P R U S i ) + h.c + j ( j tFξ R P L U F j + j qFξ L P R U F j ) + h.c . In addition, there can be interactions between composite resonances [8,23]: L int = η=L,R (k V,η ijF i γ µ (g ρ ρ µ − E µ )P η F j ) + η=L,R (S i γ µ (k A,η ij a µ + k d,η ij d µ )P η F j + h.c) . where ρ µ , a µ are massive vector resonances of the strong sector. Notice that these interactions do not enter the scalar couplings to gg and γγ at one loop because they mix different species of composite fermions [61]. In order to compute the low energy effective theory of SM fermions, we need to integrate out the composite resonances. The result, in momentum basis, up to quadratic order in the fermions, is written as L ef f = Π t Lt L/ pt L + Π t Rt R/ pt R − (Π t L t Rt L t R + h.c) (A2) The form factors are written as Π t L =Π F + h 2 f 2 Π 1F , Π t R = Π S + (1 − h 2 f 2 − η 2 f 2 )Π 1S , Π t L t R = h f 1 − h 2 f 2 − η 2 f 2 Π F S .(A3) The explicit form of the form factors in terms of the parameters in (A1) is given in [8]. Vector Resonances The Lagrangian for vector resonances is given by L = − 1 4 T r(ρ 2 µν ) + f 2 ρ 2 T r(g ρ ρ µ − E µ ) 2 − 1 4 T r(a 2 µν ) + f 2 a 2∆ 2 T r(g a a µ − ∆d µ ) 2 . (A4) General cases of vector resonances are examined in [23] and mixing between ρ and E is described in [8,23]. Similarly to the fermion case, integrating out heavy vector fields we obtain an effective Lagrangian for SM vector bosons given by, in momentum space, L = P µν T 2 (Π 0 (q 2 )Tr(A µ A ν ) + Π 1 (q 2 )Σ t A µ A ν Σ + Π X 0 (q 2 )X µ X ν ) (A5) where A µ is a spurion obtained formally gauging all the SO(5) generators. In the physical configuration where only A a L = W a , A 3 R = c X B, X = s X B are different from zero, with c X = g Y /g L and s 2 X = 1−c 2 X , the former expression reduces to L = P µν T 2 (Π 0 W a µ W a ν + Π 1 h 2 4f 2 (W 1 µ W 1 ν + W 2 µ W 2 ν ) + Π B B µ B ν + Π 1 h 2 4f 2 cos 2 θ w Z µ Z ν ) (A6) where Π B = s 2 X Π X 0 + c 2 X Π 0 and Z = cos θ w W − sin θ w B, with cos θ w = g L / g 2 L + g 2 Y . It is also customary to define Π W W =Π 0 + h 2 4f 2 Π 1 , Π BB = Π B + c 2 X h 2 4f 2 Π 1 , Π W3B = − c X h 2 4f 2 Π 1 . (A7) Dilaton Potential Unlike other Goldstone bosons, a non derivative self-interaction term for the dilaton is allowed and indeed it is expected at tree level: V tree (χ) = κ 4! χ 4 .(A8) Corrections are generated by loops of self interactions and loops of heavy resonances. The first gives V ef f =V tree + 3κ 2 32π 2 χ 4 log κχ 2 2µ 2 − 1 2 = 1 32π 2 χ 4 f 4 κ 0 log χ 2 f 2 +κ 1 (A9) wherê κ 0 =3κ 2 f 4 , κ 1 = 32π 2 κf 4 4! + 3κ 2 f 4 log κf 2 2µ 2 − 1 2 .(A10) Gauge and fermion contributions to the potential are obtained from the form factors at h = η = 0: V (χ) = d 4 p E (2π) 4 3 2 log[Π 0 Π B ] − 6 log[p 2 E Π F (Π 1S + Π S )] .(A11) Recalling the general formula d 4 p E (2π) 4 log[p 2 E + U 2 ] = 1 32π 2 U 4 (log U 2 µ 2 − 1 2 ) (A12) the result can be expressed as V (χ) 1 32π 2 χ 4 f 4 (κ 0 log χ 2 f 2 + κ 1 ) (A13) where trivially κ 0 = 2π 2 f 4 χ 3 ∂ 5 V ∂χ 5 , κ 1 = 4π 2 f 4 3 ∂ 4 V ∂χ 4 χ→f − κ 0 .(A14) We now move to study the Vacuum Expectation Value (VEV) and the mass of the dilaton. We start with the potential V ef f (h, η, χ) = χ 4 f 4 V (h, η) + 1 32π 2 χ 4 f 4 (κ 0 log χ 2 f 2 + κ 1 ) (A15) where V (h, η) is the sum of the gauge and fermion contributions. Imposing the condition χ = f we obtain κ 1 = −32π 2 V (v, 0) − κ 0 2 ,(A16) and then V ef f (h, η, χ) = χ 4 f 4 (V (h, η) − V (v, 0)) (A17) + κ 0 16π 2 χ 4 f 4 (log χ f − 1 4 ) . Therefore the mass of the dilaton is given by m 2 σ = κ 0 4π 2 f 2 (A18) and the effective potential (A17) can be rewritten as V ef f (h, η, χ) = χ 4 f 4 (V (h, η)−V (v, 0))+ m 2 σ 4 χ 4 f 2 (log χ f − 1 4 ) . (A19) We assumed κ 0 > 0 in order to have a potential bounded from below. Finally we notice that because of the tree level term the dilaton mass is model dependent and therefore in our phenomenological analysis we treat it as a free parameter. Appendix B: Decoupling of Heavy Composite Fermions We discuss here the effect of heavy fermionic resonances on the couplings of the dilaton σ to γγ and gg. They contribute entering the beta function coefficients b i IR and also circulating in triangular loops. In the limit of mass much larger than m σ /2 the two effects cancel and in the following we review this property. Indeed in extra dimensional construction heavy KK modes of bulk fermions do not generate corrections for radion couplings, as shown in [25]. We obtain the same result in a four dimensional language. We consider N F and N S heavy Dirac fermions with quantum numbers under the SM gauge group SU(N c ) × SU(2) L × U(1) Y F = (N c , 2) 7/6 ⊕(N c , 2) 1/6 ⊕(N c , 1) 2/3 , S = (N c , 1) 2/3 . (B1) They enter the Lagrangian L ⊇ α s 8π (b 3 IR,F + b 3 IR,S ) σ f G a µν G aµν + α em 8π (b em IR,F + b em IR,S ) σ f F µν F µν (B2) contributing with b 3 IR,F = − 10 3 N F , b em IR,F = N c 27 152N F , b 3 IR,S = − 2 3 N S , b em IR,S = N c 27 16N S .(B3) The second contribution comes from loop diagrams. For σgg it has the form L ef f ⊇ α s 8π ( 5N F 2 F 1/2 (x 2 F ) + N S 2 F 1/2 (x 2 S )) σ f G a µν G aµν (B4) = α s 8π ( 5N F 2 F 1/2 (x 2 F ) + N S 2 F 1/2 (x 2 S )) σ f G a µν G aµν where x F,S = 2m F,S /m σ . Note that F 1/2 (x) quickly saturates to 4/3 for x > 1. Since typical masses of heavy composite fermions are larger than m σ /2 the limit is justified and we have a perfect cancellation in the infinite mass limit. Similarly for σγγ L ef f ⊇ α em 8π N c ( 38N F 9 F 1/2 (x 2 F ) + 4N S 9 F 1/2 (x 2 S )) σ f F µν F µν (B5) = α em 8π N c (N F 38 9 F 1/2 (x 2 F ) + N S 4 9 F 1/2 (x 2 S )) σ f F µν F µν and the same cancellation is in place. Therefore we verify, at one loop, the decoupling of heavy fermions states, confirming the expectation from extra dimensional models. Appendix C: Dilaton Decay Widths Γ σ→ψψ = 3m 2 ψ (m 2 σ − 4m 2 ψ ) 3/2 8πf 2 m 2 σ , Γ σ→hh = m 2 σ − 4m 2 h (m 2 σ + 2m 2 h ) 2 32πf 2 m 2 σ , Γ σ→W W = m 2 σ − 4m 2 W (m 4 σ − 4m 2 σ m 2 W + 12m 4 W ) 16πf 2 m 2 σ , Γ σ→gg = α 2 s 32π 3 (b 3 IR − b 3 U V + 1 2 F 1/2 (x t )) 2 m 3 σ f 2 , Γ σ→γγ = α 2 256π 3 (b em IR − b em U V + 4 3 F 1/2 (x t ) − F 1 (x W )) 2 m 3 σ f 2 , Γ σ→ηη = m 2 σ − 4m 2 η (m 2 σ + 2m 2 η ) 2 32πf 2 m 2 σ . (C1) Appendix D: Annihilation Cross Sections σv ηη→W W = m 4 W m 2 η − m 2 W 32πm 3 η f 4   2 + 2m 2 η − m 2 W m 2 W 2   144m 4 η |m 2 σ − 4m 2 η + i (Π σ (4m 2 η ))| 2 + 48m 2 η m 2 W (−4m 2 η + f 2 λ hη (1 − ξ))(16m 4 η − 4m 2 η (m 2 σ + m 2 h ) + m 2 h m 2 σ + (Π σ (4m 2 η )) (Π h (4m 2 η )) |m 2 σ − 4m 2 η + i (Π σ (4m 2 η ))| 2 |m 2 h − 4m 2 η + i (Π h (4m 2 η ))| 2 +4 (4m 2 η − f 2 λ hη (1 − ξ)) 2 |m 2 h − 4m 2 η + i (Π h (4m 2 η ))| 2 ,(D1)σv ηη→σσ = m 2 η − m 2 σ 4πf 4 (2m 2 η − m 2 σ ) 2 |m 2 σ − 4m 2 η + i (Π σ (4m 2 η ))| 2 (49m 8 σ m η − 108m 4 σ m 5 η + 32m 2 σ m 7 η + 64m 9 η − 28m 6 σ m 3 η ) + m 2 η − m 2 σ 4πf 4 (2m 2 η − m 2 σ ) 2 |m 2 σ − 4m 2 η + i (Π σ (4m 2 η ))| 2 (25m 4 σ m η + 4m 5 η − 20m 2 σ m 3 η )( (Π σ (4m 2 η ))) 2 ) ,(D2)σv ηη→ψψ = 3m 2 ψ 8πm 3 η (m 2 η − m 2 ψ ) 3/2 36m 4 η f 4 |m 2 σ − 4m 2 η + i (Π σ (4m 2 η ))| 2 + 16(4m 2 η − f 2 λ hη (1 − ξ) 3/2 ) 2 (1 − 2ξ) f 4 |m 2 h − 4m 2 η + i (Π h (4m 2 η ))| 2 (1 − ξ) 3 − 48(−4m 2 η m 2 σ + 16m 4 η + m 2 σ m 2 h − 4m 2 η m 2 h + (Π σ (4m 2 η )) (Π h (4m 2 η )))(4m 2 η − f 2 λ hη (1 − ξ) 3/2 )(1 − 2ξ) |m 2 h − 4m 2 η + i (Π h (4m 2 η ))| 2 |m 2 σ − 4m 2 η + i (Π σ (4m 2 η ))| 2 (1 − ξ) 3/2 + 8(4m 2 η − m 2 h )(4m 2 η − f 2 λ hη (1 − ξ) 3/2 )(1 − 2ξ)ξ v 2 f 2 (1 − ξ) 5/2 |m 2 h − 4m 2 η + i (Π h (4m 2 η ))| 2 + + 12m 2 η (m 2 σ − 4m 2 η )ξ f 2 v 2 (1 − ξ)|m 2 σ − 4m 2 η + i (Π σ (4m 2 η ))| 2 + ξ 2 v 4 (1 − ξ) 2 ,(D3)σv ηη→σσ = m η m 2 η − m 2 σ ((−7m 4 σ + 2m 2 σ m 2 η + 8m 4 η ) 2 + (5m 2 σ − 2m 2 η ) 2 (Π σ (4m 2 η )) 2 ) 4πf 4 (m 2 σ − 2m 2 η ) 2 |m 2 σ − 4m 2 η + i (Π σ (4m 2 η ))| 2 ,(D4)σv ηη→σh = m 4 σ + (−4m 2 η + m 2 h ) 2 − 2m 2 σ (4m 2 η + m 2 h ) 128πf 2 v 2 m 4 η (1 − ξ) 4(m 2 σ + 8m 2 η − m 2 h ) 2 (m 2 h ξ − v 2 λ hη (1 − ξ)) 2 (m 2 σ + m 2 h − 4m 2 η ) 2 − 4(m 2 σ + 8m 2 η − m 2 h )(4λ hη v 2 + (m 2 σ − 4m 2 η − m 2 h − 4λ hη v 2 )ξ)(m 2 h ξ − λ hη v 2 (1 − ξ)) m 2 σ − 4m 2 η + m 2 h + (−4λ hη v 2 + (−m 2 σ + 4m 2 η + m 2 h + λ hη v 2 )ξ) 2 + (m 2 h − 4m 2 η ) 2 (λ hη v 2 − (4m 2 η + λ hη v 2 )ξ) 2 |m 2 h − 4m 2 η + i (Π h (4m 2 η ))| 2 + 4(m 2 σ − 4m 2 η )(m 2 h − 4m 2 η )(8m 2 η + m 2 σ − m 2 h )(−λ hη v 2 + (4m 2 η + λ hη )ξ)(−λ hη v 2 + (m 2 h + λ hη v 2 )ξ) (4m 2 η − m 2 σ − m 2 h )|m 2 h − 4m 2 η + i (Π h (4m 2 η ))| 2 + 2(−4λ 2 hη v 4 + λ hη v 2 ξ(−m 2 σ + 20m 2 η + m 2 h + 8λ hη v 2 ) + ξ 2 (4m 2 η + λ hη v 2 )(m 2 σ − 4m 2 η − m 2 h − 4λ hη v 2 ) (m 2 σ − 4m 2 η ) −1 (m 2 h − 4m 2 η ) −1 |m 2 h − 4m 2 η + i (Π h (4m 2 η ))| 2 ,(D5) FIG. 1 : 1DM relic density at f = 1000 GeV (right) and f = 1500 GeV (left). We contour log 10 (Ωh 2 ) < log 10 (0.12). FIG. 2 : 2Contour of values of f in GeV necessary to reproduce the observed relic density. FIG. 3 : 3Allowed region at 90% confidence level with f = 1000 GeV (left) and f = 1500 GeV (right) from direct searches. We contour the ratio of nucleon-DM cross section over LUX cross section bound. FIG. 4 : 4Excluded parameter region with f = 1000 GeV (left) and f = 1500 GeV (right), using the informations on antiproton fluxes from the Galactic gas. FIG. 5 : 5Differential antiproton spectrum per DM annihilation, computed for mη = 300 GeV, mσ = 1000 GeV and f = 1500 GeV. FIG. 6 6:p/p flux model prediction over the AMS results, computed at f = 1000 GeV (left) and f = 1500 GeV (right). FIG. 7: Antiproton to proton flux ratio. The data points are AMS-02 data, and the red line is the secondary prediction. FIG. 8 : 8Allowed parameter region at f = 1000 GeV (left) and f = 1500 GeV (right), comparing with the Fermi-LAT data at 95% confidence level. Each contour represents a different level for the value of the ratio of σb b v. over the Fermi-LAT bound.FIG. 9: Gamma ray spectrum ratio for various channels. FIG. 10 : 10Thermally averaged cross section inbb: the magenta curve is computed at f = 1000 GeV and the red curve at f = 1500 GeV, fixing mσ = 1000 GeV. The black line is the constraint forbb channel determined by the Fermi-LAT Collaboration, and the blue area is the 2σ uncertainty. FIG. 11 : 11Relic density of DM, as log 10 (Ωηh 2 ), fixing f = 1000 GeV (left) and f = 1500 GeV (right) taking into account all the constraints discussed in the text.FIG. 12: 95% CL lower bound on the symmetry breaking scale f in GeV varying the dilaton mass and the UV beta functions from searches for heavy scalars. FIG. 13 : 13χ 2 value varying f from the discussed Higgs channels. Appendix A: Details of the ModelsFermionic SectorThe Lagrangian of the fermionic sector, including composite resonances, is given by . E Katz, A E Nelson, D G E Walker, arXiv:hep-ph/0504252JHEP. 050874E. Katz, A. E. Nelson and D. G. E. Walker, JHEP 0508 (2005) 074 [arXiv:hep-ph/0504252]. . B Gripaios, A Pomarol, F Riva, J Serra, arXiv:0902.1483JHEP. 090470hep-phB. Gripaios, A. Pomarol, F. Riva and J. Serra, JHEP 0904 (2009) 070 [arXiv:0902.1483 [hep-ph]]. . J Galloway, J A Evans, M A Luty, R A Tacchi, arXiv:1001.1361JHEP. 101086hepphJ. Galloway, J. A. Evans, M. A. Luty and R. A. Tac- chi, JHEP 1010 (2010) 086 [arXiv:1001.1361 [hep- ph]]. . J Barnard, T Gherghetta, T S Ray, arXiv:1311.6562JHEP. 14022hep-phJ. Barnard, T. Gherghetta and T. S. Ray, JHEP 1402 (2014) 002 [arXiv:1311.6562 [hep-ph]]. . G Cacciapaglia, F Sannino, arXiv:1402.0233JHEP. 1404111hep-phG. Cacciapaglia and F. Sannino, JHEP 1404 (2014) 111 [arXiv:1402.0233 [hep-ph]]. . G Cacciapaglia, H Cai, A Deandrea, T Flacke, S J Lee, A Parolini, arXiv:1507.02283JHEP. 1511201hep-phG. Cacciapaglia, H. Cai, A. Deandrea, T. Flacke, S. J. Lee and A. Parolini, JHEP 1511 (2015) 201 [arXiv:1507.02283 [hep-ph]]. . M Frigerio, A Pomarol, F Riva, A Urbano, arXiv:1204.2808JHEP. 120715hep-phM. Frigerio, A. Pomarol, F. Riva and A. Urbano, JHEP 1207 (2012) 015 [arXiv:1204.2808 [hep-ph]]. . D Marzocca, A Urbano, arXiv:1404.7419JHEP. 1407107hep-phD. Marzocca and A. Urbano, JHEP 1407 (2014) 107 [arXiv:1404.7419 [hep-ph]]. . B Bellazzini, C Csaki, J Hubisz, J Serra, J Terning, arXiv:1209.3299Eur. Phys. J. C. 7322333hep-phB. Bellazzini, C. Csaki, J. Hubisz, J. Serra and J. Terning, Eur. Phys. J. C 73 (2013) 2, 2333 [arXiv:1209.3299 [hep-ph]]. . B Bellazzini, C Csaki, J Hubisz, J Serra, J Terning, arXiv:1305.3919Eur. Phys. J. C. 742790hep-thB. Bellazzini, C. Csaki, J. Hubisz, J. Serra and J. Terning, Eur. Phys. J. C 74 (2014) 2790 [arXiv:1305.3919 [hep-th]]. . F Coradeschi, P Lodone, D Pappadopulo, R Rattazzi, L Vitale, arXiv:1306.4601JHEP. 131157hep-thF. Coradeschi, P. Lodone, D. Pappadopulo, R. Rat- tazzi and L. Vitale, JHEP 1311 (2013) 057 [arXiv:1306.4601 [hep-th]]. . Z Chacko, R K Mishra, D Stolarski, arXiv:1304.1795JHEP. 1309121hep-phZ. Chacko, R. K. Mishra and D. Stolarski, JHEP 1309 (2013) 121 [arXiv:1304.1795 [hep-ph]]. . Z Chacko, R K Mishra, arXiv:1209.3022Phys. Rev. D. 8711115006hep-phZ. Chacko and R. K. Mishra, Phys. Rev. D 87 (2013) 11, 115006 [arXiv:1209.3022 [hep-ph]]. . K Blum, M Cliche, C Csaki, S J Lee, arXiv:1410.1873JHEP. 150399hep-phK. Blum, M. Cliche, C. Csaki and S. J. Lee, JHEP 1503 (2015) 099 [arXiv:1410.1873 [hep-ph]]. . A Efrati, E Kuflik, S Nussinov, Y Soreq, T Volansky, arXiv:1410.2225Phys. Rev. D. 91555034hep-phA. Efrati, E. Kuflik, S. Nussinov, Y. Soreq and T. Volansky, Phys. Rev. D 91 (2015) 5, 055034 [arXiv:1410.2225 [hep-ph]]. . S R Coleman, J Wess, B Zumino, Phys. Rev. 1772239S. R. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2239. . C G CallanJr, S R Coleman, J Wess, B Zumino, Phys. Rev. 1772247C. G. Callan, Jr., S. R. Coleman, J. Wess and B. Zu- mino, Phys. Rev. 177 (1969) 2247. . D B Kaplan, Nucl. Phys. B. 365259D. B. Kaplan, Nucl. Phys. B 365 (1991) 259. . G Ferretti, arXiv:1404.7137JHEP. 1406142hep-phG. Ferretti, JHEP 1406 (2014) 142 [arXiv:1404.7137 [hep-ph]]. . O Matsedonskyi, arXiv:1411.4638JHEP. 1502154hep-phO. Matsedonskyi, JHEP 1502 (2015) 154 [arXiv:1411.4638 [hep-ph]]. . G Cacciapaglia, H Cai, T Flacke, S J Lee, A Parolini, H Serôdio, arXiv:1501.03818JHEP. 150685hep-phG. Cacciapaglia, H. Cai, T. Flacke, S. J. Lee, A. Parolini and H. Serôdio, JHEP 1506 (2015) 085 [arXiv:1501.03818 [hep-ph]]. . R Contino, D Marzocca, D Pappadopulo, R Rattazzi, arXiv:1109.1570JHEP. 111081hep-phR. Contino, D. Marzocca, D. Pappadopulo and R. Rattazzi, JHEP 1110 (2011) 081 [arXiv:1109.1570 [hep-ph]]. . D Marzocca, M Serone, J Shu, arXiv:1205.0770JHEP. 120813hep-phD. Marzocca, M. Serone and J. Shu, JHEP 1208 (2012) 013 [arXiv:1205.0770 [hep-ph]]. . P Cox, A D Medina, T S Ray, A Spray, 10.1007/JHEP02(2014)032arXiv:1311.3663JHEP. 140232hep-phP. Cox, A. D. Medina, T. S. Ray and A. Spray, JHEP 1402 (2014) 032 doi:10.1007/JHEP02(2014)032 [arXiv:1311.3663 [hep-ph]]. . C Csaki, J Hubisz, S J Lee, arXiv:0705.3844Phys. Rev. D. 76125015hep-phC. Csaki, J. Hubisz and S. J. Lee, Phys. Rev. D 76 (2007) 125015 [arXiv:0705.3844 [hep-ph]]. . B A Kniehl, M Spira, arXiv:hep-ph/9505225Z. Phys. C. 6977B. A. Kniehl and M. Spira, Z. Phys. C 69 (1995) 77 [arXiv:hep-ph/9505225]. . Y Eshel, S J Lee, G Perez, Y Soreq, 10.1007/JHEP10(2011)015arXiv:1106.6218JHEP. 111015hep-phY. Eshel, S. J. Lee, G. Perez and Y. Soreq, JHEP 1110, 015 (2011) doi:10.1007/JHEP10(2011)015 [arXiv:1106.6218 [hep-ph]]. . M Gillioz, R Grober, C Grojean, M Muhlleitner, E Salvioni, arXiv:1206.7120JHEP. 12104hep-phM. Gillioz, R. Grober, C. Grojean, M. Muh- lleitner and E. Salvioni, JHEP 1210 (2012) 004 [arXiv:1206.7120 [hep-ph]]. . A Arbey, G Cacciapaglia, H Cai, A Deandrea, S Le Corre, F Sannino, arXiv:1502.04718hep-phA. Arbey, G. Cacciapaglia, H. Cai, A. Deandrea, S. Le Corre and F. Sannino, arXiv:1502.04718 [hep-ph]. . H Georgi, arXiv:hep-ph/9207278Phys. Lett. B. 298187H. Georgi, Phys. Lett. B 298 (1993) 187 [arXiv:hep- ph/9207278]. . Atlas The, Collaborations, ATLAS- CONF-2015-044The ATLAS and CMS Collaborations, ATLAS- CONF-2015-044. ATLAS-CONF-2013-067The ATLAS collaboration. The ATLAS collaboration [ATLAS Collaboration], ATLAS-CONF-2013-067. . S Chatrchyan, CMS CollaborationarXiv:1304.0213Eur. Phys. J. C. 732469hep-exS. Chatrchyan et al. [CMS Collaboration], Eur. Phys. J. C 73 (2013) 2469 [arXiv:1304.0213 [hep-ex]]. . V Khachatryan, CMS CollaborationarXiv:1504.00936hep-exV. Khachatryan et al. [CMS Collaboration], arXiv:1504.00936 [hep-ex]. . W L Guo, Y L Wu, arXiv:1006.2518JHEP. 101083hep-phW. L. Guo and Y. L. Wu, JHEP 1010 (2010) 083 [arXiv:1006.2518 [hep-ph]]. . J M Cline, K Kainulainen, P Scott, C Weniger, arXiv:1306.4710Phys. Rev. D. 8839906Phys. Rev. D. hep-phJ. M. Cline, K. Kainulainen, P. Scott and C. Weniger, Phys. Rev. D 88 (2013) 055025 [Phys. Rev. D 92 (2015) 3, 039906] [arXiv:1306.4710 [hep-ph]]. . B W Lee, S Weinberg, Phys. Rev. Lett. 39165B. W. Lee and S. Weinberg, Phys. Rev. Lett. 39 (1977) 165. . P A R Ade, Planck CollaborationarXiv:1303.5076Astron. Astrophys. 57116astroph.COP. A. R. Ade et al. [Planck Collaboration], Astron. Astrophys. 571 (2014) A16 [arXiv:1303.5076 [astro- ph.CO]]. . D S Akerib, LUX CollaborationarXiv:1310.8214Phys. Rev. Lett. 11291303astroph.COD. S. Akerib et al. [LUX Collaboration], Phys. Rev. Lett. 112 (2014) 091303 [arXiv:1310.8214 [astro- ph.CO]]. . D S Akerib, LUX CollaborationarXiv:1512.03506[astro-ph.COD. S. Akerib et al. [LUX Collaboration], arXiv:1512.03506 [astro-ph.CO]. . F Bishara, J Brod, P Uttayarat, J Zupan, arXiv:1504.04022hep-phF. Bishara, J. Brod, P. Uttayarat and J. Zupan, arXiv:1504.04022 [hep-ph]. . P Junnarkar, A Walker-Loud, arXiv:1301.1114Phys. Rev. D. 87114510hep-latP. Junnarkar and A. Walker-Loud, Phys. Rev. D 87 (2013) 114510 [arXiv:1301.1114 [hep-lat]]. . R Iengo, arXiv:0902.0688JHEP. 090524hep-phR. Iengo, JHEP 0905 (2009) 024 [arXiv:0902.0688 [hep-ph]]. . S Cassel, arXiv:0903.5307J. Phys. G. 37105009hep-phS. Cassel, J. Phys. G 37 (2010) 105009 [arXiv:0903.5307 [hep-ph]]. . J L Feng, M Kaplinghat, H B Yu, arXiv:1005.4678Phys. Rev. D. 8283525hep-phJ. L. Feng, M. Kaplinghat and H. B. Yu, Phys. Rev. D 82 (2010) 083525 [arXiv:1005.4678 [hep-ph]]. . T R Slatyer, arXiv:0910.5713JCAP. 100228hep-phT. R. Slatyer, JCAP 1002 (2010) 028 [arXiv:0910.5713 [hep-ph]]. . M Aguilar, AMS CollaborationPhys. Rev. Lett. 114171103M. Aguilar et al. [AMS Collaboration], Phys. Rev. Lett. 114 (2015) 17, 171103. . G Giesen, M Boudaud, Y Gnolini, V Poulin, M Cirelli, P Salati, P D Serpico, arXiv:1504.04276JCAP. 150923astro-ph.HEG. Giesen, M. Boudaud, Y. Gnolini, V. Poulin, M. Cirelli, P. Salati and P. D. Serpico, JCAP 1509 (2015) 09, 023 [arXiv:1504.04276 [astro-ph.HE]]. . K Agashe, K Blum, S J Lee, G Perez, arXiv:0912.3070Phys. Rev. D. 8175012hep-phK. Agashe, K. Blum, S. J. Lee and G. Perez, Phys. Rev. D 81 (2010) 075012 [arXiv:0912.3070 [hep-ph]]. . B Katz, K Blum, E Waxman, arXiv:0907.1686Mon. Not. Roy. Astron. Soc. 4051458astroph.HEB. Katz, K. Blum and E. Waxman, Mon. Not. Roy. Astron. Soc. 405 (2010) 1458 [arXiv:0907.1686 [astro- ph.HE]]. . P Ciafaloni, D Comelli, A Riotto, F Sala, A Strumia, A Urbano, arXiv:1009.0224JCAP. 110319hep-phP. Ciafaloni, D. Comelli, A. Riotto, F. Sala, A. Strumia and A. Urbano, JCAP 1103 (2011) 019 [arXiv:1009.0224 [hep-ph]]. . M Cirelli, arXiv:1012.4515JCAP. 110351JCAP 1210 (2012) E01. hep-phM. Cirelli et al., JCAP 1103 (2011) 051 [JCAP 1210 (2012) E01] [arXiv:1012.4515 [hep-ph]]. . G Elor, N L Rodd, T R Slatyer, arXiv:1503.01773Phys. Rev. D. 91103531hep-phG. Elor, N. L. Rodd and T. R. Slatyer, Phys. Rev. D 91 (2015) 103531 [arXiv:1503.01773 [hep-ph]]. AMS DAYS AT CERN -The Future of Cosmic Ray Physics and Latest Results. A Kounine, A. Kounine, "AMS DAYS AT CERN -The Future of Cosmic Ray Physics and Latest Results . N Arkani-Hamed, D P Finkbeiner, T R Slatyer, N Weiner, arXiv:0810.0713Phys. Rev. D. 7915014hep-phN. Arkani-Hamed, D. P. Finkbeiner, T. R. Slatyer and N. Weiner, Phys. Rev. D 79 (2009) 015014 [arXiv:0810.0713 [hep-ph]]. . A W Mcconnachie, arXiv:1204.1562Astron. J. 1444astro-ph.COA. W. McConnachie, Astron. J. 144 (2012) 4 [arXiv:1204.1562 [astro-ph.CO]]. . M Ackermann, Fermi-LAT CollaborationarXiv:1503.02641[astro-ph.HEM. Ackermann et al. [Fermi-LAT Collaboration], arXiv:1503.02641 [astro-ph.HE]. . A Abramowski, HESS CollaborationarXiv:1502.03244Phys. Rev. Lett. 11481301astro-ph.HEA. Abramowski et al. [HESS Collaboration], Phys. Rev. Lett. 114 (2015) 8, 081301 [arXiv:1502.03244 [astro-ph.HE]]. . J F Navarro, C S Frenk, S D M White, arXiv:astro-ph/9508025Astrophys. J. 462563J. F. Navarro, C. S. Frenk and S. D. M. White, As- trophys. J. 462 (1996) 563 [arXiv:astro-ph/9508025]. . M Montull, F Riva, E Salvioni, R Torre, arXiv:1308.0559Phys. Rev. D. 8895006hep-phM. Montull, F. Riva, E. Salvioni and R. Torre, Phys. Rev. D 88 (2013) 095006 [arXiv:1308.0559 [hep-ph]]. . . S Electroweak, [ Lep, Electroweak, arXiv:hep-ex/0312023Working Group and SLD Electroweak Group and SLD Heavy Flavor Group Collaborationst. S. Electroweak [LEP and ALEPH and DELPHI and L3 and OPAL and LEP Electroweak Working Group and SLD Electroweak Group and SLD Heavy Flavor Group Collaborations], arXiv:hep-ex/0312023. . A Falkowski, F Riva, A Urbano, arXiv:1303.1812JHEP. 1311111hep-phA. Falkowski, F. Riva and A. Urbano, JHEP 1311 (2013) 111 [arXiv:1303.1812 [hep-ph]]. . A Y Korchin, V A Kovalchuk, arXiv:1303.0365Phys. Rev. D. 88336009hep-phA. Y. Korchin and V. A. Kovalchuk, Phys. Rev. D 88 (2013) 3, 036009 [arXiv:1303.0365 [hep-ph]]. The couplings of η to SM fermions other than top depend on the formal embedding of the SM quarks into SO(6) representations. We fix for convenience the same couplings for all the quarks. The couplings of η to SM fermions other than top depend on the formal embedding of the SM quarks into SO(6) representations. We fix for convenience the same couplings for all the quarks. The couplings of η and h to SM fermions other than top depend on the formal embedding of the SM quarks into SO(6) representations. We fix for convenience the same couplings for all the quarks. Small changes should not make a difference our analysisThe couplings of η and h to SM fermions other than top depend on the formal embedding of the SM quarks into SO(6) representations. We fix for convenience the same couplings for all the quarks. Small changes should not make a difference our analysis.
{'fraction_non_alphanumeric': 0.0825829972932663, 'fraction_numerical': 0.07615680642250824, 'mean_word_length': 3.1331811263318112, 'pattern_counts': {'":': 0, '<': 7, '<?xml version=': 0, '>': 12, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 3, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 96, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'We study under which conditions a scalar particle is a viable WIMP Dark Matter candidate with Higgs and dilaton interactions. The theory is a composite Higgs model with top partial compositeness where both the Higgs and the Dark Matter candidate arise as pseudo Goldstone boson of the coset SO(6)/SO(5) from a new physics sector. We highlight the role of the dilaton in direct and indirect searches. We find that a Dark Matter particle with a mass around 200-400 GeV and a relatively light dilaton are a fair prediction of the model.', 'arxivid': '1602.05590', 'author': ['Manki Kim ', 'Seung J Lee ', 'Alberto Parolini ', '\nDepartment of Physics\nDepartment of Physics\nKorea Advanced Institute of Science and Technology\n335 Gwahak-ro, Yuseong-gu305-701DaejeonKorea\n', '\nSchool of Physics\nInstitute for Advanced Study\nKorea University\n136-713, 130-722Seoul, SeoulKorea, Korea, Korea\n', '\nQuantum Universe Center\nInstitute for Advanced Study\n130-722SeoulKorea, Korea\n'], 'authoraffiliation': ['Department of Physics\nDepartment of Physics\nKorea Advanced Institute of Science and Technology\n335 Gwahak-ro, Yuseong-gu305-701DaejeonKorea', 'School of Physics\nInstitute for Advanced Study\nKorea University\n136-713, 130-722Seoul, SeoulKorea, Korea, Korea', 'Quantum Universe Center\nInstitute for Advanced Study\n130-722SeoulKorea, Korea'], 'corpusid': 119283708, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 23595, 'n_tokens_neox': 18645, 'n_words': 10965, 'pdfsha': '1631664cfb422490bfa914b5ae309debc467dafb', 'pdfurls': ['https://arxiv.org/pdf/1602.05590v1.pdf'], 'title': ['WIMP Dark Matter in Composite Higgs Models and the Dilaton Portal', 'WIMP Dark Matter in Composite Higgs Models and the Dilaton Portal'], 'venue': []}
arxiv
Asymptotic variance of random symmetric digital search trees 4 Mar 2010 March 4, 2010 Hsien-Kuei Hwang Institute of Statistical Science Academia Sinica Taipei 115 Department of Applied Mathematics National Chiao Institute of Statistical Science Academia Sinica Tung University Hsinchu 300115TaipeiTaiwan, Taiwan, Taiwan Michael Fuchs Institute of Statistical Science Academia Sinica Taipei 115 Department of Applied Mathematics National Chiao Institute of Statistical Science Academia Sinica Tung University Hsinchu 300115TaipeiTaiwan, Taiwan, Taiwan Vytas Zacharovas Institute of Statistical Science Academia Sinica Taipei 115 Department of Applied Mathematics National Chiao Institute of Statistical Science Academia Sinica Tung University Hsinchu 300115TaipeiTaiwan, Taiwan, Taiwan Asymptotic variance of random symmetric digital search trees 4 Mar 2010 March 4, 2010arXiv:1001.0095v2 [math.CO] Dedicated to the 60th birthday of Philippe FlajoletDigital search treesPoisson generating functionsPoissonizationLaplace transformMellin transformsaddle-point methodColless indexweighted path-length Asymptotics of the variances of many cost measures in random digital search trees are often notoriously messy and involved to obtain. A new approach is proposed to facilitate such an analysis for several shape parameters on random symmetric digital search trees. Our approach starts from a more careful normalization at the level of Poisson generating functions, which then provides an asymptotically equivalent approximation to the variance in question. Several new ingredients are also introduced such as a combined use of the Laplace and Mellin transforms and a simple, mechanical technique for justifying the analytic de-Poissonization procedures involved. The methodology we develop can be easily adapted to many other problems with an underlying binomial distribution. In particular, the less expected and somewhat surprising n(log n) 2 -variance for certain notions of total path-length is also clarified. Introduction The variance of a distribution provides an important measure of dispersion of the distribution and plays a crucial and, in many cases, a determinantal rôle in the limit law 1 . Thus finding more effective means of computing the variance is often of considerable significance in theory and in practice. However, the calculation of the variance can be computationally or intrinsically difficult, either because of the messy procedures or cancellations involved, or because the dependence structure is too strong or simply because no simple manageable forms or reductions are available. We are concerned in this paper with random digital trees for which asymptotic approximations to the variance are often marked by heavy calculations and long, messy expressions. This paper proposes a general approach to simplify not only the analysis but also the resulting expressions, providing new insight into the methodology; furthermore, it is applicable to many other concrete situations and leads readily to discover several new results, shedding new light on the stochastic behaviors of the random splitting structures. A binomial splitting process. The analysis of many splitting procedures in computer algorithms leads naturally to a structural decomposition (in terms of the cardinalities) of the form structure of size n substructure of size B n substructure of sizeB n Here B n ≈ Binomial and B n +B n ≈ n. where B n is essentially a binomial distribution (up to truncation or small perturbations) and the sum of B n +B n is essentially n. Concrete examples in the literature include (see the books [15,28,44,50,62] and below for more detailed references) • tries, contention-resolution tree algorithms, initialization problem in distributed networks, and radix sort: B n = Binomial(n; p) andB n = n−B n , namely, P(B n = k) = n k p k q n−k (here and throughout this paper, q := 1 − p); • bucket digital search trees (DSTs), directed diffusion-limited aggregation on Bethe lattice, and Eden model: B n = Binomial(n − b; p) andB n = n − b − B n ; • Patricia tries and suffix trees: P(B n = k) = n k p k q n−k /(1 − p n − q n ) andB n = n − B n . Yet another general form arises in the analysis of multi-access broadcast channel where B n = Binomial(n; p) + Poisson(λ), B n = n − Binomial(n; p) + Poisson(λ), see [19,33]. For some other variants, see [2,6,25]. One reason of such a ubiquity of binomial distribution is simply due to the binary outcomes (either zero or one, either on or off, either positive or negative, etc.) of many practical situations, resulting in the natural adaptation of the Bernoulli distribution in the modeling. Poisson generating function and the Poisson heuristic. A very useful, standard tool for the analysis of these binomial splitting processes is the Poisson generating functioñ f (z) = e −z k≥0 a k k! z k , where {a k } is a given sequence, one distinctive feature being the Poisson heuristic, which predicts that If a n is smooth enough, then a n ∼f (n). In more precise words, if the sequence {a k } does not grow too fast (usually at most of polynomial growth) or does not fluctuate too violently, then a n is well approximated byf (n) for large n. For example, if f (z) = z m , m = 0, 1, . . . , then a n ∼ n m ; indeed, in such a simple case, a n = n(n − 1) · · · (n − m + 1). Note that the Poisson heuristic is itself a Tauberian theorem for the Borel mean in essence; an Abelian type theorem can be found in Ramanujan's Notebooks (see [3, p. 58]). From an elementary viewpoint, such a heuristic is based on the local limit theorem of the Poisson distribution (or essentially Stirling's formula for n!) n k k! e −n ∼ e −x 2 /2 √ 2πn 1 + x 3 − 3x 6 √ n + · · · (k = n + x √ n), whenever x = o(n 1/6 ). Since a n is smooth, we then expect that f (n) ≈ k=n+x √ n x=O(n ε ) a k e −x 2 /2 √ 2πn ≈ a n ∞ −∞ e −x 2 /2 √ 2π dx = a n . On the other hand, by Cauchy's integral representation, we also have a n = n! 2πi |z|=n z −n−1 e zf (z) dz ≈f (n) n! 2πi |z|=n z −n−1 e z dz =f (n), since the saddle-point z = n of the factor z −n e z is unaltered by the comparatively more smooth functioñ f (z). The Poisson-Charlier expansion. The latter analytic viewpoint provides an additional advantage of obtaining an expansion by using the Taylor expansion off at z = n, yielding a n = j≥0f (j) (n) j! τ j (n),(1) where τ j (n) := n![z n ](z − n) j e z = 0≤ℓ≤j j ℓ (−1) j−ℓ n!n j−ℓ (n − ℓ)! (j = 0, 1, . . . ), and [z n ]φ(z) denotes the coefficient of z n in the Taylor expansion of φ(z). We call such an expansion the Poisson-Charlier expansion since the τ j 's are essentially the Charlier polynomials C j (λ, n) defined by C j (λ, n) := λ −n n![z n ](z − 1) j e λz , so that τ j (n) = n j C j (n, n). For other terms used in the literature, see [28,29]. The first few terms of τ j (n) are given as follows. τ 0 (n) τ 1 (n) τ 2 (n) τ 3 (n) τ 4 (n) τ 5 (n) τ 6 (n) 1 0 −n 2n 3n(n − 2) −4n(5n − 6) −5n(3n 2 − 26n + 24) It is easily seen that τ j (n) is a polynomial in n of degree ⌊j/2⌋. The meaning of such a Poisson-Charlier expansion becomes readily clear by the following simple but extremely useful lemma. Lemma 1.1. Letf (z) := e −z k≥0 a k z k /k!. Iff is an entire function, then the Poisson-Charlier expansion (1) provides an identity for a n . Proof. Sincef is entire, we have n≥0 a n n! z n = e zf (z) = e z j≥0f (j) (n) j! (z − n) j , and the lemma follows by absolute convergence. Two specific examples are worthy of mention here as they speak volume of the difference between identity and asymptotic equivalence. Take first a n = (−1) n . Then the Poisson heuristic fails since (−1) n ∼ e −2n , but, by Lemma 1.1, we have the identity (−1) n = e −2n j≥0 (−2) j j! τ j (n). See Figure 1 for a plot of the convergence of the series to (−1) n . Now if a n = 2 n , then 2 n ∼ e n , but we still have 2 n = e n j≥0 τ j (n) j! . So when is the Poisson-Charlier expansion also an asymptotic expansion for a n , in the sense that dropping all terms with j ≥ 2ℓ introduces an error of orderf (2ℓ) n ℓ (which in typical cases is of orderf (n)n −ℓ )? Many sufficient conditions are thoroughly discussed in [36], although the terms in their expansions are expressed differently; see also [62]. Poissonized mean and variance. The majority of random variables analyzed in the algorithmic literature are at most of polynomial or sub-exponential (such as e c(log n) 2 or e cn 1/2 ) orders, and are smooth enough. Thus the Poisson generating functions of the moments are often entire functions. The use of the Poisson-Charlier expansion is then straightforward, and in many situations it remains to justify the asymptotic nature of the expansion. For convenience of discussion, letf m (z) denote the Poisson generating function of the m-th moment of the random variable in question, say X n . Then by Lemma 1.1, we have the identity E(X n ) = j≥0f (j) 1 (n) j! τ j (n), and for the second moment E(X 2 n ) = j≥0f (j) 2 (n) j! τ j (n),(2) provided only that the two Poisson generating functionsf 1 andf 2 are entire functions. These identities suggest that a good approximation to the variance of X n be given by V(X n ) = E(X 2 n ) − (E(X n )) 2 ≈f 2 (n) −f 1 (n) 2 , which holds true for many cost measures, where we can indeed replace the imprecise, approximately equal symbol "≈" by the more precise, asymptotically equivalent symbol "∼". However, for a large class of problems for which the variance is essentially linear, meaning roughly that lim n→∞ log V(X n ) log n = 1,(3) the Poissonized variancef 2 (n) −f 1 (n) 2 is not asymptotically equivalent to the variance. This is the case for the total cost of constructing random digital search trees, for example. One technical reason is that there are additional cancellations produced by dominant terms. The next question is then: can we find a better normalized function so that the variance is asymptotically equivalent to its value at n? Poissonized variance with correction. The crucial step of our approach that is needed when the variance is essentially linear is to considerṼ (z) :=f 2 (z) −f 1 (z) 2 − zf ′ 1 (z) 2 ,(4) and it then turns out that V(X n ) =Ṽ (n) + O((log n) c ), in all cases we consider for some c ≥ 0. The asymptotics of the variance is then reduced to that ofṼ (z) for large z, which satisfies, up to non-homogeneous terms, the same type of equation asf 1 (z). Thus the same tools used for analyzing the mean can be applied toṼ (z). To see how the last correction term zf ′ 1 (z) 2 appears, we writeD(z) :=f 2 (z) −f 1 (z) 2 , so thatf 2 (z) = D(z) +f 1 (z) 2 , and we obtain, by substituting this into (2), V(X n ) = E(X 2 n ) − (E(X n )) 2 = j≥0f 2 (j) (n) j! τ j (n) − j≥0f 1 (j) (n) j! τ j (n) 2 =D(n) − nf ′ 1 (n) 2 − n 2D ′′ (n) + smaller-order terms. Now takef 1 (n) ≍ n log n. Then the first term followingD(n) is generally not smaller thanD(n) because nf ′ 1 (n) 2 ≍ n(log n) 2 , whileD(n) ≍ n(log n) 2 , at least for the examples we discuss in this paper. Note that the variance is in such a case either of order n log n or of order n. Thus to get an asymptotically equivalent approximation to the variance, we need at least an additional correction term, which is exactly nf ′ 1 (n) 2 . The correction term nf ′ 1 (n) 2 already appeared in many early papers by Jacquet and Régnier (see [34]). A viewpoint from the asymptotics of the characteristic function. Most binomial recurrences of the form X n d = X Bn + X * Bn + T n ,(5) as arising from the binomial splitting processes discussed above are asymptotically normally distributed, a property partly ascribable to the highly regular behavior of the binomial distribution. Here the (X * n ) are independent copies of the (X n ) and the random or deterministic non-homogeneous part T n is often called the "toll-function," measuring the cost used to "conquer" the two subproblems. Such recurrences have been extensively studied in numerous papers; see [36,52,58,59] and the references therein. The correction term we introduced in (4) for Poissonized variance also appears naturally in the following heuristic, formal analysis, which can be justified when more properties are available. By definition and formal expansion e −z n≥0 E e Xniθ z n n! = m≥0f m (z) m! (iθ) m = exp f 1 (z)iθ −D (z) 2 θ 2 + · · · , whereD(z) :=f 2 (z) −f 1 (z) 2 , we have E e (Xn−f 1 (n))iθ ≈ n! 2πi |z|=n z −n−1 exp z + f 1 (z) −f 1 (n) iθ −D (z) 2 θ 2 + · · · dz. Observe that with z = ne it , we have the local expansion ne it − nit + f 1 (ne it ) −f 1 (n) iθ −D (ne it ) 2 θ 2 = n − nt 2 2 − nf ′ 1 (n)tθ −D (n) 2 θ 2 + · · · , for small t. It follows that E e (Xn−f 1 (n))iθ ≈ n!n −n e n 2π exp −D (n) 2 θ 2 ε −ε exp − nt 2 2 − nf ′ 1 (n)tθ dt ∼ exp − θ 2 2 D (n) − nf ′ 1 (n) 2 , by extending the integral to ±∞ and by completing the square. This again shows that nf ′ 1 (n) 2 is the right correction term for the variance. For more precise analysis of this type, see [36]. A comparison of different approaches to the asymptotic variance. What are the advantages of the Poissonized variance with correction? In the literature, a few different approaches have been adopted for computing the asymptotics of the variance of the binomial splitting processes. • Second moment approach: this is the most straightforward means and consists of first deriving asymptotic expansions of sufficient length for the expected value and for the second moment, then considering the difference E(X 2 n ) − (E(X n )) 2 , and identifying the lead terms after cancellations of dominant terms in both expansions. This approach is often computationally heavy as many terms have to be cancelled; additional complication arises from fluctuating terms, rendering the resulting expressions more messy. See below for more references. • Poissonized variance: the asymptotics of the variance is carried out through that ofD(n) =f 2 (n) − f 1 (n) 2 . The difference between this approach and the previous one is that no asymptotics off 2 (n) is derived or needed, and one always focuses directly on considering the equation (functional or differential) satisfied byD(z). As we discussed above, this does not give in many cases an asymptotically equivalent estimate for the variance, because additional cancellations have to be further taken into account; see for instance [34,35,36]. • Characteristic function approach: similar to the formal calculations we carried out above, this approach tries to derive a more precise asymptotic approximation to the characteristic function using, say complex-analytic tools, and then to identify the right normalizing term as the variance; see the survey [36] and the papers cited there. • Schachinger's differencing approach: a delicate, mostly elementary approach based on the recurrence satisfied by the variance was proposed in [58] (see also [59]). His approach is applicable to very general "toll-functions" T n in (5) but at the price of less precise expressions. The approach we use is similar to the Poissonized variance one but the difference is that the passage throughD(z) is completely avoided and we focus directly on equations satisfied byṼ (z) (defined in (4)). In contrast to Schachinger's approach, our approach, after starting from definingṼ (z), is mostly analytic. It yields then more precise expansions, but more properties of T n have to be known. The contrast here between elementary and analytic approaches is thus typical; see, for example, [7,8]. See also Appendix for a brief sketch of the asymptotic linearity of the variance by elementary arguments. Additional advantages that our approach offer include comparatively simpler forms for the resulting expressions, including Fourier series expansions, and general applicability (coupling with the introduction of several new techniques). Organization of this paper. This paper is organized as follows. We start with the variance of the total path-length of random digital search trees in the next section, which was our motivating example. We then extend the consideration to bucket DSTs for which two different notions of total path-length are distinguished, which result in very different asymptotic behaviors. The application of our approach to several other shape parameters are discussed in Section 4. Table 1 summarizes the diverse behaviors exhibited by the means and the variances of the shape parameters we consider in this paper. Shape parameters mean variance Internal PL n log n n Key-wise PL * n log n n Node-wise PL * n log n n(log n) 2 Peripheral PL n n #(leaves) n n Differential PL n n log n Weighted PL n(log n) m+1 n Applications of the approach we develop here to other classes of trees and structures, including tries, Patricia tries, bucket sort, contention resolution algorithms, etc., will be investigated in a future paper. Digital Search Trees We start in this section with a brief description of digital search trees (DSTs), list major shape parameters studied in the literature, and then focus on the total path-length. The approach we develop is also very useful for other linear shape measures, which is discussed in a more systematic form in the following sections. DSTs DSTs were first introduced by Coffman and Eve in [9] in the early 1970's under the name of sequence hash trees. They can be regarded as the bit-version of binary search trees (thus the name); see [44, p. 496 et seq.]. Given a sequence of binary strings, we place the first in the root node; those starting with "0" ("1") are directed to the left (right) subtree of the root, and are constructed recursively by the same procedure but with the removal of their first bits when comparisons are made. See Figure 2 for an illustration. While the practical usefulness of digital search trees is limited, they represent one of the simplest, fundamental, prototype models for divide-and-conquer algorithms using coin-tossing or similar random devices. Of notable interest is its close connection to the analysis of Lempel-Ziv compression scheme that has found widespread incorporation into numerous softwares. Furthermore, the mathematical analysis is often challenging and leads to intriguing phenomena. Also the splitting mechanism of DSTs appeared naturally in a few problems in other areas; some of these are mentioned in the last section. Random digital search trees. The simplest random model we discuss in this paper is the independent, Bernoulli model. In this model, we are given a sequence of n independent and identically distributed random variables, each comprising an infinity sequence of Bernoulli random variables with mean p, 0 < p < 1. The DST constructed from the given random sequence of binary strings is called a random DST. If p = 1/2, the DST is said to be symmetric; otherwise, it is asymmetric. We focus on symmetric DSTs in this paper for simplicity; extension to asymmetric DSTs is possible but much harder. Stochastic properties of many shape characteristics of random DSTs are known. Almost all of them fall into one of the two categories, according to their growth order being logarithmic or essentially linear (in the sense of (3)), which we simply refer to as "log shape measures" and "linear shape measures". Log shape measures. The two major parameters studied in this category are depth, which is the distance of the root to a randomly chosen node in the tree (each with the same probability), and height, which counts the number of nodes from the root to one of the longest paths. Both are of logarithmic order in mean. Depth provides a good indication of the typical cost needed when inserting a new key in the tree, while height measures the worst possible cost that may be needed. Depth was first studied in [45] in connection with the profile, which is the sequence of numbers, each enumerating the number of nodes with the same distance to the root. For example, the tree has the profile {1, 2, 3, 2, 3}. For other papers on the depth of random DSTs, see [11,12,13,37,38,39,44,46,47,50,55,60,61]. The height of random DSTs is addressed in [13,14,43,50,55]. Linear shape measures. These include the total internal path-length, which sums the distance between the root and every node, and the occurrences of a given pattern (leaves or nodes satisfying certain properties); see [24,26,30,31,35,40,42,44]. The profile contains generally much more information than most other shape measures, and it can to some extent be regarded as a good bridge connecting log and linear measures; see [15,17,45,46] for known properties concerning expected profile of random DSTs. Nodes of random DSTs with p = 1/2 are distributed in an extremely regular way, as shown in Figures 3 and 4. Known and new results for the total internal path-length Throughout this section, we focus on X n , the total path length of a random digital search tree built from n binary strings. By definition and by our random assumption, X n can be computed recursively by X n+1 d = X Bn + X * n−Bn + n, (n ≥ 0)(6) with the initial condition X 0 = 0, since removing the root results in a decrease of n for the total path length (each internal node below the root contributes 1). Here B n ∼ Binomial(n; 1/2), X n d = X * n , and X n , X * n , B n are independent. Known results. It is known that (see [26,30,57]) E(X n ) = (n + 1) log 2 n + n γ − 1 log 2 + 1 2 − c 1 + ̟ 1 (log 2 n) + γ − 1/2 log 2 + 5 2 − c 1 + ̟ 2 (log 2 n) + O n −1 log n ,(7) where γ denotes Euler's constant, c 1 := k≥1 (2 k − 1) −1 , and ̟ 1 (t), ̟ 2 (t) are 1-periodic functions with zero mean whose Fourier expansions are given by (χ k := 2kπi/L, L := log 2) ̟ 1 (t) = 1 L k =0 Γ (−1 − χ k ) e 2kπit ,(8)̟ 2 (t) = − 1 L k =0 1 − χ k 2 Γ(−χ k )e 2kπit , respectively. Here Γ denotes the Gamma function. Thus we see roughly that random digital search trees under the unbiased Bernoulli model are highly balanced in shape. An important feature of the periodic functions is that they are marked by very small amplitudes of fluctuation: |̟ 1 (t)| ≤ 3.4 × 10 −8 and |̟ 2 (t)| ≤ 3.4 × 10 −6 . Such a quasi-flat (or smooth) behavior may in practice be very likely to lead to wrong conclusions as they are hardly visible from simulations of moderate sample sizes. V(X n )/n E(X n )/(n + 1) − log 2 n Figure 5: A plot of E(X n )/(n + 1) − log 2 n in log-scale (the decreasing curve using the y-axis on the right-hand side), and that of V(X n )/n in log-scale (the increasing curve using the y-axis on the left-hand side). Let Q k := 1≤j≤k 1 − 1 2 j , and Q(z) := j≥1 1 − z 2 j .(9) In particular, Q(1) = Q ∞ . The variance was computed in [42] by a direct second-moment approach and the result is V(X n ) = n(C kps + ̟ kps (log 2 n)) + O(log 2 n), where ̟ kps (t) is again a 1-periodic, zero-mean function and the mean value C kps is given by (L := log 2) C kps = − 28 3L − 39 4 + π 2 2L 2 + 2 L 2 − 2Q ∞ L − 2 ℓ≥1 ℓ2 ℓ (2 ℓ − 1) 2 + 2 L ℓ≥1 1 2 ℓ − 1 − 2 L ℓ≥3 (−1) ℓ+1 (ℓ − 5) (ℓ + 1)ℓ(ℓ − 1)(2 ℓ − 1) + 2 L ℓ≥1 (−1) ℓ 2 −( ℓ+1 2 ) L(1 − 2 −ℓ+1 )/2 − 1 1 − 2 −ℓ − r≥2 (−1) r+1 r(r − 1)(2 r+ℓ − 1) + ℓ≥3 2≤r<ℓ ℓ + 1 r Q r−2 Q ℓ−r−1 2 ℓ Q ℓ j≥ℓ+1 1 2 j − 1 − 2 ̟ [1] 1 ̟ [2] 2 0 − (̟ [1] 1 ) 2 0 + 2 ℓ≥2 1 2 ℓ Q ℓ r≥0 (−1) r 2 −( r+1 2 ) Q r Q r+ℓ−2 × × − j≥1 1 2 j+r+ℓ+2 − 1 2 ℓ − ℓ − 2 + 2≤i<ℓ ℓ + 1 i 1 2 r+i−1 − 1 + 1 (1 − 2 −ℓ−r ) 2 + ℓ + 1 (1 − 2 1−ℓ−r ) 2 − 1 L(1 − 2 1−ℓ−r ) − 2≤j≤ℓ+1 ℓ + 1 j 1 2 r+j−1 − 1 + 1 L 1≤j≤ℓ+1 ℓ + 1 j 1 2 r+j − 1 + 1 L 0≤j≤ℓ+1 ℓ + 1 j i≥1 (−1) i (i + 1)(2 r+j+i − 1) . Here [̟ 1 ̟ 2 ] 0 denotes the mean value of the function ̟ 1 (t)̟ 0 (t) over the unit interval. The long expression obviously shows the complexity of the asymptotic problem. We show that this long expression can be largely simplified. Before stating our result, we mention that the asymptotic normality of X n (in the sense of convergence in distribution) was first proved in [35] by a complex-analytic approach; for other approaches, see [59] (martingale difference), [31] (method of moments), [52] (contraction method). A new asymptotic approximation to V(X n ). Define G 2 (ω) = Q ∞ j,h,ℓ≥0 (−1) j 2 −( j+1 2 )+j(ω−2) Q j Q h Q ℓ 2 h+ℓ ϕ(ω; 2 −j−h + 2 −j−ℓ ),(10) where for 0 < ℜ(ω) < 3 and x > 0 ϕ(ω; x) := ∞ 0 s ω−1 (s + 1)(s + x) 2 ds, which, by the relation ∞ 0 s ω−1 s + 1 ds = π sin(πω) = Γ(ω)Γ(1 − ω) (0 < ℜ(ω) < 1), can be represented as ϕ(ω; x) =        π (1 + x ω−2 ((ω − 2)ξ + 1 − ω) (x − 1) 2 sin(πω) , if x = 1; π(ω − 1)(ω − 2) 2 sin(πω) , if x = 1. The last expression provides indeed a meromorphic continuation of ϕ(ω; x) into the whole complex ωplane whenever x > 0. In particular, ϕ(2; x) :=      x − log x − 1 (x − 1) 2 , if x = 1; 1 2 , if x = 1. Theorem 2.1. The variance of the total path-length of random DSTs of n nodes satisfies V(X n ) = n(C kps + ̟ kps (log 2 n)) + O(1),(11) where C kps = G 2 (2) log 2 = Q ∞ log 2 j,h,ℓ≥0 (−1) j 2 −( j+1 2 ) Q j Q h Q ℓ 2 h+ℓ ϕ(2; 2 −j−h + 2 −j−ℓ ), and ̟ kps has the Fourier series expansion ̟ kps (t) = 1 log 2 k∈Z\{0} G 2 (2 + χ k ) Γ(2 + χ k ) e 2kπit , which is absolutely convergent. One can derive more precise asymptotic expansions for V(X n ) by the same approach we use. We content ourselves with (11) for convenience of presentation. Note that G 2 (2 + χ k ) Γ(2 + χ k ) = Γ(−1 − χ k )Q ∞ j,h,ℓ≥0 (−1) j 2 −( j+1 2 ) Q j Q h Q ℓ 2 h+ℓ λ k (2 −j−h + 2 −j−ℓ ), where λ k (t) :=      1 − t χ k (1 + χ k (1 − t)) (1 − t) 2 , if t = 1; χ k (χ k − 1) 2 , if t = 1. Thus the Fourier series is absolutely convergent by the order estimate (see [18]) |Γ(c + it)| = O |t| c−1/2 e −π|t|/2 (|t| → ∞).(12) Numerically, C kps ≈ 0.26600 36454 05936 . . . , in accordance with that given in [42]. Also |̟ kps (t)| ≤ 1.9 × 10 −5 . Sketch of our approach. Following the discussions in Introduction, we first prove that the Poisson-Charlier expansion for the mean and that for the second moment are not only identities but also asymptotic expansions. For that purpose, it proves very useful to introduce the following notion, which we term JSadmissible functions (following the survey paper [35]). This is reminiscent of the classical H-admissible (due to Hayman) or HS-admissible (due to Harris-Schoenfeld) functions; see [28,§VIII.5]. Once we prove the asymptotic nature of the Poisson-Charlier expansions for the mean and the second moment, it remains, according again to the discussions in Introduction, to derive more precise asymptotics for the functionṼ (as defined in (4)), for which we will use first the Laplace transforms, normalize the Laplace transform properly, and then apply the Mellin transform. Such an approach will turn out to be very effective and readily applicable to more general cases such as bucket DSTs, which is discussed in details in the next section. The approach parallels closely in essence that introduced by Flajolet and Richmond in [24], which starts from the ordinary generating function, followed by an Euler transform, a proper normalization and the Mellin transform, and then conclude by singularity analysis; see also [10]. The path we take, however, offers additional operational advantages, as will be clear later. See Figure 7 for a diagrammatic illustration of the two analytic approaches. Analytic de-Poissonization and JS-admissibility The fundamental differential-functional equations for the analysis of random DSTs is of the form f (z) +f ′ (z) = 2f (z/2) +g(z), with suitably given initial value f (0) andg. For such functions, it turns out that the asymptotic nature of the Poisson-Charlier expansions for the coefficients (or de-Poissonization) can be justified in a rather systematic way by the introduction of the notion of JS-admissible functions. Here and throughout this paper, the generic symbol ε ∈ (0, 1) always represents an arbitrarily small constant whose value is immaterial and may differ from one occurrence to another. Definition 1. An entire functionf is said to be JS-admissible, denoted byf ∈ JS , if the following two conditions hold for |z| ≥ 1. (I) There exist α, β ∈ R such that uniformly for | arg(z)| ≤ ε, f (z) = O |z| α (log + |z|) β , where log + x := log(1 + x). (O) Uniformly for ε ≤ | arg(z)| ≤ π, f (z) := e zf (z) = O e (1−ε)|z| . For convenience, we also writef ∈ JS α,β to indicate the growth order off inside the sector | arg(z)| ≤ ε. Note that iff satisfies condition (I), then, by Cauchy's integral representation for derivatives (or by Ritt's theorem; see [54, Ch. 1, § 4.3]), we have, f (k) (z) = O |w−z|=ε|z| |w| α |(log + |w|) β |w − z| k+1 | dw| = O |z| α−k (log + |z|) β . Proposition 2.2. Assumef ∈ JS α,β . Let f (z) := e zf ( z). Then the Poisson-Charlier expansion (1) of f (n) (0) is also an asymptotic expansion in the sense that a n := f (n) (0) = n![z n ]f (z) = n![z n ]e zf (z) = 0≤j<2kf (j) (n) j! τ j (n) + O n α−k (log n) β , for k = 1, 2, . . . . Proof. (Sketch) Starting from Cauchy's integral formula for the coefficients, the lemma follows from a standard application of the saddle-point method. Roughly, condition (O) guarantees that the integral over the circle with radius n and argument satisfying ε ≤ | arg(z)| ≤ π is negligible, while condition (I) implies smooth estimates for all derivatives (and thus error terms). The polynomial growth of condition (I) is sufficient for all our uses; see [36] for more general versions. The real advantage of introducing admissibility is that it opens the possibility of developing closure properties as we now discuss. Lemma 2.3. Let m be a nonnegative integer and α ∈ (0, 1). (i) z m , e −αz ∈ JS . (ii) Iff ∈ JS , thenf (αz), z mf ∈ JS . (iii) Iff ,g ∈ JS , thenf +g ∈ JS . (iv) Iff ∈ JS , then the productPf ∈ JS , whereP is a polynomial of z. (v) Iff ,g ∈ JS , thenh ∈ JS , whereh(z) :=f (αz)g((1 − α)z). (vi) Iff ∈ JS , thenf ′ ∈ JS , and thusf (m) ∈ JS . Proof. Straightforward and omitted. Specific to our need for the analysis of DSTs is the following transfer principle. Proposition 2.4. Letf (z) andg(z) be entire functions satisfying f (z) +f ′ (z) = 2f (z/2) +g(z),(13)with f (0) = 0. Theng ∈ JS if and only iff ∈ JS . Proof. Assumeg ∈ JS . We check first the condition (O) forf . Let f (z) := e zf (z) and g(z) := e zg (z). By (13), f ′ (z) = 2e z/2 f (z/2) + g(z) . Consequently, since f (0) = 0, f (z) = z 0 2e t/2 f (t/2) + g(t) dt = z 1 0 2e tz/2 f (tz/2) + g(tz) dt.(14) Now define B(r) := max z∈Cr,ε |f (z)|, where C r,ε := {z : |z| ≤ r, ε ≤ | arg(z)| ≤ π}, (r ≥ 0; 0 < ε < π/2). Then, by (14), we have B(r) ≤ r 1 0 2e tr cos(ε)/2 B(tr/2) + |g(tr)| dt = r 0 2e t cos(ε)/2 B(t/2) + O e (1−ε)t dt ≤ Ce r cos(ε)/2 B(r/2) + O e (1−ε)r , where C = 4/ cos ε > 1. This suggests that we define a majorant function K(r) of B(r) by K(r) = O(1) for r ≤ 1 and for r ≥ 1 K(r) = Ce r cos(ε)/2 K(r/2) + h(r), where h is an entire function satisfying h(r) = O(1) for r ≤ 1 and h(r) = O e (1−ε)r for r ≥ 1. Let K(r) := e −r cos(ε) K(r) andh(r) := e −r cos(ε) h(r). Then since cos ε − 1 + ε > 0 for ε ∈ (0, 1), we obtaiñ K(r) = CK(r/2) +h(r),h(r) = O(1). Thus if we choose m = ⌈log 2 r⌉ such that 2 m ≥ r and iterate m times the functional equation, then we obtain the estimateK (r) = 0≤k≤m C kh (r/2 k ) + C m+1K (r/2 m+1 ) = O   r/2 k >1 C k + C m   = O r log 2 C . Thus B(r) = O r log 2 C e r cos ε . which establishes condition (O). Our proof forf satisfying (I) proceeds in a similar manner and starts again from (14) but of the form f (z) = z 1 0 e −(1−t)z 2f (tz/2) +g(tz) dt. Now, defineB (r) := max z∈Sr,ε |f (z)|, where S r,ε := {z : |z| ≤ r, | arg(z)| ≤ ε}, (r ≥ 0; 0 < ε < π/2). ThenB (r) ≤ r 1 0 e −(1−t)r cos ε 2B(tr/2) + |g(tr)| dt = r 1 2e −(r−t) cos εB (t/2) + O e −(r−t) cos ε t α (log + t) β dt + O(1) ≤ CB(r/2) + O r α (log + r) β + 1 , where C = 2/ cos ε > 2. The same majorization argument used above for (O) then leads tõ B(r) =      O(r log 2 C ), if α < log 2 C; O(r log 2 C (log + r) β+1 ), if α = log 2 C; O r α (log + r) β , if α > log 2 C. This proves (I) forf . The necessity part follows trivially from Lemma 2.3. The estimates we derived of asymptotic-transfer type are indeed over-pessimistic when 1 ≤ α ≤ log 2 C, but they are sufficient for our use. The true orders are those with ε → 0, which can be proved by the Laplace-Mellin-de-Poissonization approach we use later. Lemma 2.3 and Proposition 2.4 provide very effective tools for justifying the de-Poissonization of functions satisfying the equation (13), which is often carried out through the use of the increasing-domain argument (see [36]). The latter argument is also inductive in nature and similar to the one we are developing here, although it is less "mechanical" and less systematic. Generating functions and integral transforms Since our approach is purely analytic and relies heavily on generating functions, we first derive in this subsection the differential-functional equations we will be working on later. Then we apply the de-Poissonization tools we developed to the Poisson generating functions of the mean and the second moment and justify the asymptotic nature of the corresponding Poisson-Charlier expansions. Then we sketch the asymptotic tools we will follow based on the Laplace and Mellin transforms. Generating functions. In terms of the moment generating function M n (y) := E(e Xny ), the recurrence (6) translates into M n+1 (y) = e ny 2 −n 0≤j≤n n j M j (y)M n−j (y), (n ≥ 0),(15) with M 0 (y) = 1. Now consider the bivariate exponential generating function F (z, y) := n≥0 M n (y) n! z n . Then by (15), ∂ ∂z F (z, y) = F e y z 2 , y 2 , and the Poisson generating functionF (z, y) := e −z F (z, y) satisfies the differential-functional equatioñ F (z, y) + ∂ ∂zF (z, y) = e (e y −1)zF e y z 2 , y 2 ,(16) withF (0, y) = 1. No exact solution of such a nonlinear differential equation is available; see [35] for an asymptotic approximation toF for y near unity. Mean and second moment. Let nowF (z, y) := m≥0f m (z) m! y m , wheref m (z) denotes the Poisson generating function of E(X m n ). Then we deduce from (16) that f 1 (z) +f ′ 1 (z) = 2f 1 (z/2) + z,(17)f 2 (z) +f ′ 2 (z) = 2f 2 (z/2) + 2f 1 (z/2) 2 + 4zf 1 (z/2) + 2zf ′ 1 (z/2) + z + z 2 ,(18) with the initial conditionsf 1 (0) =f 2 (0) = 0. Proposition 2.5. The Poisson-Charlier expansion for the mean and that for the second moment are both asymptotic expansions E(X n ) = 0≤j<2kf (j) 1 (n) j! τ j (n) + O n −k+1 , E(X 2 n ) = 0≤j<2kf (j) 2 (n) j! τ j (n) + O n −k+2 (log n) 2 , for k = 1, 2, . . . . Proof. (Sketch) By Lemma 2.3 and Proposition 2.4, we see that bothf 1 ,f 2 ∈ JS , and thus we can apply Proposition 2.2. Indeed the proof of Proposition 2.4 provides already crude bounds for the growth order off 1 ,f 2 . The more precise estimatesf 1 (z) ≍ |z|| log z| andf 2 (z) ≍ |z| 2 | log z| 2 for z inside the sector {z : | arg(z)| ≤ ε} will be provided later in the next two subsections. An asymptotic approach based on Laplace and Mellin transforms. Once the de-Poissonization steps are justified, all that remains for the proof of Theorem 2.1 is to derive more precise asymptotic approximations tof 1 andṼ (as defined in (4)). The approach we use begins with a more precise characterization off 1 (z). Bothf 1 andṼ satisfy a differential-functional equation of the form f (z) +f ′ (z) = 2f (z/2) +g(z), with the initial conditionf (0) = 0. To derive the asymptotics off for large complex z, we proceed along the following principal steps; see also [10]. Laplace transform: The Laplace transform off satisfies (s + 1)L [f ; s] = 4L [f ; 2s] + L [g; s],(19) which exists and defines an analytic function ifg grows at most polynomially for large |z|. M [L [f 1 ; s]; ω] = 1 1 − 2 2−ω M L [g; s] Q(−2s) ; ω . Inverting the process. We first derive the local behavior ofL [f ; s] for small s by the Mellin inversion (often by calculus of residues after justification of analytic properties), and then the asymptotic behavior off (z) for large z is derived by the Laplace inversion, similar to singularity analysis. Expected internal path-length of random DSTs We consider in details in this subsection the expected value µ n := E(X n ) of the total internal path-length, paving the way for the asymptotic analysis of the variance. Starting from either the equation (17) or the recurrence µ n+1 = 2 1−n 0≤j≤n n j µ j + n (n ≥ 0) with µ 0 := 0, there are several approaches to the asymptotics of µ n . We will briefly describe the one using integral representation of finite differences (or Rice's integrals) and then present the Laplace and Mellin transforms we will use, which, as will become clear, is essentially the Flajolet-Richmond approach (see [24]). Rice's integral representation. By (17), we have, withμ n := n![z n ]f 1 (z), µ n+1 = − 1 − 2 1−n μ n (n ≥ 0), withμ 0 = 0, which by iteration yields µ n = (−1) n Q n−2 , Q n := 1≤j≤n 1 − 2 −j .(20) Thus by Rice's formula ( [27]) µ n := E(X n ) = 2≤j≤n n j μ j = 1 2πi ( 3 2 ) Γ(n + 1)Γ(−s) Γ(n + 1 − s) · Q(1) (1 − 2 1−s )Q(2 1−s ) ds, where the integration path (c) is along the vertical line with real part equal to c and Q is defined in (9). We then obtain (7) by standard arguments; see [26] or [50] for details. This approach readily gives the approximation (7) for the mean and can be refined to obtain a full asymptotic expansion. However, its extension to the variance becomes extremely messy, as shown in [42]. Laplace transform. We first show that the asymptotics off 1 (z) can be derived through a direct use of the Laplace and Mellin transforms, which relies on several ad hoc steps that are not easily extended. A more general procedure will be developed below. By (17), we see that the Laplace transform of f 1 (z) satisfies the functional equation (s + 1)L [f 1 ; s] = 4L [f 1 ; 2s] + s −2 ,(21) which exists and is analytic in C \ (−∞, 0]. By dividing both sides by s + 1 and by iteration, we get L [f 1 ; s] = 1 s 2 j≥0 1 (s + 1) · · · (2 j s + 1) . On the other hand, from (20), we have L [f 1 ; s] = ∞ 0 e −sz n≥0μ n n! z n dz = n≥0 (−1) n Q n s −n−3 . This implies the identity n≥0 (−1) n Q n s n+1 = j≥0 1 (s + 1) · · · (2 j s + 1) . However, neither form is useful for our asymptotic purpose. Now by partial fraction expansion, we obtain 1 (s + 1) · · · (2 j s + 1) = 0≤ℓ≤j (−1) j−ℓ 2 −( j−ℓ+1 2 )−ℓ (s + 2 −ℓ )Q ℓ Q j−ℓ . Thus L [f 1 ; s] = 1 s 2 j≥0 0≤ℓ≤j (−1) j−ℓ 2 −( j−ℓ+1 2 )−ℓ (s + 2 −ℓ )Q ℓ Q j−ℓ = 1 s 2 ℓ≥0 1 Q ℓ (2 ℓ s + 1) j≥0 (−1) j 2 −( j+1 2 ) Q j . Note that j≥0 2 j s (s + 1) · · · (2 j s + 1) = 1. By the Euler identity j≥0 q ( j 2 ) z j (1 − q) · · · (1 − q j ) = k≥0 1 + q k z , we see that j≥0 (−1) j 2 −( j+1 2 ) Q j = Q(1) = Q ∞ ≈ 0.28878809 . . . This gives L [f 1 ; s] = Q ∞ s 2 ℓ≥0 1 Q ℓ (2 ℓ s + 1) , and thenf 1 (z) = Q ∞ ℓ≥0 2 ℓ Q ℓ e −z/2 ℓ − 1 + z 2 ℓ .(23) Consequently, µ n = Q ∞ ℓ≥0 2 ℓ Q ℓ (1 − 2 −ℓ ) n − 1 + 2 −ℓ n . Asymptotically, we have, by (23) and the identity 1 Q(z) = j≥1 1 1 − z/2 j = ℓ≥0 z ℓ Q ℓ 2 ℓ (|z| < 2),(24) the Mellin integral representatioñ f 1 (z) = 1 2πi (−3/2) Q(1)Γ(s)z −s (1 − 2 s+1 )Q(2 s+1 ) ds, from which we derive the asymptotic approximatioñ f 1 (z) = (z + 1) log 2 z + z γ − 1 log 2 + 1 2 − c 1 + ̟ 1 (log 2 z) + O(1),(25) uniformly for |z| → ∞ and | arg(z)| ≤ π/2−ε, where ̟ 1 is given in (8). (As usual, we use the asymptotic estimate (12) for the Gamma function.) Laplace and Mellin transforms. We now re-do the analysis forf 1 (z) in a more general way that can be easily extended to other cases. We again start from (21) and considerL [f 1 ; s] := L [f 1 ; s] Q(−s) , where Q(z) is defined in (9). Dividing both sides of (21) by Q(−2s) yields L [f 1 ; s] = 4L [f 1 ; 2s] + 1 Q(−2s)s 2 .(26) We now apply the Mellin transform. Note that we have, by the fact that X 0 = X 1 = 0 and the proof of Proposition 2.4,f 1 (z) = O(z 2 ), if z → 0 + ; O(z 1+ε ), if z → ∞. Then L [f 1 ; s] = O(s −2−ε ), as s → 0 + ; O(s −3 ), as s → ∞. On the other hand, by the Mellin transform, log Q(−2s) = j≥0 log 1 + s 2 j = 1 2πi (− 1 2 ) πs −w (1 − 2 w )w sin πw dw = (log s) 2 2 log 2 + log s 2 + k∈Z q k s −χ k + O(|s| −1 )(27) uniformly for |s| → ∞ and | arg(s)| ≤ π − ε, where χ k := 2kπi/ log 2, q 0 = log 2 12 + π 2 6 log 2 and q k = 1 2k sinh(2kπ/ log 2) (k = 0). This asymptotic expansion, together with the Taylor expansion Q(−2s) = 1 + O(|s|), (|s| → 0), gives rise toL [f 1 ; s] = O(s −2−ε ), as s → 0 + ; O(s −M ), as s → ∞,ω] = G 1 (ω) 1 − 2 2−ω , (ℜ(ω) > 2), where G 1 (ω) := ∞ 0 s ω−3 Q(−2s) ds = πQ(2 ω−2 ) Q(1) sin πω = Q(2 ω−2 ) Q(1) Γ(ω)Γ(1 − ω),(28) for ℜ(ω) > 2; see [24]. Inverse Mellin and inverse Laplace transforms. We can now apply successively the inverse Mellin and then Laplace transforms to derive the asymptotics off 1 (z). Observe that G 1 (ω) has a simple pole at ω = 2. By (28) or Proposition 5 in [22], we obtain |G 1 (c + it)| = O e −(π−ε)|t| , for large |t| and c ∈ R. Then by the calculus of residues, L [f 1 ; s] = 1 s 2 log 2 1 s + 1 s 2   1 2 − c 1 + 1 log 2 k∈Z\{0} G 1 (2 + χ k )s −χ k   + O(|s| −1 ), uniformly for |s| → 0 and | arg(s)| ≤ π − ε. Using the expansion Q(−s) = 1 + s + (|s| 2 ) (|s| ∼ 0), we see that L [f 1 ; s] = 1 + s s 2 log 2 1 s + 1 s 2   1 2 − c 1 + 1 log 2 k∈Z\{0} G 1 (2 + χ k )s −χ k   + O(|s| −1 ), uniformly for |s| → 0 and | arg(s)| ≤ π − ε. Finally, we consider the inverse Laplace transform. The following simple result is very useful for our purposes. Proposition 2.6. Letf (z) be a function whose Laplace transform exists and is analytic in C \ (−∞, 0]. Assume that L [f ; s] =      O (|s| −α | log |s + 1|| m ) , cs −ω (− log s) m , o(|s| −α | log |s + 1|| m ),(29) uniformly for |s| → 0 and | arg(s)| ≤ π − ε, where α ∈ R, ω ∈ C and m = 0, 1, . . . . If L [f ; s] satisfies |L [f ; s]| = O |s| −1−ε ,(30)as |s| → ∞ in | arg(s)| ≤ π − ε, theñ f (z) =          O (|z| α−1 (log |z|) m ) , cz ω−1 0≤j≤m m j (log z) m−j ∂ j ∂ω j 1 Γ(ω) , o (|z| α−1 (log |z|) m ) , respectively, where the Oand o-terms hold uniformly for |z| → ∞ and | arg(z)| ≤ π/2 − ε. Proof. LetL (s) = L [f ; s]. Then by the inverse Laplace transform, f (z) = 1 2πi (1) e zsL (s) ds = 1 2πi H e zsL (s) ds, where H is the Hankel contour consisting of the two rays te ±iε ± i/|z|, −∞ < t ≤ 0 and the semicircle exp(iϕ)/|z|, −π/2 ≤ ϕ ≤ π/2; see Figure 6. Assume from now on |z| is sufficiently large and lies in the sector with | arg(z)| ≤ π/2 − ε. We prove only the O-case, the other two cases being similar. For simplicity, we consider only the case m = 0, the other cases being easily extended. We split the above integral along H into two parts where H > comprises the two rays te iε ± i/|z|, −∞ < t ≤ −T with T > 1 a fixed constant and H ⊃ represents the remaining contour. ℜ(s) ℑ(s) ε H 1 |z| Figure 6: The contour H. The integral along H > is easily estimated 1 2πi H> e zsL (s) ds = O −T −∞ e ℜ(|z|e i arg(z) (te iε +i/|z|)) |t| −1−ε dt = O ∞ T t −1−ε e −|z|t cos(arg(z)+ε) dt = O |z| ε e −c|z|T , the O-term holding uniformly for |z| → ∞ provided that | arg(z)| + ε < π/2, where c > 0 is a suitable constant. For the second integral, we use (29). Then the integral along the semicircle is bounded as follows. 2π|z| π/2 −π/2 e ze iθ /|z|+iθL (e iθ /|z|) dθ = O |z| α−1 , uniformly for |z| → ∞. For the remaining part t ± i/|z|, −T < t ≤ 0, we have 1 2πi 0 −T e z(t±i/|z|)L (t ± i/|z|) dt = O |z| α 0 −T e c|z|t (|z| 2 t 2 + 1) α/2 dt = O |z| α−1 ∞ 0 e −cu (u 2 + 1) α/2 du = O |z| α−1 , uniformly for |z| → ∞, where c > 0 is a suitable constant. This completes the proof. Note that the inverse Laplace transform of s −2 log(1/s) is z log z − (1 − γ)z. This, together with a combined use of Proposition 2.6, leads to (25). The justification of the estimate (30) is easily performed by using the relation (31) below. The Flajolet-Richmond approach [24]. Instead of the Poisson generating function, this approach starts from the ordinary generating function A(z) := n µ n z n . M [Ā; ω] = G 1 (ω) 1 − 2 2−ω , where G 1 (ω) is as defined in (28). Then invert the process by considering first the Mellin inversion, deriving asymptotics of A(s) = 1 2πi (5/2) s −ω G 1 (ω) 1 − 2 2−ω dω, as s → 0 in C. Then deduce asymptotics of A(z) = 1 z 1 z − 1 , as z → 1. Finally, apply singularity analysis (see [23]) to conclude the asymptotics of µ n . The crucial reason why the two approaches are identical at certain steps is that the Laplace transform of a Poisson generating function is essentially equal to the Euler transform of an ordinary generating function; or formally, ∞ 0 e −sz n≥0 a n n! z n dz = n≥0 a n (s + 1) −n−1 = 1 s + 1 A 1 s + 1 .(31) Thus the simple result in Proposition 2.6 closely parallels that in singularity analysis. While identical at certain steps, the two approaches diverge in their final treatment of the coefficients, and the distinction here is typically that between the saddle-point method and the singularity analysis, a situation reminiscent of the use before and after Lagrange's inversion formula; see for instance [28]. The relation (31) implies that the order estimate (30) for the Laplace transform at infinity can be easily justified for all the generating functions we consider in this paper since A(0) = 0, implying that A(z) = O(|z|) as |z| → 0. This comparison also suggests the possibility of developing de-Poissonization tools by singularity analysis, which will be investigated in details elsewhere. Variance of the internal path-length In this section, we apply the Laplace-Mellin-de-Poissonization approach to the Poissonized variance with correctionṼ (z) :=f 2 (z) −f 1 (z) 2 − zf ′ 1 (z) 2 , aiming at proving Theorem 2.1. The starting point of focusing onṼ instead of onf 2 removes all heavy cancellations involved when dealing with the variance, a key step differing from all previous approaches. Laplace and Mellin transform. The following lemma will be useful. Lemma 2.7. If f 1 (z) +f ′ 1 (z) = 2f 1 (z/2) +h 1 (z), f 2 (z) +f ′ 2 (z) = 2f 2 (z/2) +h 2 (z), where all functions are entire withf 1 (0) =f 2 (0) = 0, then the functionṼ (z) : =f 2 (z) −f 1 (z) 2 − zf ′ 1 (z) 2 satisfiesṼ (z) +Ṽ ′ (z) = 2Ṽ (z/2) +g(z), withṼ (0) = 0, wherẽ g(z) = zf ′′ 1 (z) 2 +h 2 (z) −h 1 (z) 2 − zh ′ 1 (z) 2 − 4h 1 (z)f 1 (z/2) − 2zh ′ 1 (z)f ′ 1 (z/2) − 2f 1 (z/2) 2 . Proof. Straightforward and omitted. By using the differential-functional equations (17) and (18) forf 1 (z) andf 2 (z), we see, by Lemma 2.7, thatṼ (z) +Ṽ ′ (z) = 2Ṽ (z/2) + zf ′′ 1 (z) 2 ,(32) withṼ (0) = 0. Before applying the integral transforms, we need rough estimates ofṼ (z) near z = 0 and z = ∞. We haveṼ (z) = O (z 2 ) , as z → 0 + ; O(z 1+ε ), as z → ∞.(33) These estimates follow from zf ′′ 1 (z) 2 = O(|z|), as |z| → 0; O(|z| −1 ), as |z| → ∞,(34) which in turn result from X 0 = X 1 = 0 and (25) (by the proof of condition (I) of Proposition 2.4). Indeed, the proof there shows that the same bounds hold uniformly for z ∈ C with | arg(z)| ≤ π/2 − ε. We now apply the Laplace transform to both sides of (32). First, observe that the Laplace transform of V (z) exists and is analytic in C \ (−∞, 0]. Then, by (32), ; ω = ∞ 0 s ω−1 Q(−2s) ∞ 0 e −zs zf ′′ 1 (z) 2 dz ds.(35) By (23), we have zf ′′ 1 (z) 2 = Q 2 ∞ h,ℓ≥0 1 Q h Q ℓ 2 h+ℓ ze −z/2 h −z/2 ℓ . Substituting this and the partial fraction expansion 1 Q(−2s) = 1 Q ∞ j≥0 (−1) j 2 −( j 2 ) Q j (s + 2 −j ) , into (35), we obtain (10). Inverse Mellin and inverse Laplace transforms. For the Mellin inversion, we need more precise analytic properties of G 2 (ω). By (34), we deduce that the Laplace transformg ⋆ (s) of zf ′′ 1 (z) 2 satisfies g ⋆ (s) = O(| log s|), as |s| → 0; O(|s| −2 ), as |s| → ∞ uniformly in the cone | arg(s)| ≤ π − ε. Thus, by the asymptotic expansion (27) for Q(−2s) and Proposition 5 in [22], we have |G 2 (c + it)| = O e −(π−ε)|t| , for large |t| and c > 0. Also the Mellin transform G 2 ofg ⋆ (s)/Q(−2s) exists in the half-plane ℜ(ω) > 0. Consequently, by standard calculus of residues, L [Ṽ ; s] = 1 log 2 k∈Z G 2 (2 + χ k )s −2−χ k + O(|s| −ε ), uniformly for |s| → 0 and | arg(s)| ≤ π − ε. This in turn yields the following expansion for L [Ṽ ; s] L [Ṽ ; s] = 1 log 2 k∈Z G 2 (2 + χ k )s −2−χ k + 1 log 2 k∈Z G 2 (2 + χ k )s −1−χ k + O(|s| −ε ), again uniformly for |s| → 0 and | arg(s)| ≤ π − ε. Finally, standard Laplace inversion gives V (z) = z log 2 k∈Z G 2 (2 + χ k ) Γ(2 + χ k ) z χ k + 1 log 2 k∈Z G 2 (1 + χ k ) Γ(1 + χ k ) z χ k + O(|z| ε−1 ),(36) uniformly for |z| → ∞ and | arg(z)| ≤ π/2 − ε. Sincef 2 (z) =Ṽ (z) +f 1 (z) 2 + zf ′ 1 (z) 2 , we see from (36) and (25) that f 2 (z) ≍f 1 (z) 2 ≍ |z| 2 log 2 |z| (| arg(z)| ≤ π/2 − ε). This proves Proposition 2.5 and Theorem 2.1 by straightforward expansion. More refined calculations give V(X n ) =Ṽ (n) − n 2Ṽ ′′ (n) − n 2 2f ′′ 1 (n) 2 + O(n −1 ), the two terms followingṼ (n) being both O(1) and periodic in nature. It is possible to further extend the same idea and derive a full asymptotic expansion, which has also its identity nature; details will be presented in a future paper. Bucket Digital Search Trees In this section, we extend the same approach to bucket digital search trees (b-DSTs) in which each node can hold up to b keys. The construction rule is the same as DSTs, except that keys keep staying in a node as long as its capacity remains less than b; see Figure 8 for a simple example with b = 2. DSTs correspond to b = 1. Note that when b ≥ 2 we can distinguish two different types of total path-length: the total path-length of all keys (summing the distance between each key to the root over all keys), which will be referred to as the total key-wise path-length (KPL) and the total path-length of all nodes (summing the distance between each node to the root over all nodes, regardless of the number of keys in each node), referred to as the total node-wise path-length (NPL). When b = 1 the two total path-lengths coincide. For simplicity, we will use KPL and NPL, dropping the collective adjective "total". While the expected values of both TPLs are of order n log n under the same independent Bernoulli model, their variances surprisingly turn out to exhibit very different behavior; see Table 1. Key-wise path-length (KPL) We assume the same independent Bernoulli model for the input strings. Let X n denote the KPL in a random b-DST built from n random stings. Then by definition and the independence model assumption X n+b d = X Bn + X * n−Bn + n, (n ≥ 0)(37) with the initial conditions X 0 = · · · = X b−1 = 0. Here B n ∼ Binomial(n, 1/2), X n d = X * n , and X n , X * n , B n are independent. Known and new results. Hubalek [30] showed, by the Flajolet-Richmond approach, that the mean satisfies E(X n ) = (n + b) log 2 n + n (c 2 + ̟ 3 (log 2 n)) + c 3 + ̟ 4 (log 2 n) + O n −1 log n , where c 2 , c 3 are effectively computable constants and ̟ 3 and ̟ 4 are very smooth periodic functions. He also proved that the variance is asymptotically linear V(X n ) = n (C h + ̟ h (log 2 n)) + O((log n) 2 ), where C h is expressed in terms of a very long, involved expression and ̟ h is a periodic function. We improve this estimate by deriving a much simpler expression for the periodic function, including its average value C h . To state our result, we define the following functions. Let g(z) := 0≤j≤b b j f (j) 1 (z) 2 + z 0≤j≤b b j f (j+1) 1 (z) 2 − 0≤j≤b b j f 2 1 (z) (j) + zf ′ 1 (z) 2 (j) .(38) It is easily seen thatg(z) is of the form g(z) = 2≤i 1 ,i 2 ≤bg i 1 ,i 2f (i 1 ) 1 (z)f (i 2 ) 1 (z) + z 2≤i 1 ,i 2 ≤b+1g ′ i 1 ,i 2f (i 1 ) 1 (z)f (i 2 ) 1 (z),(39) whereg i 1 ,i 2 ,g ′ i 1 ,i 2 ≥ 0 are given explicitly bỹ g i 1 ,i 2 = b i 1 b i 2 − b i 1 b − i 1 i 2 − (b − i 1 + 1) b i 1 − 1 b − i 1 i 2 − 1 , g ′ i 1 ,i 2 = b i 1 − 1 b i 2 − 1 − b i 1 − 1 b − i 1 + 1 i 2 − 1 , both coefficients being symmetric in i 1 and i 2 . Define G 2 (ω) = ∞ 0 s ω−1 Q(−2s) b ∞ 0 e −zsg (z) dz ds, which is well-defined for ℜ(ω) > 0, as we will see later. Theorem 3.1. The variance of the total key-wise path-length of random b-DSTs of n strings satisfies V(X n ) = n (C h + ̟ h (log 2 n)) + O(1),(40) where C h = G 2 (2) log 2 = 1 log 2 ∞ 0 s Q(−2s) b ∞ 0 e −zsg (z) dz ds, and ̟ h (t) = 1 log 2 k∈Z\{0} G 2 (2 + χ k ) Γ(2 + χ k ) e 2kπit . By straightforward truncations, expansions and approximations, we obtain the following numerical values for b = 1, . . . , 5. withF (0, y) = 1. From this form, the asymptotic analysis of the mean value and that of the variance proceed along exactly the same line we developed in the previous section. Thus we briefly sketch the principal steps of the analysis, leaving the details to the interested reader. The expected value of X n . From (41), we derive the following differential-functional equation for the Poisson generating function of the mean 0≤j≤b b j f (j) 1 (z) = 2f 1 (z/2) + z, with the initial conditionsf (j) 1 (0) = 0 for 0 ≤ j < b. Before applying the Laplace-Mellin approach, we need first a transfer-type result similar to Proposition 2.4. Proposition 3.2. Letf (z) andg(z) be entire functions satisfying 0≤j≤b b j f (j) (z) = 2f (z/2) +g(z),(42) with f (0) = 0. Theng ∈ JS ⇐⇒f ∈ JS . Proof. (Sketch) The same proof as that for Proposition 2.4 applies mutatis mutandis to (42). The only difference is that we now have f (b) (z) = 2e z/2 f (z/2) + g(z), where f (z) := e zf (z) and g(z) := e zg (z), so that (14) has the extended representation f (z) = 1 (b − 1)! z 0 (z − t) b−1 2e t/2 f (t/2) + g(t) dt = z b (b − 1)! 1 0 (1 − t) b−1 2e tz/2 f (tz/2) + g(tz) dt, andf (z) = z b (b − 1)! 1 0 (1 − t) b e −(1−t)z 2f (tz/2) +g(z) dt. All required estimates can be derived by the same arguments used there. The Laplace transform off 1 now satisfies the functional equation (s + 1) b L [f 1 ; s] = 4L [f 1 ; 2s] + s −2 , for ℜ(s) > 0. From this equation, we obtain L [f 1 ; s] = 1 s 2 j≥0 1 (s + 1) b · · · (1 + 2 j s) b , which extends (22). From this series and partial fraction expansions, we can derive a close-form expression forf 1 (z), which becomes messy especially for large b. Define as beforeL [f 1 ; s] := L [f 1 ; s]/Q(−s) b . Then we obtainL [f 1 ; s] = 4L [f 1 ; 2s] + 1 Q(−2s) b s 2 . This relation is almost the same as (26). Thus the same Mellin analysis given there carries over and we deduce that L [f 1 ; s] = 1 s 2 log 2 1 s + 1 s 2   1 2 + c 4 log 2 + 1 log 2 k∈Z\{0} G 1 (2 + χ k )s −χ k   + b s log 1 s + O(|s| −1 ), uniformly for |s| → 0 and | arg(s)| ≤ π − ε, where G 1 (ω) := ∞ 0 s ω−3 Q(−2s) b ds, and c 4 := lim ω→2 G 1 (ω) − 1 ω − 2 = 1 0 1 s 1 Q(−2s) b − 1 ds + ∞ 1 1 sQ(−2s) b ds. Consequently, by the Laplace inversion, f 1 (z) = (z + b) log 2 z + z   1 2 + G 1 (2) + γ − 1 log 2 + 1 log 2 k∈Z\{0} G 1 (2 + χ k ) Γ(2 + χ k ) z χ k   + O(1),(43) uniformly for |z| → ∞ and | arg(z)| ≤ π/2 − ε. From this and Propositions 3.2 and 2.2, we obtain E(X n ) = 0≤j<2kf (j) (n) j! τ j (n) + O n −1+k , for any k = 1, 2, . . . . Finally, E(X n ) = (n + b) log 2 n + n   1 2 + G 1 (2) + γ − 1 log 2 + 1 log 2 k∈Z\{0} G 1 (2 + χ k ) Γ(2 + χ k ) n χ k   + O(1). Variance of X n . The analysis here is again similar to that for the mean. Letf 2 (z) denote the Poisson generating function of the second moment E(X 2 n ). Then, by (41), 0≤j≤b b j f (j) 2 (z) = 2f 2 (z/2) + 2f 1 (z/2) 2 + 4zf 1 (z/2) + 2zf ′ 1 (z/2) + z + z 2 , with the first b Taylor coefficients zero. Define agaiñ V (z) =f 2 (z) −f 1 (z) 2 − zf ′ 1 (z) 2 . ThenṼ (z) satisfies 0≤j≤b b j Ṽ (j) (z) = 2Ṽ (z/2) +g(z), whereg(z) is given in (38). By the representations (39) and (43), we havẽ g(z) = O(|z|), as |z| → 0; O(|z| −1 ), as |z| → ∞, uniformly in the sector | arg(z)| ≤ π/2 − ε. This is similar to the corresponding estimate (34) in the analysis of the variance in the previous section. The same procedure there applies and we deduce (40). Node-wise path-length (NPL) We consider in this section the total node-wise path-length (NPL). Under the same independent Bernoulli model, we still use X n to denote the NPL in a random b-DST of n binary strings with node capacity b ≥ 2. Also let N n stand for the total number of nodes (space requirement) in random b-DST of n strings. Despite its being one of the most natural shape measures for b-DSTs, the consideration of X n here seems to be new. For N n , it is known that the distribution is asymptotically normal with the mean and the variance both asymptotically n times a different smooth periodic function; see [31]. In contrast to (40) for the variance of KPL, what is unexpected and surprising here is that the variance of X n is of order n(log n) 2 . Theorem 3.3. Assume b ≥ 2. The mean of N n and that of X n satisfy the following asymptotic relations. E(N n ) = nP 1,0 (log 2 n) + O(1), E(X n ) = n(log 2 n)P 1,0 (log 2 n) + nP [2] 0,1 (log 2 n) + (log 2 n)P [3] 0,1 (log 2 n) + O(1); (44) and the variances of N n and X n satisfy            V(N n ) = nP 2,0 (log 2 n) + O(1), Cov(N n , X n ) = n(log 2 n)P 2,0 (log 2 n) + nP [2] 1,1 (log 2 n) + (log n)P [3] 1,1 (log 2 n) + O(1), V(X n ) = n(log 2 n) 2 P 2,0 (log 2 n) + n(log 2 n)P [2] 0,2 (log 2 n) + nP [3] 0,2 (log 2 n) + (log n) 2 P [4] 0,2 (log 2 n) + (log 2 n)P [5] 0,2 (log 2 n) + O(1), (45) where the P ·,· 's are all computable, smooth, 1-periodic functions. Intuitively, that the variance of NPL is larger than that of KPL can be seen from the definition of NPL, which depends on the random variable N n (see (46)), while on the other hand, KPL depends on n only (in addition to on the two subtrees). The following figure shows the first few values of the variance of NPL and that of KPL. NPL KPL We see that the variance of NPL increases faster than that of KPL. Note that the periodic functions of the dominant terms are all equal, implying that the correlation coefficient of N n and X n is asymptotically 1. On the other hand, the mean value c 1,0 of P 1,0 (t) is given by c 1,0 = 1 log 2 ∞ 0 (s + 1) b−1 Q(−2s) b ds; numerical approximations to c 1,0 for the first few b are given as follows. by (28), which is consistent with the fact that N n ≡ n in this case. When b = 2, we see that about 42.5% of nodes on average contain two keys and 14% of nodes a single key. The storage utilization is thus not very bad. From (44) and these numerical values, we see that, in contrast to the expected KPL, which is asymptotic to n log 2 n for all b, the expected NPL provides a better indication of the "shape variation" of random b-DSTs. Our analysis is based on the following straightforward distributional recurrences N n+b d = N Bn + N * n−Bn + 1, X n+b d = X Bn + X * n−Bn + N Bn + N * n−Bn , (n ≥ 0),(46) with the initial conditions N 0 = 0, N 1 = · · · = N b−1 = 1 and X 0 = · · · = X b−1 = 0. Here again B n ∼ Binomial(n, 1/2), N n d = N * n , X n d = X * n and X n , X * n , B n as well as N n , N * n , B n are independent. Generating functions. Define M n (u, v) = E(e Nnu+Xnv ). Then (46) translates into the recurrence M n+b (u, v) = e u 2 −n n j=0 n j M j (u + v, v)M n−j (u + v, v), (n ≥ 0), with M 0 (u, v) = 1, M 1 (u, v) = · · · = M b−1 (u, v) = e u . Next, let F (z, u, v) := n≥0 M n (u, v) n! z n . Then the recurrence relation gives ∂ b ∂z b F (z, u, v) = e u F z 2 , u + v, v 2 , and the Poisson generating functionF (z, u, v) := e −z F (z, u, v) satisfies 0≤j≤b b j ∂ j ∂z jF (z, u, v) = e uF z 2 , u + v, v 2 ,(47) with the initial conditionsF (z, u, v) = 1 + (e u − 1) 1≤j<b (−1) j−1 z j /j! + · · · . For the moments, if we expandF (z, u, v) in terms of u and v, F (z, u, v) = m≥0 1 m! 0≤j≤m m j f j,m−j (z)u j v m−j , thenf j,m−j (z) is the Poisson generating function of E(N j n X m−j n ). Thus all moments of X n and N n or their products can be computed by taking suitable derivatives of (47) with respect to u and v and then substituting u = v = 0. Expected number of nodes and expected node-wise path length. By taking first derivatives of (47), we obtain          0≤j≤b b j f (j) 1,0 (z) = 2f 1,0 (z/2) + 1, 0≤j≤b b j f (j) 0,1 (z) = 2f 0,1 (z/2) + 2f 1,0 (z/2),(48) the initial conditions beingf 1,0 (0) = 0,f 1,0 (0) = (−1) j−1 for 1 ≤ j < b andf (j) 0,1 (0) = 0 for 0 ≤ j < b.(j) We can apply the Laplace-Mellin approach as before, starting from the mean of N n . Note that L [f (j) ; s] = s j L [f ; s] − 0≤ℓ<j s ℓf (j−1−ℓ) (0) (j = 0, 1, . . . ), provided that the Laplace transform exists for ℜ(s) > 0. This gives (s + 1) b L [f 1,0 ; s] = 4L [f 1,0 ; 2s] +g ⋆ 1,0 (s), whereg ⋆ 1,0 (s) := 1 s + 0≤ℓ≤b−2 s ℓ ℓ≤j≤b−2 b j + 2 f (j+1−ℓ) 1,0 (0) = 1 s + 1≤j<b b − 1 j s j−1 = s −1 (s + 1) b−1 . Unlike all previous cases, iterating this functional equation leads to a divergent series. Although this problem can be solved by subtracting a sufficient number of initial terms off 1,0 (z), the approach we use does not rely on this and avoids completely such a consideration. LetL [f 1,0 ; s] := L [f 1,0 ; s]/Q(−s) b . Then M [L [f 1,0 ; s]; ω] = G 1,0 (ω) 1 − 2 2−ω , (ℜ(ω) > 2), where G 1,0 (ω) := ∞ 0 s ω−2 Q(−2s) b (s + 1) b−1 ds, for ℜ(ω) > 1. From this, we deduce thatf 1,0 (z) = zP 1,0 (log 2 z) + O(1),(49) uniformly for |z| → ∞ and | arg(z)| ≤ π/2 − ε, where P 1,0 (t) is a periodic function with the Fourier series representation P 1,0 (t) := 1 log 2 k∈Z G 1,0 (2 + χ k ) Γ(2 + χ k ) e 2kπit , the series being absolutely convergent. From this we deduce the first approximation of (44). We now turn to the expected NPL E(X n ). By (48), we have Then M [f 0,1 ; ω] = 2 2−ω G 1,0 (ω) (1 − 2 2−ω ) 2 , (ℜ(ω) > 2). From this we deduce that f 0,1 (z) = z(log 2 z)P [1] 0,1 (log 2 z) + zP [2] 0,1 (log 2 z) + (log 2 z)P [4] 0,1 (log 2 z) + O(1),(50) uniformly for |z| → ∞ and | arg(z)| ≤ π/2 − ε, where P [1] 0,1 (t), P [2] 0,1 (t), P [4] 0,1 (t) are smooth, 1-periodic functions whose Fourier coefficients are given by P [1] 0,1 (t) = P 1,0 (t) = 1 log 2 k∈Z G 1,0 (2 + χ k ) Γ(2 + χ k ) e 2kπit , P [2] 0,1 (t) = − 1 (log 2) 2 k∈Z G ′ 1,0 (2 + χ k )ψ(2 + χ k ) − G 1,0 (2 + χ k ) Γ(2 + χ k ) e 2kπit , P [4] 0,1 (t) = b log 2 k∈Z G 1,0 (2 + χ k ) Γ(1 + χ k ) e 2kπit . Here ψ(z) denotes the derivative of log Γ(z) and all series are absolutely convergent. This proves (44). Variance. Taking second derivatives in (47) and substituting u = v = 0 gives                      0≤j≤b b j f (j) 2,0 (z) = 2f 2,0 (z/2) + 2f 1,0 (z/2) 2 + 4f 1,0 (z/2) + 1, 0≤j≤b b j f (j) 1,1 (z) = 2f 1,1 (z/2) + 2f 2,0 (z/2) + 2(f 1,0 (z/2) +f 0,1 (z/2))(f 1,0 (z/2) + 1), 0≤j≤b b j f (j) 0,2 (z) = 2f 0,2 (z/2) + 4f 1,1 (z/2) + 2f 0,2 (z/2) + 2(f 1,0 (z/2) +f 0,1 (z/2)) 2 , with the initial conditionsf (j) 2,0 (0) = (−1) j−1 for 1 ≤ j < b andf 2,0 (0) =f (j) 1,1 (0) =f (j) 0,2 (0) = 0, for 0 ≤ j < b. The remaining calculations follow the same pattern of proof we used above but become much more involved. We begin with     Ṽ (z) =f 2,0 (z) −f 1,0 (z) 2 − zf ′ 1,0 (z) 2 , U (z) =f 1,1 (z) −f 1,0 (z)f 0,1 (z) − zf ′ 1,0 (z)f ′ 0,1 (z), W (z) =f 0,2 (z) −f 0,1 (z) 2 − zf ′ 0,1 (z) 2 . Then we deduce                      0≤j≤b b j Ṽ (j) (z) = 2Ṽ (z/2) +g 2,0 (z), 0≤j≤b b j Ũ (j) (z) = 2Ũ(z/2) +g 1,1 (z), 0≤j≤b b j W (j) (z) = 2W (z/2) +g 0,2 (z), where                                                                       g 2,0 (z) = 0≤j≤b b j f (j) 1,0 (z) 2 + z 0≤j≤b b j f (j+1) 1,0 (z) 2 − 0≤j≤b b j f 1,0 (z) 2 + zf ′ 1,0 (z) 2 (j) , g 1,1 (z) = 2Ṽ (z/2) + 0≤j≤b b j f (j) 1,0 (z) 0≤j≤b b j f (j) 0,1 (z) + z 0≤j≤b b j f (j+1) 1,0 (z) 0≤j≤b b j f (j+1) 0,1 (z) − 0≤j≤b b j f 1,0 (z)f 0,1 (z) + zf ′ 1,0 (z)f ′ 0,1 (z) (j) , g 0,2 (z) = 4Ũ (z/2) + 2Ṽ (z/2) + 0≤j≤b b j f (j) 0,1 (z) 2 + z 0≤j≤b b j f (j+1) 0,1 (z) 2 − 0≤j≤b b j f 0,1 (z) 2 + zf ′ 0,1 (z) 2 (j) . The initial conditions areṼ (0) =Ũ (j) (0) =W (j) (0) = 0 for 0 ≤ j < b and V (j) (0) = (−1) j 1 + (j − 2)2 j−1 , (1 ≤ j ≤ b). From (49), (50) and Ritt's theorem (see [54]), we have Then we obtain, for ℜ(ω) > 2,     g 2,0 (z) = O |z| −1 , g 1,1 (z) − 2Ṽ (z/2) = O |z| −1 ,                 M [L [Ṽ ; s]; ω] = G 2,0 (ω) 1 − 2 2−ω , M [L [Ũ ; s]; ω] = 2 2−ω G 2,0 (ω) (1 − 2 2−ω ) 2 + G 1,1 (ω) 1 − 2 2−ω , M [L [W ; s]; ω] = 2 5−2ω G 2,0 (ω) (1 − 2 2−ω ) 3 + 2 2−ω (2G 1,1 (ω) + G 2,0 (ω)) (1 − 2 2−ω ) 2 + G 1,1 (ω) + G 0,2 (ω) 1 − 2 2−ω , where                  G 2,0 (ω) := ∞ 0 s ω−1 Q(−2s) b L [g 2,0 ; s] + (s + 1) b−1 − (−1) b (2b − 3 + (b − 1)s) (s + 2) 2 ds, G 1,1 (ω) := ∞ 0 s ω−1 Q(−2s) b ∞ 0 e −sz g 1,1 (z) − 2Ṽ (z/2) dz ds, Digital search trees. II. More shape parameters. We consider in this section four additional examples on DSTs whose variances are essentially linear. The same tools we use readily apply to b-DSTs, but we focus on DSTs because the results are easier to state and the asymptotic behaviors do not differ in essence with those for the more general b-DSTs the corresponding expressions of which are however much messier. The first parameter we consider is the so-called w-parameter (see [16]), which is the sum of the subtreesize of the parent-node of each leaf (over all leaves) 3 . Instead of w-parameter, we call it the total peripheral path-length (PPL), since it measures to some extent the fringe ampleness of the trees. Also this is in consistency with the two previous notions of path-length we distinguished. Then we consider the number of leaves, which has previously been studied in details in [26,31,39] and which is well connected to PPL. Our expression for the variance simplifies known ones. Yet another notion of path-length we consider here is the so-called Colless index in phylogenetics, which is the sum of the absolute difference of the two subtree-sizes of each node (over all nodes). We call this index the total differential path-length (DPL) as it clearly indicates the balance or symmetry of the tree. Another widely used measure of imbalance in phylogenetics is the Sackin index, which is nothing but the external path-length. The last example we consider is the weighted path-length (WPL), which often arises in coding, optimization and many related problems. The orders of the means and the variances exhibited by all the shape parameters we study in this paper are listed in Table 1. Peripheral path-length (PPL) The PPL (or w-parameter) was introduced in [16], the motivations arising from the analysis of compression algorithms. We start from the fringe-size of a leaf node λ, which is defined to be the size of the subtree rooted at its parent-node; see Figure 9. The PPL of a tree is then defined to be the sum of the fringe-sizes of all leaf-nodes. Let X n denote the PPL in a DST built from n random binary strings under our usual independent Bernoulli model. Drmota et al. showed in [16] that E(X n ) = n (C w + ̟ w (log 2 n)) + o(n),(51) where C w := ℓ≥0 (ℓ + 1)(ℓ − 2) Q ℓ 2 ℓ k≥1 1 2 ℓ+k − 1 − 1 + 1 log 2 ℓ≥0 2ℓ − 1 Q ℓ 2 ℓ . Note that by (24), we have the identities ℓ≥0 (ℓ + 1)(ℓ − 2) Q ℓ 2 ℓ = 1 Q ∞   j≥1 1 (2 j − 1) 2 + j≥1 1 2 j + 1 2 − 2   , ℓ≥0 2ℓ − 1 Q ℓ 2 ℓ = 1 Q ∞ j≥1 2 2 j − 1 − 1 . The asymptotic behavior (51) is to be compared with the n log n-order exhibited by most other log-trees such as binary search trees and recursive trees; see [16]. It reflects that most fringes of random DSTs are small in size; see Figure 3. Indeed, since the expected number of leaves is also asymptotic to n times a periodic function, the result (51) implies that the average size of a fringe in random DSTs is bounded. We show that the standard deviation is also small. Defineg z 16 e −z z 4 + 4z 3 + 16z 2 − 8z + 64 − z 4 e −z/2 4(z + 4)f 1 (z/2) − 2(z 2 + 2z + 8)f ′ 1 (z/2) − (z + 2)(z + 8) ,(52)E(X n ) = n (C w + ̟ w (log 2 n)) + O(1), V(X n ) = nP w (log 2 n) + O(1),(53) where P w (t) is a smooth, 1-periodic function with the Fourier series expansion P w (t) = 1 log 2 k∈Z G 2 (2 + χ k ) Γ(2 + χ k ) e 2kπit , the series being absolutely convergent. We provide only the major steps of the proof since it follows the same approach we developed above. Recurrence and generating functions. By definition and by conditioning on the size of one of the subtrees of the root, we have the following different configurations n − 1 n − 1 n − 2 n − 2 k n − 1 − k from which we derive the recurrence for the PPL X n d =      X n−1 , with probability 2 2−n ; n + X n−2 , with probability (n − 1)2 2−n ; X k + X * n−1−k , with probability 2 1−n n−1 k , 2 ≤ k ≤ n − 3, where X 0 = X 1 = 0, X 2 = 2 and X 3 has the distribution X 3 = 6, with probability 1/2; 2, with probability 1/2. From this recurrence, it follows that the bivariate Poisson generating functioñ F (z, y) := e −z n≥0 E(e Xny ) n! z n satisfies the nonlinear equatioñ F (z, y) + ∂ ∂zF (z, y) =F z 2 , y 2 + ze 2y+e y z/2−zF e y z 2 , y − ze −z/2F z 2 , y + z 2 4 e −z e 3y − 1 2 ,(54) with the initial conditionF (0, y) = 1. The expected PPL. By (54), we obtain the differential-functional equation forf 1 (z) by taking derivative with respect to y and then substituting y = 1, giving f 1 (z) +f ′ 1 (z) = 2f 1 (z/2) + z(2 + z/2)e −z/2 ,(55) with f 1 (0) = 0. The Laplace transform off 1 satisfies L [f 1 ; s] = 4 s + 1 L [f 1 ; s] + 16 (1 + 2s) 3 = 16 k≥0 4 k (s + 1) · · · (2 k−1 s + 1)(2 k+1 s + 1) 3 . Then a straightforward application of the Laplace-Mellin-de-Poissonization approach yields E(X n ) = n log 2 k∈Z G 1 (2 + χ k ) Γ(2 + χ k ) n χ k + O(1), where G 1 (ω) := 16 ∞ 0 s ω−1 Q(−s)(2s + 1) 3 ds (ℜ(ω) > 0). The O(1)-term can be further refined by the same analysis. In particular, we get an alternative expression for C w C w = G 1 (2) log 2 = 16 log 2 ∞ 0 s Q(−s)(2s + 1) 3 ds ≈ 1.10302 66959 · · · . That the two expressions of C w are identical can be proved by standard calculus of residues; see [24] for similar details. The variance of the PPL. Again from (54), we derive the equation for the Poisson generating functioñ f 2 (z) of the second moment of X ñ f 2 (z) +f ′ 2 (z) = 2f 2 (z/2) + 2f 1 (z/2) 2 + 9 2 z 2 e −z + ze −z/2 (z + 4)f 1 (z/2) + zf ′ 1 (z/2) + z 2 + 10z + 16 4 ,(56)withf 2 (0) = 0. LetṼ (z) =f 2 (z) −f 1 (z) 2 − zf ′ 1 (z) 2 . Then, by (55), (56) and Lemma 2.7, V (z) +Ṽ ′ (z) = 2Ṽ (z/2) +g 2 (z), withṼ (0) = 0, whereg 2 is defined in (52). Applying again the Laplace-Mellin-de-Poissonization approach, we deduce (53). In particular, the mean value of the periodic function P w is given by G 2 (2) log 2 = 1 log 2 ∞ 0 s Q(−2s) ∞ 0 e −zsg 2 (z) dz ds. The number of leaves The leaves of a tree are the locations where the nodes holding new-coming keys will be connected; thus different types of data fields can be used to save memory, notably for b-DSTs. The number of leaves then provides a quick and simpler look at the "fringes" of a tree. Such nodes are sometimes referred to as the external-internal nodes or internal endnodes in the literature; see [16,26,41,56]. Let X n denote the number of leaves in a random DST of n keys. Then X n satisfies the recurrence X n+1 d = X Bn + X * n−Bn (n ≥ 1),(57) with X 0 = 0 and X 1 = 1, where B n ∼ Binomial(n; 1/2). Flajolet and Sedgewick [26], solving an open question raised by Knuth, showed that E(X n ) = n (C fs + ̟ fs (log 2 n)) + O(n 1/2 ), where ̟ fs (t) is a smooth, 1-periodic function and C fs = 1 + k≥1 k Q k 2 k 1≤j≤k 1 2 j − 1 − 1 Q ∞   1 log 2 + k≥1 1 2 k − 1 2 − k≥1 1 2 k − 1   ≈ 0.37204 86812 · · · . A finer approximation, together with the alternative (and numerically better) expression C fs = 1 + k≥1 1 2 k − 1 − 1 Q ∞ 1 log 2 + k≥1 (−1) k k Q k (2 k − 1)2 k(k+1)/2 , was derived by Kirschenhofer and Prodinger [39]; see also [56]. They proved additionally the asymptotic linearity of the variance V(X n ) ∼ n (C kp + ̟ kp (log 2 n)) , where ̟ kp is a smooth, 1-periodic function with mean zero and a long, complicated expression is given for the leading constant C kp . We derive different forms for these two asymptotic approximations. Defineg 2 (z) = zf ′′ 1 (z) 2 + e −z 1 − e −z (1 + z) + 2zf ′ 1 (z/2) − 4f 1 (z/2) ,(58) wheref 1 (z) := e −z n≥0 E(X n )z n /n!. Theorem 4.2. The mean and the variance of the number of leaves are both asymptotically linear with the approximations E(X n ) = n log 2 k∈Z G 1 (2 + χ k ) Γ(1 + χ k ) n χ k + O(1), V(X n ) = n log 2 k∈Z G 2 (2 + χ k ) Γ(2 + χ k ) n χ k + O(1), where the two series are absolutely convergent with G 1 , G 2 defined by G 1 (ω) = ∞ 0 s ω−1 (s + 1)Q(−2s) ds, G 2 (ω) = ∞ 0 s ω−1 Q(−2s) ∞ 0 e −zsg 2 (z) dz ds, for ℜ(ω) > 0. We see in particular that C fs = 1 log 2 ∞ 0 s (s + 1)Q(−2s) ds, C kp = 1 log 2 ∞ 0 s Q(−2s) ∞ 0 e −zsg 2 (z) dz ds.(59) Sketch of proof. From (57), we derive the equation for the bivariate generating functionF (z, y) := e −z n≥0 E(e Xny )z n /n!F (z, y) + ∂ ∂zF (z, y) =F z 2 , y 2 + (e y − 1) e −z , withF (0, y) = 1. Then the Poisson generating functions of the first two moments satisfỹ f 1 (z) +f ′ 1 (z) = 2f 1 (z/2) + e −z ,(60)f 2 (z) +f ′ 2 (z) = 2f 2 (z/2) + 2f 1 (z/2) 2 + e −z ,withf 1 (0) =f 2 (0). Consequently, the functionṼ (z) :=f 2 (z) −f 1 (z) 2 − zf ′ 1 (z) 2 satisfies V (z) +Ṽ ′ (z) = 2Ṽ (z/2) +g 2 (z), withṼ (0) = 0, whereg 2 is given in (58). The remaining analysis follows the same pattern as above and is omitted. We provide instead some details for the numerical evaluation of the constant C kp as defined in (59), which is similar to the case of internal path-length of DSTs. By applying the Laplace transform to both sides of (60) and by iteration, we get L [f 1 ; s] = k≥0 4 k (s + 1)(2s + 1) · · · (2 k−1 s + 1)(2 k s + 1) 2 . Since the inverse Laplace transform derived from the partial fraction expansion of this series is divergent, we consider the functionf 1 (z) :=f 1 (z) − z + z 2 /2 for which the equation (60) becomeŝ f 1 (z) +f ′ 1 (z) = 2f 1 (z/2) − 1 + z + z 2 4 + e −z , withf 1 (0) = 0, and we have L [f 1 ; s] = 1 2s 3 k≥0 3 · 2 k s + 1 2 k (s + 1) · · · (2 k−1 s + 1)(2 k s + 1) 2 . Then by the partial fraction expansion 3 · 2 k s + 1 (s + 1) · · · (2 k−1 s + 1)(2 k s + 1) 2 = 0≤ℓ<k (−1) k−ℓ (3 · 2 k−ℓ − 1)2 −( k−ℓ+1 2 ) (2 k−ℓ − 1)Q ℓ Q k−ℓ · 1 2 ℓ s + 1 + 1 Q k 3 + 2 1≤j≤k 1 2 j − 1 1 2 k s + 1 − 2 Q k (2 k s + 1) 2 , we obtain L [f 1 ; s] = 1 2s 3 ℓ≥0 1 2 ℓ Q ℓ δ ℓ 2 ℓ s + 1 − 2 (2 ℓ s + 1) 2 , where δ ℓ = 3 + 2 1≤j≤ℓ 1 2 j − 1 + j≥1 (−1) j (3 · 2 j − 1)2 −( j+1 2 ) (2 j − 1)2 j Q j . Obviously, lim ℓ→∞ δ ℓ = 4. Now, by the inverse Laplace transform, f 1 (z) = 1 2 ℓ≥0 1 Q ℓ 2 ℓ δ ℓ 1 − z 2 ℓ + z 2 2 2ℓ+1 − e −z/2 ℓ − 2 ℓ+1 3 − z 2 ℓ−1 + z 2 2 2ℓ+1 − 3e −z/2 ℓ + 2ze −z/2 ℓ , which converges for all z; also from [26] we havê f 1 (z) = n≥3 (−1) n−1 z n n! Q(n − 2) 0≤j≤n−2 1 Q(j) . with M 0 (y) = 1. Let alsõ g 2 (z) := zf ′′ 1 (z) 2 + z −h 1 (z) 2 − zh ′ 1 (z) 2 − 4h 1 (z)f (z/2) − 2zh ′ 1 (z)f ′ 1 (z/2) + 4h c (z), wheref 1 (z) is the Poisson generating function of E(X n ) andh c (z) is defined bỹ h c (z) := e −z n≥0 (z/2) n n! 0≤k≤n n k E(X k )|n − 2k|.(62) Theorem 4.3. The mean and the variance of the DPL of random DSTs satisfy the asymptotic relations E(X n ) = nP d,µ (log 2 n) − √ 2n √ π( √ 2 − 1) + O(1),(63) V(X n ) = 1 − 2 π n log 2 n + nP d,σ (log 2 n) + O(n 1/2 ),(64) where P d,µ and P d,σ are explicitly computable, smooth, 1-periodic functions. These results are to be compared with the known results for random binary search trees for which the DPL has mean of order n log n and variance of order n 2 ; see [4]. Expected DPL. The approach we follow here for deriving the differential-functional equations satisfied by the Poisson generating functions of the first two moments is slightly different from the one we used since the corresponding nonlinear equation for the bivariate generating function F (z, y) := n≥0 M n (y)z n /n! is very involved as given below. ∂ ∂z F (z, y) − 1 = F e y z 2 , y F e −y z 2 , y + 1 2πi |w|=r>0 F wz 2 , y F (e y z/2, y) − w −1 e −y F (z/(2w), y) w − e −y − F (e −y z/2, y) − w −1 e y F (z/(2w), y) w − e y dw, with F (0, y) = 1. We use instead a more elementary argument. From the recurrence (61), we obtain, with µ n := E(X n ), µ n+1 = 2 1−n 0≤k≤n n k µ k + 2 −n 0≤k≤n n k |n − 2k| (n ≥ 1), the initial condition being µ 0 = 0. Then the Poisson generating function of X n satisfies the equatioñ f 1 (z) +f ′ 1 (z) = 2f 1 (z/2) +h 1 (z), withf 1 (0) = 0, whereh 1 is given bỹ h 1 (z) = e −z n≥0 (z/2) n n! 0≤k≤n n k |n − 2k| = ze −z (I 0 (z) + I 1 (z)) , where we used the identity 0≤k≤n n k |n − 2k| = 2n! ⌊n/2⌋!(⌈n/2⌉ − 1)! (n ≥ 1), and I α (z) denotes the modified Bessel functions I α (z) := n≥0 (z/2) 2n+α n!Γ(n + α + 1) . It is known (see [63]) that, as |z| → ∞, I α (z) =    e z √ 2πz 1 + O(|z| −1 ) , if | arg(z)| ≤ π/2 − ε, O |z| −1/2 e ℜ(z) + e −ℜ(z) , if | arg(z)| ≤ π, ,(65) the O-term holding uniformly in z in each case. Thus, by (65),h 1 ∈ JS and h 1 (z) = 2z π 1 + O(|z| −1 ) , for |z| → ∞ in | arg(z)| ≤ π/2 − ε. Also L [h 1 ; s] = (s + 2) −1/2 s −3/2 (ℜ(s) > 0). Thus we can apply the same approach and deduce that E(X n ) = n log 2 k∈Z G 1 (2 + χ k ) Γ(2 + χ k ) n χ k − √ 2n √ 2π( √ 2 − 1) + O(1), where G 1 (ω) is the Mellin transform of L [h 1 ; s]/Q(−2s) G 1 (ω) = ∞ 0 s ω−5/2 Q(−2s) √ s + 2 ds (ℜ(ω) > 3/2). This proves (63). Numerically, the mean value of the dominant periodic function is G 1 (2)/ log 2 ≈ 1.33907 46494. The variance of DPL. Again from (61), we have the recurrence for the second moment s n := E(X 2 n ) s n+1 = 2 −n 0≤k≤n n k s k + s n−k + (n − 2k) 2 + 2µ k µ n−k + 4µ k |n − 2k| , for n ≥ 1 with s 0 = s 1 = 0. Since 2 −n 0≤k≤n n k (n − 2k) 2 = n,(66) we see that the Poisson generating function of s n satisfies the nonlinear equatioñ f 2 (z) +f ′ 2 (z) = 2f 2 (z/2) + 2f 1 (z/2) 2 + z + 4h c (z), withf 2 (0) = 0, whereh c (z) is defined in (62). Lemma 4.4. The functionh c is JS-admissible and satisfies in their incoming order) to the root and w j the weight attached to the j-th node. The calculation of W n in the case of random DSTs can be carried out recursively by W n+1 d = W Bn + W * n−Bn + 2≤j≤n+1 w j , assuming that the root is labelled 1. We consider in this section the case when w j = (log j) m , m ≥ 1. From a technical point of view, it suffices to consider the random variables X n+1 d = X Bn + X * n−Bn + (n + 1)(log(n + 1)) m (n ≥ 0), with X 0 = 0, since the partial sum 2≤j≤n (log j) m is nothing but 2≤j≤n (log j) m = [z n ] L 0,m (z) 1 − z , where L a,m (z) := k≥1 n −a (log k) m z m (a = 1, 2, . . . ), on whose analytic properties our analytic approach heavily relies. The random variables X n represent the sole example on DSTs we discuss in this paper with nonintegral values; they also exhibit an interesting phenomenon in that the mean is of order n(log n) m+1 but the variance is asymptotic to n times a periodic function, in contrast to the orders of DPL. That the variance is linear is well-predicted by the deep theorem of Schachinger derived in [58] since the second difference of the sequence n(log n) m is o(n −1/2−ε ). Our approach has the advantage of providing more precise approximations. The new ingredient we need is incorporated in the following lemma. Indeed, the tools developed in [21] can also be easily extended to similar "toll-functions" such as nH m n . Details are left for the interested readers. Conclusions and extensions We showed in this paper, through many shape parameters on random DSTs that the crucial use of the normalizationṼ (z) :=f 2 (z) −f 1 (z) 2 − zf ′ (z) 2 at the level of Poisson generating function is extremely helpful in simplifying the asymptotic analysis of the variance as well as the resulting expressions. The same idea can be applied to a large number of concrete problems with a binomial splitting procedure. These and some related topics and extensions will be pursued elsewhere. We briefly mention in this final section some extensions and related properties. Central limit theorems. All shape parameters we considered in this paper are asymptotically normally distributed in the sense of convergence in distribution. We describe the results in this section and merely indicate the methods of proofs. The only case that requires a separate study is NPL of random b-DSTs with b ≥ 2 (a bivariate consideration of the limit laws is needed), details being given in a future paper. Theorem 5.1. The internal path-length, the peripheral path-length, the number of leaves, the differential path-length, the weighted path-length of random DSTs, and the key-wise path-length of random b-DSTs with b ≥ 2 are all asymptotically normally distributed X n − E(X n ) V(X n ) d −→ N (0, 1), where X n denotes any of these shape parameters, d −→ stands for convergence in distribution, and N (0, 1) is a standard normal distribution with zero mean and unit variance. See Figure 10 for a plot of the histograms of DPL. The method of moments applies to all these cases and establishes the central limit theorems; similar details are given as in [31] (the asymptotic normality of the number of leaves being already proved there as a special case). In a parallel way, contraction method also works well for all these shape parameters; see [51,52,53]. On the other hand, Schachinger's asymptotic normality results cover the IPL, PPL, number of leaves and WPL, but not PPL and KPL on b-DSTs, although his approach may be modified for that purpose. Finally, the complex-analytic approach used in [35] for internal path-length may be extended to prove some of these cases, but the proofs are messy, although the results established are often stronger (for example, with convergence rate). The depth. The asymptotic analysis we used in this paper can also be extended to the depth (the distance between a randomly chosen internal node and the root) although it is of logarithmic order. Let X n denote the depth of a random DST of n nodes. The starting point is to consider the expected profile polynomial P n (y) := 0≤k<n nP(X n = k)y k , where nP(X n = k) is nothing but the expected number of internal nodes at distance k to the root. Then we have the recurrence P n+1 (y) = 1 + y2 −n 0≤k≤n n k (P k (y) + P n−k (y)) (n ≥ 0), for |y − 1| ≤ ε. More precisely, if t ∈ C lies in a small neighborhood of the origin, then E(e Xnt ) = P n (e t ) n = (e t − 1)Q(e t ) Q(1) log 2 k∈Z Γ −1 − t log 2 − χ k n t/ log 2 +χ k 1 + O n −1 + O(n −1 ), uniformly for |t| ≤ ε. Alternatively, one can also apply the Laplace-Mellin-de-Poissonization approach and obtain the same type of result for not only DSTs but also for more general b-DSTs. See [48,49] for a more general and detailed treatment (by a different approach). The estimate (68) leads to effective asymptotic estimates for all moments of X n − log 2 n by standard arguments; see [32]. In particular, we obtain E(X n ) = log 2 n + γ − 1 log 2 + 1 2 − k≥1 1 2 k − 1 + ̟ 1 (log 2 n) + O n −1 log n , V(X n ) = 1 12 + 1 (log 2) 2 1 + π 2 6 − k≥1 2 k (2 k − 1) 2 + ̟ 5 (log 2 n) + O n −1 log 2 n , where the estimate for the mean is exactly (7) with ̟ 1 given in (8) and ̟ 5 is a smooth periodic function. An analytic extension. From a purely analytic viewpoint, the underlying differential-functional equation (13) for the moments can be extended to an equation of the form 0≤j≤b b j f (j) (z) = αf z β +g(z) (α > 0; β > 1), for which our approach still applies, leading to the functional equation for the Laplace transform , and the corresponding Laplace-Mellin asymptotic analysis is similar. In particular, the case when α = β = m corresponds to a straightforward extension of binary DSTs to m-ary DSTS (and the binary unbiased Bernoulli random variable to the uniform distribution over {0, 1, . . . , m−1}). The stochastic behaviors of all shape parameters on such trees follow the same patterns as showed in this paper. Yet another concrete instance arises in the so-called Eden model studied by Dean and Majumdar [10], which corresponds to α = m and β > 1. The model is constructed in the following way. We start at time t = 0 at which we have an empty node. Then at time t = T , where T ∼ Exponential(1), we fill the empty node and attach to it m different empty nodes. The process then continues independently for each empty node by the following recursive rule. Once an empty node of depth j is attached to a tree at time t = t ′ , it is then filled at time point t ′ + T , where T ∼ E(β j ), and m new empty nodes are attached to it. The mean and the variance of the number of filled nodes at a large time of such trees are studied in details in [10]. Since the model is continuous, there is no need to de-Poissonize to derive the asymptotics of the coefficient; as a consequence, no correction term as we used in this paper is required for the asymptotics of the variance. Other DST-type recurrences. While the technique of Poissonized variance with correction remains useful for the natural case when the Bernoulli random variable is no longer symmetric, the Laplace-Mellin approach does not apply directly. Other asymptotic ingredients are needed such as a direct manipulation of the Mellin transforms; see [49] and the references therein. DST-type structures and recurrences also arise in other statistical physical models such as the diffusionlimited aggregation; see [1,5]. where π n,k := P(B n = k) = n k 2 −n (0 ≤ k ≤ n). The starting point is to consider the recurrence satisfied by the variance v n := V(X n ) v n+1 = 0≤k≤n π n,k (v k + v n−k ) + u n + V(T n ), where µ k := E(X n ) and u n := 0≤k≤n π n,k (µ k + µ n−k − µ n+1 + E(T n )) 2 . In most cases, we have the estimate µ k =f 1 (k) + O(k ε ). This, together with the Gaussian approximation of the binomial distribution, implies that u n ≈ |k−n/2|=o(n 2/3 ) k=n/2+x √ n/2 π n,k f 1 n 2 + x 2 √ n +f 1 n 2 − x 2 √ n −f 1 (n + 1) + E(T n ) 2 ≈ |k−n/2|=o(n 2/3 ) k=n/2+x √ n/2 π n,k 2f 1 n 2 −f 1 (n) −f ′ 1 (n) + E(T n ) 2 ≈ 2f 1 n 2 −f 1 (n) −f ′ 1 (n) + E(T n ) 2 . But then (see (13) below) The order of the difference E(T n ) −h 1 (n) ≈ n|h ′′ 1 (n)| are expected to be small, roughly O(n ε ) in all cases we consider here. Consequently, the variance is asymptotically linear; see [31,58] for more precise details. We see clearly that the smallness of the variance results naturally from the high concentration of the binomial distribution near its mean. Figure 1 : 1Convergence of e −2n j≤k (−2) j τ j (n)/j! to (−1) n for n = 10 (left) and n = 11 (right) for increasing k. Figure 2 : 2A digital search tree of nine binary strings. Figure 3 : 3Two typical random DSTs. Figure 4 : 4Two random DSTs of 1000 nodes rendered differently. For more graphical renderings of random DSTs, see the first author's webpage algo.stat.sinica.edu.tw. where M > 0 0is an arbitrary real number. Consequently, the Mellin transform ofL [f 1 ; s], denoted by M [L ; ω], exists in the half-plane ℜ(ω) ≥ 2 + ε. Then by applying the Mellin transform to (26), we obtain M [L ; - The Mellin transform ofĀ satisfies (ℜ(ω) > 2) Figure 7 : 7A diagrammatic comparison of the major steps used in the Laplace-Mellin (left-half) approach and the Flajolet-Richmond (right-half) approach. Here EGF denotes "exponential generating function", OGF stands for "ordinary generating function" and de-Poi is the abbreviation for de-Poissonization. L Ṽ ; s] = 4L [Ṽ ; 2s] +g ⋆ (s), whereg ⋆ (s) := L [zf ′′ 1 ; s]. Next the normalized Laplace transformL [Ṽ ; s] := L [Ṽ ; [Ṽ ; s] = O(s −2−ε ), as s → 0 + ; O(s −3 ),as s → ∞.From this and the asymptotic expansion(27) of Q(−2s), it follows that the Mellin transform ofL [Ṽ ; s] exists in the half-plane ℜ(ω) ≥ 2 + ε. Consequently, Figure 8 : 8A 2-DST with nine keys. The total key-wise path-length is equal to 4 × 1 + 3 × 2 = 10 and the total node-wise path-length equals 2 × 1 + 3 × 2 = 8. means are needed to be developed if more degree of precision is required.Generating functions. From(37), it follows that the moment generating function M n (y) := E(e Xny ) can be recursively computed by the relation M n+b (yM n (y) = 1 for 0 ≤ n < b. The bivariate exponential generating function F (z, y) then satisfies the equation∂ b ∂z b F (z, y) = F e y z 2 , y 2 ,with F (j) (0, y) = 1 for 0 ≤ j < b, and we have the nonlinear equation for the Poisson generating functioñ F (z, y) := e −z F (z, (s + 1 ) 1b L [f 0,1 ; s] = 4L [f 0,1 ; 2s] + 4L [f 1,0 ; 2s]. LetL [f 0,1 ; s] := L [f 0,1 ; s]/Q(−s) b . g 0,2 (z) − 4Ũ(z/2) − 2Ṽ (z/2) = O |z| −1 , uniformly for |z| → ∞ and | arg(z)| ≤ π/2−ε. LetL [Ã; s] := L [Ã; s]/Q(−s) b , whereà ∈ {Ṽ ,Ũ,W }. e −sz g 0,2 (z) − 2Ṽ (z/2) − 4Ũ (z/2) dz ds, with all functions analytic for ℜ(ω) > 0. Consequently, we deduce(45). Figure 9 : 9The two possible configurations of the fringe of a leaf: the fringe-size (or w-parameter) equals |T | + 2. Note that T may be empty. wheref 1 (z) represents as usual the Poisson generating function of E(X n ). Let G 2 (ω) denote the Mellin transform of L [g 2 ; s]/Q(−2s). Theorem 4. 1 . 1The mean and the variance of the total PPL X n of random DSTs of n strings satisfy Theorem 4. 5 . 5The mean and the variance of the weighted path-length X n are asymptotic toE(X n ) = n(log n) m+1 (m + 1) log 2 + n 1≤j≤m c m,j (log n) j + nP w,µ (log 2 n) + O (log n) m+1 , V(X n ) = nP w,σ (log 2 n) + O (log n) 2m+2 ,respectively, where the c m,j 's are constants depending on m, and P w,µ and P w,σ are 1-periodic, smooth functions. Lemma 4. 6 6([21]). The function L a,m (z) can analytically be continued into the cut-plane C \ [1, ∞) with a sole singularity at z = 1 near which it admits the asymptotic approximationL a,m (e −s ) = Γ(1 − a)s a−1 (− log s) m + O(1), the O-term holding uniformly for | arg(s)| ≤ π − ε. Figure 10 := 0 . 100The histograms of DPL for n = 20, 30, 40 and 50, normalized by their standard deviations.with P 0 (y) = 0. From this relation, we obtain the equation for the Poisson generating functionF (z, y) of P n (It follows, by taking coefficients of z n on both sides and by solving the resulting recurrence, that 44, p. 504] for a different proof. Asymptotic approximation to P n (y) can then be obtained by Rice' (s + 1 ) 1b L [f ; s] = αβL [f ; βs] + L [g; s].The natural normalizing function is then provided by 1 (n) −f ′ 1 (n) + E(T n ) = E(T n ) −h 1 Table 1 : 1Orders of the means and the variances of all shape parameters in this paper; those marked with an * are for b-DSTs with b ≥ 2. Here PL denotes path-length and m ≥ 0. Normalizing factor: Dividing both sides of (19) by Q(−2s) = j≥0 (1+s/2 j ) gives a functional equation whereL [f ; s] := L [f ; s]/Q(−s). transform: The Mellin transform ofL then satisfiesof the formL [f ; s] = 4L [f ; 2s] + L [g; s] Q(−2s) , Mellin The first formal use of the term "variance" in its statistical sense is generally attributed to R. A. Fisher in his 1918 paper (see[20] or Wikipedia's webpage on variance), although its practical use in diverse scientific disciplines predated this by a few centuries (including closely-defined terms such as mean-squared errors and standard deviations). For a better comparison with the approach we use, our differs from the usual Euler transform by a factor of s. The leaves or leaf-nodes of a tree are nodes without any descendants. (z) := zf ′′ 1 (z) 2 − AcknowledgementWe thank the referee for opportune helpful comments and the more precise title.Then the first and the second derivatives are given bŷNow the constant C kp can be expressed in terms of the integrals off 1 as follows.And we get C kp ≈ 0.034203 · · · .A general weighted sum of node-types for b-DSTs. For b ≥ 2, we can consider X[j]n , 1 ≤ j ≤ b, the number of leaves containing j records in a random b-DST with bucket capacity b built from n records. Let also X[b+1] n be the number of internal (non-leaf) nodes. Definewhere a 1 , . . . , a b+1 are arbitrary real numbers. By a straightforward computation 0≤j≤b b j ∂ j ∂z jF (z, y) = e a b+1 yF z 2 , y 2 + e −z (e a b y − e a b+1 y ) , withF (0, y) = 1. Then our approach can be applied and leads to the same type of results as Theorem 4.2 with different G 1 and G 2 ; the resulting expressions for the variance are more explicit and simpler than those given in[31].Colless index: the differential path-length (DPL)The DPL of a tree is defined to be the sum over all nodes of the absolute difference of the two subtree-sizes of each node as depicted below.Properties of such a path length in random binary search trees have long been investigated in the systematic biology literature; see[4]and the references therein.Let X n denote the DPL of a random DST of n input-strings. Then by definition and by our independence assumption, we have the recurrence for the moment generating functionin the sector | arg(z)| ≤ π/2 − ε.Proof. Observe first thatOn the other hand, since f 1 (z) = n≥0 µ n z n /n!, we have, by the same argument,To prove condition (O), we start with changes of variables, givingwhere the first integration circle is indented to the right to avoid the polar singularity w = z, and the second to the left. By splitting each integration contour into two parts, we obtainwhere the integration contour ⊃ is any path connecting the two endpoints |z|e ±iε and indented to the right, and ⊂ denotes the corresponding symmetric contour with respect to w = z (and indented to the left). Sincẽ f 1 ∈ JS , condition (O) forh c (z) is readily checked.For condition (I), it suffices to prove (67). For that purpose, we use the representatioñis analytic at w = z. The error term ofh c (z) −h 1 (z)f 1 (z/2) can be estimated by a similar argument as that used for checking condition (O). This completes the proof of the Lemma.The remaining analysis is now routine. LetṼ (z) :where, by Lemma 2.7,for | arg(z)| ≤ π/2 − ε. From this and the analytic properties of the functions involved, we deduce (64).Remark. The same approach can be extended to more general differential path-length of the form all nodes |T left − T right | m with m ≥ 2. Interestingly, when m = 2, the mean is identical to the total internal path-length in view of (66) and the variance is asymptotic to 4n 2 . For m > 2, the mean and the variance are asymptotic torespectively.A weighted path-length (WPL)Weighted path-lengths of the form W n := 1≤j≤n w j ℓ j appear often in applications, where ℓ j denotes the distance of the j-th node (arranged in an appropriate manner, say first level-wise and then left-to-right orAppendix. An Elementary Approach to the Asymptotic Linearity of the Variance.We describe briefly here a direct elementary approach to the variance of random variables satisfying the recurrence X n+1 d = X Bn + X * n−Bn + T n , A diffusion limit for a class of randomly-growing binary trees. D Aldous, P Shields, Probab. Theory Related Fields. 794D. Aldous and P. Shields. A diffusion limit for a class of randomly-growing binary trees. Probab. Theory Related Fields, 79(4):509-542, 1988. Limit theorems for the number of maxima in random samples from planar regions. Z.-D Bai, H.-K Hwang, W.-Q Liang, T.-H Tsai, Electron. J. Probab. 63Z.-D. Bai, H.-K. Hwang, W.-Q. Liang, and T.-H. Tsai. Limit theorems for the number of maxima in random samples from planar regions. Electron. J. Probab., 6:no. 3, 41 pp. (electronic), 2001. Ramanujan's notebooks. Part I. B C Berndt, S. ChandrasekharSpringer-VerlagNew YorkB. C. Berndt. Ramanujan's notebooks. Part I. Springer-Verlag, New York, 1985. With a foreword by S. Chandrasekhar. The mean, variance and limiting distribution of two statistics sensitive to phylogenetic tree balance. M G B Blum, O François, S Janson, Ann. Appl. Probab. 164M. G. B. Blum, O. François, and S. Janson. The mean, variance and limiting distribution of two statistics sensitive to phylogenetic tree balance. Ann. Appl. Probab., 16(4):2195-2214, 2006. Directed aggregation on the bethe lattice: Scaling, mappings, and universality. R M Bradley, P N Strenski, Phys. Rev. B. 317R. M. Bradley and P. N. Strenski. Directed aggregation on the bethe lattice: Scaling, mappings, and universality. Phys. Rev. B, 31(7):4319-4328, Apr 1985. Analysis in distribution of two randomized algorithms for finding the maximum in a broadcast communication model. W.-M Chen, H.-K Hwang, J. Algorithms. 462W.-M. Chen and H.-K. Hwang. Analysis in distribution of two randomized algorithms for finding the maximum in a broadcast communication model. J. Algorithms, 46(2):140-177, 2003. Phase changes in random point quadtrees. H.-H Chern, M Fuchs, H.-K Hwang, Art. 12ACM Trans. Algorithms. 3251H.-H. Chern, M. Fuchs, and H.-K. Hwang. Phase changes in random point quadtrees. ACM Trans. Algorithms, 3(2):Art. 12, 51, 2007. An asymptotic theory for Cauchy-Euler differential equations with applications to the analysis of algorithms. H.-H Chern, H.-K Hwang, T.-H Tsai, J. Algorithms. 441Analysis of algorithmsH.-H. Chern, H.-K. Hwang, and T.-H. Tsai. An asymptotic theory for Cauchy-Euler differential equa- tions with applications to the analysis of algorithms. J. Algorithms, 44(1):177-225, 2002. Analysis of algorithms. File structures using hashing functions. E G Coffman, Jr , J Eve, Commun. ACM. 137E. G. Coffman, Jr. and J. Eve. File structures using hashing functions. Commun. ACM, 13(7):427- 432, 1970. Phase transition in a generalized Eden growth model on a tree. D S Dean, S N Majumdar, J. Stat. Phys. 1246D. S. Dean and S. N. Majumdar. Phase transition in a generalized Eden growth model on a tree. J. Stat. Phys., 124(6):1351-1376, 2006. Renewals for exponentially increasing lifetimes, with an application to digital search trees. F Dennert, R Grübel, Ann. Appl. Probab. 172F. Dennert and R. Grübel. Renewals for exponentially increasing lifetimes, with an application to digital search trees. Ann. Appl. Probab., 17(2):676-687, 2007. A study of trie-like structures under the density model. L Devroye, Ann. Appl. Probab. 22L. Devroye. A study of trie-like structures under the density model. Ann. Appl. Probab., 2(2):402- 434, 1992. Universal limit laws for depths in random trees. L Devroye, SIAM J. Comput. 282L. Devroye. Universal limit laws for depths in random trees. SIAM J. Comput., 28(2):409-432 (electronic), 1999. The variance of the height of digital search trees. M Drmota, Acta Inform. 384M. Drmota. The variance of the height of digital search trees. Acta Inform., 38(4):261-276, 2002. Random trees. SpringerWienNewYork, Vienna, 2009. An interplay between combinatorics and probability. M Drmota, M. Drmota. Random trees. SpringerWienNewYork, Vienna, 2009. An interplay between combina- torics and probability. On the shape of the fringe of various types of random trees. M Drmota, B Gittenberger, A Panholzer, H Prodinger, M D Ward, Math. Methods Appl. Sci. 3210M. Drmota, B. Gittenberger, A. Panholzer, H. Prodinger, and M. D. Ward. On the shape of the fringe of various types of random trees. Math. Methods Appl. Sci., 32(10):1207-1245, 2009. Un)expected behavior of digital search tree profile. M Drmota, W Szpankowski, SODA. M. Drmota and W. Szpankowski. (Un)expected behavior of digital search tree profile. In SODA, pages 130-138, 2009. A Erdélyi, W Magnus, F Oberhettinger, F Tricomi, Higher transcendental functions. I. McGraw-HillNew YorkA. Erdélyi, W. Magnus, F. Oberhettinger, and F. Tricomi. Higher transcendental functions. Vol. I. McGraw-Hill(New York), 1953. On a functional equation arising in the analysis of a protocol for a multi-access broadcast channel. G Fayolle, P Flajolet, M Hofri, Adv. in Appl. Probab. 182G. Fayolle, P. Flajolet, and M. Hofri. On a functional equation arising in the analysis of a protocol for a multi-access broadcast channel. Adv. in Appl. Probab., 18(2):441-472, 1986. The correlation between relatives on the supposition of mendelian inheritance. Philosophical Transactions of the. R A Fisher, Royal Society of Edinburgh. 52R. A. Fisher. The correlation between relatives on the supposition of mendelian inheritance. Philo- sophical Transactions of the Royal Society of Edinburgh, 52:399-433, 1918. Singularity analysis and asymptotics of Bernoulli sums. P Flajolet, Theoret. Comput. Sci. 2151-2P. Flajolet. Singularity analysis and asymptotics of Bernoulli sums. Theoret. Comput. Sci., 215(1- 2):371-381, 1999. Mellin transforms and asymptotics: harmonic sums. P Flajolet, X Gourdon, P Dumas, Theoret. Comput. Sci. 1441-2Special volume on mathematical analysis of algorithmsP. Flajolet, X. Gourdon, and P. Dumas. Mellin transforms and asymptotics: harmonic sums. Theoret. Comput. Sci., 144(1-2):3-58, 1995. Special volume on mathematical analysis of algorithms. Singularity analysis of generating functions. P Flajolet, A Odlyzko, SIAM J. Discrete Math. 32P. Flajolet and A. Odlyzko. Singularity analysis of generating functions. SIAM J. Discrete Math., 3(2):216-240, 1990. Generalized digital trees and their difference-differential equations. P Flajolet, B Richmond, Random Structures Algorithms. 33P. Flajolet and B. Richmond. Generalized digital trees and their difference-differential equations. Random Structures Algorithms, 3(3):305-320, 1992. The complexity of generating an exponentially distributed variate. P Flajolet, N Saheb, J. Algorithms. 74P. Flajolet and N. Saheb. The complexity of generating an exponentially distributed variate. J. Algorithms, 7(4):463-488, 1986. Digital search trees revisited. P Flajolet, R Sedgewick, SIAM J. Comput. 153P. Flajolet and R. Sedgewick. Digital search trees revisited. SIAM J. Comput., 15(3):748-767, 1986. Mellin transforms and asymptotics: finite differences and Rice's integrals. P Flajolet, R Sedgewick, Theoret. Comput. Sci. 1441-2Special volume on mathematical analysis of algorithmsP. Flajolet and R. Sedgewick. Mellin transforms and asymptotics: finite differences and Rice's inte- grals. Theoret. Comput. Sci., 144(1-2):101-124, 1995. Special volume on mathematical analysis of algorithms. Analytic combinatorics. P Flajolet, R Sedgewick, Cambridge University PressCambridgeP. Flajolet and R. Sedgewick. Analytic combinatorics. Cambridge University Press, Cambridge, 2009. On the history of series expansions of frequency functions and sampling distributions, 1873-1944. Matematisk-Fysiske Meddelelser. 49. Copenhagen: The Royal Danish Academy of Sciences and Letters. A Hald, A. Hald. On the history of series expansions of frequency functions and sampling distributions, 1873- 1944. Matematisk-Fysiske Meddelelser. 49. Copenhagen: The Royal Danish Academy of Sciences and Letters., 2002. On the variance of the internal path length of generalized digital trees -the Mellin convolution approach. F Hubalek, Theoret. Comput. Sci. 2421-2F. Hubalek. On the variance of the internal path length of generalized digital trees -the Mellin convolution approach. Theoret. Comput. Sci., 242(1-2):143-168, 2000. A multivariate view of random bucket digital search trees. F Hubalek, H.-K Hwang, W Lew, H Mahmoud, H Prodinger, J. Algorithms. 441Analysis of algorithmsF. Hubalek, H.-K. Hwang, W. Lew, H. Mahmoud, and H. Prodinger. A multivariate view of random bucket digital search trees. J. Algorithms, 44(1):121-158, 2002. Analysis of algorithms. On convergence rates in the central limit theorems for combinatorial structures. H.-K Hwang, European J. Combin. 193H.-K. Hwang. On convergence rates in the central limit theorems for combinatorial structures. Eu- ropean J. Combin., 19(3):329-343, 1998. Analysis of a stack algorithm for csma-cd random length packet communication. P Jacquet, E Merle, IEEE Transactions on Information Theory. 362P. Jacquet and E. Merle. Analysis of a stack algorithm for csma-cd random length packet communi- cation. IEEE Transactions on Information Theory, 36(2):420-426, 1990. Normal limiting distribution of the size of tries. P Jacquet, M Régnier, Performance'87 (Brussels, 1987). North-Holland, AmsterdamP. Jacquet and M. Régnier. Normal limiting distribution of the size of tries. In Performance'87 (Brussels, 1987), pages 209-223. North-Holland, Amsterdam, 1988. Asymptotic behavior of the Lempel-Ziv parsing scheme and [in] digital search trees. P Jacquet, W Szpankowski, Theoret. Comput. Sci. 1441-2Special volume on mathematical analysis of algorithmsP. Jacquet and W. Szpankowski. Asymptotic behavior of the Lempel-Ziv parsing scheme and [in] digital search trees. Theoret. Comput. Sci., 144(1-2):161-197, 1995. Special volume on mathematical analysis of algorithms. Analytical de-Poissonization and its applications. P Jacquet, W Szpankowski, Theoret. Comput. Sci. 2011-2P. Jacquet and W. Szpankowski. Analytical de-Poissonization and its applications. Theoret. Comput. Sci., 201(1-2):1-62, 1998. Average profile of the Lempel-Ziv parsing scheme for a Markovian source. P Jacquet, W Szpankowski, J Tang, Algorithmica. 313Mathematical analysis of algorithmsP. Jacquet, W. Szpankowski, and J. Tang. Average profile of the Lempel-Ziv parsing scheme for a Markovian source. Algorithmica, 31(3):318-360, 2001. Mathematical analysis of algorithms. Rounding of continuous random variables and oscillatory asymptotics. S Janson, Ann. Probab. 345S. Janson. Rounding of continuous random variables and oscillatory asymptotics. Ann. Probab., 34(5):1807-1826, 2006. Eine Anwendung der Theorie der Modulfunktionen in der Informatik. P Kirschenhofer, H Prodinger, Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II4-7P. Kirschenhofer and H. Prodinger. Eine Anwendung der Theorie der Modulfunktionen in der Infor- matik.Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II, 197(4-7):339-366, 1988. Further results on digital search trees. P Kirschenhofer, H Prodinger, Thirteenth International Colloquium on Automata, Languages and Programming. Rennes58P. Kirschenhofer and H. Prodinger. Further results on digital search trees. Theoret. Comput. Sci., 58(1-3):143-154, 1988. Thirteenth International Colloquium on Automata, Languages and Pro- gramming (Rennes, 1986). On some applications of formulae of Ramanujan in the analysis of algorithms. P Kirschenhofer, H Prodinger, Mathematika. 381P. Kirschenhofer and H. Prodinger. On some applications of formulae of Ramanujan in the analysis of algorithms. Mathematika, 38(1):14-33, 1991. Digital search trees again revisited: the internal path length perspective. P Kirschenhofer, H Prodinger, W Szpankowski, SIAM J. Comput. 233P. Kirschenhofer, H. Prodinger, and W. Szpankowski. Digital search trees again revisited: the internal path length perspective. SIAM J. Comput., 23(3):598-616, 1994. Asymptotic behavior of the height in a digital search tree and the longest phrase of the Lempel-Ziv scheme. C Knessl, W Szpankowski, SIAM J. Comput. 303C. Knessl and W. Szpankowski. Asymptotic behavior of the height in a digital search tree and the longest phrase of the Lempel-Ziv scheme. SIAM J. Comput., 30(3):923-964 (electronic), 2000. The art of computer programming. D E Knuth, Addison-Wesley Publishing Co3Reading, MassSorting and searching. second editionD. E. Knuth. The art of computer programming. Volume 3: Sorting and searching. Addison-Wesley Publishing Co., Reading, Mass., second edition, 1998. A note on growing binary trees. A G Konheim, D J Newman, Discrete Math. 4A. G. Konheim and D. J. Newman. A note on growing binary trees. Discrete Math., 4:57-63, 1973. Exact and asymptotic distributions in digital and binary search trees. G Louchard, RAIRO Inform. Théor. Appl. 214G. Louchard. Exact and asymptotic distributions in digital and binary search trees. RAIRO Inform. Théor. Appl., 21(4):479-495, 1987. Digital search trees revisited. G Louchard, Cahiers CentreÉtudes Rech. Opér. 36Hommageà Simone HuyberechtsG. Louchard. Digital search trees revisited. Cahiers CentreÉtudes Rech. Opér., 36:259-278, 1994. Hommageà Simone Huyberechts. Average profile and limiting distribution for a phrase size in the Lempel-Ziv parsing algorithm. G Louchard, W Szpankowski, IEEE Trans. Inform. Theory. 412G. Louchard and W. Szpankowski. Average profile and limiting distribution for a phrase size in the Lempel-Ziv parsing algorithm. IEEE Trans. Inform. Theory, 41(2):478-488, 1995. Average profile of the generalized digital search tree and the generalized Lempel-Ziv algorithm. G Louchard, W Szpankowski, J Tang, SIAM J. Comput. 283G. Louchard, W. Szpankowski, and J. Tang. Average profile of the generalized digital search tree and the generalized Lempel-Ziv algorithm. SIAM J. Comput., 28(3):904-934 (electronic), 1999. Evolution of random search trees. H M Mahmoud, Wiley-Interscience Series in Discrete Mathematics and Optimization. New YorkWiley-Interscience PublicationH. M. Mahmoud. Evolution of random search trees. Wiley-Interscience Series in Discrete Mathemat- ics and Optimization. John Wiley & Sons Inc., New York, 1992. A Wiley-Interscience Publication. On a multivariate contraction method for random recursive structures with applications to Quicksort. R Neininger, Analysis of algorithms. Krynica Morska19R. Neininger. On a multivariate contraction method for random recursive structures with applica- tions to Quicksort. Random Structures Algorithms, 19(3-4):498-524, 2001. Analysis of algorithms (Krynica Morska, 2000). A general limit theorem for recursive algorithms and combinatorial structures. R Neininger, L Rüschendorf, Ann. Appl. Probab. 141R. Neininger and L. Rüschendorf. A general limit theorem for recursive algorithms and combinatorial structures. Ann. Appl. Probab., 14(1):378-418, 2004. A survey of multivariate aspects of the contraction method. R Neininger, L Rüschendorf, Discrete Math. Theor. Comput. Sci. 81R. Neininger and L. Rüschendorf. A survey of multivariate aspects of the contraction method. Dis- crete Math. Theor. Comput. Sci., 8(1):31-56 (electronic), 2006. Asymptotics and special functions. F W J Olver, Computer Science and Applied Mathematics. Academic PressF. W. J. Olver. Asymptotics and special functions. Academic Press, 1974. Computer Science and Applied Mathematics. Paths in a random digital tree: limiting distributions. B Pittel, Adv. in Appl. Probab. 181B. Pittel. Paths in a random digital tree: limiting distributions. Adv. in Appl. Probab., 18(1):139-155, 1986. External internal nodes in digital search trees via Mellin transforms. H Prodinger, SIAM J. Comput. 216H. Prodinger. External internal nodes in digital search trees via Mellin transforms. SIAM J. Comput., 21(6):1180-1183, 1992. Hypothetical analyses: approximate counting in the style of Knuth, path length in the style of Flajolet. H Prodinger, Theoret. Comput. Sci. 1001H. Prodinger. Hypothetical analyses: approximate counting in the style of Knuth, path length in the style of Flajolet. Theoret. Comput. Sci., 100(1):243-251, 1992. On the variance of a class of inductive valuations of data structures for digital search. W Schachinger, Theoret. Comput. Sci. 1441-2Special volume on mathematical analysis of algorithmsW. Schachinger. On the variance of a class of inductive valuations of data structures for digital search. Theoret. Comput. Sci., 144(1-2):251-275, 1995. Special volume on mathematical analysis of algorithms. Asymptotic normality of recursive algorithms via martingale difference arrays. W Schachinger, Discrete Math. Theor. Comput. Sci. 42W. Schachinger. Asymptotic normality of recursive algorithms via martingale difference arrays. Discrete Math. Theor. Comput. Sci., 4(2):363-397 (electronic), 2001. The evaluation of an alternative sum with applications to the analysis of some data structures. W Szpankowski, Inform. Process. Lett. 281W. Szpankowski. The evaluation of an alternative sum with applications to the analysis of some data structures. Inform. Process. Lett., 28(1):13-19, 1988. A characterization of digital search trees from the successful search viewpoint. W Szpankowski, Algorithms Automat. Complexity Games. 851Theoret. Comput. Sci.W. Szpankowski. A characterization of digital search trees from the successful search viewpoint. Theoret. Comput. Sci., 85(1, Algorithms Automat. Complexity Games):117-134, 1991. Average case analysis of algorithms on sequences. W Szpankowski, Wiley-Interscience Series in Discrete Mathematics and Optimization. Philippe FlajoletNew YorkWiley-InterscienceW. Szpankowski. Average case analysis of algorithms on sequences. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience, New York, 2001. With a foreword by Philippe Flajolet. A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions. E T Whittaker, G N Watson, Cambridge University PressCambridge Mathematical Library; Cambridgewith an account of the principal transcendental functions. fourth editionE. T. Whittaker and G. N. Watson. A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions. Cambridge Mathematical Library. Cambridge University Press, Cambridge, fourth edition, 1927.
{'fraction_non_alphanumeric': 0.10406091370558376, 'fraction_numerical': 0.04238401192715913, 'mean_word_length': 3.4120986687548944, 'pattern_counts': {'":': 0, '<': 35, '<?xml version=': 0, '>': 36, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 133, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'Asymptotics of the variances of many cost measures in random digital search trees are often notoriously messy and involved to obtain. A new approach is proposed to facilitate such an analysis for several shape parameters on random symmetric digital search trees. Our approach starts from a more careful normalization at the level of Poisson generating functions, which then provides an asymptotically equivalent approximation to the variance in question. Several new ingredients are also introduced such as a combined use of the Laplace and Mellin transforms and a simple, mechanical technique for justifying the analytic de-Poissonization procedures involved. The methodology we develop can be easily adapted to many other problems with an underlying binomial distribution. In particular, the less expected and somewhat surprising n(log n) 2 -variance for certain notions of total path-length is also clarified.', 'arxivid': '1001.0095', 'author': ['Hsien-Kuei Hwang \nInstitute of Statistical Science Academia Sinica Taipei 115\nDepartment of Applied Mathematics National Chiao\nInstitute of Statistical Science Academia Sinica\nTung University\nHsinchu 300115TaipeiTaiwan, Taiwan, Taiwan\n', 'Michael Fuchs \nInstitute of Statistical Science Academia Sinica Taipei 115\nDepartment of Applied Mathematics National Chiao\nInstitute of Statistical Science Academia Sinica\nTung University\nHsinchu 300115TaipeiTaiwan, Taiwan, Taiwan\n', 'Vytas Zacharovas \nInstitute of Statistical Science Academia Sinica Taipei 115\nDepartment of Applied Mathematics National Chiao\nInstitute of Statistical Science Academia Sinica\nTung University\nHsinchu 300115TaipeiTaiwan, Taiwan, Taiwan\n'], 'authoraffiliation': ['Institute of Statistical Science Academia Sinica Taipei 115\nDepartment of Applied Mathematics National Chiao\nInstitute of Statistical Science Academia Sinica\nTung University\nHsinchu 300115TaipeiTaiwan, Taiwan, Taiwan', 'Institute of Statistical Science Academia Sinica Taipei 115\nDepartment of Applied Mathematics National Chiao\nInstitute of Statistical Science Academia Sinica\nTung University\nHsinchu 300115TaipeiTaiwan, Taiwan, Taiwan', 'Institute of Statistical Science Academia Sinica Taipei 115\nDepartment of Applied Mathematics National Chiao\nInstitute of Statistical Science Academia Sinica\nTung University\nHsinchu 300115TaipeiTaiwan, Taiwan, Taiwan'], 'corpusid': 13025114, 'doi': '10.46298/dmtcs.498', 'github_urls': [], 'n_tokens_mistral': 43271, 'n_tokens_neox': 38028, 'n_words': 21359, 'pdfsha': 'b5de918bd9098a3bb04b399eadc5868928734d0b', 'pdfurls': ['https://arxiv.org/pdf/1001.0095v2.pdf'], 'title': ['Asymptotic variance of random symmetric digital search trees', 'Asymptotic variance of random symmetric digital search trees'], 'venue': []}
arxiv
7 May 2012 MAY 2012 William J Demeo Dissertation Committee ChairpersonRalph Freese Dissertation Committee William Lampe Dissertation Committee J B Nation Dissertation Committee Peter Jipsen Dissertation Committee Nick Kaiser Dissertation Committee William J Demeo Dissertation Committee Susan Hasegawa Dissertation Committee Shirley Kikiloi Dissertation Committee Troy Ludwick Dissertation Committee 7 May 2012 MAY 2012arXiv:1204.4305v3 [math.GR] CONGRUENCE LATTICES OF FINITE ALGEBRAS A DISSERTATION SUBMITTED TO THE GRADUATE DIVISION OF THE UNIVERSITY OF HAWAI'I AT MĀNOA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS ACKNOWLEDGMENTSFirst, I would like to thank my advisor, Ralph Freese, for his patience, support, and expert guidance, without which I could not have completed this dissertation. Next, I thank the members of my dissertation committee, Peter Jipsen, Bill Lampe, and J.B. Nation. All have made significant contributions to this work. Bill Lampe, in particular, is responsible for introducing me to the beautiful subject of universal algebra.I thank Nick Kaiser for agreeing to act as the University Representative on my dissertation committee, and for enduring long meetings about topics unrelated to his area of expertise (though I suspect he understands far more than he lets on).A number of other professors played a significant role in my mathematical training. Among them, I would especially like to thank Ron Brown, Tom Craven, Erik Guentner, Bjørn Kjos-Hanssen, Tom Ramsey, and Wayne Smith. Mike Hilden was kind enough to administer my French language exam, and I thank him for his help with this minor hurdle, and for not setting the bar too high.The Mathematics Department at the University of Hawai'i has generously supported me through the doctoral program, and for that I am grateful. I would also like to thank other members of the department who have played vital roles in my progress through the program; in particular, I thank LIST OF SYMBOLS the subuniverse of A generated by the set X ⊆ A Cg A (X) the congruence of A generated by the set X ⊆ A × A Eq(X) the lattice of equivalence relations on the set X X X the set of unary maps from a set X into itself ker f the kernel of f , {(x, y) | f (x) = f (y)} ID(X) the idempotent decreasing functions in X X ⊑ the partial order defined on ID(X) by f ⊑ g ⇔ ker f ker g K a class of algebras H(K ) the class of homomorphic images of algebras in K S(K ) the class of subalgebras of algebras in K P(K ) the class of direct products of algebras in K P fi (K ) the class of finite direct products of algebras in K V a variety, or equational class, of algebras V(A) the variety generated by A (thus V(A) = HSP(A) V(K ) the variety generated by the class K F V (X) the free algebra in the variety V over the generating set X L 0 the class of finite lattices L 1 the class of lattices isomorphic to sublattices of finite partition lattices L 2 the class of lattices isomorphic to strong congruence lattices of finite partial algebras L 3 the class of lattices isomorphic to congruence lattices of finite algebras L 4 the class of lattices isomorphic to intervals in subgroup lattices of finite groups L 5 the class of lattices isomorphic to subgroup lattices of finite groups x Part I Background 1 INTRODUCTION We begin with an informal overview of some of the basic objects of study. This will help to fix notation and motivate our discussion. (Italicized terms are defined more formally in later sections or in the appendix.) Then we introduce the problem that is the main focus of this dissertation, the finite lattice representation problem (FLRP). In subsequent sections, we give further notational and algebraic prerequisites and summarize the well known results surrounding the FLRP. In the final section of this chapter we provide a list of the new results of this thesis. Motivation and problem statement Among the most basic objects of study in all of mathematics are algebras. An algebra A = A, F consists of a nonempty set A and a collection F of operations; the most important examples are lattices, groups, rings, and modules. To understand a particular algebra, A, we often study its representations, which are homomorphisms from A into some other algebra B. A very important feature of such a homomorphism ϕ is its kernel, which we define as the set {(x, y) ∈ A 2 | ϕ(x) = ϕ(y)}. This is a congruence relation of the algebra A which tells us how A is "reduced" when represented by its image under ϕ in B. Thus, every homomorphism gives rise to a congruence relation, and the set Con A of all congruence relations of the algebra A forms a lattice. For example, if A happens to be a group, Con A is isomorphic to the lattice of normal subgroups of A. 1 To each congruence θ ∈ Con A there corresponds the natural homomorphism of A onto A/θ which has θ as its kernel. Thus, there is a one-to-one correspondence between Con A and the natural homomorphisms, and the shape of Con A provides useful information about the algebra and its representations. For instance, Con A tells us whether and how A can be decomposed as, or embedded in, a product of simpler algebras. Given an arbitrary algebra, then, we ought to know whether there are, a priori, any restrictions on the possible shape of its congruence lattice. A celebrated result of Grätzer and Schmidt says that there are (essentially) no such restrictions. Indeed, in [18] it is proved that every (algebraic) lattice is the congruence lattice of some algebra. Moreover, as Jiří Tůma proves in [45], the Grätzer-Schmidt Theorem still holds if we restrict ourselves to intervals in subgroup lattices. That is, every algebraic lattice is isomorphic to an interval in the subgroup lattice of an (infinite) group. Now, suppose we restrict our attention to finite algebras. Given an arbitrary finite algebra, it is natural to ask whether there are any restrictions (besides finiteness) on the shape of its congruence lattice. If it turns out that, given an arbitrary finite lattice L, we can always find a finite algebra A that has L as its congruence lattice, then apparently there are no such restrictions. We call a lattice finitely representable, or simply representable, if it is isomorphic to the congruence lattice of a finite algebra, and deciding whether every finite lattice is representable is known as the finite lattice representation problem (FLRP). For the reasons mentioned above, this is a fundamental question of modern algebra, and the fact that it remains unanswered is quite remarkable. Universal algebra preliminaries We now describe in greater detail some of the algebraic objects that are central to our work. A more complete introduction to this material can be found in the books and articles listed in the bibliography. In particular, the following are the main references for this work: [26], [32], [12], [38], and [20]. Two excellent survey articles on the finite lattice representation problem are [29] and [30]. First, a few words about notation. When discussing universal algebras, such as A = A, F , we denote the algebras using bold symbols, as in A, B, . . . , and reserve the symbols A, B, . . . for the universes of these algebras. However, this convention becomes tiresome and inconvenient if strictly adhered to for all algebras, and we often find ourselves referring to an algebra by its universe. For example, we frequently use L when referring to the lattice L = L, ∨, ∧ , and we usually refer to "the lattice of congruence relations Con A, F ," even though it would be more precise to call Con A, F the universe (a set) and use Con A = Con A, F , ∧, ∨ to denote the lattice (an algebra). Certainly we will feel free to commit this sort of abuse when speaking about groups, preferring to use G when referring to the group G = G, ·, −1 , 1 . Sometimes we use the more precise notation Eq(X) to denote the lattice of equivalence relations on the set X, but more frequently we will refer to this lattice by its universe, Eq(X). This has never been a source of confusion. An operation symbol f is an object that has an associated arity, which we denote by a(f ). A set of operation symbols F is called a similarity type. An algebra of similarity type F is a pair A = A, F A consisting of a set A, which we call the universe of A, and a set F A = {f A : f ∈ F } of operations on A, which are functions f A : A a(f ) → A of arity a(f ). Occasionally the set of operations only enters the discussion abstractly, and it becomes unnecessary to refer to specific operation symbols. In such instances, we often denote the algebra by A, . . . . Note that the symbol f -like the operation symbol + that is used to denote addition in some algebras -is an abstract operation symbol which, apart from its arity, has no specific meaning attached to it. We use the notation f A to signify that we have given the operation symbol a specific interpretation as an operation in the algebra A. Having said that, when there is only one algebra under consideration, it seems pedantic to attach the superscript A to every operation. In such cases, when no confusion can arise, we allow the operation symbol f to denote a specific operation interpreted in the algebra. Also, if F is the set of operations (or operation symbols) of A, we let F n ⊆ F denote the n-ary operations (or operation symbols) of A. Let A and B be sets and let ϕ : A → B be any mapping. We say that a pair (a 0 , a 1 ) ∈ A 2 belongs to the kernel of ϕ, and we write (a 0 , a 1 ) ∈ ker ϕ, provided ϕ(a 0 ) = ϕ(a 1 ). It is easily verified that ker ϕ is an equivalence relation on the set A. If θ is an equivalence relation on a set A, then a/θ denotes the equivalence class containing a; that is, a/θ := {a ′ ∈ A | (a, a ′ ) ∈ θ}. The set of all equivalence classes of θ in A is denoted A/θ. That is, A/θ = {a/θ | a ∈ A}. Let A = A, F A and B = B, F B be algebras of the same similarity type. A homomorphism from A to B is a function ϕ : A → B that respects the interpretation of the operation symbols. That is, if f ∈ F with, say, n = a(f ), and if a 1 , . . . , a n ∈ A, then ϕ(f A (a 1 , . . . , a n )) = f B (ϕ(a 1 ), . . . , ϕ(a n )). A congruence relation of A is the kernel of a homomorphism defined on A. We denote the set of all congruence relations of A by Con A. Thus, θ ∈ Con A if and only if θ = ker ϕ for some homomorphism ϕ : A → B. It is easy to check that this is equivalent to the following: θ ∈ Con A if and only if θ ∈ Eq(A) and for all n (a i , a ′ i ) ∈ θ (0 i < n) ⇒ (f (a 0 , . . . , a n−1 ), f (a ′ 0 , . . . , a ′ n−1 )) ∈ θ, (1.2.1) for all f ∈ F n and all a 0 , . . . , a n−1 , a ′ 0 , . . . , a ′ n−1 ∈ A. Equivalently, Con A = Eq(A) ∩ Sub(A × A). Given a congruence relation θ ∈ Con A, the quotient algebra A/θ is the algebra with universe A/θ = {a/θ | a ∈ A} and operations {f A/θ | f ∈ F } defined as follows: f A/θ (a 1 /θ, . . . , a n /θ) = f A (a 1 , . . . , a n )/θ, where n = a(f ). A partial algebra is a set A (the universe) along with a set of partial operations, that is, operations which may be defined on only part of the universe. A strong congruence relation of a partial algebra A is an equivalence relation θ ∈ Eq(A) with the following property: for each (partial) operation f of A, if f is k-ary, if (x i , y i ) ∈ θ (1 i k) , and if f (x 1 , . . . , x k ) exists, then f (y 1 , . . . , y k ) exists, and (f (x 1 , . . . , x k ), f (y 1 , . . . , y k )) ∈ θ. We will have very little to say about partial algebras, but they appear below in our overview of significant results related to the FLRP. By a unary algebra we mean an algebra with any number of unary operations. 2 In our work, as we are primarily concerned with congruence lattices, we may restrict our attention to unary algebras whenever helpful or convenient, as the next result shows (cf. Theorem 4.18 of [26]). algebraic if and only if every element is a join of compact elements, we see that subgroup lattices are always algebraic. We mention these facts because of their general importance, but we remind the reader that all groups in this work are finite. Overview of well known results Major inroads toward a solution to the FLRP have been made by many prominent researchers, including Michael Aschbacher, Walter Feit, Hans Kurzweil, Adrea Lucchini, Ralph McKenzie, Raimund Netter, Péter Pálfy, Pavel Pudlák, John Snow, and Jiří Tůma, to name a few. We will have occasion to discuss and apply a number of their results in the sequel. Here we merely mention some of the highlights, in roughly chronological order. In his 1968 book Universal Algebra [19], George Grätzer defines the following classes of lattices: • L 0 = the class of finite lattices; • L 1 = the class of lattices isomorphic to sublattices of finite partition lattices; • L 2 = the class of lattices isomorphic to strong congruence lattices of finite partial algebras; • L 3 = the class of lattices isomorphic to congruence lattices of finite algebras. Clearly L 0 ⊇ L 1 ⊇ L 2 ⊇ L 3 . Grätzer asks ( [19] prob. 13, p. 116) whether equality holds in each case. Whether L 0 = L 1 is the finite version of a question Garrett Birkhoff had asked by 1935. In [6] Birkhoff asks whether every lattice is isomorphic to a sublattice of some partition lattice. Whitman [47] answered this affirmatively in 1946, but his proof embeds every finite lattice in a countably infinite partition lattice. Still, the result of Whitman also proves that there is no non-trivial law that holds in the subgroup lattice of every group. That is, Theorem 1.3.1 (Whitman [47]). Every lattice is isomorphic to a sublattice of the subgroup lattice of some group. Confirmation that L 0 = L 1 did not come until the late 1970's, when Pavel Pudlák and Jiří Tůma published [35], in which they prove that every finite lattice can be embedded in a finite partition lattice, thus settling this important and long-standing open question. This result also yields the following finite analogue of Whitman's result: Theorem 1.3.2 (Pudlák-Tůma [35]). Every finite lattice is isomorphic to a sublattice of the subgroup lattice of some finite group. If we confine ourselves to distributive lattices, the analogue of the FLRP is relatively easy. By the 1930's it was already known to Robert Dilworth that every finite distributive lattice is the congruence lattice of a finite lattice. 3 (In fact, if we allow representations by infinite algebras -which, as a rule in this work, we do not -then the congruence lattices of modular lattices already account for all distributive lattices. This is shown by E.T. Schmidt in [40], and extended by Ralph Freese who shows in [15] that finitely generated modular lattices suffice.) 4 A lattice L is called strongly representable if, whenever L is isomorphic to a spanning sublattice 5 L 0 Eq(X) for some X, then there is an algebra X, . . . whose congruence lattice is L 0 . Theorem 1.3.3 (Berman [5], Quackenbush and Wolk [36]). Every finite distributive lattice is strongly representable. (We give a short proof of this result in Section 3.3.3.) Berman also proves that if A p is a finite partial unary algebra with strong congruence lattice Con s A p , then there is a finite unary algebra A with Con A ∼ = Con s A p . Therefore, by Lemma 1.2.1, L 2 = L 3 . As our focus is mainly on whether L 0 = L 3 , we will not say more about partial algebras except to note that the results of Pudlák, Tůma, and Berman imply that L 0 = L 3 holds if and only if L 1 = L 2 holds. Next, we mention another deep result of Pudlák and Tůma, which proves the existence of congruence lattice representations for a large class of lattices. Theorem 1.3.4 (Pudlák and Tůma [34]). Let L be a finite lattice such that both L and its congruence lattice have the same number of join irreducible elements. Then L is representable. Notice that finite distributive lattices satisfy the assumption of Theorem 1.3.4, so this provides yet another proof that such lattices are representable. We now turn to subgroup lattices of finite groups and their connection with the FLRP. The study of subgroup lattices has a long history, starting with Richard Dedekind's work [10] in 1877, including Ada Rottlaender's paper [39] 4 It turns out that the finite distributive lattices are representable as congruence lattices of other restricted classes of algebras. We will say a bit more about this below, but we refer the reader to [28] for more details. 5 By a spanning sublattice of a bounded lattice L 0 , we mean a sublattice L L 0 that has the same top and bottom as L 0 . That is 1 L = 1 L 0 and 0 L = 0 L 0 . That is [H, G] is the lattice of subgroups of G that contain H. 6 We define the following classes of lattices: • L 4 = the class of lattices isomorphic to intervals in subgroup lattices of finite groups; • L 5 = the class of lattices isomorphic to subgroup lattices of finite groups. Recall that L 3 , the class of all lattices isomorphic to congruence lattices of finite algebras, is known as the class of representable lattices. We adhere to this convention throughout and, moreover, we will call a lattice group representable if it belongs to L 4 . Clearly, L 4 ⊇ L 5 , since Sub(G) is itself the interval [1, G]. Moreover, it's easy to find a lattice that is in L 4 but not it L 5 , so the inclusion is strict. For example, there is no group G for which Sub(G) is isomorphic to the lattice shown below. To see this, note that if G has a unique maximal subgroup H, then there exists g ∈ G \ H and we must have g = G. Thus, if Sub(G) has a unique coatom, then G is cyclic, and subgroup lattices of cyclic groups are self-dual, unlike the lattice shown above. However, this lattice belongs to L 4 . For example, it is the filter above H = C 3 in the subgroup lattice of G = C 3 × (C 3 ⋊ C 4 ). We will have a lot more to say about intervals in subgroup lattices throughout this thesis. Perhaps the most useful fact for our work is the following: Every interval in a subgroup lattice is the congruence lattice of a finite algebra. (1.3.1) In particular, as we explain below in Chapter 4, if G/H, G is the algebra consisting of the group G acting on the left (right) cosets of a subgroup H G by left (right) multiplication, then Con G/H, G ∼ = [H, G]. Thus, we see that L 3 ⊇ L 4 . Whether the converse of (1.3.1) holds -and thus whether L 3 = L 4 -is an open question. In other words, it is not known whether every congruence lattice of a finite algebra is isomorphic to an 6 The reader may anticipate confusion arising from the conflict between our notation and the well-established notation for the commutator subgroup, [H, G] := {hgh −1 g −1 | h ∈ H, g ∈ G} , which we will also have occasion to use. However, we have found that context always makes clear which meaning is intended. In any case, we often refer to "the interval [H, G]" or "the commutator [H, G]." interval in the subgroup lattice of a finite group. However, a surprising and deep result related to this question was proved in 1980 by Péter Pálfy and Pavel Pudlák. In [32], they prove Theorem 1.3.5. The following statements are equivalent: (i) Every finite lattice is isomorphic to the congruence lattice of a finite algebra. (ii) Every finite lattice is isomorphic to the congruence lattice of a finite transitive G-set. As we will see later (Theorem 4.1.2), statement (ii) is equivalent to (ii) ′ Every finite lattice is isomorphic to an interval in the subgroup lattice of a finite group. It is important to note that Theorem 1.3.5 does not say L 3 = L 4 . Rather, it says that L 0 = L 3 if and only if L 0 = L 4 . Moreover, this result implies that if we prove the existence of a lattice which is not isomorphic to an interval in a subgroup lattice of a finite group, then we have solved the FLRP. It is surprising that a problem about general algebras can be reduced to a problem about such a special class of algebras -finite transitive G-sets. Also surprising, in view of all that we know about finite groups and their actions, is that we have yet to determine whether these statements are true or false. To put it another way, given an arbitrary finite lattice L, it is unknown whether there must be a finite group having this lattice as an interval in its lattice of subgroups. We pause for a moment to consider the L 3 = L 4 question in the restricted case of finite distributive lattices (which we know are strongly representable). Silcock [42] and Pálfy [28] prove that every finite distributive lattice is an interval in the subgroup lattice of some finite solvable group. The main result is stated below as Theorem 1.3.7, and this can be combined with the following easy lemma to establish the claim. Beyond those mentioned in this brief introduction, many other results surrounding the FLRP have been proven. Some of these are not as relevant to our work, and others will be discussed in detail in Chapter 2. A more complete overview of the FLRP with an emphasis on group theory can be found in the articles by Pálfy, [29] and [30]. Lemma 1.3.6. If D = {(g, g) ∈ G × G | g ∈ G} then the interval [D, G × G] is isomorphic to the lattice of normal subgroups of G. CHAPTER 2 AN OVERVIEW OF FINITE LATTICE REPRESENTATIONS In this chapter we give a brief overview of various known methods for representing a given lattice as the congruence lattice of a finite algebra or proving that such a representation exists. In later chapters we describe these methods in greater detail and show how to apply them. In particular, in Section 6.2, we use them along with some new methods to show that, with one possible exception, every lattice with no more than seven elements is isomorphic to the congruence lattice of a finite algebra. Throughout this chapter, we continue to use L 3 to denote the class of finite lattices that are isomorphic to congruence lattices of finite algebras. Again, we call the lattices that belong to L 3 representable lattices. Closure properties of the class of representable lattices This section concerns closure properties of the class L 3 . More precisely, if O is an operation that can be applied to a lattice or collection of lattices, we say that L 3 is closed under O provided O(K ) ⊆ L 3 for all K ⊆ L 3 . For example, if S(K ) = {all sublattices of lattices in K }, then it is clearly unknown whether L 3 is closed under S, for otherwise the FLRP would be solved. (Clearly, Eq(X) ∈ L 3 for every finite set X -take the algebra to be the set X with no operations. Then Con X, ∅ = Eq(X). So, if L 3 were closed under S, then L 3 would contain all finite lattices, by the result of Pudlák and Tůma mentioned above; that is, L 0 = L 1 .) The following is a list of known closure properties of L 3 and the names of those who first (or independently) proved them. We discuss some of these results in greater detail later in this section. The class L 3 of lattices isomorphic to congruence lattices of finite algebras is closed under 1. lattice duals 1 (Hans Kurzweil [23] and Raimund Netter [27], 1986), 2. interval sublattices (follows from Kurzweil-Netter), 3. direct products (Jiří Tůma [45], 1986), 4. ordinal sums (Ralph McKenzie [25], 1984; John Snow [43], 2000), 5. parallel sums (John Snow [43], 2000), 6. certain sublattices of lattices in L 3 -namely, those which are obtained as a union of a filter and an ideal of a lattice in L 3 (John Snow [43], 2000). L 1 L 2 L 1 L 2 α β Figure 2.1: The ordinal (left) and parallel (middle) sum of the lattices L 1 and L 2 ; a sublattice obtained as a union of a filter α ↑ and an ideal β ↓ (right). Remarks. 1. The first result says that if L is representable then so is the dual of L. 2. It follows from item 1. that any interval sublattice of a representable lattice is representable. For, let [α, β] := {θ ∈ L | α θ β} be an interval in the representable lattice L = Con A. 3. Of course, by direct products we mean finite direct products. 6. The property in item 6. is very useful and we discuss it further in Section 2.3 below, where we present a very short proof of this result. It will come up again in Section 6 when we prove the existence of representations of small lattices. Whether the class L 3 is closed under homomorphic images seems to be an open question. Lattice duals: the theorem of Kurzweil and Netter As mentioned above, the class L 3 -the lattices isomorphic to congruence lattices of finite algebrasis closed under dualization. That is, if L is representable, then so is the dual of L. This was proved in 1986 by Raimund Netter [27], generalizing the idea of his advisor, Hans Kurzweil [23]. Though Kurzweil's article did appear (in German), it is unclear whether Netter's article was ever published. In this section we present a proof of their result. The argument requires a fair bit of machinery, but it is a nice idea and well worth the effort. 2 If G is a group and X a set, then the set {f | X → G} of functions from X into G is denoted by G X . This is a group with binary operation (f, g) → f ·g, where, for each x ∈ X, (f ·g)(x) = f (x)g(x) is simply multiplication in the group G. The identity of the group G X is of course the constant map f (x) = 1 G for all x ∈ X. Let X be a finite totally ordered set, with order relation , and consider the set X X of functions mapping X into itself. The subset of X X consisting of functions that are both idempotent and decreasing 3 will be denoted by ID(X). That is, ID(X) = {f ∈ X X | f 2 = f and ∀x f (x) x}. Define a partial order ⊑ on the set ID(X) by f ⊑ g ⇔ ker f ker g, (2.2.1) where ker f = {(x, y) | f (x) = f (y)}. It is easy to see that f ⊑ g holds if and only if gf = g. Moreover, under this partial ordering ID(X) is a lattice which is isomorphic to Eq(X) (viz. the map Θ : Eq(X) → ID(X) given by Θ(α) = f α , where f α (x) = min{y ∈ X | (x, y) ∈ α}.) Suppose S is a finite nonabelian simple group, and consider S n , the direct power of n copies of S. An element of S n may be viewed as a map from the set n = {0, 1, . . . , n − 1} into S. Thus, if x = (x 0 , x 1 , . . . , x n−1 ) ∈ S n , then by ker x we mean the relation (i, j) ∈ ker x if and only if x i = x j . The set of constant maps is a subgroup D < S n , sometimes called the diagonal subgroup; that is, D = {(s, s, . . . , s) | s ∈ S} S n . 2 We learned of the main argument used in the proof from slides of a series of three lectures given by Péter Pálfy in 2009 [31]. Pálfy gives credit for the argument to Kurzweil and Netter. 3 When we say that the map f is decreasing we mean f (x) x for all x. (We do not mean x y implies f (y) x.) For each f ∈ ID(n), define K f = {(x f (0) , x f (1) , . . . , x f (n−1) ) | x f (i) ∈ S, i = 0, 1, . . . , n − 1}. Then D K f S n , and K f is the set of maps K f = {xf ∈ S n | x ∈ S n }; i.e., compositions of the given map f ∈ n n , followed by any x ∈ S n . Thus, K f = {y ∈ S n | ker f ker y}. For example, if f = (0, 0, 2, 3, 2) ∈ ID (5), then ker f = |0, 1|2, 4|3| and K f is the subgroup of all (y 0 , y 1 , . . . , y 4 ) ∈ S 5 having y 0 = y 1 and y 2 = y 4 . That is, K f = {(x 0 , x 0 , x 2 , x 3 , x 2 ) | x ∈ S 5 }. Lemma 2.2.1. The map f → K f is a dual lattice isomorphism from Eq(n) onto the interval sublattice [D, S n ] Sub(S n ). Proof. This is clear since ID(n) is ordered by ( [27]). If the finite lattice L is representable (as the congruence lattice of a finite algebra), then so is the dual lattice L ′ . Proof. Without loss of generality, we assume that L is concretely represented as L = Con n, F . By Lemma 1.2.1, we can further assume that F consists of unary operations: F ⊆ n n . As above, let S be a nonabelian simple group and let D be the diagonal subgroup of S n . Then the unary algebra S n /D, S n is a transitive S n -set which (by Theorem 4.1.2 below) has congruence lattice isomorphic to the interval [D, S n ]. By Lemma 2.2.1, this is the dual of the lattice Eq(n). That is, Con S n /D, S n ∼ = (Eq(n)) ′ . Now, each operation ϕ ∈ F gives rise to an operation on S n by composition: ϕ(s) =φ(s 0 , s 1 . . . , s n−1 ) = (s ϕ(0) , s ϕ(1) . . . , s ϕ(n−1) ). Thus, ϕ induces an operation on S n /D since, for d = (d, d, . . . , d) ∈ D and s ∈ S n we have sd = (s 0 d, s 1 d, . . . , s n−1 d) andφ(sd) = (s ϕ(0) d, s ϕ(1) d, . . . , s ϕ(n−1) d) =φ(s)d, soφ(sD) =φ(s)D. Finally, add the set of operationsF = {φ | ϕ ∈ F } to S n /D, S n , yielding the new algebra S n /D, S n ∪F , and observe that a congruence θ ∈ Con S n /D, S n remains a congruence of S n /D, S n ∪F if and only if it correponds to a partition on n that is invariant under F . Union of a filter and ideal The lemma in this section was originally proved by John Snow using primitive positive formulas. Since it provides such a useful tool for proving that certain finite lattices are representable as congruence lattices, we give our own direct proof of the result below. In Chapter 6 we use this lemma to prove the existence of representations of a number of small lattices. Before stating the lemma, we need a couple of definitions. (These will be discussed in greater detail in Section 3.2.) Given a relation θ ⊆ X × X, we say that the map f : X n → X respects θ and we write f (θ) ⊆ θ provided (x i , y i ) ∈ θ implies (f (x 1 , . . . , x n ), f (y 1 , . . . , y n )) ∈ θ. For a set L ⊆ Eq(X) of equivalence relations we define λ(L) = {f ∈ X X : (∀θ ∈ L) f (θ) ⊆ θ}, which is the set of all unary maps on X which respect all relations in L. Lemma 2.3.1. Let X be a finite set. If L Eq(X) is representable and L 0 L is a sublattice with universe α ↑ ∪ β ↓ where α ↑ = {x ∈ L | α x} and β ↓ = {x ∈ L | x β} for some α, β ∈ L, then L 0 is representable. θ L 0 L α β Proof. Assume L 0 ≇ 2, otherwise the result holds trivially. Since L Eq(X) is representable, we have L = Con X, λ(L) (cf. Section 3.2). Take an arbitrary θ ∈ L \ L 0 . Since θ / ∈ α ↑ , there is a pair (a, b) ∈ α \ θ. Since θ / ∈ β ↓ , there is a pair (u, v) ∈ θ \ β. Define h ∈ X X as follows: h(x) =        a, x ∈ u/β, b, otherwise. Then, β ker h = (u/β) 2 ∪((u/β) c ) 2 , where (u/β) c denotes the complement of the β class containing u. Therefore, h respects every γ β. Furthermore, (a, b) ∈ γ for all γ α, so h respects every γ above α. This proves that h ∈ λ(L 0 ). Now, θ was arbitrary, so we have proved that for every θ ∈ L \ L 0 there exists a function in λ(L 0 ) which respects every γ ∈ α ↑ ∪ β ↓ = L 0 , but violates θ. Finally, since L 0 L, we have λ(L) ⊆ λ(L 0 ). Combining these observations, we see that every θ ∈ Eq(X) \ L 0 is violated by some function in λ(L 0 ). Therefore, L 0 = Con X, λ(L 0 ) . Ordinal sums The following theorem is a consequence of McKenzie's shift product construction [25]. A more direct proof of Theorem 2.4.1 follows the argument given by John Snow in [43]. As noted above, Jiří Tůma proved that the class of finite representable lattices is closed under direct products. Thus, if L 1 and L 2 are representable, then so is L 1 × L 2 . Now note that the adjoined ordinal sum of L 1 and L 2 is the union, α ↑ ∪ β ↓ , of a filter and ideal in the lattice L 1 × L 2 , where α = β = 1 L1 × 0 L2 . Therefore, Part II Finite Lattice Representations In this chapter we introduce a strategy that has proven very useful for showing that a given lattice is representable as a congruence lattice of a finite algebra. We call it the closure method, and it has become especially useful with the advent of powerful computers which can search for such representations. Here, as above, Eq(X) denotes the lattice of equivalence relations on X. Sometimes we abuse notation and take Eq(X) to mean the lattice of partitions of the set X. This has never caused problems because these two lattices are isomorphic. Concrete versus abstract representations As Bjarni Jónsson explains in [21], there are two types of representation problems for congruence lattices, the concrete and the abstract. The concrete representation problem asks whether a specific family of equivalence relations on a set A is equal to Con A for some algebra A with universe A. The abstract representation problem asks whether a given lattice is isomorphic to Con A for some algebra A. These two problems are closely related, and have become even more so since the publication in 1980 of [35], in which Pavel Pudlák and Jiří Tůma prove that every finite lattice can be embedded as a spanning sublattice 1 of the lattice Eq(X) of equivalence relations on a finite set X. Given this result, we see that even if our goal is to solve the abstract representation problem for some (abstract) lattice L, then we can embed L into Eq(X) as L ∼ = L 0 Eq(X), for some finite set X, and then try to solve the concrete representation problem for L 0 . A point of clarification is in order here. The term representation has become a bit overused in the literature about the finite lattice representation problem. On the one hand, given a finite lattice L, if there is a finite algebra A such that L ∼ = Con A, then L is called a representable lattice. On the other hand, given a sublattice L 0 Eq(X), if L 0 ∼ = L, then L 0 is sometimes called a concrete representation of the lattice L (whether or not it is the congruence lattice of an algebra). Below we will define the notion of a closed concrete representation, and if we have this special kind of concrete representation of a give lattice, then that lattice is indeed representable in the first sense. As we will see below, there are many examples in which a particular concrete representation L 0 Eq(X) of L is not a congruence lattice of a finite algebra. (In fact, we will describe general situations in which we can guarantee that there are no non-trivial 2 operations which respect the equivalence relations of L 0 .) This does not imply that L / ∈ L 3 . It may simply mean that L 0 is not the "right" concrete representation of L, and perhaps we can find some other L ∼ = L 1 Eq(X) such that L 1 = Con X, λ(L 1 ) . The closure method The idea described in this section first appeared in Topics in Universal Algebra [21], pages 174- Let X X denote the set of all (unary) maps from the set X to itself, and let Eq(X) denote the lattice of equivalence relations on the set X. If θ ∈ Eq(X) and h ∈ X X , we write h(θ) ⊆ θ and say that "h respects θ" if and only if for all (x, y) ∈ X 2 (x, y) ∈ θ implies (h(x), h(y)) ∈ θ. If h(θ) θ, we sometimes say that "h violates θ." For L ⊆ Eq(X) define λ(L) = {h ∈ X X : (∀θ ∈ L) h(θ) ⊆ θ}. For H ⊆ X X define ρ(H) = {θ ∈ Eq(X) | (∀h ∈ H) h(θ) ⊆ θ}. The map ρλ is a closure operator on Sub[Eq(X)]. That is, ρλ is • idempotent: 3 ρλρλ = ρλ; • extensive: L ⊆ ρλ(L) for every L Eq(X); • order preserving: ρλ(L) ρλ(L 0 ) if L L 0 . Given L Eq(X), if ρλ(L) = L, then we say L is a closed sublattice of Eq(X), in which case we 2 By a non-trivial function we mean a function that is not constant and not the identity. 3 In fact, ρλρ = ρ and λρλ = λ. clearly have L = Con X, λ(L) . This suggests the following strategy for solving the representation problem for a given abstract finite lattice L: search for a concrete representation L ∼ = L 0 Eq(X), compute λ(L 0 ), compute ρλ(L 0 ), and determine whether ρλ(L 0 ) = L 0 . If so, then we have solved the abstract representation problem for L, by finding a closed concrete representation, or simply closed representation, of L 0 . We call this strategy the closure method. We now state without proof a well known theorem which shows that the finite lattice representation problem can be formulated in terms of closed concrete representations (cf. [21]). In the remaining sections of this chapter, we consider various aspects of the closure method and prove some results about it. Later, in Section 6.2, we apply it to the problem of finding closed representations of all lattices of small order. Before proceeding, however, we introduce a slightly different set-up than the one introduced above that we have found particularly useful for implementing the closure method on a computer. Instead of considering the set of equivalence relations on a finite set, we work with the set of idempotent decreasing maps. These were introduced above in Section 2.2, but we briefly review the definitions here for convenience. Given a totally ordered set X, let the set ID(X) = {f ∈ X X : f 2 = f and f (x) x} be partially ordered by ⊑ as follows: f ⊑ g ⇔ ker f ker g. As noted above, this makes ID(X) into a lattice that is isomorphic to Eq(X). Define a relation R on X X × ID(X) as follows: (h, f ) ∈ R ⇔ (∀(x, y) ∈ ker f ) (h(x), h(y)) ∈ ker f. If h R f , we say that h respects f . Let F = P(ID(X)) and H = P(X X ) be partially ordered by set inclusion, and define the maps λ : F → H and ρ : H → F as follows: λ(F ) = {h ∈ X X : ∀f ∈ F, h R f } (F ∈ F ) ρ(H) = {f ∈ ID(X) : ∀h ∈ H, h R f } (H ∈ H ) The pair (λ, ρ) defines a Galois correspondence between ID(X) and X X . That is, λ and ρ are antitone maps such that λρ id H and ρλ id F . In particular, for any set F ∈ F we have F ⊆ ρλ(F ). These statements are all trivial verifications, and a couple of easy consequences are: 1. ρλρ = ρ and λρλ = λ, 2. ρλ and λρ are idempotent. Since the map ρλ from F to itself is idempotent, extensive, and order preserving, it is a closure operator on F , and we say a set F ∈ F is closed if and only if ρλ(F ) = F . Equivalently, F is closed if and only if F = ρ(H) for some H ∈ H . Superbad representations In this section we describe what is in some sense the worst kind of concrete representation. Given an abstract finite lattice L, it may happen that, upon computing the closure of a particular representation L ∼ = L 0 Eq(X), we find that ρλ(L 0 ) is all of Eq(X). We call such an L 0 a dense sublattice of Eq(X), or more colloquially, a superbad representation of L. More generally, if A and B are subsets of ID(X), we say that A is dense in B if and only if ρλ(A) ⊇ B. If L is a finite lattice and there exists an embedding L ∼ = L 0 Eq(X) such that ρλ(L 0 ) = Eq(X), we say that L can be densely embedded in Eq(X). Density One of the first questions we asked concerned the 5-element modular lattice, denoted M 3 (sometimes called the diamond; see Figure 3.3.1). We asked for which sets X does the lattice of equivalence relations on X contain a dense M 3 sublattice. The answer is given by This basically says that, when |X| 5, the lattice of equivalences on X contains a spanning diamond L with the property that every non-trivial operation in X X violates some equivalence relation in the universe L of L. Thus, the closure ρλ(L) is all of Eq(X). John Snow proved this for |X| odd. Using the same technique (and some rather tedious calculations), we verified that the result holds for |X| even as well. Before moving on to the next result, we note that the necessity part of the proposition above is obvious. For, if |X| 2, then Eq(X) has no M 3 sublattice. If |X| = 3, then Eq(X) is itself M 3 . It can be checked directly (by computing all possibilities) that, when |X| = 4, Eq(X) has one closed M 3 sublattice and five M 3 sublattices that are neither closed nor dense. For ease of notation, let Eq(n) denote the set of equivalence relations on an n-element set, and let M n denote the (n + 2)-element lattice of height two (Figure 3.2). Thus, every M n can be densely embedded in Eq(X) for some finite set X. Proof. (sketch) We begin with Snow's example of a dense M 3 sublattice of Eq(X), where X = results in a dense M 5 in Eq(X). Proceeding inductively, when |X| = 2n+1 there are n+1 partitions of the form α i = |x i0 |x i1 , x i2 | · · · |x i2n−1 , x i2n |, and one of the form α n+2 = |evens|odds|, with the following properties: 1. α i ∧ α j = 0 X , 2. α i ∨ α j = 1 X , 3. the lattice generated by α n+2 and at least two other α i is dense in Eq(X). Non-density The results in this section give sufficient conditions under which a lattice cannot be densely embedded in a lattice of equivalence relations. These results require some standard terminology that we have not yet introduced, so we begin the section with these preliminaries. As always, we will only deal with finite lattices L = L, ∧, ∨ , and we use 0 L = L to denote the bottom of L and 1 L = L to denote the top. If L = L, ∧, ∨ is a lattice, a non-empty subset I ⊆ L is called an ideal of L if (i) I is a down-set: if α ∈ I and β α, then β ∈ I; (ii) I is closed under finite joins: α, β ∈ I implies α ∨ β ∈ I. A filter of a lattice is defined dually as a non-empty up-set that is closed under finite meets. An ideal or filter is said to be proper if it is not equal to all of L. The smallest ideal that contains a given element α is a principal ideal and α is said to be a principal element or generator of the ideal in this situation. The principal ideal generated by α is defined and denoted by α ↓ = {θ ∈ L | θ α}. Similarly, α ↑ = {θ ∈ L | θ α} is the principal filter generated by α. An ideal I called a prime ideal provided α ∧ β ∈ I implies α ∈ I or β ∈ I for all α, β ∈ L. Equivalently, a prime ideal is an ideal whose set-theoretic complement is a filter. Since we require ideals (filters) to be non-empty, every prime filter (ideal) is necessarily proper. An element is called meet prime if it is the generator of a principal prime ideal. Equivalently, α ∈ L \ {1 L } is meet prime if for all β, γ ∈ L we have β ∧ γ α implies β α or γ α. Join prime is defined dually. (i) There is an element α ∈ L \ {0 L } such that {γ ∈ L : γ α} < 1 L . (ii) There is an element α ∈ L \ {1 L } such that {γ ∈ L : γ α} > 0 L . (iii) L is the union of a proper principal ideal and a proper principal filter. Proof. (i) ⇒ (ii): Suppose α ∈ L \ {0 L } is such that the element α ′ = {γ : γ α} is strictly below 1 L , and consider {γ : γ α ′ }. If β α ′ , then β / ∈ {γ : γ α} so β α. Therefore, {γ : γ α ′ } α > 0 L . Thus α ′ ∈ L \ {1 L } is such that {γ : γ α ′ } > 0 L so (ii) holds. (ii) ⇒ (iii): Let α < 1 L be such that β = {γ : γ α} > 0 L . Then, L = α ↓ ∪ β ↑ satisfies (iii). (iii) ⇒ (i): Suppose L = α ↑ ∪ β ↓ for some α > 0 L , β < 1 L . Then {γ ∈ L : γ α} ⊆ β ↓ ; i.e. γ α ⇒ γ β. Therefore, {γ : γ α} β < 1 L , so (i) holds. Proof. Suppose L ≇ 2 is a sublattice of Eq(X) which satisfies condition (i) of the lemma. We must show that there is a non-trivial (i.e. non-constant, non-identity) h ∈ X X which respects every θ ∈ L. By condition (i), there is an element α ∈ L \ {0 L } such that β = {γ ∈ L : γ α} is strictly below 1 L . Since α > 0 L , there is a pair (u, v) of distinct elements of X that are α related. Since β < 1 L , there is a β equivalence class B X. Define h ∈ X X as follows: Then h is not constant, since ∅ = B = X; h is not the identity, since L ≇ 2; h respects everything above α and everything below β, and therefore, h ∈ λ(α ↑ ∪ β ↓ ) = λ(L). Proof. The theorem says that, for any embedding L ∼ = L 0 Eq(X) of such a lattice, L 0 is not dense in Eq(X); i.e. ρλ(L 0 ) Eq(X). To prove that this follows from Lemma 3.3.4, we must verify the following statement: If 2 ≇ L Eq(X) and if there is a non-trivial unary function h ∈ λ(L), then ρλ(L) Eq(X). h(x) =        u, x ∈ B, v, x / ∈ B. If h ∈ X X is any non-trivial unary function, then there are elements {x, y, u, v} of X such that x = y and h(x) = u = v = h(y). We can assume X has at least three distinct elements since L ≇ 2. There are two cases to consider. In the first, h simply permutes x and y. In this case, x = v and y = u, and h(v) = u, h(u) = v. There must be a third element of X, say, w / ∈ {u, v}. If h(w) = u, then h violates any equivalence that puts v, w in the same block and puts u and h(w) in separate blocks. If h(w) = v, then h violates any equivalence that puts u, w in the same block and v and h(w) in separate blocks. In the second case to consider, {x, u, v} are three distinct elements. In this case, h violates every relation that puts x, y in the same block and puts u and v in separate blocks. We have thus proved that ρλ(L) Eq(X) whenever λ(L) contains a non-trivial unary function. Corollary 3.3.6. If L ≇ 2 is a finite lattice with a meet prime element and X is any set, then L cannot be densely embedded in Eq(X). Remark. The same result holds if we assume the lattice has a join prime element. Proof. It is clear by the definition of meet prime that a lattice satisfying the hypotheses of the corollary also satisfies the conditions of Lemma 3.3.3, so the result follows from Theorem 3.3.5. A lattice is called meet-semidistributive if it satisfies the meet-semidistributive law, SD ∧ : α ∧ β = α ∧ γ ⇒ α ∧ (β ∨ γ) = α ∧ β. Corollary 3.3.7. If L ≇ 2 is a finite meet-semidistributive lattice and X is any set, then L cannot be densely embedded in Eq(X). Proof. We prove that every finite meet-semidistributive lattice L contains a meet prime element. The result will then follow by Corollary 3.3.6. Since L is finite, there exists an atom α ∈ L. If α is the only atom, then α ↑ is trivially prime. Suppose β ∨ γ ∈ α ↑ . Then (β ∨ γ) ∧ α = α, and β ∧ α α implies β ∧ α ∈ {0 L , α}. Similarly for γ. If both β ∧ α = 0 L = γ ∧ α then SD ∧ implies (β ∨ γ) ∧ α = 0 L , Distributive lattices A lattice L is called strongly representable as a congruence lattice if whenever L ∼ = L 0 Eq(X) for some X then there is an algebra based on X whose congruence lattice is L 0 . Theorem 3.3.8 (Berman [5], Quackenbush and Wolk [36]). Every finite distributive lattice is strongly representable. Remark: By Theorem 3.2.1 above, the result of Berman, Quackenbush and Wolk says, if L is a finite distributive lattice then every embedding L ∼ = L 0 Eq(X) is closed. The following proof is only slightly shorter than to the original in [36], and the methods are similar. Proof. Without loss of generality, suppose L Eq(X). Fix θ ∈ Eq(X) \ L and define θ * = {γ ∈ L | γ θ} and θ * = {γ ∈ L | γ θ}. Let α be a join irreducible in L below θ * and not below θ * . Note that α is not below θ. Let β = {γ ∈ L | γ α}. If β were above θ, then β would be above θ * , and so β would be above α. But α is join prime, so β is not above θ. Choose (u, v) ∈ α \ θ and note that u = v. Choose (x, y) ∈ θ \ β and note that x = y. Let B be the β block of y and define h ∈ X X as in (3.3.1). Then it is clear that h violates θ, h respects all elements in the sets α ↑ = {γ ∈ L : α γ} and β ↓ = {γ ∈ L : γ β}, and L = α ↑ ∪ β ↓ . Since θ was an arbitrary element of Eq(X) \ L, we can construct such an h = h θ for each θ ∈ Eq(X) \ L. Let H = {h θ : θ ∈ Eq(X) \ L} and let A be the algebra X, H . Then, L = Con (A). In general, we might ask the following: Are there closed sublattices of dense embeddings? Conclusions and open questions Another question we have not answered is whether the converse of Theorem 3.3.5 is true, but this seems unlikely. Rather, we expect there exists a finite lattice that is neither densely embeddable nor the union of a proper principal ideal and a proper principal filter. Finally, we mention that even if we restrict ourselves to one of the smaller classes of finite lattices mentioned above -those satisfying the conditions of Lemma 3.3.3 or Corollary 3.3.6, or the finite meet-semidistributive lattices -it is still unknown whether every lattice is this class is representable as the congruence lattice of a finite algebra. CHAPTER 4 CONGRUENCE LATTICES OF GROUP ACTIONS Let X be a finite set and consider the set X X of all maps from X to itself, which, when endowed with composition of maps and the identity mapping, forms a monoid, X X , •, id X . The submonoid S X of all bijective maps in X X is a group, the symmetric group on X. When the underlying set is more complicated, or for emphasis, we denote the symmetric group on X by Sym(X). When the underlying set isn't important, we usually write S n to denote the symmetric group on an n-element set. If we have defined some set F of basic operations on X, so that X = X, F is an algebra, then two other important submonoids of X X are End(X), the set of maps in X X which respect all operations in F , and Aut(X), the set of bijective maps in X X which respect all operations in F . It is apparent from the definition that Aut(X) = S X ∩ End(X), and Aut(X) is a submonoid of End(X) and a subgroup of S X . These four fundamental monoids associated with the algebra X, and their relative ordering under inclusion, are shown in the diagram below. Aut(X) End(X) S X X X Given a finite group G, and an algebra X = X, F , a representation of G on X is a group homomorphism from G into Aut(X). That is, a representation of G is a mapping ϕ : G → Aut(X) which satisfies ϕ(g 1 g 2 ) = ϕ(g 1 ) • ϕ(g 2 ), where (as above) • denotes composition of maps in Aut(X). Transitive G-sets From the foregoing, we see that a representation defines an action by G on the set X, as follows: gx = ϕ(g)(x). IfḠ = ϕ[G] Aut(X) denotes the image of G under ϕ, we call the algebra X,Ḡ a G-set. 1 The action is called transitive if for each pair x, y ∈ X there is some g ∈ G such thatḡx = y. The representation ϕ is called faithful if it is a monomorphism, in which case G is isomorphic to its image under ϕ, which is a subgroup of Aut(X). We also say, in this case, that the group acts faithfully, and call it a permutation group. A group which acts transitively on some set is called a transitive group. Without specifying the set, however, this term is meaningless, since every group acts transitively on some sets and intransitively on others. A representation ϕ is called transitive if the resulting action is transitive. Finally, we define degree of a group action on a set X to be the cardinality of X. Two special cases are almost always what one means when one speaks of a representation of a finite group. These are the so called • linear representations, where X = X, +, •, −, 0, 1, F is a finite dimensional vector space over a field F, so Aut(X) is the set of invertible matrices with entries from F; • permutation representations, where X = X is just a set, so Aut(X) = S X . For us the most important representation of a group G is its action on a set of cosets of a subgroup. That is, for any subgroup H G, we define a transitive permutation representation of G, which we will denote byλ H . Specifically,λ H is a group homomorphism from G into the symmetric The action is simply left multiplication by elements of G. That is,λ H (g)(xH) = gxH. Clearly, λ H (g 1 g 2 ) =λ H (g 1 )λ H (g 2 ) for all g 1 , g 2 ∈ G, soλ H is a homomorphism. Each xH is a point in the set G/H, and the point stabilizer of xH in G is defined by G xH = {g ∈ G | gxH = xH}. Notice that G xH = {g ∈ G | x −1 gxH = H} = xG H x −1 = xHx −1 = H x , where G H = {g ∈ G | gH = H} is the point stabilizer of H in G. Thus, the kernel of the homomorphismλ H is kerλ H = {g ∈ G | ∀x ∈ G, gxH = xH} = x∈G G xH = x∈G xHx −1 = x∈G H x . Note that kerλ H is the largest normal subgroup of G contained in H, also known as the core of H in G, which we denote by core G (H) = x∈G H x . If the subgroup H happens to be core-free, that is, core G (H) = 1, thenλ H : G ֒→ Sym(G/H) is an embedding, soλ H is a faithful representation; G acts faithfully on G/H. Hence the group G, being isomorphic to a subgroup of Sym(G/H), is itself a permutation group. Other definitions relating to G-sets will be introduced as needed and in the appendix, and we assume the reader is already familiar with these. However, we mention one more important concept before proceeding, as it is a potential source of confusion. By a primitive group we mean a group that contains a core-free maximal subgroup. This definition is not the typical one found in group theory textbooks, but we feel it is better. (See the appendix Section A.1 for justification.) G-set isomorphism theorems We have seen above that the action of a group on cosets of a subgroup H is a transitive permutation representation, and the representation is faithful when H is core-free. The first theorem in this section states that every transitive permutation representation is of this form. (In fact, as we will see in Lemma 4.2.1 below, every permutation representation, whether transitive or not, can be viewed as an action on cosets.) First, we need some more notation. Given a G-set A = A, G and any element a ∈ A, the set G a = {g ∈ G | ga = a} of all elements of G which fix a is a subgroup of G, called the stabilizer of a in G. Theorem 4.1.1 (1st G-set Isomorphism Theorem). If A = A,Ḡ is a transitive G-set, then A is isomorphic to the G-set Γ := G/G a , {λ g : g ∈ G} for any a ∈ A. Proof. Suppose A = A,Ḡ is a transitive G-set, so A = {ḡa | g ∈ G} for any a ∈ A. The operations of the G-set Γ are defined, for each g ∈ G and each coset xG a ∈ G/G a , byλ g (xG a ) = gxG a . Let G Λ denote the G-set G, {λ g : g ∈ G} , that is, the group G acting on itself by left multiplication. Fix a ∈ A, and define ϕ a : G → A by ϕ a (x) = x(a) for each x ∈ G. Then ϕ a is a homomorphism from G Λ into A -that is, ϕ a respects operations: 2 ϕ a (λ g (x)) = ϕ a (gx) = gx(a) =ḡ · x(a) =ḡϕ a (x). Moreover, since A is transitive, ϕ a (G) = {ḡa | g ∈ G} = A, so ϕ a is an epimorphism. Therefore, G Λ / ker ϕ a ∼ = A. To complete the proof, one simply checks that the two algebras G Λ / ker ϕ a and Γ are identical. 3 The next theorem shows why intervals of subgroup lattices are so important for our work. Since G is transitive, it is equivalent to (b, c) ∈ θ. Therefore, θ H θ = θ. Finally, H θ H ϕ if and only if θ ϕ, so θ → H θ is an isomorphism between Con A and [G a , G]. Theorem 4.1.2 (2nd G-set Isomorphism Theorem). Let A = A, G be a transitive G-set and fix a ∈ A. Then the lattice Con A is isomorphic to the interval [G a , G] in the subgroup lattice of G. Proof. For each θ ∈ Con A, let H θ = {g ∈ G | (g(a), a) ∈ θ}, and for each H ∈ [G a , G], let (b, c) ∈ θ H mean there exist g ∈ G and h ∈ H such that gh(a) = b and g(a) = c. If g 1 , g 2 ∈ H θ , then (g 2 (a), a) ∈ θ ⇒ (g −1 2 g 2 (a), g −1 2 (a)) = (a, g −1 2 (a)) ∈ θ, so (g −1 2 (a), a) ∈ θ, by symmetry. Therefore, (g 1 g −1 2 (a), g 1 (a)) ∈ θ, so (g 1 g −1 2 (a), (a)) ∈ θ, Since the foregoing theorem is so central to our work, we provide an alternative statement of it. This is the version typically found in group theory textbooks (e.g., [12]). Keeping these two alternative perspectives in mind can be useful. 2 In general, if A = A, F and B = B, F are two algebras of the same similarity type, then ϕ : A → B is a homomorphism provided ϕ(f A (a 1 , . . . , an)) = f B (ϕ(a 1 ), . . . , ϕ(an)) whenever f A is an n-ary operation of A, f B is the corresponding n-ary operation of B, and a 1 , . . . , an are arbitrary elements of A. (Note that a one-to-one correspondence between the operations of two algebras of the same similarity type is assumed, and required for the definition of homomorphism to make sense. ) 3 Indeed, ker ϕa = {(x, y) ∈ G 2 | ϕa(x) = ϕa(y)} and the universe of G Λ / ker ϕa is G/ ker ϕa = {x/ ker ϕa | x ∈ G}. where for each x ∈ G x/ ker ϕa = {y ∈ G | (x, y) ∈ ker ϕa} = {y ∈ G | ϕa(x) = ϕa(y)} = {y ∈ G | x(a) = y(a)} = {y ∈ G | id A (a) = x −1 y(a)} = {y ∈ G | x −1 y ∈ Ga} = xGa. These are precisely the elements of G/Ga, so the universes of G Λ / ker ϕa and Γ are the same, as are their operations (left multiplication by g ∈ G). with inverse mapping Φ : [G a , G] → B given by Φ(H) = Ha = {ha | h ∈ H}. The mapping Ψ is order-preserving in the sense that if B 1 , B 2 ∈ B then B 1 ⊆ B 2 ⇔ Ψ(B 1 ) Ψ(B 2 ). Briefly, the poset B, ⊆ is order-isomorphic to the poset [G a , G], . The point stabilizers of the actionλ H described above are the conjugates of H in G. Therefore, the lemma implies that, for any two subgroups H, K G, the representationsλ H andλ K are equivalent precisely when K = xHx −1 for some x ∈ G. Hence, the transitive permutation representations of G are given, up to equivalence, byλ Ki as K i runs over a set of representatives of conjugacy classes of subgroups of G. An M-set isomorphism theorem It is natural to ask whether the two theorems of the previous subsection hold more generally for a unary algebra X, M , where M is a monoid (rather than a permutation group). We call such an algebra X, M an M -set, and although we will see that there is no analogue to the 2nd G-set Isomorphism Theorem, we do have Theorem 4.1.6 (1st M -set Isomorphism Theorem). If X, M is a transitive M -set, then for any fixed x ∈ X, the map ϕ x : M → X defined by ϕ x (m) = mx is an M -set epimorphism. Moreover, the (transitive) M -set M/ ker ϕ x , M is isomorphic to X, M . Proof. By transitivity, for each y ∈ X, there is an m ∈ M such that ϕ x (m) = mx = y, so ϕ x is onto. Also, ϕ x is a homomorphism of the M -set M, M onto the M -set X, M , since for all m, m 1 ∈ M , ϕ x (m • m 1 ) = m(m 1 x) = mϕ x (m 1 ). By the usual isomorphism theorem, M/ ker ϕ x , M ∼ = X, M (4.1.1) where ker ϕ x = {(m 1 , m 2 ) ∈ M 2 | ϕ x (m 1 ) = ϕ x (m 2 )} = {(m 1 , m 2 ) ∈ M 2 | m 1 x = m 2 x}. Note that, since X, M is a transitive M -set, the M -set M/ ker ϕ x , M must also be transitive, otherwise (4.1.1) would fail. Just to be sure, let's verify that M/ ker ϕ x , M is indeed transitive. Let m 1 / ker ϕ x , m 2 / ker ϕ x be any two ker ϕ x -classes of M . We must show there exists m 3 ∈ M such that m 3 [m 1 / ker ϕ x ] = m 2 / ker ϕ x . Let ϕ x (m 1 ) = y 1 and ϕ x (m 2 ) = y 2 . Let m 3 ∈ M be a map which takes y 1 to y 2 , (guaranteed to exist by transitivity of X, M ). Then for all m ∈ m 1 / ker ϕ x , we have m 3 mx = m 3 y 1 = y 2 , so m 3 m ∈ m 2 / ker ϕ x . Therefore,m 3 [m 1 / ker ϕ x ] ⊆ m 2 / ker ϕ x . By the same argument, there is m ′ 3 ∈ M such that m ′ 3 [m 2 / ker ϕ x ] ⊆ m 1 / ker ϕ x . By cardinality, m 3 [m 1 / ker ϕ x ] = m 2 / ker ϕ x . An analogue to the 2nd G-set Isomorphism Theorem for monoids would be that [M x , M ] ∼ = Con X, M should hold for a transitive M -set X, M . By the following counter-example, we see that this is false: Consider the monoid M consisting of the identity and constant maps. Of course, X, M is a transitive M -set, and Con X, M = Eq(X). However, for x ∈ X, the stabilizer is M x = {m ∈ M : mx = x} Intransitive G-sets The problem of characterizing congruence lattices of intransitive G-sets seems open. In this section we prove a couple of results which help determine the shape of congruence lattices of intransitive G-sets. In [11] we use these and other results to show that for many lattices a minimal representation as the congruence lattice of an intransitive G-set is not possible. 5 In the previous section we considered transitive, or one-generated, G-sets. In Theorem 4.1.1, we presented the well known result that a transitive G-set Ω, G , with universe Ω, is isomorphic to the G-set G/H, G , where the universe is now the collection of cosets of a subgroup H = G ω -the stabilizer of a point ω ∈ Ω. Then, Theorem 4.1.2 gave us a precise description of the shape of the congruence lattice: Con G/H, G ∼ = [H, G]. It is natural to ask whether results analogous to these hold for intransitive G-sets. In this section, we first prove that an arbitrary (intransitive) G-set Ω, G is isomorphic to a G-set of the form G 1 /H 1 ∪ · · · ∪ G r /H r , G , where H i G i ∼ = G. This result is well known, and appears as Theorem 3.4 in [26]. Nonetheless we present a short proof and describe the G-set isomorphism explicitly. 6 Thereafter, we prove lemma which, along with the first, gives a characterization of the congruence lattice of an arbitrary G-set. It is almost certain that this simple result is also well known, but to my knowledge it does not appear in print elsewhere. 7 5 In other words, if there exists a representation of such a lattice as the congruence lattice of an algebra (of minimal cardinality), then the algebra must be a transitive G-set. 6 Such an explicit description is useful when we are working with such algebras on the computer, using the Universal Algebra Calculator or GAP, for example. 7 I thank Alexander Hulpke for alerting me to the special case, described below, of the second lemma. Throughout this section, we adhere to the convention that groups act on the left, so we will denote the action of g ∈ G on an element ω ∈ Ω by g : ω → gω, and we use Gω to denote the orbit of ω under this action, that is, Gω = {gω | g ∈ G}. Finally, we remind the reader that all groups under consideration are finite. Our first lemma shows that, even in the intransitive case, we can take the universe of an arbitrary G-set to be a collection cosets of the group G. Lemma 4.2.1. Every G-set Ω, G is isomorphic to a G-set on a universe of the form G 1 /H 1 ∪ · · · ∪ G r /H r , where H i G i ∼ = G and G i /H i is the set of left cosets of H i in G i , for each 1 i r, Proof. Suppose Ω = Ω, G is an arbitrary G-set, and let Ω i , G , 1 i r, be the minimal subalgebras of Ω. That is, each Ω i is an orbit, say, Ω i = Gω i , and Ω = Gω 1 ∪ · · · ∪ Gω r is a disjoint union. For each 1 i r, let G i be an isomorphic copy of G, with, say, ϕ i : G i ∼ = G as the isomorphism. Clearly, H i := {x ∈ G i | ϕ i (x)ω i = ω i } ∼ = {g ∈ G | gω i = ω i } = G ωi . Note that G i /H i , G ∼ = Gω i , G , where G acts on G i /H i as one expects: for g ∈ G and xH i ∈ G i /H i , the action is g : xH i → ϕ −1 i (g)xH i . Define ψ : G 1 /H 1 ∪ · · · ∪ G r /H r → Ω by ψ(xH i ) = ϕ i (x)ω i . This map is well-defined. For, if xH i = x ′ H j , then i = j and x −1 x ′ ∈ H i , and it is easy to verify that x −1 x ′ ∈ H i holds if and only if ϕ i (x ′ )ω i = ϕ i (x)ω i . Thus, ψ(xH i ) = ψ(x ′ H j ). Now consider the G-set G 1 /H 1 ∪ · · · ∪ G r /H r , G with the same action as above: g(xH i ) = ϕ −1 i (g)(xH i ). We claim that ψ is a G-set isomorphism of G 1 /H 1 ∪ · · · ∪ G r /H r , G onto Ω, G . It is clearly a bijection. 8 We check that ψ respects the interpretation of the action of G: Fix g ∈ G and x ∈ G i . Then, since ϕ i is a homomorphism, ψ(ϕ −1 i (g)(xH i )) = ϕ i (ϕ −1 i (g)x)ω i = ϕ i (ϕ −1 i (g))ϕ i (x)ω i = gψ(xH i ). The foregoing lemma shows that we can always take the universe of an intransitive G-set to be a 8 Define ζ : Ω → G 1 /H 1 ∪ · · · ∪ Gr/Hr by ζ(gω i ) = ϕ −1 i (g)H i , check that this map is well-defined, and note that ψζ = id Ω , and ζψ is the identity on G 1 /H 1 ∪ · · · ∪ Gr/Hr. disjoint union of sets of cosets of stabilizer subgroups. We now use this fact to describe the structure of the congruence lattice of an arbitrary G-set. As above, let Ω = Ω, G be a G-set with universe Ω = Gω 1 ∪ · · · ∪ Gω r , where each Gω i , G is a minimal subalgebra. Consider the partition τ ∈ Eq(Ω), given by τ = |Gω 1 |Gω 2 | · · · |Gω r |. Clearly, this is a congruence relation, since the action of every g ∈ G fixes each block. We call τ the intransitivity congruence. It's clear that we can join two or more blocks of τ and the new larger block will still be preserved by every g ∈ G. Thus, the interval above τ in the congruence lattice Ω is isomorphic to the lattice of partitions of a set of size r. That is, [τ, 1 Ω ] := {θ ∈ Con Ω | τ θ 1 Ω } ∼ = Eq(r). (4.2.1) Another obvious fact is that the interval below τ in Con Ω is [0 Ω , τ ] ∼ = r i=1 Con ( Gω i , G ). (4.2.2) Since each minimal algebra Gω i , G ∼ = G i /H i , G is transitive, we have Con ( Gω i , G ) ∼ = [H i , G i ]. Thus, the structure of that part of Con Ω that is comparable with the intransitivity congruence is explicitly described by (4.2.1) and (4.2.2). Our next result describes the congruences that are incomparable with the intransitivity congruence. The description is in terms of the blocks of congruences below the intransitivity congruence. Thus, the lemma does not give a nice abstract characterization of the shape of the Con Ω in terms of the shape of Sub(G), as we had in the transitive case. However, besides being useful for computing the congruences, this result can be used in certain situations to draw conclusions about the general shape of Con Ω, based on the subgroup structure of G (for example, using combinatorial arguments involving the index of subgroups of G). We will say more about this below. Though the proof of Lemma 4.2.2 is elementary, it gets a bit complicated when presented in full generality. Therefore, we begin by discussing the simplest special case of an intransitive G-set, that is, one which has just two minimal subalgebras. Suppose Ω = Ω, G = Ω 1 ∪ Ω 2 , G is a G-set with Ω i = Gω i for some ω i ∈ Ω i , i = 1, 2. For each subset Λ ⊆ Ω, for each g ∈ G, let gΛ := {gω | ω ∈ Λ}, and define the set-wise stabilizer of Λ in G to be the subgroup Stab G (Λ) := {g ∈ G | gω ∈ Λ for all ω ∈ Λ}. As above, we call the congruence τ = |Ω 1 |Ω 2 | the intransitivity congruence. Fix a congruence τ 0 strictly below τ , and for each i = 1, 2 let Λ i = ω i /τ 0 denote the block of τ 0 containing ω i . Then there is a congruence θ above τ 0 with a block Λ 1 ∪ Λ 2 if and only if Stab G (Λ 1 ) = Stab G (Λ 2 ). (We will verify this claim below when we prove it more generally in Lemma 4.2.2.) This characterizes all congruences in Con Ω that are incomparable with the intransitivity congruence, τ , in terms of the congruences below τ . Let Ω = Ω 1 ∪ · · · ∪ Ω r , G be a G-set with minimal subalgebras Ω i = Gω i , for some ω i ∈ Ω i , 1 i r. Let τ = |Ω 1 |Ω 2 | · · · |Ω r | be the intransitivity congruence and fix τ 0 < τ in Con Ω. For each 1 i r, let Λ i = ω i /τ 0 denote the block of τ 0 containing ω i , and let T i = {g i,0 =1, g i,1 , . . . , g i,ni } be a transversal of G/Stab G (Λ i ). 9 It is important to note that the blocks of τ 0 are g i,k Λ i , where 1 i r and 0 k n i . This is illustrated in the following diagram, where the blocks of τ 0 appear below the blocks of τ to which they belong. τ = Ω 1 Ω 2 · · · Ω r τ 0 = Λ 1 |g 1,1 Λ 1 | · · · |g 1,n1 Λ 1 Λ 2 |g 2,1 Λ 2 | · · · |g 2,n2 Λ 2 · · · Λ r |g r,1 Λ r | · · · |g r,nr Λ r It should be obvious that the blocks of τ 0 are as given above, but since this plays such an important role in the lemma below, we check it explicitly: If Λ i ⊆ Ω i is a block of τ 0 , then so is gΛ i for all g ∈ G, and either gΛ i ∩ Λ i = ∅ or gΛ i = Λ i . If Λ ′ ⊆ Ω i is also a block of τ 0 , then Λ ′ = g ′ Λ i for some g ′ ∈ G = Stab G (Λ i ) ∪ g i,1 Stab G (Λ i ) ∪ g i,ni Stab G (Λ i ), say g ′ ∈ g i,j Stab G (Λ i ). Then, g −1 i,j g ′ ∈ Stab G (Λ i ), so g −1 i,j g ′ Λ i = Λ i . Therefore, g ′ Λ i = g i,j Λ i . Another obvious but important consequence: If T 1 = {g 1,0 =1, g 1,1 , . . . , g 1,n1 } is a transversal of G/Stab(Λ 1 ), and if Stab(Λ 1 ) = Stab(Λ j ), then T 1 is also a transversal of G/Stab(Λ j ) , so the blocks of τ 0 in Ω j may be written as g 1,k Λ j , where 0 k n 1 . Lemma 4.2.2. Given a subset {i 1 , . . . , i m } ⊆ {1, . . . , r}, there exists θ ∈ Con Ω with block Λ i1 ∪ · · · ∪ Λ im if and only if Stab G (Λ i1 ) = · · · = Stab G (Λ im ). For example, θ = τ 0 ∪ ni 1 k=0 (g i1k Λ i1 ∪ · · · ∪ g i1k Λ im ) 2 . (4.2.3) Remarks. The index set {i 1 , . . . , i m } identifies the subalgebras from which to choose blocks that will be joined in the new congruence θ. The number of blocks of τ 0 which intersect the subalgebra Ω ij is n ij , which is the length of the transversal of G/Stab G (Λ ij ). Therefore, n ij = |G : Stab G (Λ ij )|. As noted above, if Stab G (Λ i1 ) = Stab G (Λ im ), then we can assume the transversals T 1 = {g i11 , . . . , g i1ni 1 } and T m = {g im1 , . . . , g imni m } are the same. In the proof below, we will use T to denote this common transversal. Proof. (⇒) Assume there is a congruence θ ∈ Con Ω with block Λ i1 ∪ · · · ∪ Λ im . Suppose there exists 1 j < k m such that Stab G (Λ ij ) = Stab G (Λ i k ) . Without loss of generality, assume g ∈ Stab G (Λ ij ) \ Stab G (Λ i k ), so gΛ ij = Λ ij and there is an x ∈ Λ i k such that gx / ∈ Λ i k . Of course, gΩ i k = Ω i k , so we must have gx / ∈ Λ i1 ∪ · · · ∪ Λ im . Thus, choosing any y ∈ Λ ij , we have (x, y) ∈ θ while (gx, gy) / ∈ θ, contradicting θ ∈ Con Ω. Therefore, it must be the case that Stab G (Λ i1 ) = · · · = Stab G (Λ im ). (⇐) Suppose Stab G (Λ i1 ) = · · · = Stab G (Λ im ) . Let θ be the relation defined in (4.2.3). We will prove θ ∈ Con Ω. It is easy to see that θ is an equivalence relation, so we just need to check gθ ⊆ θ; that is, we prove (∀ (x, y) ∈ θ) (∀ g ∈ G) (gx, gy) ∈ θ. Fix (x, y) ∈ θ, say, x ∈ g i1k Λ ij and y ∈ g i1k Λ i ℓ , for some 0 k n i1 , 1 j < ℓ m. For each g ∈ G we have g g i1k Λ ij = g i1s Λ ij for some g i1s ∈ T . Thus, g −1 i1s g g i1k ∈ Stab G (Λ ij ). Similarly, g g i1k Λ i ℓ = g i1t Λ i ℓ for some g i1t ∈ T , so g −1 i1t g g i1k ∈ Stab G (Λ i ℓ ) . This and the hypothesis Stab G (Λ ij ) = Stab G (Λ i ℓ ) together imply g i1s Stab G (Λ ij ) = g i1t Stab G (Λ ij ), so g i1s = g i1t , since they are both elements of the transversal of Stab G (Λ ij ). We have thus shown that the action of g ∈ G maps pairs of blocks with equal stabilizers to the same block of θ; that is, g g i1k Λ ij = g i1s Λ ij θ g i1t Λ i ℓ = g g i1k Λ i ℓ . INTERVAL SUBLATTICE ENFORCEABLE PROPERTIES 5.1 Introduction Given a finite lattice L, the expression L ∼ = [H, G] means "there exist finite groups H < G such that L is isomorphic to the interval {K | H K G} in the subgroup lattice of G." A group G is called almost simple if G has a normal subgroup S G which is nonabelian, simple, and has trivial centralizer, C G (S) = 1. If H G, then the core of H in G, denoted core G (H), is the largest normal subgroup of G contained in H; it is given by core G (H) = g∈G gHg −1 . A subgroup H G for which core G (H) = 1 is called core-free in G. If every finite lattice can be represented as the congruence lattice of a finite algebra, we say that the FLRP has a positive answer. If we assume that the FLRP has a positive answer, then for every finite lattice L there is a finite group G having L as an upper interval in Sub(G). In this chapter we consider the following question: Given a finite lattice L, what can we say about a finite group G that has L as an upper interval in its subgroup lattice? Taking this a step further, we consider certain finite collections of finite lattices ask what sort of properties we can prove about a group G if we assume it has all of these lattices as upper intervals in its subgroup lattice. In this and the next section, we address these questions somewhat informally in order to motivate this approach. In Section 5.3 we introduce a new formalism for interval sublattice enforceable properties of groups. One easy consequence that comes out of this investigation is the following observation: L i ∼ = [H i , G] Sub(G), with H i core-free in G. By the "parachute" construction described in the next section, we will see that the only nontrivial part of this proposition is the conclusion that all the H i be core-free in G. However, this will follow easily from Lemma 5.2.4 below. Before proceeding, it might be worth pausing to consider what seems like a striking consequence of the proposition above: If the FLRP has a positive answer, then no matter what we take as our finite collection L -for example, we might take L to be all finite lattices with at most N elements for some large N < ω -we can always find a single finite group G such that every lattice in L is an upper interval in Sub(G); moreover, (by Lemma 5.2.4) we can assume the subgroup H i at the bottom of each interval is core-free. As a result, the single finite group G must have so many faithful representations, G ֒→ Sym(G/H i ) with Con G/H i , G ∼ = L i , one such representation for each distinct L i ∈ L . Parachute lattices As mentioned above, in 1980 Pálfy and Pudlák published the following striking result: [32]). The following statements are equivalent: Theorem 5.2.1 (Pálfy-Pudlák (A) Every finite lattice is isomorphic to the congruence lattice of a finite algebra. (B) Every finite lattice is isomorphic to an interval in the subgroup lattice of a finite group. Also noted in [32] is the important fact that (B) is equivalent to: (B') Every finite lattice is isomorphic to the congruence lattice of a finite transitive G-set. There are a number of examples in the literature of the following situation: a specific finite lattice is considered, and it is shown that if such a lattice is an interval in the subgroup lattice of a finite group, then this group must be of a certain form or have certain properties. As the number of such results grows, it becomes increasingly useful to keep in mind the following simple observation: Lemma 5.2.2. Let G 1 , . . . , G n be classes of groups and suppose that for each i ∈ {1, . . . , n} there exists a finite lattice L i such that L i ∼ = [H, G] only if G ∈ G i . Then (B) is equivalent to (C) For each finite lattice L, there is a finite group G ∈ n i=1 G i such that L ∼ = [H, G]. Proof. Obviously, (C) implies (B). Assume (B) holds and let L be any finite lattice. Suppose since L i ∼ = [K i , G], we must have G ∈ G i , by hypothesis. This is true for all 1 i n, so G ∈ n i=1 G i , which proves that (B) implies (C). L L 1 L 2 L n . . . (a) L L 1 L 2 L n . . . (b) G K K 1 K 2 K n H Examples. As usual, we let A n and S n denote the alternating and symmetric groups on n letters. In addition, the following notation will be useful: • G = the class of all finite groups; • S = the class of all finite solvable groups; • Gi = n<ω {A n , S n } = the alternating or symmetric groups, also known as the "giant" groups. It is easy to find a lattice L with the property that L ∼ = [H, G] implies G / ∈ S. We will see an example of such a lattice in Section 6.3. (For another example, see [29].) In his thesis [4], Alberto We would like to generalize Lemma 5.2.2 because it is much easier and more common to find a 1 Recall, Mn denotes the (n + 2)-element lattice with n atoms. class of groups G i and a lattice L i with the following property: If L i ∼ = [H, G] with H core-free in G, then G ∈ G i . (⋆) This leads naturally to the following question: Given a class of groups G and a finite lattice L satisfying (⋆), when can we safely drop the caveat "with H core-free in G" and get back to the hypothesis of Lemma 5.2.2? There is a very simple sufficient condition involving the class G c := G i . This will follow from (⋆) once we prove that each K i is core-free in G. We now give an easy direct proof this fact, but we note that it also follows from Lemma 5.4.3 below, as well as from a more general result about L-P lattices. (See, e.g., {G ∈ G | G / ∈ G }. (Recall, if K isL ∼ = [H, G] with H core-free ⇒ G ∈ G , (5.2.1) and suppose H(G c ) = G c . Then, L ∼ = [H, G] ⇒ G ∈ G . Börner [8].) For each i ∈ {1, . . . , n}, let N i = core G (K i ). We prove that N i = 1 for all i. Suppose, on the contrary, that N i = 1 for some i, and consider any K j with j = i. 5 A sketch of the part of the subgroup lattice under consideration is shown in Figure 5 L i L j G K i K j H N i N i ∩ K j.2. Notice that N i K j = G. For, N i is not below H, since H is core-free, so N i H = K i , so N i K j is above both K i and K j . Now, clearly, N i ∩ K j K j , and the standard isomorphism theorem implies K j /(N i ∩ K j ) ∼ = N i K j /N i = G/N i . In particular, under this correspondence we have, [N i ∩ K j , K j ] ∋ H → N i H = K i ∈ [N i , G], and it follows that the intervals [K i , G] and [H, K j ] must be isomorphic as lattices. However, by construction, H is a maximal subgroup of K j , so we have [H, K j ] ∼ = 2 ≇ L i ∼ = [K i , G] . This contradiction proves that core G (K i ) = 1 for all 1 i n, as claimed. ISLE properties of groups The previous section motivates the study of what we call interval sublattice enforceable (ISLE) properties of groups. In this section we formalize this concept, as well as some of the concepts introduced above, and we summarize what we have proved about them. We conclude with some conjectures that will provide the basis for future research. By a group theoretical class, or class of groups, we mean a collection G of groups that is closed under isomorphism: if G 0 ∈ G and G 1 ∼ = G 0 , then G 1 ∈ G . A group theoretical property, or simply property of groups, is a property P such that if a group G 0 has property P and G 1 ∼ = G 0 , then G 1 has property P. 6 Thus if G P denotes the collection of groups with group theoretical property P, then G P is a class of groups, and belonging to a class of groups is a group theoretical property. Therefore, we need not distinguish between a property of groups and the class of groups which possess that property. A group in the class G is called a G -group, and a group with property P is called a P-group. Occasionally we write G P to indicate that G is a P-group. with H core-free in G, then G is a P-group. Clearly, if P is ISLE, then it is also cf-ISLE, and Lemma 5.2.3 above gives a sufficient condition for the converse to hold. We restate this formally as follows: Lemma 5.2.3 ′ . If P is cf-ISLE and if G c P = {G ∈ G | G P} is closed under homomorphic images, H(G c P ) = G c P , then P is ISLE. As we noted in the previous section, two examples of ISLE classes are • G 0 = S c = the finite non-solvable groups; • G 1 = (Gi) c = the finite non-giant groups, {G ∈ G | (∀n < ω) (G = A n and G = S n )}; The following classes are at least cf-ISLE: 7 • G 2 = the finite subdirectly irreducible groups; • G 3 = the finite groups having no nontrivial abelian normal subgroups. • G 4 = {G ∈ G | C G (M ) = 1 for a minimal normal subgroup M G} 6 It seems there is no single standard definition of group theoretical class. While some authors (e.g., [13], [3]) use the definition given here, others (e.g. [37], [38]) require that a group theoretical class contain groups of order 1. 7 The symbols we use to denote these classes are not standard. Note that G 4 ⊂ G 2 ∩ G 3 ⊂ G 0 . Given two (group theoretical) properties P 1 , P 2 , we write P 1 → P 2 to denote that property P 1 implies property P 2 . In other words, G P 1 only if G P 2 . Thus → provides a natural partial order on any given set of properties, as follows: P 1 P 2 ⇔ P 1 → P 2 ⇔ G P1 ⊆ G P2 , where G Pi = {G ∈ G | G P i }. The following is an obvious corollary of the parachute construction. Corollary 5.3.1. If P = {P i | i ∈ I } is a collection of (cf-)ISLE properties, then P is (cf-)ISLE. Note: the conjunction P corresponds to the class {G ∈ G | (∀i ∈ I ) G P i }. that H × K is not core-free, so a more interesting question to ask might be whether solvability is a cf-ISLE property. The following lemma proves that this is not the case. Assuming D 1 ⋊Ū is core-free in W = S m ⋊Ū , then, it follows by the original hypothesis that W must be a P-group. To complete the proof, we check that starting with a core-free subgroup H G in the Kurzweil construction just described results in a core-free subgroup D ⋊Ḡ U . Let N = core U (D ⋊Ḡ). Then, for all n = (d, . . . , d, x) ∈ N and for all u = (t 1 , . . . , t n , g) ∈ U , we have unu −1 ∈ N . In particular, we are free to choose t 1 = t 2 , all other t k distinct, and g = 1. Then unu −1 = (t 1 , . . . , t n , 1)(d, . . . , d, x)(t −1 1 , . . . , t −1 n , 1) = (t 1 d t −1 x(1) , . . . , t n d t −1 x(n) , 1) ∈ N. Therefore, t 1 d t −1 x(1) = · · · = t n d t −1 x(n) . With t 1 = t 2 and all other t k distinct, it's clear that x must stabilize the set {1, 2}. Of course, the same argument applies in case t 1 = t 3 with all other t k distinct, 9 so we conclude that x stabilizes the set {1, 3} as well. Therefore, x(i) = i, for i = 1, 2, 3. Since the same argument works for all i, we see that n = (d, . . . , d, x) ∈ N implies x ∈ ker ϕ = 1. This puts N below D × 1, and the only normal subgroup of U that lies below D × 1 is the trivial subgroup. The foregoing result enables us to conclude that any class of groups that does not include wreath products of the form S ≀ G for all finite simple groups S cannot be a cf-ISLE class. We conclude this section with the following two equivalent conjectures: Conjecture 5.1. If P is a (cf-)ISLE property, then ¬P is not a (cf-)ISLE property. Conjecture 5.2. If G is a (cf-)ISLE class, then G c is not a (cf-)ISLE class. A pair of lattices witnessing the failure of either of these conjectures would solve the FLRP. More precisely, if G is a class and L 0 and L 1 are lattices such that L 0 ∼ = [H, G] ⇒ G ∈ G and L 1 ∼ = [H, G] ⇒ G ∈ G c Then the parachute lattice P(L 0 , L 1 ) is not an interval in the subgroup lattice of a finite group. Dedekind's rule We prove a few more lemmas which lead to additional constraints on any group which has a nontrivial parachute lattice as an upper interval in its subgroup lattice. We will need the following standard theorem 10 which we refer to as Dedekind's rule: 9 Note that we can be sure |G : H| = n > 2, since |G : H| = 2 would imply H G, which contradicts that H is core-free in G. 10 See, for example, page 122 of Rose, A Course on Group Theory [38]. Our next lemma (Lemma 5.4.2) is a slight variation on a standard result that we find very useful. The standard result is essentially part (ii) of Lemma 5.4.2. Surely part (i) of the lemma is also well known, though we have not seen it elsewhere. We will see that the standard result is powerful enough to answer all of our questions about parachute lattices, but later, in Section 6.3, we make use of (i) in a situation where (ii) does not apply. To state Lemma 5.4.2, we need some new notation. Let U and H be subgroups of a group, let If H normalizes U (which implies U H = HU ), then we define [U 0 , U ] H := {V ∈ [U 0 , U ] | H N G (V )},(5.(i) [H, U H] ∼ = [U 0 , U ] H [U 0 , U ]. (ii) If U U H, then [U 0 , U ] H = [U 0 , U ] H [U 0 , U ]. (iii) If H U H, then [U 0 , U ] H = [U 0 , U ] H = [U 0 , U ]. Remarks. Since G = U H is a group, the hypothesis of (ii) is equivalent to H N G (U ), and the hypothesis of (iii) is equivalent to U N G (H). Part (i) of the lemma says that when two subgroups permute, we can identify the interval above either one of them with the sublattice of subgroups below the other that permute with the first. Part (ii) is similar except we identify the interval above H with the sublattice of H-invariant subgroups below U . Once we have proved (i), the proof of (iii) follows trivially from the standard isomorphism theorem for groups, so we omit the details. Proof. To prove (i), we show that the following maps are inverse order isomorphisms: ϕ : [H, U H] ∋ X → U ∩ X ∈ [U 0 , U ] H (5.4.5) ψ : [U 0 , U ] H ∋ V → V H ∈ [H, U H]. Then we show that [U 0 , U ] H is a sublattice of [U 0 , U ], that is, [U 0 , U ] H [U 0 , U ]. Fix X ∈ [H, U H]. We claim that U ∩ X ∈ [U 0 , U ] H . Indeed, (U ∩ X)H = U H ∩ X = HU ∩ X = H(U ∩ X).If V ∈ [U 0 , U ] H , then V H = HV implies V H ∈ [H, U H]. Also, ϕ • ψ is the identity on [U 0 , U ] H , since ϕ • ψ(V ) = V H ∩ U = V (H ∩ U ) = V U 0 = V ,. Fix x ∈ V 1 ∩ V 2 and h ∈ H. We show xh = h ′ x ′ for some h ′ ∈ H, x ′ ∈ V 1 ∩ V 2 . Since V 1 and V 2 permute with H, we have xh = h 1 v 1 and xh = h 2 v 2 for some h 1 , h 2 ∈ H, v 1 ∈ V 1 , v 2 ∈ V 2 . Therefore, h 1 v 1 = h 2 v 2 , which implies v 1 = h −1 1 h 2 v 2 ∈ HV 2 , so v 1 belongs to V 1 ∩ HV 2 . Note that V 1 ∩ HV 2 is below both V 1 and U ∩ HV 2 = ϕψ(V 2 ) = V 2 . Therefore, v 1 ∈ V 1 ∩ HV 2 V 1 ∩ V 2 , and we have proved that xh = h 1 v 1 for h 1 ∈ H and v 1 ∈ V 1 ∩ V 2 , as desired. To prove (ii), assuming U G, we show that if U 0 V U , then V H = HV if and only if H N G (V ). If H N G (V ), then V H = HV (even when U G). Suppose V H = HV . We must show (∀v ∈ V ) (∀h ∈ H) hvh −1 ∈ V . Fix v ∈ V, h ∈ H. Then, hv = v ′ h ′ for some v ′ ∈ V, h ′ ∈ H, since V H = HV . Therefore, v ′ h ′ h −1 = hvh −1 = u for some u ∈ U , since H N G (U ). This proves that hvh −1 ∈ V H ∩ U = V (H ∩ U ) = V U 0 = V , as desired. Next we prove that any group which has a nontrivial parachute lattice as an upper interval in its subgroup lattice must have some rather special properties. (iii) G is subdirectly irreducible. (iv) G is not solvable. Remark. If a subgroup M G is abelian, then M C G (M ), so (ii) implies that a minimal normal subgroup (hence, every normal subgroup) of G must be nonabelian. Finally, we note that Theorem 4.3.A of Dixon and Mortimer [12] describes the structure of the unique minimal normal subgroup as follows: This forms the basis and motivation for the idea of (cf-)ISLE properties, as discussed in Section 5.3. LATTICES WITH AT MOST SEVEN ELEMENTS (iii) M = T 0 × · · · × T r−1 , Introduction In the spring of 2011, our research seminar was fortunate enough to have as a visitor Peter Jipsen, who initiated the project of cataloging every small finite lattice L for which there is a known finite algebra A with Con A ∼ = L. It is well known that all lattices with at most six elements are representable. In fact, these can be found as intervals in subgroup lattices of finite groups, but this fact was not known until recently. By 1996, Yasuo Watatani had found each six-element lattice, except for the two lattices appearing below, as intervals in subgroup lattices of finite groups. See [46]. Then, in 2008, Michael Aschbacher showed in [1] how to construct some (very large) twisted wreath product groups that have the lattices above as intervals in their subgroup lattices. Note that, although it was apparently quite difficult to find group representations of the lattices shown above, it is quite easy to represent them concretely as the lattices of congruences of very small finite algebras. Take, for example, the set X = {0, 1, . . . , 6} and consider the lattice L Eq(X) generated by the partitions |0, 3, 4|1, 6|2, 5| and |0, 6|1, 5|2|3|4| |0, 6|1, 4, 5|2|3| |0, 6|1, 4, 5|2, 3|. This concrete representation of the lattice on the left above happens to be closed: ρλ(L) = L, so it is equal to the congruence lattice Con X, λ(L) . We prove two main results in this chapter. The first is Theorem 6.1.1. Every finite lattice with at most seven elements, with one possible exception, is representable as the congruence lattice of a finite algebra. The second result concerns the one possible exception of this theorem, a seven element lattice, which we call L 7 . It is the focus of Section 6.3. As we explain below, if L 7 is representable as the congruence lattice of a finite algebra, then it must appear as an interval in the subgroup lattice of a finite group. 1 Our main result, Theorem 6.3.1, places some fairly strong restrictions on such a group. Our motivation is to apply this new theorem, along with some well known theorems classifying finite groups, to eventually either find such a group or prove that none exists. This application will be the focus of future research. Seven element lattices In this section we show that, with one possible exception (discussed in the next section), every lattice with at most seven elements is representable as a congruence lattice of a finite algebra. There are 53 lattices with at most seven elements. Using these methods, it was not hard to find, or at least prove the existence of, congruence lattice representations of all seven element lattices except for the seven lattices appearing in Figure 6.2, 1 Note that the result of Pálfy and Pudlák does not say that every representable lattice is isomorphic to an interval in a subgroup lattice of a finite group. Rather, it is a statement about the whole class of representable lattices. However, for certain lattices, such as the one described in Section 6.3, we can prove that it belongs to L 3 if and only if it belongs to L 4 . 2 The Hasse diagrams of all lattices with at most seven elements are shown here http://db.tt/2qJUkoaG or alternatively here http://math.chapman.edu/~jipsen/mathposters/lattices7.pdf (courtesy of Peter Jipsen). plus their duals. Four of these seven are self-dual, so there are ten lattices in total for which a representation is not relatively easy to find. We now prove the existence of congruence lattice representations for all but the last of these. The first two, L 19 and L 20 were found using the closure method with the help of Sage by searching for closed concrete representations in the partition lattice Eq (8). As for L 17 , recall that the lattice Sub(A 4 ) of subgroups of the group A 4 (the group of all even permutations of a four element set) is the lattice shown below. 3 The names of these lattices do not conform to any well established naming convention. The lattice L 13 is an interval in a subgroup lattice. Specifically, a GAP search reveals that the Note that the filter plus ideal method only adds operations to the algebra of which the original lattice was the congruence lattice, leaving the universe fixed. Thus, the filter-ideal sublattice is the congruence lattice of an algebra with the same number of elements as the original algebra. 5 Incidentally, since L 17 is also representable as an interval above a subgroup (of index 48), we could apply the Kurzweil-Netter method using this representation instead. Then we would obtain a group representation of the dual (namely, an upper interval in a group of the form S 48 ⋊ G, where G = (A 4 × A 4 ) ⋊ C 2 ). 6 In GAP this is SmallGroup(960,11358). 7 Q 8 denotes the eight element quaternion group. lattice, so if there exists a subgroup K ≻ 1, below β and not below γ, then L 11 ∼ = K ↓ ∪ H ↑ . Indeed, there is such a subgroup K. group 6 G = (C 2 × C 2 × C 2 × C 2 ) ⋊ A Apart from the easy cases, which we only briefly covered at the start of this section, there remain just two seven element lattices for which we have not yet described a representation. These are the lattices at the bottom of Figure 6.2. Finding a representation of L 9 , dubbed the "triplewing pentagon," was quite challenging. It sparked the idea of expanding finite algebras, which we describe at length in the next chapter (Ch. 7). Here we only mention the basic idea as it applies to this particular lattice. As the goal is to find an algebra with congruence lattice L 9 , we start with an algebra having an M 4 congruence lattice -that is, a six element lattice of height two with four atoms (which are also coatoms). Then we expand the algebra by adding elements to the universe and adding certain operations so that the newly expanded algebra has almost the same congruence lattice as the original, except one of the atoms has been doubled. That is, the resulting congruence lattice is isomorphic to L 9 . This example and the powerful techniques that grew out of it are described in Chapter 7. It is still unknown whether the final lattice appearing in Figure 6.2 is representable as the congruence lattice of a finite algebra. Thus, L 7 is the unique smallest lattice for which there is no known representation. It is the subject of the next section. The exceptional seven element lattice In this section we consider L 7 , the last seven element lattice appearing in Figure 6.2. As yet, we are unable to find a finite algebra which has a congruence lattice isomorphic to L 7 , and this is the smallest lattice for which we have not found such a representation. Suppose A is a finite algebra with Con A ∼ = L 7 , and suppose A is of minimal cardinality among those algebras having a congruence lattice isomorphic to L 7 . Then A must be isomorphic to a transitive G-set. (This fact is proved in a forthcoming article, [11].) Therefore, if L 7 is representable, we can assume there is a finite group G with a core-free 8 subgroup H < G such that L 7 is isomorphic to the interval sublattice [H, G] Sub(G). In this section we present some restrictions on the possible groups for which this can occur. The first restriction, which is the easiest to observe, is that G must act primitively on the cosets of one of its maximal subgroups. This suggests the possibility of describing G in terms of the O'Nan-Scott Theorem which characterizes primitive permutation groups. The goal is to eventually find enough restrictions on G so as to rule out all finite groups. As yet, we have not achieved this goal. However, the new results in this section reduce the possibilities to very special subclasses of the O'Nan-Scott classification theorem. This paves the way for future studies to focus on these subclasses when searching for a group representation of L 7 , or proving that none exists. The main result of this section is the following: Then the following hold. (i) G is a primitive permutation group. (ii) If N ⊳ G, then C G (N ) = 1. (iii) G contains no non-trivial abelian normal subgroup. (iv) G is not solvable. (v) G is subdirectly irreducible. Remark. It is obvious that (ii) ⇒ (iii) ⇒ (iv), and (ii) ⇒ (v), but we include these easy consequences in the statement of the result for emphasis; for, although the hard work will be in proving (ii) and We now prove the foregoing theorem through a series of claims. The first thing to notice about the interval [H, G] is that K is a non-modular element of the interval. This means that there is a spanning pentagonal (N 5 ) sublattice of the interval with K as the incomparable proper element. (See the diagram below, for example.) H K J 2 M 1 G Using this non-modularity property of K, it is easy to prove the following Claim 6.1. K is a core-free subgroup of G. Proof. Let N := core G (K). If N X for some X ∈ {M 1 , M 2 , J 1 , J 2 }, then N < X ∩ K = H, so N = 1 (since H is core-free). If N X for all X ∈ {M 1 , M 2 , J 1 , J 2 }, then N J 2 = G. But then Dedekind's rule leads to the following contradiction: J 2 M 1 ⇒ J 2 = J 2 (N ∩ M 1 ) = J 2 N ∩ M 1 = G ∩ M 1 = M 1 . Therefore, N = 1. Note that (i) of the theorem follows from Claim 6.1. Since K is core-free, G acts faithfully on the cosets G/K by right multiplication. Since K is a maximal subgroup, the action is primitive. The next claim is only slightly harder than the previous one as it requires the more general consequence of Dedekind's rule that we established above in Lemma 5.4.2 (i). Recall, for subgroups X and Y of a group G, we define the sets XY = {xy | x ∈ X, y ∈ Y }, and Y X = {yx | x ∈ X, y ∈ Y }, and we say that X and Y are permuting subgroups (or that X and Y permute, or that X permutes with Y ) provided the two sets XY and Y X coincide, in which case the set forms a group: XY = X, Y = Y X. Now that we know K, J 1 , J 2 are each core-free in G, we use this information to prove that at least one of the other maximal subgroups, M 1 or M 2 , is core-free in G, thereby establishing (vi) of the theorem. We will also see that G is subdirectly irreducible, proving (v). The proof of (ii) will then follow from the same argument used to prove Lemma 5.4.2 (ii), which we repeat below. We now prove that the alternative, C G (N ) = 1, does not occur. This case is a bit more challenging and must be split up into further subcases, each of which leads to a contradiction. Throughout, the assumption 1 = N M 2 is in force, and it helps to keep in mind the diagram in Figure 6.4. Of course, it's also normalized by J 1 , so N ∩ J 1 is normalized by the set M 1 J 1 , so it's normalized by the group generated by that set, which is M 1 , J 1 = G. 11 The conclusion is that N ∩ J 1 ⊳ G. Since J 1 is core-free, N ∩ J 1 = 1. But this contradicts the (by now familiar) consequence of Dedekind's rule: H < J 1 < M 2 ⇒ N ∩ H < N ∩ J 1 < N ∩ M 2 . Therefore, C G (N )H = M 1 does not occur. H < J 2 < M 2 ⇒ N ∩ H < N ∩ J 2 < N ∩ M 2 , (6.3.1) while the latter equality (N ∩ M 1 = N ) implies that N M 1 ∩ M 2 = J 2 which contradicts core G (J 2 ) = 1. We have proved that either M 1 or M 2 is core-free in G, and we have shown that, if M 2 has non-trivial core, then G is subdirectly irreducible. In fact, we proved that C G (N ) = 1 for the unique minimal normal subgroup N in this case. It remains to prove that G is subdirectly irreducible in case M 1 has non-trivial core. The argument is similar to the foregoing, and we omit some of the details that can be checked exactly as above. Claim 6.4. If M 1 has non-trivial core and N ⊳ G is contained in M 1 , then C G (N ) = 1 and G is subdirectly irreducible. Proof. If M 1 has non-trivial core, then there is a minimal normal subgroup N ⊳ G contained in M 1 . We proved above that M 2 must be core-free in this case, so either C G (N Finally, we note that the claims above taken together prove (ii), and thereby complete the proof of the theorem. For if G is subdirectly irreducible with unique minimal normal subgroup N , and if C G (N ) = 1, then all normal subgroups (which necessarily lie above N ) must have trivial centralizers. Conclusion We conclude this chapter with a final observation which helps us describe the O'Nan-Scott type of a group which has L 7 as an interval in its subgroup lattice. We end with a conjecture that should be the subject of future research. By what we have proved above, G acts primitively on the cosets of K, and it also acts primitively on the cosets of at least one of M 1 or M 2 . Suppose M 1 is core-free so that G is a primitive permutation group in its action on cosets of M 1 and let N be the minimal normal subgroup of G. As we have seen, N has trivial centralizer, so it is nonabelian and is the unique minimal normal subgroup of G. By the following elementary result (see, e.g., [20]) we see that the action of N on the cosets of the core-free maximal subgroup M i is not regular. 12 Consequently, G is characterized by case 2 of the version of the O'Nan-Scott Theorem given in the appendix, Section A.2. 12 Recall, a transitive permutation group N is acts regularly on a set Ω provided the stabilizer subgroup of N is trivial. Equivalently, every non-identity element of N is fixed-point-free. Equivalently, N is regular on Ω if and only if for each ω 1 , ω 2 ∈ Ω there is a unique n ∈ N such that nω 1 = ω 2 . In particular, |N | = |Ω|. CHAPTER 7 EXPANSIONS OF FINITE ALGEBRAS Background and motivation In this chapter we present a novel approach to the construction of new finite algebras and describe the congruence lattices of these algebras. Given a finite algebra B, . . . , let B 1 , B 2 has the second of these as its congruence lattice. The idea is to start with an algebra B = B, . . . having congruence lattice Con B ∼ = M 4 , expand the universe to the larger set A = B ∪ B 1 ∪ B 2 , and then define the right set F A of operations on A so that the congruence lattice of A = A, F F will be an M 4 with one atom "doubled" -that is, Con A will be the second lattice in figure 7.1. In this chapter we formalize this approach and extend it in four ways. The first is a straightforward generalization of the original overalgebra construction, and the second is a further expansion of these overalgebras. The third is a construction based on one suggested by Bill Lampe which addresses a basic limitation of the original procedure. Finally, we give a generalization of the third construction. For each of these constructions we prove results which allow us to describe the congruence lattices of the resulting overalgebras. Here is a brief outline of the remaining sections of this chapter: In Section 7.2 we prove a lemma which greatly simplifies the analysis of the structure of the newly enlarged congruence lattice and its relation to the original congruence lattice. In Section 7.3 we define overalgebra and in Section 7. A residuation lemma β = {(x, y) ∈ A 2 | (ef (x), ef (y)) ∈ β for all f ∈ Pol 1 (A)}. It is not hard to see that maps Con B into Con A. For example, if (x, y) ∈ β and g ∈ Pol 1 (A), then for all f ∈ Pol 1 (A) we have (ef g(x), ef g(y)) ∈ β, so (g(x), g(y)) ∈ β. For each β ∈ Con B, let β * = Cg A (β). That is, * : Con B → Con A is the congruence generation operator restricted to the set Con B. The following lemma concerns the three mappings, | B , , and * . The third statement of the lemma, which follows from the first two, will be useful in the later sections of this chapter. (i) * : Con B → Con A is a residuated mapping with residual | B . (ii) | B : Con A → Con B is a residuated mapping with residual . (iii) For all α ∈ Con A, β ∈ Con B, β = α| B ⇔ β * α β. In particular, β * | B = β = β| B . Proof. We first recall the definition of residuated mapping. If X and Y are partially ordered sets, and if f : X → Y and g : Y → X are order preserving maps, then the following are equivalent: (a) f : X → Y is a residuated mapping with residual g : Y → X; (b) for all x ∈ X, y ∈ Y , f (x) y iff x g(y); (c) g • f id X and f • g id Y . The definition says that for each y ∈ Y there is a unique x ∈ X that is maximal with respect to the property f (x) y, and the maximum x is given by g(y). Thus, (i) is equivalent to β * α ⇔ β α| B (∀ α ∈ Con A, ∀ β ∈ Con B). (7.2.1) This is easily verified, as follows: If β * α and (x, y) ∈ β, then (x, y) ∈ β * α and (x, y) ∈ B 2 , so (x, y) ∈ α| B . If β α| B then β * (α| B ) * Cg A (α) = α. Statement (ii) is equivalent to α| B β ⇔ α β (∀ α ∈ Con A, ∀ β ∈ Con B). (7.2.2) This is also easy to check. For, suppose α| B β and (x, y) ∈ α. Then (ef (x), ef (y)) ∈ α for all f ∈ Pol 1 (A) and (ef (x), ef (y)) ∈ B 2 , therefore, (ef (x), ef (y)) ∈ α| B β, so (x, y) ∈ β. Suppose α β and (x, y) ∈ α| B . Then (x, y) ∈ α β, so (ef (x), ef (y)) ∈ β for all f ∈ Pol 1 (A), including f = id A , so (e(x), e(y)) ∈ β. But (x, y) ∈ B 2 , so (x, y) = (e(x), e(y)) ∈ β. Combining (7.2.1) and (7.2.2), we obtain statement (iii) of the lemma. The lemma above was inspired by the two approaches to proving Lemma 1 of [32]. In the original paper * is used, while McKenzie uses the operator. Both β * and β are mapped onto β by the restriction map | B , so the restriction map is indeed onto Con B. However, our lemma emphasizes the fact that the interval [β * , β] = {α ∈ Con A | β * α β} is precisely the set of congruences for which α| B = β. In other words, the inverse image of β under | B is β| −1 B = [β * , β] . This fact plays a central rôle in the theory developed below. Nonetheless, for the sake of completeness, we conclude this section by verifying that Lemma 1 of [32] can be obtained from the lemma above. Corollary 7.2.2. | B : Con A → Con B is onto and preserves meets and joins. Proof. Given β ∈ Con B, each θ ∈ Con A in the interval [β * , β] is mapped to θ| B = β, so | B is clearly onto. That | B preserves meets is obvious. To see that | B is join preserving, note that for all η, θ ∈ Con A, we have η| B ∨ θ| B (η ∨ θ)| B since | B is order preserving. The opposite inequality follows from (7.2.2) above. For, (η ∨ θ)| B η| B ∨ θ| B ⇔ η ∨ θ η| B ∨ θ| B , and the second inequality holds since, by (7.2.2) again, η η| B ∨ θ| B ⇔ η| B η| B ∨ θ| B and θ η| B ∨ θ| B ⇔ θ| B η| B ∨ θ| B . Remark. This approach to proving Lemma 1 of [32], which is similar to the proof given in [24], does not reveal any information about the permutability of the congruences of A, unlike the more direct proof given in [32]. Overalgebras In the previous section, we started with an algebra A and considered a subreduct B with universe B = e(A), the image of an idempotent unary polynomial of A. In this section, we start with a fixed finite algebra B = B, . . . and consider various ways to construct an overalgebra, that is, an algebra A = A, F A having B as a subreduct where B = e(A) for some idempotent e ∈ F A . Beginning with a specific finite algebra B, our goal is to understand what (finitely representable) congruence lattices Con A can be built up from Con B by expanding the algebra B in this way. Overalgebras I Let B be a finite set, say, B = {b 1 , b 2 . . . , b n }, let F ⊆ B B be a set of unary maps taking B into itself, and consider the unary algebra B = B, F , with universe B and basic operations F . When clarity demands it, we call this collection of operations F B . Let B 1 , B 2 , . . . , B K be sets of the same cardinality as B, which intersect B at exactly one point, as follows: B = {b 1 , b 2 , b 3 , . . . , b n } B 1 = {b 1 , b 1 2 , b 1 3 , . . . , b 1 n } B 2 = {b 2 1 , b 2 , b 2 3 , . . . , b 2 n } B 3 = {b 3 1 , b 3 2 , b 3 , . . . , b 3 n } . . . (7.3.1) B K = {b K 1 , . . . , b K K−1 , b K , b K K+1 , . . . , b K n }. That is, for all 1 i < j K, we have |B i | = n K, B ∩ B i = {b i }, and B i ∩ B j = ∅. Sometimes it is notationally convenient to use the label B 0 := B. π i : B, F ∼ = B i , F i B ∋ b → b i ∈ B i F ∋ f → f πi ∈ F i To say that π i is an isomorphism of two non-indexed algebras is to say that π i is a bijection of the universes which respects the interpretation of the basic operations; that is, π i f (b) = f πi (π i b). In the present case, this holds by construction: 2 π i f (b) = π i f (π −1 i π i b) = f πi (π i b). Let A = K i=0 B i and define the following unary maps on A: • e k : A → A is e k (b j i ) = b k i (1 i n; 0 j, k K); • s : A → A is s(x) =        x, if x ∈ B 0 , b i , if x ∈ B i . Let F A := {f e 0 : f ∈ F } ∪ {e k : 0 k K} ∪ {s}, and define the unary algebra A := A, F A . Throughout, the map is defined in essentially the same way as it is in McKenzie's paper [24]. That is, given two algebras A = A, . . . and B = B, . . . with B = e(A) for some idempotent e ∈ Pol 1 (A), we define : Con B → Con A by 2 This generalizes to k-ary operations if we adopt the following convention: β = {(x, y) ∈ A 2 | (ef (x), ef (y)) ∈ β, ∀ f ∈ Pol 1 (A)} (β ∈ Con B).f π i (a 1 , . . . , a k ) = π i f (π −1 i (a 1 ), . . . , π −1 i (a k )). 3 In fact, there are infinitely many, but apart from those involving S 3 , C 3 × C 3 , and (C 3 × C 3 ) ⋊ C 3 , they are quite large. The next smallest G-set with M 4 congruence lattice that we know of comes from the group G = We prefer to use "0-offset" notation, and define the universe of the S 3 -set described above to be {0, 1, . . . , 5} instead of {1, 2, . . . , 6}. As such, the nontrivial congruence relations of this algebra are, gap> for b in AllBlocks(G) do Print(Orbit(G,b,OnSets)-1, "\n"); od; [ 3,4,5 ] ] [ 2,5 ], [ 1,4 ] ] [ 2,3 ], [ 1,5 ] ] [ 2,4 ], [ 1, 3 ] ] Next, we create an algebra in UACalc format using the two generators of the group as basic opera- We now construct an overalgebra which "doubles" the congruence α = Cg B (0, 2) = |0, 1, 2|3, 4, 5| by choosing intersection points 0 and 2. The GAP function Overalgebra carries out the construction, and is invoked as follows: 6 gap> Read("Overalgebras.g"); [((C 3 × C 3 ) ⋊ C 2 ) × ((C 3 × C 3 ) ⋊ C 2 )] ⋊ C[ [ 0, 1, 2 ],[ [ 0, 3 ],[ [ 0, 4 ],[ [ 0, 5 ], gap> Overalgebra([G, [0,2]]); 5 The GAP routine gap2uacalc.g is available at www.uacalc.org. 6 The GAP file Overalgebras.g is available at http://dl.dropbox.com/u/17739547/diss/Overalgebras.g. If F A = {e 0 , e 1 , e 2 , s, g 0 e 0 , g 1 e 0 }, then the algebra A, F A has the congruence lattice shown in We now prove two theorems which describe the basic structure of the congruence of an overalgebra constructed as described at the outset of this section. In particular, the theorems explain why the interval [α * , α] ∼ = 2 appears in the first example above, while [β * , β] ∼ = 2 × 2 appears in the second. This gives an overalgebra with universe Figure 7.3. α * α β * γ * δ * 1 A 0 Aα * β β ε β ε ′ β * γ * δ * 1 A 0 A Given a congruence relation β ∈ Con B, let {b β(1) , . . . , b β(m) } denote a transversal of β; i.e. a full set of β-class representatives. Thus, as a partition of the set B, β has m classes, or blocks. (Using the notation β(r) for the indices of the representatives helps us to remember that b β(r) is a representative of the r-th block of the congruence β.) By the isomorphisms π i defined above, to each β ∈ Con B there corresponds a congruence relation β Bi ∈ Con B i , and if {b β(1) , . . . , b β(m) } is a transversal of β, then the map π i also gives a transversal of β Bi , namely {π i (b β(1) ), . . . , π i (b β(m) )} = {b i β(1) , . . . , b i β(m) }. Thus, the r-th block of β Bi is b i β(r) /β Bi . Let T = {b 1 , b 2 , . . . , b K } be the set of tie-points, that is, the points at which the sets B i (1 i K) intersect the set B. Let T r = {b ∈ T | (b, b β(r) ) ∈ β} be the set of those tie-points that are in the r-th congruence class of β. Theorem 7.3.2. For each β ∈ Con B, Cg A (β) = K k=0 β B k ∪ m r=1   b β(r) /β ∪ bj ∈Tr b j /β Bj   2 . (7.3.2) Remark. Before proceeding to the proof, we advise the reader to consider the small example il- Proof. Let β * denote the right-hand side of (7.3.2). We first check that β * ∈ Con A. It is easy to see that β * is an equivalence relation, so we need only show f (β * ) ⊆ β * for all 7 f ∈ F A , where F A := {f e 0 : f ∈ F } ∪ {e k : 0 k K} ∪ {s}. In other words, we prove: if (x, y) ∈ β * and f ∈ F A , then (f (x), f (y)) ∈ β * . Case 1: (x, y) ∈ β B k for some 0 k K. Then, (e i (x), e i (y)) ∈ β Bi ⊆ β * for all 0 i K, and (f e 0 (x), f e 0 (y)) ∈ β ⊆ β * for all f ∈ F B . Also, (s(x), s(y)) =        (x, y), if k = 0 (b k , b k ), if k = 0 belongs to β * . Thus, (f (x), f (y)) ∈ β * for all f ∈ F A . Case 2: (x, y) ∈ b β(r) /β ∪ bj ∈Tr b j /β Bj 2 for some 1 r m. Assume x ∈ b j /β Bj and y ∈ b k /β B k for some b j , b k ∈ T r . Then (e 0 (x), b j ) ∈ β, (e 0 (y), b k ) ∈ β, and and b j β b β(r) β b k so (e 0 (x), e 0 (y)) ∈ β. (7.3.3) Thus, for all 0 ℓ K we have (e ℓ e 0 (x), e ℓ e 0 (y)) ∈ β B ℓ . But note that e ℓ e 0 = e ℓ . It also follows from (7.3. 3) that (f e 0 (x), f e 0 (y)) ∈ β for all f ∈ F B . Finally, (s(x), s(y)) = (b j , b k ) ∈ β. The only remaining possibility for case 2 is x ∈ b β(r) /β and y ∈ b j /β Bj for some b j ∈ T r . Since b j ∈ T r , we have (b j , b β(r) ) ∈ β, so (e 0 (y), b j ) ∈ β, so (e 0 (y), b β(r) ) ∈ β, so (e 0 (x), x) = (e 0 (y), e 0 (x)) ∈ β. Therefore, (e ℓ (y), e ℓ (x)) ∈ β B ℓ for all 0 ℓ K and (f e 0 (y), f e 0 (x)) ∈ β for all f ∈ F B . Finally, s(x) = x β b β(r) β b j = s(y), so (s(x), s(y)) ∈ β. We have established that f (β * ) ⊆ β * for all f ∈ F A . To complete the proof of Theorem 7.3.2, we must show that β ⊆ η ∈ Con A implies β * η. If β ⊆ η ∈ Con A, then β B k ⊆ η, since (x, y) ∈ β implies (e k (x), e k (y)) ∈ β B k for all 0 k K. To see that the second term of (7.3.2) belongs to η, let (x, y) be an arbitrary element of that term, say, (x, b i ) ∈ β Bi and (y, b j ) ∈ β Bj . As we just observed, β, β Bi , and β Bj are subsets of η, and (b i , b j ) ∈ β, so x β Bi b i β b j β Bj y, so (x, y) ∈ η. As above, for a given β ∈ Con B with transversal {b β(1) , . . . , b β(m) }, we denote the set of tie-points contained in the r-th block of β by T r ; that is, T r = {b ∈ T | (b, b β(r) ) ∈ β} = K k=1 B k ∩ b β(r) /β. Suppose this set is T r = {b i1 , b i2 , . . . , b i |Tr | } and let I r = {i 1 , i 2 , . . . , i |Tr| } be the indices of these tie-points. Also, we define β * = Cg A (β), for β ∈ Con B. β = |b 0 , b 1 , b 2 | b 3 , b 4 , b 5 | b 6 , b 7 , b 8 |, and two blocks of β contain two tie-points each. In particular, the set of tie-points in the first block of β is T 1 = {b 0 , b 2 }. For the second and third blocks, T 2 = ∅ and T 3 = {b 6 , b 8 }. Figure 7.5: The universe A = B 0 ∪ · · · ∪ B 4 for a simple example; dotted lines surround each congruence class of β. B 0 → B 1 b 0 b 1 1 b 1 2 b 1 4 b 1 5 b 1 7 b 1 8 b 1 3 b 1 6 b 1 B 2 b 2 b 2 1 b 2 0 b 2 4 b 2 3 b 2 7 b 2 6 b 2 5 b 2 8 b 3 b 4 b 5 b 6 b 7 b 8 B 3 b 3 7 b 3 8 b 3 4 b 3 5 b 3 1 b 3 2 b 3 3 b 3 0 B 4 b 3 3 b 3 4 b 3 5 b 3 0 b 3 1 b 3 2 b 3 6 b 3 7β * β b 0 b 1 1 b 1 2 b 1 4 b 1 5 b 1 7 b 1 8 b 1 3 b 1 6 b 1 b 2 b 2 1 b 2 0 b 2 4 b 2 3 b 2 7 b 2 6 b 2 5 b 2 8 b 3 b 4 b 5 b 6 b 7 b 8 b 3 7 b 3 8 b 3 4 b 3 5 b 3 1 b 3 2 b 3 3 b 3 0 b 3 3 b 3 4 b 3 5 b 3 0 b 3 1 b 3 2 b 3 6 b 3 7 b 0 b 1 1 b 1 2 b 1 4 b 1 5 b 1 7 b 1 8 b 1 3 b 1 6 b 1 b 2 b 2 1 b 2 0 b 2 4 b 2 3 b 2 7 b 2 6 b 2 5 b 2 8 b 3 b 4 b 5 b 6 b 7 b 8 b 3 7 b 3 8 b 3 4 b 3 5 b 3 1 b 3 2 b 3 3 b 3 0 b 3 3 b 3 4 b 3 5 b 3 0 b 3 1 b 3 2 b 3 6 b 3 7β = β * ∪ m r=1 m ℓ=1 ℓ =r (j,k)∈I 2 r b j β(ℓ) /β Bj ∪ b k β(ℓ) /β B k 2 . (7.3.4) Moreover, the interval [β * , β] of Con A contains every equivalence relation of A between β * and β, and is isomorphic to (Eq|T r |) m−1 ; that is, [β * , β] = {θ ∈ Eq(A) | β * ⊆ θ ⊆ β} ∼ = m r=1 (Eq|T r |) m−1 . (7.3.5) Remark. Blocks containing only one tie-point, i.e. those for which |T r | = 1, contribute nothing to the direct product in (7.3.5). Also, for some 1 r m we may have T r = ∅, in which case we agree to let Eq|T r | = Eq(0) := 1. Proof. Let β denote the right-hand side of (7.3.4). It is easy to see that β is an equivalence relation on A. To see that it is also a congruence relation, we will prove f ( β) ⊆ β for all f ∈ F A . Fix (x, y) ∈ β. If (x, y) ∈ β * , then (f (x), f (y)) ∈ β * holds for all f ∈ F A , as in Theorem 7.3.2. Suppose (x, y) / ∈ β * , say, x ∈ b j β(ℓ) /β Bj and y ∈ b k β(ℓ) /β B k for some j, k ∈ I r , 1 r m, and ℓ = r. Then x and y are in the ℓ-th blocks of their respective subreduct universes, B j and B k , so for each 0 i K, (e i (x), e i (y)) ∈ β Bi . In particular, (e 0 (x), e 0 (y)) ∈ β, so (ge 0 (x), ge 0 (y)) ∈ β for all g ∈ F B . Also, (s(x), s(y)) = (b j , b k ) ∈ T 2 r ⊆ β. This proves that for each f ∈ F A we have (f (x), f (y)) ∈ β. (In fact, (f (x), f (y)) ∈ β * .) Whence β ∈ Con A. Now notice that β| B = β. Therefore, by the residuation lemma of Section 7.2, we have β β. To prove the reverse inclusion, we suppose (x, y) / ∈ β and show (x, y) / ∈ β. Without loss of generality, assume x ∈ b j β(p) /β Bj and y ∈ b k β(q) /β B k , for some 1 p, q m and 1 j, k K + 1. If p = q, then (j, k) / ∈ I 2 r for all 1 r m (otherwise (x, y) ∈ β), so (e 0 s(x), e 0 s(y)) = (e 0 (b j ), e 0 (b k )) = (b j , b k ) / ∈ β, so (x, y) / ∈ β. If p = q, then e 0 (x) ∈ b β(p) /β and e 0 (y) ∈ b β(q) /β -distinct β classes - so (e 0 (x), e 0 (y)) / ∈ β, so (x, y) / ∈ β. To prove (7.3.5), we first note that every equivalence relation θ on A with β * ⊆ θ ⊆ β satisfies f (θ) ⊆ θ for all f ∈ F A , and is therefore a congruence of A. Indeed, in proving β = β above, we saw that f ( β) ⊆ β * for all f ∈ F A , so, a fortiori, f (θ) ⊆ β * for all equivalence relations θ ⊆ β. Therefore, [β * , β] = {θ ∈ Eq(A) | β * ⊆ θ ⊆ β}. To complete the proof, we must show that this interval is isomorphic to the lattice m r=1 (Eq|T r |) m−1 . Consider, β/β * = {(x/β * , y/β * ) ∈ (A/β * ) 2 | (x, y) ∈ β}. Let N be the number of blocks of β/β * (which, of course, is the same as the number of blocks of β). For 1 k N , let x k /β * be a representative of the k-th block of β/β * . Let B k = (x k /β * )/( β/β * ) denote this block; that is, B k = {y/β * ∈ A/β * | (x k /β * , y/β * ) ∈ β/β * }. Then, N k=1 Eq(B k ) ∼ = {θ ∈ Eq(A) | β * ⊆ θ ⊆ β} = [β * , β]. The isomorphism is given by the maps, N k=1 Eq(B k ) ∋ η → N k=1 η k ∈ [β * , β] [β * , β] ∋ θ → N k=1 θ ∩ B 2 k ∈ N k=1 Eq(B k ), where η k denotes the projection of η onto its k-th coordinate. Now, the r-th β-class of B 0 , denoted b β(r) /β, has |T r | tie-points, so there are |T r | sets, B i1 , B i2 , . . . , B i |Tr | , each of which intersects B 0 at a distinct tie-point in b β(r) /β; that is, B ij ∩ b β(r) /β = {b ij } (b ij ∈ T r ). (See Figure 7.6.) A block B k of β/β * has a single element when it contains b β(r) /β. Otherwise, it has |T r | elements, namely, b i1 β (ℓ) /β Bi 2 , b i2 β (ℓ) /β Bi 2 , . . . , b i |Tr | β (ℓ) /β Bi |Tr | , for some 1 ℓ m; ℓ = r. Thus, for each 1 r m, we have m − 1 such |T r |-element blocks, so N k=1 Eq(B k ) ∼ = m r=1 (Eq|T r |) m−1 . We now describe the situation in which the foregoing construction is most useful. Here and in the sequel, instead of Eq(2), we usually write 2 to denote the two element lattice. Given a finite congruence lattice Con B and a pair (x, y) ∈ B 2 , let β ∈ Con B be the unique smallest congruence containing (x, y). Then β = Cg B (x, y), and if we build an overalgebra as described above using {x, y} as tie-points, then, by Theorem 7.3.3, the interval of all θ ∈ Con A for which θ| B = β will be [β * , β] ∼ = Eq(2) m−1 = 2 m−1 , where m is the number of congruence classes in β. Also, since β is the smallest congruence containing (x, y) we can be sure that, for all θ β, the interval [θ * , θ] is trivial; that is, θ * = θ. Finally, for each θ > β, we will have [θ * , θ] ∼ = 2 r−1 , where r is the number of congruence classes of θ. This is easy to achieve by adding more operations in the overalgebra construction described above. In fact, it is possible to introduce additional operations so that, if β = Cg B (x, y), then θ * = θ for all θ ∈ Con B with θ β. We now describe these operations and state this claim more formally as Proposition 7.3.5 below. We start with the overalgebra construction described above. Suppose β = Cg B (x, y) has transversal {b β(1) , . . . , b β(m) }, and for each 1 r m, let T r = {b ∈ T | (b, b β(r) ) ∈ β} = {b i1 , b i2 , . . . , b i |Tr | } be the tie-points contained in the r-th block of β, as above. Let I r = {i 1 , i 2 , . . . , i |Tr| } be the indices of these tie-points. Then {B i : i ∈ I r } is the collection of subreduct universes which intersect the r-th β block of B. For each 1 r m, define the operation s r : A → A as follows: s r (x) =        b i if x ∈ B i for some i ∈ I r , x otherwise. Define all other operations as above and let 1. if θ ∧ β = 0 B , then θ * = θ; 2. if θ β, then [θ * , θ] ∼ = n r=1 (Eq|T ∩ b θ(r) /θ|) n−1 , where n m is the number of congruence classes of θ. The first part of the proposition is easy to prove, given the additional operations s r , 1 r m. The second part follows from Theorem 7.3.3. Note that T r was defined above to be T ∩ b β(r) /β, so T = m r=1 T r is a partition of the tiepoints, and it is on this partition that our definition of the additional operations s r is based. A modified version of the GAP function used above to construct overalgebras allows the user to specify an arbitrary partition of the tie-points, and the extra operations will be defined accordingly. For example, to base the selection and partition of the tie-points on the congruence β in the example above, we invoke the following command: [[0,3], [2,5]] ]); gap> OveralgebraXO([ G, The resulting overalgebra has congruence lattice isomorphic to the lattice in Figure 7 We close this subsection with a result which describes one way to add even more operations to the overalgebra in case we wish to eliminate some of the congruences in [β * , β] without affecting congruences outside that interval. In the following claim we assume the base algebra B = B, G is a transitive G-set. Then, for each θ ∈ Con A, g(θ) θ only if β * < θ < β. (7.3.6) Of course, these g maps may not be the only functions in A A which have the property stated in (7.3.6). Also, in general, even with the whole collection of maps g defined above, we may not be able to eliminate every β * < θ < β. In fact, it's easy to construct examples in which there exist β * < θ < β such that g(θ) ⊆ θ for every every g ∈ A A . Overalgebras II In the previous section we described a procedure for building an overalgebra A of B such that for some principal congruence β ∈ Con B and for all β θ < 1 B , the inverse image θ| −1 B = [θ * , θ] Con A is non-trivial. In this section, we start with a non-principal congruence β ∈ Con B and ask if it is possible to construct an overalgebra A such that θ| −1 B 0 ∩ B 1 = {a 1 } = {a 1 1 }, B i ∩ B i+1 = {b i i } = {a i+1 i+1 } for 1 i < K, B K ∩ B K+1 = {b K K } = {a K+1 1 }. All other intersections are empty. (See Figure 7.10.) Figure 7.10: The universe of the overalgebra. B B1 B2 B3 · · · BK BK+1 a1 = a 1 1 b 1 1 = a 2 2 b 2 2 = a 3 3 b K−1 K−1 = a K K b K K = a K+1 1 For 0 i, j K + 1, let S i,j : B i → B j be the bijection S i,j (x i ) = x j . Put A := B 0 ∪ · · · ∪ B K+1 , and define the following functions in A A : e 0 (x) =                x, x ∈ B 0 , a 1 , x ∈ B j , 1 j K, S K+1,0 (x), x ∈ B K+1 ; e i (x) =                a i i , x ∈ B j , j < i, x, x ∈ B i , b i i , x ∈ B j , j > i; (1 i K), e K+1 (x) =                S 0,K+1 (x), x ∈ B 0 , a K+1 1 , x ∈ B j , 1 j K, x, x ∈ B K+1 . Using these maps we define the set F A of operations on A as follows: let q i,j = S i,j • e i for 0 i, j K + 1 and define 8 1 , b 1 ), . . . , (a K , b K )), as described above, and define β * = K+1 j=0 β Bj ∪ (a 1 /β ∪ a 1 1 /β B1 ∪ a 2 2 /β B2 ∪ · · · ∪ a K K /β BK ∪ a K+1 1 /β BK+1 ) 2 . Then, β * = Cg A (β). If β has transversal {a 1 , c 1 , c 2 , . . . , c m−1 }, then β = β * ∪ m−1 i=1 (c i /β ∪ c K+1 i /β BK+1 ) 2 . (7.3.7) Moreover, [β * , β] ∼ = 2 m−1 . Proof. It is clear that β * is an equivalence relation on A, so we first check that f (β * ) ⊆ β * for all f ∈ F A . This will establish that β * ∈ Con A. Thereafter we show that β ⊆ η ∈ Con A implies β * η, which will prove that β * is the smallest congruence of A containing β, as claimed in the first part of the theorem. Fix (x, y) ∈ β * . To show (f (x), f (y)) ∈ β * we consider two possible cases. Case 1: (x, y) ∈ β Bj for some 0 j K + 1. In this case it is easy to verify that (q i,0 (x), q i,0 (y)) ∈ β and (q 0,i (x), q 0,i (y)) ∈ β Bi for all 0 i K + 1. For example, if (x, y) ∈ β Bj with 1 j K, then (q 0,i (x), q 0,i (y)) = (a i 1 , a i 1 ) and (q i,0 (x), q i,0 (y)) is either (b i , b i ) or (a i , a i ) depending on whether i is below or above j, respectively. If i = j, then (q i,0 (x), q i,0 (y)) is the pair in B 2 corresponding to (x, y) ∈ β Bj , so (q i,0 (x), q i,0 (y)) ∈ β. A special case is (q 0,0 (x), q 0,0 (y)) ∈ β. Now, since q 0,0 = e 0 , we have (f e 0 (x), f e 0 (y)) ∈ β for all f ∈ F B . Altogether, the foregoing implies that (f (x), f (y)) ∈ β * for all f ∈ F A . Case 2: (x, y) ∈ B 2 where B := a 1 /β ∪ a 1 1 /β B1 ∪ · · · ∪ a K K /β BK ∪ a K+1 1 /β BK+1 . Note that e 0 (B) = a 1 /β. Therefore, (e 0 (x), e 0 (y)) ∈ β, so (f e 0 (x), f e 0 (y)) ∈ β for all f ∈ F B . Also, q 0,k (B) = S 0,k e 0 (B) = S 0,k (a 1 /β) = a k 1 /β B k , which is a single block of β * . Similarly, e k (B) = a k k /β B k , so q k,0 (B) = S k,0 e k (B) = S k,0 (a k k /β B k ) = a k /β. Whence, (x, y) ∈ B 2 implies (f (x), f (y)) ∈ β * for all f ∈ F A . We have thus established that β * is a congruence of A which contains β. We now show that it is the smallest such congruence. Indeed, suppose β ⊆ η ∈ Con A, and fix (x, y) ∈ β * . If (x, y) ∈ β Bj for some 0 j K + 1, then (q j,0 (x), q j,0 (y)) ∈ β ⊆ η, so (x, y) = (q 0,j q j,0 (x), q 0,j q j,0 (y)) ∈ η. If, instead of (x, y) ∈ β Bj , we have (x, y) ∈ B 2 , then without loss of generality x ∈ a i i /β Bi and y ∈ a j j /β Bj for some 0 i < j K + 1. We only discuss the case 1 i < j K, as the other cases can be handled similarly. Since x ∈ a i i /β Bi = b i i /β Bi , we have (q i,0 (x), b i ) ∈ β. Similarly, (a j , q j,0 (y)) ∈ β. Therefore, we obtain the following diagram 9 q i,0 (x) β b i a i+1 β b i+1 a i+2 b j−1 a j β q j,0 (y) b j−1 j−1 = a j j x b i i = a i+1 i+1 b i+1 i+1 = a i+2 i+2 y q 0,i q 0,i+1 q 0,i+2 . . . . . . q 0,j−1 q 0,j Since β ⊆ η ∈ Con A, and since q 0,k ∈ F A for each k, the diagram makes it clear that (x, y) must belong to η. To prove (7.3.7), let β denote the right-hand side. That is, β := β * ∪ m−1 i=1 (c i /β ∪ c K+1 i /β BK+1 ) 2 . It is clear that β ∈ Eq(A), so we verify β ∈ Con A by proving that f ( β) ⊆ β for all f ∈ F A . Fix (x, y) ∈ β. If (x, y) ∈ β * , then (f (x), f (y)) ∈ β * for all f ∈ F A , by the first part of the theorem. So suppose (x, y) ∈ (c i /β ∪ c K+1 i /β BK+1 ) 2 , for some 1 i m − 1. For ease of notation, define C i := c i /β ∪ c K+1 i /β BK+1 . Then, since e 0 (C i ) = c i /β, we have (e 0 (x), e 0 (y)) ∈ β, so (f e 0 (x), f e 0 (y)) ∈ β for all f ∈ F B . Also, for 0 k K + 1, we have 10 q 0,k (C i ) = S 0,k (c i /β) = c k i /β B k . Therefore, q 0,k (C i ) is in a single block of β * , so (q 0,k (x), q 0,k (y)) ∈ β * . Also, for 1 k K, we have e k (c i /β) = {a k k } and e k (c K+1 i /β BK+1 ) = {b k k }, so q k,0 (C i ) = S k,0 ({a k k , b k k }) = {a k , b k } ⊆ a k /β, while, for k = K + 1, we have e K+1 (C i ) = c K+1 i /β BK+1 , so q K+1,0 (C i ) = S K+1,0 (c K+1 i /β BK+1 ) = c i /β. Also, the cases involving i = 0 and/or j = K + 1 can be handled similarly. 10 By c 0 i /β B 0 we mean, of course, c i /β. Thus, for all 0 k K + 1, we have (q k,0 (x), q k,0 (y)) ∈ β * . This proves that (f (x), f (y)) ∈ β * ⊆ β holds for all f ∈ F A , so β ∈ Con A. Next, note that β| B = β, so by the residuation lemma of Section 7.2, β β. Thus, to prove (7.3.7), it suffices to show that (x, y) / ∈ β implies (x, y) / ∈ β. This is straight-forward, and similar to the argument we used to check the analogous fact in the proof of Theorem 7.3.3. Nonetheless, we verify most of the cases, and omit only a few special cases which are easy to check. Suppose (x, y) / ∈ β, and suppose x ∈ c j p /β Bj and y ∈ c k q /β B k for some 0 j k K + 1 and 1 p, q m − 1. If j = 0 and k = K + 1, then p = q (otherwise, (x, y) ∈ β). Therefore, e 0 (x) ∈ c p /β and e 0 (y) ∈ c q /β, so (e 0 (x), e 0 (y)) / ∈ β, so (x, y) / ∈ β. If p = q, then j = k (otherwise, (x, y) ∈ β). Thus, (e j (x), e j (y)) = (x, b j j ) ⇒ (q j,0 (x), q j,0 (y)) = (q j,0 (x), b j ); (e k (x), e k (y)) = (a k k , y) ⇒ (q k,0 (x), q k,0 (y)) = (a k , q k,0 (y)). One of the pairs on the right is not in β. For if both are in β, then x = q 0,j q j,0 (x) β * q 0,j (b j ) = b j j = a j+1 j+1 β * · · · · · · β * a k k = q 0,k (a k ) β * q 0,k q k,0 (y) = y, which contradicts (x, y) / ∈ β, so we must have either (q j,0 (x), q j,0 (y)) / ∈ β or (q k,0 (x), q k,0 (y)) / ∈ β. Therefore, since e 0 q i,0 = q i,0 , we see that (x, y) / ∈ β. The other cases, e.g. x ∈ a 1 /β, y ∈ c k q /β B k , can be checked similarly. It remains to prove that [β * , β] ∼ = 2 m−1 , but this follows easily from the first part of the proof, where we saw that (f (x), f (y)) ∈ β * for all f ∈ F A and for all (x, y) ∈ β. This implies that all equivalence relations on A that are above β * and below β are, in fact, congruence relations of A. The shape of this interval of equivalence relations is even simpler than the shape of the analogous interval we found in Theorem 7.3.3. In the present case, we have [β * , β] = {θ ∈ Eq(A) | β * ⊆ θ ⊆ β} ∼ = 2 m−1 . Before stating the next result, we remind the reader that θ * = Cg A (θ) for each θ ∈ Con B. Lemma 7.3.7. If η ∈ Con A satisfies η| B = θ, and if (x, y) ∈ η \ θ * for some x ∈ B i , y ∈ B j , then i = 0, j = K + 1, and θ β. In other words, unless i = 0 and j = K + 1, the congruence η doesn't join blocks of B i with blocks of B j (except for those already joined by θ * ). Proof. We rule out all 0 i j K + 1 except for i = 0 and j = K + 1 by showing that, in each of the following cases, we arrive at the contradiction (x, y) ∈ θ * := Cg A (θ). Case 1: i = j. If (x, y) ∈ B 2 i for some 0 i K + 1, then (q i,0 (x), q i,0 (y)) ∈ η| B = θ θ * , so (x, y) = (q 0,i q i,0 (x), q 0,i q i,0 (y)) ∈ θ * . Case 2: 1 i < j K. In this case, (q i,0 (x), q i,0 (y)) = (q i,0 (x), b i ) ∈ θ, (q j,0 (x), q j,0 (y)) = (a j , q j,0 (y)) ∈ θ, When j = i + 1, we obtain x = q 0,i q i,0 (x) θ * q 0,i (b i ) = b i i = a j j = q 0,j (a j ) θ * q 0,j q j,0 (y) = y, (7.3.8) so (x, y) ∈ θ * . This can be seen more transparently in a diagram. q i,0 (x) b i θ a j θ q j,0 (y) x b i i = a j j y q 0,i q 0,j If j > i+1, then (q k,0 (x), q k,0 (y)) = (a k , b k ) ∈ θ for all i < k < j, and we have the following diagram: q i,0 (x) θ b i a i+1 θ b i+1 a i+2 b j−1 a j θ q j,0 (y) b j−1 j−1 = a j j x b i i = a i+1 i+1 b i+1 i+1 = a i+2 i+2 y q 0,i q 0,i+1 q 0,i+2 . . . . . . q 0,j−1 q 0,j Here too we could write out a line analogous to (7.3.8), but it is obvious from the diagram that (x, y) ∈ θ * . The case i = 0; 1 j K, as well as the case 1 i K; j = K + 1, can be handled with diagrams similar to those used above, and the proofs are almost identical, so we omit them. The only remaining possibility is x ∈ B 0 and y ∈ B K+1 . In this case we have (q k,0 (x), q k,0 (y)) = (a k , b k ) ∈ θ, for all 1 k K. Therefore, θ β = Cg A ((a 1 , b 1 ), . . . , (a K , b K )). We now consider an example of a congruence lattice having a coatom β that is not principal, and we use the method described in this section to construct an overalgebra A for which β * < β in Con A, and θ * = θ for all θ β in Con B. However, we also have γ 3 = Cg B (0, 3), a congruence with 6 classes, so again by Theorem 7.3.3, [γ * 3 , γ 3 ] ∼ = 2 5 . Thus, using this method it is not possible to obtain a non-trivial interval [β * , β] while preserving the original congruence lattice structure below β. This is true no matter which pair (x, y) ∈ β we choose as tie-points, since, in every case, the pair will belong to a congruence below β. The procedure described in this subsection does not have the same limitation. Indeed, if we set (a 1 , b 1 ) = (0, 3) and (a 2 , b 2 ) = (8,11) in this construction, then the universe of the overalgebra is A = Arranging the subreduct universes as in Figure 7.12 reveals the congruences above β * . In fact, the four congruences in the interval [β * , β] can be read off directly from the diagram. For example, the congruence classes of β * are shown in Figure 7.13, while the congruence β, in addition to these relations, joins blocks |4, 5, 6, 7| and |37, 38, 39, 40|, as well as blocks |8, 9, 10, 11| and |41, 42, 43, 44|. As for the congruences β ε , β ε ′ , one joins |4, 5, 6, 7| and |37, 38, 39, 40|, while the other joins |8, 9, 10, 11| and |41, 42, 43, 44|. The full congruence lattice, Con A, appears in Figure 7.14. Overalgebras III In Section 7.3.1 we constructed an algebra A with a congruence lattice Con A having interval sublattices [β * , β] that are isomorphic to products of powers of partition lattices. We saw that the construction has two main limitations. First, the size of the partition lattices is limited by the size of the congruence classes of β ∈ Con B. Second, when β is non-principal, it is impossible with this construction to obtain a nontrivial inverse image [β * , β] without also having nontrivial inverse images [θ * ,θ] for some θ β. In Section 7.3.2, we presented a construction which resolves the second limitation. However, the first limitation is even more severe in that the resulting intervals [β * , β] are simply powers of 2 -i.e., Boolean algebras. In this section, we present a generalization of the previous constructions which overcomes both of the limitations mentioned above. 0 A 1 Aβ β ε β ε ′ β * α * γ * 1 γ * 2 γ * 3 Let B = B, F be a finite algebra, and suppose β = Cg B ((a 1 , b 1 ), . . . , (a K−1 , b K−1 )) for some a 1 , . . . , a K−1 , b 1 , . . . , b K−1 ∈ B. Define B 0 = B and, for some fixed Q 0, let B 1 , B 2 , . . . , B (2Q+1)K be sets of cardinality |B| = n. As above, we use the label x i to denote the element of B i which corresponds to x ∈ B under the bijection. For ease of notation, let M := (2Q + 1). We arrange the sets so that they intersect as follows: B 0 ∩ B 1 = {a 1 } = {a 1 1 }, B 1 ∩ B 2 = {b 1 1 } = {a 2 2 }, B 2 ∩ B 3 = {b 2 2 } = {a 3 3 }, . . . B K−2 ∩ B K−1 = {b K−2 K−2 } = {a K−1 K−1 }, B K−1 ∩ B K = B K ∩ B K+1 = {b K−1 K−1 } = {b K K−1 } = {b K+1 K−1 }, B K+1 ∩ B K+2 = {a K+1 K−1 } = {b K+2 K−2 }, B K+2 ∩ B K+3 = {a K+2 K−2 } = {b K+3 K−3 }, . . . . . . , B 2K−2 ∩ B 2K−1 = {a 2K−2 2 } = {b 2K−1 1 }, B 2K−1 ∩ B 2K = B 2K ∩ B 2K+1 = {a 2K−1 1 } = {a 2K 1 } = {a 2K+1 1 }, B 2K+1 ∩ B 2K+2 = {b 2K+1 1 } = {b 2K+2 2 }, B 2K+2 ∩ B 2K+3 = {b 2K+2 2 } = {b 2K+3 3 }, . . . B MK−2 ∩ B MK−1 = {b K−2 MK−2 } = {a K−1 MK−1 }, B MK−1 ∩ B MK = {b MK−1 K−1 } = {b MK K−1 }. All other intersections are empty. (See Figure 7.15.) Figure 7.15: The universe of the overalgebra. B B1 B2 · · · BK−2 BK−1 BK BK+1 BK+2 · · · B2K−1 B2K B2K+1 · · · a1=a 1 1 b 1 1 =a 2 2 b 2 2 =a 3 3 b K−2 K−2 =a K−1 K−1 b K−1 K−1 =b K K−1 =b K+1 K−1 a K+1 K−1 =b K+1 K−2 b 2K−1 1 a 2K−1 1 =a 2K 1 =a 2K+1 1 b 2K+1 1 As usual, we put A := B 0 ∪ · · · ∪ B MK , and we proceed to define some unary operations on A. e ℓ (x) =        S j,ℓ (x), if x ∈ B j for some j ∈ E , a ℓ 1 , otherwise. and, for 0 < i < K, e ℓ+i (x) =                a ℓ+i i , if x ∈ B j for some j < ℓ + i, x, if x ∈ B ℓ+i , b ℓ+i i , if x ∈ B j for some j > ℓ + i. For each ℓ ∈ O, let e ℓ (x) =        S j,ℓ (x), if x ∈ B j for some j ∈ O, b ℓ K−1 , otherwise. and, for 0 < i < K, We then consider the overalgebra A := A, F A . This overalgebra is, once again, based on the specific congruence β = Cg B ((a 1 , b 1 ), . . . , (a K−1 , b K−1 )) ∈ Con B, and the following theorem describes the inverse image of β under | B -that is, the interval [β * , β] in Con A. Theorem 7.3.10. Let A = A, F A be the overalgebra described above, and, for each 0 i M K, let t i denote a tie-point of the set B i . Define e ℓ+i (x) =                b ℓ+i K−i , if x ∈ B j for some j < ℓ + i, x, if x ∈ B ℓ+i , a ℓ+i K−i , if x ∈ B j for some j > ℓ + i,β * = MK j=0 β Bj ∪ MK i=0 t i /β Bi 2 . Then, β * = Cg A (β). If β has transversal {a 1 , c 1 , c 2 , . . . , c m−1 }, then β = β * ∪ m−1 i=1 ℓ∈E c ℓ i /β B ℓ 2 ∪ m−1 i=1 ℓ∈O c ℓ i /β B ℓ 2 . (7.3.9) Moreover, [β * , β] ∼ = (Eq|E |) m−1 × (Eq|O|) m−1 . Remark. Recall that m is the number of congruence classes in β. Proof of Theorem 7.3.10. It is easy to check that β * is an equivalence relation on A, so we first check that f (β * ) ⊆ β * for all f ∈ F A . This will establish that β * ∈ Con A. Thereafter we show that β ⊆ η ∈ Con A implies β * η, which will prove that β * is the smallest congruence of A containing β, as claimed in the first part of the theorem. Fix (x, y) ∈ β * . To show (f (x), f (y)) ∈ β * we consider two possible cases. Case 1: (x, y) ∈ β Bj for some 0 j (2q + 1)K. In this case it is easy to verify that (q i,0 (x), q i,0 (y)) ∈ β and (q 0,i (x), q 0,i (y)) ∈ β Bi for all 0 i K + 1. For example, if (x, y) ∈ β Bj with 1 j K, then (q 0,i (x), q 0,i (y)) = (a i 1 , a i 1 ) and (q i,0 (x), q i,0 (y)) is either (b i , b i ) or (a i , a i ) depending on whether i is below or above j, respectively. If i = j, then (q i,0 (x), q i,0 (y)) is the pair in B 2 corresponding to (x, y) ∈ β Bj , so (q i,0 (x), q i,0 (y)) ∈ β. A special case is (q 0,0 (x), q 0,0 (y)) ∈ β. Therefore, q 0,0 = e 0 , implies (f e 0 (x), f e 0 (y)) ∈ β for all f ∈ F B . Altogether, we have proved that (f (x), f (y)) ∈ β * for all f ∈ F A . Case 2: (x, y) ∈ B 2 where B := MK i=0 t i /β Bi . Note that e 0 (B) = a 1 /β. Therefore, (e 0 (x), e 0 (y)) ∈ β, so (f e 0 (x), f e 0 (y)) ∈ β for all f ∈ F B . Also, q 0,k (B) = S 0,k e 0 (B) = S 0,k (a 1 /β) = a k 1 /β B k , which is a single block of β * . Similarly, e k (B) = t k /β B k , so q k,0 (B) = S k,0 e k (B) = S k,0 (t k /β B k ) = S k,0 (t k )/β, a single block of β * . Whence, (x, y) ∈ B 2 implies (f (x), f (y)) ∈ β * for all f ∈ F A . We have thus established that β * is a congruence of A which contains β. We now show that β * is the smallest such congruence. Indeed, suppose β ⊆ η ∈ Con A, and fix (x, y) ∈ β * . If (x, y) ∈ β Bj for some 0 j M K, then (q j,0 (x), q j,0 (y)) = (S j,0 e j (x), S j,0 e j (y)) = (S j,0 (x), S j,0 (y)) ∈ β ⊆ η, so (x, y) = (q 0,j q j,0 (x), q 0,j q j,0 (y)) ∈ η. If, instead of (x, y) ∈ β Bj , we have (x, y) ∈ B 2 , then without loss of generality x ∈ a i i /β Bi and y ∈ a j j /β Bj for some 0 i < j K + 1. Then, (q i,0 (x), q i,0 (t i )) ∈ β and (q j,0 (t j ), q j,0 (y)) ∈ β and, since i < j, there is a sequence of tie points c i i , d i+1 i+1 , c i+1 i+1 , d i+2 i+2 , c i+2 i+2 , . . . , c j j (where {c, d} = {a, b}) such that t i β Bi c i i = d i+1 i+1 β Bi+1 c i+1 i+1 = d i+2 i+2 β Bi+2 c i+2 i+2 = · · · = c j j β Bj t j . (7.3.10) We could sketch a diagram similar to the one given in the proof of Theorem 7.3.6, but it should be obvious by now that the relations (7.3.10) imply (t i , t j ) ∈ η. Therefore, β * = Cg A (β). Next we prove equation (7.3.9). Let β denote the right hand side of (7.3.9). We first show β ∈ Con A. Let C E i := ℓ∈E c ℓ i /β B ℓ and C O i := ℓ∈O c ℓ i /β B ℓ . Note that C E i is the join of the corresponding (i-th) β blocks in the up-pointing sets in Figure 7.15. Thus, C E i can be visualized as a single slice through all of the up-pointing sets. Similarly, C O i is the join of corresponding blocks in the down-pointing sets in Figure 7.15. If 0 < i < K and ℓ ∈ E , then e ℓ+i (C E i ) = e ℓ+i (C O i ) = {a ℓ+i i , b ℓ+i i }. Thus, for each such k = ℓ + i we have q k0 (C E i ) = S k0 e k (C E i ) = S k0 e k (C O i ) = q k0 (C O i ) = {a i , b i }, a single block of β. Similarly, if 0 < i < K and ℓ ∈ O, then e ℓ+i (C E i ) = e ℓ+i (C O i ) = {a ℓ+i K−i , b ℓ+i K−i }. Thus, for each such k = ℓ + i we have q k0 (C E i ) = S k0 e k (C E i ) = S k0 e k (C O i ) = q k0 (C O i ) = {a K−i , b K−i }, which is also a single block of β. It follows that q k0 ( β) ⊆ β for all k / ∈ E ∪ O. If k ∈ E , then e k (C E i ) = c k i /β B k and e k (C O i ) = a k 1 , so q k0 (C E i ) = c i /β and q k0 (C O i ) = a 1 . Thus, q k0 ( β) ⊆ β. If This completes the proof that f ( β) ⊆ β for all f ∈ F A . Since the restriction of β to B is clearly β| B = β, the residuation lemma yields β β, and we now prove β β. Indeed, it is easy to see that, for each (x, y) / ∈ β, there is an operation f ∈ Pol 1 (A) such that (e 0 f (x), e 0 f (y)) / ∈ β, and thus (x, y) / ∈ β. Verification of this statement is trivial. For example, if x ∈ c ℓ i /β B ℓ for some 1 i < m, ℓ ∈ E and y / ∈ C E i , then e 0 (x) ∈ c i /β and e 0 (y) / ∈ c i /β, so (e 0 (x), e 0 (y)) / ∈ β. To take a slightly less trivial case, suppose x ∈ c ℓ i /β B ℓ for some 1 i < m, ℓ ∈ O and y / ∈ C O i . Then (e ℓ (x), e ℓ (y)) / ∈ β B ℓ , so (e 0 q ℓ0 (x), e 0 q ℓ0 (y)) = (q ℓ0 (x), q ℓ0 (y)) / ∈ β. The few remaining cases are even easier to verify, so we omit them. This completes the proof of (7.3.9). It remains to prove [β * , β] ∼ = (Eq|E |) m−1 × (Eq|O|) m−1 . This follows trivially from what we have proved above. For, in proving that β is a congruence, we showed that, in fact, each operation f ∈ F A maps blocks of β (= β) into blocks of β * . That is, each operation collapses the interval [β * , β]. Therefore, every equivalence relation on the set A that lies between β * and β is respected by every operation of A. In other words, [β * , β] = {θ ∈ Eq(A) : β * θ β}. In view of the configuration of the universe of A, as shown in Figure 7.15, it is clear that the interval sublattice {θ ∈ Eq(A) : β * θ β} is isomorphic to (Eq|E |) m−1 × (Eq|O|) m−1 . Conclusions We have described an approach to building new finite algebras out of old which is useful in the following situation: given an algebra B with a congruence lattice Con B of a particular shape, we seek an algebra A with congruence lattice Con A which has Con B as a (non-trivial) homomorphic image; specifically, we construct A so that | B : Con A → Con B is a lattice epimorphism. We described the original example -the "triple-winged pentagon" shown on the right of Figure 7 We mainly focused on a few specific overalgebra constructions. In each case, the congruence lattice that results has the same basic shape as the one with which we started, except that some congruences are replaced with intervals that are direct products of powers of partition lattices. Thus we have identified a broad new class of finitely representable lattices. However, the fact that the new intervals in these lattices must be products of partition lattices seems quite limiting, and this is the first limitation that we think future research might aim to overcome. We envision potential variations on the constructions described herein, which might bring us closer toward the goal of replacing certain congruences β ∈ Con B with an more general finite lattices, L ∼ = [β * , β] Con A. Using the constructions described above, we have found examples of overalgebras for which it is not possible to simply add operations in order to eliminate all relations strictly contained in the interval (β * , β). Nonetheless, we remain encouraged by the success of a very modest example in this direction, which we now describe. Such an algebra exists by the theorem of Berman [5], and Quackenbush and Wolk [36]. Let B 1 , B 2 , . . . , B N be sets of size 2N which intersect B as follows: for all 1 i < j K, B 0 ∩ B i = {b i }, and B i ∩ B j = ∅. If A = A, F A is the overalgebra constructed as in Section 7.3.1, then Con A is isomorphic to the lattice in Figure 7.16, but with L C replaced with Eq(C). Now expand the set F A of operations on A as follows: for each f ∈ F C , define f 0 : B → B by f 0 (a i ) = a f (i) and f 0 (b i ) = b f (i) , and definê f : A → A byf (x) = f 0 (s(x)). Defining F + A = F A ∪ {f : f ∈ F C }, we claim that the congruence lattice of the algebra A, F + A is (isomorphic to) the lattice appearing in Figure 7.16. As a final remark, we call attention to another obvious limitation of the methods describe in this chapter -they cannot be used to find an algebra with congruence lattice isomorphic to the lattice L 7 , which is the subject of Section 6.3. This lattice is simple, so it is certainly not the inverse image under | B of some smaller lattice. properties, besides simplicity, describing lattices that cannot be the congruence lattice of an overalgebra? 10. Is the seven element lattice L 11 group representable? (Recall, we proved that L 11 is representable in Section 6.2 using the filter+ideal method which necessarily results in a non-permutational algebra.) 11. Is every lattice with at most seven elements group representable? (In Section 6.2 we described the seven element lattices which are the most challenging to represent. These appear in Figure 7.1. We saw that both L 13 and L 17 are group representable. Though we did not mention it above, we have also found the lattice L 9 (which motivated the invention of overalgebras) as an interval in the subgroup lattice of A 10 . At the bottom of this interval is a subgroup of index 25,400. So the smallest G-set we have found with congruence lattice isomorphic to L 9 is on 25,400 elements. Clearly this is not the minimal representation of L 9 . Indeed, in Example 7.3.1 we constructed an overalgebra with 16 elements that has a congruence lattice isomorphic to L 9 . We suspect it will not be very difficult to prove that the lattices L 19 and L 20 are group representable. Of the lattices appearing in Figure 7.1 then, L 7 may not be representable, and L 11 , though representable, seems difficult to find as an interval in a subgroup lattice of a finite group.) APPENDIX A GROUP THEORY BACKGROUND In this section we review some aspects of group theory that are relevant to our problem of representing a finite lattice as the congruence lattice of a finite algebra. A.1 Group actions and permutation groups Let G be a group, A = A,Ḡ a G-set, and let Sym(A) denote the group of permutations of A. For a ∈ A, the one-generated subalgebra a ∈ Sub(A) is called the orbit of a in A. It is easily verified that a is the setḠa := {ḡa | g ∈ G}, and we often use the more suggestiveḠa when referring to this orbit. The orbits of the G-set A partition the set A into disjoint equivalence classes. The equivalence relation ∼ is defined on A 2 as follows: x ∼ y if and only ifḡx = y for some g ∈ G. In fact, ∼ is a congruence relation of the algebra A since, x ∼ y impliesḡx ∼ḡy. Thus, as mentioned above, each orbit is indeed a subalgebra of A. Keep in mind that A is the disjoint union of the orbits. That is, if {a 1 , . . . , a r } is a full set of ∼-class representatives, then A = r i=1Ḡ a i is a disjoint union. A G-set with only one orbit is called transitive. Equivalently, A,Ḡ is a transitive G-set if and only if (∀a, b ∈ A)(∃g ∈ G)(ḡa = b). In this case, we say that G acts transitively on A, and occasionally we refer to the group G itself as a transitive group of degree |A|. For a ∈ A, the set Stab G (a) := {g ∈ G |ḡa = a} is called the stabilizer of a. It is easy to verify that Stab G (a) is a subgroup of G. An alternative notation for the stabilizer is G a := Stab G (a). Let λ : G →Ḡ Sym(A) denote the permutation representation of G; that is, λ(g) =ḡ. Then ker λ = {g ∈ G |ḡa = a for all a ∈ A} = a∈A Stab G (a) = a∈A G a . (A.1.1) Therefore, G/ ker λ ∼ = λ[G] Sym(A). We say that the representation λ of G is faithful, or that G acts faithfully on A, just in case ker λ = 1. In this case λ : G ֒→ Sym(A), so G itself is isomorphic to a subgroup of Sym(A), and we call G a permutation group. If H G are groups, the core of H in G, denoted core G (H), is the largest normal subgroup of G that is contained in H. It is easy to see that core G (H) = g∈G gHg −1 . A subgroup H is called core-free provided core G (H) = 1. Elements in the same orbit of a G-set have conjugate stabilizers. Specifically, if a, b ∈ A and g ∈ G are such thatḡa = b, then G b = Gḡ a = g G a g −1 . If the G-set happens to be transitive, then it is faithful if and only if the stabilizer G a is core-free in G. For, ker λ = a∈A G a = g∈G Gḡ a = g∈G g G a g −1 . Thus G a is core-free if and only if ker λ = 1 if and only if G acts faithfully on A. In case G is a transitive permutation group, we say that G is regular (or that G acts regularly on A, or that λ : G →Ḡ is a regular representation) provided G a = 1 for each a ∈ A; i.e., every non-identity element of G is fixed-point-free. 1 Equivalently, G is regular on A if and only if for each a, b ∈ A there is a unique g ∈ G such thatḡa = b. In particular, |G| = |A|. A block system for G is a partition of A that is preserved by the action of G. In other words, a block system is a congruence relation of the algebra A = A,Ḡ . The trivial block systems are 0 A = |a 1 |a 2 | · · · |a i | · · · and 1 A = |a 1 a 2 · · · a i · · · |. The non-trivial block systems are called systems of imprimitivity. A nonempty subset B ⊆ A is a block for A if for each g ∈ G eitherḡB = B orḡB ∩ B = ∅. Let A = A,Ḡ be a transitive G-set. In most group theory textbooks one finds the following definition: a group G is called primitive if A has no systems of imprimitivity; otherwise G is called imprimitive. In other words, G is primitive if and only if the transitive G-set A,Ḡ is a simple algebra -that is, Con A,Ḡ ∼ = 2. In the author's view, this definition of primitive is meaningless and is the source of unnecessary confusion. Clearly every finite group acts transitively on the cosets of a maximal subgroup H and the resulting G-set has Con G/H,Ḡ ∼ = [H, G] ∼ = 2. This means that, according to the usual definition, every finite group is primitive. To make the definition more meaningful, we should require that a primitive group be isomorphic to a permutation group. That is, we call a transitive permutation group primitive if the induced algebra is simple. To see the distinction, take an arbitrary group G acting on the cosets of a subgroup H. This action is faithful, and G is a permutation group, if and only if H is core-free. If, in addition, H is a maximal subgroup, then the induced algebra G/H,Ḡ is simple. For these reasons, we will call a group primitive if and only if it has a core-free maximal subgroup. (Note that the terms "primitive" and "imprimitive" are used only with reference to transitive G-sets.) A.2 Classifying permutation groups A permutation group is either transitive or is a subdirect product of transitive groups, while a transitive group is either primitive or is a subgroup of an iterated wreath product of primitive groups. (See, e.g., Praeger [33].) Hence primitive groups can be viewed as the building blocks of all permutations groups and their classification helps us to better understand the structure of permutation groups in general. The socle of a group G is the subgroup generated by the minimal normal subgroups of G and is denoted by Soc(G). By [12], Corollary 4.3B, the socle of a finite primitive group is isomorphic to the direct product of one or more copies of a simple group T . The O'Nan-Scott Theorem classifies the primitive permutation groups according to the structure of their socles. The following version of the theorem seems to be among the most useful, and it appears for example in the Ph.D. thesis of Hannah Coutts [9]. {0 , {01}, or the two element lattice 3 {0, 1, 2}, or the three element lattice n the set {0, 1, . . . , n − 1}, or the n element chain ω the natural numbers, {0, 1, 2, . . . } Z the integers, {. . . , −1, 0, 1, . . . } Q the rational numbers F an arbitrary field A, B, C, . . . universal algebras A = A, F an algebra with universe A and operations F Clo(A) the clone of term operations of A Pol(A) the clone of polynomial operations of A Pol n (A) the set of n-ary members of Pol(A) Aut(A) the group of automorphisms of A Inn(A) the inner automorphisms of A Out(A) the outer of automorphisms of A End(A) the monoid of endomorphisms of A Hom(A, B) the set of homomorphisms from A into B Con (A) the lattice of congruence relations of A Sub(A) the lattice of subalgebras of A Sg A (X) A = A, . . . be an algebra with congruence lattice Con A, . . . . Recall that a clone on a non-void set A is a set of operations on A that contains the projection operations and is closed under compositions. The clone of term operations of the algebra A, denoted by Clo(A), is the smallest clone on A containing the basic operations of A. The clone of polynomial operations of A, denoted by Pol(A), is the clone generated by the basic operations of A and the constant unary maps on A. The set of n-ary members of Pol(A) is denoted by Pol n (A). Lemma 1 .2. 1 . 11If F is a set of operations on A, then Con A, F = Con A, F ′ , where F ′ is any of Pol(A), Pol 1 (A), or the set of basic translations (operations in Pol 1 (A) obtained from F by fixing all but one coordinate). The lattice formed by all subgroups of a group G, denoted Sub(G), is called the subgroup lattice of G. It is a complete lattice: any number of subgroups H i have a meet (greatest lower bound) H i , namely their intersection H i , and a join (least upper bound) H i , namely the subgroup generated by the union of them. We denote the group generated by the subgroups {H i : i ∈ I} by H i : i ∈ I when I is infinite, and by H 0 , H 1 , . . . , H n−1 , otherwise. Since a complete lattice is Theorem 1.3. 7 . 7Every finite distributive lattice is isomorphic to the lattice of normal subgroups of a finite solvable group. 4.- 5 . 5By the ordinal (parallel) sum of two lattices L 1 , L 2 , we mean the lattice on the left (middle) of Figure 2.1. Theorem 2 .4. 1 . 21If L 1 , . . . , L n ∈ L 3 is a collection of representable lattices, then the ordinal sum and the adjoined ordinal sum, shown in Figure 2.4, are representable. by Lemma 2.3.1, the adjoined ordinal sum is representable. A trivial induction argument proves the result for adjoined ordinal sums of n lattices. The same result for ordinal sums (Figure 2.4 left) follows since the two element lattice is obviously representable. Figure 2 . 2 : 22The ordinal sum (left) and the adjoined ordinal sum (right) of the lattices L 1 , . . . , L n . Theorem 3 .2. 1 . 31If L Eq(X), then L = Con A for some algebra A = X, F if and only if L is closed. Proposition 3 .3. 1 . 31The lattice Eq(X) contains a proper dense M 3 sublattice if and only if |X| 5. M 3 Figure 3 . 1 : 331The 5-element non-distributive lattice, M 3 . Figure 3 . 2 :. 2 . 322The (n + 2)-element lattice of height 2, M n . For n 1, Eq(2n + 1) contains a dense M n+2 . = |0, 1|2, 3|4|, α 2 = |0|1, 2|3, 4|, α 3 = |0, 2, 4|1, 3|, let L = {0 X , α 1 , α 2 , α 3 , 1 X } and let L = L, ∧, ∨ denote the sublattice of Eq(X) generated by the three equivalences α 1 , α 2 , α 3(Figure 3.3). Figure 3 . 3 : 33The lattice L = {0 X , α 1 , α 2 , α 3 , 1 X };Obviously L ∼ = M 3 , and it is not hard to show that the only unary maps which respect all equivalences in L are the constants and the identity. In other words, the set λ(L) ⊆ X X consists of the six trivial maps in X X . Therefore, ρλ(L) = Eq(X). Now notice that if we adjoin the equivalence α 4 = |0, 3|1, 4|2| to L we get an M 4 , which we denote by L(α 4 ). Obviously, λ(L) ⊇ λ(L(α 4 )), as adding more equivalences only shrinks the set of functions respecting all equivalences. Therefore, Eq(X) = ρλ(L) ⊆ ρλ(L(α 4 )), so L(α 4 ) is a dense M 4 sublattice of Eq(5).Similarly, letting X = {0, 1, . . . , 6} and α 1 = |0, 1|2, 3|4, 5|6|, α 2 = |0|1, 2|3, 4|5, 6|, α 3 = |0, 2, 4, 6|1, 3, 5|, the sublattice L = {0 X , α 1 , α 2 , α 3 , 1 X }, ∧, ∨ is a dense M 3 in Eq(X). Adjoining the partitions α 4 = |0, 3|2, 5|1, 6|4| and α 5 = |0, 5|1, 4|3, 6|2| Lemma 3 .3. 3 . 33Suppose L = L, ∧, ∨ is a complete 0, 1-lattice. Then the following are equivalent: Lemma 3 .3. 4 . 34If L ≇ 2 is a sublattice of Eq(X) satisfying the conditions of Lemma 3.3.3, then λ(L) contains a non-trivial unary function. Theorem 3 .3. 5 . 35If L ≇ 2 is a lattice satisfying the conditions of Lemma 3.3.3 and X is any set, then L cannot be densely embedded in Eq(X). Figure 3 . 4 : 34The lattice M 3,3 . J.B. Nation has found examples of densely embedded double-winged pentagons none of whose sublattices are densely embedded. John Snow then asked if any of the sublattices are closed embeddings. group Sym(G/H) of permutations on the set G/H = {H, x 1 H, x 2 H, . . . } of left cosets of H in G. by transitivity. Thus H θ is a subgroup of G, and clearly G a H θ . It is also easy to see that θ H is a congruence of A. The equality H θH = H trivially follows from the definitions. On the other hand (b, c) ∈ θ H θ if and only if there exist g, h ∈ G for which (h(a), a) ∈ θ and b = gh(a), and c = g(a). 2nd G-set Isomorphism Theorem, version 2). Let A = A,Ḡ be a transitive G-set and let a ∈ A. Let B be the set of all blocks B with a ∈ B. Let [G a , G] ⊆ Sub(G) denote the set of all subgroups of G containing G a . Then there is a bijection Ψ : B → [G a , G] given by Ψ(B) = G(B), . Let G act transitively on a set with at least two points. Then G is primitive if and only if each stabilizer G a is a maximal subgroup of G.Since the point stabilizers of a transitive group are all conjugate, one stabilizer is maximal only when all of the stabilizers are maximal. In particular, a regular permutation group is primitive if and only if it has prime degree.Next we describe (up to equivalence) all transitive permutation representations of a given group G. We call two representations (or actions) equivalent provided the associated G-sets are isomorphic.The foregoing implies that every transitive permutation representation of G is equivalent toλ H for some subgroup H G. The following lemma 4 shows that we need only consider a single representative H from each of the conjugacy classes of subgroups. Lemma 4 .1. 5 . 45Suppose G acts transitively on two sets, A and B. Fix a ∈ A and let G a be the stabilizer of a (under the first action). Then the two actions are equivalent if and only if the subgroup G a is also a stabilizer under the second action of some point b ∈ B. which is the set containing the identity map on X and the constant function that maps all points to x. So the lattice [M x , M ] of submonoids of M above M x is just the lattice of subsets of M which contain the identity and the constant map x. This is a distributive lattice, so it cannot be isomorphic to Con X, M = Eq(X). . Let L be a finite collection of finite lattices. If the FLRP has a positive answer, then there exists a finite group G such that each lattice L i ∈ L is an upper interval G 1 1, . . . , G n and L 1 , . . . , L n satisfy the hypothesis of the lemma. Construct a new lattice P = P(L, L 1 , . . . , L n ) as shown in Figure 5.1 (a). By (B), there exist finite groups H G with P ∼ = [H, G]. Let K, K 1 , . . . , K n be the subgroups of G which cover H and satisfy L ∼ = [K, G], and L i ∼ = [K i , G], i = 1, . . . , n (Figure 5.1 (b)). Thus, L is an interval in the subgroup lattice of G, and, Figure 5 . 1 : 51The parachute construction. Basile proves a result which implies that 1 M 6 ∼ = [H, G] only if G / ∈ Gi. Given these examples and Lemma 5.2.2, it is clear that (B) holds if and only if for each finite lattice L there exist finite groups H G such that L ∼ = [H, G] and G is not solvable, not alternating, and not symmetric. Now, if our goal is to solve the finite lattice representation problem, Lemma 5.2.2 suggests the following path to a negative solution: Find examples of lattices L i which place restrictions on the G for which L i ∼ = [H, G] can hold, say G ∈ G i , and eventually reach i G i = ∅ (at which point we are done). a class of algebras, then H(K ) is the class of homomorphic images of members of K .) Lemma 5.2.3. Let G be a class of groups and L a finite lattice such that Suppose L satisfies (5.2.1) and H(G c ) = G c , that is, G c is closed under homomorphic images. (For groups this means if G ∈ G c and N G, then G/N ∈ G c .) If (5.2.2) fails, then there is a finite group G ∈ G c with L ∼ = [H, G]. Let N = core G (H). Then L ∼ = [H/N, G/N ] and H/N is core-free in G/N so, by hypothesis (5.2.1), G/N ∈ G . But G/N ∈ G c , since G c is closed under homomorphic images. Examples. As mentioned above, there is a lattice L with the property that L ∼ = [H, G] implies G is not solvable, so let G = S c . Then G c = S is closed under homomorphic images. For the second example above, we have G = Gi c , so G c = n<ω {A n , S n }. This class is also closed under homomorphic images. It follows from Lemma 5.2.3 that these examples do not require the core-free hypothesis. In contrast, consider the following result of Köhler [22]: If n − 1 is not a power of a prime, then 2 M n ∼ = [H, G] with H core-free ⇒ G is subdirectly irreducible. Lemma 5.2.3 does not apply in this case since G c , the class of subdirectly reducible groups, is obviously not closed under homomorphic images. 3 Though Lemma 5.2.3 seems like a useful observation, the last example above shows that a generalized version of Lemma 5.2.2 -a version based on hypothesis (⋆) -would be more powerful, as it would allow us to impose greater restrictions on G, such as those implied by the results of Köhler and others. Fortunately, the "parachute" construction used in the proof of Lemma 5.2.2 works in the more general case, with only a trivial modification to the hypotheses -namely, the lattices L i should not be two-element chains (which almost goes without saying in the present context). (Recall, 2 denotes the two-element chain.) Lemma 5.2.4. Let G 1 , . . . , G n be classes of groups and suppose that for each i ∈ {1, . . . , n} there is a finite lattice L i ≇ 2 which satisfies the following: If L i ∼ = [H, G] and H is core-free in G, then G ∈ G i . (⋆) Then (B) is equivalent to (C) For every finite lattice L, there is a finite group G ∈ n i=1 G i such that L ∼ = [H, G]. Proof. Obviously, (C) implies (B). Assume (B) and let L be any finite lattice. Suppose G 1 , . . . , G n and L 1 , . . . , L n satisfy (⋆) and L i ≇ 2 for all i. Note that there is no loss of generality in assuming that n 2. For if n = 1, just throw in one of the examples above to make n = 2. Call this additional class of groups G 2 . Then, at the end of the argument, we'll have G ∈ G 1 ∩ G 2 , and therefore, G ∈ G 1 , which is the stated conclusion of the theorem in case n = 1. Construct the lattice P = P(L, L 1 , . . . , L n ) as in the proof of Lemma 5.2.2. By (B) there exist finite groups H G with P ∼ = [H, G], and we can assume without loss of generality that H is core-free 4 in G. Let K, K 1 , . . . , K n be the subgroups of G which cover H and satisfy L ∼ = [K, G], and L i ∼ = [K i , G], 1 i n, as in Figure 5.1 (b). Thus, L is an upper interval in the subgroup lattice of G, and it remains to show that G ∈ n i=1 Figure 5 . 2 : 52The impossibility of a non-trivial core, N i = core G (K i ), in a parachute lattice. We say that a group theoretical property (or class) P is interval sublattice enforceable (ISLE) if there exists a lattice L such that L ∼ = [H, G] implies G is a P-group. (By the convention agreed upon at the outset of this chapter, it is implicit in the notation L ∼ = [H, G] that G is a finite group; thus the class G of all finite groups is trivially an ISLE class.) We say that the property (or class) P is core-free interval sublattice enforceable (cf-ISLE) if there exists a lattice L such that if L ∼ = [H, G] Lemma 5.3. 2 .Figure 5 . 3 : 253Let P be a cf-ISLE property, and let L be a finite lattice such that L ∼ = [H, G] with H core-free implies G P. Also, suppose there exists a group G witnessing this; that is, G has a core-free subgroup H with L ∼ = [H, G]. Then, for any finite nonabelian simple group S, there exists a wreath product group of the form W = S ≀Ū that is also a P-group.Proof. We apply the idea of Kurzweil twice (cf. Theorem 2.2.2). Fix a finite nonabelian simple group S, and suppose the index of H in G is |G : H| = n. Then the action of G on the cosets of H induces an automorphism of the group S n by permutation of coordinates. Denote this representation by ϕ : G → Aut(S n ), and let the image of G be ϕ(G) =Ḡ Aut(S n ). The semidirect product (or wreath product) under this action is the groupU := S ≀ ϕ G = S n ⋊ ϕ G = S n ⋊Ḡ = S ≀Ḡ,with multiplication given by (s 1 , . . . , s n , x)(t 1 , . . . , t n , y) = (s 1 t x(1) , . . . , s n t x(n) , xy), for s i , t i ∈ S and x, y ∈Ḡ. An illustration of the subgroup lattice of such a wreath product appears in Figure 5.3. The dual lattice L ′ is an upper interval in the subgroup lattice of this group, namely, Representation of the dual of a group representable lattice. L ′ ∼ = [D ⋊Ḡ, U ]. (As usual, D denotes the diagonal subgroup of S n .) It is important to note that if H is core-free in G -equivalently, if ker ϕ = 1 -then the foregoing construction results in the subgroup D ⋊Ḡ being core-free in U . (We postpone the proof of this fact.) Now if we repeat the foregoing procedure, with H 1 := D ⋊Ḡ denoting the (core-free) subgroup of U such that L ′ ∼ = [H 1 , U ], then we find that L = L ′′ ∼ = [D 1 ⋊Ū , S m ⋊Ū ], where m = |U : H 1 |. 8 Theorem 5.4. 1 ( 1Dedekind's rule). Let G be a group and let A, B and C be subgroups of G with A B. Then, A(C ∩ B) = AC ∩ B, and (5.4.1) (C ∩ B)A = CA ∩ B. (5.4.2) U 0 : 0= U ∩ H, and consider the interval [U 0 , U ] := {V | U 0 V U }. In general, when we write U H we mean the set {uh | u ∈ U, h ∈ H}, and we write U ∨ V or U, H to mean the group generated by U and H. Clearly U H ⊆ U, H . Equality holds if and only if U and H permute, that is, U H = HU . In any case, it is often helpful to visualize part of the subgroup lattice of U, H , as shown below. Recall that the usual isomorphism theorem for groups implies that if H is a normal subgroup of U, H , then the interval [H, U, H ] is isomorphic to the interval [U ∩ H, U ]. The purpose of the next lemma is to relate these two intervals in cases where we drop the assumption H U, H and add the assumption U H = U, H . If the two subgroups U and H permute, then we define [U 0 , U ] H := {V ∈ [U 0 , U ] | V H = HV }, (5.4.3) which consists of those subgroups V in [U 0 , U ] that permute with H. The first equality holds by(5.4.2) since H X, the second holds by assumption, and the third by(5.4.1). This proves U ∩ X ∈ [U 0 , U ] H . Moreover, by the first equality, ψ • ϕ(X) = (U ∩ X)H = U H ∩ X = X, so ψ • ϕ is the identity on [H, U H]. Lemma 5.4. 3 . 3Let P = P(L 1 , . . . , L n ) with n 2 and |L i | > 2 for all i, and suppose P ∼ = [H, G], with H core-free in G.(i) If 1 = N G, then N H = G. (ii) If M is a minimal normal subgroup of G, then C G (M ) = 1. Proof. (i)Let 1 = N G. Then N H, since H is core-free in G. Therefore, H < N H. As in Section 5.2, we let K i denote the subgroups of G corresponding to the atoms of P. Then H is covered by each K i , so K j N H for some 1 j n. Suppose, by way of contradiction, that N H < G. By assumption, n 2 and |L i | > 2. Thus for any i = j we have K i Y < Z < G for some subgroups Y and Z which satisfy (N H) ∩ Z = H and (N H) ∨ Y = G. Also, (N H)Y = N Y is a group, so (N H)Y = N H ∨ Y = G. But then, by Dedekind's rule, we haveY = HY = ((N H) ∩ Z)Y = (N H)Y ∩ Z = G ∩ Z = Z,contrary to Y < Z. This contradiction proves that N H = G.(ii) If C G (M ) = 1, then (i) implies C G (M )H = G, since C G (M ) N G (M ) = G.Consider any H < K < G. Then 1 < M ∩ K < M (strictly, by Lemma 5.4.2). Now M ∩ K is normalized by H and centralized (hence normalized) by C G (M ). (Indeed, C G (M ) centralizes every subgroup of M .) Therefore, M ∩ K C G (M )H = G, contradicting the minimality of M . (iii) We prove that G has a unique minimal normal subgroup. Let M be a minimal 11 normal subgroup of G and let N G be any normal subgroup not containing M . We show that N = 1. Since both subgroups are normal, the commutator 12 of M and N lies in the intersection M ∩ N , which is trivial by the minimality of M . Thus, M and N centralize each other. In particular, N C G (M ) = 1, by (ii). (iv) Let M ′ denote the commutator of M . As remarked above, M is nonabelian, so M ′ = 1. Also, M ′ M G, and M ′ is a characteristic subgroup of M (i.e., M ′ invariant under Aut(M )). Therefore, M ′ G, and, as M is a minimal normal subgroup of G, we have M ′ = M . Thus, M is not solvable, so G is not solvable. Remark. It follows from (i) that, if P is a nontrivial parachute lattice with P ∼ = [H, G], where H is core-free, then core G (X) = 1 for every H X < G. This gives a second way to complete the proof of Lemma 5.2.4. To summarize what we have thus far, the lemmas above imply that (B) holds if and only if every finite lattice is an interval [H, G], with H core-free in G, where (i) G is not solvable, not alternating, and not symmetric; (ii) G has a unique minimal normal subgroup M which satisfies M H = G and C G (M ) = 1; in particular, M is nonabelian and core G (X) = 1 for all H X < G. where T i are simple minimal normal subgroups of M which are conjugate (under conjugation by elements of G). Thus, M is a direct power of a simple group T .In fact, when C G (M ) = 1, as in our application, we can specify these conjugates more precisely. Let T be any minimal normal subgroup of M . Note that T is simple.Let N = N H (T ) = {h ∈ H | T h = T }be the normalizer of T in H. Then the proof of the following lemma is routine, so we omit it.Lemma 5.4.4. If H/N = {N, h 1 N, . . . , h k−1 N } is a full set of left cosets of N in H, then k = r and M = T 0 × · · · × T r−1 = T × T h1 × T hr−1 .We conclude this chapter by noting that other researchers, such as Baddeley, Börner, and Lucchini, have proved similar results for the more general case of quasiprimitive permutation groups. In particular, our proof of Lemma 5.4.3 (i) uses the same argument as the one in [8], where it is used to prove Lemma 2.4: if L ∼ = [H, G] is an LP-lattice, 13 then G must be a quasiprimitive permutation group. We remark that parachute lattices, in which each panel L i has |L i | > 2, are LP-lattices, so Lemma 5.4.3 follows from theorems of Baddeley, Börner, Lucchini, et al. (cf. [2], [8]).However, the main purpose of the parachute construction, besides providing a quick route toLemma 5.4.3, is to demonstrate a natural way to insert arbitrary finite lattices L i as upper intervals[K i , G] in Sub[G], with K i core-free in G. Then, once we prove special properties of groups G for which L i = [K i , G] (K i core-free), it follows that every finite lattice L must be an upper interval L = [K, G] for some G satisfying all of these properties, assuming the FLRP has a positive answer. 2 2Representations for most of these lattices can be found quite easily by applying the methods described in previous chapters. The easiest, of course, are the distributive lattices, which we know are representable by Theorem 1.3.3. Some others are found to be representable by searching (with a computer) for closed concrete representations L Eq(X) over some small set X, say |X| < 8. Still others are found by checking that they are obtained by applying operations under which L 3 is closed ( § 2.1). For example, the lattice on the left inFigure 6.1 is the ordinal sum of two copies of the distributive lattice 2 × 2. On the right of the same figure is the parallel sum of the distributive lattices 2 and 3. Figure 6 . 1 : 61The ordinal sum of 2 × 2 with itself (left) and the parallel sum of 2 and 3 (right). Figure 6 . 2 : 62Seven element lattices with no obvious congruence lattice representation. Here V 4 4denotes the Klein four subgroup and P marks one of the four Sylow 3 subgroups of A 4 . Of course, Sub(A 4 ) is the congruence lattice of the permutational algebra consisting of A 4 acting regularly on itself by multiplication. Now note that L 17 ∼ = P ↑ ∪ V ↓ 4 , the union of a filter and ideal of a representable lattice. Therefore, L 17 is representable. The question of whether the existence of such a "filter-idea representation" implies that the lattice in question is also an interval in a subgroup lattice seems open. Although, in the present case, we have found that L 17 has a group representation. Indeed, the group G = (A 4 × A 4 ) ⋊ C 2 has a subgroup H ∼ = S 3 such that [H, G] ∼ = L 17 . Now, by the Kurzweil-Netter result, the dual of L 17 is also representable. Explicitly, since L 17 is representable on a 12-element set (the elements of A 4 ) via the filter-ideal method, 4 the dual of L 17 can be embedded above diagonal subgroup of the 12-th power of a simple group: L ′ 17 ֒→ [D, S 12 ] ∼ = (Eq(12)) ′ . Then, adding the operations from the original representation of L 17 as described in the proof of Theorem 2.2.2, we have an algebra with universe S 12 /D and congruence lattice isomorphic to L ′ 17 . 5 5 has a subgroup H ∼ = A 4 such that [H, G] ∼ = L 13 . The index is |G : H| = 80, so the action of G on the cosets G/H is an algebra on an 80 element universe. Though we have not found L 11 as an interval in a subgroup lattice, we have found that the pentagon N 5 is an upper interval in the subgroup lattice of the groups G = ((C 3 × C 3 ) ⋊ Q 8 ) ⋊ C 3 and G = (A 4 × A 4 ) ⋊ C 2 . 7 In each of these groups, there exists a subgroup H < G (of index 36) with [H, G] ∼ = N 5 . Let [H, G] = {H, α, β, γ, G} ∼ = N 5 . (See Figure 6.3.) Of course, Sub(G) is a congruence4 Figure 6 . 3 : 63The lattice L 11 represented as the union of a filter and ideal in the subgroup lattice of the group G. Two choices for G that work are SmallGroup(216,153) = ((C 3 × C 3 ) ⋊ Q 8 ) ⋊ C 3 and SmallGroup(288,1025) = (A 4 × A 4 ) ⋊ C 2 . . Suppose H < G are finite groups with core G (H) = 1 and suppose L 7 ∼ = [H, G]. ( vi) With the possible exception of at most one maximal subgroup, all proper subgroups in the interval [H, G] are core-free. (K vi), our main goal is the pair of restrictions (iii) and (v), which allow us to rule out a number of the O'Nan-Scott types describing primitive permutation groups. (Section A.2.1 includes a detailed description of these types.) Assume the hypotheses of the theorem above. In particular, throughout this section all groups are finite, H is a core-free subgroup of G, and [H, G] ∼ = L 7 . Label the seven subgroups of G in the interval [H, G] as in the following diagram: The labels are chosen with the intention of helping us remember to which subgroups they refer: the maximal subgroup M 2 covers two subgroups in the interval [H, G], while J 2 is covered by two subgroups of G. Claim 6. 2 . 2J 1 and J 2 are core-free subgroups of G. Proof. First note that if N G then the subgroup N H permutes 9 with any subgroup containing H. To see this, let H X G and note that N HX = N X = XN = XHN = XN H, since H X and N G. Suppose 1 = 1N J 1 for some N ⊳ G. Then N H = J 1 , so J 1 and K are permuting subgroups. Since J 1 K = G and J 1 ∩ K = H, Lemma 5.4.2 yields [J 1 , G] ∼ = [H, K] J1 := {X ∈ [H, K] | J 1 X = XJ 1 }. But this is impossible since [H, K] J1 [H, K] ∼ = 2, while [J 1 , G] ∼ = 3. This proves that core G (J 1 ) = 1.The intervals involved in the argument are drawn with bold lines in the following diagram. Figure 6 . 4 : 64Hasse diagram illustrating the cases in which M 2 has non-trivial core: 1 = N M 2 for some N ⊳ G. The proof that J 2 is core-free is similar. Suppose 1 = N J 2 where N ⊳ G. Then N H = J 2 and the subgroups J 2 and K permute. Therefore, [H, K] J2 ∼ = [J 2 , G], by Lemma 5.4.2, which is a contradiction since [H, K] J2 [H, K] ∼ = 2, while [J 2 , G] ∼ = 2 × 2. Claim 6. 3 . 3Either M 1 or M 2 is core-free in G. If M 2 has non-trivial core and N ⊳ G is contained in M 2 , then C G (N ) = 1 and G is subdirectly irreducible.Proof. Suppose M 2 has non-trivial core. Then there is a minimal normal subgroup 1 = N ⊳ G contained in M 2 . Since H, J 1 , J 2 are core-free, N H = M 2 . Consider the centralizer, C G (N ), of N in G. Of course, this is a normal subgroup of G.10 If C G (N ) = 1, then, since minimal normal subgroups centralize each other, N must be the unique minimal normal subgroup of G. Furthermore, M 1 must be core-free in this case. Otherwise N M 1 ∩ M 2 = J 2 , contradicting core G (J 2 ) = 1. Therefore, in case C G (N ) = 1 we conclude that G is subdirectly irreducible and M 1 is core-free. 10 The centralizer of a normal subgroup N G is itself normal in G. For, it is the kernel of the conjugation action of G on N . Thus, C G (N ) N G (N ) = G. Suppose C G (N ) = 1. Then, since C G (N ) G, and since H, J 1 , J 2 , K are core-free, it's clear thatC G (N )H ∈ {G, M 1 , M 2 }.We consider each case separately.Case 1: Suppose C G (N )H = G. Note that N ∩ H < N ∩ J 1 < N (strictly). The subgroup N ∩ J 1 is normalized by J 1 and by C G (N ), and so it is normal in C G (N )J 1 C G (N )H = G, contradicting the minimality of N . Thus, the case C G (N )H = G does not occur. Case 2: Suppose C G (N )H = M 1 . The subgroup N ∩ J 1 is normalized by both H and C G (N ). For, C G (N ) centralizes, hence normalizes, every subgroup of N . Therefore, N ∩ J 1 is normalized by C G (N )H = M 1 . Case 3 : 3Suppose C G (N )H = M 2 . The subgroup N ∩ M 1 is normalized by both H and C G (N ). Therefore, N ∩ M 1 is normalized by C G (N )H = M 2 . Of course, it's also normalized by M 1 , so N ∩ M 1 is normalized by M 1 , M 2 = G. The conclusion is that N ∩ M 1 ⊳ G. By minimality of the normal subgroup N , we must have either N ∩ M 1 = 1 or N ∩ M 1 = N . The former equality implies N ∩ J 2 = 1, which contradicts the strict inequalities of Dedekind's rule, )H = G, C G (N )H = M 1 , or C G (N ) = 1. The first case is easily ruled out exactly as in Case 1 above. The second case is handled by the argument we used in Case 3. Indeed, if we suppose C G (N )H = M 1 , then N ∩ M 2 is normalized by both H and C G (N ), hence by M 1 . It is also normalized by M 2 , so N ∩ M 2 ⊳ G. Thus, by minimality of N , and since M 2 is core-free, N ∩ M 2 = 1. But then N ∩ J 2 = 1, leading to a contradiction similar to (6.3.1) but with M 1 replacing M 2 . Therefore, the case C G (N )H = M 1 does not occur, and we have proved C G (N ) = 1. So far we have proved that all intermediate proper subgroups in the interval [H, G] are core-free except possibly at most one of M 1 or M 2 . Moreover, we proved that if one of the maximal subgroups has non-trivial core, then there is a unique minimal normal subgroup N ⊳ G with trivial centralizer, C G (N ) = 1. As explained above, G is subdirectly irreducible in this case, since minimal normal subgroups centralize each other. In order to prove (ii), there remains only one case left to check, and the argument is by now very familiar. Claim 6.5. If each H X < G is core-free and N is a minimal normal subgroup of G, then C G (N ) = 1. Proof. Let N be a minimal normal subgroup of G. Then, by the core-free hypothesis we have N H = G. Fix a subgroup H < X < G. Then N ∩ H < N ∩ X < N . The subgroup N ∩ X is normalized by H and by C G (N ). If C G (N ) = 1, then C G (N )H = G, by the core-free hypothesis, so N ∩ X ⊳ G, contradicting the minimality of N . Therefore, C G (N ) = 1. Lemma 6 .4. 1 . 61If G acts transitively on a set Ω with stabilizer G ω , then a subgroup N G acts transitively on Ω if and only if N G ω = G. Also, N is regular if and only if in addition N ∩ G ω = 1. , . . . , B K be sets which intersect B at specific points. We construct an overalgebra A, F A , by which we mean an expansion of B, . . . with universe A := B ∪ B 1 ∪ · · · ∪ B K , and a certain set F A of unary operations which include idempotent mappings e and e i satisfying e(A) = B and e i (A) = B i . We explore a number of such constructions and prove results about the shape of the new congruence lattices Con A, F A that result. Thus, descriptions of some new classes of finitely representable lattices is one of our primary contributions. Another, perhaps more significant contribution is the announcement of a novel approach to the discovery of new classes of representable lattices. Our main contribution is the description and analysis of a new procedure for generating finite lattices which are, by construction, finitely representable. Roughly speaking, we start with an arbitrary finite algebra B := B, . . . , with known congruence lattice Con B, and we let B 1 , B 2 , . . . , B K be sets which intersect B at certain points. The choice of intersection points plays an important rôle which we describe in detail later. We then construct an overalgebra A := A, F A , by which we mean an expansion of B with universe A = B ∪ B 1 ∪ · · · ∪ B K , and a certain set F A of unary operations which include idempotent mappings e and e i satisfying e(A) = B and e i (A) = B i .Given our interest in the problem mentioned above, the important consequence of this procedure is the new (finitely representable) lattice Con A that it produces. The shape of this lattice is, of course, determined by the shape of Con B, the choice of intersection points of the B i , and the unary operations chosen for inclusion in F A . In this chapter, we describe a number of constructions of this type and prove some results about the shape of the congruence lattices of the resulting overalgebras.Before giving an overview of this chapter, we give a bit of background about the original example which provided the impetus for this work. In the spring of 2011, our research seminar was fortunate enough to have as a visitor Peter Jipsen, who initiated the ambitious project of cataloging every small finite lattice L for which there is a known finite algebra A with Con A ∼ = L. Before long, we had identified such finite representations for all lattices of order seven or less, except for the two lattices appearing inFigure 7.1. (Section 6.2 describes some of the methods we used to find representations of the other seven-element lattices.) Ralph Freese then discovered a way to construct an algebra whichFigure 7.1: Lattices of order 7 with no obvious finite algebraic representation. a formal description of the original construction mentioned above. We then describe the original example in detail before proving some general results about the congruence lattices of such overalgebras. At the end of Section 7.3.1 we describe a further expansion of the set of operations defined in the first construction, and we conclude the section with an example demonstrating the utility of these additional operations. Section 7.3.2 presents a second overalgebra construction which overcomes a basic limitation of the first. We then prove a result about the structure of the congruence lattices of these overalgebras, and close the section with some further examples which illustrate the procedure and demonstrate its utility. In Section 7.3.3 we describe a construction that further generalizes the one in Section 7.3.2. The last section discusses the impact that our results have on the main problem -the finite congruence lattice representation problem -as well as the inherent limitations of this approach, and concludes with some open questions and suggestions for further research. Let e 2 2= e ∈ Pol 1 (A) be an idempotent unary polynomial, define B := e(A) and F B := {ef | B | f ∈ Pol 1 (A)}, and consider the unary 1 algebra B := B, F B . Pálfy and Pudlák prove in Lemma 1 of [32] that the restriction mapping | B , defined on Con A by α| B = α∩B 2 , is a lattice epimorphism of Con A onto Con B. In [24], McKenzie, taking Lemma 1 as a starting point, develops the foundations of what would become tame congruence theory. In reproving the Pálfy-Pudlák congruence lattice epimorphism lemma, McKenzie introduces the mapping defined on Con B by Let π i : B → B i be given by π i (b j ) = b i j , for i = 0, 1, 2, . . . , n and j = 1, 2, . . . , K. (It is convenient to include i = 0 in this definition, in which case we let π 0 (b j ) = b 0 j := b j .) The map π i and the operations F induce a set F i of unary operations on B i , as follows: to each f ∈ F corresponds the operation f πi : B i → B i defined by f πi = π i f π −1 i . Thus, for each i, B i := B i , F i and B = B, F are isomorphic algebras. That is, for all i = 1, . . . , K, we have Example 7 .3. 1 . 71Before proving some results about the basic structure of the congruence lattice of an overalgebra, we present the original example, discovered by Ralph Freese, of a finite algebra with a congruence lattice isomorphic to the second lattice inFigure 7.1. Consider a finite permutational algebra B = B, F with congruence lattice Con B ∼ = M 4 .(Figure 7.2) There are only a few small algebras to choose from.3 We consider the right regular S 3 -set -i.e. the algebra S 3 acting on itself by right multiplication. In GAP, 2 acting on right cosets of H = D 8 . The index in this case is |G : H| = 81. (In GAP, G:=SmallGroup(648,725), and H is found to be the fourth maximal subgroup class representative of the fourth maximal subgroup class representative of G.) 4 All of the computational experiments we describe in this chapter rely on two open source programs, GAP [17] and the Universal Algebra Calculator [16] (UACalc). To conduct our experiments, we have written a small collection of GAP functions; these are available at http://math.hawaii.edu/~williamdemeo/Overalgebras.html. Figure 7 Figure 7 . 2 : 772This creates a UACalc file specifying an algebra with universe B = {0, 1, . . . , 5} and two basic unary operations g 0 the S 3 -set appearing in the GAP output above. Congruence lattice of the right regular S 3 -set, where α = |0, 1, 2|3, 4, 5|, β = |0, 3|2, 5|1, 4|, γ = |0, 4|2, 3|1, 5|, δ = |0, 5|2, 4|1, 3|. A = B 0 ∪ B 1 ∪ B 2 = 012{0 Figure 7 . 3 :β 73Congruence lattice of the overalgebra of the S 3 -set with intersection points 0 and 2. The congruence relations in Figure 7.3 are as follows: * = |0, 3, 8|1, 4|2, 5, 15|6, 9|7, 10|11, 13|12, 14| γ * = |0, 4, 9|1, 5|2, 3, 13|6, 10|7, 8|11, 14|12, 15| δ * = |0, 5, 10|1, 3|2, 4, 14|6, 8|7, 9, 11, 15|12, 13|. It is important to note that the resulting congruence lattice depends on our choice of which congruence to "expand," which is controlled by our specification of the intersection points of the overalgebra. For example, suppose we want one of the congruences having three blocks, say, β = Cg B (0, 3) = |0, 3|2, 5|1, 4|, to have a non-trivial inverse image β| −1 B = [β * , β]. Then we would select This produces an overalgebra with universe A = B 0 ∪ B 1 ∪ B 2 = {0, 1, 2, 3, 4, 5} ∪ {0, 6, 7, 8, 9, 10} ∪ {11, 12, 13, 3, 14, 15} and congruence lattice shown in figure 7.4. Figure 7 . 4 : 74Congruence lattice of the overalgebra of the S 3 -set with intersection points 0 and 3. where α * = |0, 1, 2, 6, 7|3, 4, 5, 14, 15|8, 9, 10|11, 12, 13| β = |0, 3, 8, 11|1, 4|2, 5|6, 9, 12, 14|7, 10, 13, 15| β ε = |0, 3, 8, 11|1, 4|2, 5|6, 9, 12, 14|7, 10|13, 15| β ε ′ = |0, 3, 8, 11|1, 4|2, 5|6, 9|7, 10, 13, 15|12, 14| β * = |0, 3, 8, 11|1, 4|2, 5|6, 9|7, 10|12, 14|13, 15| γ * = |0, 4, 9|1, 5|2, 3, 13|6, 10|7, 8|11, 14|12, 15| δ * = |0, 5, 10|1, 3, 12|2, 4|6, 8|7, 9|11, 15|13, 14|. lustrated in Figures 7.5 and 7.6. Identifying the objects on the right of equation (7.3.2) in these figures will make the proof of the theorem easier to follow. In particular, as the figures make clear, transitivity requires that β Bj classes which are linked together by tie-points must end up in the same class of Cg A (β). This is the purpose of the m r=1 (·) 2 term. Figures 7. 5 5and 7.6 illustrate these objects for a simple example in which B 0 = {b 0 , b 1 , . . . , b 8 }, Figure 7 . 6 : 76Solid lines show the congruence classes of β * (left) and β (right); dotted lines delineate the sets B i . Theorem 7.3.3. For each β ∈ Con B, Figure 7 . 7 :Figure 7 . 8 : 7778With the theorems above, we can explain the shapes of the congruence lattices of Example 7.3.1. Returning to that example, with base algebra B equal to the right regular S 3 -set, we now show some other congruence lattices that result by simply changing the set of tie-points,T . Recall, the relations in Con B are α = |0, 1, 2|3, 4, 5|, β = |0, 3|2, 5|1, 4|, γ = |0, 4|2, 3|1, 5|, and δ = |0, 5|2, 4|1, 3|. As Theorems 7.3.2 and 7.3.3 make clear, choosing T to be {0, 1}, {0, 1, 2}, or {0, 2, 3} yields the congruence lattices appearing in Figure 7.7. Figure 7.8 shows the congruences lattices resulting from the choices T = {0, 1, 2, 3} and T = {0, 2, 3, 5}. Congruence lattices of overalgebras of the S 3 -set for various choices of T , the set of tie-points. Congruence lattices of overalgebras of the S 3 -set for various choices of T ; L ∼ = 2 2 × 2 2 .Since β = |0, 3|2, 5|1, 4|, when T = {0, 2, 3, 5}, the interval [β * , β] is 2 2 × 2 2 . InFigure 7.8, we denote this abstractly by L, instead of drawing all 16 points of this interval.Next, consider the situation depicted in the last congruence lattice ofFigure 7.8, where L ∼ = 2 2 × 2 2 , and suppose we prefer that all the other | B -inverse images be trivial: [β * , β] ∼ = 2 2 × 2 2 ; α * = α; γ * = γ; δ * = δ. In other words, we seek a finite algebraic representation of the lattice inFigure 7.9. LFigure 7 . 79: A lattice which motivates further expansion of the set of basic operations in the overalgebra. F A := {f e 0 : f ∈ F } ∪ {e k : 0 k K} ∪ {s r : 0 r m}, where s 0 := s was defined earlier. Finally, let A := A, F A , and define θ * and θ as above. Proposition 7.3.5. For each θ ∈ Con B, overalgebra with congruence lattice isomorphic to the one in Figure 7.9, but with L ∼ = Eq(3) × Eq(3). Incidentally, with the additional operations s r , we are not limited with respect to how many terms appear in the direct product. For example, gap> OveralgebraXO([ G, [[0,1,2], [0,1,2], [3,4,5]] ]); produces an overalgebra with a 130 element congruence lattice like the one in Figure 7.9, with L ∼ = Eq(3) × Eq(3) × Eq(3), while gap> OveralgebraXO([ G, [[0,3], [0,3], [0,3], [0,3]] ]);gives a 261 element congruence lattice with L ∼ = 216 . Claim 7. 1 . 1Consider the collection of maps g : A → A defined for each g ∈ Stab G T := {g ∈ G | gb = b ∀b ∈ T } by the rules g| B i = e g(bi) ge 0 (i = 1, . . . , n). B Con A is non-trivial if and only if β θ < 1 B . To answer this question, we now describe an overalgebra construction that is based on a construction proposed by Bill Lampe.Let B = B; F be a finite algebra, and suppose β = Cg B((a 1 , b 1), . . . , (a K , b K ))for some a 1 , . . . , a K , b 1 , . . . , b K ∈ B. Let B = B 0 , B 1 , B 2 , . . . , B K+1 be sets of cardinality |B| = n which intersect as follows: F A := {f e 0 : f ∈ F } ∪ {q i,0 : 0 i K + 1} ∪ {q 0,j : 1 j K + 1}. The overalgebra in this section is defined to be the unary algebra A := A, F A . Theorem 7.3.6. Suppose A = A, F A is the overalgebra based on the congruence relation β = Cg B ((a Theorem 7 .3. 8 . 78Suppose A = A, F A is the overalgebra based on the congruence relation β = Cg B ((a 1 , b 1 ), . . . , (a K , b K )), as described above. Then, θ * < θ if and only if β θ < 1 B , in which case [θ * , θ] ∼ = 2 r−1 ,where r is the number of congruence classes of θ.Consequently, if θ β, then θ = θ * .Proof. Lemma 7.3.7 implies that θ * < θ only if β θ < 1 B . On the other hand, if β θ < 1 B , then we obtain [θ * , θ] ∼ = 2 r−1 by the same argument used to prove [β * , β] ∼ = 2 m−1 in Theorem 7.3.6. Example 7 .3. 9 .γ 1 =Figure 7 . 7917Let G be the group C 2 × A 4 defined in GAP as follows:11 gap> G:=Group([ (9,10)(11,12)This is a group of order 24 which acts transitively on the set {1, 2, . . . , 12}. (If we let H denote the stabilizer of a point, say H := G 1 ∼ = C 2 , then the group acts transitively by right multiplication on the set G/H of right cosets. These two G-sets are of course isomorphic.) The congruence lattice of this algebra (which is isomorphic to the interval from H up to G in the subgroup lattice of G) is shown inFigure 7.11. After relabeling the elements to conform to our 0-offset notation, the universe is B := {0, 1, . . . , 11}, and the non-trivial congruences are as follows: |0, 1|2, 3|4, 5|6, 7|8, 9|10, 11|γ 2 = |0, 2|1, 3|4, 7|5, 6|8, 11|9, 10| γ 3 = |0, 3|1, 2|4, 6|5, 7|8, 10|9, 11|. 11: The congruence lattice of the permutational algebra B, G , where B = {0, 1, . . . , 11} and G ∼ = C 2 × A 4 .Clearly, the coatom β is not principal. It is generated by {(0, 3), (8, 11)}, for example. If our goal is to construct an overalgebra which has β > β * in Con A, and θ * = θ for all θ β in Con B, it is clear that the method described in the Section 7.3.1 will not work. For, if we base the overalgebra on tie-points {0, 3}, then the universe isA = B ∪ B 1 ∪ B 2 , where B ∩ B 1 = {0}, B ∩ B 2 = {3},and B 1 ∩ B 2 = ∅, and the operations are F A := {ge 0 : g ∈ G} ∪ {e 0 , e 1 , e 2 , s}. Since β has three congruence classes, by Theorem 7.3.3 the interval of all θ ∈ Con A for which θ| B = β is [β * , β] ∼ = 2 2 . Figure 7 . 12 : 712The universe of the overalgebra of the (C 2 × A 4 )-set, arranged to reveal the congruences above β * . Figure 7 . 713: The universe of the overalgebra; solid lines delineate the congruence classes of β * . Figure 7 . 714: The congruence lattice of the overalgebra A, F A of B, G , where B = {0, 1, . . . , 11} and G ∼ = C 2 × A 4 . , let S i,j : B i → B j be the bijection S i,j (x i ) = x j , and note that S i,i = id Bi . Define the following subsets of even and odd multiples of K, respectively:E = {2qK : q = 0, 1, . . . , Q} and O = {(2q + 1)K : q = 0, 1, . . . , Q}. For each ℓ ∈ E , let In other words, if ℓ ∈ E , then e ℓ maps each up-pointing set inFigure 7.15 bijectively onto the up-pointing set B ℓ , and maps all other points of A to the tie-point a ℓ 1 ∈ B ℓ ; if ℓ ∈ O, then e ℓ maps each down-pointing set in the figure onto the down-pointing set B ℓ , and maps all other points to the tie-point b ℓ K−1 . For each set B ℓ+i in between -represented in the figure by an ellipse with horizontal major axis -there corresponds a map e ℓ+i which act as the identity on B ℓ+i and maps all points in A left of B ℓ+i to the left tie-point of B ℓ+i and all points to the right of B ℓ+i to the right tie-point of B ℓ+i . Finally, for 0 i, j M K, we define q i,j = S i,j • e i and take the set of basic operations on A to be F A := {f e 0 : f ∈ F } ∪ {q i,0 : 0 i M K} ∪ {q 0,j : 1 j M K}. k ∈ O, then e k (C E i ) = b k K−1 and e k (C O i ) = c k i /β B k , so q k0 (C E i ) = b K−1 and q k0 (C O i ) = c i /β. Thus, q k0 ( β) ⊆ β.Finally, e 0 (C E i ) = c i /β and e 0 (C O i ) = a 1 , so, for each f ∈ F B , the operation f e 0 takes all of C E i to a single β class, and all of C O i to a single beta class. That is, f e 0 ( β) ⊆ β for all f ∈ F B . Ralph Freese, which motivated us to develop a general procedure for finding such finite algebraic representations. Example 7 .4. 1 . 12 LCFigure 7 . 71127Suppose C, . . . is an arbitrary finite algebra with congruence lattice L C := Con C, . . . . Relabel the elements so that C = {1, 2, . . . , N }. We show how to use the overalgebra construction described in Section 7.3.1 to obtain a finite algebra with congruence lattice appearing in Figure 7.16. 16: L C an arbitrary finitely representable lattice. Let B = B, F B be a unary algebra with universe B = {a 1 , a 2 , . . . , a N , b 1 , b 2 , . . . , b N }, and congruence lattice Con B = {0 B , α, β, 1 B } ∼ = 2 × 2, where α = |a 1 , b 1 |a 2 , b 2 | · · · |a N , b N | and β = |a 1 , a 2 , . . . , a N |b 1 , b 2 , . . . , b N |. A.2. 1 1The O'Nan-Scott Theorem Theorem A.2.1 (O'Nan-Scott Theorem). Let G be a primitive permutation group of degree d, and let N := Soc(G) ∼ = T m with m 1. Then one of the following holds.1. N is regular and (a) Affine type T is cyclic of order p, so |N | = p m . Then d = p m and G is permutation isomorphic to a subgroup of the affine general linear group AGL(m, p). We call G a group of affine type.(b) Twisted wreath product type m 6, the group T is nonabelian and G is a group of twisted wreath product type, with d = |T | m . 2. N is non-regular and non-abelian and (a) Almost simple m = 1 and T G Aut(T ).INDEX F n , the n-ary operations in F , 4 Clo(A), see clone of term operations Pol(A), see clone of polynomial operations Pol n ( from 1928, and later numerous important contributions by Reinhold Baer, Øystein Ore, Kenkichi Iwasawa, Leonid Efimovich Sadovskii, Michio Suzuki, Giovanni Zacher, Mario Curzio, Federico Menegazzo, Roland Schmidt, Stewart Stonehewer, Giorgio Busetto, and many others. The book [41] by Roland Schmidt gives a comprehensive account of this work. Suppose H is a subgroup of G (denoted H G). By the interval sublattice [H, G] we mean thesublattice of Sub(G) given by: [H, G] := {K | H K G}, Then [α, 1 A ] ∼ = Con A/α. By 1., the dual of ℓ := [α, 1 A ] is representable. Now take the filter above β ′ in ℓ ′ (where β ′ is the image of β under dualization) and we obtain a representation of a lattice isomorphic to the dual of [α, β]. Apply 1. again and we have the desired representation of [α, β]. 2.2.1), and we have f ⊑ h if and only if K h = {y ∈ S n | ker h ker y} {y ∈ S n | ker f ker y} = K f .Theorem 2.2.2 (Kurzweil [23], Netter which is a contradiction.The converse of Corollary 3.3.6 is false. That is, there exists a finite lattice L ≇ 2 with no meet prime element that cannot be densely embedded in some Eq(X). The lattice M 3,3 shown below is an example. It has no meet prime element but it does satisfy the conditions of Lemma 3.3.3. Thus, by Theorem 3.3.5, M 3,3 is not densely embeddable. It is clear from the foregoing that if solvability were an ISLE property then we would have a solution to the FLRP. But solvability is obviously not ISLE. For, if L ∼ = [H, G] then for any nonsolvable group K we have L ∼ = [H × K, G × K], and of course G × K is not solvable. Notice, however, where G := U H. This is the set consisting of those subgroups V in [U 0 , U ] that are normalized by H. The latter are sometimes called H-invariant subgroups. Notice that to even define [U 0 , U ] H Suppose U and H are permuting subgroups of a group. Let U 0 := U ∩ H. Then4.4) we must have H N G (U ), and in this case, as we will see below, the two sublattices coincide: [U 0 , U ] H = [U 0 , U ] H . We are finally ready to state the main result relating the sets defined in (5.4.3) and (5.4.4) (when they exist) to the interval [H, U H]. Lemma 5.4.2. To complete the proof of (i), we show that [U 0 , U ] H is a sublattice of [U 0 , U ]. Suppose V 1 and V 2 are subgroups in [U 0 , U ] which permute with H. It is easy to see that their join V 1 ∨ V 2 = V 1 , V 2 also permutes with H, so we just check that their intersection permutes with Hby (5.4.1). This proves that ϕ and ψ are inverses of each other on the sets indicated, and it's easy to see that they are order preserving: X Y implies U ∩ X U ∩ Y , and V W implies V H W H. Therefore, ϕ and ψ are inverse order isomorphisms. The number of up-pointing sets in Figure 7.15 is |E |, while |O| counts the number of down-pointing sets. In our construction, we took |E | = |O| = Q + 1, but, apart from being notationally convenient, this choice was arbitrary; in fact, there's no reason E and O should be equal in number, and they could even be empty. Choosing O = ∅, for example, would result in the interval [β * , β] ∼ = (Eq|E |) m−1 . Thus, for any N , we can construct an algebra A that has (EqN ) m−1 ∼ = [β * , β] < Con A. Suzuki, Michio, 7 symmetric group on, 28 systems of imprimitivity, 105 trivial block systems, 105Twisted wreath product type, 106L 5 , 8 P-group, 45 abstract representation problem, 17 adjoined ordinal sum, 15 Affine type, 106 algebra, 2, 3 Almost simple, 106 arity, 3 Aschbacher, Michael, 6, 54 Aschbacher-O'Nan-Scott Theorem, 59, 106 Börder, Ferdinand, 53 Baddeley, Robert, 53 Baer, Reinhold, 7 Basile, Alberto, 41 Berman, Joel, 7, 26 Birkhoff, Garrett, 6 block, 105 block system, 105 cf-ISLE, 45 class of groups, 45 clone, 5 of polynomial operations, 5 of term operations, 5 closed concrete representation, 17, 19 closed sublattice, 18 closure method, 17, 19 closure operator, 18, 20 closure properties, 10 applied, 55 commutator subgroup, 8 complete lattice, 5 concrete representation, 17 congruence relation, 2, 4 core, 29, 104 core-free, 30, 59 Coutts, Hannah, 106 Dedekind's rule, 48, 48-52 Dedekind, Richard, 7 degree, 29 dense, 20 dense sublattice, 20 densely embedded, 20 diagonal subgroup, 12 Diagonal type, 107 diamond, 20 Dilworth, Robert, 6 down-set, 23 dual, 11, 57 dual of a lattice, 10 equivalent representations, 32 faithful, 104 action, 30 representation, 29, 30 Feit, Walter, 6, 101 filter, 23 finite lattice representation problem, 3 finitely representable, 3 FLRP, 2 Freese, Ralph, 7, 67, 72, 98 G-set, 28 Galois correspondence, 20 Grätzer, George, 2, 6 group representable, 8 group representable lattice, 101 group theoretical class, 45 group theoretical property, 45 homomorphism, 4 ideal, 23 imprimitive, 105 interval sublattice, 7 interval sublattice enforceable (ISLE), 45 intransitivity congruence, 36 invariant subgroup, 49 ISLE, 45 Iwasawa, Kenkichi, 7 Jónsson, Bjarni, 17 Jipsen, Peter, 54, 66 join, 5 Join prime, 23 Köhler, 42 kernel, 2, 4 Kurzweil, Hans, 6, 10, 12, 46, 57 Kurzweil-Netter Theorem, 13 Lampe, William, 67 lattice, 2 linear representations, 29 LP-lattice, 53 Lucchini, Adrea, 6, 53 McKenzie, Ralph, 6, 10, 15, 68 meet, 5 meet prime, 23 meet-semidistributive, 25 multi-unary algebra, 5 Netter, Raimund, 6, 10, 12, 57 non-modular element, 60 non-trivial function, 18 O'Nan-Scott Theorem, 59, 106 operation symbol, 3 operations, 3 orbit, 104 ordinal sum, 10, 11, 15 applied, 55 Ore, Øystein, 7 Pálfy, Péter, 6, 9, 12, 40 parallel sum, 11 applied, 55 partial algebra, 5 partial operations, 5 permutation group, 29, 104 permutation representations, 29 permuting subgroups, 61 point stabilizer, 29 prime ideal, 23 primitive, 105, 106 primitive group, 30 principal element, 23 principal ideal, 23 Product action, 107 property of groups, 45 Pudlák, Pavel, 6, 7, 9, 10, 17, 40 Quackenbush, R., 7, 26 quotient algebra, 4 regular, 105 representable lattice, 3, 17, 101 representation, 17 of a finite group, 28 Rottlaender, Ada, 7 Schmidt, E.T., 2, 7 Schmidt, Roland, 7 set-wise stabilizer, 36 similarity type, 3 simple algebra, 105 Snow, John, 6, 11, 15, 99, 101 socle, 106 spanning sublattice, 7, 17 stabilizer, 104 stabilizer subgroup, 30 strong congruence relation, 5 strongly representable, 7, 26 subgroup lattice, 5 superbad representation, 20 Tůma, Jiří, 6, 7, 10, 15, 17 transitive, 104 action, 28 group, 29 transversal, 37 unary algebra, 5 universe, 3 up-set, 23 Watatani, Yasuo, 54 Whitman, P.M., 6 Wolk, B., 7, 26 wreath product, 46 In this context, by "kernel" of a homomorphism ϕ one typically means the normal subgroup {a ∈ A | ϕ(a) = e}, whereas this is a single congruence class of the kernel as we have defined it. Note that some authors reserve this term for algebras with a single unary operation, and use the term multi-unary algebra when referring to what we call unary algebra. This is mentioned in[7] without proof. Recall, the dual of a lattice is simply the lattice turned on its head, that is, the lattice obtained by reversing the partial order of the original lattice. Recall, by a spanning sublattice of a bounded lattice L 0 , we mean a sublattice L L 0 that has the same top and bottom as L 0 . That is 1 L = 1 L 0 and 0 L = 0 L 0 . More generally, a G-set is sometimes defined to be a pair (X, ϕ), where ϕ is a homomorphism from a group into the symmetric group S X , see e.g.[44]. Lemma 1.6B of[12]. Here G/Stab G (Λ i ) denotes the set of right cosets of Stab G (Λ i ) in G, and a transversal is a set containing one element from each coset. Recall, for groups, subdirectly irreducible is equivalent to having a unique minimal normal subgroup. Every algebra, and in particular every group G, has a subdirect decomposition into subdirectly irreducibles, G G/N 1 × · · · × G/Nn. Thus, there will always be homomorphic images, G/N i , which are subdirectly irreducible.4 This is standard. For, if P ∼ = [H, G] with N := core G (H) = 1, then P ∼ = [H/N, G/N ]. This is where we use n 2; though, if n = 1, we could have used K instead of K j , but then we would need to assume L ≇ 2. Here we use D 1 to denote the diagonal subgroup of S m to distinguish it from D, the diagonal subgroup of S n . If G is simple, then M = G; "minimal" assumes nontrivial.12 The commutator of M and N is the subgroup generated by the set {mnm −1 n −1 | m ∈ M, n ∈ N }. The commutator of M is the subgroup generated by {aba −1 b −1 : a, b ∈ M }. The n th degree commutator of M , denoted M (n) , is defined recursively as the commutator of M (n−1) . A group M is solvable if M (n) = 1 for some n ∈ N. An LP-lattice is one in which every element except 0 and 1 is a non-modular element. Recall that the core of a subgroup X in G is the largest normal subgroup of G contained in X. This is denoted by core G (X). We say the X is core-free in G provided core G (X) = 1. Actually, the set is already a group in this case since M 1 J 1 = C G (N )HJ 1 = J 1 C G (N )H = J 1 M 1 . In the definition of F B , we could have used Pol(A) instead of Pol 1 (A), and then our discussion would not be limited to unary algebras. However, as we are mainly concerned with congruence lattices, we lose nothing by restricting the scope in this way. Also, later sections of this chapter will be solely concerned with unary algebras, so for consistency we define B to be unary in this section as well. Note that β B 0 = β. If we were to include q i,j for all 0 i, j K + 1, the resulting overalgebra would have the same congruence lattice as A, F A , but using a reduced set of operations simplifies our proofs. The diagram illustrates the case 1 i < j K where i + 1 < j. In case j = i + 1, the diagram is even simpler. The GAP command TransitiveGroup(12,7) also gives a group isomorphic to C 2 × A 4 , but by defining it explicitly in terms of certain generators, we obtain more attractive partitions in the congruence lattice. i=0 B i where B 0 = {0, 1, . . . , 11}, B 1 = {0, 12, 13, . . . , 22}, B 2 = {23, 24, . . . , 29, 30, 14, 31, 32, 33}, and B 3 = {33, 34, . . . , 44}. (See Figure 7.12.) John Snow has already proved that "parallel sums" of finitely representable lattices are finitely representable (See Lemmas 3.9 and 3.10 of[43]). The action of a regular permutation group is sometimes called a "free" action. Appendix algebras in general and finite groups in particular. It is the author's view that such progress will undoubtedly lead to a solution to the FLRP in the very near future. algebras in general and finite groups in particular. It is the author's view that such progress will undoubtedly lead to a solution to the FLRP in the very near future. Let L 3 denote the class of representable lattices; that is, L ∈ L 3 if and only if L ∼ = Con A for some finite algebra A. Let H(K ) denote the class of homomorphic images of a class K of algebrasLet H(K ) denote the class of homomorphic images of a class K of algebras. Let L 3 denote the class of representable lattices; that is, L ∈ L 3 if and only if L ∼ = Con A for some finite algebra A. Is H(L 3 ) = L 3 true?. Is H(L 3 ) = L 3 true? Is L 3 = L 4 true? In other words, if L is the congruence lattice of a finite algebra, is L (isomorphic to) the congruence lattice of a transitive G-set? Equivalently, is every congruence lattice of a finite algebra (isomorphic to) an interval in the subgroup lattice of a finite group?. Is L 3 = L 4 true? In other words, if L is the congruence lattice of a finite algebra, is L (isomorphic to) the congruence lattice of a transitive G-set? Equivalently, is every congruence lattice of a finite algebra (isomorphic to) an interval in the subgroup lattice of a finite group? It is true that, L 0 = {x ∈ L | x α or β x} ∈ L 4 for all α, β ∈ L? Note that. L Suppose, by the result of John Snow (Lemma 2.3.1) this is true if we replace L 4 with L 3Suppose L ∈ L 4 . It is true that, L 0 = {x ∈ L | x α or β x} ∈ L 4 for all α, β ∈ L? Note that, by the result of John Snow (Lemma 2.3.1) this is true if we replace L 4 with L 3 . What other properties of groups, in addition to those described in Chapter 5, are interval sublattice enforceable (ISLE) properties?. What other properties of groups, in addition to those described in Chapter 5, are interval sublattice enforceable (ISLE) properties? lattice of an algebra of cardinality less than 30!/10?. Walter Feit finds M 7 ∼ =. Is the lattice M 7 the congruence. In [14. H, A 31 ], where |H| = 31 · 5, so M 7 is the congruence lattice of a transitive G-set on |A 31 : H| = 30!/10 elementsIs the lattice M 7 the congruence lattice of an algebra of cardinality less than 30!/10? (In [14], Walter Feit finds M 7 ∼ = [H, A 31 ], where |H| = 31 · 5, so M 7 is the congruence lattice of a transitive G-set on |A 31 : H| = 30!/10 elements.) Is there a general characterization of the class of finite lattices that occur as congruence lattices of overalgebras? As we pointed. out in Section 7.4.1, a simple lattice is not the congruence lattice of a (non-trivial) expansion of the type described in Chapter. Are there other such (b) Product action m 2 and G is permutation isomorphic to a subgroup of the productIs there a general characterization of the class of finite lattices that occur as congruence lattices of overalgebras? As we pointed out in Section 7.4.1, a simple lattice is not the congruence lattice of a (non-trivial) expansion of the type described in Chapter 7. Are there other such (b) Product action m 2 and G is permutation isomorphic to a subgroup of the product Diagonal type m 2 and T m G T m .(Out(T ) × S m ), with the diagonal action. The degree d = |T | m−1. ) Diagonal type m 2 and T m G T m .(Out(T ) × S m ), with the diagonal action. The degree d = |T | m−1 . Note that this definition of product action groups is more restrictive than that given by some authors. This is in order to make the O'Nan-Scott classes disjoint. 46656 billion)(= 46.656 billion). Note that this definition of product action groups is more restrictive than that given by some authors. This is in order to make the O'Nan-Scott classes disjoint. BIBLIOGRAPHY On intervals in subgroup lattices of finite groups. Michael Aschbacher, 10.1090/S0894-0347-08-00602-4J. Amer. Math. Soc. 213Michael Aschbacher. On intervals in subgroup lattices of finite groups. J. Amer. Math. Soc., 21(3):809-830, 2008. doi:10.1090/S0894-0347-08-00602-4. On representing finite lattices as intervals in subgroup lattices of finite groups. Robert Baddeley, Andrea Lucchini, 10.1006/jabr.1997.7069J. Algebra. 1961Robert Baddeley and Andrea Lucchini. On representing finite lattices as intervals in subgroup lattices of finite groups. J. Algebra, 196(1):1-100, 1997. doi:10.1006/jabr.1997.7069. Classes of finite groups. Adolfo Ballester, - Bolinches, Luis M Ezquerro, Mathematics and Its Applications. 584SpringerAdolfo Ballester-Bolinches and Luis M. Ezquerro. Classes of finite groups, volume 584 of Math- ematics and Its Applications (Springer). Springer, Dordrecht, 2006. Second maximal subgroups of the finite alternating and symmetric groups. Alberto Basile, CanberraAustralian National UniversityPhD thesisAlberto Basile. Second maximal subgroups of the finite alternating and symmetric groups. PhD thesis, Australian National University, Canberra, April 2001. Congruence lattices of finite universal algebras. Joel Berman, University of WashingtonPhD thesisJoel Berman. Congruence lattices of finite universal algebras. PhD thesis, University of Wash- ington, 1970. Available from: http://db.tt/mXUVTzSr. On the structure of abstract algebras. Garrett Birkhoff, Proc. Cmabridge Phil. Soc. 31Garrett Birkhoff. On the structure of abstract algebras. Proc. Cmabridge Phil. Soc., 31:433-454, 1935. Lattice Theory. Garrett Birkhoff, American Mathematical SocietyNew YorkGarrett Birkhoff. Lattice Theory. American Mathematical Society, New York, 1940. A remark on the finite lattice representation problem. Ferdinand Börner, Contributions to general algebra. Olomouc/Velké Karlovice; KlagenfurtHeyn11Ferdinand Börner. A remark on the finite lattice representation problem. In Contributions to general algebra, 11 (Olomouc/Velké Karlovice, 1998), pages 5-38, Klagenfurt, 1999. Heyn. Topics in Computational Group Theory: Primitive permutation groups and matrix group normalisers. Hannah Coutts, University of St. AndrewsPhD thesisHannah Coutts. Topics in Computational Group Theory: Primitive permutation groups and matrix group normalisers. PhD thesis, University of St. Andrews, 2010. Available from: http://www-circa.mcs.st-and.ac.uk/Theses/HCoutts_thesis.pdf. Über die Anzahl der Ideal-classen in den verschiedenen Ordnungen eines endlichen Körpers. Richard Dedekind, Festschrift zur Saecularfeier des Geburtstages von C. F. Gauss. BraunschweigViewegsee Ges. WerkeRichard Dedekind.Über die Anzahl der Ideal-classen in den verschiedenen Ordnungen eines endlichen Körpers. In Festschrift zur Saecularfeier des Geburtstages von C. F. Gauss, pages 1-55. Vieweg, Braunschweig, 1877. see Ges. Werke, Band I, 1930, 105-157. Congruence lattices of intransitive G-sets. William Demeo, Ralph Freese, preprintWilliam DeMeo and Ralph Freese. Congruence lattices of intransitive G-sets. preprint, 2012. Available from: http://db.tt/tNzQsZl9. Permutation groups. John D Dixon, Brian Mortimer, Graduate Texts in Mathematics. 163Springer-VerlagJohn D. Dixon and Brian Mortimer. Permutation groups, volume 163 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1996. Finite soluble groups. Klaus Doerk, Trevor Hawkes, de Gruyter Expositions in Mathematics. 4Walter de Gruyter & CoKlaus Doerk and Trevor Hawkes. Finite soluble groups, volume 4 of de Gruyter Expositions in Mathematics. Walter de Gruyter & Co., Berlin, 1992. An interval in the subgroup lattice of a finite group which is isomorphic to M 7 . Algebra Universalis. Walter Feit, 10.1007/BF0119453217Walter Feit. An interval in the subgroup lattice of a finite group which is isomorphic to M 7 . Algebra Universalis, 17(2):220-221, 1983. doi:10.1007/BF01194532. Congruence lattices of finitely generated modular lattices. Ralph Freese, Proceedings of the Lattice Theory Conference. the Lattice Theory ConferenceUlm; Ulm; UlmUnivRalph Freese. Congruence lattices of finitely generated modular lattices. In Proceedings of the Lattice Theory Conference (Ulm, 1975), pages 62-70, Ulm, 1975. Univ. Ulm. Universal Algebra Calculator. Ralph Freese, Emil Kiss, Matthew Valeriote, Ralph Freese, Emil Kiss, and Matthew Valeriote. Universal Algebra Calculator, 2008. Available from: http://www.uacalc.org. GAP -Groups, Algorithms, and Programming. Gap The, Group, Ver. 4.4.12The GAP Group. GAP -Groups, Algorithms, and Programming, Ver. 4.4.12, 2008. Available from: http://www.gap-system.org. Characterizations of congruence lattices of abstract algebras. G Grätzer, E T Schmidt, Acta Sci. Math. (Szeged). 24G. Grätzer and E. T. Schmidt. Characterizations of congruence lattices of abstract algebras. Acta Sci. Math. (Szeged), 24:34-59, 1963. Universal algebra. D. George Grätzer, Van Nostrand Co., IncPrinceton, N.J.-Toronto, Ont.-LondonGeorge Grätzer. Universal algebra. D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.- London, 1968. Finite group theory. I , Martin Isaacs, Graduate Studies in Mathematics. 92American Mathematical SocietyI. Martin Isaacs. Finite group theory, volume 92 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2008. Topics in universal algebra. Bjarni Jónsson, Lecture Notes in Mathematics. 250Springer-VerlagBjarni Jónsson. Topics in universal algebra. Lecture Notes in Mathematics, Vol. 250. Springer- Verlag, Berlin, 1972. M 7 as an interval in a subgroup lattice. Peter Köhler, 10.1007/BF01194535Algebra Universalis. 173Peter Köhler. M 7 as an interval in a subgroup lattice. Algebra Universalis, 17(3):263-266, 1983. doi:10.1007/BF01194535. Endliche Gruppen mit vielen Untergruppen. Hans Kurzweil, 10.1515/crll.1985.356.140J. Reine Angew. Math. 356Hans Kurzweil. Endliche Gruppen mit vielen Untergruppen. J. Reine Angew. Math., 356:140- 160, 1985. doi:10.1515/crll.1985.356.140. Finite forbidden lattices. Ralph Mckenzie, Universal Algebra and Lattice Theory. Puebla; BerlinSpringer1004Ralph McKenzie. Finite forbidden lattices. In Universal Algebra and Lattice Theory (Puebla, 1982), volume 1004 of Lecture Notes in Math., pages 176-205, Berlin, 1983. Springer. A new product of algebras and a type reduction theorem. Algebra Universalis. Ralph Mckenzie, 10.1007/BF0118224718Ralph McKenzie. A new product of algebras and a type reduction theorem. Algebra Universalis, 18(1):29-69, 1984. doi:10.1007/BF01182247. Ralph N Mckenzie, George F Mcnulty, Walter F Taylor, Algebras, lattices, varieties. I. Wadsworth & Brooks/ColeMonterey, CARalph N. McKenzie, George F. McNulty, and Walter F. Taylor. Algebras, lattices, varieties. Vol. I. Wadsworth & Brooks/Cole, Monterey, CA, 1987. Eine bemerkung zu kongruenzverbanden. R Netter, preprintR. Netter. Eine bemerkung zu kongruenzverbanden. preprint, 1986. Distributive congruence lattices of finite algebras. P P Pálfy, Acta Sci. Math. (Szeged). 511-2P. P. Pálfy. Distributive congruence lattices of finite algebras. Acta Sci. Math. (Szeged), 51(1- 2):153-162, 1987. Intervals in subgroup lattices of finite groups. Pálfy Péter Pál, Groups '93 Galway/St. Péter Pál Pálfy. Intervals in subgroup lattices of finite groups. In Groups '93 Galway/St. . Andrews, 10.1017/CBO9780511629297.014London Math. Soc. Lecture Note Ser. 2Cambridge Univ. PressAndrews, Vol. 2, volume 212 of London Math. Soc. Lecture Note Ser., pages 482-494. Cambridge Univ. Press, Cambridge, 1995. doi:10.1017/CBO9780511629297.014. Groups and lattices. Pálfy Péter Pál, 10.1017/CBO9780511542787.014Groups St. Andrews. Oxford; CambridgeCambridge Univ. PressIIPéter Pál Pálfy. Groups and lattices. In Groups St. Andrews 2001 in Oxford. Vol. II, volume 305 of London Math. Soc. Lecture Note Ser., pages 428-454, Cambridge, 2003. Cambridge Univ. Press. doi:10.1017/CBO9780511542787.014. Summer School on General Algebra and Ordered Sets Stará Lesná. Pálfy Péter Pál, 6The finite congruence lattice problemPéter Pál Pálfy. The finite congruence lattice problem, September 2009. Summer School on Gen- eral Algebra and Ordered Sets Stará Lesná, 6, 2009. Available from: http://db.tt/DydVmisY. Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups. Algebra Universalis. Pál Péter, Pavel Pálfy, Pudlák, 10.1007/BF0248308011Péter Pál Pálfy and Pavel Pudlák. Congruence lattices of finite algebras and in- tervals in subgroup lattices of finite groups. Algebra Universalis, 11(1):22-27, 1980. doi:10.1007/BF02483080. Seminormal and subnormal subgroup lattices for transitive permutation groups. Cheryl E Praeger, 10.1017/S144678870001137XJ. Aust. Math. Soc. 801Cheryl E. Praeger. Seminormal and subnormal subgroup lattices for transitive permutation groups. J. Aust. Math. Soc., 80(1):45-63, 2006. doi:10.1017/S144678870001137X. Yeast graphs and fermentation of algebraic lattices. P Pudlák, J Tuma, Lattice theory (Proc. Colloq. Szeged; AmsterdamNorth-Holland14P. Pudlák and J. Tuma. Yeast graphs and fermentation of algebraic lattices. In Lattice theory (Proc. Colloq., Szeged, 1974), pages 301-341. Colloq. Math. Soc. János Bolyai, Vol. 14. North- Holland, Amsterdam, 1976. Every finite lattice can be embedded in a finite partition lattice. Algebra Universalis. Pavel Pudlák, Jiří Tuma, 10.1007/BF0248289310Pavel Pudlák and Jiří Tuma. Every finite lattice can be embedded in a finite partition lattice. Algebra Universalis, 10(1):74-95, 1980. doi:10.1007/BF02482893. Strong representation of congruence lattices. R Quackenbush, B Wolk, Algebra Universalis. 1R. Quackenbush and B. Wolk. Strong representation of congruence lattices. Algebra Universalis, 1:165-166, 1971/72. A course in the theory of groups. J S Derek, Robinson, Graduate Texts in Mathematics. 80Springer-Verlagsecond editionDerek J. S. Robinson. A course in the theory of groups, volume 80 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1996. A course on group theory. John S Rose, Cambridge University PressNew YorkReprint of the 1978 original. MR0498810 (58 #16847)John S. Rose. A course on group theory. Dover Publications Inc., New York, 1994. Reprint of the 1978 original [Cambridge University Press; MR0498810 (58 #16847)]. Nachweis der Existenz nicht-isomorpher Gruppen von gleicher Situation der Untergruppen. Ada Rottlaender, 10.1007/BF01181188Math. Z. 281Ada Rottlaender. Nachweis der Existenz nicht-isomorpher Gruppen von gleicher Situation der Untergruppen. Math. Z., 28(1):641-653, 1928. doi:10.1007/BF01181188. A survey on congruence lattice representations. E , Tamás Schmidt, Teubner-Texte zur Mathematik. BSB B. G. Teubner Verlagsgesellschaft42LeipzigE. Tamás Schmidt. A survey on congruence lattice representations, volume 42 of Teubner-Texte zur Mathematik. BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1982. Subgroup lattices of groups, volume 14 of de Gruyter Expositions in Mathematics. Roland Schmidt, Walter de Gruyter & CoBerlinRoland Schmidt. Subgroup lattices of groups, volume 14 of de Gruyter Expositions in Mathe- matics. Walter de Gruyter & Co., Berlin, 1994. Generalized wreath products and the lattice of normal subgroups of a group. Howard L Silcock, Algebra Universalis. 73Howard L. Silcock. Generalized wreath products and the lattice of normal subgroups of a group. Algebra Universalis, 7(3):361-372, 1977. A constructive approach to the finite congruence lattice representation problem. John W Snow, 10.1007/s000120050159Algebra Universalis. 432-3John W. Snow. A constructive approach to the finite congruence lattice representation problem. Algebra Universalis, 43(2-3):279-293, 2000. doi:10.1007/s000120050159. Group theory. I, volume 247 of Grundlehren der Mathematischen Wissenschaften. Michio Suzuki, Fundamental Principles of Mathematical SciencesMichio Suzuki. Group theory. I, volume 247 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. . Springer-Verlag, BerlinTranslated from the Japanese by the authorSpringer-Verlag, Berlin, 1982. Translated from the Japanese by the author. Some finite congruence lattices. I. Jiří Tůma, Czechoslovak Math. J. 36111Jiří Tůma. Some finite congruence lattices. I. Czechoslovak Math. J., 36(111)(2):298-330, 1986. Lattices of intermediate subfactors. Yasuo Watatani, 10.1006/jfan.1996.0110J. Funct. Anal. 1402Yasuo Watatani. Lattices of intermediate subfactors. J. Funct. Anal., 140(2):312-334, 1996. doi:10.1006/jfan.1996.0110. Lattices, equivalence relations, and subgroups. Philip M Whitman, Bull. Amer. Math. Soc. 52Philip M. Whitman. Lattices, equivalence relations, and subgroups. Bull. Amer. Math. Soc., 52:507-522, 1946.
{'fraction_non_alphanumeric': 0.08412521849674241, 'fraction_numerical': 0.028498785498626594, 'mean_word_length': 3.098319781180805, 'pattern_counts': {'":': 0, '<': 86, '<?xml version=': 0, '>': 23, 'https://': 0, 'lorem ipsum': 0, 'www.': 4, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 354, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': "ACKNOWLEDGMENTSFirst, I would like to thank my advisor, Ralph Freese, for his patience, support, and expert guidance, without which I could not have completed this dissertation. Next, I thank the members of my dissertation committee, Peter Jipsen, Bill Lampe, and J.B. Nation. All have made significant contributions to this work. Bill Lampe, in particular, is responsible for introducing me to the beautiful subject of universal algebra.I thank Nick Kaiser for agreeing to act as the University Representative on my dissertation committee, and for enduring long meetings about topics unrelated to his area of expertise (though I suspect he understands far more than he lets on).A number of other professors played a significant role in my mathematical training. Among them, I would especially like to thank Ron Brown, Tom Craven, Erik Guentner, Bjørn Kjos-Hanssen, Tom Ramsey, and Wayne Smith. Mike Hilden was kind enough to administer my French language exam, and I thank him for his help with this minor hurdle, and for not setting the bar too high.The Mathematics Department at the University of Hawai'i has generously supported me through the doctoral program, and for that I am grateful. I would also like to thank other members of the department who have played vital roles in my progress through the program; in particular, I thank", 'arxivid': '1204.4305', 'author': ['William J Demeo \nDissertation Committee\n\n', 'ChairpersonRalph Freese \nDissertation Committee\n\n', 'William Lampe \nDissertation Committee\n\n', 'J B Nation \nDissertation Committee\n\n', 'Peter Jipsen \nDissertation Committee\n\n', 'Nick Kaiser \nDissertation Committee\n\n', 'William J Demeo \nDissertation Committee\n\n', 'Susan Hasegawa \nDissertation Committee\n\n', 'Shirley Kikiloi \nDissertation Committee\n\n', 'Troy Ludwick \nDissertation Committee\n\n'], 'authoraffiliation': ['Dissertation Committee\n', 'Dissertation Committee\n', 'Dissertation Committee\n', 'Dissertation Committee\n', 'Dissertation Committee\n', 'Dissertation Committee\n', 'Dissertation Committee\n', 'Dissertation Committee\n', 'Dissertation Committee\n', 'Dissertation Committee\n'], 'corpusid': 116903773, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 79791, 'n_tokens_neox': 70082, 'n_words': 44249, 'pdfsha': '454736d396176b7573e93c4f1e2c240b7d828e6c', 'pdfurls': ['https://arxiv.org/pdf/1204.4305v3.pdf'], 'title': [], 'venue': []}
arxiv
Københavns Universitet String-like dual models for scalar theories Published for SISSA by Springer String-like dual models for scalar theories OnlineCopyright OnlinePublished: December 6, 2016(2016)019 Christian ; Baadsgaard Emil ; Bjerrum-Bohr Jacob Bourjaily [email protected] ; Lewis Poul Damgaard Henrik Christian Baadsgaard Department of Physics Princeton University 08544PrincetonNJU.S.A N E J Bjerrum-Bohr Niels Bohr International Academy and Discovery Center University of Copenhagen The Niels Bohr Institute Blegdamsvej 17DK-2100CopenhagenDenmark Jacob Bourjaily Niels Bohr International Academy and Discovery Center University of Copenhagen The Niels Bohr Institute Blegdamsvej 17DK-2100CopenhagenDenmark Poul H Damgaard Niels Bohr International Academy and Discovery Center University of Copenhagen The Niels Bohr Institute Blegdamsvej 17DK-2100CopenhagenDenmark Københavns Universitet String-like dual models for scalar theories Published for SISSA by Springer String-like dual models for scalar theories Journal of High Energy Physics OnlinePublished: December 6, 2016(2016)01910.1007/JHEP12(2016)019Publication date: 2016 Download date: 21. Jul. 2018 JHEP12(2016)019 Received: October 20, 2016 Accepted: November 29, 2016Published in: Document Version Publisher's PDF, also known as Version of record Citation for published version (APA): Baadsgaard, C., Bjerrum-Bohr, N. E. J., Bourjaily, J., & Damgaard, P. H. (2016). String-like dual models for scalar theories. DOI: 10.1007/JHEP12(2016)019 JHEP12(2016)019 Contents 1 Introduction 1Scattering Amplitudes, String Duality ArXiv ePrint: 161004228 We show that all tree-level amplitudes in ϕ p scalar field theory can be represented as the α → 0 limit of an SL(2, R)-invariant, string-theory-like dual model integral. These dual models are constructed according to constraints that admit families of solutions. We derive these dual models, and give closed formulae for all tree-level amplitudes of any ϕ p scalar field theory. Introduction String theory has been extremely useful in providing alternative avenues for representing amplitudes in quantum field theory. One reason for this is string theory's totally different organization of terms in the scattering amplitudes, and the manner in which it is completely detached from conventional Feynman diagram evaluations. Irrespective of string theory's potential as a unified description of nature, the formalism has provided us with novel tools with which field theory amplitudes can be understood. This was noted early on [1] and the whole field of string-based rules for scattering amplitudes in field theory [2] illustrates this. Other examples include the simple and unified proof of field theory identities such as Kleiss-Kuijf relations [3] and BCJ relations [4] in terms of monodromy in string theory [5,6]. Much additional information about field theory amplitudes can indeed be obtained from string theory in this way [7,8]. Related ideas continue to provide new insight into amplitude calculations [9]. Another striking case is the KLT relations [10] between graviton amplitudes and gauge field amplitudes, the field theory momentum kernel of which [11] follows immediately from the more general momentum kernel at the level of string theory [12]. The CHY formalism [13][14][15] based on scattering equations provides another interesting example of how string theory, and modifications thereof [16][17][18][19][20][21], can provide new insight into the computation of field theory amplitudes. These more recent examples suggest that there is still much more to gain from exploring the way string theory(-like) amplitudes can be computed, even if one is only interested in the field theory limit. An obvious clue comes from duality. This was evident already from the beginning of string theory, where the duality of the Veneziano amplitude [22] shows the possibility of treating different scattering channels in a unified manner. A multitude of field theory diagrams can correspond to a single string theory diagram. The Koba-Nielsen formula [23] JHEP12(2016)019 generalizes this to an n-particle scattering amplitude. From various directions, one is led towards an interpretation reminiscent of Feynman diagrams in field theory [24][25][26][27][28]. Indeed, in modern language this can be seen as an alternative way of generating ϕ 3 -theory amplitudes if one lets the Regge slope α approach zero [29]. This unusual way of producing tree-level amplitudes for scalar ϕ 3 -theory immediately raises the question of whether other types of scalar interactions can be generated in a similar way. We wish to answer that question here. The original motivation for the present study came from our derivation of how the CHY formalism of scattering equations could be understood in terms of Feynman diagrams at both tree level and loop level [30][31][32][33]. If a suitable string-theory-like integration measure could be established for more general scalar field theories, the transcription between string integrands and CHY integrands [19] would then possibly provide the compact prescription for generating general scalar field theories based on scattering equations. As will be explained below, the situation is slightly more complicated, both from the perspective of string theory (or, more appropriately, generalized dual models) and scattering equations. A first step towards understanding scalar field theories beyond ϕ 3 -theory in the scattering equation formalism has been taken by Cachazo, He and Yuan [34] using an elegant dimensional reduction argument for Yang-Mills theory and making a corresponding judicious choice of polarization vectors that projects dimensionally reduced Yang-Mills gauge connections onto just one scalar degree of freedom. The quartic Yang-Mills vertex then yields the sought-for scalar ϕ 4 interaction vertex. 1 At its simplest level, this procedure is therefore obviously limited to scalar interactions of ϕ 4 type. To go beyond, one might again gain insight from string theory and consider the next terms in the α -expansion. This is not straightforward, because in order to use the map [19] between string theory integrands and CHY integrands these integrands must be manifestly tachyon-free, and the leading α -correction (which could potentially yield ϕ 6 -vertices) vanishes for the superstring. An alternative way of generating higher ϕ p -theories using the CHY measure was presented in ref. [31]. The essential mechanism there was the generation of clusters of vertices with any number of legs using basic ϕ 3 vertices and corresponding summation over propagators. This is clearly a rather indirect prescription. Here, instead, we return to the question of generating such theories in the context of (generalized) dual models, leaving a potential description in terms of CHY integrands as an open problem. This paper is organized as follows. In section 2 we review how amplitudes in scalar ϕ 3theory can be obtained in the α → 0 limit of string theory, and present the generalized dual models we have found. The derivation of these new models will be described in section 3, where we will take care to describe their generalizations. We will conclude with some forward-looking remarks about the possible interpretation of these models in section 4. String-like, dual models for scalar field theories The string-like dual models we have found are natural generalizations of the way in which scalar amplitudes in scalar ϕ 3 -theory are represented in string theory in the α → 0 limit. 1 Suggestions for generating ϕ 4 -theory as a limit of string theory have been considered in [35,36]. JHEP12(2016)019 Therefore, it will be useful to briefly review this well-known story. Color-ordered scattering amplitudes in scalar ϕ 3 field theory arise in the α → 0 limit of string theory in following form: A ϕ 3 n = lim α →0 (α ) n−3 dΩ Λ n (α , k, z) I 3 n (z) with I 3 n ≡ n i=1 1 (z i − z i+1 ) , (2.1) where the auxiliary variables z i are cyclically ordered (with z n+1 = z 1 understood), Λ denotes the Koba-Nielsen factor [23], Λ n (α , k, z) ≡ i<j (z i − z j ) α s ij where s ij ≡ (k i + k j ) 2 ,(2.2) and dΩ is the integration measure of string theory, which we may define as: dΩ ≡ δ(z a − z 0 a )δ(z b − z 0 b )δ(z c − z 0 c ) × (z a − z b )(z b − z c )(z c − z a ) i dz i θ(z i − z i+1 ), (2.3) where the Heaviside functions, denoted θ(z) above, encode the ranges of integration. As is well known from string theory, the formula (2.1) is SL(2, R)-invariant, which ensures that we may choose z a , z b , z c as well as their gauge-fixings, z 0 a , z 0 b , z 0 c , freely. In order to illustrate our new string-like formulae, it will be useful to define: P j n (z) ≡ n i=1 (z i − z i+j ). (2.4) In terms of this, we can give a formula for any (connected) n-point amplitude in ϕ ptheory as, A ϕ p n ≡ lim α →0 (α ) (n−p) (p−2) γ p,0 (2−n) (p−2) dΩ Λ n (α , k, z) I p,0 n (z), (2.5) in terms of the integrand I p,0 n (z), defined according to, I p,0 n (z) ≡ 1 P q n n−2 p−2 /2 j=1 P (p−2)j+1 n 2 P (p−2)j n P (p−2)j+4−p n , q ≡    n/2 (n−2) (p−2) ∈ (2Z), (n − p + 4)/2 else,(2.6) and for which γ p,0 is a momentum-independent constant, γ p,0 ≡ dΩ I p,0 p (z) = π p− 7 2 (p − 2) Γ p−2 2 Γ p−1 2 . (2.7) The first few values of this constant are: γ 3,0 = γ 4,0 = 1, γ 5,0 = π 2 /6, γ 6,0 = π 2 /3, γ 7,0 = 3π 4 /40 . (2.8) As the reader may infer, we will later find it possible to generalize the expression (2.5) to a family parameterized by x, with an integrand I q,x n (z) and constant γ p,x n . It should be clear that the integrand in (2.6) is SL(2, R)-invariant, has the correct weights, and the power of α provides the correct scaling dimensions. While not completely obvious, it is a simple exercise to see that when p = 3, the representation (2.5) matches the string-theory expression (2.1) exactly. JHEP12(2016)019 It is worthwhile to illustrate this formula in a few particular instances. For example, the 6-particle amplitude in ϕ 4 -theory would be given by, I 4,0 6 = P 3 6 P 2 6 2 = (z 1 − z 4 ) 2 (z 2 − z 5 ) 2 (z 3 − z 6 ) 2 (z 1 − z 3 ) 2 (z 2 − z 4 ) 2 (z 3 − z 5 ) 2 (z 4 − z 6 ) 2 (z 5 − z 1 ) 2 (z 6 − z 2 ) 2 , (2.9) while the 10-particle amplitude would be given by, I 4,0 10 = P 3 10 2 P 5 10 P 2 10 P 4 10 2 = 10 i=1 (z i − z i+3 ) 2 (z i − z i+5 ) (z i − z i+2 ) 2 (z i − z i+4 ) 2 . (2.10) In fact, ϕ 4 -theory is simple enough for us to write down a relatively compact expression for any multiplicity. Expanding the general expression (2.6), we find: I 4,0 n = 1 P n/2 n (n−1)/4 j=1 P 2j+1 n P 2j n 2 , =              n i=1 (z i − z i+3 ) 2 · · · (z i − z i+n/2−2 ) 2 (z i − z i+n/2 ) (z i − z i+2 ) 2 (z i − z i+4 ) 2 · · · (z i − z i+n/2−1 ) 2 n 2 ∈ (2Z + 1) ; n i=1 (z i − z i+3 ) 2 (z i − z i+5 ) 2 · · · (z i − z i+n/2−1 ) 2 (z i − z i+2 ) 2 · · · (z i − z i+n/2−2 ) 2 (z i − z i+n/2 ) n 2 ∈ (2Z) . (2.11) It is not hard to generate corresponding expressions from (2.6) for any particular case of interest. Just for the sake of illustration, let us give a few more concrete examples. The 8-point amplitude in ϕ 5 -theory would be given by, I 5,0 8 = P 4 8 P 2 8 P 3 8 = 8 i=1 (z i − z i+4 ) (z i − z i+2 )(z i − z i+3 ) ; (2.12) and the 11-point amplitude by the integrand, I 5,0 11 = P 4 11 2 P 2 11 P 3 11 P 5 11 = 11 i=1 (z i − z i+4 ) 2 (z i − z i+2 )(z i − z i+3 )(z i − z i+5 ) . (2.13) Similarly, the 14-particle amplitude in ϕ 6 -theory would be generated by the following integrand, I 6,0 14 = (P 5 14 ) 2 P 2 14 P 4 14 P 6 14 = 14 i=1 (z i − z i+5 ) 2 (z i − z i+2 )(z i − z i+4 )(z i − z i+6 ) ; (2.14) and the 22-point amplitude in ϕ 7 -theory by I 7,0 22 = (P 6 22 ) 2 P 11 22 P 2 22 P 5 22 P 7 22 P 10 22 = 22 i=1 (z i − z i+6 ) 2 (z i − z i+11 ) (z i − z i+2 )(z i − z i+5 )(z i − z i+7 )(z i − z i+10 ) . (2.15) In the next section, we derive the formula for I p,0 n (z) in (2.6), discuss the origins of the constant prefactors γ p,0 n , show how these dual models arise from basic principles, and can be generalized in several interesting ways. JHEP12(2016)019 3 Derivation and generalizations In this section, we derive the formula (2.6), explain the meaning of the prefactor γ p,0 n , and see how it can be naturally be generalized in a number of ways. Let us begin with some general (fairly trivial) considerations that will allow us to define some important notation, and see how the basic ingredients in (2.6) emerge. General considerations and notation We seek to construct string-theory-like dual models that generalize the representation of ϕ 3 -theory according to (2.1). Recall that connected tree amplitudes in ϕ p -theory are only non-zero when n = L(p − 2) + 2 for some integer L ∈ Z, where L − 1 provides the number of propagators of the amplitude. We are interested in expressing such non-zero amplitudes as integrals of the following form: (α ) L−1 γ dΩ Λ n (α , k, z) I n (z). (3.1) We have allowed for a kinematic-independent prefactor γ in part to emphasize that we are going to be careful about such things (up to an overall sign); but also because, as we will see later on, that the form of γ required to reproduce scattering amplitudes precisely will turn out to be quite interesting. The possible integrands I(z) appearing in (3.1) should be required to be SL(2, R)invariant. This implies that it must be constructed out of products of differences (z i − z j ), I n (z) = i<j (z i − z j ) c ij ,(3.2) for some numbers c ij (which we do not assume to be integers). Moreover, upon including the integration measure dΩ, SL(2, R)-invariance requires that the weight of any z i must be −2: j =i c ij = −2 ∀i, where c ij ≡ c ji . (3.3) In order for I(z)'s constructed in this way to give rise to color-ordered amplitudes, it is necessary that the factors be cyclically-invariant. This requires that c i,i+q = c i−q,i ∀ i, q. (3.4) As a consequence we see that we may in fact define cyclic exponents, e j ≡ c i,i+j . (3.5) Notice that (3.4) immediately implies that e j = e n−j for all j. Because of this, we can always without loss of generality restrict our attention to e j with j ≤ n/2 . Using the exponents {e i }, we can rewrite the integrand (3.2) as, In order to reproduce scattering amplitudes in the α → 0 limit of (3.1), it must be that divergences arise in order to cancel the vanishing power of α . The regions where the integrand develops divergences sufficient to contribute something non-vanishing in this limit are quite combinatorial in nature, and give rise to poles involving propagators. Thus, in order to reproduce scattering amplitudes, the exponents e j must be chosen carefully. Let us describe how this can be done presently. I n (z) ≡ n/2 j=1 P j n e j ,(3. 3.2 Analysis of divergences in the limit α → 0 The way string theory reproduces the correct (dimensionful) poles of field theory amplitudes in the α → 0 limit must necessarily be associated with a corresponding divergence in inverse powers of α since only the dimensionless quantities α s ij appear in the initial expression. The prefactor of the integral provides the canceling powers of α . This means that we need to understand the rate at which the integral itself (e.g., the integral without its prefactor) diverges in the α → 0 limit. It is this degree of divergence together with regions within the integration region where divergences occurs that will tell us how we can generate general scalar tree-level amplitudes in the α → 0 limit. Because the measure dΩ enforces an ordering of the variables in (3.1)-that is, z i < z i−1 -a divergence in inverse powers of α in the integral can only come about in regions of the domain where subsets of consecutive variables z i , z i+1 , . . . , z i+m tend to the same value. We can check whether an integral of the form (3.1) has such a divergence, by letting τ ≡ {i, i + 1, . . . , i + m}, defining, y j ≡ z i − z j and y j ≡ y j /y i+m for j ∈ τ , and then considering the ≡ y i+m → 0 region of the integral over y i+m . In changing variables from z i , i ∈ τ , to variables z i , y i+1 , . . . , y i+m−1 , : • from the measure dΩ, we pick up a factor of m−1 ; • from Λ n (α , k, z), we pick up a factor of α sτ where s τ ≡ j∈τ k j 2 ; • from I n (z), we pick up a factor of e l for each factor of (z j − z j+l ) e l with j, j + l ∈ τ . The variables z i to z i+m contain m pairs of neighboring variables, (m − 1) pairs of next-nearest neighbors, and so forth, down to one pair of variables m steps away. Consequently, the total factor we pick up from I n (z) is Σm where Σ m ≡ m e 1 + (m − 1)e 2 + · · · + 2e m−1 + e m . (3.8) In total then, the integral over reduces to: So a divergence arising from variables {z j |j ∈ τ }, tending to the same value, results in the propagator carrying the external momenta {k j |j ∈ τ }. The remaining integrals factor in two: integrals over z j with j / ∈τ \{i} and integrals over y j with j ∈ {i + 1, i + 2, . . . , i + m − 1}. Iteratively repeating the above reasoning to the remaining integrations, one can determine the overall degree of divergence in inverse powers of α . In general an integral of the form (3.1) can have several distinct divergent regions of the integration domain, and it is necessary to sum over all of them to get the leading term in α . We refer the reader to ref. [30] for more details. z i 0 d m−1+α sτ +Σm 1 + O( ) . Matching divergences to Feynman diagrams Turning our attention to ϕ p -theory, recall that the number of particles involved in any (connected) amplitude must be a multiple of (p − 2). In order to construct an integral expression for the full amplitude, there has to be a divergent region of the integration domain for each such possible propagator. In other words, equation Σ m = −m + 1 + x m (3.14) when m is not a multiple of p − 2. As long as each x m is greater than minus one, the integral (3.9) converges, and we get no incorrect propagators. 2 JHEP12(2016)019 In summary, we impose the following conditions on the exponents e i : Σ i = −i 0 = i mod (p − 2) , x i − i + 1 else . (3.15) These equations are fairly straightforward to solve. If we adopt the convention that Σ i = 0 for i < 0, then, for ϕ 4 -theory: (3.16) and for ϕ p -theory with p > 4: e i = Σ i − 2Σ i−1 + Σ i−2 =        x 1 i = 1 −2 − 2x i−1 i ∈ (2Z) 2 + x i + x i−2 else,e i = Σ i − 2Σ i−1 + Σ i−2 =                  x 1 i = 1 x i − 2x i−1 − 1 2 = i mod (p − 2) , x i−2 − 2x i−1 − 1 0 = i mod (p − 2) , x i + x i−2 + 2 i > 1, 1 = i mod (p − 2) , x i − 2x i−1 + x i−2 else . (3.17) The solutions (3.16) and (3.17) apply only to the first n/2 exponents e i , but as explained above, this suffices. The remaining exponents can be found from the relation e n−i = e i . Residual integrations As long as the exponents e i satisfy (3.16) or (3.17), the integral (3.1) will contain divergences corresponding to all the factorization channels of the ϕ p tree-amplitude. But in order to identify a full amplitude with the corresponding integral in the α → 0 limit, we must ensure that all Feynman diagrams obtained from that integral come dressed with the same numerical prefactor, which we can then cancel with the overall normalization factor γ. The prefactors arise due to the fact that after carrying out all the integrations that lead to divergences in inverse powers of α , following the reasoning of section 3.2, what remains is the product of L residual integrals that are finite as α tend to zero. To leading order in α , we can therefore set Λ n (α , k, z) = 1 for those integrals so that all momentumdependence disappears. Feynman diagrams related by cyclic interchanges of the external momenta will necessarily carry the same numerical prefactor. The same may not be true in general for Feynman diagrams of different topologies (corresponding to the polygon graphs in table 1 of [31]). To ensure that the prefactors match, we must impose additional conditions on the exponents e i . Consider the factorization mentioned in the end of section 3.2. After performing the -integral, the integral over z j , j ∈ Z n , has factored into an integral over z j , j ∈ {1, 2, . . . , i, i + m + 1, . . . , n} and an integral over y j , j ∈ {i + 1, i + 2, . . . , i + m − 1}. So in the JHEP12(2016)019 z-integral, the variables z i−1 and z i+m+1 are now next-nearest neighbors. In order for the z-and y-integrals to match, we must therefore require that e 2 = e m+2 . Because all the original variables z i to z i+m have all merged to the same value z i , which is now the nearest-neighbor of z i−1 , we must also demand that e 1 = m+1 l=1 e l . Extending the above considerations beyond nearest and next-nearest neighbors, we find that the full list of (not-independent) requirements of these two types can be stated thus: for any multiple m of (p − 2) we require that: These conditions will be satisfied if we equate all the parameters x i in the expressions (3.16) and (3.17) for the exponents. This being done, the L residual integrations all evaluate to the same momentum-independent value. The overall normalization constant must therefore be chosen as: e j =       γ ≡ dΩ I p (z) −L . (3.19) Generalized dual models for all scalar field theories We see that the requirement that all the correct poles arise results in a family of solutions for e i , each of which have extra free parameters x i . The requirement that all terms arise from integrations with identical coefficients implies that all these parameters must be identical: x ≡ x i for all i. Thus, we arrive at a one-parameter family of dual models for any scalar tree-amplitude with p > 3. For ϕ 4 -theory the exponents are given by, e j ≡      x j = 1; −2(1 + x) j ∈ (2Z); 2(1 + x) else. (3.20) When p > 4 we have the following exponents: e j ≡          x j = 1; −(1 + x) j mod (p − 2) = 0, 2 ; 2(1 + x) j > 1 , 1 = j mod (p − 2); 0 else. (3.21) By using these exponents together with equations (3.1) and (3.6), we arrive at the following generalized dual model for all n-point amplitudes in ϕ p field theory: (3.22) where the generalized integrand I p,x n (z) is given by, A ϕ p n ≡ lim α →0 (α ) L−1 (γ p,x ) −L dΩ Λ n (α , k, z) I p,x n (z),I p,x n (z) ≡ P 1 n x I p,0 n 1+x ,(3. JHEP12(2016)019 4 Conclusions and discussion The well-known relationship between string theory and cubic graphs in its field theory limit immediately leads to the question: can we construct scalar field theories based on quartic or quintic or higher-order vertices that similarly follow from one single integral representation? To deviate as little as possible from the original string theory setting we have here considered a scenario as similar to open string theory as possible. We have introduced an ordered set of n real parameters z i integrated on the real line. We have insisted that the integrand be SL(2, R)-invariant, and we have defined the integration measure with the conventional Koba-Nielsen term times SL(2, R)-invariant factors that, if successful, should define for us the different kinds of scalar field theories in the α → 0 limit. We have found that cubic graphs play a special role in that the conditions that need to be fulfilled for the integral to generate ϕ 3 -theory in the α → 0 limit have a unique solution, the one already known. This sheds some new light on the observations of Scherk in the classic paper on dual models [29]. Surprisingly, we have found that all other scalar field theories based on ϕ p vertices can be generated as well. These theories require a bit more care, and the conditions needed to determine their integrands do not lead to unique solutions. Nevertheless, special solutions can be found in which, for given n and given p, we can write down the corresponding n-point amplitude as the α → 0 limit of a single string-like integral. The succinct expressions automatically generate the sum over all color-ordered Feynman diagrams of ϕ p -theory for that n-point amplitude. Our original motivation for this study was an aim towards establishing a CHY-type prescription for arbitrary scalar field theories. Let us therefore include some comments on this program. First, for ϕ 3 -theory the string-like construction is unique, and the integrand is just in the form for which the transcription between string theory integrands and CHY integrands is well established [19,30]: ending up as the product of two 'cycles'. Interestingly, even Yang-Mills theory can in CHY form be described entirely in terms of integrands composed of products of such cycles [37], again indicating the fundamental nature of cubic graphs underneath the formalism. Higher-order scalar field theories, as constructed in this paper, are not provided in that form. There are numerator factors that spoil an immediate transcription into CHY language. Since the specific case of ϕ 4 -theory, for which there does exist a CHY construction [34], appears as intractable as any other ϕ p -theory with p > 4 there is still hope that it may be possible to rewrite our integrands suitably (perhaps by partial fractioning) so as to end up with expressions that can transcribed to CHY form. This we leave as an open problem. A constructive solution can always be provided by brute-force expressions of arbitrary scalar graphs in terms of underlying cubic graphs and correspondingly canceling numerator factors that eliminate unwanted propagators. This program can be carried through systematically in the CHY formalism, as described in detail in ref. [31]. String-like dual models and the CHY formalism bear a close resemblance. As a matter of fact, the scattering equations, which constitute the cornerstone of the CHY formalism, made their first appearance in the literature as a modified version of the Virasoro conditions as an attempt to derive a tachyon-free Veneziano model. In the CHY formalism, amplitudes JHEP12(2016)019 are computed via integrals completely localized by δ-functions supported on the scattering equations. This is very much similar to stringy dual models, where integrals are localized by taking a limit that makes the integral blow up while its pre-factor tends to zero such that the non-divergent parts of the integration domain are killed off. But despite the analogous mechanisms by which the two formalism ensure the correct factorization properties, their mathematical frameworks are entirely distinct: the CHY formalism relies on multivariate complex residues, while everything in the stringy formalism is couched in terms of purely real integrals. And the analogy starts to fade when considering expressions that cannot straightforwardly be reduced to trivalent graphs, and which involve higher order poles in the CHY formalism and integrals not converging (in the sense of Riemann-Stieltjes) but requiring analytical extension. Furthermore, since the δ-functions supported on the scattering equations always carry a fixed mass dimension, the CHY formalism does not readily lend itself to theories with amplitudes of any dimension, unlike the string-like case where a single prefactor can be tuned to match the mass dimension of any ϕ p amplitude. It seems unlikely that the integral constructions of this paper can be related to some sort of open string theory: how could they be produced by a world-sheet path integral on the disc with vertex operator insertions? This question is especially interesting in the context of e.g. ref. [38]. It is intriguing that dual models of this kind can be constructed so that tree-level amplitudes of arbitrary ϕ p -theories fall out in the α → 0 limit. What kind of deformation parameter could this α correspond to? Only its 'point-like' α → 0 limit plays any role here. We leave these questions for future work. (3.10) must be satisfied for m equal to (p − 2), 2(p − 2), . . . , and (L − 1)(p − 2). (Recall that L ≡ (n − 2)/(p − 2).) At the same time, we must ensure that equation (3.10) is not satisfied for values of m that are not divisible by p − 2. Given these conditions, the requirement (3.7) of SL(2, R)-invariance is equivalent to equation (3.10) with m ≡ L(p − 2). The conditions on the set {e i } can therefore be summarized by the following set of (not all independent) equations Σ l(p−2) = −l(p − 2), for l ∈ {1, 2, . . . , L}. (3.12) For ϕ 3 -theory, these equations read −1 = e 1 −2 = 2e 1 + e 2 −3 = 3e 1 + 2e 2 + e 3 . . . n − 2 = (n − 2)e 1 + (n − 3)e 2 + · · · + e n−2 , (3.13)and have the unique solution e 1 = −1 and e i = 0 for i = 1. We recognize these as the standard dual model exponents (2.1) that lead to ϕ 3 -theory. For p > 3, the conditions (3.12) provide an under-determined set of equations. We can parametrize the solution space by introducing parameters {x m } and demanding j = 1, 2, . . . , p − 3, e j+m j = 2, . . . , p − 3 . One could also consider the case xm < −1, in which case the integral (3.9), after analytically continuing, remains finite in the α → 0 limit. However, it will no longer be possible to take the α → 0 limit before evaluating the integral, and the residual integrations discussed in the next section will no longer yield momentum independent-factors. AcknowledgmentsWe thank Paolo Di Vecchia and Peter Goddard for discussions and helpful suggestions. We would also like to thank Bo Feng for numerous enlightening discussions on related issues. This work was supported in part by the Danish National Research Foundation (DNRF91) and by a MOBILEX research grant from the Danish Council for Independent Research (JLB).Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. Duality and multi-gluon scattering. M L Mangano, S J Parke, Z Xu, 10.1016/0550-3213(88)90001-6Nucl. Phys. B. 298653INSPIREM.L. Mangano, S.J. Parke and Z. Xu, Duality and multi-gluon scattering, Nucl. Phys. B 298 (1988) 653 [INSPIRE]. A new approach to one loop calculations in gauge theories. Z Bern, D A Kosower, 10.1103/PhysRevD.38.1888Phys. Rev. D. 381888INSPIREZ. Bern and D.A. Kosower, A new approach to one loop calculations in gauge theories, Phys. Rev. D 38 (1988) 1888 [INSPIRE]. Multi-gluon cross-sections and five jet production at hadron colliders. R Kleiss, H Kuijf, 10.1016/0550-3213(89)90574-9Nucl. Phys. B. 312616INSPIRER. Kleiss and H. Kuijf, Multi-gluon cross-sections and five jet production at hadron colliders, Nucl. Phys. B 312 (1989) 616 [INSPIRE]. New relations for gauge-theory amplitudes. Z Bern, J J M Carrasco, H Johansson, 10.1103/PhysRevD.78.085011arXiv:0805.3993Phys. Rev. D. 7885011INSPIREZ. Bern, J.J.M. Carrasco and H. Johansson, New relations for gauge-theory amplitudes, Phys. Rev. D 78 (2008) 085011 [arXiv:0805.3993] [INSPIRE]. Minimal basis for gauge theory amplitudes. N E J Bjerrum-Bohr, P H Damgaard, P Vanhove, 10.1103/PhysRevLett.103.161602arXiv:0907.1425Phys. Rev. Lett. 103161602INSPIREN.E.J. Bjerrum-Bohr, P.H. Damgaard and P. Vanhove, Minimal basis for gauge theory amplitudes, Phys. Rev. Lett. 103 (2009) 161602 [arXiv:0907.1425] [INSPIRE]. S Stieberger, arXiv:0907.2211Open & closed vs. pure open string disk amplitudes. INSPIRES. Stieberger, Open & closed vs. pure open string disk amplitudes, arXiv:0907.2211 [INSPIRE]. Monodromy and Jacobi-like relations for color-ordered amplitudes. N E J Bjerrum-Bohr, P H Damgaard, T Sondergaard, P Vanhove, 10.1007/JHEP06(2010)003arXiv:1003.2403JHEP. 063INSPIREN.E.J. Bjerrum-Bohr, P.H. Damgaard, T. Sondergaard and P. Vanhove, Monodromy and Jacobi-like relations for color-ordered amplitudes, JHEP 06 (2010) 003 [arXiv:1003.2403] [INSPIRE]. Explicit BCJ numerators from pure spinors. C R Mafra, O Schlotterer, S Stieberger, 10.1007/JHEP07(2011)092arXiv:1104.5224JHEP. 0792INSPIREC.R. Mafra, O. Schlotterer and S. Stieberger, Explicit BCJ numerators from pure spinors, JHEP 07 (2011) 092 [arXiv:1104.5224] [INSPIRE]. Analytic representations of Yang-Mills amplitudes. N E J Bjerrum-Bohr, J L Bourjaily, P H Damgaard, B Feng, 10.1016/j.nuclphysb.2016.10.012arXiv:1605.06501Nucl. Phys. B. 913964INSPIREN.E.J. Bjerrum-Bohr, J.L. Bourjaily, P.H. Damgaard and B. Feng, Analytic representations of Yang-Mills amplitudes, Nucl. Phys. B 913 (2016) 964 [arXiv:1605.06501] [INSPIRE]. A relation between tree amplitudes of closed and open strings. H Kawai, D C Lewellen, S H H Tye, 10.1016/0550-3213(86)90362-7Nucl. Phys. B. 2691INSPIREH. Kawai, D.C. Lewellen and S.H.H. Tye, A relation between tree amplitudes of closed and open strings, Nucl. Phys. B 269 (1986) 1 [INSPIRE]. Gravity and Yang-Mills amplitude relations. N E J Bjerrum-Bohr, P H Damgaard, B Feng, T Sondergaard, 10.1103/PhysRevD.82.107702arXiv:1005.4367Phys. Rev. D. 82107702INSPIREN.E.J. Bjerrum-Bohr, P.H. Damgaard, B. Feng and T. Sondergaard, Gravity and Yang-Mills amplitude relations, Phys. Rev. D 82 (2010) 107702 [arXiv:1005.4367] [INSPIRE]. The momentum Kernel of gauge and gravity theories. N E J Bjerrum-Bohr, P H Damgaard, T Sondergaard, P Vanhove, 10.1007/JHEP01(2011)001arXiv:1010.3933JHEP. 011INSPIREN.E.J. Bjerrum-Bohr, P.H. Damgaard, T. Sondergaard and P. Vanhove, The momentum Kernel of gauge and gravity theories, JHEP 01 (2011) 001 [arXiv:1010.3933] [INSPIRE]. Scattering equations and Kawai-Lewellen-Tye orthogonality. F Cachazo, S He, E Y Yuan, 10.1103/PhysRevD.90.065001arXiv:1306.6575Phys. Rev. D. 9065001INSPIREF. Cachazo, S. He and E.Y. Yuan, Scattering equations and Kawai-Lewellen-Tye orthogonality, Phys. Rev. D 90 (2014) 065001 [arXiv:1306.6575] [INSPIRE]. Scattering of massless particles in arbitrary dimensions. F Cachazo, S He, E Y Yuan, 10.1103/PhysRevLett.113.171601arXiv:1307.2199Phys. Rev. Lett. 113171601INSPIREF. Cachazo, S. He and E.Y. Yuan, Scattering of massless particles in arbitrary dimensions, Phys. Rev. Lett. 113 (2014) 171601 [arXiv:1307.2199] [INSPIRE]. Scattering of massless particles: scalars, gluons and gravitons. F Cachazo, S He, E Y Yuan, 10.1007/JHEP07(2014)033arXiv:1309.0885JHEP. 0733INSPIREF. Cachazo, S. He and E.Y. Yuan, Scattering of massless particles: scalars, gluons and gravitons, JHEP 07 (2014) 033 [arXiv:1309.0885] [INSPIRE]. Ambitwistor strings and the scattering equations. L Mason, D Skinner, 10.1007/JHEP07(2014)048arXiv:1311.2564JHEP. 0748INSPIREL. Mason and D. Skinner, Ambitwistor strings and the scattering equations, JHEP 07 (2014) 048 [arXiv:1311.2564] [INSPIRE]. Infinite tension limit of the pure spinor superstring. N Berkovits, 10.1007/JHEP03(2014)017arXiv:1311.4156JHEP. 0317INSPIREN. Berkovits, Infinite tension limit of the pure spinor superstring, JHEP 03 (2014) 017 [arXiv:1311.4156] [INSPIRE]. Ambitwistor strings and the scattering equations at one loop. T Adamo, E Casali, D Skinner, 10.1007/JHEP04(2014)104arXiv:1312.3828JHEP. 04104INSPIRET. Adamo, E. Casali and D. Skinner, Ambitwistor strings and the scattering equations at one loop, JHEP 04 (2014) 104 [arXiv:1312.3828] [INSPIRE]. Scattering equations and string theory amplitudes. N E J Bjerrum-Bohr, P H Damgaard, P Tourkine, P Vanhove, 10.1103/PhysRevD.90.106002arXiv:1403.4553Phys. Rev. D. 90106002INSPIREN.E.J. Bjerrum-Bohr, P.H. Damgaard, P. Tourkine and P. Vanhove, Scattering equations and string theory amplitudes, Phys. Rev. D 90 (2014) 106002 [arXiv:1403.4553] [INSPIRE]. Loop integrands for scattering amplitudes from the Riemann sphere. Y Geyer, L Mason, R Monteiro, P Tourkine, 10.1103/PhysRevLett.115.121603arXiv:1507.00321Phys. Rev. Lett. 115121603INSPIREY. Geyer, L. Mason, R. Monteiro and P. Tourkine, Loop integrands for scattering amplitudes from the Riemann sphere, Phys. Rev. Lett. 115 (2015) 121603 [arXiv:1507.00321] [INSPIRE]. Y Geyer, L Mason, R Monteiro, P Tourkine, arXiv:1607.08887Two-loop scattering amplitudes from the Riemann sphere. INSPIREY. Geyer, L. Mason, R. Monteiro and P. Tourkine, Two-loop scattering amplitudes from the Riemann sphere, arXiv:1607.08887 [INSPIRE]. Construction of a crossing-symmetric, Regge behaved amplitude for linearly rising trajectories. G Veneziano, 10.1007/BF02824451Nuovo Cim. A. 57190INSPIREG. Veneziano, Construction of a crossing-symmetric, Regge behaved amplitude for linearly rising trajectories, Nuovo Cim. A 57 (1968) 190 [INSPIRE]. Reaction amplitude for n mesons: a generalization of the Veneziano-Bardakci-Ruegg-Virasora model. Z Koba, H B Nielsen, 10.1016/0550-3213(69)90331-9Nucl. Phys. B. 10633INSPIREZ. Koba and H.B. Nielsen, Reaction amplitude for n mesons: a generalization of the Veneziano-Bardakci-Ruegg-Virasora model, Nucl. Phys. B 10 (1969) 633 [INSPIRE]. Feynman-like diagrams compatible with duality. I: Planar diagrams. K Kikkawa, B Sakita, M A Virasoro, 10.1103/PhysRev.184.1701Phys. Rev. 1841701INSPIREK. Kikkawa, B. Sakita and M.A. Virasoro, Feynman-like diagrams compatible with duality. I: Planar diagrams, Phys. Rev. 184 (1969) 1701 [INSPIRE]. Feynman-like diagrams compatible with duality. II: General discussion including nonplanar diagrams. K Kikkawa, B Sakita, M A Virasoro, S A Klein, 10.1103/PhysRevD.1.3258Phys. Rev. D. 13258INSPIREK. Kikkawa, B. Sakita, M.A. Virasoro and S.A. Klein, Feynman-like diagrams compatible with duality. II: General discussion including nonplanar diagrams, Phys. Rev. D 1 (1970) 3258 [INSPIRE]. The primitive graphs of dual resonance models. D J Gross, A Neveu, J Scherk, J H Schwarz, 10.1016/0370-2693(70)90703-3Phys. Lett. B. 31592INSPIRED.J. Gross, A. Neveu, J. Scherk and J.H. Schwarz, The primitive graphs of dual resonance models, Phys. Lett. B 31 (1970) 592 [INSPIRE]. Dynamical model of dual amplitudes. B Sakita, M A Virasoro, 10.1103/PhysRevLett.24.1146Phys. Rev. Lett. 241146INSPIREB. Sakita and M.A. Virasoro, Dynamical model of dual amplitudes, Phys. Rev. Lett. 24 (1970) 1146 [INSPIRE]. A parton view on dual amplitudes. H B Nielsen, P Olesen, 10.1016/0370-2693(70)90474-0Phys. Lett. B. 32203INSPIREH.B. Nielsen and P. Olesen, A parton view on dual amplitudes, Phys. Lett. B 32 (1970) 203 [INSPIRE]. Zero-slope limit of the dual resonance model. J Scherk, 10.1016/0550-3213(71)90227-6Nucl. Phys. B. 31222INSPIREJ. Scherk, Zero-slope limit of the dual resonance model, Nucl. Phys. B 31 (1971) 222 [INSPIRE]. Integration rules for scattering equations. C Baadsgaard, N E J Bjerrum-Bohr, J L Bourjaily, P H Damgaard, 10.1007/JHEP09(2015)129arXiv:1506.06137JHEP. 09129INSPIREC. Baadsgaard, N.E.J. Bjerrum-Bohr, J.L. Bourjaily and P.H. Damgaard, Integration rules for scattering equations, JHEP 09 (2015) 129 [arXiv:1506.06137] [INSPIRE]. Scattering equations and Feynman diagrams. C Baadsgaard, N E J Bjerrum-Bohr, J L Bourjaily, P H Damgaard, 10.1007/JHEP09(2015)136arXiv:1507.00997JHEP. 09136INSPIREC. Baadsgaard, N.E.J. Bjerrum-Bohr, J.L. Bourjaily and P.H. Damgaard, Scattering equations and Feynman diagrams, JHEP 09 (2015) 136 [arXiv:1507.00997] [INSPIRE]. Integration rules for loop scattering equations. C Baadsgaard, N E J Bjerrum-Bohr, J L Bourjaily, P H Damgaard, B Feng, 10.1007/JHEP11(2015)080arXiv:1508.03627JHEP. 1180INSPIREC. Baadsgaard, N.E.J. Bjerrum-Bohr, J.L. Bourjaily, P.H. Damgaard and B. Feng, Integration rules for loop scattering equations, JHEP 11 (2015) 080 [arXiv:1508.03627] [INSPIRE]. New representations of the perturbative S-matrix. C Baadsgaard, N E J Bjerrum-Bohr, J L Bourjaily, S Caron-Huot, P H Damgaard, B Feng, 10.1103/PhysRevLett.116.061601arXiv:1509.02169Phys. Rev. Lett. 11661601INSPIREC. Baadsgaard, N.E.J. Bjerrum-Bohr, J.L. Bourjaily, S. Caron-Huot, P.H. Damgaard and B. Feng, New representations of the perturbative S-matrix, Phys. Rev. Lett. 116 (2016) 061601 [arXiv:1509.02169] [INSPIRE]. Einstein-Yang-Mills scattering amplitudes from scattering equations. F Cachazo, S He, E Y Yuan, 10.1007/JHEP01(2015)121arXiv:1409.8256JHEP. 01121INSPIREF. Cachazo, S. He and E.Y. Yuan, Einstein-Yang-Mills scattering amplitudes from scattering equations, JHEP 01 (2015) 121 [arXiv:1409.8256] [INSPIRE]. Scalar field theory limits of bosonic string amplitudes. A Frizzo, L Magnea, R Russo, 10.1016/S0550-3213(00)00200-5hep-th/9912183Nucl. Phys. B. 579379INSPIREA. Frizzo, L. Magnea and R. Russo, Scalar field theory limits of bosonic string amplitudes, Nucl. Phys. B 579 (2000) 379 [hep-th/9912183] [INSPIRE]. Two loop ϕ 4 diagrams from string theory. R Marotta, F Pezzella, 10.1103/PhysRevD.61.106006hep-th/9912158Phys. Rev. D. 61106006INSPIRER. Marotta and F. Pezzella, Two loop ϕ 4 diagrams from string theory, Phys. Rev. D 61 (2000) 106006 [hep-th/9912158] [INSPIRE]. Manifesting color-kinematics duality in the scattering equation formalism. N E J Bjerrum-Bohr, J L Bourjaily, P H Damgaard, B Feng, 10.1007/JHEP09(2016)094arXiv:1608.00006JHEP. 0994INSPIREN.E.J. Bjerrum-Bohr, J.L. Bourjaily, P.H. Damgaard and B. Feng, Manifesting color-kinematics duality in the scattering equation formalism, JHEP 09 (2016) 094 [arXiv:1608.00006] [INSPIRE]. Axiomatic conformal field theory. M R Gaberdiel, P Goddard, 10.1007/s002200050031hep-th/9810019Commun. Math. Phys. 209549INSPIREM.R. Gaberdiel and P. Goddard, Axiomatic conformal field theory, Commun. Math. Phys. 209 (2000) 549 [hep-th/9810019] [INSPIRE].
{'fraction_non_alphanumeric': 0.0751164596273292, 'fraction_numerical': 0.06189343944099379, 'mean_word_length': 3.947425279078142, 'pattern_counts': {'":': 0, '<': 5, '<?xml version=': 0, '>': 7, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 23, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'abstract': 'We show that all tree-level amplitudes in ϕ p scalar field theory can be represented as the α → 0 limit of an SL(2, R)-invariant, string-theory-like dual model integral. These dual models are constructed according to constraints that admit families of solutions. We derive these dual models, and give closed formulae for all tree-level amplitudes of any ϕ p scalar field theory.', 'arxivid': '1610.04228', 'author': ['Christian ; Baadsgaard ', 'Emil ; Bjerrum-Bohr ', 'Jacob Bourjaily [email protected] ', '; Lewis ', 'Poul Damgaard ', 'Henrik ', 'Christian Baadsgaard \nDepartment of Physics\nPrinceton University\n08544PrincetonNJU.S.A\n', 'N E J Bjerrum-Bohr \nNiels Bohr International Academy and Discovery Center\nUniversity of Copenhagen\nThe Niels Bohr Institute\nBlegdamsvej 17DK-2100CopenhagenDenmark\n', 'Jacob Bourjaily \nNiels Bohr International Academy and Discovery Center\nUniversity of Copenhagen\nThe Niels Bohr Institute\nBlegdamsvej 17DK-2100CopenhagenDenmark\n', 'Poul H Damgaard \nNiels Bohr International Academy and Discovery Center\nUniversity of Copenhagen\nThe Niels Bohr Institute\nBlegdamsvej 17DK-2100CopenhagenDenmark\n'], 'authoraffiliation': ['Department of Physics\nPrinceton University\n08544PrincetonNJU.S.A', 'Niels Bohr International Academy and Discovery Center\nUniversity of Copenhagen\nThe Niels Bohr Institute\nBlegdamsvej 17DK-2100CopenhagenDenmark', 'Niels Bohr International Academy and Discovery Center\nUniversity of Copenhagen\nThe Niels Bohr Institute\nBlegdamsvej 17DK-2100CopenhagenDenmark', 'Niels Bohr International Academy and Discovery Center\nUniversity of Copenhagen\nThe Niels Bohr Institute\nBlegdamsvej 17DK-2100CopenhagenDenmark'], 'corpusid': 3154184, 'doi': '10.1007/jhep12(2016)019', 'github_urls': [], 'n_tokens_mistral': 14985, 'n_tokens_neox': 12352, 'n_words': 6833, 'pdfsha': 'e9af909e2c379a2cff2f5688736f76e0f0a77dcd', 'pdfurls': ['https://web.archive.org/web/20180721121721/https:/curis.ku.dk/ws/files/171791439/art_10.1007_JHEP12_2016_019_1_.pdf'], 'title': ['Københavns Universitet String-like dual models for scalar theories Published for SISSA by Springer String-like dual models for scalar theories', 'Københavns Universitet String-like dual models for scalar theories Published for SISSA by Springer String-like dual models for scalar theories'], 'venue': ['Journal of High Energy Physics']}